[{"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Euclidean/Basic.lean", "full_name": "EuclideanGeometry.dist_orthogonalProjection_eq_zero_iff", "start": [416, 1], "end": [418, 65], "traced_tactics": [{"tactic": "rw [dist_comm, dist_eq_zero, orthogonalProjection_eq_self_iff]", "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : InnerProductSpace \u211d V\ninst\u271d\u00b3 : MetricSpace P\ninst\u271d\u00b2 : NormedAddTorsor V P\ns : AffineSubspace \u211d P\ninst\u271d\u00b9 : Nonempty { x // x \u2208 s }\ninst\u271d : CompleteSpace { x // x \u2208 direction s }\np : P\n\u22a2 dist p \u2191(\u2191(orthogonalProjection s) p) = 0 \u2194 p \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/NAry.lean", "full_name": "Set.image2_insert_right", "start": [209, 1], "end": [210, 61], "traced_tactics": [{"tactic": "rw [insert_eq, image2_union_right, image2_singleton_right]", "state_before": "\u03b1 : Type u_2\n\u03b1' : Type ?u.28640\n\u03b2 : Type u_3\n\u03b2' : Type ?u.28646\n\u03b3 : Type u_1\n\u03b3' : Type ?u.28652\n\u03b4 : Type ?u.28655\n\u03b4' : Type ?u.28658\n\u03b5 : Type ?u.28661\n\u03b5' : Type ?u.28664\n\u03b6 : Type ?u.28667\n\u03b6' : Type ?u.28670\n\u03bd : Type ?u.28673\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\n\u22a2 image2 f s (insert b t) = (fun a => f a b) '' s \u222a image2 f s t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/ContDiff.lean", "full_name": "ContDiffAt.fst", "start": [772, 1], "end": [774, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "full_name": "MeasurableSet.inv", "start": [503, 1], "end": [504, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Cast/Basic.lean", "full_name": "ofDual_natCast", "start": [358, 1], "end": [359, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subsemigroup/Operations.lean", "full_name": "Subsemigroup.coe_comap", "start": [194, 1], "end": [195, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finsupp/Defs.lean", "full_name": "Finsupp.support_onFinset_subset", "start": [705, 1], "end": [707, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/ContinuousFunction/CocompactMap.lean", "full_name": "CocompactMap.comp_assoc", "start": [160, 1], "end": [162, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/Independent.lean", "full_name": "AffineIndependent.of_set_of_injective", "start": [351, 1], "end": [356, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/LaurentSeries.lean", "full_name": "PowerSeries.coe_sub", "start": [203, 1], "end": [204, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Hom/Units.lean", "full_name": "divp_eq_div", "start": [144, 1], "end": [145, 53], "traced_tactics": [{"tactic": "rw [div_eq_mul_inv, divp, u.val_inv_eq_inv_val]", "state_before": "\u03b1 : Type u_1\nM : Type u\nN : Type v\nP : Type w\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Monoid N\ninst\u271d\u00b9 : Monoid P\ninst\u271d : DivisionMonoid \u03b1\na : \u03b1\nu : \u03b1\u02e3\n\u22a2 a /\u209a u = a / \u2191u", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Combination.lean", "full_name": "Finset.centerMass_segment'", "start": [86, 1], "end": [93, 77], "traced_tactics": [{"tactic": "rw [s.centerMass_eq_of_sum_1 _ hws, t.centerMass_eq_of_sum_1 _ hwt, smul_sum, smul_sum, \u2190\n  Finset.sum_sum_elim, Finset.centerMass_eq_of_sum_1]", "state_before": "R : Type u_3\nE : Type u_4\nF : Type ?u.41233\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type ?u.41242\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset \u03b9\nt : Finset \u03b9'\nws : \u03b9 \u2192 R\nzs : \u03b9 \u2192 E\nwt : \u03b9' \u2192 R\nzt : \u03b9' \u2192 E\nhws : \u2211 i in s, ws i = 1\nhwt : \u2211 i in t, wt i = 1\na b : R\nhab : a + b = 1\n\u22a2 a \u2022 centerMass s ws zs + b \u2022 centerMass t wt zt =\n    centerMass (disjSum s t) (Sum.elim (fun i => a * ws i) fun j => b * wt j) (Sum.elim zs zt)", "state_after": "R : Type u_3\nE : Type u_4\nF : Type ?u.41233\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type ?u.41242\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset \u03b9\nt : Finset \u03b9'\nws : \u03b9 \u2192 R\nzs : \u03b9 \u2192 E\nwt : \u03b9' \u2192 R\nzt : \u03b9' \u2192 E\nhws : \u2211 i in s, ws i = 1\nhwt : \u2211 i in t, wt i = 1\na b : R\nhab : a + b = 1\n\u22a2 \u2211 x in disjSum s t, Sum.elim (fun x => a \u2022 ws x \u2022 zs x) (fun x => b \u2022 wt x \u2022 zt x) x =\n    \u2211 i in disjSum s t, Sum.elim (fun i => a * ws i) (fun j => b * wt j) i \u2022 Sum.elim zs zt i\n\ncase hw\nR : Type u_3\nE : Type u_4\nF : Type ?u.41233\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type ?u.41242\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset \u03b9\nt : Finset \u03b9'\nws : \u03b9 \u2192 R\nzs : \u03b9 \u2192 E\nwt : \u03b9' \u2192 R\nzt : \u03b9' \u2192 E\nhws : \u2211 i in s, ws i = 1\nhwt : \u2211 i in t, wt i = 1\na b : R\nhab : a + b = 1\n\u22a2 \u2211 i in disjSum s t, Sum.elim (fun i => a * ws i) (fun j => b * wt j) i = 1"}, {"tactic": "congr with \u27e8\u27e9 <;> simp only [Sum.elim_inl, Sum.elim_inr, mul_smul]", "state_before": "R : Type u_3\nE : Type u_4\nF : Type ?u.41233\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type ?u.41242\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset \u03b9\nt : Finset \u03b9'\nws : \u03b9 \u2192 R\nzs : \u03b9 \u2192 E\nwt : \u03b9' \u2192 R\nzt : \u03b9' \u2192 E\nhws : \u2211 i in s, ws i = 1\nhwt : \u2211 i in t, wt i = 1\na b : R\nhab : a + b = 1\n\u22a2 \u2211 x in disjSum s t, Sum.elim (fun x => a \u2022 ws x \u2022 zs x) (fun x => b \u2022 wt x \u2022 zt x) x =\n    \u2211 i in disjSum s t, Sum.elim (fun i => a * ws i) (fun j => b * wt j) i \u2022 Sum.elim zs zt i", "state_after": "no goals"}, {"tactic": "rw [sum_sum_elim, \u2190 mul_sum, \u2190 mul_sum, hws, hwt, mul_one, mul_one, hab]", "state_before": "case hw\nR : Type u_3\nE : Type u_4\nF : Type ?u.41233\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type ?u.41242\ninst\u271d\u2077 : LinearOrderedField R\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt\u271d : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset \u03b9\nt : Finset \u03b9'\nws : \u03b9 \u2192 R\nzs : \u03b9 \u2192 E\nwt : \u03b9' \u2192 R\nzt : \u03b9' \u2192 E\nhws : \u2211 i in s, ws i = 1\nhwt : \u2211 i in t, wt i = 1\na b : R\nhab : a + b = 1\n\u22a2 \u2211 i in disjSum s t, Sum.elim (fun i => a * ws i) (fun j => b * wt j) i = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fin/VecNotation.lean", "full_name": "Matrix.cons_val_succ", "start": [137, 1], "end": [138, 17], "traced_tactics": [{"tactic": "simp [vecCons]", "state_before": "\u03b1 : Type u\nm n o : \u2115\nm' : Type ?u.14351\nn' : Type ?u.14354\no' : Type ?u.14357\nx : \u03b1\nu : Fin m \u2192 \u03b1\ni : Fin m\n\u22a2 vecCons x u (Fin.succ i) = u i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Dfinsupp/Basic.lean", "full_name": "Dfinsupp.addHom_ext", "start": [1015, 1], "end": [1020, 10], "traced_tactics": [{"tactic": "refine' AddMonoidHom.eq_of_eqOn_denseM add_closure_iUnion_range_single fun f hf => _", "state_before": "\u03b9 : Type u\n\u03b3\u271d : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\n\u03b3 : Type w\ninst\u271d : AddZeroClass \u03b3\nf g : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192+ \u03b3\nH : \u2200 (i : \u03b9) (y : \u03b2 i), \u2191f (single i y) = \u2191g (single i y)\n\u22a2 f = g", "state_after": "\u03b9 : Type u\n\u03b3\u271d : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\n\u03b3 : Type w\ninst\u271d : AddZeroClass \u03b3\nf\u271d g : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192+ \u03b3\nH : \u2200 (i : \u03b9) (y : \u03b2 i), \u2191f\u271d (single i y) = \u2191g (single i y)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nhf : f \u2208 \u22c3 (i : \u03b9), Set.range (single i)\n\u22a2 \u2191f\u271d f = \u2191g f"}, {"tactic": "simp only [Set.mem_iUnion, Set.mem_range] at hf", "state_before": "\u03b9 : Type u\n\u03b3\u271d : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\n\u03b3 : Type w\ninst\u271d : AddZeroClass \u03b3\nf\u271d g : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192+ \u03b3\nH : \u2200 (i : \u03b9) (y : \u03b2 i), \u2191f\u271d (single i y) = \u2191g (single i y)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nhf : f \u2208 \u22c3 (i : \u03b9), Set.range (single i)\n\u22a2 \u2191f\u271d f = \u2191g f", "state_after": "\u03b9 : Type u\n\u03b3\u271d : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\n\u03b3 : Type w\ninst\u271d : AddZeroClass \u03b3\nf\u271d g : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192+ \u03b3\nH : \u2200 (i : \u03b9) (y : \u03b2 i), \u2191f\u271d (single i y) = \u2191g (single i y)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nhf : \u2203 i y, single i y = f\n\u22a2 \u2191f\u271d f = \u2191g f"}, {"tactic": "rcases hf with \u27e8x, y, rfl\u27e9", "state_before": "\u03b9 : Type u\n\u03b3\u271d : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\n\u03b3 : Type w\ninst\u271d : AddZeroClass \u03b3\nf\u271d g : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192+ \u03b3\nH : \u2200 (i : \u03b9) (y : \u03b2 i), \u2191f\u271d (single i y) = \u2191g (single i y)\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nhf : \u2203 i y, single i y = f\n\u22a2 \u2191f\u271d f = \u2191g f", "state_after": "case intro.intro\n\u03b9 : Type u\n\u03b3\u271d : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\n\u03b3 : Type w\ninst\u271d : AddZeroClass \u03b3\nf g : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192+ \u03b3\nH : \u2200 (i : \u03b9) (y : \u03b2 i), \u2191f (single i y) = \u2191g (single i y)\nx : \u03b9\ny : \u03b2 x\n\u22a2 \u2191f (single x y) = \u2191g (single x y)"}, {"tactic": "apply H", "state_before": "case intro.intro\n\u03b9 : Type u\n\u03b3\u271d : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\n\u03b3 : Type w\ninst\u271d : AddZeroClass \u03b3\nf g : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192+ \u03b3\nH : \u2200 (i : \u03b9) (y : \u03b2 i), \u2191f (single i y) = \u2191g (single i y)\nx : \u03b9\ny : \u03b2 x\n\u22a2 \u2191f (single x y) = \u2191g (single x y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Basic.lean", "full_name": "SimpleGraph.sdiff_eq_deleteEdges", "start": [1115, 1], "end": [1117, 7], "traced_tactics": [{"tactic": "ext", "state_before": "\u03b9 : Sort ?u.154284\n\ud835\udd5c : Type ?u.154287\nV : Type u\nW : Type v\nX : Type w\nG\u271d : SimpleGraph V\nG'\u271d : SimpleGraph W\na b c u v w : V\ne : Sym2 V\nG G' : SimpleGraph V\n\u22a2 G \\ G' = deleteEdges G (edgeSet G')", "state_after": "case Adj.h.h.a\n\u03b9 : Sort ?u.154284\n\ud835\udd5c : Type ?u.154287\nV : Type u\nW : Type v\nX : Type w\nG\u271d : SimpleGraph V\nG'\u271d : SimpleGraph W\na b c u v w : V\ne : Sym2 V\nG G' : SimpleGraph V\nx\u271d\u00b9 x\u271d : V\n\u22a2 Adj (G \\ G') x\u271d\u00b9 x\u271d \u2194 Adj (deleteEdges G (edgeSet G')) x\u271d\u00b9 x\u271d"}, {"tactic": "simp", "state_before": "case Adj.h.h.a\n\u03b9 : Sort ?u.154284\n\ud835\udd5c : Type ?u.154287\nV : Type u\nW : Type v\nX : Type w\nG\u271d : SimpleGraph V\nG'\u271d : SimpleGraph W\na b c u v w : V\ne : Sym2 V\nG G' : SimpleGraph V\nx\u271d\u00b9 x\u271d : V\n\u22a2 Adj (G \\ G') x\u271d\u00b9 x\u271d \u2194 Adj (deleteEdges G (edgeSet G')) x\u271d\u00b9 x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Valuation/Basic.lean", "full_name": "Valuation.isEquiv_iff_val_lt_one", "start": [486, 1], "end": [513, 60], "traced_tactics": [{"tactic": "constructor", "state_before": "K : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\n\u22a2 IsEquiv v v' \u2194 \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1", "state_after": "case mp\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\n\u22a2 IsEquiv v v' \u2192 \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\n\ncase mpr\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\n\u22a2 (\u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1) \u2192 IsEquiv v v'"}, {"tactic": "intro h x", "state_before": "case mp\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\n\u22a2 IsEquiv v v' \u2192 \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1", "state_after": "case mp\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : IsEquiv v v'\nx : K\n\u22a2 \u2191v x < 1 \u2194 \u2191v' x < 1"}, {"tactic": "simp only [lt_iff_le_and_ne,\n  and_congr ((isEquiv_iff_val_le_one _ _).1 h) ((isEquiv_iff_val_eq_one _ _).1 h).not]", "state_before": "case mp\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : IsEquiv v v'\nx : K\n\u22a2 \u2191v x < 1 \u2194 \u2191v' x < 1", "state_after": "no goals"}, {"tactic": "rw [isEquiv_iff_val_eq_one]", "state_before": "case mpr\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\n\u22a2 (\u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1) \u2192 IsEquiv v v'", "state_after": "case mpr\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\n\u22a2 (\u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1) \u2192 \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1"}, {"tactic": "intro h x", "state_before": "case mpr\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\n\u22a2 (\u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1) \u2192 \u2200 {x : K}, \u2191v x = 1 \u2194 \u2191v' x = 1", "state_after": "case mpr\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\n\u22a2 \u2191v x = 1 \u2194 \u2191v' x = 1"}, {"tactic": "by_cases hx : x = 0", "state_before": "case mpr\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\n\u22a2 \u2191v x = 1 \u2194 \u2191v' x = 1", "state_after": "case pos\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : x = 0\n\u22a2 \u2191v x = 1 \u2194 \u2191v' x = 1\n\ncase neg\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\n\u22a2 \u2191v x = 1 \u2194 \u2191v' x = 1"}, {"tactic": "constructor", "state_before": "case neg\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\n\u22a2 \u2191v x = 1 \u2194 \u2191v' x = 1", "state_after": "case neg.mp\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\n\u22a2 \u2191v x = 1 \u2192 \u2191v' x = 1\n\ncase neg.mpr\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\n\u22a2 \u2191v' x = 1 \u2192 \u2191v x = 1"}, {"tactic": "rw [(zero_iff _).2 hx, (zero_iff _).2 hx]", "state_before": "case pos\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : x = 0\n\u22a2 \u2191v x = 1 \u2194 \u2191v' x = 1", "state_after": "case pos\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : x = 0\n\u22a2 0 = 1 \u2194 0 = 1"}, {"tactic": "simp only [zero_ne_one]", "state_before": "case pos\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : x = 0\n\u22a2 0 = 1 \u2194 0 = 1", "state_after": "no goals"}, {"tactic": "intro hh", "state_before": "case neg.mp\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\n\u22a2 \u2191v x = 1 \u2192 \u2191v' x = 1", "state_after": "case neg.mp\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v x = 1\n\u22a2 \u2191v' x = 1"}, {"tactic": "by_contra h_1", "state_before": "case neg.mp\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v x = 1\n\u22a2 \u2191v' x = 1", "state_after": "case neg.mp\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v x = 1\nh_1 : \u00ac\u2191v' x = 1\n\u22a2 False"}, {"tactic": "cases ne_iff_lt_or_gt.1 h_1 with\n| inl h_2 => simpa [hh, lt_self_iff_false] using h.2 h_2\n| inr h_2 =>\n    rw [\u2190 inv_one, \u2190inv_eq_iff_eq_inv, \u2190 map_inv\u2080] at hh\n    exact hh.not_lt (h.2 ((one_lt_val_iff v' hx).1 h_2))", "state_before": "case neg.mp\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v x = 1\nh_1 : \u00ac\u2191v' x = 1\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simpa [hh, lt_self_iff_false] using h.2 h_2", "state_before": "case neg.mp.inl\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v x = 1\nh_1 : \u00ac\u2191v' x = 1\nh_2 : \u2191v' x < 1\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rw [\u2190 inv_one, \u2190inv_eq_iff_eq_inv, \u2190 map_inv\u2080] at hh", "state_before": "case neg.mp.inr\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v x = 1\nh_1 : \u00ac\u2191v' x = 1\nh_2 : \u2191v' x > 1\n\u22a2 False", "state_after": "case neg.mp.inr\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v x\u207b\u00b9 = 1\nh_1 : \u00ac\u2191v' x = 1\nh_2 : \u2191v' x > 1\n\u22a2 False"}, {"tactic": "exact hh.not_lt (h.2 ((one_lt_val_iff v' hx).1 h_2))", "state_before": "case neg.mp.inr\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v x\u207b\u00b9 = 1\nh_1 : \u00ac\u2191v' x = 1\nh_2 : \u2191v' x > 1\n\u22a2 False", "state_after": "no goals"}, {"tactic": "intro hh", "state_before": "case neg.mpr\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\n\u22a2 \u2191v' x = 1 \u2192 \u2191v x = 1", "state_after": "case neg.mpr\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v' x = 1\n\u22a2 \u2191v x = 1"}, {"tactic": "by_contra h_1", "state_before": "case neg.mpr\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v' x = 1\n\u22a2 \u2191v x = 1", "state_after": "case neg.mpr\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v' x = 1\nh_1 : \u00ac\u2191v x = 1\n\u22a2 False"}, {"tactic": "cases ne_iff_lt_or_gt.1 h_1 with\n| inl h_2 => simpa [hh, lt_self_iff_false] using h.1 h_2\n| inr h_2 =>\n  rw [\u2190 inv_one, \u2190 inv_eq_iff_eq_inv, \u2190 map_inv\u2080] at hh\n  exact hh.not_lt (h.1 ((one_lt_val_iff v hx).1 h_2))", "state_before": "case neg.mpr\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v' x = 1\nh_1 : \u00ac\u2191v x = 1\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simpa [hh, lt_self_iff_false] using h.1 h_2", "state_before": "case neg.mpr.inl\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v' x = 1\nh_1 : \u00ac\u2191v x = 1\nh_2 : \u2191v x < 1\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rw [\u2190 inv_one, \u2190 inv_eq_iff_eq_inv, \u2190 map_inv\u2080] at hh", "state_before": "case neg.mpr.inr\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v' x = 1\nh_1 : \u00ac\u2191v x = 1\nh_2 : \u2191v x > 1\n\u22a2 False", "state_after": "case neg.mpr.inr\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v' x\u207b\u00b9 = 1\nh_1 : \u00ac\u2191v x = 1\nh_2 : \u2191v x > 1\n\u22a2 False"}, {"tactic": "exact hh.not_lt (h.1 ((one_lt_val_iff v hx).1 h_2))", "state_before": "case neg.mpr.inr\nK : Type u_3\nF : Type ?u.3380888\nR : Type ?u.3380891\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_1\n\u0393'\u2080 : Type u_2\n\u0393''\u2080 : Type ?u.3380903\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : LinearOrderedCommGroupWithZero \u0393'\u2080\nv : Valuation K \u0393\u2080\nv' : Valuation K \u0393'\u2080\nh : \u2200 {x : K}, \u2191v x < 1 \u2194 \u2191v' x < 1\nx : K\nhx : \u00acx = 0\nhh : \u2191v' x\u207b\u00b9 = 1\nh_1 : \u00ac\u2191v x = 1\nh_2 : \u2191v x > 1\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Derivative.lean", "full_name": "Polynomial.iterate_derivative_map", "start": [348, 1], "end": [352, 90], "traced_tactics": [{"tactic": "induction' k with k ih generalizing p", "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nf : R \u2192+* S\nk : \u2115\n\u22a2 (\u2191derivative^[k]) (map f p) = map f ((\u2191derivative^[k]) p)", "state_after": "case zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np\u271d : R[X]\nf : R \u2192+* S\np : R[X]\n\u22a2 (\u2191derivative^[Nat.zero]) (map f p) = map f ((\u2191derivative^[Nat.zero]) p)\n\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np\u271d : R[X]\nf : R \u2192+* S\nk : \u2115\nih : \u2200 (p : R[X]), (\u2191derivative^[k]) (map f p) = map f ((\u2191derivative^[k]) p)\np : R[X]\n\u22a2 (\u2191derivative^[Nat.succ k]) (map f p) = map f ((\u2191derivative^[Nat.succ k]) p)"}, {"tactic": "simp", "state_before": "case zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np\u271d : R[X]\nf : R \u2192+* S\np : R[X]\n\u22a2 (\u2191derivative^[Nat.zero]) (map f p) = map f ((\u2191derivative^[Nat.zero]) p)", "state_after": "no goals"}, {"tactic": "simp only [ih, Function.iterate_succ, Polynomial.derivative_map, Function.comp_apply]", "state_before": "case succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np\u271d : R[X]\nf : R \u2192+* S\nk : \u2115\nih : \u2200 (p : R[X]), (\u2191derivative^[k]) (map f p) = map f ((\u2191derivative^[k]) p)\np : R[X]\n\u22a2 (\u2191derivative^[Nat.succ k]) (map f p) = map f ((\u2191derivative^[Nat.succ k]) p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Basic.lean", "full_name": "MulOpposite.algebraMap_apply", "start": [612, 1], "end": [613, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean", "full_name": "InnerProductGeometry.angle_comm", "start": [85, 1], "end": [87, 33], "traced_tactics": [{"tactic": "unfold angle", "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\n\u22a2 angle x y = angle y x", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\n\u22a2 arccos (inner x y / (\u2016x\u2016 * \u2016y\u2016)) = arccos (inner y x / (\u2016y\u2016 * \u2016x\u2016))"}, {"tactic": "rw [real_inner_comm, mul_comm]", "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\n\u22a2 arccos (inner x y / (\u2016x\u2016 * \u2016y\u2016)) = arccos (inner y x / (\u2016y\u2016 * \u2016x\u2016))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.mem_filter", "start": [1978, 1], "end": [1979, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Int/Lemmas.lean", "full_name": "Int.min_def", "start": [685, 11], "end": [685, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/ContinuousFunction/Algebra.lean", "full_name": "ContinuousMap.coe_star", "start": [974, 1], "end": [975, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Game/PGame.lean", "full_name": "PGame.lf_of_equiv_of_lf", "start": [835, 1], "end": [836, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean", "full_name": "measurable_limit_of_tendsto_metrizable_ae", "start": [150, 1], "end": [177, 50], "traced_tactics": [{"tactic": "inhabit \u03b9", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))"}, {"tactic": "rcases eq_or_ne L \u22a5 with (rfl | hL)", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))", "state_after": "case inl\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\ninhabited_h : Inhabited \u03b9\ninst\u271d : IsCountablyGenerated \u22a5\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) \u22a5 (\ud835\udcdd l)\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) \u22a5 (\ud835\udcdd (f_lim x))\n\ncase inr\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))"}, {"tactic": "haveI : NeBot L := \u27e8hL\u27e9", "state_before": "case inr\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))", "state_after": "case inr\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))"}, {"tactic": "let p : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l : \u03b2, Tendsto (fun n => f' n) L (\ud835\udcdd l)", "state_before": "case inr\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))", "state_after": "case inr\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))"}, {"tactic": "have hp_mem : \u2200 x \u2208 aeSeqSet hf p, p x fun n => f n x := fun x hx =>\n  aeSeq.fun_prop_of_mem_aeSeqSet hf hx", "state_before": "case inr\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))", "state_after": "case inr\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))"}, {"tactic": "have h_ae_eq : \u2200\u1d50 x \u2202\u03bc, \u2200 n, aeSeq hf p n x = f n x := aeSeq.aeSeq_eq_fun_ae hf h_ae_tendsto", "state_before": "case inr\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))", "state_after": "case inr\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))"}, {"tactic": "set f_lim : \u03b1 \u2192 \u03b2 := fun x => dite (x \u2208 aeSeqSet hf p) (fun h => (hp_mem x h).choose)\n  fun h => (\u27e8f default x\u27e9 : Nonempty \u03b2).some", "state_before": "case inr\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))", "state_after": "case inr\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n  fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))"}, {"tactic": "have h_ae_tendsto_f_lim : \u2200\u1d50 x \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x)) :=\n  h_ae_eq.mono fun x hx => (hf_lim x).congr hx", "state_before": "case inr\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n  fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nhf_lim : \u2200 (x : \u03b1), Tendsto (fun n => aeSeq hf p n x) L (\ud835\udcdd (f_lim x))\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))", "state_after": "case inr\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n  fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nhf_lim : \u2200 (x : \u03b1), Tendsto (fun n => aeSeq hf p n x) L (\ud835\udcdd (f_lim x))\nh_ae_tendsto_f_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))"}, {"tactic": "have h_f_lim_meas : Measurable f_lim :=\n  measurable_of_tendsto_metrizable' L (aeSeq.measurable hf p)\n    (tendsto_pi_nhds.mpr fun x => hf_lim x)", "state_before": "case inr\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n  fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nhf_lim : \u2200 (x : \u03b1), Tendsto (fun n => aeSeq hf p n x) L (\ud835\udcdd (f_lim x))\nh_ae_tendsto_f_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))", "state_after": "case inr\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n  fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nhf_lim : \u2200 (x : \u03b1), Tendsto (fun n => aeSeq hf p n x) L (\ud835\udcdd (f_lim x))\nh_ae_tendsto_f_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))\nh_f_lim_meas : Measurable f_lim\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))"}, {"tactic": "exact \u27e8f_lim, h_f_lim_meas, h_ae_tendsto_f_lim\u27e9", "state_before": "case inr\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n  fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nhf_lim : \u2200 (x : \u03b1), Tendsto (fun n => aeSeq hf p n x) L (\ud835\udcdd (f_lim x))\nh_ae_tendsto_f_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))\nh_f_lim_meas : Measurable f_lim\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) L (\ud835\udcdd (f_lim x))", "state_after": "no goals"}, {"tactic": "exact \u27e8(hf default).mk _, (hf default).measurable_mk, eventually_of_forall fun x => tendsto_bot\u27e9", "state_before": "case inl\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\ninhabited_h : Inhabited \u03b9\ninst\u271d : IsCountablyGenerated \u22a5\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) \u22a5 (\ud835\udcdd l)\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) \u22a5 (\ud835\udcdd (f_lim x))", "state_after": "no goals"}, {"tactic": "intro x", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n  fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\n\u22a2 \u2200 (x : \u03b1), Tendsto (fun n => aeSeq hf p n x) L (\ud835\udcdd (f_lim x))", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n  fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nx : \u03b1\n\u22a2 Tendsto (fun n => aeSeq hf p n x) L (\ud835\udcdd (f_lim x))"}, {"tactic": "simp only [aeSeq]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n  fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nx : \u03b1\n\u22a2 Tendsto (fun n => aeSeq hf p n x) L (\ud835\udcdd (f_lim x))", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n  fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nx : \u03b1\n\u22a2 Tendsto\n    (fun n =>\n      if x \u2208 aeSeqSet hf fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l) then\n        AEMeasurable.mk (f n) (_ : AEMeasurable (f n)) x\n      else Nonempty.some (_ : Nonempty \u03b2))\n    L\n    (\ud835\udcdd\n      (if h : x \u2208 aeSeqSet hf fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l) then\n        Exists.choose (_ : p x fun n => f n x)\n      else Nonempty.some (_ : Nonempty \u03b2)))"}, {"tactic": "split_ifs with h", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n  fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nx : \u03b1\n\u22a2 Tendsto\n    (fun n =>\n      if x \u2208 aeSeqSet hf fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l) then\n        AEMeasurable.mk (f n) (_ : AEMeasurable (f n)) x\n      else Nonempty.some (_ : Nonempty \u03b2))\n    L\n    (\ud835\udcdd\n      (if h : x \u2208 aeSeqSet hf fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l) then\n        Exists.choose (_ : p x fun n => f n x)\n      else Nonempty.some (_ : Nonempty \u03b2)))", "state_after": "case inl\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n  fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nx : \u03b1\nh : x \u2208 aeSeqSet hf fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\n\u22a2 Tendsto (fun n => AEMeasurable.mk (f n) (_ : AEMeasurable (f n)) x) L (\ud835\udcdd (Exists.choose (_ : p x fun n => f n x)))\n\ncase inr\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n  fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nx : \u03b1\nh : \u00acx \u2208 aeSeqSet hf fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\n\u22a2 Tendsto (fun n => Nonempty.some (_ : Nonempty \u03b2)) L (\ud835\udcdd (Nonempty.some (_ : Nonempty \u03b2)))"}, {"tactic": "refine' (hp_mem x h).choose_spec.congr fun n => _", "state_before": "case inl\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n  fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nx : \u03b1\nh : x \u2208 aeSeqSet hf fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\n\u22a2 Tendsto (fun n => AEMeasurable.mk (f n) (_ : AEMeasurable (f n)) x) L (\ud835\udcdd (Exists.choose (_ : p x fun n => f n x)))", "state_after": "case inl\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n  fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nx : \u03b1\nh : x \u2208 aeSeqSet hf fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nn : \u03b9\n\u22a2 (fun n => f n x) n = AEMeasurable.mk (f n) (_ : AEMeasurable (f n)) x"}, {"tactic": "exact (aeSeq.mk_eq_fun_of_mem_aeSeqSet hf h n).symm", "state_before": "case inl\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n  fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nx : \u03b1\nh : x \u2208 aeSeqSet hf fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nn : \u03b9\n\u22a2 (fun n => f n x) n = AEMeasurable.mk (f n) (_ : AEMeasurable (f n)) x", "state_after": "no goals"}, {"tactic": "exact tendsto_const_nhds", "state_before": "case inr\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\n\u03b9 : Type u_1\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03bc : MeasureTheory.Measure \u03b1\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\nhf : \u2200 (n : \u03b9), AEMeasurable (f n)\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) L (\ud835\udcdd l)\ninhabited_h : Inhabited \u03b9\nhL : L \u2260 \u22a5\nthis : NeBot L\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\nhp_mem : \u2200 (x : \u03b1), x \u2208 aeSeqSet hf p \u2192 p x fun n => f n x\nh_ae_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200 (n : \u03b9), aeSeq hf p n x = f n x\nf_lim : \u03b1 \u2192 \u03b2 :=\n  fun x => if h : x \u2208 aeSeqSet hf p then Exists.choose (_ : p x fun n => f n x) else Nonempty.some (_ : Nonempty \u03b2)\nx : \u03b1\nh : \u00acx \u2208 aeSeqSet hf fun x f' => \u2203 l, Tendsto (fun n => f' n) L (\ud835\udcdd l)\n\u22a2 Tendsto (fun n => Nonempty.some (_ : Nonempty \u03b2)) L (\ud835\udcdd (Nonempty.some (_ : Nonempty \u03b2)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Dual.lean", "full_name": "Subspace.dual_finrank_eq", "start": [1109, 1], "end": [1110, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "full_name": "CategoryTheory.Limits.colimit.hom_ext", "start": [837, 1], "end": [839, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Digits.lean", "full_name": "Nat.coe_int_ofDigits", "start": [216, 1], "end": [219, 43], "traced_tactics": [{"tactic": "induction' L with d L _", "state_before": "n b : \u2115\nL : List \u2115\n\u22a2 \u2191(ofDigits b L) = ofDigits (\u2191b) L", "state_after": "case nil\nn b : \u2115\n\u22a2 \u2191(ofDigits b []) = ofDigits \u2191b []\n\ncase cons\nn b d : \u2115\nL : List \u2115\ntail_ih\u271d : \u2191(ofDigits b L) = ofDigits (\u2191b) L\n\u22a2 \u2191(ofDigits b (d :: L)) = ofDigits (\u2191b) (d :: L)"}, {"tactic": "rfl", "state_before": "case nil\nn b : \u2115\n\u22a2 \u2191(ofDigits b []) = ofDigits \u2191b []", "state_after": "no goals"}, {"tactic": "dsimp [ofDigits]", "state_before": "case cons\nn b d : \u2115\nL : List \u2115\ntail_ih\u271d : \u2191(ofDigits b L) = ofDigits (\u2191b) L\n\u22a2 \u2191(ofDigits b (d :: L)) = ofDigits (\u2191b) (d :: L)", "state_after": "case cons\nn b d : \u2115\nL : List \u2115\ntail_ih\u271d : \u2191(ofDigits b L) = ofDigits (\u2191b) L\n\u22a2 \u2191(d + b * ofDigits b L) = \u2191d + \u2191b * ofDigits (\u2191b) L"}, {"tactic": "push_cast", "state_before": "case cons\nn b d : \u2115\nL : List \u2115\ntail_ih\u271d : \u2191(ofDigits b L) = ofDigits (\u2191b) L\n\u22a2 \u2191(d + b * ofDigits b L) = \u2191d + \u2191b * ofDigits (\u2191b) L", 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[460, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/SymmDiff.lean", "full_name": "symmDiff_comm", "start": [117, 1], "end": [117, 75], "traced_tactics": [{"tactic": "simp only [symmDiff, sup_comm]", "state_before": "\u03b9 : Type ?u.17458\n\u03b1 : Type u_1\n\u03b2 : Type ?u.17464\n\u03c0 : \u03b9 \u2192 Type ?u.17469\ninst\u271d : GeneralizedCoheytingAlgebra \u03b1\na b c d : \u03b1\n\u22a2 a \u2206 b = b \u2206 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Valuation/Basic.lean", "full_name": "AddValuation.map_pow", "start": [715, 1], "end": [716, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Group/Prod.lean", "full_name": "MeasureTheory.measurePreserving_prod_mul_right", "start": [368, 1], "end": [371, 98], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Monoid/MinMax.lean", "full_name": "le_or_le_of_mul_le_mul", "start": [111, 1], "end": [115, 48], "traced_tactics": [{"tactic": "contrapose!", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.6020\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\n\u22a2 a\u2081 * b\u2081 \u2264 a\u2082 * b\u2082 \u2192 a\u2081 \u2264 a\u2082 \u2228 b\u2081 \u2264 b\u2082", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.6020\ninst\u271d\u00b3 : 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"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type ?u.304347\nM\u2086 : Type ?u.304350\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 map (fst R M M\u2082) (prod p q) = p", "state_after": "case h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type ?u.304347\nM\u2086 : Type ?u.304350\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx : M\n\u22a2 x \u2208 map (fst R M M\u2082) (prod p q) 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Type ?u.568854\n\u03b1 : Type u_1\n\u03b2 : Type ?u.568860\n\u03b3 : Type ?u.568863\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DivisionMonoid \u03b1\ns t : Finset \u03b1\n\u22a2 IsUnit \u2191s \u2194 IsUnit s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Dynamics/FixedPoints/Basic.lean", "full_name": "Function.IsFixedPt.perm_pow", "start": [112, 11], "end": [114, 20], "traced_tactics": [{"tactic": "rw [Equiv.Perm.coe_pow]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf fa g : \u03b1 \u2192 \u03b1\nx y : \u03b1\nfb : \u03b2 \u2192 \u03b2\nm n\u271d k : \u2115\ne : Perm \u03b1\nh : IsFixedPt (\u2191e) x\nn : \u2115\n\u22a2 IsFixedPt (\u2191(e ^ n)) x", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nf fa g : \u03b1 \u2192 \u03b1\nx y : \u03b1\nfb : \u03b2 \u2192 \u03b2\nm n\u271d k : \u2115\ne : Perm \u03b1\nh : IsFixedPt (\u2191e) x\nn : \u2115\n\u22a2 IsFixedPt (\u2191e^[n]) x"}, {"tactic": "exact h.iterate _", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf fa g : \u03b1 \u2192 \u03b1\nx y : \u03b1\nfb : \u03b2 \u2192 \u03b2\nm n\u271d k : \u2115\ne : Perm \u03b1\nh : IsFixedPt (\u2191e) x\nn : \u2115\n\u22a2 IsFixedPt (\u2191e^[n]) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/DiscreteQuotient.lean", "full_name": "DiscreteQuotient.LEComap.mono", "start": [306, 1], "end": [307, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/CharP/ExpChar.lean", "full_name": "expChar_is_prime_or_one", "start": [106, 1], "end": [109, 35], "traced_tactics": [{"tactic": "cases hq", "state_before": "R : Type u\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nq : \u2115\nhq : ExpChar R q\n\u22a2 Nat.Prime q \u2228 q = 1", "state_after": "case zero\nR : Type u\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\n\u22a2 Nat.Prime 1 \u2228 1 = 1\n\ncase prime\nR : Type u\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nq : \u2115\nhprime\u271d : Nat.Prime q\nhchar\u271d : CharP R q\n\u22a2 Nat.Prime q \u2228 q = 1"}, {"tactic": "case zero => exact .inr rfl", "state_before": "case zero\nR : Type u\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\n\u22a2 Nat.Prime 1 \u2228 1 = 1\n\ncase prime\nR : Type u\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nq : \u2115\nhprime\u271d : Nat.Prime q\nhchar\u271d : CharP R q\n\u22a2 Nat.Prime q \u2228 q = 1", "state_after": "case prime\nR : Type u\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nq : \u2115\nhprime\u271d : Nat.Prime q\nhchar\u271d : CharP R q\n\u22a2 Nat.Prime q \u2228 q = 1"}, {"tactic": "case prime hp _ => exact .inl hp", "state_before": "case prime\nR : Type u\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nq : \u2115\nhprime\u271d : Nat.Prime q\nhchar\u271d : CharP R q\n\u22a2 Nat.Prime q \u2228 q = 1", "state_after": "no goals"}, {"tactic": "exact .inr rfl", "state_before": "R : Type u\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : Nontrivial R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\n\u22a2 Nat.Prime 1 \u2228 1 = 1", "state_after": "no goals"}, {"tactic": "exact .inl hp", "state_before": "R : Type u\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Nontrivial R\ninst\u271d : NoZeroDivisors R\nq : \u2115\nhp : Nat.Prime q\nhchar\u271d : CharP R q\n\u22a2 Nat.Prime q \u2228 q = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Analytic/Basic.lean", "full_name": "FormalMultilinearSeries.norm_le_div_pow_of_pos_of_lt_radius", "start": [256, 1], "end": [259, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Field/Basic.lean", "full_name": "half_pos", "start": [506, 1], "end": [507, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Type.lean", "full_name": "Equiv.Perm.cycleType_one", "start": [86, 1], "end": [86, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Associated.lean", "full_name": "Prime.dvd_or_dvd", "start": [50, 1], "end": [51, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "full_name": "norm_of_subsingleton'", "start": [528, 1], "end": [529, 40], "traced_tactics": [{"tactic": "rw [Subsingleton.elim a 1, norm_one']", "state_before": "\ud835\udcd5 : Type ?u.74246\n\ud835\udd5c : Type ?u.74249\n\u03b1 : Type ?u.74252\n\u03b9 : Type ?u.74255\n\u03ba : Type ?u.74258\nE : Type u_1\nF : Type ?u.74264\nG : Type ?u.74267\ninst\u271d\u00b3 : SeminormedGroup E\ninst\u271d\u00b2 : SeminormedGroup F\ninst\u271d\u00b9 : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : Subsingleton E\na : E\n\u22a2 \u2016a\u2016 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/LiminfLimsup.lean", "full_name": "Filter.IsBoundedUnder.bddBelow_range_of_cofinite", "start": [150, 1], "end": [152, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/LinearIndependent.lean", "full_name": "Fintype.linearIndependent_iff", "start": [154, 1], "end": [161, 26], "traced_tactics": [{"tactic": "refine'\n  \u27e8fun H g => by simpa using linearIndependent_iff'.1 H Finset.univ g, fun H =>\n    linearIndependent_iff''.2 fun s g hg hs i => H _ _ _\u27e9", "state_before": "\u03b9 : Type u'\n\u03b9' : Type ?u.29646\nR : Type u_1\nK : Type ?u.29652\nM : Type u_2\nM' : Type ?u.29658\nM'' : Type ?u.29661\nV : Type u\nV' : Type ?u.29666\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Fintype \u03b9\n\u22a2 LinearIndependent R v \u2194 \u2200 (g : \u03b9 \u2192 R), \u2211 i : \u03b9, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0", "state_after": "\u03b9 : Type u'\n\u03b9' : Type ?u.29646\nR : Type u_1\nK : Type ?u.29652\nM : Type u_2\nM' : Type ?u.29658\nM'' : Type ?u.29661\nV : Type u\nV' : Type ?u.29666\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Fintype \u03b9\nH : \u2200 (g : \u03b9 \u2192 R), \u2211 i : \u03b9, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhg : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhs : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\n\u22a2 \u2211 i : \u03b9, g i \u2022 v i = 0"}, {"tactic": "rw [\u2190 hs]", "state_before": "\u03b9 : Type u'\n\u03b9' : Type ?u.29646\nR : Type u_1\nK : Type ?u.29652\nM : Type u_2\nM' : Type ?u.29658\nM'' : Type ?u.29661\nV : Type u\nV' : Type ?u.29666\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Fintype \u03b9\nH : \u2200 (g : \u03b9 \u2192 R), \u2211 i : \u03b9, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhg : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhs : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\n\u22a2 \u2211 i : \u03b9, g i \u2022 v i = 0", "state_after": "\u03b9 : Type u'\n\u03b9' : Type ?u.29646\nR : Type u_1\nK : Type ?u.29652\nM : Type u_2\nM' : Type ?u.29658\nM'' : Type ?u.29661\nV : Type u\nV' : Type ?u.29666\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Fintype \u03b9\nH : \u2200 (g : \u03b9 \u2192 R), \u2211 i : \u03b9, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhg : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhs : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\n\u22a2 \u2211 i : \u03b9, g i \u2022 v i = \u2211 i in s, g i \u2022 v i"}, {"tactic": "refine' (Finset.sum_subset (Finset.subset_univ _) fun i _ hi => _).symm", "state_before": "\u03b9 : Type u'\n\u03b9' : Type ?u.29646\nR : Type u_1\nK : Type ?u.29652\nM : Type u_2\nM' : Type ?u.29658\nM'' : Type ?u.29661\nV : Type u\nV' : Type ?u.29666\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Fintype \u03b9\nH : \u2200 (g : \u03b9 \u2192 R), \u2211 i : \u03b9, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhg : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhs : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\n\u22a2 \u2211 i : \u03b9, g i \u2022 v i = \u2211 i in s, g i \u2022 v i", "state_after": "\u03b9 : Type u'\n\u03b9' : Type ?u.29646\nR : Type u_1\nK : Type ?u.29652\nM : Type u_2\nM' : Type ?u.29658\nM'' : Type ?u.29661\nV : Type u\nV' : Type ?u.29666\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Fintype \u03b9\nH : \u2200 (g : \u03b9 \u2192 R), \u2211 i : \u03b9, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhg : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhs : \u2211 i in s, g i \u2022 v i = 0\ni\u271d i : \u03b9\nx\u271d : i \u2208 Finset.univ\nhi : \u00aci \u2208 s\n\u22a2 g i \u2022 v i = 0"}, {"tactic": "rw [hg i hi, zero_smul]", "state_before": "\u03b9 : Type u'\n\u03b9' : Type ?u.29646\nR : Type u_1\nK : Type ?u.29652\nM : Type u_2\nM' : Type ?u.29658\nM'' : Type ?u.29661\nV : Type u\nV' : Type ?u.29666\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Fintype \u03b9\nH : \u2200 (g : \u03b9 \u2192 R), \u2211 i : \u03b9, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhg : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhs : \u2211 i in s, g i \u2022 v i = 0\ni\u271d i : \u03b9\nx\u271d : i \u2208 Finset.univ\nhi : \u00aci \u2208 s\n\u22a2 g i \u2022 v i = 0", "state_after": "no goals"}, {"tactic": "simpa using linearIndependent_iff'.1 H Finset.univ g", "state_before": "\u03b9 : Type u'\n\u03b9' : Type ?u.29646\nR : Type u_1\nK : Type ?u.29652\nM : Type u_2\nM' : Type ?u.29658\nM'' : Type ?u.29661\nV : Type u\nV' : Type ?u.29666\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Fintype \u03b9\nH : LinearIndependent R v\ng : \u03b9 \u2192 R\n\u22a2 \u2211 i : \u03b9, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Polynomial/BigOperators.lean", "full_name": "Polynomial.degree_multiset_prod", "start": [335, 1], "end": [336, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/Padics/PadicVal.lean", "full_name": "padicValNat.zero", "start": [68, 11], "end": [68, 70], "traced_tactics": [{"tactic": "simp [padicValNat]", "state_before": "p : \u2115\n\u22a2 padicValNat p 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Types.lean", "full_name": "CategoryTheory.types_ext", "start": [61, 8], "end": [63, 12], "traced_tactics": [{"tactic": "funext x", "state_before": "\u03b1 \u03b2 : Type u\nf g : \u03b1 \u27f6 \u03b2\nh : \u2200 (a : \u03b1), f a = g a\n\u22a2 f = g", "state_after": "case h\n\u03b1 \u03b2 : Type u\nf g : \u03b1 \u27f6 \u03b2\nh : \u2200 (a : \u03b1), f a = g a\nx : \u03b1\n\u22a2 f x = g x"}, {"tactic": "exact h x", "state_before": "case h\n\u03b1 \u03b2 : Type u\nf g : \u03b1 \u27f6 \u03b2\nh : \u2200 (a : \u03b1), f a = g a\nx : \u03b1\n\u22a2 f x = g x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "full_name": "dist_div_eq_dist_mul_left", "start": [446, 1], "end": [447, 47], "traced_tactics": [{"tactic": "rw [\u2190 dist_mul_right _ _ b, div_mul_cancel']", "state_before": "\ud835\udcd5 : Type ?u.58803\n\ud835\udd5c : Type ?u.58806\n\u03b1 : Type ?u.58809\n\u03b9 : Type ?u.58812\n\u03ba : Type ?u.58815\nE : Type u_1\nF : Type ?u.58821\nG : Type ?u.58824\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na b c : E\n\u22a2 dist (a / b) c = dist a (c * b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/BooleanAlgebra.lean", "full_name": "sdiff_eq", "start": [575, 1], "end": [576, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.inv_coe_unit", "start": [707, 1], "end": [715, 6], "traced_tactics": [{"tactic": "have := congr_arg ((\u2191) : \u2115 \u2192 ZMod n) (val_coe_unit_coprime u)", "state_before": "n : \u2115\nu : (ZMod n)\u02e3\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9", "state_after": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191(Nat.gcd (val \u2191u) n) = \u21911\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9"}, {"tactic": "rw [\u2190 mul_inv_eq_gcd, Nat.cast_one] at this", "state_before": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191(Nat.gcd (val \u2191u) n) = \u21911\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9", "state_after": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9"}, {"tactic": "let u' : (ZMod n)\u02e3 := \u27e8u, (u : ZMod n)\u207b\u00b9, this, by rwa [mul_comm]\u27e9", "state_before": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9", "state_after": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9"}, {"tactic": "have h : u = u' := by\n  apply Units.ext\n  rfl", "state_before": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9", "state_after": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\nh : u = u'\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9"}, {"tactic": "rw [h]", "state_before": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\nh : u = u'\n\u22a2 (\u2191u)\u207b\u00b9 = \u2191u\u207b\u00b9", "state_after": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\nh : u = u'\n\u22a2 (\u2191u')\u207b\u00b9 = \u2191u'\u207b\u00b9"}, {"tactic": "rfl", "state_before": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\nh : u = u'\n\u22a2 (\u2191u')\u207b\u00b9 = \u2191u'\u207b\u00b9", "state_after": "no goals"}, {"tactic": "rwa [mul_comm]", "state_before": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\n\u22a2 (\u2191u)\u207b\u00b9 * \u2191u = 1", "state_after": "no goals"}, {"tactic": "apply Units.ext", "state_before": "n : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\n\u22a2 u = u'", "state_after": "case a\nn : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\n\u22a2 \u2191u = \u2191u'"}, {"tactic": "rfl", "state_before": "case a\nn : \u2115\nu : (ZMod n)\u02e3\nthis : \u2191u * (\u2191u)\u207b\u00b9 = 1\nu' : (ZMod n)\u02e3 := { val := \u2191u, inv := (\u2191u)\u207b\u00b9, val_inv := this, inv_val := (_ : (\u2191u)\u207b\u00b9 * \u2191u = 1) }\n\u22a2 \u2191u = \u2191u'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "full_name": "Zsqrtd.gcd_eq_zero_iff", "start": [389, 1], "end": [390, 75], "traced_tactics": [{"tactic": "simp only [Int.gcd_eq_zero_iff, ext, eq_self_iff_true, zero_im, zero_re]", "state_before": "d : \u2124\na : \u2124\u221ad\n\u22a2 Int.gcd a.re a.im = 0 \u2194 a = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Quaternion.lean", "full_name": "QuaternionAlgebra.sub_self_re", "start": [307, 1], "end": [308, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Iso.lean", "full_name": "CategoryTheory.Iso.symm_inv", "start": [115, 1], "end": [116, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/SuccPred/Limit.lean", "full_name": "Order.isPredLimit_iff_of_noMin", "start": [324, 1], "end": [325, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "full_name": "MeasureTheory.JordanDecomposition.toSignedMeasure_neg", "start": [183, 1], "end": [187, 69], "traced_tactics": [{"tactic": "ext1 i hi", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.17598\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\n\u22a2 toSignedMeasure (-j) = -toSignedMeasure j", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.17598\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(toSignedMeasure (-j)) i = \u2191(-toSignedMeasure j) i"}, {"tactic": "rw [neg_apply, toSignedMeasure, toSignedMeasure, toSignedMeasure_sub_apply hi,\n  toSignedMeasure_sub_apply hi, neg_sub, neg_posPart, neg_negPart]", "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.17598\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(toSignedMeasure (-j)) i = \u2191(-toSignedMeasure j) i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/BumpFunctionInner.lean", "full_name": "ContDiffWithinAt.contDiffBump", "start": [436, 11], "end": [446, 91], "traced_tactics": [{"tactic": "change ContDiffWithinAt \u211d n (uncurry (someContDiffBumpBase E).toFun \u2218 fun x : X =>\n  ((f x).rOut / (f x).rIn, (f x).rIn\u207b\u00b9 \u2022 (g x - c x))) s x", "state_before": "E : Type u_2\nX : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \u211d X\ninst\u271d : HasContDiffBump E\nc\u271d : E\nf\u271d : ContDiffBump c\u271d\nx\u271d : E\nn : \u2115\u221e\nc g : X \u2192 E\ns : Set X\nf : (x : X) \u2192 ContDiffBump (c x)\nx : X\nhc : ContDiffWithinAt \u211d n c s x\nhr : ContDiffWithinAt \u211d n (fun x => (f x).rIn) s x\nhR : ContDiffWithinAt \u211d n (fun x => (f x).rOut) s x\nhg : ContDiffWithinAt \u211d n g s x\n\u22a2 ContDiffWithinAt \u211d n (fun x => \u2191(f x) (g x)) s x", "state_after": "E : Type u_2\nX : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \u211d X\ninst\u271d : HasContDiffBump E\nc\u271d : E\nf\u271d : ContDiffBump c\u271d\nx\u271d : E\nn : \u2115\u221e\nc g : X \u2192 E\ns : Set X\nf : (x : X) \u2192 ContDiffBump (c x)\nx : X\nhc : ContDiffWithinAt \u211d n c s x\nhr : ContDiffWithinAt \u211d n (fun x => (f x).rIn) s x\nhR : ContDiffWithinAt \u211d n (fun x => (f x).rOut) s x\nhg : ContDiffWithinAt \u211d n g s x\n\u22a2 ContDiffWithinAt \u211d n\n    (uncurry (someContDiffBumpBase E).toFun \u2218 fun x => ((f x).rOut / (f x).rIn, (f x).rIn\u207b\u00b9 \u2022 (g x - c x))) s x"}, {"tactic": "refine (((someContDiffBumpBase E).smooth.contDiffAt ?_).of_le le_top).comp_contDiffWithinAt x ?_", "state_before": "E : Type u_2\nX : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \u211d X\ninst\u271d : HasContDiffBump E\nc\u271d : E\nf\u271d : ContDiffBump c\u271d\nx\u271d : E\nn : \u2115\u221e\nc g : X \u2192 E\ns : Set X\nf : (x : X) \u2192 ContDiffBump (c x)\nx : X\nhc : ContDiffWithinAt \u211d n c s x\nhr : ContDiffWithinAt \u211d n (fun x => (f x).rIn) s x\nhR : ContDiffWithinAt \u211d n (fun x => (f x).rOut) s x\nhg : ContDiffWithinAt \u211d n g s x\n\u22a2 ContDiffWithinAt \u211d n\n    (uncurry (someContDiffBumpBase E).toFun \u2218 fun x => ((f x).rOut / (f x).rIn, (f x).rIn\u207b\u00b9 \u2022 (g x - c x))) s x", "state_after": "case refine_1\nE : Type u_2\nX : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \u211d X\ninst\u271d : HasContDiffBump E\nc\u271d : E\nf\u271d : ContDiffBump c\u271d\nx\u271d : E\nn : \u2115\u221e\nc g : X \u2192 E\ns : Set X\nf : (x : X) \u2192 ContDiffBump (c x)\nx : X\nhc : ContDiffWithinAt \u211d n c s x\nhr : ContDiffWithinAt \u211d n (fun x => (f x).rIn) s x\nhR : ContDiffWithinAt \u211d n (fun x => (f x).rOut) s x\nhg : ContDiffWithinAt \u211d n g s x\n\u22a2 Ioi 1 \u00d7\u02e2 univ \u2208 \ud835\udcdd ((f x).rOut / (f x).rIn, (f x).rIn\u207b\u00b9 \u2022 (g x - c x))\n\ncase refine_2\nE : Type u_2\nX : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \u211d X\ninst\u271d : HasContDiffBump E\nc\u271d : E\nf\u271d : ContDiffBump c\u271d\nx\u271d : E\nn : \u2115\u221e\nc g : X \u2192 E\ns : Set X\nf : (x : X) \u2192 ContDiffBump (c x)\nx : X\nhc : ContDiffWithinAt \u211d n c s x\nhr : ContDiffWithinAt \u211d n (fun x => (f x).rIn) s x\nhR : ContDiffWithinAt \u211d n (fun x => (f x).rOut) s x\nhg : ContDiffWithinAt \u211d n g s x\n\u22a2 ContDiffWithinAt \u211d n (fun x => ((f x).rOut / (f x).rIn, (f x).rIn\u207b\u00b9 \u2022 (g x - c x))) s x"}, {"tactic": "exact prod_mem_nhds (Ioi_mem_nhds (f x).one_lt_rOut_div_rIn) univ_mem", "state_before": "case refine_1\nE : Type u_2\nX : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \u211d X\ninst\u271d : HasContDiffBump E\nc\u271d : E\nf\u271d : ContDiffBump c\u271d\nx\u271d : E\nn : \u2115\u221e\nc g : X \u2192 E\ns : Set X\nf : (x : X) \u2192 ContDiffBump (c x)\nx : X\nhc : ContDiffWithinAt \u211d n c s x\nhr : ContDiffWithinAt \u211d n (fun x => (f x).rIn) s x\nhR : ContDiffWithinAt \u211d n (fun x => (f x).rOut) s x\nhg : ContDiffWithinAt \u211d n g s x\n\u22a2 Ioi 1 \u00d7\u02e2 univ \u2208 \ud835\udcdd ((f x).rOut / (f x).rIn, (f x).rIn\u207b\u00b9 \u2022 (g x - c x))", "state_after": "no goals"}, {"tactic": "exact (hR.div hr (f x).rIn_pos.ne').prod ((hr.inv (f x).rIn_pos.ne').smul (hg.sub hc))", "state_before": "case refine_2\nE : Type u_2\nX : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \u211d X\ninst\u271d : HasContDiffBump E\nc\u271d : E\nf\u271d : ContDiffBump c\u271d\nx\u271d : E\nn : \u2115\u221e\nc g : X \u2192 E\ns : Set X\nf : (x : X) \u2192 ContDiffBump (c x)\nx : X\nhc : ContDiffWithinAt \u211d n c s x\nhr : ContDiffWithinAt \u211d n (fun x => (f x).rIn) s x\nhR : ContDiffWithinAt \u211d n (fun x => (f x).rOut) s x\nhg : ContDiffWithinAt \u211d n g s x\n\u22a2 ContDiffWithinAt \u211d n (fun x => ((f x).rOut / (f x).rIn, (f x).rIn\u207b\u00b9 \u2022 (g x - c x))) s x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "full_name": "WithBot.sSup_eq", "start": [79, 1], "end": [81, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Lie/Submodule.lean", "full_name": "LieHom.map_le_idealRange", "start": [941, 1], "end": [943, 33], "traced_tactics": [{"tactic": "rw [f.idealRange_eq_map]", "state_before": "R : Type u\nL : Type v\nL' : Type w\u2082\nM : Type w\nM' : Type w\u2081\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : LieRing L\ninst\u271d\u00b9\u2070 : LieAlgebra R L\ninst\u271d\u2079 : LieRing L'\ninst\u271d\u2078 : LieAlgebra R L'\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nf : L \u2192\u2097\u2045R\u2046 L'\nI : LieIdeal R L\nJ : LieIdeal R L'\n\u22a2 LieIdeal.map f I \u2264 idealRange f", "state_after": "R : Type u\nL : Type v\nL' : Type w\u2082\nM : Type w\nM' : Type w\u2081\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : LieRing L\ninst\u271d\u00b9\u2070 : LieAlgebra R L\ninst\u271d\u2079 : LieRing L'\ninst\u271d\u2078 : LieAlgebra R L'\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nf : L \u2192\u2097\u2045R\u2046 L'\nI : LieIdeal R L\nJ : LieIdeal R L'\n\u22a2 LieIdeal.map f I \u2264 LieIdeal.map f \u22a4"}, {"tactic": "exact LieIdeal.map_mono le_top", "state_before": "R : Type u\nL : Type v\nL' : Type w\u2082\nM : Type w\nM' : Type w\u2081\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : LieRing L\ninst\u271d\u00b9\u2070 : LieAlgebra R L\ninst\u271d\u2079 : LieRing L'\ninst\u271d\u2078 : LieAlgebra R L'\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nf : L \u2192\u2097\u2045R\u2046 L'\nI : LieIdeal R L\nJ : LieIdeal R L'\n\u22a2 LieIdeal.map f I \u2264 LieIdeal.map f \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/FreeMonoid/Count.lean", "full_name": "FreeAddMonoid.countp_apply", "start": [39, 1], "end": [39, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/QuotientGroup.lean", "full_name": "QuotientGroup.comap_comap_center", "start": [680, 1], "end": [685, 28], "traced_tactics": [{"tactic": "ext x", "state_before": "G : Type u\ninst\u271d\u00b3 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b2 : Group H\n\u03c6 : G \u2192* H\nH\u2081 : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal H\u2081\nH\u2082 : Subgroup (G \u29f8 H\u2081)\ninst\u271d : Subgroup.Normal H\u2082\n\u22a2 Subgroup.comap (mk' H\u2081) (Subgroup.comap (mk' H\u2082) (Subgroup.center ((G \u29f8 H\u2081) \u29f8 H\u2082))) =\n    Subgroup.comap (mk' (Subgroup.comap (mk' H\u2081) H\u2082)) (Subgroup.center (G \u29f8 Subgroup.comap (mk' H\u2081) H\u2082))", "state_after": "case h\nG : Type u\ninst\u271d\u00b3 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b2 : Group H\n\u03c6 : G \u2192* H\nH\u2081 : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal H\u2081\nH\u2082 : Subgroup (G \u29f8 H\u2081)\ninst\u271d : Subgroup.Normal H\u2082\nx : G\n\u22a2 x \u2208 Subgroup.comap (mk' H\u2081) (Subgroup.comap (mk' H\u2082) (Subgroup.center ((G \u29f8 H\u2081) \u29f8 H\u2082))) \u2194\n    x \u2208 Subgroup.comap (mk' (Subgroup.comap (mk' H\u2081) H\u2082)) (Subgroup.center (G \u29f8 Subgroup.comap (mk' H\u2081) H\u2082))"}, {"tactic": "simp only [mk'_apply, Subgroup.mem_comap, Subgroup.mem_center_iff, forall_mk, \u2190 mk_mul,\n  eq_iff_div_mem, mk_div]", "state_before": "case h\nG : Type u\ninst\u271d\u00b3 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b2 : Group H\n\u03c6 : G \u2192* H\nH\u2081 : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal H\u2081\nH\u2082 : Subgroup (G \u29f8 H\u2081)\ninst\u271d : Subgroup.Normal H\u2082\nx : G\n\u22a2 x \u2208 Subgroup.comap (mk' H\u2081) (Subgroup.comap (mk' H\u2082) (Subgroup.center ((G \u29f8 H\u2081) \u29f8 H\u2082))) \u2194\n    x \u2208 Subgroup.comap (mk' (Subgroup.comap (mk' H\u2081) H\u2082)) (Subgroup.center (G \u29f8 Subgroup.comap (mk' H\u2081) H\u2082))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Order.lean", "full_name": "Finset.exists_le", "start": [32, 1], "end": [34, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "full_name": "ENNReal.essSup_eq_zero_iff", "start": [314, 1], "end": [315, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/AList.lean", "full_name": "AList.insert_empty", "start": [284, 1], "end": [285, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/Symmetric.lean", "full_name": "Matrix.IsSymm.smul", "start": [117, 1], "end": [118, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Factorization/Basic.lean", "full_name": "Nat.factorization_pow", "start": [260, 1], "end": [265, 13], "traced_tactics": [{"tactic": "induction' k with k ih", "state_before": "n k : \u2115\n\u22a2 factorization (n ^ k) = k \u2022 factorization n", "state_after": "case zero\nn : \u2115\n\u22a2 factorization (n ^ zero) = zero \u2022 factorization n\n\ncase succ\nn k : \u2115\nih : factorization (n ^ k) = k \u2022 factorization n\n\u22a2 factorization (n ^ succ k) = succ k \u2022 factorization n"}, {"tactic": "rcases eq_or_ne n 0 with (rfl | hn)", "state_before": "case succ\nn k : \u2115\nih : factorization (n ^ k) = k \u2022 factorization n\n\u22a2 factorization (n ^ succ k) = succ k \u2022 factorization n", "state_after": "case succ.inl\nk : \u2115\nih : factorization (0 ^ k) = k \u2022 factorization 0\n\u22a2 factorization (0 ^ succ k) = succ k \u2022 factorization 0\n\ncase succ.inr\nn k : \u2115\nih : factorization (n ^ k) = k \u2022 factorization n\nhn : n \u2260 0\n\u22a2 factorization (n ^ succ k) = succ k \u2022 factorization n"}, {"tactic": "rw [pow_succ, mul_comm, factorization_mul hn (pow_ne_zero _ hn), ih, succ_eq_one_add, add_smul,\n one_smul]", "state_before": "case succ.inr\nn k : \u2115\nih : factorization (n ^ k) = k \u2022 factorization n\nhn : n \u2260 0\n\u22a2 factorization (n ^ succ k) = succ k \u2022 factorization n", "state_after": "no goals"}, {"tactic": "simp", "state_before": "case zero\nn : \u2115\n\u22a2 factorization (n ^ zero) = zero \u2022 factorization n", "state_after": "no goals"}, {"tactic": "simp", "state_before": "case succ.inl\nk : \u2115\nih : factorization (0 ^ k) = k \u2022 factorization 0\n\u22a2 factorization (0 ^ succ k) = succ k \u2022 factorization 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/FreeProduct.lean", "full_name": "FreeProduct.NeWord.of_word", "start": [569, 1], "end": [587, 33], "traced_tactics": [{"tactic": "suffices : \u2203 (i j : _)(w' : NeWord M i j), w'.toWord.toList = w.toList", "state_before": "\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nw : Word M\nh : w \u2260 empty\n\u22a2 \u2203 i j w', toWord w' = w", "state_after": "\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nw : Word M\nh : w \u2260 empty\nthis : \u2203 i j w', (toWord w').toList = w.toList\n\u22a2 \u2203 i j w', toWord w' = w\n\ncase this\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nw : Word M\nh : w \u2260 empty\n\u22a2 \u2203 i j w', (toWord w').toList = w.toList"}, {"tactic": "cases' w with l hnot1 hchain", "state_before": "case this\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nw : Word M\nh : w \u2260 empty\n\u22a2 \u2203 i j w', (toWord w').toList = w.toList", "state_after": "case this.mk\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl : List ((i : \u03b9) \u00d7 M i)\nhnot1 : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 l \u2192 l_1.snd \u2260 1\nhchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\nh : { toList := l, ne_one := hnot1, chain_ne := hchain } \u2260 empty\n\u22a2 \u2203 i j w', (toWord w').toList = { toList := l, ne_one := hnot1, chain_ne := hchain }.toList"}, {"tactic": "induction' l with x l hi", "state_before": "case this.mk\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl : List ((i : \u03b9) \u00d7 M i)\nhnot1 : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 l \u2192 l_1.snd \u2260 1\nhchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\nh : { toList := l, ne_one := hnot1, chain_ne := hchain } \u2260 empty\n\u22a2 \u2203 i j w', (toWord w').toList = { toList := l, ne_one := hnot1, chain_ne := hchain }.toList", "state_after": "case this.mk.nil\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 l \u2192 l_1.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\nh\u271d : { toList := l, ne_one := hnot1\u271d, chain_ne := hchain\u271d } \u2260 empty\nhnot1 : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 [] \u2192 l.snd \u2260 1\nhchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) []\nh : { toList := [], ne_one := hnot1, chain_ne := hchain } \u2260 empty\n\u22a2 \u2203 i j w', (toWord w').toList = { toList := [], ne_one := hnot1, chain_ne := hchain }.toList\n\ncase this.mk.cons\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl\u271d : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 l\u271d \u2192 l.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\u271d\nh\u271d : { toList := l\u271d, ne_one := hnot1\u271d, chain_ne := hchain\u271d } \u2260 empty\nx : (i : \u03b9) \u00d7 M i\nl : List ((i : \u03b9) \u00d7 M i)\nhi :\n  \u2200 (hnot1 : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 l \u2192 l_1.snd \u2260 1) (hchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l),\n    { toList := l, ne_one := hnot1, chain_ne := hchain } \u2260 empty \u2192\n      \u2203 i j w', (toWord w').toList = { toList := l, ne_one := hnot1, chain_ne := hchain }.toList\nhnot1 : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 x :: l \u2192 l_1.snd \u2260 1\nhchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (x :: l)\nh : { toList := x :: l, ne_one := hnot1, chain_ne := hchain } \u2260 empty\n\u22a2 \u2203 i j w', (toWord w').toList = { toList := x :: l, ne_one := hnot1, chain_ne := hchain }.toList"}, {"tactic": "rcases this with \u27e8i, j, w, h\u27e9", "state_before": "\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nw : Word M\nh : w \u2260 empty\nthis : \u2203 i j w', (toWord w').toList = w.toList\n\u22a2 \u2203 i j w', toWord w' = w", "state_after": "case intro.intro.intro\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nw\u271d : Word M\nh\u271d : w\u271d \u2260 empty\ni j : \u03b9\nw : NeWord M i j\nh : (toWord w).toList = w\u271d.toList\n\u22a2 \u2203 i j w', toWord w' = w\u271d"}, {"tactic": "refine' \u27e8i, j, w, _\u27e9", "state_before": "case intro.intro.intro\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nw\u271d : Word M\nh\u271d : w\u271d \u2260 empty\ni j : \u03b9\nw : NeWord M i j\nh : (toWord w).toList = w\u271d.toList\n\u22a2 \u2203 i j w', toWord w' = w\u271d", "state_after": "case intro.intro.intro\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nw\u271d : Word M\nh\u271d : w\u271d \u2260 empty\ni j : \u03b9\nw : NeWord M i j\nh : (toWord w).toList = w\u271d.toList\n\u22a2 toWord w = w\u271d"}, {"tactic": "ext", "state_before": "case intro.intro.intro\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nw\u271d : Word M\nh\u271d : w\u271d \u2260 empty\ni j : \u03b9\nw : NeWord M i j\nh : (toWord w).toList = w\u271d.toList\n\u22a2 toWord w = w\u271d", "state_after": "case intro.intro.intro.toList.a.a\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nw\u271d : Word M\nh\u271d : w\u271d \u2260 empty\ni j : \u03b9\nw : NeWord M i j\nh : (toWord w).toList = w\u271d.toList\nn\u271d : \u2115\na\u271d : (i : \u03b9) \u00d7 M i\n\u22a2 a\u271d \u2208 List.get? (toWord w).toList n\u271d \u2194 a\u271d \u2208 List.get? w\u271d.toList n\u271d"}, {"tactic": "rw [h]", "state_before": "case intro.intro.intro.toList.a.a\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nw\u271d : Word M\nh\u271d : w\u271d \u2260 empty\ni j : \u03b9\nw : NeWord M i j\nh : (toWord w).toList = w\u271d.toList\nn\u271d : \u2115\na\u271d : (i : \u03b9) \u00d7 M i\n\u22a2 a\u271d \u2208 List.get? (toWord w).toList n\u271d \u2194 a\u271d \u2208 List.get? w\u271d.toList n\u271d", "state_after": "no goals"}, {"tactic": "contradiction", "state_before": "case this.mk.nil\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 l \u2192 l_1.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\nh\u271d : { toList := l, ne_one := hnot1\u271d, chain_ne := hchain\u271d } \u2260 empty\nhnot1 : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 [] \u2192 l.snd \u2260 1\nhchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) []\nh : { toList := [], ne_one := hnot1, chain_ne := hchain } \u2260 empty\n\u22a2 \u2203 i j w', (toWord w').toList = { toList := [], ne_one := hnot1, chain_ne := hchain }.toList", "state_after": "no goals"}, {"tactic": "rw [List.forall_mem_cons] at hnot1", "state_before": "case this.mk.cons\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl\u271d : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 l\u271d \u2192 l.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\u271d\nh\u271d : { toList := l\u271d, ne_one := hnot1\u271d, chain_ne := hchain\u271d } \u2260 empty\nx : (i : \u03b9) \u00d7 M i\nl : List ((i : \u03b9) \u00d7 M i)\nhi :\n  \u2200 (hnot1 : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 l \u2192 l_1.snd \u2260 1) (hchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l),\n    { toList := l, ne_one := hnot1, chain_ne := hchain } \u2260 empty \u2192\n      \u2203 i j w', (toWord w').toList = { toList := l, ne_one := hnot1, chain_ne := hchain }.toList\nhnot1 : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 x :: l \u2192 l_1.snd \u2260 1\nhchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (x :: l)\nh : { toList := x :: l, ne_one := hnot1, chain_ne := hchain } \u2260 empty\n\u22a2 \u2203 i j w', (toWord w').toList = { toList := x :: l, ne_one := hnot1, chain_ne := hchain }.toList", "state_after": "case this.mk.cons\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl\u271d : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d\u00b9 : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 l\u271d \u2192 l.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\u271d\nh\u271d : { toList := l\u271d, ne_one := hnot1\u271d\u00b9, chain_ne := hchain\u271d } \u2260 empty\nx : (i : \u03b9) \u00d7 M i\nl : List ((i : \u03b9) \u00d7 M i)\nhi :\n  \u2200 (hnot1 : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 l \u2192 l_1.snd \u2260 1) (hchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l),\n    { toList := l, ne_one := hnot1, chain_ne := hchain } \u2260 empty \u2192\n      \u2203 i j w', (toWord w').toList = { toList := l, ne_one := hnot1, chain_ne := hchain }.toList\nhnot1\u271d : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 x :: l \u2192 l_1.snd \u2260 1\nhnot1 : x.snd \u2260 1 \u2227 \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 l \u2192 x.snd \u2260 1\nhchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (x :: l)\nh : { toList := x :: l, ne_one := hnot1\u271d, chain_ne := hchain } \u2260 empty\n\u22a2 \u2203 i j w', (toWord w').toList = { toList := x :: l, ne_one := hnot1\u271d, chain_ne := hchain }.toList"}, {"tactic": "cases' l with y l", "state_before": "case this.mk.cons\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl\u271d : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d\u00b9 : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 l\u271d \u2192 l.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\u271d\nh\u271d : { toList := l\u271d, ne_one := hnot1\u271d\u00b9, chain_ne := hchain\u271d } \u2260 empty\nx : (i : \u03b9) \u00d7 M i\nl : List ((i : \u03b9) \u00d7 M i)\nhi :\n  \u2200 (hnot1 : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 l \u2192 l_1.snd \u2260 1) (hchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l),\n    { toList := l, ne_one := hnot1, chain_ne := hchain } \u2260 empty \u2192\n      \u2203 i j w', (toWord w').toList = { toList := l, ne_one := hnot1, chain_ne := hchain }.toList\nhnot1\u271d : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 x :: l \u2192 l_1.snd \u2260 1\nhnot1 : x.snd \u2260 1 \u2227 \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 l \u2192 x.snd \u2260 1\nhchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (x :: l)\nh : { toList := x :: l, ne_one := hnot1\u271d, chain_ne := hchain } \u2260 empty\n\u22a2 \u2203 i j w', (toWord w').toList = { toList := x :: l, ne_one := hnot1\u271d, chain_ne := hchain }.toList", "state_after": "case this.mk.cons.nil\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d\u00b9 : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 l \u2192 l_1.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\nh\u271d : { toList := l, ne_one := hnot1\u271d\u00b9, chain_ne := hchain\u271d } \u2260 empty\nx : (i : \u03b9) \u00d7 M i\nhi :\n  \u2200 (hnot1 : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 [] \u2192 l.snd \u2260 1) (hchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) []),\n    { toList := [], ne_one := hnot1, chain_ne := hchain } \u2260 empty \u2192\n      \u2203 i j w', (toWord w').toList = { toList := [], ne_one := hnot1, chain_ne := hchain }.toList\nhnot1\u271d : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 [x] \u2192 l.snd \u2260 1\nhnot1 : x.snd \u2260 1 \u2227 \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 [] \u2192 x.snd \u2260 1\nhchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) [x]\nh : { toList := [x], ne_one := hnot1\u271d, chain_ne := hchain } \u2260 empty\n\u22a2 \u2203 i j w', (toWord w').toList = { toList := [x], ne_one := hnot1\u271d, chain_ne := hchain }.toList\n\ncase this.mk.cons.cons\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl\u271d : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d\u00b9 : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 l\u271d \u2192 l.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\u271d\nh\u271d : { toList := l\u271d, ne_one := hnot1\u271d\u00b9, chain_ne := hchain\u271d } \u2260 empty\nx y : (i : \u03b9) \u00d7 M i\nl : List ((i : \u03b9) \u00d7 M i)\nhi :\n  \u2200 (hnot1 : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 y :: l \u2192 l_1.snd \u2260 1)\n    (hchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (y :: l)),\n    { toList := y :: l, ne_one := hnot1, chain_ne := hchain } \u2260 empty \u2192\n      \u2203 i j w', (toWord w').toList = { toList := y :: l, ne_one := hnot1, chain_ne := hchain }.toList\nhnot1\u271d : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 x :: y :: l \u2192 l_1.snd \u2260 1\nhnot1 : x.snd \u2260 1 \u2227 \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 y :: l \u2192 x.snd \u2260 1\nhchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (x :: y :: l)\nh : { toList := x :: y :: l, ne_one := hnot1\u271d, chain_ne := hchain } \u2260 empty\n\u22a2 \u2203 i j w', (toWord w').toList = { toList := x :: y :: l, ne_one := hnot1\u271d, chain_ne := hchain }.toList"}, {"tactic": "refine' \u27e8x.1, x.1, singleton x.2 hnot1.1, _\u27e9", "state_before": "case this.mk.cons.nil\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d\u00b9 : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 l \u2192 l_1.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\nh\u271d : { toList := l, ne_one := hnot1\u271d\u00b9, chain_ne := hchain\u271d } \u2260 empty\nx : (i : \u03b9) \u00d7 M i\nhi :\n  \u2200 (hnot1 : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 [] \u2192 l.snd \u2260 1) (hchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) []),\n    { toList := [], ne_one := hnot1, chain_ne := hchain } \u2260 empty \u2192\n      \u2203 i j w', (toWord w').toList = { toList := [], ne_one := hnot1, chain_ne := hchain }.toList\nhnot1\u271d : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 [x] \u2192 l.snd \u2260 1\nhnot1 : x.snd \u2260 1 \u2227 \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 [] \u2192 x.snd \u2260 1\nhchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) [x]\nh : { toList := [x], ne_one := hnot1\u271d, chain_ne := hchain } \u2260 empty\n\u22a2 \u2203 i j w', (toWord w').toList = { toList := [x], ne_one := hnot1\u271d, chain_ne := hchain }.toList", "state_after": "case this.mk.cons.nil\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d\u00b9 : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 l \u2192 l_1.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\nh\u271d : { toList := l, ne_one := hnot1\u271d\u00b9, chain_ne := hchain\u271d } \u2260 empty\nx : (i : \u03b9) \u00d7 M i\nhi :\n  \u2200 (hnot1 : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 [] \u2192 l.snd \u2260 1) (hchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) []),\n    { toList := [], ne_one := hnot1, chain_ne := hchain } \u2260 empty \u2192\n      \u2203 i j w', (toWord w').toList = { toList := [], ne_one := hnot1, chain_ne := hchain }.toList\nhnot1\u271d : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 [x] \u2192 l.snd \u2260 1\nhnot1 : x.snd \u2260 1 \u2227 \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 [] \u2192 x.snd \u2260 1\nhchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) [x]\nh : { toList := [x], ne_one := hnot1\u271d, chain_ne := hchain } \u2260 empty\n\u22a2 (toWord (singleton x.snd (_ : x.snd \u2260 1))).toList = { toList := [x], ne_one := hnot1\u271d, chain_ne := hchain }.toList"}, {"tactic": "simp [toWord]", "state_before": "case this.mk.cons.nil\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d\u00b9 : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 l \u2192 l_1.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\nh\u271d : { toList := l, ne_one := hnot1\u271d\u00b9, chain_ne := hchain\u271d } \u2260 empty\nx : (i : \u03b9) \u00d7 M i\nhi :\n  \u2200 (hnot1 : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 [] \u2192 l.snd \u2260 1) (hchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) []),\n    { toList := [], ne_one := hnot1, chain_ne := hchain } \u2260 empty \u2192\n      \u2203 i j w', (toWord w').toList = { toList := [], ne_one := hnot1, chain_ne := hchain }.toList\nhnot1\u271d : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 [x] \u2192 l.snd \u2260 1\nhnot1 : x.snd \u2260 1 \u2227 \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 [] \u2192 x.snd \u2260 1\nhchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) [x]\nh : { toList := [x], ne_one := hnot1\u271d, chain_ne := hchain } \u2260 empty\n\u22a2 (toWord (singleton x.snd (_ : x.snd \u2260 1))).toList = { toList := [x], ne_one := hnot1\u271d, chain_ne := hchain }.toList", "state_after": "no goals"}, {"tactic": "rw [List.chain'_cons] at hchain", "state_before": "case this.mk.cons.cons\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl\u271d : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d\u00b9 : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 l\u271d \u2192 l.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\u271d\nh\u271d : { toList := l\u271d, ne_one := hnot1\u271d\u00b9, chain_ne := hchain\u271d } \u2260 empty\nx y : (i : \u03b9) \u00d7 M i\nl : List ((i : \u03b9) \u00d7 M i)\nhi :\n  \u2200 (hnot1 : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 y :: l \u2192 l_1.snd \u2260 1)\n    (hchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (y :: l)),\n    { toList := y :: l, ne_one := hnot1, chain_ne := hchain } \u2260 empty \u2192\n      \u2203 i j w', (toWord w').toList = { toList := y :: l, ne_one := hnot1, chain_ne := hchain }.toList\nhnot1\u271d : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 x :: y :: l \u2192 l_1.snd \u2260 1\nhnot1 : x.snd \u2260 1 \u2227 \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 y :: l \u2192 x.snd \u2260 1\nhchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (x :: y :: l)\nh : { toList := x :: y :: l, ne_one := hnot1\u271d, chain_ne := hchain } \u2260 empty\n\u22a2 \u2203 i j w', (toWord w').toList = { toList := x :: y :: l, ne_one := hnot1\u271d, chain_ne := hchain }.toList", "state_after": "case this.mk.cons.cons\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl\u271d : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d\u00b9 : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 l\u271d \u2192 l.snd \u2260 1\nhchain\u271d\u00b9 : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\u271d\nh\u271d : { toList := l\u271d, ne_one := hnot1\u271d\u00b9, chain_ne := hchain\u271d\u00b9 } \u2260 empty\nx y : (i : \u03b9) \u00d7 M i\nl : List ((i : \u03b9) \u00d7 M i)\nhi :\n  \u2200 (hnot1 : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 y :: l \u2192 l_1.snd \u2260 1)\n    (hchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (y :: l)),\n    { toList := y :: l, ne_one := hnot1, chain_ne := hchain } \u2260 empty \u2192\n      \u2203 i j w', (toWord w').toList = { toList := y :: l, ne_one := hnot1, chain_ne := hchain }.toList\nhnot1\u271d : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 x :: y :: l \u2192 l_1.snd \u2260 1\nhnot1 : x.snd \u2260 1 \u2227 \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 y :: l \u2192 x.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (x :: y :: l)\nhchain : x.fst \u2260 y.fst \u2227 List.Chain' (fun l l' => l.fst \u2260 l'.fst) (y :: l)\nh : { toList := x :: y :: l, ne_one := hnot1\u271d, chain_ne := hchain\u271d } \u2260 empty\n\u22a2 \u2203 i j w', (toWord w').toList = { toList := x :: y :: l, ne_one := hnot1\u271d, chain_ne := hchain\u271d }.toList"}, {"tactic": "specialize hi hnot1.2 hchain.2 (by rintro \u27e8rfl\u27e9)", "state_before": "case this.mk.cons.cons\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl\u271d : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d\u00b9 : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 l\u271d \u2192 l.snd \u2260 1\nhchain\u271d\u00b9 : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\u271d\nh\u271d : { toList := l\u271d, ne_one := hnot1\u271d\u00b9, chain_ne := hchain\u271d\u00b9 } \u2260 empty\nx y : (i : \u03b9) \u00d7 M i\nl : List ((i : \u03b9) \u00d7 M i)\nhi :\n  \u2200 (hnot1 : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 y :: l \u2192 l_1.snd \u2260 1)\n    (hchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (y :: l)),\n    { toList := y :: l, ne_one := hnot1, chain_ne := hchain } \u2260 empty \u2192\n      \u2203 i j w', (toWord w').toList = { toList := y :: l, ne_one := hnot1, chain_ne := hchain }.toList\nhnot1\u271d : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 x :: y :: l \u2192 l_1.snd \u2260 1\nhnot1 : x.snd \u2260 1 \u2227 \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 y :: l \u2192 x.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (x :: y :: l)\nhchain : x.fst \u2260 y.fst \u2227 List.Chain' (fun l l' => l.fst \u2260 l'.fst) (y :: l)\nh : { toList := x :: y :: l, ne_one := hnot1\u271d, chain_ne := hchain\u271d } \u2260 empty\n\u22a2 \u2203 i j w', (toWord w').toList = { toList := x :: y :: l, ne_one := hnot1\u271d, chain_ne := hchain\u271d }.toList", "state_after": "case this.mk.cons.cons\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl\u271d : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d\u00b9 : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 l\u271d \u2192 l.snd \u2260 1\nhchain\u271d\u00b9 : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\u271d\nh\u271d : { toList := l\u271d, ne_one := hnot1\u271d\u00b9, chain_ne := hchain\u271d\u00b9 } \u2260 empty\nx y : (i : \u03b9) \u00d7 M i\nl : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 x :: y :: l \u2192 l_1.snd \u2260 1\nhnot1 : x.snd \u2260 1 \u2227 \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 y :: l \u2192 x.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (x :: y :: l)\nhchain : x.fst \u2260 y.fst \u2227 List.Chain' (fun l l' => l.fst \u2260 l'.fst) (y :: l)\nh : { toList := x :: y :: l, ne_one := hnot1\u271d, chain_ne := hchain\u271d } \u2260 empty\nhi :\n  \u2203 i j w',\n    (toWord w').toList =\n      { toList := y :: l, ne_one := (_ : \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 y :: l \u2192 x.snd \u2260 1),\n          chain_ne := (_ : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (y :: l)) }.toList\n\u22a2 \u2203 i j w', (toWord w').toList = { toList := x :: y :: l, ne_one := hnot1\u271d, chain_ne := hchain\u271d }.toList"}, {"tactic": "obtain \u27e8i, j, w', hw' : w'.toList = y::l\u27e9 := hi", "state_before": "case this.mk.cons.cons\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl\u271d : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d\u00b9 : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 l\u271d \u2192 l.snd \u2260 1\nhchain\u271d\u00b9 : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\u271d\nh\u271d : { toList := l\u271d, ne_one := hnot1\u271d\u00b9, chain_ne := hchain\u271d\u00b9 } \u2260 empty\nx y : (i : \u03b9) \u00d7 M i\nl : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 x :: y :: l \u2192 l_1.snd \u2260 1\nhnot1 : x.snd \u2260 1 \u2227 \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 y :: l \u2192 x.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (x :: y :: l)\nhchain : x.fst \u2260 y.fst \u2227 List.Chain' (fun l l' => l.fst \u2260 l'.fst) (y :: l)\nh : { toList := x :: y :: l, ne_one := hnot1\u271d, chain_ne := hchain\u271d } \u2260 empty\nhi :\n  \u2203 i j w',\n    (toWord w').toList =\n      { toList := y :: l, ne_one := (_ : \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 y :: l \u2192 x.snd \u2260 1),\n          chain_ne := (_ : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (y :: l)) }.toList\n\u22a2 \u2203 i j w', (toWord w').toList = { toList := x :: y :: l, ne_one := hnot1\u271d, chain_ne := hchain\u271d }.toList", "state_after": "case this.mk.cons.cons.intro.intro.intro\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl\u271d : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d\u00b9 : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 l\u271d \u2192 l.snd \u2260 1\nhchain\u271d\u00b9 : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\u271d\nh\u271d : { toList := l\u271d, ne_one := hnot1\u271d\u00b9, chain_ne := hchain\u271d\u00b9 } \u2260 empty\nx y : (i : \u03b9) \u00d7 M i\nl : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 x :: y :: l \u2192 l_1.snd \u2260 1\nhnot1 : x.snd \u2260 1 \u2227 \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 y :: l \u2192 x.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (x :: y :: l)\nhchain : x.fst \u2260 y.fst \u2227 List.Chain' (fun l l' => l.fst \u2260 l'.fst) (y :: l)\nh : { toList := x :: y :: l, ne_one := hnot1\u271d, chain_ne := hchain\u271d } \u2260 empty\ni j : \u03b9\nw' : NeWord M i j\nhw' : toList w' = y :: l\n\u22a2 \u2203 i j w', (toWord w').toList = { toList := x :: y :: l, ne_one := hnot1\u271d, chain_ne := hchain\u271d }.toList"}, {"tactic": "obtain rfl : y = \u27e8i, w'.head\u27e9 := by simpa [hw'] using w'.toList_head?", "state_before": "case this.mk.cons.cons.intro.intro.intro\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl\u271d : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d\u00b9 : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 l\u271d \u2192 l.snd \u2260 1\nhchain\u271d\u00b9 : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\u271d\nh\u271d : { toList := l\u271d, ne_one := hnot1\u271d\u00b9, chain_ne := hchain\u271d\u00b9 } \u2260 empty\nx y : (i : \u03b9) \u00d7 M i\nl : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 x :: y :: l \u2192 l_1.snd \u2260 1\nhnot1 : x.snd \u2260 1 \u2227 \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 y :: l \u2192 x.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (x :: y :: l)\nhchain : x.fst \u2260 y.fst \u2227 List.Chain' (fun l l' => l.fst \u2260 l'.fst) (y :: l)\nh : { toList := x :: y :: l, ne_one := hnot1\u271d, chain_ne := hchain\u271d } \u2260 empty\ni j : \u03b9\nw' : NeWord M i j\nhw' : toList w' = y :: l\n\u22a2 \u2203 i j w', (toWord w').toList = { toList := x :: y :: l, ne_one := hnot1\u271d, chain_ne := hchain\u271d }.toList", "state_after": "case this.mk.cons.cons.intro.intro.intro\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl\u271d : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d\u00b9 : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 l\u271d \u2192 l.snd \u2260 1\nhchain\u271d\u00b9 : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\u271d\nh\u271d : { toList := l\u271d, ne_one := hnot1\u271d\u00b9, chain_ne := hchain\u271d\u00b9 } \u2260 empty\nx : (i : \u03b9) \u00d7 M i\nl : List ((i : \u03b9) \u00d7 M i)\ni j : \u03b9\nw' : NeWord M i j\nhnot1\u271d : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 x :: { fst := i, snd := head w' } :: l \u2192 l_1.snd \u2260 1\nhnot1 : x.snd \u2260 1 \u2227 \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 { fst := i, snd := head w' } :: l \u2192 x.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (x :: { fst := i, snd := head w' } :: l)\nhchain :\n  x.fst \u2260 { fst := i, snd := head w' }.fst \u2227\n    List.Chain' (fun l l' => l.fst \u2260 l'.fst) ({ fst := i, snd := head w' } :: l)\nh : { toList := x :: { fst := i, snd := head w' } :: l, ne_one := hnot1\u271d, chain_ne := hchain\u271d } \u2260 empty\nhw' : toList w' = { fst := i, snd := head w' } :: l\n\u22a2 \u2203 i_1 j_1 w'_1,\n    (toWord w'_1).toList =\n      { toList := x :: { fst := i, snd := head w' } :: l, ne_one := hnot1\u271d, chain_ne := hchain\u271d }.toList"}, {"tactic": "refine' \u27e8x.1, j, append (singleton x.2 hnot1.1) hchain.1 w', _\u27e9", "state_before": "case this.mk.cons.cons.intro.intro.intro\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl\u271d : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d\u00b9 : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 l\u271d \u2192 l.snd \u2260 1\nhchain\u271d\u00b9 : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\u271d\nh\u271d : { toList := l\u271d, ne_one := hnot1\u271d\u00b9, chain_ne := hchain\u271d\u00b9 } \u2260 empty\nx : (i : \u03b9) \u00d7 M i\nl : List ((i : \u03b9) \u00d7 M i)\ni j : \u03b9\nw' : NeWord M i j\nhnot1\u271d : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 x :: { fst := i, snd := head w' } :: l \u2192 l_1.snd \u2260 1\nhnot1 : x.snd \u2260 1 \u2227 \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 { fst := i, snd := head w' } :: l \u2192 x.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (x :: { fst := i, snd := head w' } :: l)\nhchain :\n  x.fst \u2260 { fst := i, snd := head w' }.fst \u2227\n    List.Chain' (fun l l' => l.fst \u2260 l'.fst) ({ fst := i, snd := head w' } :: l)\nh : { toList := x :: { fst := i, snd := head w' } :: l, ne_one := hnot1\u271d, chain_ne := hchain\u271d } \u2260 empty\nhw' : toList w' = { fst := i, snd := head w' } :: l\n\u22a2 \u2203 i_1 j_1 w'_1,\n    (toWord w'_1).toList =\n      { toList := x :: { fst := i, snd := head w' } :: l, ne_one := hnot1\u271d, chain_ne := hchain\u271d }.toList", "state_after": "case this.mk.cons.cons.intro.intro.intro\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl\u271d : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d\u00b9 : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 l\u271d \u2192 l.snd \u2260 1\nhchain\u271d\u00b9 : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\u271d\nh\u271d : { toList := l\u271d, ne_one := hnot1\u271d\u00b9, chain_ne := hchain\u271d\u00b9 } \u2260 empty\nx : (i : \u03b9) \u00d7 M i\nl : List ((i : \u03b9) \u00d7 M i)\ni j : \u03b9\nw' : NeWord M i j\nhnot1\u271d : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 x :: { fst := i, snd := head w' } :: l \u2192 l_1.snd \u2260 1\nhnot1 : x.snd \u2260 1 \u2227 \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 { fst := i, snd := head w' } :: l \u2192 x.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (x :: { fst := i, snd := head w' } :: l)\nhchain :\n  x.fst \u2260 { fst := i, snd := head w' }.fst \u2227\n    List.Chain' (fun l l' => l.fst \u2260 l'.fst) ({ fst := i, snd := head w' } :: l)\nh : { toList := x :: { fst := i, snd := head w' } :: l, ne_one := hnot1\u271d, chain_ne := hchain\u271d } \u2260 empty\nhw' : toList w' = { fst := i, snd := head w' } :: l\n\u22a2 (toWord (append (singleton x.snd (_ : x.snd \u2260 1)) (_ : x.fst \u2260 { fst := i, snd := head w' }.fst) w')).toList =\n    { toList := x :: { fst := i, snd := head w' } :: l, ne_one := hnot1\u271d, chain_ne := hchain\u271d }.toList"}, {"tactic": "rintro \u27e8rfl\u27e9", "state_before": "\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl\u271d : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d\u00b9 : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 l\u271d \u2192 l.snd \u2260 1\nhchain\u271d\u00b9 : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\u271d\nh\u271d : { toList := l\u271d, ne_one := hnot1\u271d\u00b9, chain_ne := hchain\u271d\u00b9 } \u2260 empty\nx y : (i : \u03b9) \u00d7 M i\nl : List ((i : \u03b9) \u00d7 M i)\nhi :\n  \u2200 (hnot1 : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 y :: l \u2192 l_1.snd \u2260 1)\n    (hchain : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (y :: l)),\n    { toList := y :: l, ne_one := hnot1, chain_ne := hchain } \u2260 empty \u2192\n      \u2203 i j w', (toWord w').toList = { toList := y :: l, ne_one := hnot1, chain_ne := hchain }.toList\nhnot1\u271d : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 x :: y :: l \u2192 l_1.snd \u2260 1\nhnot1 : x.snd \u2260 1 \u2227 \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 y :: l \u2192 x.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (x :: y :: l)\nhchain : x.fst \u2260 y.fst \u2227 List.Chain' (fun l l' => l.fst \u2260 l'.fst) (y :: l)\nh : { toList := x :: y :: l, ne_one := hnot1\u271d, chain_ne := hchain\u271d } \u2260 empty\n\u22a2 { toList := y :: l, ne_one := (_ : \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 y :: l \u2192 x.snd \u2260 1),\n      chain_ne := (_ : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (y :: l)) } \u2260\n    empty", "state_after": "no goals"}, {"tactic": "simpa [hw'] using w'.toList_head?", "state_before": "\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl\u271d : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d\u00b9 : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 l\u271d \u2192 l.snd \u2260 1\nhchain\u271d\u00b9 : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\u271d\nh\u271d : { toList := l\u271d, ne_one := hnot1\u271d\u00b9, chain_ne := hchain\u271d\u00b9 } \u2260 empty\nx y : (i : \u03b9) \u00d7 M i\nl : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 x :: y :: l \u2192 l_1.snd \u2260 1\nhnot1 : x.snd \u2260 1 \u2227 \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 y :: l \u2192 x.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (x :: y :: l)\nhchain : x.fst \u2260 y.fst \u2227 List.Chain' (fun l l' => l.fst \u2260 l'.fst) (y :: l)\nh : { toList := x :: y :: l, ne_one := hnot1\u271d, chain_ne := hchain\u271d } \u2260 empty\ni j : \u03b9\nw' : NeWord M i j\nhw' : toList w' = y :: l\n\u22a2 y = { fst := i, snd := head w' }", "state_after": "no goals"}, {"tactic": "simpa [toWord] using hw'", "state_before": "case this.mk.cons.cons.intro.intro.intro\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type ?u.593472\ninst\u271d : Monoid N\nl\u271d : List ((i : \u03b9) \u00d7 M i)\nhnot1\u271d\u00b9 : \u2200 (l : (i : \u03b9) \u00d7 M i), l \u2208 l\u271d \u2192 l.snd \u2260 1\nhchain\u271d\u00b9 : List.Chain' (fun l l' => l.fst \u2260 l'.fst) l\u271d\nh\u271d : { toList := l\u271d, ne_one := hnot1\u271d\u00b9, chain_ne := hchain\u271d\u00b9 } \u2260 empty\nx : (i : \u03b9) \u00d7 M i\nl : List ((i : \u03b9) \u00d7 M i)\ni j : \u03b9\nw' : NeWord M i j\nhnot1\u271d : \u2200 (l_1 : (i : \u03b9) \u00d7 M i), l_1 \u2208 x :: { fst := i, snd := head w' } :: l \u2192 l_1.snd \u2260 1\nhnot1 : x.snd \u2260 1 \u2227 \u2200 (x : (i : \u03b9) \u00d7 M i), x \u2208 { fst := i, snd := head w' } :: l \u2192 x.snd \u2260 1\nhchain\u271d : List.Chain' (fun l l' => l.fst \u2260 l'.fst) (x :: { fst := i, snd := head w' } :: l)\nhchain :\n  x.fst \u2260 { fst := i, snd := head w' }.fst \u2227\n    List.Chain' (fun l l' => l.fst \u2260 l'.fst) ({ fst := i, snd := head w' } :: l)\nh : { toList := x :: { fst := i, snd := head w' } :: l, ne_one := hnot1\u271d, chain_ne := hchain\u271d } \u2260 empty\nhw' : toList w' = { fst := i, snd := head w' } :: l\n\u22a2 (toWord (append (singleton x.snd (_ : x.snd \u2260 1)) (_ : x.fst \u2260 { fst := i, snd := head w' }.fst) w')).toList =\n    { toList := x :: { fst := i, snd := head w' } :: l, ne_one := hnot1\u271d, chain_ne := hchain\u271d }.toList", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/FractionalIdeal.lean", "full_name": "FractionalIdeal.spanSingleton_ne_zero_iff", "start": [1354, 1], "end": [1355, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "full_name": "Metric.infDist_le_infDist_add_hausdorffDist", "start": [824, 1], "end": [829, 55], "traced_tactics": [{"tactic": "refine toReal_le_add' infEdist_le_infEdist_add_hausdorffEdist (fun h \u21a6 ?_) (flip absurd fin)", "state_before": "\u03b9 : Sort ?u.82341\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nfin : hausdorffEdist s t \u2260 \u22a4\n\u22a2 infDist x t \u2264 infDist x s + hausdorffDist s t", "state_after": "\u03b9 : Sort ?u.82341\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nfin : hausdorffEdist s t \u2260 \u22a4\nh : infEdist x s = \u22a4\n\u22a2 infEdist x t = \u22a4"}, {"tactic": "rw [infEdist_eq_top_iff, \u2190 not_nonempty_iff_eq_empty] at h \u22a2", "state_before": "\u03b9 : Sort ?u.82341\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nfin : hausdorffEdist s t \u2260 \u22a4\nh : infEdist x s = \u22a4\n\u22a2 infEdist x t = \u22a4", "state_after": "\u03b9 : Sort ?u.82341\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace 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FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 (\u2200 (n : \u2115\u221e), ContDiff \ud835\udd5c n f) \u2194 \u2200 (n : \u2115), ContDiff \ud835\udd5c (\u2191n) f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean", "full_name": "BoxIntegral.TaggedPrepartition.IsPartition.biUnionPrepartition", "start": [192, 1], "end": [195, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GroupPower/Order.lean", "full_name": "one_le_zpow", "start": [373, 1], "end": [376, 32], "traced_tactics": [{"tactic": "lift n to \u2115 using hn", "state_before": "\u03b2 : Type ?u.197165\nA : Type ?u.197168\nG : Type u_1\nM : Type ?u.197174\nR : Type ?u.197177\ninst\u271d\u00b2 : DivInvMonoid G\ninst\u271d\u00b9 : Preorder G\ninst\u271d : CovariantClass G G (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : G\nH : 1 \u2264 x\nn : \u2124\nhn : 0 \u2264 n\n\u22a2 1 \u2264 x ^ n", "state_after": "case intro\n\u03b2 : Type ?u.197165\nA : Type ?u.197168\nG : Type u_1\nM : Type ?u.197174\nR : Type ?u.197177\ninst\u271d\u00b2 : DivInvMonoid G\ninst\u271d\u00b9 : Preorder G\ninst\u271d : CovariantClass G G (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : G\nH : 1 \u2264 x\nn : \u2115\n\u22a2 1 \u2264 x ^ \u2191n"}, {"tactic": "rw [zpow_ofNat]", "state_before": "case intro\n\u03b2 : Type ?u.197165\nA : Type ?u.197168\nG : Type u_1\nM : Type ?u.197174\nR : Type ?u.197177\ninst\u271d\u00b2 : DivInvMonoid G\ninst\u271d\u00b9 : Preorder G\ninst\u271d : CovariantClass G G (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : G\nH : 1 \u2264 x\nn : \u2115\n\u22a2 1 \u2264 x ^ \u2191n", "state_after": "case intro\n\u03b2 : Type ?u.197165\nA : Type ?u.197168\nG : Type u_1\nM : Type ?u.197174\nR : Type ?u.197177\ninst\u271d\u00b2 : DivInvMonoid G\ninst\u271d\u00b9 : Preorder G\ninst\u271d : CovariantClass G G (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : G\nH : 1 \u2264 x\nn : \u2115\n\u22a2 1 \u2264 x ^ n"}, {"tactic": "apply one_le_pow_of_one_le' H", "state_before": "case intro\n\u03b2 : Type ?u.197165\nA : Type ?u.197168\nG : Type u_1\nM : Type ?u.197174\nR : Type ?u.197177\ninst\u271d\u00b2 : DivInvMonoid G\ninst\u271d\u00b9 : Preorder G\ninst\u271d : CovariantClass G G (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : G\nH : 1 \u2264 x\nn : \u2115\n\u22a2 1 \u2264 x ^ n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Log/Base.lean", "full_name": "Real.tendsto_logb_atTop", "start": [243, 1], "end": [244, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Sort.lean", "full_name": "List.mergeSort_singleton", "start": [472, 1], "end": [472, 86], "traced_tactics": [{"tactic": "rw [List.mergeSort]", "state_before": "\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na : \u03b1\n\u22a2 mergeSort r [a] = [a]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.disjoint_Ioi_Iio", "start": [517, 1], "end": [518, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finsupp/Multiset.lean", "full_name": "Finsupp.sum_id_lt_of_lt", "start": [223, 1], "end": [226, 32], "traced_tactics": [{"tactic": "rw [\u2190 card_toMultiset, \u2190 card_toMultiset]", "state_before": "\u03b1 : Type ?u.101435\n\u03b2 : Type ?u.101438\n\u03b9 : Type u_1\nm n : \u03b9 \u2192\u2080 \u2115\nh : m < n\n\u22a2 (sum m fun x => id) < sum n fun x => id", "state_after": "\u03b1 : Type ?u.101435\n\u03b2 : Type ?u.101438\n\u03b9 : Type u_1\nm n : \u03b9 \u2192\u2080 \u2115\nh : m < n\n\u22a2 \u2191Multiset.card (\u2191toMultiset m) < \u2191Multiset.card (\u2191toMultiset n)"}, {"tactic": "apply Multiset.card_lt_of_lt", "state_before": "\u03b1 : Type ?u.101435\n\u03b2 : Type ?u.101438\n\u03b9 : Type u_1\nm n : \u03b9 \u2192\u2080 \u2115\nh : m < n\n\u22a2 \u2191Multiset.card (\u2191toMultiset m) < \u2191Multiset.card (\u2191toMultiset n)", "state_after": "case h\n\u03b1 : Type ?u.101435\n\u03b2 : Type ?u.101438\n\u03b9 : Type u_1\nm n : \u03b9 \u2192\u2080 \u2115\nh : m < n\n\u22a2 \u2191toMultiset m < \u2191toMultiset n"}, {"tactic": "exact toMultiset_strictMono h", "state_before": "case h\n\u03b1 : Type ?u.101435\n\u03b2 : Type ?u.101438\n\u03b9 : Type u_1\nm n : \u03b9 \u2192\u2080 \u2115\nh : m < n\n\u22a2 \u2191toMultiset m < \u2191toMultiset n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Analytic/Linear.lean", "full_name": "ContinuousLinearMap.hasFPowerSeriesAt_bilinear", "start": [111, 11], "end": [113, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.le_pure_iff", "start": [2607, 1], "end": [2608, 47], "traced_tactics": [{"tactic": "rw [\u2190 principal_singleton, le_principal_iff]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.289304\n\u03b9 : Sort x\nf : Filter \u03b1\na : \u03b1\n\u22a2 f \u2264 pure a \u2194 {a} \u2208 f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/PathConnected.lean", "full_name": "Path.coe_mk", "start": [147, 1], "end": [148, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/String/Basic.lean", "full_name": "String.ltb_cons_addChar", "start": [66, 1], "end": [80, 46], "traced_tactics": [{"tactic": "apply ltb.inductionOn \u27e8\u27e8cs\u2081\u27e9, i\u2081\u27e9 \u27e8\u27e8cs\u2082\u27e9, i\u2082\u27e9 (motive := fun \u27e8\u27e8cs\u2081\u27e9, i\u2081\u27e9 \u27e8\u27e8cs\u2082\u27e9, i\u2082\u27e9 \u21a6\n  ltb \u27e8\u27e8c :: cs\u2081\u27e9, i\u2081 + c\u27e9 \u27e8\u27e8c :: cs\u2082\u27e9, i\u2082 + c\u27e9 =\n  ltb \u27e8\u27e8cs\u2081\u27e9, i\u2081\u27e9 \u27e8\u27e8cs\u2082\u27e9, i\u2082\u27e9) <;> simp <;>\nintro \u27e8cs\u2081\u27e9 \u27e8cs\u2082\u27e9 i\u2081 i\u2082 <;>\nintros <;>\n(conv => lhs; rw [ltb]) <;> (conv => rhs; rw [ltb]) <;>\nsimp [Iterator.hasNext_cons_addChar, *]", "state_before": "c : Char\ncs\u2081 cs\u2082 : List Char\ni\u2081 i\u2082 : Pos\n\u22a2 ltb { s := { data := c :: cs\u2081 }, i := i\u2081 + c } { s := { data := c :: cs\u2082 }, i := i\u2082 + c } =\n    ltb { s := { data := cs\u2081 }, i := i\u2081 } { s := { data := cs\u2082 }, i := i\u2082 }", "state_after": "case ind\nc : Char\ncs\u2081\u271d cs\u2082\u271d : List Char\ni\u2081\u271d i\u2082\u271d : Pos\ncs\u2081 cs\u2082 : List Char\ni\u2081 i\u2082 : Pos\na\u271d\u00b3 : Iterator.hasNext { s := { data := cs\u2082 }, i := i\u2082 } = true\na\u271d\u00b2 : Iterator.hasNext { s := { data := cs\u2081 }, i := i\u2081 } = true\na\u271d\u00b9 : get { data := cs\u2081 } i\u2081 = get { data := cs\u2082 } i\u2082\na\u271d :\n  ltb\n      { s := { data := c :: (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }).1.data },\n        i := (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }).i + c }\n      { s := { data := c :: (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 }).1.data },\n        i := (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 }).i + c } =\n    ltb\n      { s := { data := (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }).1.data },\n        i := (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }).i }\n      { s := { data := (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 }).1.data },\n        i := (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 }).i }\n\u22a2 (if\n        Iterator.curr { s := { data := c :: cs\u2081 }, i := i\u2081 + c } =\n          Iterator.curr { s := { data := c :: cs\u2082 }, i := i\u2082 + c } then\n      ltb (Iterator.next { s := { data := c :: cs\u2081 }, i := i\u2081 + c })\n        (Iterator.next { s := { data := c :: cs\u2082 }, i := i\u2082 + c })\n    else\n      decide\n        (Iterator.curr { s := { data := c :: cs\u2081 }, i := i\u2081 + c } <\n          Iterator.curr { s := { data := c :: cs\u2082 }, i := i\u2082 + c })) =\n    if Iterator.curr { s := { data := cs\u2081 }, i := i\u2081 } = Iterator.curr { s := { data := cs\u2082 }, i := i\u2082 } then\n      ltb (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }) (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 })\n    else decide (Iterator.curr { s := { data := cs\u2081 }, i := i\u2081 } < Iterator.curr { s := { data := cs\u2082 }, i := i\u2082 })\n\ncase eq\nc : Char\ncs\u2081\u271d cs\u2082\u271d : List Char\ni\u2081\u271d i\u2082\u271d : Pos\ncs\u2081 cs\u2082 : List Char\ni\u2081 i\u2082 : Pos\na\u271d\u00b2 : Iterator.hasNext { s := { data := cs\u2082 }, i := i\u2082 } = true\na\u271d\u00b9 : Iterator.hasNext { s := { data := cs\u2081 }, i := i\u2081 } = true\na\u271d : \u00acget { data := cs\u2081 } i\u2081 = get { data := cs\u2082 } i\u2082\n\u22a2 (if\n        Iterator.curr { s := { data := c :: cs\u2081 }, i := i\u2081 + c } =\n          Iterator.curr { s := { data := c :: cs\u2082 }, i := i\u2082 + c } then\n      ltb (Iterator.next { s := { data := c :: cs\u2081 }, i := i\u2081 + c })\n        (Iterator.next { s := { data := c :: cs\u2082 }, i := i\u2082 + c })\n    else\n      decide\n        (Iterator.curr { s := { data := c :: cs\u2081 }, i := i\u2081 + c } <\n          Iterator.curr { s := { data := c :: cs\u2082 }, i := i\u2082 + c })) =\n    if Iterator.curr { s := { data := cs\u2081 }, i := i\u2081 } = Iterator.curr { s := { data := cs\u2082 }, i := i\u2082 } then\n      ltb (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }) (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 })\n    else decide (Iterator.curr { s := { data := cs\u2081 }, i := i\u2081 } < Iterator.curr { s := { data := cs\u2082 }, i := i\u2082 })"}, {"tactic": "conv => lhs; rw [ltb]", "state_before": "case base\u2082\nc : Char\ncs\u2081\u271d cs\u2082\u271d : List Char\ni\u2081\u271d i\u2082\u271d : Pos\ncs\u2081 cs\u2082 : List Char\ni\u2081 i\u2082 : Pos\na\u271d : Iterator.hasNext { s := { data := cs\u2082 }, i := i\u2082 } = false\n\u22a2 ltb { s := { data := c :: { data := cs\u2081 }.data }, i := i\u2081 + c }\n      { s := { data := c :: { data := cs\u2082 }.data }, i := i\u2082 + c } =\n    ltb { s := { data := { data := cs\u2081 }.data }, i := i\u2081 } { s := { data := { data := cs\u2082 }.data }, i := i\u2082 }", "state_after": "case base\u2082\nc : Char\ncs\u2081\u271d cs\u2082\u271d : List Char\ni\u2081\u271d i\u2082\u271d : Pos\ncs\u2081 cs\u2082 : List Char\ni\u2081 i\u2082 : Pos\na\u271d : Iterator.hasNext { s := { data := cs\u2082 }, i := i\u2082 } = false\n\u22a2 (if Iterator.hasNext { s := { data := c :: { data := cs\u2082 }.data }, i := i\u2082 + c } = true then\n      if Iterator.hasNext { s := { data := c :: { data := cs\u2081 }.data }, i := i\u2081 + c } = true then\n        if\n            Iterator.curr { s := { data := c :: { data := cs\u2081 }.data }, i := i\u2081 + c } =\n              Iterator.curr { s := { data := c :: { data := cs\u2082 }.data }, i := i\u2082 + c } then\n          ltb (Iterator.next { s := { data := c :: { data := cs\u2081 }.data }, i := i\u2081 + c })\n            (Iterator.next { s := { data := c :: { data := cs\u2082 }.data }, i := i\u2082 + c })\n        else\n          decide\n            (Iterator.curr { s := { data := c :: { data := cs\u2081 }.data }, i := i\u2081 + c } <\n              Iterator.curr { s := { data := c :: { data := cs\u2082 }.data }, i := i\u2082 + c })\n      else true\n    else false) =\n    ltb { s := { data := { data := cs\u2081 }.data }, i := i\u2081 } { s := { data := { data := cs\u2082 }.data }, i := i\u2082 }"}, {"tactic": "conv => rhs; rw [ltb]", "state_before": "case base\u2082\nc : Char\ncs\u2081\u271d cs\u2082\u271d : List Char\ni\u2081\u271d i\u2082\u271d : Pos\ncs\u2081 cs\u2082 : List Char\ni\u2081 i\u2082 : Pos\na\u271d : Iterator.hasNext { s := { data := cs\u2082 }, i := i\u2082 } = false\n\u22a2 (if Iterator.hasNext { s := { data := c :: { data := cs\u2082 }.data }, i := i\u2082 + c } = true then\n      if Iterator.hasNext { s := { data := c :: { data := cs\u2081 }.data }, i := i\u2081 + c } = true then\n        if\n            Iterator.curr { s := { data := c :: { data := cs\u2081 }.data }, i := i\u2081 + c } =\n              Iterator.curr { s := { data := c :: { data := cs\u2082 }.data }, i := i\u2082 + c } then\n          ltb (Iterator.next { s := { data := c :: { data := cs\u2081 }.data }, i := i\u2081 + c })\n            (Iterator.next { s := { data := c :: { data := cs\u2082 }.data }, i := i\u2082 + c })\n        else\n          decide\n            (Iterator.curr { s := { data := c :: { data := cs\u2081 }.data }, i := i\u2081 + c } <\n              Iterator.curr { s := { data := c :: { data := cs\u2082 }.data }, i := i\u2082 + c })\n      else true\n    else false) =\n    ltb { s := { data := { data := cs\u2081 }.data }, i := i\u2081 } { s := { data := { data := cs\u2082 }.data }, i := i\u2082 }", "state_after": "case base\u2082\nc : Char\ncs\u2081\u271d cs\u2082\u271d : List Char\ni\u2081\u271d i\u2082\u271d : Pos\ncs\u2081 cs\u2082 : List Char\ni\u2081 i\u2082 : Pos\na\u271d : Iterator.hasNext { s := { data := cs\u2082 }, i := i\u2082 } = false\n\u22a2 (if Iterator.hasNext { s := { data := c :: { data := cs\u2082 }.data }, i := i\u2082 + c } = true then\n      if Iterator.hasNext { s := { data := c :: { data := cs\u2081 }.data }, i := i\u2081 + c } = true then\n        if\n            Iterator.curr { s := { data := c :: { data := cs\u2081 }.data }, i := i\u2081 + c } =\n              Iterator.curr { s := { data := c :: { data := cs\u2082 }.data }, i := i\u2082 + c } then\n          ltb (Iterator.next { s := { data := c :: { data := cs\u2081 }.data }, i := i\u2081 + c })\n            (Iterator.next { s := { data := c :: { data := cs\u2082 }.data }, i := i\u2082 + c })\n        else\n          decide\n            (Iterator.curr { s := { data := c :: { data := cs\u2081 }.data }, i := i\u2081 + c } <\n              Iterator.curr { s := { data := c :: { data := cs\u2082 }.data }, i := i\u2082 + c })\n      else true\n    else false) =\n    if Iterator.hasNext { s := { data := { data := cs\u2082 }.data }, i := i\u2082 } = true then\n      if Iterator.hasNext { s := { data := { data := cs\u2081 }.data }, i := i\u2081 } = true then\n        if\n            Iterator.curr { s := { data := { data := cs\u2081 }.data }, i := i\u2081 } =\n              Iterator.curr { s := { data := { data := cs\u2082 }.data }, i := i\u2082 } then\n          ltb (Iterator.next { s := { data := { data := cs\u2081 }.data }, i := i\u2081 })\n            (Iterator.next { s := { data := { data := cs\u2082 }.data }, i := i\u2082 })\n        else\n          decide\n            (Iterator.curr { s := { data := { data := cs\u2081 }.data }, i := i\u2081 } <\n              Iterator.curr { s := { data := { data := cs\u2082 }.data }, i := i\u2082 })\n      else true\n    else false"}, {"tactic": "rename_i h\u2082 h\u2081 heq ih", "state_before": "case ind\nc : Char\ncs\u2081\u271d cs\u2082\u271d : List Char\ni\u2081\u271d i\u2082\u271d : Pos\ncs\u2081 cs\u2082 : List Char\ni\u2081 i\u2082 : Pos\na\u271d\u00b3 : Iterator.hasNext { s := { data := cs\u2082 }, i := i\u2082 } = true\na\u271d\u00b2 : Iterator.hasNext { s := { data := cs\u2081 }, i := i\u2081 } = true\na\u271d\u00b9 : get { data := cs\u2081 } i\u2081 = get { data := cs\u2082 } i\u2082\na\u271d :\n  ltb\n      { s := { data := c :: (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }).1.data },\n        i := (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }).i + c }\n      { s := { data := c :: (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 }).1.data },\n        i := (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 }).i + c } =\n    ltb\n      { s := { data := (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }).1.data },\n        i := (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }).i }\n      { s := { data := (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 }).1.data },\n        i := (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 }).i }\n\u22a2 (if\n        Iterator.curr { s := { data := c :: cs\u2081 }, i := i\u2081 + c } =\n          Iterator.curr { s := { data := c :: cs\u2082 }, i := i\u2082 + c } then\n      ltb (Iterator.next { s := { data := c :: cs\u2081 }, i := i\u2081 + c })\n        (Iterator.next { s := { data := c :: cs\u2082 }, i := i\u2082 + c })\n    else\n      decide\n        (Iterator.curr { s := { data := c :: cs\u2081 }, i := i\u2081 + c } <\n          Iterator.curr { s := { data := c :: cs\u2082 }, i := i\u2082 + c })) =\n    if Iterator.curr { s := { data := cs\u2081 }, i := i\u2081 } = Iterator.curr { s := { data := cs\u2082 }, i := i\u2082 } then\n      ltb (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }) (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 })\n    else decide (Iterator.curr { s := { data := cs\u2081 }, i := i\u2081 } < Iterator.curr { s := { data := cs\u2082 }, i := i\u2082 })", "state_after": "case ind\nc : Char\ncs\u2081\u271d cs\u2082\u271d : List Char\ni\u2081\u271d i\u2082\u271d : Pos\ncs\u2081 cs\u2082 : List Char\ni\u2081 i\u2082 : Pos\nh\u2082 : Iterator.hasNext { s := { data := cs\u2082 }, i := i\u2082 } = true\nh\u2081 : Iterator.hasNext { s := { data := cs\u2081 }, i := i\u2081 } = true\nheq : get { data := cs\u2081 } i\u2081 = get { data := cs\u2082 } i\u2082\nih :\n  ltb\n      { s := { data := c :: (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }).1.data },\n        i := (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }).i + c }\n      { s := { data := c :: (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 }).1.data },\n        i := (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 }).i + c } =\n    ltb\n      { s := { data := (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }).1.data },\n        i := (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }).i }\n      { s := { data := (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 }).1.data },\n        i := (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 }).i }\n\u22a2 (if\n        Iterator.curr { s := { data := c :: cs\u2081 }, i := i\u2081 + c } =\n          Iterator.curr { s := { data := c :: cs\u2082 }, i := i\u2082 + c } then\n      ltb (Iterator.next { s := { data := c :: cs\u2081 }, i := i\u2081 + c })\n        (Iterator.next { s := { data := c :: cs\u2082 }, i := i\u2082 + c })\n    else\n      decide\n        (Iterator.curr { s := { data := c :: cs\u2081 }, i := i\u2081 + c } <\n          Iterator.curr { s := { data := c :: cs\u2082 }, i := i\u2082 + c })) =\n    if Iterator.curr { s := { data := cs\u2081 }, i := i\u2081 } = Iterator.curr { s := { data := cs\u2082 }, i := i\u2082 } then\n      ltb (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }) (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 })\n    else decide (Iterator.curr { s := { data := cs\u2081 }, i := i\u2081 } < Iterator.curr { s := { data := cs\u2082 }, i := i\u2082 })"}, {"tactic": "simp [Iterator.curr, get_cons_addChar, Iterator.next, next, *] at *", "state_before": "case ind\nc : Char\ncs\u2081\u271d cs\u2082\u271d : List Char\ni\u2081\u271d i\u2082\u271d : Pos\ncs\u2081 cs\u2082 : List Char\ni\u2081 i\u2082 : Pos\nh\u2082 : Iterator.hasNext { s := { data := cs\u2082 }, i := i\u2082 } = true\nh\u2081 : Iterator.hasNext { s := { data := cs\u2081 }, i := i\u2081 } = true\nheq : get { data := cs\u2081 } i\u2081 = get { data := cs\u2082 } i\u2082\nih :\n  ltb\n      { s := { data := c :: (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }).1.data },\n        i := (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }).i + c }\n      { s := { data := c :: (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 }).1.data },\n        i := (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 }).i + c } =\n    ltb\n      { s := { data := (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }).1.data },\n        i := (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }).i }\n      { s := { data := (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 }).1.data },\n        i := (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 }).i }\n\u22a2 (if\n        Iterator.curr { s := { data := c :: cs\u2081 }, i := i\u2081 + c } =\n          Iterator.curr { s := { data := c :: cs\u2082 }, i := i\u2082 + c } then\n      ltb (Iterator.next { s := { data := c :: cs\u2081 }, i := i\u2081 + c })\n        (Iterator.next { s := { data := c :: cs\u2082 }, i := i\u2082 + c })\n    else\n      decide\n        (Iterator.curr { s := { data := c :: cs\u2081 }, i := i\u2081 + c } <\n          Iterator.curr { s := { data := c :: cs\u2082 }, i := i\u2082 + c })) =\n    if Iterator.curr { s := { data := cs\u2081 }, i := i\u2081 } = Iterator.curr { s := { data := cs\u2082 }, i := i\u2082 } then\n      ltb (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }) (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 })\n    else decide (Iterator.curr { s := { data := cs\u2081 }, i := i\u2081 } < Iterator.curr { s := { data := cs\u2082 }, i := i\u2082 })", "state_after": "case ind\nc : Char\ncs\u2081\u271d cs\u2082\u271d : List Char\ni\u2081\u271d i\u2082\u271d : Pos\ncs\u2081 cs\u2082 : List Char\ni\u2081 i\u2082 : Pos\nh\u2082 : Iterator.hasNext { s := { data := cs\u2082 }, i := i\u2082 } = true\nh\u2081 : Iterator.hasNext { s := { data := cs\u2081 }, i := i\u2081 } = true\nheq : get { data := cs\u2081 } i\u2081 = get { data := cs\u2082 } i\u2082\nih :\n  ltb { s := { data := c :: cs\u2081 }, i := i\u2081 + get { data := cs\u2082 } i\u2082 + c }\n      { s := { data := c :: cs\u2082 }, i := i\u2082 + get { data := cs\u2082 } i\u2082 + c } =\n    ltb { s := { data := cs\u2081 }, i := i\u2081 + get { data := cs\u2082 } i\u2082 }\n      { s := { data := cs\u2082 }, i := i\u2082 + get { data := cs\u2082 } i\u2082 }\n\u22a2 ltb { s := { data := c :: cs\u2081 }, i := i\u2081 + c + get { data := cs\u2082 } i\u2082 }\n      { s := { data := c :: cs\u2082 }, i := i\u2082 + c + get { data := cs\u2082 } i\u2082 } =\n    ltb { s := { data := cs\u2081 }, i := i\u2081 + get { data := cs\u2082 } i\u2082 }\n      { s := { data := cs\u2082 }, i := i\u2082 + get { data := cs\u2082 } i\u2082 }"}, {"tactic": "repeat rw [Pos.addChar_right_comm _ c]", "state_before": "case ind\nc : Char\ncs\u2081\u271d cs\u2082\u271d : List Char\ni\u2081\u271d i\u2082\u271d : Pos\ncs\u2081 cs\u2082 : List Char\ni\u2081 i\u2082 : Pos\nh\u2082 : Iterator.hasNext { s := { data := cs\u2082 }, i := i\u2082 } = true\nh\u2081 : Iterator.hasNext { s := { data := cs\u2081 }, i := i\u2081 } = true\nheq : get { data := cs\u2081 } i\u2081 = get { data := cs\u2082 } i\u2082\nih :\n  ltb { s := { data := c :: cs\u2081 }, i := i\u2081 + get { data := cs\u2082 } i\u2082 + c }\n      { s := { data := c :: cs\u2082 }, i := i\u2082 + get { data := cs\u2082 } i\u2082 + c } =\n    ltb { s := { data := cs\u2081 }, i := i\u2081 + get { data := cs\u2082 } i\u2082 }\n      { s := { data := cs\u2082 }, i := i\u2082 + get { data := cs\u2082 } i\u2082 }\n\u22a2 ltb { s := { data := c :: cs\u2081 }, i := i\u2081 + c + get { data := cs\u2082 } i\u2082 }\n      { s := { data := c :: cs\u2082 }, i := i\u2082 + c + get { data := cs\u2082 } i\u2082 } =\n    ltb { s := { data := cs\u2081 }, i := i\u2081 + get { data := cs\u2082 } i\u2082 }\n      { s := { data := cs\u2082 }, i := i\u2082 + get { data := cs\u2082 } i\u2082 }", "state_after": "case ind\nc : Char\ncs\u2081\u271d cs\u2082\u271d : List Char\ni\u2081\u271d i\u2082\u271d : Pos\ncs\u2081 cs\u2082 : List Char\ni\u2081 i\u2082 : Pos\nh\u2082 : Iterator.hasNext { s := { data := cs\u2082 }, i := i\u2082 } = true\nh\u2081 : Iterator.hasNext { s := { data := cs\u2081 }, i := i\u2081 } = true\nheq : get { data := cs\u2081 } i\u2081 = get { data := cs\u2082 } i\u2082\nih :\n  ltb { s := { data := c :: cs\u2081 }, i := i\u2081 + get { data := cs\u2082 } i\u2082 + c }\n      { s := { data := c :: cs\u2082 }, i := i\u2082 + get { data := cs\u2082 } i\u2082 + c } =\n    ltb { s := { data := cs\u2081 }, i := i\u2081 + get { data := cs\u2082 } i\u2082 }\n      { s := { data := cs\u2082 }, i := i\u2082 + get { data := cs\u2082 } i\u2082 }\n\u22a2 ltb { s := { data := c :: cs\u2081 }, i := i\u2081 + get { data := cs\u2082 } i\u2082 + c }\n      { s := { data := c :: cs\u2082 }, i := i\u2082 + get { data := cs\u2082 } i\u2082 + c } =\n    ltb { s := { data := cs\u2081 }, i := i\u2081 + get { data := cs\u2082 } i\u2082 }\n      { s := { data := cs\u2082 }, i := i\u2082 + get { data := cs\u2082 } i\u2082 }"}, {"tactic": "exact ih", "state_before": "case ind\nc : Char\ncs\u2081\u271d cs\u2082\u271d : List Char\ni\u2081\u271d i\u2082\u271d : Pos\ncs\u2081 cs\u2082 : List Char\ni\u2081 i\u2082 : Pos\nh\u2082 : Iterator.hasNext { s := { data := cs\u2082 }, i := i\u2082 } = true\nh\u2081 : Iterator.hasNext { s := { data := cs\u2081 }, i := i\u2081 } = true\nheq : get { data := cs\u2081 } i\u2081 = get { data := cs\u2082 } i\u2082\nih :\n  ltb { s := { data := c :: cs\u2081 }, i := i\u2081 + get { data := cs\u2082 } i\u2082 + c }\n      { s := { data := c :: cs\u2082 }, i := i\u2082 + get { data := cs\u2082 } i\u2082 + c } =\n    ltb { s := { data := cs\u2081 }, i := i\u2081 + get { data := cs\u2082 } i\u2082 }\n      { s := { data := cs\u2082 }, i := i\u2082 + get { data := cs\u2082 } i\u2082 }\n\u22a2 ltb { s := { data := c :: cs\u2081 }, i := i\u2081 + get { data := cs\u2082 } i\u2082 + c }\n      { s := { data := c :: cs\u2082 }, i := i\u2082 + get { data := cs\u2082 } i\u2082 + c } =\n    ltb { s := { data := cs\u2081 }, i := i\u2081 + get { data := cs\u2082 } i\u2082 }\n      { s := { data := cs\u2082 }, i := i\u2082 + get { data := cs\u2082 } i\u2082 }", "state_after": "no goals"}, {"tactic": "rw [Pos.addChar_right_comm _ c]", "state_before": "case ind\nc : Char\ncs\u2081\u271d cs\u2082\u271d : List Char\ni\u2081\u271d i\u2082\u271d : Pos\ncs\u2081 cs\u2082 : List Char\ni\u2081 i\u2082 : Pos\nh\u2082 : Iterator.hasNext { s := { data := cs\u2082 }, i := i\u2082 } = true\nh\u2081 : Iterator.hasNext { s := { data := cs\u2081 }, i := i\u2081 } = true\nheq : get { data := cs\u2081 } i\u2081 = get { data := cs\u2082 } i\u2082\nih :\n  ltb { s := { data := c :: cs\u2081 }, i := i\u2081 + get { data := cs\u2082 } i\u2082 + c }\n      { s := { data := c :: cs\u2082 }, i := i\u2082 + get { data := cs\u2082 } i\u2082 + c } =\n    ltb { s := { data := cs\u2081 }, i := i\u2081 + get { data := cs\u2082 } i\u2082 }\n      { s := { data := cs\u2082 }, i := i\u2082 + get { data := cs\u2082 } i\u2082 }\n\u22a2 ltb { s := { data := c :: cs\u2081 }, i := i\u2081 + get { data := cs\u2082 } i\u2082 + c }\n      { s := { data := c :: cs\u2082 }, i := i\u2082 + get { data := cs\u2082 } i\u2082 + c } =\n    ltb { s := { data := cs\u2081 }, i := i\u2081 + get { data := cs\u2082 } i\u2082 }\n      { s := { data := cs\u2082 }, i := i\u2082 + get { data := cs\u2082 } i\u2082 }", "state_after": "case ind\nc : Char\ncs\u2081\u271d cs\u2082\u271d : List Char\ni\u2081\u271d i\u2082\u271d : Pos\ncs\u2081 cs\u2082 : List Char\ni\u2081 i\u2082 : Pos\nh\u2082 : Iterator.hasNext { s := { data := cs\u2082 }, i := i\u2082 } = true\nh\u2081 : Iterator.hasNext { s := { data := cs\u2081 }, i := i\u2081 } = true\nheq : get { data := cs\u2081 } i\u2081 = get { data := cs\u2082 } i\u2082\nih :\n  ltb { s := { data := c :: cs\u2081 }, i := i\u2081 + get { data := cs\u2082 } i\u2082 + c }\n      { s := { data := c :: cs\u2082 }, i := i\u2082 + get { data := cs\u2082 } i\u2082 + c } =\n    ltb { s := { data := cs\u2081 }, i := i\u2081 + get { data := cs\u2082 } i\u2082 }\n      { s := { data := cs\u2082 }, i := i\u2082 + get { data := cs\u2082 } i\u2082 }\n\u22a2 ltb { s := { data := c :: cs\u2081 }, i := i\u2081 + get { data := cs\u2082 } i\u2082 + c }\n      { s := { data := c :: cs\u2082 }, i := i\u2082 + get { data := cs\u2082 } i\u2082 + c } =\n    ltb { s := { data := cs\u2081 }, i := i\u2081 + get { data := cs\u2082 } i\u2082 }\n      { s := { data := cs\u2082 }, i := i\u2082 + get { data := cs\u2082 } i\u2082 }"}, {"tactic": "rename_i h\u2082 h\u2081 hne", "state_before": "case eq\nc : Char\ncs\u2081\u271d cs\u2082\u271d : List Char\ni\u2081\u271d i\u2082\u271d : Pos\ncs\u2081 cs\u2082 : List Char\ni\u2081 i\u2082 : Pos\na\u271d\u00b2 : Iterator.hasNext { s := { data := cs\u2082 }, i := i\u2082 } = true\na\u271d\u00b9 : Iterator.hasNext { s := { data := cs\u2081 }, i := i\u2081 } = true\na\u271d : \u00acget { data := cs\u2081 } i\u2081 = get { data := cs\u2082 } i\u2082\n\u22a2 (if\n        Iterator.curr { s := { data := c :: cs\u2081 }, i := i\u2081 + c } =\n          Iterator.curr { s := { data := c :: cs\u2082 }, i := i\u2082 + c } then\n      ltb (Iterator.next { s := { data := c :: cs\u2081 }, i := i\u2081 + c })\n        (Iterator.next { s := { data := c :: cs\u2082 }, i := i\u2082 + c })\n    else\n      decide\n        (Iterator.curr { s := { data := c :: cs\u2081 }, i := i\u2081 + c } <\n          Iterator.curr { s := { data := c :: cs\u2082 }, i := i\u2082 + c })) =\n    if Iterator.curr { s := { data := cs\u2081 }, i := i\u2081 } = Iterator.curr { s := { data := cs\u2082 }, i := i\u2082 } then\n      ltb (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }) (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 })\n    else decide (Iterator.curr { s := { data := cs\u2081 }, i := i\u2081 } < Iterator.curr { s := { data := cs\u2082 }, i := i\u2082 })", "state_after": "case eq\nc : Char\ncs\u2081\u271d cs\u2082\u271d : List Char\ni\u2081\u271d i\u2082\u271d : Pos\ncs\u2081 cs\u2082 : List Char\ni\u2081 i\u2082 : Pos\nh\u2082 : Iterator.hasNext { s := { data := cs\u2082 }, i := i\u2082 } = true\nh\u2081 : Iterator.hasNext { s := { data := cs\u2081 }, i := i\u2081 } = true\nhne : \u00acget { data := cs\u2081 } i\u2081 = get { data := cs\u2082 } i\u2082\n\u22a2 (if\n        Iterator.curr { s := { data := c :: cs\u2081 }, i := i\u2081 + c } =\n          Iterator.curr { s := { data := c :: cs\u2082 }, i := i\u2082 + c } then\n      ltb (Iterator.next { s := { data := c :: cs\u2081 }, i := i\u2081 + c })\n        (Iterator.next { s := { data := c :: cs\u2082 }, i := i\u2082 + c })\n    else\n      decide\n        (Iterator.curr { s := { data := c :: cs\u2081 }, i := i\u2081 + c } <\n          Iterator.curr { s := { data := c :: cs\u2082 }, i := i\u2082 + c })) =\n    if Iterator.curr { s := { data := cs\u2081 }, i := i\u2081 } = Iterator.curr { s := { data := cs\u2082 }, i := i\u2082 } then\n      ltb (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }) (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 })\n    else decide (Iterator.curr { s := { data := cs\u2081 }, i := i\u2081 } < Iterator.curr { s := { data := cs\u2082 }, i := i\u2082 })"}, {"tactic": "simp [Iterator.curr, get_cons_addChar, *]", "state_before": "case eq\nc : Char\ncs\u2081\u271d cs\u2082\u271d : List Char\ni\u2081\u271d i\u2082\u271d : Pos\ncs\u2081 cs\u2082 : List Char\ni\u2081 i\u2082 : Pos\nh\u2082 : Iterator.hasNext { s := { data := cs\u2082 }, i := i\u2082 } = true\nh\u2081 : Iterator.hasNext { s := { data := cs\u2081 }, i := i\u2081 } = true\nhne : \u00acget { data := cs\u2081 } i\u2081 = get { data := cs\u2082 } i\u2082\n\u22a2 (if\n        Iterator.curr { s := { data := c :: cs\u2081 }, i := i\u2081 + c } =\n          Iterator.curr { s := { data := c :: cs\u2082 }, i := i\u2082 + c } then\n      ltb (Iterator.next { s := { data := c :: cs\u2081 }, i := i\u2081 + c })\n        (Iterator.next { s := { data := c :: cs\u2082 }, i := i\u2082 + c })\n    else\n      decide\n        (Iterator.curr { s := { data := c :: cs\u2081 }, i := i\u2081 + c } <\n          Iterator.curr { s := { data := c :: cs\u2082 }, i := i\u2082 + c })) =\n    if Iterator.curr { s := { data := cs\u2081 }, i := i\u2081 } = Iterator.curr { s := { data := cs\u2082 }, i := i\u2082 } then\n      ltb (Iterator.next { s := { data := cs\u2081 }, i := i\u2081 }) (Iterator.next { s := { data := cs\u2082 }, i := i\u2082 })\n    else decide (Iterator.curr { s := { data := cs\u2081 }, i := i\u2081 } < Iterator.curr { s := { data := cs\u2082 }, i := i\u2082 })", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Abelian/Homology.lean", "full_name": "homology.map_\u03b9", "start": [313, 1], "end": [320, 71], "traced_tactics": [{"tactic": "simp [h, \u03b2.w.symm]", "state_before": "A : Type u\ninst\u271d\u00b9 : Category A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 f \u226b \u03b2.left = \u03b1.left \u226b f'", "state_after": "no goals"}, {"tactic": "rw [map_eq_lift_desc'_left, lift_\u03b9]", "state_before": "A : Type u\ninst\u271d\u00b9 : Category A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 map w w' \u03b1 \u03b2 h \u226b \u03b9 f' g' w' = \u03b9 f g w \u226b cokernel.map f f' \u03b1.left \u03b2.left (_ : f \u226b \u03b2.left = \u03b1.left \u226b f')", "state_after": "A : Type u\ninst\u271d\u00b9 : Category A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 desc' f g w (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f' = 0) =\n    \u03b9 f g w \u226b cokernel.map f f' \u03b1.left \u03b2.left (_ : f \u226b \u03b2.left = \u03b1.left \u226b f')"}, {"tactic": "apply homology.hom_from_ext", "state_before": "A : Type u\ninst\u271d\u00b9 : Category A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 desc' f g w (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f' = 0) =\n    \u03b9 f g w \u226b cokernel.map f f' \u03b1.left \u03b2.left (_ : f \u226b \u03b2.left = \u03b1.left \u226b f')", "state_after": "case h\nA : Type u\ninst\u271d\u00b9 : Category A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 \u03c0' f g w \u226b\n      desc' f g w (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n        (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f' = 0) =\n    \u03c0' f g w \u226b \u03b9 f g w \u226b cokernel.map f f' \u03b1.left \u03b2.left (_ : f \u226b \u03b2.left = \u03b1.left \u226b f')"}, {"tactic": "simp only [\u2190 Category.assoc]", "state_before": "case h\nA : Type u\ninst\u271d\u00b9 : Category A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 \u03c0' f g w \u226b\n      desc' f g w (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n        (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f' = 0) =\n    \u03c0' f g w \u226b \u03b9 f g w \u226b cokernel.map f f' \u03b1.left \u03b2.left (_ : f \u226b \u03b2.left = \u03b1.left \u226b f')", "state_after": "case h\nA : Type u\ninst\u271d\u00b9 : Category A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 \u03c0' f g w \u226b\n      desc' f g w ((kernel.\u03b9 g \u226b \u03b2.left) \u226b cokernel.\u03c0 f')\n        (_ : kernel.lift g f w \u226b (kernel.\u03b9 g \u226b \u03b2.left) \u226b cokernel.\u03c0 f' = 0) =\n    (\u03c0' f g w \u226b \u03b9 f g w) \u226b cokernel.map f f' \u03b1.left \u03b2.left (_ : f \u226b \u03b2.left = \u03b1.left \u226b f')"}, {"tactic": "rw [\u03c0'_\u03b9, \u03c0'_desc', Category.assoc, Category.assoc, cokernel.\u03c0_desc]", "state_before": "case h\nA : Type u\ninst\u271d\u00b9 : Category A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 \u03c0' f g w \u226b\n      desc' f g w ((kernel.\u03b9 g \u226b \u03b2.left) \u226b cokernel.\u03c0 f')\n        (_ : kernel.lift g f w \u226b (kernel.\u03b9 g \u226b \u03b2.left) \u226b cokernel.\u03c0 f' = 0) =\n    (\u03c0' f g w \u226b \u03b9 f g w) \u226b cokernel.map f f' \u03b1.left \u03b2.left (_ : f \u226b \u03b2.left = \u03b1.left \u226b f')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Hom/Iterate.lean", "full_name": "SemiconjBy.function_semiconj_mul_right_swap", "start": [252, 1], "end": [253, 89], "traced_tactics": [{"tactic": "simp_rw [mul_assoc, \u2190 h.eq]", "state_before": "M : Type ?u.89651\nN : Type ?u.89654\nG : Type u_1\nH : Type ?u.89660\ninst\u271d : Semigroup G\na b c : G\nh : SemiconjBy a b c\nj : G\n\u22a2 (fun x => x * a) ((fun x => x * c) j) = (fun x => x * b) ((fun x => x * a) j)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/WithBot.lean", "full_name": "WithTop.lt_ofDual_iff", "start": [904, 1], "end": [906, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Sups.lean", "full_name": "Finset.Nonempty.of_infs_left", "start": [300, 1], "end": [301, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Transfer.lean", "full_name": "MonoidHom.transferCenterPow_apply", "start": [209, 1], "end": [211, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Field/Basic.lean", "full_name": "inv_le_inv_of_le", "start": [253, 1], "end": [254, 91], "traced_tactics": [{"tactic": "rwa [\u2190 one_div a, le_div_iff' ha, \u2190 div_eq_mul_inv, div_le_iff (ha.trans_le h), one_mul]", "state_before": "\u03b9 : Type ?u.45557\n\u03b1 : Type u_1\n\u03b2 : Type ?u.45563\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\nh : a \u2264 b\n\u22a2 b\u207b\u00b9 \u2264 a\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Span.lean", "full_name": "LinearMap.ker_toSpanSingleton", "start": [982, 1], "end": [983, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Finsupp.lean", "full_name": "Finsupp.ker_lsingle", "start": [132, 1], "end": [133, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.unbounded_of_tendsto_atBot'", "start": [1711, 1], "end": [1713, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Semicontinuous.lean", "full_name": "lowerSemicontinuousOn_ciSup", "start": [595, 1], "end": [598, 92], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "full_name": "BoxIntegral.Box.Ioo_subset_coe", "start": [450, 1], "end": [451, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Archimedean.lean", "full_name": "exists_pos_rat_lt", "start": [315, 1], "end": [316, 53], "traced_tactics": [{"tactic": "simpa only [Rat.cast_pos] using exists_rat_btwn x0", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y \u03b5 x : \u03b1\nx0 : 0 < x\n\u22a2 \u2203 q, 0 < q \u2227 \u2191q < x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "full_name": "MeasureTheory.Measure.isOpenPosMeasure_smul", "start": [69, 1], "end": [70, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean", "full_name": "lipschitzWith_lineMap", "start": [88, 1], "end": [90, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Monotone/Basic.lean", "full_name": "StrictMono.minimal_of_minimal_image", "start": [858, 1], "end": [860, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Cycle.lean", "full_name": "List.prev_mem", "start": [252, 1], "end": [263, 41], "traced_tactics": [{"tactic": "cases' l with hd tl", "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nh : x \u2208 l\n\u22a2 prev l x h \u2208 l", "state_after": "case nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nh : x \u2208 []\n\u22a2 prev [] x h \u2208 []\n\ncase cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd : \u03b1\ntl : List \u03b1\nh : x \u2208 hd :: tl\n\u22a2 prev (hd :: tl) x h \u2208 hd :: tl"}, {"tactic": "induction' tl with hd' tl hl generalizing hd", "state_before": "case cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd : \u03b1\ntl : List \u03b1\nh : x \u2208 hd :: tl\n\u22a2 prev (hd :: tl) x h \u2208 hd :: tl", "state_after": "case cons.nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\nhd : \u03b1\nh : x \u2208 [hd]\n\u22a2 prev [hd] x h \u2208 [hd]\n\ncase cons.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl\u271d : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\u271d\nhd' : \u03b1\ntl : List \u03b1\nhl : \u2200 (hd : \u03b1) (h : x \u2208 hd :: tl), prev (hd :: tl) x h \u2208 hd :: tl\nhd : \u03b1\nh : x \u2208 hd :: hd' :: tl\n\u22a2 prev (hd :: hd' :: tl) x h \u2208 hd :: hd' :: tl"}, {"tactic": "simp at h", "state_before": "case nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx : \u03b1\nh : x \u2208 []\n\u22a2 prev [] x h \u2208 []", "state_after": "no goals"}, {"tactic": "simp", "state_before": "case cons.nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\nhd : \u03b1\nh : x \u2208 [hd]\n\u22a2 prev [hd] x h \u2208 [hd]", "state_after": "no goals"}, {"tactic": "by_cases hx : x = hd", "state_before": "case cons.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl\u271d : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\u271d\nhd' : \u03b1\ntl : List \u03b1\nhl : \u2200 (hd : \u03b1) (h : x \u2208 hd :: tl), prev (hd :: tl) x h \u2208 hd :: tl\nhd : \u03b1\nh : x \u2208 hd :: hd' :: tl\n\u22a2 prev (hd :: hd' :: tl) x h \u2208 hd :: hd' :: tl", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl\u271d : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\u271d\nhd' : \u03b1\ntl : List \u03b1\nhl : \u2200 (hd : \u03b1) (h : x \u2208 hd :: tl), prev (hd :: tl) x h \u2208 hd :: tl\nhd : \u03b1\nh : x \u2208 hd :: hd' :: tl\nhx : x = hd\n\u22a2 prev (hd :: hd' :: tl) x h \u2208 hd :: hd' :: tl\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl\u271d : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\u271d\nhd' : \u03b1\ntl : List \u03b1\nhl : \u2200 (hd : \u03b1) (h : x \u2208 hd :: tl), prev (hd :: tl) x h \u2208 hd :: tl\nhd : \u03b1\nh : x \u2208 hd :: hd' :: tl\nhx : \u00acx = hd\n\u22a2 prev (hd :: hd' :: tl) x h \u2208 hd :: hd' :: tl"}, {"tactic": "simp only [hx, prev_cons_cons_eq]", "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl\u271d : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\u271d\nhd' : \u03b1\ntl : List \u03b1\nhl : \u2200 (hd : \u03b1) (h : x \u2208 hd :: tl), prev (hd :: tl) x h \u2208 hd :: tl\nhd : \u03b1\nh : x \u2208 hd :: hd' :: tl\nhx : x = hd\n\u22a2 prev (hd :: hd' :: tl) x h \u2208 hd :: hd' :: tl", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl\u271d : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\u271d\nhd' : \u03b1\ntl : List \u03b1\nhl : \u2200 (hd : \u03b1) (h : x \u2208 hd :: tl), prev (hd :: tl) x h \u2208 hd :: tl\nhd : \u03b1\nh : x \u2208 hd :: hd' :: tl\nhx : x = hd\n\u22a2 getLast (hd' :: tl) (_ : hd' :: tl \u2260 []) \u2208 hd :: hd' :: tl"}, {"tactic": "exact mem_cons_of_mem _ (getLast_mem _)", "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl\u271d : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\u271d\nhd' : \u03b1\ntl : List \u03b1\nhl : \u2200 (hd : \u03b1) (h : x \u2208 hd :: tl), prev (hd :: tl) x h \u2208 hd :: tl\nhd : \u03b1\nh : x \u2208 hd :: hd' :: tl\nhx : x = hd\n\u22a2 getLast (hd' :: tl) (_ : hd' :: tl \u2260 []) \u2208 hd :: hd' :: tl", "state_after": "no goals"}, {"tactic": "rw [prev, dif_neg hx]", "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl\u271d : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\u271d\nhd' : \u03b1\ntl : List \u03b1\nhl : \u2200 (hd : \u03b1) (h : x \u2208 hd :: tl), prev (hd :: tl) x h \u2208 hd :: tl\nhd : \u03b1\nh : x \u2208 hd :: hd' :: tl\nhx : \u00acx = hd\n\u22a2 prev (hd :: hd' :: tl) x h \u2208 hd :: hd' :: tl", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl\u271d : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\u271d\nhd' : \u03b1\ntl : List \u03b1\nhl : \u2200 (hd : \u03b1) (h : x \u2208 hd :: tl), prev (hd :: tl) x h \u2208 hd :: tl\nhd : \u03b1\nh : x \u2208 hd :: hd' :: tl\nhx : \u00acx = hd\n\u22a2 (if x = hd' then hd else prev (hd' :: tl) x (_ : x \u2208 hd' :: tl)) \u2208 hd :: hd' :: tl"}, {"tactic": "split_ifs with hm", "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl\u271d : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\u271d\nhd' : \u03b1\ntl : List \u03b1\nhl : \u2200 (hd : \u03b1) (h : x \u2208 hd :: tl), prev (hd :: tl) x h \u2208 hd :: tl\nhd : \u03b1\nh : x \u2208 hd :: hd' :: tl\nhx : \u00acx = hd\n\u22a2 (if x = hd' then hd else prev (hd' :: tl) x (_ : x \u2208 hd' :: tl)) \u2208 hd :: hd' :: tl", "state_after": "case neg.inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl\u271d : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\u271d\nhd' : \u03b1\ntl : List \u03b1\nhl : \u2200 (hd : \u03b1) (h : x \u2208 hd :: tl), prev (hd :: tl) x h \u2208 hd :: tl\nhd : \u03b1\nh : x \u2208 hd :: hd' :: tl\nhx : \u00acx = hd\nhm : x = hd'\n\u22a2 hd \u2208 hd :: hd' :: tl\n\ncase neg.inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl\u271d : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\u271d\nhd' : \u03b1\ntl : List \u03b1\nhl : \u2200 (hd : \u03b1) (h : x \u2208 hd :: tl), prev (hd :: tl) x h \u2208 hd :: tl\nhd : \u03b1\nh : x \u2208 hd :: hd' :: tl\nhx : \u00acx = hd\nhm : \u00acx = hd'\n\u22a2 prev (hd' :: tl) x (_ : x \u2208 hd' :: tl) \u2208 hd :: hd' :: tl"}, {"tactic": "exact mem_cons_self _ _", "state_before": "case neg.inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl\u271d : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\u271d\nhd' : \u03b1\ntl : List \u03b1\nhl : \u2200 (hd : \u03b1) (h : x \u2208 hd :: tl), prev (hd :: tl) x h \u2208 hd :: tl\nhd : \u03b1\nh : x \u2208 hd :: hd' :: tl\nhx : \u00acx = hd\nhm : x = hd'\n\u22a2 hd \u2208 hd :: hd' :: tl", "state_after": "no goals"}, {"tactic": "exact mem_cons_of_mem _ (hl _ _)", "state_before": "case neg.inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx hd\u271d : \u03b1\ntl\u271d : List \u03b1\nh\u271d : x \u2208 hd\u271d :: tl\u271d\nhd' : \u03b1\ntl : List \u03b1\nhl : \u2200 (hd : \u03b1) (h : x \u2208 hd :: tl), prev (hd :: tl) x h \u2208 hd :: tl\nhd : \u03b1\nh : x \u2208 hd :: hd' :: tl\nhx : \u00acx = hd\nhm : \u00acx = hd'\n\u22a2 prev (hd' :: tl) x (_ : x \u2208 hd' :: tl) \u2208 hd :: hd' :: tl", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Log/Base.lean", "full_name": "Real.logb_injOn_pos", "start": [231, 1], "end": [232, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "full_name": "cmp_mul_neg_left", "start": [1158, 1], "end": [1159, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Complex/Basic.lean", "full_name": "Complex.normSq_eq_abs", "start": [1117, 1], "end": [1118, 64], "traced_tactics": [{"tactic": "simp [abs, sq, abs_def, Real.mul_self_sqrt (normSq_nonneg _)]", "state_before": "x : \u2102\n\u22a2 \u2191normSq x = \u2191abs x ^ 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.tsum_nnnorm_sub_ae_lt_top", "start": [1398, 9], "end": [1412, 66], "traced_tactics": [{"tactic": "have hp_pos : 0 < p := zero_lt_one.trans_le hp1", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.8184309\nG : Type ?u.8184312\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : (\u2211' (i : \u2115), B i) \u2260 \u22a4\nh : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (\u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a) < \u22a4", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.8184309\nG : Type ?u.8184312\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : (\u2211' (i : \u2115), B i) \u2260 \u22a4\nh : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nhp_pos : 0 < p\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (\u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a) < \u22a4"}, {"tactic": "have h_integral : (\u222b\u207b a, (\u2211' i, \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) < \u221e := by\n  have h_tsum_lt_top : (\u2211' i, B i) ^ p < \u221e := ENNReal.rpow_lt_top_of_nonneg hp_pos.le hB\n  refine' lt_of_le_of_lt _ h_tsum_lt_top\n  rwa [\u2190 ENNReal.le_rpow_one_div_iff (by simp [hp_pos] : 0 < 1 / p), one_div_one_div] at h", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.8184309\nG : Type ?u.8184312\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : (\u2211' (i : \u2115), B i) \u2260 \u22a4\nh : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nhp_pos : 0 < p\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (\u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a) < \u22a4", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.8184309\nG : Type ?u.8184312\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : (\u2211' (i : \u2115), B i) \u2260 \u22a4\nh : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nhp_pos : 0 < p\nh_integral : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (\u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a) < \u22a4"}, {"tactic": "have rpow_ae_lt_top : \u2200\u1d50 x \u2202\u03bc, (\u2211' i, \u2016f (i + 1) x - f i x\u2016\u208a : \u211d\u22650\u221e) ^ p < \u221e := by\n  refine' ae_lt_top' (AEMeasurable.pow_const _ _) h_integral.ne\n  exact AEMeasurable.ennreal_tsum fun n => ((hf (n + 1)).sub (hf n)).ennnorm", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.8184309\nG : Type ?u.8184312\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : (\u2211' (i : \u2115), B i) \u2260 \u22a4\nh : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nhp_pos : 0 < p\nh_integral : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (\u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a) < \u22a4", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.8184309\nG : Type ?u.8184312\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : (\u2211' (i : \u2115), B i) \u2260 \u22a4\nh : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nhp_pos : 0 < p\nh_integral : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) < \u22a4\nrpow_ae_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (\u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a) ^ p < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (\u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a) < \u22a4"}, {"tactic": "refine' rpow_ae_lt_top.mono fun x hx => _", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.8184309\nG : Type ?u.8184312\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : (\u2211' (i : \u2115), B i) \u2260 \u22a4\nh : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nhp_pos : 0 < p\nh_integral : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) < \u22a4\nrpow_ae_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (\u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a) ^ p < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (\u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a) < \u22a4", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.8184309\nG : Type ?u.8184312\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : (\u2211' (i : \u2115), B i) \u2260 \u22a4\nh : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nhp_pos : 0 < p\nh_integral : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) < \u22a4\nrpow_ae_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (\u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a) ^ p < \u22a4\nx : \u03b1\nhx : (\u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a) ^ p < \u22a4\n\u22a2 (\u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a) < \u22a4"}, {"tactic": "rwa [\u2190 ENNReal.lt_rpow_one_div_iff hp_pos,\n  ENNReal.top_rpow_of_pos (by simp [hp_pos] : 0 < 1 / p)] at hx", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.8184309\nG : Type ?u.8184312\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : (\u2211' (i : \u2115), B i) \u2260 \u22a4\nh : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nhp_pos : 0 < p\nh_integral : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) < \u22a4\nrpow_ae_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (\u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a) ^ p < \u22a4\nx : \u03b1\nhx : (\u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a) ^ p < \u22a4\n\u22a2 (\u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a) < \u22a4", "state_after": "no goals"}, {"tactic": "have h_tsum_lt_top : (\u2211' i, B i) ^ p < \u221e := ENNReal.rpow_lt_top_of_nonneg hp_pos.le hB", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.8184309\nG : Type ?u.8184312\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : (\u2211' (i : \u2115), B i) \u2260 \u22a4\nh : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nhp_pos : 0 < p\n\u22a2 (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) < \u22a4", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.8184309\nG : Type ?u.8184312\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : (\u2211' (i : \u2115), B i) \u2260 \u22a4\nh : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nhp_pos : 0 < p\nh_tsum_lt_top : (\u2211' (i : \u2115), B i) ^ p < \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) < \u22a4"}, {"tactic": "refine' lt_of_le_of_lt _ h_tsum_lt_top", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.8184309\nG : Type ?u.8184312\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : (\u2211' (i : \u2115), B i) \u2260 \u22a4\nh : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nhp_pos : 0 < p\nh_tsum_lt_top : (\u2211' (i : \u2115), B i) ^ p < \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) < \u22a4", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.8184309\nG : Type ?u.8184312\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : (\u2211' (i : \u2115), B i) \u2260 \u22a4\nh : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nhp_pos : 0 < p\nh_tsum_lt_top : (\u2211' (i : \u2115), B i) ^ p < \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) \u2264 (\u2211' (i : \u2115), B i) ^ p"}, {"tactic": "rwa [\u2190 ENNReal.le_rpow_one_div_iff (by simp [hp_pos] : 0 < 1 / p), one_div_one_div] at h", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.8184309\nG : Type ?u.8184312\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : (\u2211' (i : \u2115), B i) \u2260 \u22a4\nh : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nhp_pos : 0 < p\nh_tsum_lt_top : (\u2211' (i : \u2115), B i) ^ p < \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) \u2264 (\u2211' (i : \u2115), B i) ^ p", "state_after": "no goals"}, {"tactic": "simp [hp_pos]", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.8184309\nG : Type ?u.8184312\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : (\u2211' (i : \u2115), B i) \u2260 \u22a4\nh : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nhp_pos : 0 < p\nh_tsum_lt_top : (\u2211' (i : \u2115), B i) ^ p < \u22a4\n\u22a2 0 < 1 / p", "state_after": "no goals"}, {"tactic": "refine' ae_lt_top' (AEMeasurable.pow_const _ _) h_integral.ne", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.8184309\nG : Type ?u.8184312\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : (\u2211' (i : \u2115), B i) \u2260 \u22a4\nh : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nhp_pos : 0 < p\nh_integral : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (\u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a) ^ p < \u22a4", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.8184309\nG : Type ?u.8184312\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : (\u2211' (i : \u2115), B i) \u2260 \u22a4\nh : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nhp_pos : 0 < p\nh_integral : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) < \u22a4\n\u22a2 AEMeasurable fun x => \u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a"}, {"tactic": "exact AEMeasurable.ennreal_tsum fun n => ((hf (n + 1)).sub (hf n)).ennnorm", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.8184309\nG : Type ?u.8184312\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : (\u2211' (i : \u2115), B i) \u2260 \u22a4\nh : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nhp_pos : 0 < p\nh_integral : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) < \u22a4\n\u22a2 AEMeasurable fun x => \u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a", "state_after": "no goals"}, {"tactic": "simp [hp_pos]", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.8184309\nG : Type ?u.8184312\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\np : \u211d\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : (\u2211' (i : \u2115), B i) \u2260 \u22a4\nh : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nhp_pos : 0 < p\nh_integral : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) < \u22a4\nrpow_ae_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (\u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a) ^ p < \u22a4\nx : \u03b1\nhx : (\u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a) < \u22a4 ^ (1 / p)\n\u22a2 0 < 1 / p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "Complex.ofReal_cosh", "start": [679, 1], "end": [680, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Homology/ComplexShape.lean", "full_name": "ComplexShape.symm_symm", "start": [103, 1], "end": [105, 7], "traced_tactics": [{"tactic": "ext", "state_before": "\u03b9 : Type u_1\nc : ComplexShape \u03b9\n\u22a2 symm (symm c) = c", "state_after": "case Rel.h.h.a\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nx\u271d\u00b9 x\u271d : \u03b9\n\u22a2 Rel (symm (symm c)) x\u271d\u00b9 x\u271d \u2194 Rel c x\u271d\u00b9 x\u271d"}, {"tactic": "simp", "state_before": "case Rel.h.h.a\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nx\u271d\u00b9 x\u271d : \u03b9\n\u22a2 Rel (symm (symm c)) x\u271d\u00b9 x\u271d \u2194 Rel c x\u271d\u00b9 x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "full_name": "MonoidHom.liftOfRightInverseAux_comp_apply", "start": [3287, 1], "end": [3293, 19], "traced_tactics": [{"tactic": "dsimp [liftOfRightInverseAux]", "state_before": "G : Type ?u.644961\nG' : Type ?u.644964\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\nA : Type ?u.644973\ninst\u271d\u00b3 : AddGroup A\nG\u2081 : Type u_1\nG\u2082 : Type u_2\nG\u2083 : Type u_3\ninst\u271d\u00b2 : Group G\u2081\ninst\u271d\u00b9 : Group G\u2082\ninst\u271d : Group G\u2083\nf : G\u2081 \u2192* G\u2082\nf_inv : G\u2082 \u2192 G\u2081\nhf : Function.RightInverse f_inv \u2191f\ng : G\u2081 \u2192* G\u2083\nhg : ker f \u2264 ker g\nx : G\u2081\n\u22a2 \u2191(liftOfRightInverseAux f f_inv hf g hg) (\u2191f x) = \u2191g x", "state_after": "G : Type ?u.644961\nG' : Type ?u.644964\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\nA : Type ?u.644973\ninst\u271d\u00b3 : AddGroup A\nG\u2081 : Type u_1\nG\u2082 : Type u_2\nG\u2083 : Type u_3\ninst\u271d\u00b2 : Group G\u2081\ninst\u271d\u00b9 : Group G\u2082\ninst\u271d : Group G\u2083\nf : G\u2081 \u2192* G\u2082\nf_inv : G\u2082 \u2192 G\u2081\nhf : Function.RightInverse f_inv \u2191f\ng : G\u2081 \u2192* G\u2083\nhg : ker f \u2264 ker g\nx : G\u2081\n\u22a2 \u2191g (f_inv (\u2191f x)) = \u2191g x"}, {"tactic": "rw [\u2190 mul_inv_eq_one, \u2190 g.map_inv, \u2190 g.map_mul, \u2190 g.mem_ker]", "state_before": "G : Type ?u.644961\nG' : Type ?u.644964\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\nA : Type ?u.644973\ninst\u271d\u00b3 : AddGroup A\nG\u2081 : Type u_1\nG\u2082 : Type u_2\nG\u2083 : Type u_3\ninst\u271d\u00b2 : Group G\u2081\ninst\u271d\u00b9 : Group G\u2082\ninst\u271d : Group G\u2083\nf : G\u2081 \u2192* G\u2082\nf_inv : G\u2082 \u2192 G\u2081\nhf : Function.RightInverse f_inv \u2191f\ng : G\u2081 \u2192* G\u2083\nhg : ker f \u2264 ker g\nx : G\u2081\n\u22a2 \u2191g (f_inv (\u2191f x)) = \u2191g x", "state_after": "G : Type ?u.644961\nG' : Type ?u.644964\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\nA : Type ?u.644973\ninst\u271d\u00b3 : AddGroup A\nG\u2081 : Type u_1\nG\u2082 : Type u_2\nG\u2083 : Type u_3\ninst\u271d\u00b2 : Group G\u2081\ninst\u271d\u00b9 : Group G\u2082\ninst\u271d : Group G\u2083\nf : G\u2081 \u2192* G\u2082\nf_inv : G\u2082 \u2192 G\u2081\nhf : Function.RightInverse f_inv \u2191f\ng : G\u2081 \u2192* G\u2083\nhg : ker f \u2264 ker g\nx : G\u2081\n\u22a2 f_inv (\u2191f x) * x\u207b\u00b9 \u2208 ker g"}, {"tactic": "apply hg", "state_before": "G : Type ?u.644961\nG' : Type ?u.644964\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\nA : Type ?u.644973\ninst\u271d\u00b3 : AddGroup A\nG\u2081 : Type u_1\nG\u2082 : Type u_2\nG\u2083 : Type u_3\ninst\u271d\u00b2 : Group G\u2081\ninst\u271d\u00b9 : Group G\u2082\ninst\u271d : Group G\u2083\nf : G\u2081 \u2192* G\u2082\nf_inv : G\u2082 \u2192 G\u2081\nhf : Function.RightInverse f_inv \u2191f\ng : G\u2081 \u2192* G\u2083\nhg : ker f \u2264 ker g\nx : G\u2081\n\u22a2 f_inv (\u2191f x) * x\u207b\u00b9 \u2208 ker g", "state_after": "case a\nG : Type ?u.644961\nG' : Type ?u.644964\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\nA : Type ?u.644973\ninst\u271d\u00b3 : AddGroup A\nG\u2081 : Type u_1\nG\u2082 : Type u_2\nG\u2083 : Type u_3\ninst\u271d\u00b2 : Group G\u2081\ninst\u271d\u00b9 : Group G\u2082\ninst\u271d : Group G\u2083\nf : G\u2081 \u2192* G\u2082\nf_inv : G\u2082 \u2192 G\u2081\nhf : Function.RightInverse f_inv \u2191f\ng : G\u2081 \u2192* G\u2083\nhg : ker f \u2264 ker g\nx : G\u2081\n\u22a2 f_inv (\u2191f x) * x\u207b\u00b9 \u2208 ker f"}, {"tactic": "rw [f.mem_ker, f.map_mul, f.map_inv, mul_inv_eq_one]", "state_before": "case a\nG : Type ?u.644961\nG' : Type ?u.644964\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\nA : Type ?u.644973\ninst\u271d\u00b3 : AddGroup A\nG\u2081 : Type u_1\nG\u2082 : Type u_2\nG\u2083 : Type u_3\ninst\u271d\u00b2 : Group G\u2081\ninst\u271d\u00b9 : Group G\u2082\ninst\u271d : Group G\u2083\nf : G\u2081 \u2192* G\u2082\nf_inv : G\u2082 \u2192 G\u2081\nhf : Function.RightInverse f_inv \u2191f\ng : G\u2081 \u2192* G\u2083\nhg : ker f \u2264 ker g\nx : G\u2081\n\u22a2 f_inv (\u2191f x) * x\u207b\u00b9 \u2208 ker f", "state_after": "case a\nG : Type ?u.644961\nG' : Type ?u.644964\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\nA : Type ?u.644973\ninst\u271d\u00b3 : AddGroup A\nG\u2081 : Type u_1\nG\u2082 : Type u_2\nG\u2083 : Type u_3\ninst\u271d\u00b2 : Group G\u2081\ninst\u271d\u00b9 : Group G\u2082\ninst\u271d : Group G\u2083\nf : G\u2081 \u2192* G\u2082\nf_inv : G\u2082 \u2192 G\u2081\nhf : Function.RightInverse f_inv \u2191f\ng : G\u2081 \u2192* G\u2083\nhg : ker f \u2264 ker g\nx : G\u2081\n\u22a2 \u2191f (f_inv (\u2191f x)) = \u2191f x"}, {"tactic": "simp only [hf _]", "state_before": "case a\nG : Type ?u.644961\nG' : Type ?u.644964\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\nA : Type ?u.644973\ninst\u271d\u00b3 : AddGroup A\nG\u2081 : Type u_1\nG\u2082 : Type u_2\nG\u2083 : Type u_3\ninst\u271d\u00b2 : Group G\u2081\ninst\u271d\u00b9 : Group G\u2082\ninst\u271d : Group G\u2083\nf : G\u2081 \u2192* G\u2082\nf_inv : G\u2082 \u2192 G\u2081\nhf : Function.RightInverse f_inv \u2191f\ng : G\u2081 \u2192* G\u2083\nhg : ker f \u2264 ker g\nx : G\u2081\n\u22a2 \u2191f (f_inv (\u2191f x)) = \u2191f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Hom/NonUnitalAlg.lean", "full_name": "NonUnitalAlgHom.comp_apply", "start": [288, 1], "end": [289, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Deprecated/Group.lean", "full_name": "IsUnit.map'", "start": [438, 1], "end": [439, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/CharP/Basic.lean", "full_name": "CharP.natCast_eq_natCast", "start": [149, 1], "end": [152, 67], "traced_tactics": [{"tactic": "rw [\u2190 Int.cast_ofNat, \u2190 Int.cast_ofNat b]", "state_before": "R : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\np : \u2115\ninst\u271d : CharP R p\na b : \u2115\n\u22a2 \u2191a = \u2191b \u2194 a \u2261 b [MOD p]", "state_after": "R : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\np : \u2115\ninst\u271d : CharP R p\na b : \u2115\n\u22a2 \u2191\u2191a = \u2191\u2191b \u2194 a \u2261 b [MOD p]"}, {"tactic": "exact (CharP.intCast_eq_intCast _ _).trans Int.coe_nat_modEq_iff", "state_before": "R : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\np : \u2115\ninst\u271d : CharP R p\na b : \u2115\n\u22a2 \u2191\u2191a = \u2191\u2191b \u2194 a \u2261 b [MOD p]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.inter_union_self", "start": [1644, 1], "end": [1645, 44], "traced_tactics": [{"tactic": "rw [inter_comm, union_inter_cancel_right]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.183892\n\u03b3 : Type ?u.183895\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t\u271d t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\n\u22a2 s \u2229 (t \u222a s) = s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "full_name": "nndist_mul_mul_le", "start": [1662, 1], "end": [1664, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/Independent.lean", "full_name": "exists_affineIndependent", "start": [587, 1], "end": [608, 81], "traced_tactics": [{"tactic": "rcases s.eq_empty_or_nonempty with (rfl | \u27e8p, hp\u27e9)", "state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\n\u22a2 \u2203 t x, affineSpan k t = affineSpan k s \u2227 AffineIndependent k Subtype.val", "state_after": "case inl\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\n\u22a2 \u2203 t x, affineSpan k t = affineSpan k \u2205 \u2227 AffineIndependent k Subtype.val\n\ncase inr.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\n\u22a2 \u2203 t x, affineSpan k t = affineSpan k s \u2227 AffineIndependent k Subtype.val"}, {"tactic": "obtain \u27e8b, hb\u2081, hb\u2082, hb\u2083\u27e9 := exists_linearIndependent k ((Equiv.vaddConst p).symm '' s)", "state_before": "case inr.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\n\u22a2 \u2203 t x, affineSpan k t = affineSpan k s \u2227 AffineIndependent k Subtype.val", "state_after": "case inr.intro.intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = Submodule.span k (\u2191(Equiv.vaddConst p).symm '' s)\nhb\u2083 : LinearIndependent k Subtype.val\n\u22a2 \u2203 t x, affineSpan k t = affineSpan k s \u2227 AffineIndependent k Subtype.val"}, {"tactic": "have hb\u2080 : \u2200 v : V, v \u2208 b \u2192 v \u2260 0 := fun v hv => hb\u2083.ne_zero (\u27e8v, hv\u27e9 : b)", "state_before": "case inr.intro.intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = Submodule.span k (\u2191(Equiv.vaddConst p).symm '' s)\nhb\u2083 : LinearIndependent k Subtype.val\n\u22a2 \u2203 t x, affineSpan k t = affineSpan k s \u2227 AffineIndependent k Subtype.val", "state_after": "case inr.intro.intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = Submodule.span k (\u2191(Equiv.vaddConst p).symm '' s)\nhb\u2083 : LinearIndependent k Subtype.val\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 \u2203 t x, affineSpan k t = affineSpan k s \u2227 AffineIndependent k Subtype.val"}, {"tactic": "rw [linearIndependent_set_iff_affineIndependent_vadd_union_singleton k hb\u2080 p] at hb\u2083", "state_before": "case inr.intro.intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = Submodule.span k (\u2191(Equiv.vaddConst p).symm '' s)\nhb\u2083 : LinearIndependent k Subtype.val\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 \u2203 t x, affineSpan k t = affineSpan k s \u2227 AffineIndependent k Subtype.val", "state_after": "case inr.intro.intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = Submodule.span k (\u2191(Equiv.vaddConst p).symm '' s)\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 \u2203 t x, affineSpan k t = affineSpan k s \u2227 AffineIndependent k Subtype.val"}, {"tactic": "refine' \u27e8{p} \u222a Equiv.vaddConst p '' b, _, _, hb\u2083\u27e9", "state_before": "case inr.intro.intro.intro.intro\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = Submodule.span k (\u2191(Equiv.vaddConst p).symm '' s)\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 \u2203 t x, affineSpan k t = affineSpan k s \u2227 AffineIndependent k Subtype.val", "state_after": "case inr.intro.intro.intro.intro.refine'_1\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = Submodule.span k (\u2191(Equiv.vaddConst p).symm '' s)\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 {p} \u222a \u2191(Equiv.vaddConst p) '' b \u2286 s\n\ncase inr.intro.intro.intro.intro.refine'_2\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = Submodule.span k (\u2191(Equiv.vaddConst p).symm '' s)\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 affineSpan k ({p} \u222a \u2191(Equiv.vaddConst p) '' b) = affineSpan k s"}, {"tactic": "exact \u27e8\u2205, Set.empty_subset \u2205, rfl, affineIndependent_of_subsingleton k _\u27e9", "state_before": "case inl\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\n\u22a2 \u2203 t x, affineSpan k t = affineSpan k \u2205 \u2227 AffineIndependent k Subtype.val", "state_after": "no goals"}, {"tactic": "apply Set.union_subset (Set.singleton_subset_iff.mpr hp)", "state_before": "case inr.intro.intro.intro.intro.refine'_1\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = Submodule.span k (\u2191(Equiv.vaddConst p).symm '' s)\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 {p} \u222a \u2191(Equiv.vaddConst p) '' b \u2286 s", "state_after": "case inr.intro.intro.intro.intro.refine'_1\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = Submodule.span k (\u2191(Equiv.vaddConst p).symm '' s)\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 \u2191(Equiv.vaddConst p) '' b \u2286 s"}, {"tactic": "rwa [\u2190 (Equiv.vaddConst p).subset_image' b s]", "state_before": "case inr.intro.intro.intro.intro.refine'_1\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = Submodule.span k (\u2191(Equiv.vaddConst p).symm '' s)\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 \u2191(Equiv.vaddConst p) '' b \u2286 s", "state_after": "no goals"}, {"tactic": "rw [Equiv.coe_vaddConst_symm, \u2190 vectorSpan_eq_span_vsub_set_right k hp] at hb\u2082", "state_before": "case inr.intro.intro.intro.intro.refine'_2\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = Submodule.span k (\u2191(Equiv.vaddConst p).symm '' s)\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 affineSpan k ({p} \u222a \u2191(Equiv.vaddConst p) '' b) = affineSpan k s", "state_after": "case inr.intro.intro.intro.intro.refine'_2\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 affineSpan k ({p} \u222a \u2191(Equiv.vaddConst p) '' b) = affineSpan k s"}, {"tactic": "apply AffineSubspace.ext_of_direction_eq", "state_before": "case inr.intro.intro.intro.intro.refine'_2\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 affineSpan k ({p} \u222a \u2191(Equiv.vaddConst p) '' b) = affineSpan k s", "state_after": "case inr.intro.intro.intro.intro.refine'_2.hd\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 AffineSubspace.direction (affineSpan k ({p} \u222a \u2191(Equiv.vaddConst p) '' b)) = AffineSubspace.direction (affineSpan k s)\n\ncase inr.intro.intro.intro.intro.refine'_2.hn\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 Set.Nonempty (\u2191(affineSpan k ({p} \u222a \u2191(Equiv.vaddConst p) '' b)) \u2229 \u2191(affineSpan k s))"}, {"tactic": "have : Submodule.span k b = Submodule.span k (insert 0 b) := by simp", "state_before": "case inr.intro.intro.intro.intro.refine'_2.hd\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 AffineSubspace.direction (affineSpan k ({p} \u222a \u2191(Equiv.vaddConst p) '' b)) = AffineSubspace.direction (affineSpan k s)", "state_after": "case inr.intro.intro.intro.intro.refine'_2.hd\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\nthis : Submodule.span k b = Submodule.span k (insert 0 b)\n\u22a2 AffineSubspace.direction (affineSpan k ({p} \u222a \u2191(Equiv.vaddConst p) '' b)) = AffineSubspace.direction (affineSpan k s)"}, {"tactic": "simp only [direction_affineSpan, \u2190 hb\u2082, Equiv.coe_vaddConst, Set.singleton_union,\n  vectorSpan_eq_span_vsub_set_right k (Set.mem_insert p _), this]", "state_before": "case inr.intro.intro.intro.intro.refine'_2.hd\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\nthis : Submodule.span k b = Submodule.span k (insert 0 b)\n\u22a2 AffineSubspace.direction (affineSpan k ({p} \u222a \u2191(Equiv.vaddConst p) '' b)) = AffineSubspace.direction (affineSpan k s)", "state_after": "case inr.intro.intro.intro.intro.refine'_2.hd\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\nthis : Submodule.span k b = Submodule.span k (insert 0 b)\n\u22a2 Submodule.span k ((fun x => x -\u1d65 p) '' insert p ((fun a => a +\u1d65 p) '' b)) = Submodule.span k (insert 0 b)"}, {"tactic": "congr", "state_before": "case inr.intro.intro.intro.intro.refine'_2.hd\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\nthis : Submodule.span k b = Submodule.span k (insert 0 b)\n\u22a2 Submodule.span k ((fun x => x -\u1d65 p) '' insert p ((fun a => a +\u1d65 p) '' b)) = Submodule.span k (insert 0 b)", "state_after": "case inr.intro.intro.intro.intro.refine'_2.hd.e_s\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\nthis : Submodule.span k b = Submodule.span k (insert 0 b)\n\u22a2 (fun x => x -\u1d65 p) '' insert p ((fun a => a +\u1d65 p) '' b) = insert 0 b"}, {"tactic": "change (Equiv.vaddConst p).symm '' insert p (Equiv.vaddConst p '' b) = _", "state_before": "case inr.intro.intro.intro.intro.refine'_2.hd.e_s\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\nthis : Submodule.span k b = Submodule.span k (insert 0 b)\n\u22a2 (fun x => x -\u1d65 p) '' insert p ((fun a => a +\u1d65 p) '' b) = insert 0 b", "state_after": "case inr.intro.intro.intro.intro.refine'_2.hd.e_s\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\nthis : Submodule.span k b = Submodule.span k (insert 0 b)\n\u22a2 \u2191(Equiv.vaddConst p).symm '' insert p (\u2191(Equiv.vaddConst p) '' b) = insert 0 b"}, {"tactic": "rw [Set.image_insert_eq, \u2190 Set.image_comp]", "state_before": "case inr.intro.intro.intro.intro.refine'_2.hd.e_s\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\nthis : Submodule.span k b = Submodule.span k (insert 0 b)\n\u22a2 \u2191(Equiv.vaddConst p).symm '' insert p (\u2191(Equiv.vaddConst p) '' b) = insert 0 b", "state_after": "case inr.intro.intro.intro.intro.refine'_2.hd.e_s\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\nthis : Submodule.span k b = Submodule.span k (insert 0 b)\n\u22a2 insert (\u2191(Equiv.vaddConst p).symm p) (\u2191(Equiv.vaddConst p).symm \u2218 \u2191(Equiv.vaddConst p) '' b) = insert 0 b"}, {"tactic": "simp", "state_before": "case inr.intro.intro.intro.intro.refine'_2.hd.e_s\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\nthis : Submodule.span k b = Submodule.span k (insert 0 b)\n\u22a2 insert (\u2191(Equiv.vaddConst p).symm p) (\u2191(Equiv.vaddConst p).symm \u2218 \u2191(Equiv.vaddConst p) '' b) = insert 0 b", "state_after": "no goals"}, {"tactic": "simp", "state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 Submodule.span k b = Submodule.span k (insert 0 b)", "state_after": "no goals"}, {"tactic": "use p", "state_before": "case inr.intro.intro.intro.intro.refine'_2.hn\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 Set.Nonempty (\u2191(affineSpan k ({p} \u222a \u2191(Equiv.vaddConst p) '' b)) \u2229 \u2191(affineSpan k s))", "state_after": "case inr.intro.intro.intro.intro.refine'_2.hn\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 p \u2208 \u2191(affineSpan k ({p} \u222a \u2191(Equiv.vaddConst p) '' b)) \u2229 \u2191(affineSpan k s)"}, {"tactic": "simp only [Equiv.coe_vaddConst, Set.singleton_union, Set.mem_inter_iff, coe_affineSpan]", "state_before": "case inr.intro.intro.intro.intro.refine'_2.hn\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 p \u2208 \u2191(affineSpan k ({p} \u222a \u2191(Equiv.vaddConst p) '' b)) \u2229 \u2191(affineSpan k s)", "state_after": "case inr.intro.intro.intro.intro.refine'_2.hn\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 p \u2208 spanPoints k (insert p ((fun a => a +\u1d65 p) '' b)) \u2227 p \u2208 spanPoints k s"}, {"tactic": "exact \u27e8mem_spanPoints k _ _ (Set.mem_insert p _), mem_spanPoints k _ _ hp\u27e9", "state_before": "case inr.intro.intro.intro.intro.refine'_2.hn\nk : Type u_2\nV : Type u_3\nP : Type u_1\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type ?u.401520\ns : Set P\np : P\nhp : p \u2208 s\nb : Set V\nhb\u2081 : b \u2286 \u2191(Equiv.vaddConst p).symm '' s\nhb\u2082 : Submodule.span k b = vectorSpan k s\nhb\u2083 : AffineIndependent k fun p_1 => \u2191p_1\nhb\u2080 : \u2200 (v : V), v \u2208 b \u2192 v \u2260 0\n\u22a2 p \u2208 spanPoints k (insert p ((fun a => a +\u1d65 p) '' b)) \u2227 p \u2208 spanPoints k s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Sum/Order.lean", "full_name": "Sum.Lex.toLex_lt_toLex", "start": [334, 1], "end": [336, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Analytic/Basic.lean", "full_name": "HasFPowerSeriesOnBall.r_eq_top_of_exists", "start": [1044, 1], "end": [1056, 99], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean", "full_name": "HasFDerivAt.rpow", "start": [420, 1], "end": [423, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/InformationTheory/Hamming.lean", "full_name": "hammingNorm_smul_le_hammingNorm", "start": [225, 1], "end": [227, 93], "traced_tactics": [{"tactic": "simp_rw [smul_zero]", "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_3\n\u03b2 : \u03b9 \u2192 Type u_2\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : (i : \u03b9) \u2192 DecidableEq (\u03b2 i)\n\u03b3 : \u03b9 \u2192 Type ?u.30405\ninst\u271d\u2074 : (i : \u03b9) \u2192 DecidableEq (\u03b3 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b3 i)\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : (i : \u03b9) \u2192 SMulWithZero \u03b1 (\u03b2 i)\nk : \u03b1\nx : (i : \u03b9) \u2192 \u03b2 i\ni : \u03b9\n\u22a2 (fun i c => k \u2022 c) i 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean", "full_name": "continuousAt_const_cpow", "start": [73, 1], "end": [78, 95], "traced_tactics": [{"tactic": "have cpow_eq : (fun x : \u2102 => a ^ x) = fun x => exp (log a * x) := by\n  ext1 b\n  rw [cpow_def_of_ne_zero ha]", "state_before": "\u03b1 : Type ?u.195507\na b : \u2102\nha : a \u2260 0\n\u22a2 ContinuousAt (fun x => a ^ x) b", "state_after": "\u03b1 : Type ?u.195507\na b : \u2102\nha : a \u2260 0\ncpow_eq : (fun x => a ^ x) = fun x => Complex.exp (Complex.log a * x)\n\u22a2 ContinuousAt (fun x => a ^ x) b"}, {"tactic": "rw [cpow_eq]", "state_before": "\u03b1 : Type ?u.195507\na b : \u2102\nha : a \u2260 0\ncpow_eq : (fun x => a ^ x) = fun x => Complex.exp (Complex.log a * x)\n\u22a2 ContinuousAt (fun x => a ^ x) b", "state_after": "\u03b1 : Type ?u.195507\na b : \u2102\nha : a \u2260 0\ncpow_eq : (fun x => a ^ x) = fun x => Complex.exp (Complex.log a * x)\n\u22a2 ContinuousAt (fun x => Complex.exp (Complex.log a * x)) b"}, {"tactic": "exact continuous_exp.continuousAt.comp (ContinuousAt.mul continuousAt_const continuousAt_id)", "state_before": "\u03b1 : Type ?u.195507\na b : \u2102\nha : a \u2260 0\ncpow_eq : (fun x => a ^ x) = fun x => Complex.exp (Complex.log a * x)\n\u22a2 ContinuousAt (fun x => Complex.exp (Complex.log a * x)) b", "state_after": "no goals"}, {"tactic": "ext1 b", "state_before": "\u03b1 : Type ?u.195507\na b : \u2102\nha : a \u2260 0\n\u22a2 (fun x => a ^ x) = fun x => Complex.exp (Complex.log a * x)", "state_after": "case h\n\u03b1 : Type ?u.195507\na b\u271d : \u2102\nha : a \u2260 0\nb : \u2102\n\u22a2 a ^ b = Complex.exp (Complex.log a * b)"}, {"tactic": "rw [cpow_def_of_ne_zero ha]", "state_before": "case h\n\u03b1 : Type ?u.195507\na b\u271d : \u2102\nha : a \u2260 0\nb : \u2102\n\u22a2 a ^ b = Complex.exp (Complex.log a * b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finsupp/Multiset.lean", "full_name": "Finsupp.toMultiset_toFinsupp", "start": [188, 1], "end": [190, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "full_name": "CategoryTheory.Limits.cospanExt_inv_app_one", "start": [440, 1], "end": [441, 20], "traced_tactics": [{"tactic": "dsimp [cospanExt]", "state_before": "C : Type u\ninst\u271d\u00b9 : Category C\nD : Type u\u2082\ninst\u271d : Category D\nX Y Z X' Y' Z' : C\niX : X \u2245 X'\niY : Y \u2245 Y'\niZ : Z \u2245 Z'\nf : X \u27f6 Z\ng : Y \u27f6 Z\nf' : X' \u27f6 Z'\ng' : Y' \u27f6 Z'\nwf : iX.hom \u226b f' = f \u226b iZ.hom\nwg : iY.hom \u226b g' = g \u226b iZ.hom\n\u22a2 (cospanExt iX iY iZ wf wg).inv.app WalkingCospan.one = iZ.inv", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/Defs.lean", "full_name": "Equiv.injective", "start": [200, 11], "end": [200, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/Completion.lean", "full_name": "UniformSpace.Completion.continuous_extension", "start": [549, 1], "end": [550, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Star.lean", "full_name": "starConvex_iff_ordConnected", "start": [448, 1], "end": [450, 97], "traced_tactics": [{"tactic": "simp_rw [ordConnected_iff_uIcc_subset_left hx, starConvex_iff_segment_subset, segment_eq_uIcc]", "state_before": "\ud835\udd5c : Type u_1\nE : Type ?u.199834\nF : Type ?u.199837\ninst\u271d : LinearOrderedField \ud835\udd5c\nx : \ud835\udd5c\ns : Set \ud835\udd5c\nhx : x \u2208 s\n\u22a2 StarConvex \ud835\udd5c x s \u2194 OrdConnected s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/EMetricSpace.lean", "full_name": "zero_eq_edist", "start": [1011, 1], "end": [1011, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/Sub.lean", "full_name": "MeasureTheory.Measure.sub_add_cancel_of_le", "start": [102, 1], "end": [104, 79], "traced_tactics": [{"tactic": "ext1 s h_s_meas", "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : \u03bd \u2264 \u03bc\n\u22a2 \u03bc - \u03bd + \u03bd = \u03bc", "state_after": "case h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns\u271d : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : \u03bd \u2264 \u03bc\ns : Set \u03b1\nh_s_meas : MeasurableSet s\n\u22a2 \u2191\u2191(\u03bc - \u03bd + \u03bd) s = \u2191\u2191\u03bc s"}, {"tactic": "rw [add_apply, sub_apply h_s_meas h\u2081, tsub_add_cancel_of_le (h\u2081 s h_s_meas)]", "state_before": "case h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns\u271d : Set \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nh\u2081 : \u03bd \u2264 \u03bc\ns : Set \u03b1\nh_s_meas : MeasurableSet s\n\u22a2 \u2191\u2191(\u03bc - \u03bd + \u03bd) s = \u2191\u2191\u03bc s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Cone/Basic.lean", "full_name": "ConvexCone.mem_iInf", "start": [164, 1], "end": [165, 31], "traced_tactics": [{"tactic": "simp", "state_before": "\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type ?u.13551\nG : Type ?u.13554\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\nS T : ConvexCone \ud835\udd5c E\n\u03b9 : Sort u_1\nx : E\nf : \u03b9 \u2192 ConvexCone \ud835\udd5c E\n\u22a2 (\u2200 (i : ConvexCone \ud835\udd5c E), i \u2208 Set.range f \u2192 x \u2208 \u2191i) \u2194 \u2200 (i : \u03b9), x \u2208 f i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "full_name": "PowerSeries.order_eq_top", "start": [2280, 1], "end": [2287, 21], "traced_tactics": [{"tactic": "constructor", "state_before": "R : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 : PowerSeries R\n\u22a2 order \u03c6 = \u22a4 \u2194 \u03c6 = 0", "state_after": "case mp\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 : PowerSeries R\n\u22a2 order \u03c6 = \u22a4 \u2192 \u03c6 = 0\n\ncase mpr\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 : PowerSeries R\n\u22a2 \u03c6 = 0 \u2192 order \u03c6 = \u22a4"}, {"tactic": "intro h", "state_before": "case mp\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 : PowerSeries R\n\u22a2 order \u03c6 = \u22a4 \u2192 \u03c6 = 0", "state_after": "case mp\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 : PowerSeries R\nh : order \u03c6 = \u22a4\n\u22a2 \u03c6 = 0"}, {"tactic": "ext n", "state_before": "case mp\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 : PowerSeries R\nh : order \u03c6 = \u22a4\n\u22a2 \u03c6 = 0", "state_after": "case mp.h\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 : PowerSeries R\nh : order \u03c6 = \u22a4\nn : \u2115\n\u22a2 \u2191(coeff R n) \u03c6 = \u2191(coeff R n) 0"}, {"tactic": "rw [(coeff R n).map_zero, coeff_of_lt_order]", "state_before": "case mp.h\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 : PowerSeries R\nh : order \u03c6 = \u22a4\nn : \u2115\n\u22a2 \u2191(coeff R n) \u03c6 = \u2191(coeff R n) 0", "state_after": "case mp.h.h\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 : PowerSeries R\nh : order \u03c6 = \u22a4\nn : \u2115\n\u22a2 \u2191n < order \u03c6"}, {"tactic": "simp [h]", "state_before": "case mp.h.h\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 : PowerSeries R\nh : order \u03c6 = \u22a4\nn : \u2115\n\u22a2 \u2191n < order \u03c6", "state_after": "no goals"}, {"tactic": "rintro rfl", "state_before": "case mpr\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 : PowerSeries R\n\u22a2 \u03c6 = 0 \u2192 order \u03c6 = \u22a4", "state_after": "case mpr\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6 : PowerSeries R\n\u22a2 order 0 = \u22a4"}, {"tactic": "exact order_zero", "state_before": "case mpr\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6 : PowerSeries R\n\u22a2 order 0 = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/MonoidLocalization.lean", "full_name": "Localization.r_of_eq", "start": [425, 1], "end": [426, 32], "traced_tactics": [{"tactic": "rw [h]", "state_before": "M : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type ?u.126394\ninst\u271d\u00b9 : CommMonoid N\nP : Type ?u.126400\ninst\u271d : CommMonoid P\nx y : M \u00d7 { x // x \u2208 S }\nh : \u2191y.snd * x.fst = \u2191x.snd * y.fst\n\u22a2 \u21911 * (\u2191y.snd * x.fst) = \u21911 * (\u2191x.snd * y.fst)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.toList_eq_singleton_iff", "start": [3393, 1], "end": [3394, 70], "traced_tactics": [{"tactic": "rw [toList, Multiset.toList_eq_singleton_iff, val_eq_singleton_iff]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.481437\n\u03b3 : Type ?u.481440\na : \u03b1\ns : Finset \u03b1\n\u22a2 toList s = [a] \u2194 s = {a}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Matrix/Notation.lean", "full_name": "Matrix.mul_fin_three", "start": [445, 1], "end": [456, 96], "traced_tactics": [{"tactic": "ext (i j)", "state_before": "\u03b1 : Type u\no n m : \u2115\nm' : Type u\u2098\nn' : Type u\u2099\no' : Type u\u2092\na b : \u2115\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : Mul \u03b1\na\u2081\u2081 a\u2081\u2082 a\u2081\u2083 a\u2082\u2081 a\u2082\u2082 a\u2082\u2083 a\u2083\u2081 a\u2083\u2082 a\u2083\u2083 b\u2081\u2081 b\u2081\u2082 b\u2081\u2083 b\u2082\u2081 b\u2082\u2082 b\u2082\u2083 b\u2083\u2081 b\u2083\u2082 b\u2083\u2083 : \u03b1\n\u22a2 \u2191of ![![a\u2081\u2081, a\u2081\u2082, a\u2081\u2083], ![a\u2082\u2081, a\u2082\u2082, a\u2082\u2083], ![a\u2083\u2081, a\u2083\u2082, a\u2083\u2083]] \u2b1d\n      \u2191of ![![b\u2081\u2081, b\u2081\u2082, b\u2081\u2083], ![b\u2082\u2081, b\u2082\u2082, b\u2082\u2083], ![b\u2083\u2081, b\u2083\u2082, b\u2083\u2083]] =\n    \u2191of\n      ![![a\u2081\u2081 * b\u2081\u2081 + a\u2081\u2082 * b\u2082\u2081 + a\u2081\u2083 * b\u2083\u2081, a\u2081\u2081 * b\u2081\u2082 + a\u2081\u2082 * b\u2082\u2082 + a\u2081\u2083 * b\u2083\u2082, a\u2081\u2081 * b\u2081\u2083 + a\u2081\u2082 * b\u2082\u2083 + a\u2081\u2083 * b\u2083\u2083],\n        ![a\u2082\u2081 * b\u2081\u2081 + a\u2082\u2082 * b\u2082\u2081 + a\u2082\u2083 * b\u2083\u2081, a\u2082\u2081 * b\u2081\u2082 + a\u2082\u2082 * b\u2082\u2082 + a\u2082\u2083 * b\u2083\u2082, a\u2082\u2081 * b\u2081\u2083 + a\u2082\u2082 * b\u2082\u2083 + a\u2082\u2083 * b\u2083\u2083],\n        ![a\u2083\u2081 * b\u2081\u2081 + a\u2083\u2082 * b\u2082\u2081 + a\u2083\u2083 * b\u2083\u2081, a\u2083\u2081 * b\u2081\u2082 + a\u2083\u2082 * b\u2082\u2082 + a\u2083\u2083 * b\u2083\u2082, a\u2083\u2081 * b\u2081\u2083 + a\u2083\u2082 * b\u2082\u2083 + a\u2083\u2083 * b\u2083\u2083]]", "state_after": "case a.h\n\u03b1 : Type u\no n m : \u2115\nm' : Type u\u2098\nn' : Type u\u2099\no' : Type u\u2092\na b : \u2115\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : Mul \u03b1\na\u2081\u2081 a\u2081\u2082 a\u2081\u2083 a\u2082\u2081 a\u2082\u2082 a\u2082\u2083 a\u2083\u2081 a\u2083\u2082 a\u2083\u2083 b\u2081\u2081 b\u2081\u2082 b\u2081\u2083 b\u2082\u2081 b\u2082\u2082 b\u2082\u2083 b\u2083\u2081 b\u2083\u2082 b\u2083\u2083 : \u03b1\ni j : Fin 3\n\u22a2 (\u2191of ![![a\u2081\u2081, a\u2081\u2082, a\u2081\u2083], ![a\u2082\u2081, a\u2082\u2082, a\u2082\u2083], ![a\u2083\u2081, a\u2083\u2082, a\u2083\u2083]] \u2b1d\n        \u2191of ![![b\u2081\u2081, b\u2081\u2082, b\u2081\u2083], ![b\u2082\u2081, b\u2082\u2082, b\u2082\u2083], ![b\u2083\u2081, b\u2083\u2082, b\u2083\u2083]])\n      i j =\n    \u2191of\n      ![![a\u2081\u2081 * b\u2081\u2081 + a\u2081\u2082 * b\u2082\u2081 + a\u2081\u2083 * b\u2083\u2081, a\u2081\u2081 * b\u2081\u2082 + a\u2081\u2082 * b\u2082\u2082 + a\u2081\u2083 * b\u2083\u2082, a\u2081\u2081 * b\u2081\u2083 + a\u2081\u2082 * b\u2082\u2083 + a\u2081\u2083 * b\u2083\u2083],\n        ![a\u2082\u2081 * b\u2081\u2081 + a\u2082\u2082 * b\u2082\u2081 + a\u2082\u2083 * b\u2083\u2081, a\u2082\u2081 * b\u2081\u2082 + a\u2082\u2082 * b\u2082\u2082 + a\u2082\u2083 * b\u2083\u2082, a\u2082\u2081 * b\u2081\u2083 + a\u2082\u2082 * b\u2082\u2083 + a\u2082\u2083 * b\u2083\u2083],\n        ![a\u2083\u2081 * b\u2081\u2081 + a\u2083\u2082 * b\u2082\u2081 + a\u2083\u2083 * b\u2083\u2081, a\u2083\u2081 * b\u2081\u2082 + a\u2083\u2082 * b\u2082\u2082 + a\u2083\u2083 * b\u2083\u2082, a\u2083\u2081 * b\u2081\u2083 + a\u2083\u2082 * b\u2082\u2083 + a\u2083\u2083 * b\u2083\u2083]]\n      i j"}, {"tactic": "fin_cases i <;> fin_cases j <;> simp [Matrix.mul, dotProduct, Fin.sum_univ_succ, \u2190 add_assoc]", "state_before": "case a.h\n\u03b1 : Type u\no n m : \u2115\nm' : Type u\u2098\nn' : Type u\u2099\no' : Type u\u2092\na b : \u2115\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : Mul \u03b1\na\u2081\u2081 a\u2081\u2082 a\u2081\u2083 a\u2082\u2081 a\u2082\u2082 a\u2082\u2083 a\u2083\u2081 a\u2083\u2082 a\u2083\u2083 b\u2081\u2081 b\u2081\u2082 b\u2081\u2083 b\u2082\u2081 b\u2082\u2082 b\u2082\u2083 b\u2083\u2081 b\u2083\u2082 b\u2083\u2083 : \u03b1\ni j : Fin 3\n\u22a2 (\u2191of ![![a\u2081\u2081, a\u2081\u2082, a\u2081\u2083], ![a\u2082\u2081, a\u2082\u2082, a\u2082\u2083], ![a\u2083\u2081, a\u2083\u2082, a\u2083\u2083]] \u2b1d\n        \u2191of ![![b\u2081\u2081, b\u2081\u2082, b\u2081\u2083], ![b\u2082\u2081, b\u2082\u2082, b\u2082\u2083], ![b\u2083\u2081, b\u2083\u2082, b\u2083\u2083]])\n      i j =\n    \u2191of\n      ![![a\u2081\u2081 * b\u2081\u2081 + a\u2081\u2082 * b\u2082\u2081 + a\u2081\u2083 * b\u2083\u2081, a\u2081\u2081 * b\u2081\u2082 + a\u2081\u2082 * b\u2082\u2082 + a\u2081\u2083 * b\u2083\u2082, a\u2081\u2081 * b\u2081\u2083 + a\u2081\u2082 * b\u2082\u2083 + a\u2081\u2083 * b\u2083\u2083],\n        ![a\u2082\u2081 * b\u2081\u2081 + a\u2082\u2082 * b\u2082\u2081 + a\u2082\u2083 * b\u2083\u2081, a\u2082\u2081 * b\u2081\u2082 + a\u2082\u2082 * b\u2082\u2082 + a\u2082\u2083 * b\u2083\u2082, a\u2082\u2081 * b\u2081\u2083 + a\u2082\u2082 * b\u2082\u2083 + a\u2082\u2083 * b\u2083\u2083],\n        ![a\u2083\u2081 * b\u2081\u2081 + a\u2083\u2082 * b\u2082\u2081 + a\u2083\u2083 * b\u2083\u2081, a\u2083\u2081 * b\u2081\u2082 + a\u2083\u2082 * b\u2082\u2082 + a\u2083\u2083 * b\u2083\u2082, a\u2083\u2081 * b\u2081\u2083 + a\u2083\u2082 * b\u2082\u2083 + a\u2083\u2083 * b\u2083\u2083]]\n      i j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Interval.lean", "full_name": "Interval.dual_map", "start": [432, 1], "end": [435, 33], "traced_tactics": [{"tactic": "cases s", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.20428\n\u03b4 : Type ?u.20431\n\u03b9 : Sort ?u.20434\n\u03ba : \u03b9 \u2192 Sort ?u.20439\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192o \u03b2\ns : Interval \u03b1\n\u22a2 \u2191dual (map f s) = map (\u2191OrderHom.dual f) (\u2191dual s)", "state_after": "case none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.20428\n\u03b4 : Type ?u.20431\n\u03b9 : Sort ?u.20434\n\u03ba : \u03b9 \u2192 Sort ?u.20439\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192o \u03b2\n\u22a2 \u2191dual (map f none) = map (\u2191OrderHom.dual f) (\u2191dual none)\n\ncase some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.20428\n\u03b4 : Type ?u.20431\n\u03b9 : Sort ?u.20434\n\u03ba : \u03b9 \u2192 Sort ?u.20439\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192o \u03b2\nval\u271d : NonemptyInterval \u03b1\n\u22a2 \u2191dual (map f (some val\u271d)) = map (\u2191OrderHom.dual f) (\u2191dual (some val\u271d))"}, {"tactic": "rfl", "state_before": "case none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.20428\n\u03b4 : Type ?u.20431\n\u03b9 : Sort ?u.20434\n\u03ba : \u03b9 \u2192 Sort ?u.20439\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192o \u03b2\n\u22a2 \u2191dual (map f none) = map (\u2191OrderHom.dual f) (\u2191dual none)", "state_after": "no goals"}, {"tactic": "exact WithBot.map_comm rfl _", "state_before": "case some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.20428\n\u03b4 : Type ?u.20431\n\u03b9 : Sort ?u.20434\n\u03ba : \u03b9 \u2192 Sort ?u.20439\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192o \u03b2\nval\u271d : NonemptyInterval \u03b1\n\u22a2 \u2191dual (map f (some val\u271d)) = map (\u2191OrderHom.dual f) (\u2191dual (some val\u271d))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Tropical/Basic.lean", "full_name": "Tropical.untrop_inj_iff", "start": [91, 1], "end": [92, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Operations.lean", "full_name": "Submodule.mul_one", "start": [226, 11], "end": [228, 40], "traced_tactics": [{"tactic": "conv_lhs => rw [one_eq_span, \u2190 span_eq M]", "state_before": "\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n\u22a2 M * 1 = M", "state_after": "\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n\u22a2 span R \u2191M * span R {1} = M"}, {"tactic": "erw [span_mul_span, mul_one, span_eq]", "state_before": "\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n\u22a2 span R \u2191M * span R {1} = M", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/Hyperreal.lean", "full_name": "Hyperreal.infinite_iff_abs_lt_abs", "start": [533, 1], "end": [535, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Submonoid/Operations.lean", "full_name": "MonoidHom.mclosure_preimage_le", "start": [1083, 1], "end": [1084, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "full_name": "UniqueFactorizationMonoid.normalizedFactors_one", "start": [655, 1], "end": [664, 47], "traced_tactics": [{"tactic": "cases' subsingleton_or_nontrivial \u03b1 with h h", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\n\u22a2 normalizedFactors 1 = 0", "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Subsingleton \u03b1\n\u22a2 normalizedFactors 1 = 0\n\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\n\u22a2 normalizedFactors 1 = 0"}, {"tactic": "dsimp [normalizedFactors, factors]", "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Subsingleton \u03b1\n\u22a2 normalizedFactors 1 = 0", "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Subsingleton \u03b1\n\u22a2 Multiset.map (\u2191normalize)\n      (if h : 1 = 0 then 0 else Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 1)) =\n    0"}, {"tactic": "simp [Subsingleton.elim (1:\u03b1) 0]", "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Subsingleton \u03b1\n\u22a2 Multiset.map (\u2191normalize)\n      (if h : 1 = 0 then 0 else Classical.choose (_ : \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Prime b) \u2227 Multiset.prod f ~\u1d64 1)) =\n    0", "state_after": "no goals"}, {"tactic": "rw [\u2190 Multiset.rel_zero_right]", "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\n\u22a2 normalizedFactors 1 = 0", "state_after": "case inr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\n\u22a2 Multiset.Rel ?m.597280 (normalizedFactors 1) 0\n\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\n\u22a2 Type ?u.597277\n\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\n\u22a2 \u03b1 \u2192 ?m.597279 \u2192 Prop"}, {"tactic": "apply factors_unique irreducible_of_normalized_factor", "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\n\u22a2 Multiset.Rel ?m.597280 (normalizedFactors 1) 0\n\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\n\u22a2 Type ?u.597277\n\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\n\u22a2 \u03b1 \u2192 ?m.597279 \u2192 Prop", "state_after": "case inr.hg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\n\u22a2 \u2200 (x : \u03b1), x \u2208 0 \u2192 Irreducible x\n\ncase inr.h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\n\u22a2 Multiset.prod (normalizedFactors 1) ~\u1d64 Multiset.prod 0"}, {"tactic": "intro x hx", "state_before": "case inr.hg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\n\u22a2 \u2200 (x : \u03b1), x \u2208 0 \u2192 Irreducible x", "state_after": "case inr.hg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\nx : \u03b1\nhx : x \u2208 0\n\u22a2 Irreducible x"}, {"tactic": "exfalso", "state_before": "case inr.hg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\nx : \u03b1\nhx : x \u2208 0\n\u22a2 Irreducible x", "state_after": "case inr.hg.h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\nx : \u03b1\nhx : x \u2208 0\n\u22a2 False"}, {"tactic": "apply Multiset.not_mem_zero x hx", "state_before": "case inr.hg.h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\nx : \u03b1\nhx : x \u2208 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "apply normalizedFactors_prod one_ne_zero", "state_before": "case inr.h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nh : Nontrivial \u03b1\n\u22a2 Multiset.prod (normalizedFactors 1) ~\u1d64 Multiset.prod 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Stream/Init.lean", "full_name": "Stream'.tail_drop", "start": [78, 1], "end": [78, 91], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\ns : Stream' \u03b1\n\u22a2 tail (drop n s) = drop n (tail s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/IsROrC.lean", "full_name": "ContinuousLinearMap.op_norm_bound_of_ball_bound", "start": [88, 1], "end": [96, 27], "traced_tactics": [{"tactic": "apply ContinuousLinearMap.op_norm_le_bound", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nr_pos : 0 < r\nc : \u211d\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nh : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c\n\u22a2 \u2016f\u2016 \u2264 c / r", "state_after": "case hMp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nr_pos : 0 < r\nc : \u211d\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nh : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c\n\u22a2 0 \u2264 c / r\n\ncase hM\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nr_pos : 0 < r\nc : \u211d\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nh : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c\n\u22a2 \u2200 (x : E), \u2016\u2191f x\u2016 \u2264 c / r * \u2016x\u2016"}, {"tactic": "apply LinearMap.bound_of_ball_bound' r_pos", "state_before": "case hM\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nr_pos : 0 < r\nc : \u211d\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nh : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c\n\u22a2 \u2200 (x : E), \u2016\u2191f x\u2016 \u2264 c / r * \u2016x\u2016", "state_after": "case hM.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nr_pos : 0 < r\nc : \u211d\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nh : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c\n\u22a2 \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191\u2191f z\u2016 \u2264 c"}, {"tactic": "exact fun z hz => h z hz", "state_before": "case hM.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nr_pos : 0 < r\nc : \u211d\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nh : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c\n\u22a2 \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191\u2191f z\u2016 \u2264 c", "state_after": "no goals"}, {"tactic": "apply div_nonneg _ r_pos.le", "state_before": "case hMp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nr_pos : 0 < r\nc : \u211d\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nh : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c\n\u22a2 0 \u2264 c / r", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nr_pos : 0 < r\nc : \u211d\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nh : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c\n\u22a2 0 \u2264 c"}, {"tactic": "exact\n  (norm_nonneg _).trans\n    (h 0 (by simp only [norm_zero, mem_closedBall, dist_zero_left, r_pos.le]))", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nr_pos : 0 < r\nc : \u211d\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nh : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c\n\u22a2 0 \u2264 c", "state_after": "no goals"}, {"tactic": "simp only [norm_zero, mem_closedBall, dist_zero_left, r_pos.le]", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nr_pos : 0 < r\nc : \u211d\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nh : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c\n\u22a2 0 \u2208 closedBall 0 r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Submonoid/Basic.lean", "full_name": "Submonoid.closure_eq_of_le", "start": [435, 1], "end": [436, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.average_union", "start": [151, 1], "end": [157, 100], "traced_tactics": [{"tactic": "haveI := Fact.mk hs\u03bc.lt_top", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.213640\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc : Measure \u03b1\ns\u271d : Set E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\n\u22a2 (\u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc) =\n    ((ENNReal.toReal (\u2191\u2191\u03bc s) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in s, f x \u2202\u03bc) +\n      (ENNReal.toReal (\u2191\u2191\u03bc t) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in t, f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.213640\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc : Measure \u03b1\ns\u271d : Set E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 (\u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc) =\n    ((ENNReal.toReal (\u2191\u2191\u03bc s) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in s, f x \u2202\u03bc) +\n      (ENNReal.toReal (\u2191\u2191\u03bc t) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in t, f x \u2202\u03bc"}, {"tactic": "haveI := Fact.mk ht\u03bc.lt_top", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.213640\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc : Measure \u03b1\ns\u271d : Set E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\nthis : Fact (\u2191\u2191\u03bc s < \u22a4)\n\u22a2 (\u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc) =\n    ((ENNReal.toReal (\u2191\u2191\u03bc s) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in s, f x \u2202\u03bc) +\n      (ENNReal.toReal (\u2191\u2191\u03bc t) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in t, f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.213640\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc : Measure \u03b1\ns\u271d : Set E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\nthis\u271d : Fact (\u2191\u2191\u03bc s < \u22a4)\nthis : Fact (\u2191\u2191\u03bc t < \u22a4)\n\u22a2 (\u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc) =\n    ((ENNReal.toReal (\u2191\u2191\u03bc s) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in s, f x \u2202\u03bc) +\n      (ENNReal.toReal (\u2191\u2191\u03bc t) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in t, f x \u2202\u03bc"}, {"tactic": "rw [restrict_union\u2080 hd ht, average_add_measure hfs hft, restrict_apply_univ, restrict_apply_univ]", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.213640\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\n\u03bc : Measure \u03b1\ns\u271d : Set E\nf : \u03b1 \u2192 E\ns t : Set \u03b1\nhd : AEDisjoint \u03bc s t\nht : NullMeasurableSet t\nhs\u03bc : \u2191\u2191\u03bc s \u2260 \u22a4\nht\u03bc : \u2191\u2191\u03bc t \u2260 \u22a4\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\nthis\u271d : Fact (\u2191\u2191\u03bc s < \u22a4)\nthis : Fact (\u2191\u2191\u03bc t < \u22a4)\n\u22a2 (\u2a0d (x : \u03b1) in s \u222a t, f x \u2202\u03bc) =\n    ((ENNReal.toReal (\u2191\u2191\u03bc s) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in s, f x \u2202\u03bc) +\n      (ENNReal.toReal (\u2191\u2191\u03bc t) / (ENNReal.toReal (\u2191\u2191\u03bc s) + ENNReal.toReal (\u2191\u2191\u03bc t))) \u2022 \u2a0d (x : \u03b1) in t, f x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Int/Order/Basic.lean", "full_name": "Int.sign_eq_ediv_abs", "start": [413, 11], "end": [415, 85], "traced_tactics": [{"tactic": "simp [az]", "state_before": "a\u271d b : \u2124\nn : \u2115\na : \u2124\naz : a = 0\n\u22a2 sign a = a / abs a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Finrank.lean", "full_name": "Subalgebra.rank_top", "start": [552, 1], "end": [554, 28], "traced_tactics": [{"tactic": "rw [subalgebra_top_rank_eq_submodule_top_rank]", "state_before": "K : Type u\nV : Type v\nF : Type u_2\nE : Type u_1\ninst\u271d\u00b2 : CommRing F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\n\u22a2 Module.rank F { x // x \u2208 \u22a4 } = Module.rank F E", "state_after": "K : Type u\nV : Type v\nF : Type u_2\nE : Type u_1\ninst\u271d\u00b2 : CommRing F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\n\u22a2 Module.rank F { x // x \u2208 \u22a4 } = Module.rank F E"}, {"tactic": "exact _root_.rank_top F E", "state_before": "K : Type u\nV : Type v\nF : Type u_2\nE : Type u_1\ninst\u271d\u00b2 : CommRing F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\n\u22a2 Module.rank F { x // x \u2208 \u22a4 } = Module.rank F E", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "full_name": "chartedSpaceSelf_prod", "start": [729, 1], "end": [734, 8], "traced_tactics": [{"tactic": "ext1", "state_before": "H : Type u\nH' : Type u_1\nM : Type ?u.51637\nM' : Type ?u.51640\nM'' : Type ?u.51643\ninst\u271d\u2075 : TopologicalSpace H\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace H M\ninst\u271d\u00b2 : TopologicalSpace H'\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H' M'\nx : M \u00d7 M'\n\u22a2 prodChartedSpace H H H' H' = chartedSpaceSelf (H \u00d7 H')", "state_after": "case atlas\nH : Type u\nH' : Type u_1\nM : Type ?u.51637\nM' : Type ?u.51640\nM'' : Type ?u.51643\ninst\u271d\u2075 : TopologicalSpace H\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace H M\ninst\u271d\u00b2 : TopologicalSpace H'\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H' M'\nx : M \u00d7 M'\n\u22a2 ChartedSpace.atlas = ChartedSpace.atlas\n\ncase chartAt\nH : Type u\nH' : Type u_1\nM : Type ?u.51637\nM' : Type ?u.51640\nM'' : Type ?u.51643\ninst\u271d\u2075 : TopologicalSpace H\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace H M\ninst\u271d\u00b2 : TopologicalSpace H'\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H' M'\nx : M \u00d7 M'\n\u22a2 ChartedSpace.chartAt = ChartedSpace.chartAt"}, {"tactic": "simp [prodChartedSpace, atlas, ChartedSpace.atlas]", "state_before": "case atlas\nH : Type u\nH' : Type u_1\nM : Type ?u.51637\nM' : Type ?u.51640\nM'' : Type ?u.51643\ninst\u271d\u2075 : TopologicalSpace H\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace H M\ninst\u271d\u00b2 : TopologicalSpace H'\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H' M'\nx : M \u00d7 M'\n\u22a2 ChartedSpace.atlas = ChartedSpace.atlas", "state_after": "no goals"}, {"tactic": "ext1", "state_before": "case chartAt\nH : Type u\nH' : Type u_1\nM : Type ?u.51637\nM' : Type ?u.51640\nM'' : Type ?u.51643\ninst\u271d\u2075 : TopologicalSpace H\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace H M\ninst\u271d\u00b2 : TopologicalSpace H'\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H' M'\nx : M \u00d7 M'\n\u22a2 ChartedSpace.chartAt = ChartedSpace.chartAt", "state_after": "case chartAt.h\nH : Type u\nH' : Type u_1\nM : Type ?u.51637\nM' : Type ?u.51640\nM'' : Type ?u.51643\ninst\u271d\u2075 : TopologicalSpace H\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace H M\ninst\u271d\u00b2 : TopologicalSpace H'\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H' M'\nx : M \u00d7 M'\nx\u271d : H \u00d7 H'\n\u22a2 ChartedSpace.chartAt x\u271d = ChartedSpace.chartAt x\u271d"}, {"tactic": "simp only [prodChartedSpace_chartAt, chartAt_self_eq, refl_prod_refl]", "state_before": "case chartAt.h\nH : Type u\nH' : Type u_1\nM : Type ?u.51637\nM' : Type ?u.51640\nM'' : Type ?u.51643\ninst\u271d\u2075 : TopologicalSpace H\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace H M\ninst\u271d\u00b2 : TopologicalSpace H'\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H' M'\nx : M \u00d7 M'\nx\u271d : H \u00d7 H'\n\u22a2 ChartedSpace.chartAt x\u271d = ChartedSpace.chartAt x\u271d", "state_after": "case chartAt.h\nH : Type u\nH' : Type u_1\nM : Type ?u.51637\nM' : Type ?u.51640\nM'' : Type ?u.51643\ninst\u271d\u2075 : TopologicalSpace H\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace H M\ninst\u271d\u00b2 : TopologicalSpace H'\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H' M'\nx : M \u00d7 M'\nx\u271d : H \u00d7 H'\n\u22a2 LocalHomeomorph.refl (H \u00d7 H') = LocalHomeomorph.refl (ModelProd H H')"}, {"tactic": "rfl", "state_before": "case chartAt.h\nH : Type u\nH' : Type u_1\nM : Type ?u.51637\nM' : Type ?u.51640\nM'' : Type ?u.51643\ninst\u271d\u2075 : TopologicalSpace H\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace H M\ninst\u271d\u00b2 : TopologicalSpace H'\ninst\u271d\u00b9 : TopologicalSpace M'\ninst\u271d : ChartedSpace H' M'\nx : M \u00d7 M'\nx\u271d : H \u00d7 H'\n\u22a2 LocalHomeomorph.refl (H \u00d7 H') = LocalHomeomorph.refl (ModelProd H H')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.coe_subset", "start": [388, 1], "end": [389, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "full_name": "Matrix.transvection_mul_apply_of_ne", "start": [135, 1], "end": [136, 87], "traced_tactics": [{"tactic": "simp [transvection, Matrix.add_mul, ha]", "state_before": "n : Type u_1\np : Type ?u.16503\nR : Type u\u2082\n\ud835\udd5c : Type ?u.16508\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\na b : n\nha : a \u2260 i\nc : R\nM : Matrix n n R\n\u22a2 (transvection i j c \u2b1d M) a b = M a b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Laurent.lean", "full_name": "Polynomial.toLaurent_ne_zero", "start": [389, 1], "end": [390, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Opposite.lean", "full_name": "Set.op_unop", "start": [56, 1], "end": [56, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/List/Init/Lemmas.lean", "full_name": "List.and_nil", "start": [40, 9], "end": [40, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "full_name": "Matrix.det_units_conj", "start": [219, 1], "end": [221, 78], "traced_tactics": [{"tactic": "rw [det_mul_right_comm, \u2190 mul_eq_mul, \u2190 mul_eq_mul, Units.mul_inv, one_mul]", "state_before": "m : Type u_1\nn : Type ?u.907427\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM : (Matrix m m R)\u02e3\nN : Matrix m m R\n\u22a2 det (\u2191M \u2b1d N \u2b1d \u2191M\u207b\u00b9) = det N", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Ray.lean", "full_name": "Module.Ray.units_smul_of_neg", "start": [490, 1], "end": [492, 41], "traced_tactics": [{"tactic": "rw [\u2190 neg_inj, neg_neg, \u2190 neg_units_smul, units_smul_of_pos]", "state_before": "R : Type u_1\ninst\u271d\u2074 : StrictOrderedCommRing R\nM : Type u_2\nN : Type ?u.193734\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nx y : M\nu : R\u02e3\nhu : \u2191u < 0\nv : Ray R M\n\u22a2 u \u2022 v = -v", "state_after": "case hu\nR : Type u_1\ninst\u271d\u2074 : StrictOrderedCommRing R\nM : Type u_2\nN : Type ?u.193734\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nx y : M\nu : R\u02e3\nhu : \u2191u < 0\nv : Ray R M\n\u22a2 0 < \u2191(-u)"}, {"tactic": "rwa [Units.val_neg, Right.neg_pos_iff]", "state_before": "case hu\nR : Type u_1\ninst\u271d\u2074 : StrictOrderedCommRing R\nM : Type u_2\nN : Type ?u.193734\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nx y : M\nu : R\u02e3\nhu : \u2191u < 0\nv : Ray R M\n\u22a2 0 < \u2191(-u)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Bornology/Basic.lean", "full_name": "Bornology.cobounded_eq_bot", "start": [353, 1], "end": [354, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subgroup/Simple.lean", "full_name": "IsSimpleGroup.isSimpleGroup_of_surjective", "start": [69, 1], "end": [75, 34], "traced_tactics": [{"tactic": "refine' (iH.comap f).eq_bot_or_eq_top.imp (fun h => _) fun h => _", "state_before": "G : Type u_2\ninst\u271d\u2074 : Group G\nA : Type ?u.1231\ninst\u271d\u00b3 : AddGroup A\nH\u271d : Type u_1\ninst\u271d\u00b2 : Group H\u271d\ninst\u271d\u00b9 : IsSimpleGroup G\ninst\u271d : Nontrivial H\u271d\nf : G \u2192* H\u271d\nhf : Function.Surjective \u2191f\nH : Subgroup H\u271d\niH : Normal H\n\u22a2 H = \u22a5 \u2228 H = \u22a4", "state_after": "case refine'_1\nG : Type u_2\ninst\u271d\u2074 : Group G\nA : Type ?u.1231\ninst\u271d\u00b3 : AddGroup A\nH\u271d : Type u_1\ninst\u271d\u00b2 : Group H\u271d\ninst\u271d\u00b9 : IsSimpleGroup G\ninst\u271d : Nontrivial H\u271d\nf : G \u2192* H\u271d\nhf : Function.Surjective \u2191f\nH : Subgroup H\u271d\niH : Normal H\nh : comap f H = \u22a5\n\u22a2 H = \u22a5\n\ncase refine'_2\nG : Type u_2\ninst\u271d\u2074 : Group G\nA : Type ?u.1231\ninst\u271d\u00b3 : AddGroup A\nH\u271d : Type u_1\ninst\u271d\u00b2 : Group H\u271d\ninst\u271d\u00b9 : IsSimpleGroup G\ninst\u271d : Nontrivial H\u271d\nf : G \u2192* H\u271d\nhf : Function.Surjective \u2191f\nH : Subgroup H\u271d\niH : Normal H\nh : comap f H = \u22a4\n\u22a2 H = \u22a4"}, {"tactic": "rw [\u2190 map_bot f, \u2190 h, map_comap_eq_self_of_surjective hf]", "state_before": "case refine'_1\nG : Type u_2\ninst\u271d\u2074 : Group G\nA : Type ?u.1231\ninst\u271d\u00b3 : AddGroup A\nH\u271d : Type u_1\ninst\u271d\u00b2 : Group H\u271d\ninst\u271d\u00b9 : IsSimpleGroup G\ninst\u271d : Nontrivial H\u271d\nf : G \u2192* H\u271d\nhf : Function.Surjective \u2191f\nH : Subgroup H\u271d\niH : Normal H\nh : comap f H = \u22a5\n\u22a2 H = \u22a5", "state_after": "no goals"}, {"tactic": "rw [\u2190 comap_top f] at h", "state_before": "case refine'_2\nG : Type u_2\ninst\u271d\u2074 : Group G\nA : Type ?u.1231\ninst\u271d\u00b3 : AddGroup A\nH\u271d : Type u_1\ninst\u271d\u00b2 : Group H\u271d\ninst\u271d\u00b9 : IsSimpleGroup G\ninst\u271d : Nontrivial H\u271d\nf : G \u2192* H\u271d\nhf : Function.Surjective \u2191f\nH : Subgroup H\u271d\niH : Normal H\nh : comap f H = \u22a4\n\u22a2 H = \u22a4", "state_after": "case refine'_2\nG : Type u_2\ninst\u271d\u2074 : Group G\nA : Type ?u.1231\ninst\u271d\u00b3 : AddGroup A\nH\u271d : Type u_1\ninst\u271d\u00b2 : Group H\u271d\ninst\u271d\u00b9 : IsSimpleGroup G\ninst\u271d : Nontrivial H\u271d\nf : G \u2192* H\u271d\nhf : Function.Surjective \u2191f\nH : Subgroup H\u271d\niH : Normal H\nh : comap f H = comap f \u22a4\n\u22a2 H = \u22a4"}, {"tactic": "exact comap_injective hf h", "state_before": "case refine'_2\nG : Type u_2\ninst\u271d\u2074 : Group G\nA : Type ?u.1231\ninst\u271d\u00b3 : AddGroup A\nH\u271d : Type u_1\ninst\u271d\u00b2 : Group H\u271d\ninst\u271d\u00b9 : IsSimpleGroup G\ninst\u271d : Nontrivial H\u271d\nf : G \u2192* H\u271d\nhf : Function.Surjective \u2191f\nH : Subgroup H\u271d\niH : Normal H\nh : comap f H = comap f \u22a4\n\u22a2 H = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Ring/Divisibility.lean", "full_name": "dvd_add", "start": [27, 1], "end": [28, 101], "traced_tactics": [{"tactic": "simp [left_distrib, hd, he]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.23\ninst\u271d\u00b2 : Add \u03b1\ninst\u271d\u00b9 : Semigroup \u03b1\ninst\u271d : LeftDistribClass \u03b1\na b c : \u03b1\nh\u2081 : a \u2223 b\nh\u2082 : a \u2223 c\nd : \u03b1\nhd : b = a * d\ne : \u03b1\nhe : c = a * e\n\u22a2 a * (d + e) = b + c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Prelude.lean", "full_name": "Nat.ble_self_eq_true", "start": [1680, 1], "end": [1682, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Monoid.lean", "full_name": "Set.Ioc_add_bij", "start": [61, 1], "end": [65, 34], "traced_tactics": [{"tactic": "rw [\u2190 Ioi_inter_Iic, \u2190 Ioi_inter_Iic]", "state_before": "M : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 BijOn (fun x => x + d) (Ioc a b) (Ioc (a + d) (b + d))", "state_after": "M : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 BijOn (fun x => x + d) (Ioi a \u2229 Iic b) (Ioi (a + d) \u2229 Iic (b + d))"}, {"tactic": "exact\n  (Ioi_add_bij a d).inter_mapsTo (fun x hx => add_le_add_right hx _) fun x hx =>\n    le_of_add_le_add_right hx.2", "state_before": "M : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 BijOn (fun x => x + d) (Ioi a \u2229 Iic b) (Ioi (a + d) \u2229 Iic (b + d))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/CompactOperator.lean", "full_name": "IsCompactOperator.isCompact_closure_image_of_isVonNBounded", "start": [115, 1], "end": [119, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Range.lean", "full_name": "List.enum_eq_zip_range", "start": [181, 1], "end": [182, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Rat/Lemmas.lean", "full_name": "Rat.add_den_dvd", "start": [77, 1], "end": [80, 26], "traced_tactics": [{"tactic": "rw [add_def, normalize_eq]", "state_before": "q\u2081 q\u2082 : \u211a\n\u22a2 (q\u2081 + q\u2082).den \u2223 q\u2081.den * q\u2082.den", "state_after": "q\u2081 q\u2082 : \u211a\n\u22a2 (mk'\n        ((q\u2081.num * \u2191q\u2082.den + q\u2082.num * \u2191q\u2081.den) /\n          \u2191(Nat.gcd (Int.natAbs (q\u2081.num * \u2191q\u2082.den + q\u2082.num * \u2191q\u2081.den)) (q\u2081.den * q\u2082.den)))\n        (q\u2081.den * q\u2082.den / Nat.gcd (Int.natAbs (q\u2081.num * \u2191q\u2082.den + q\u2082.num * \u2191q\u2081.den)) (q\u2081.den * q\u2082.den))).den \u2223\n    q\u2081.den * q\u2082.den"}, {"tactic": "apply Nat.div_dvd_of_dvd", "state_before": "q\u2081 q\u2082 : \u211a\n\u22a2 (mk'\n        ((q\u2081.num * \u2191q\u2082.den + q\u2082.num * \u2191q\u2081.den) /\n          \u2191(Nat.gcd (Int.natAbs (q\u2081.num * \u2191q\u2082.den + q\u2082.num * \u2191q\u2081.den)) (q\u2081.den * q\u2082.den)))\n        (q\u2081.den * q\u2082.den / Nat.gcd (Int.natAbs (q\u2081.num * \u2191q\u2082.den + q\u2082.num * \u2191q\u2081.den)) (q\u2081.den * q\u2082.den))).den \u2223\n    q\u2081.den * q\u2082.den", "state_after": "case h\nq\u2081 q\u2082 : \u211a\n\u22a2 Nat.gcd (Int.natAbs (q\u2081.num * \u2191q\u2082.den + q\u2082.num * \u2191q\u2081.den)) (q\u2081.den * q\u2082.den) \u2223 q\u2081.den * q\u2082.den"}, {"tactic": "apply Nat.gcd_dvd_right", "state_before": "case h\nq\u2081 q\u2082 : \u211a\n\u22a2 Nat.gcd (Int.natAbs (q\u2081.num * \u2191q\u2082.den + q\u2082.num * \u2191q\u2081.den)) (q\u2081.den * q\u2082.den) \u2223 q\u2081.den * q\u2082.den", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/Halting.lean", "full_name": "Nat.Partrec'.Vec.prim", "start": [347, 8], "end": [347, 96], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "full_name": "Set.preimage_const_sub_Ico", "start": [246, 1], "end": [247, 54], "traced_tactics": [{"tactic": "simp [\u2190 Ioi_inter_Iic, \u2190 Ici_inter_Iio, inter_comm]", "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedAddCommGroup \u03b1\na b c : \u03b1\n\u22a2 (fun x => a - x) \u207b\u00b9' Ico b c = Ioc (a - c) (a - b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Game/PGame.lean", "full_name": "PGame.one_rightMoves", "start": [370, 1], "end": [371, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "full_name": "mul_eq_mul_of_div_eq_div", "start": [183, 1], "end": [185, 94], "traced_tactics": [{"tactic": "rw [\u2190 mul_one a, \u2190 div_self hb, \u2190 mul_comm_div, h, div_mul_eq_mul_div, div_mul_cancel _ hd]", "state_before": "\u03b1 : Type ?u.12337\nM\u2080 : Type ?u.12340\nG\u2080 : Type u_1\nM\u2080' : Type ?u.12346\nG\u2080' : Type ?u.12349\nF : Type ?u.12352\nF' : Type ?u.12355\ninst\u271d\u00b9 : MonoidWithZero M\u2080\ninst\u271d : CommGroupWithZero G\u2080\na\u271d b\u271d c\u271d d\u271d a b c d : G\u2080\nhb : b \u2260 0\nhd : d \u2260 0\nh : a / b = c / d\n\u22a2 a * d = c * b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/Stieltjes.lean", "full_name": "iInf_Ioi_eq_iInf_rat_gt", "start": [38, 1], "end": [60, 16], "traced_tactics": [{"tactic": "refine' le_antisymm _ _", "state_before": "f : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\n\u22a2 (\u2a05 (r : \u2191(Ioi x)), f \u2191r) = \u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q", "state_after": "case refine'_1\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\n\u22a2 (\u2a05 (r : \u2191(Ioi x)), f \u2191r) \u2264 \u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q\n\ncase refine'_2\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\n\u22a2 (\u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q) \u2264 \u2a05 (r : \u2191(Ioi x)), f \u2191r"}, {"tactic": "have : Nonempty { r' : \u211a // x < \u2191r' } := by\n  obtain \u27e8r, hrx\u27e9 := exists_rat_gt x\n  exact \u27e8\u27e8r, hrx\u27e9\u27e9", "state_before": "case refine'_1\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\n\u22a2 (\u2a05 (r : \u2191(Ioi x)), f \u2191r) \u2264 \u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q", "state_after": "case refine'_1\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nthis : Nonempty { r' // x < \u2191r' }\n\u22a2 (\u2a05 (r : \u2191(Ioi x)), f \u2191r) \u2264 \u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q"}, {"tactic": "refine' le_ciInf fun r => _", "state_before": "case refine'_1\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nthis : Nonempty { r' // x < \u2191r' }\n\u22a2 (\u2a05 (r : \u2191(Ioi x)), f \u2191r) \u2264 \u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q", "state_after": "case refine'_1\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nthis : Nonempty { r' // x < \u2191r' }\nr : { q' // x < \u2191q' }\n\u22a2 (\u2a05 (r : \u2191(Ioi x)), f \u2191r) \u2264 f \u2191\u2191r"}, {"tactic": "obtain \u27e8y, hxy, hyr\u27e9 := exists_rat_btwn r.prop", "state_before": "case refine'_1\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nthis : Nonempty { r' // x < \u2191r' }\nr : { q' // x < \u2191q' }\n\u22a2 (\u2a05 (r : \u2191(Ioi x)), f \u2191r) \u2264 f \u2191\u2191r", "state_after": "case refine'_1.intro.intro\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nthis : Nonempty { r' // x < \u2191r' }\nr : { q' // x < \u2191q' }\ny : \u211a\nhxy : x < \u2191y\nhyr : \u2191y < \u2191\u2191r\n\u22a2 (\u2a05 (r : \u2191(Ioi x)), f \u2191r) \u2264 f \u2191\u2191r"}, {"tactic": "refine' ciInf_set_le hf (hxy.trans _)", "state_before": "case refine'_1.intro.intro\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nthis : Nonempty { r' // x < \u2191r' }\nr : { q' // x < \u2191q' }\ny : \u211a\nhxy : x < \u2191y\nhyr : \u2191y < \u2191\u2191r\n\u22a2 (\u2a05 (r : \u2191(Ioi x)), f \u2191r) \u2264 f \u2191\u2191r", "state_after": "case refine'_1.intro.intro\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nthis : Nonempty { r' // x < \u2191r' }\nr : { q' // x < \u2191q' }\ny : \u211a\nhxy : x < \u2191y\nhyr : \u2191y < \u2191\u2191r\n\u22a2 \u2191y < \u2191\u2191r"}, {"tactic": "exact_mod_cast hyr", "state_before": "case refine'_1.intro.intro\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nthis : Nonempty { r' // x < \u2191r' }\nr : { q' // x < \u2191q' }\ny : \u211a\nhxy : x < \u2191y\nhyr : \u2191y < \u2191\u2191r\n\u22a2 \u2191y < \u2191\u2191r", "state_after": "no goals"}, {"tactic": "obtain \u27e8r, hrx\u27e9 := exists_rat_gt x", "state_before": "f : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\n\u22a2 Nonempty { r' // x < \u2191r' }", "state_after": "case intro\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nr : \u211a\nhrx : x < \u2191r\n\u22a2 Nonempty { r' // x < \u2191r' }"}, {"tactic": "exact \u27e8\u27e8r, hrx\u27e9\u27e9", "state_before": "case intro\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nr : \u211a\nhrx : x < \u2191r\n\u22a2 Nonempty { r' // x < \u2191r' }", "state_after": "no goals"}, {"tactic": "refine' le_ciInf fun q => _", "state_before": "case refine'_2\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\n\u22a2 (\u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q) \u2264 \u2a05 (r : \u2191(Ioi x)), f \u2191r", "state_after": "case refine'_2\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\n\u22a2 (\u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q) \u2264 f \u2191q"}, {"tactic": "have hq := q.prop", "state_before": "case refine'_2\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\n\u22a2 (\u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q) \u2264 f \u2191q", "state_after": "case refine'_2\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : \u2191q \u2208 Ioi x\n\u22a2 (\u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q) \u2264 f \u2191q"}, {"tactic": "rw [mem_Ioi] at hq", "state_before": "case refine'_2\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : \u2191q \u2208 Ioi x\n\u22a2 (\u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q) \u2264 f \u2191q", "state_after": "case refine'_2\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\n\u22a2 (\u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q) \u2264 f \u2191q"}, {"tactic": "obtain \u27e8y, hxy, hyq\u27e9 := exists_rat_btwn hq", "state_before": "case refine'_2\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\n\u22a2 (\u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q) \u2264 f \u2191q", "state_after": "case refine'_2.intro.intro\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\n\u22a2 (\u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q) \u2264 f \u2191q"}, {"tactic": "refine' (ciInf_le _ _).trans _", "state_before": "case refine'_2.intro.intro\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\n\u22a2 (\u2a05 (q : { q' // x < \u2191q' }), f \u2191\u2191q) \u2264 f \u2191q", "state_after": "case refine'_2.intro.intro.refine'_1\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\n\u22a2 BddBelow (range fun q => f \u2191\u2191q)\n\ncase refine'_2.intro.intro.refine'_2\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\n\u22a2 { q' // x < \u2191q' }\n\ncase refine'_2.intro.intro.refine'_3\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\n\u22a2 f \u2191\u2191?refine'_2.intro.intro.refine'_2 \u2264 f \u2191q"}, {"tactic": "refine' \u27e8hf.some, fun z => _\u27e9", "state_before": "case refine'_2.intro.intro.refine'_1\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\n\u22a2 BddBelow (range fun q => f \u2191\u2191q)", "state_after": "case refine'_2.intro.intro.refine'_1\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\nz : \u211d\n\u22a2 (z \u2208 range fun q => f \u2191\u2191q) \u2192 Set.Nonempty.some hf \u2264 z"}, {"tactic": "rintro \u27e8u, rfl\u27e9", "state_before": "case refine'_2.intro.intro.refine'_1\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\nz : \u211d\n\u22a2 (z \u2208 range fun q => f \u2191\u2191q) \u2192 Set.Nonempty.some hf \u2264 z", "state_after": "case refine'_2.intro.intro.refine'_1.intro\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\nu : { q' // x < \u2191q' }\n\u22a2 Set.Nonempty.some hf \u2264 (fun q => f \u2191\u2191q) u"}, {"tactic": "suffices hfu : f u \u2208 f '' Ioi x", "state_before": "case refine'_2.intro.intro.refine'_1.intro\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\nu : { q' // x < \u2191q' }\n\u22a2 Set.Nonempty.some hf \u2264 (fun q => f \u2191\u2191q) u", "state_after": "case refine'_2.intro.intro.refine'_1.intro\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\nu : { q' // x < \u2191q' }\nhfu : f \u2191\u2191u \u2208 f '' Ioi x\n\u22a2 Set.Nonempty.some hf \u2264 (fun q => f \u2191\u2191q) u\n\ncase hfu\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\nu : { q' // x < \u2191q' }\n\u22a2 f \u2191\u2191u \u2208 f '' Ioi x"}, {"tactic": "exact hf.choose_spec hfu", "state_before": "case refine'_2.intro.intro.refine'_1.intro\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\nu : { q' // x < \u2191q' }\nhfu : f \u2191\u2191u \u2208 f '' Ioi x\n\u22a2 Set.Nonempty.some hf \u2264 (fun q => f \u2191\u2191q) u\n\ncase hfu\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\nu : { q' // x < \u2191q' }\n\u22a2 f \u2191\u2191u \u2208 f '' Ioi x", "state_after": "case hfu\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\nu : { q' // x < \u2191q' }\n\u22a2 f \u2191\u2191u \u2208 f '' Ioi x"}, {"tactic": "exact \u27e8u, u.prop, rfl\u27e9", "state_before": "case hfu\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\nu : { q' // x < \u2191q' }\n\u22a2 f \u2191\u2191u \u2208 f '' Ioi x", "state_after": "no goals"}, {"tactic": "exact \u27e8y, hxy\u27e9", "state_before": "case refine'_2.intro.intro.refine'_2\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\n\u22a2 { q' // x < \u2191q' }", "state_after": "no goals"}, {"tactic": "refine' hf_mono (le_trans _ hyq.le)", "state_before": "case refine'_2.intro.intro.refine'_3\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\n\u22a2 f \u2191\u2191{ val := y, property := hxy } \u2264 f \u2191q", "state_after": "case refine'_2.intro.intro.refine'_3\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\n\u22a2 \u2191\u2191{ val := y, property := hxy } \u2264 \u2191y"}, {"tactic": "norm_cast", "state_before": "case refine'_2.intro.intro.refine'_3\nf : \u211d \u2192 \u211d\nx : \u211d\nhf : BddBelow (f '' Ioi x)\nhf_mono : Monotone f\nq : \u2191(Ioi x)\nhq : x < \u2191q\ny : \u211a\nhxy : x < \u2191y\nhyq : \u2191y < \u2191q\n\u22a2 \u2191\u2191{ val := y, property := hxy } \u2264 \u2191y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Hom/GroupAction.lean", "full_name": "DistribMulActionHom.id_comp", "start": [377, 1], "end": [378, 44], "traced_tactics": [{"tactic": "rw [comp_apply, id_apply]", "state_before": "M' : Type ?u.176380\nX : Type ?u.176383\ninst\u271d\u00b2\u00b3 : SMul M' X\nY : Type ?u.176390\ninst\u271d\u00b2\u00b2 : SMul M' Y\nZ : Type ?u.176397\ninst\u271d\u00b2\u00b9 : SMul M' Z\nM : Type u_1\ninst\u271d\u00b2\u2070 : Monoid M\nA : Type u_2\ninst\u271d\u00b9\u2079 : AddMonoid A\ninst\u271d\u00b9\u2078 : DistribMulAction M A\nA' : Type ?u.176438\ninst\u271d\u00b9\u2077 : AddGroup A'\ninst\u271d\u00b9\u2076 : DistribMulAction M A'\nB : Type u_3\ninst\u271d\u00b9\u2075 : AddMonoid B\ninst\u271d\u00b9\u2074 : DistribMulAction M B\nB' : Type ?u.176740\ninst\u271d\u00b9\u00b3 : AddGroup B'\ninst\u271d\u00b9\u00b2 : DistribMulAction M B'\nC : Type ?u.177016\ninst\u271d\u00b9\u00b9 : AddMonoid C\ninst\u271d\u00b9\u2070 : DistribMulAction M C\nR : Type ?u.177042\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : MulSemiringAction M R\nR' : Type ?u.177069\ninst\u271d\u2077 : Ring R'\ninst\u271d\u2076 : MulSemiringAction M R'\nS : Type ?u.177265\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : MulSemiringAction M S\nS' : Type ?u.177292\ninst\u271d\u00b3 : Ring S'\ninst\u271d\u00b2 : MulSemiringAction M S'\nT : Type ?u.177488\ninst\u271d\u00b9 : Semiring T\ninst\u271d : MulSemiringAction M T\nf : A \u2192+[M] B\nx : A\n\u22a2 \u2191(comp (DistribMulActionHom.id M) f) x = \u2191f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "full_name": "MvQPF.supp_map", "start": [254, 1], "end": [258, 6], "traced_tactics": [{"tactic": "rw [\u2190 abs_repr x]", "state_before": "n : \u2115\nF : TypeVec n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : IsUniform\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\nx : F \u03b1\ni : Fin2 n\n\u22a2 supp (g <$$> x) i = g i '' supp x i", "state_after": "n : \u2115\nF : TypeVec n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : IsUniform\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\nx : F \u03b1\ni : Fin2 n\n\u22a2 supp (g <$$> abs (repr x)) i = g i '' supp (abs (repr x)) i"}, {"tactic": "cases' repr x with a f", "state_before": "n : \u2115\nF : TypeVec n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : IsUniform\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\nx : F \u03b1\ni : Fin2 n\n\u22a2 supp (g <$$> abs (repr x)) i = g i '' supp (abs (repr x)) i", "state_after": "case mk\nn : \u2115\nF : TypeVec n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : IsUniform\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\nx : F \u03b1\ni : Fin2 n\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\n\u22a2 supp (g <$$> abs { fst := a, snd := f }) i = g i '' supp (abs { fst := a, snd := f }) i"}, {"tactic": "rw [\u2190 abs_map, MvPFunctor.map_eq]", "state_before": "case mk\nn : \u2115\nF : TypeVec n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : IsUniform\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\nx : F \u03b1\ni : Fin2 n\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\n\u22a2 supp (g <$$> abs { fst := a, snd := f }) i = g i '' supp (abs { fst := a, snd := f }) i", "state_after": "case mk\nn : \u2115\nF : TypeVec n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : IsUniform\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\nx : F \u03b1\ni : Fin2 n\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\n\u22a2 supp (abs { fst := a, snd := g \u229a f }) i = g i '' supp (abs { fst := a, snd := f }) i"}, {"tactic": "rw [supp_eq_of_isUniform h, supp_eq_of_isUniform h, \u2190 image_comp]", "state_before": "case mk\nn : \u2115\nF : TypeVec n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : IsUniform\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\nx : F \u03b1\ni : Fin2 n\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\n\u22a2 supp (abs { fst := a, snd := g \u229a f }) i = g i '' supp (abs { fst := a, snd := f }) i", "state_after": "case mk\nn : \u2115\nF : TypeVec n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : IsUniform\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\nx : F \u03b1\ni : Fin2 n\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\n\u22a2 (g \u229a f) i '' univ = g i \u2218 f i '' univ"}, {"tactic": "rfl", "state_before": "case mk\nn : \u2115\nF : TypeVec n \u2192 Type u_1\ninst\u271d : MvFunctor F\nq : MvQPF F\nh : IsUniform\n\u03b1 \u03b2 : TypeVec n\ng : \u03b1 \u27f9 \u03b2\nx : F \u03b1\ni : Fin2 n\na : (P F).A\nf : MvPFunctor.B (P F) a \u27f9 \u03b1\n\u22a2 (g \u229a f) i '' univ = g i \u2218 f i '' univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Associated.lean", "full_name": "IsSquare.not_prime", "start": [364, 1], "end": [364, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Separation.lean", "full_name": "nhdsWithin_insert_of_ne", "start": [646, 1], "end": [652, 69], "traced_tactics": [{"tactic": "refine' le_antisymm (Filter.le_def.2 fun t ht => _) (nhdsWithin_mono x <| subset_insert y s)", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : T1Space \u03b1\nx y : \u03b1\ns : Set \u03b1\nhxy : x \u2260 y\n\u22a2 \ud835\udcdd[insert y s] x = \ud835\udcdd[s] x", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : T1Space \u03b1\nx y : \u03b1\ns : Set \u03b1\nhxy : x \u2260 y\nt : Set \u03b1\nht : t \u2208 \ud835\udcdd[s] x\n\u22a2 t \u2208 \ud835\udcdd[insert y s] x"}, {"tactic": "obtain \u27e8o, ho, hxo, host\u27e9 := mem_nhdsWithin.mp ht", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : T1Space \u03b1\nx y : \u03b1\ns : Set \u03b1\nhxy : x \u2260 y\nt : Set \u03b1\nht : t \u2208 \ud835\udcdd[s] x\n\u22a2 t \u2208 \ud835\udcdd[insert y s] x", "state_after": "case intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : T1Space \u03b1\nx y : \u03b1\ns : Set \u03b1\nhxy : x \u2260 y\nt : Set \u03b1\nht : t \u2208 \ud835\udcdd[s] x\no : Set \u03b1\nho : IsOpen o\nhxo : x \u2208 o\nhost : o \u2229 s \u2286 t\n\u22a2 t \u2208 \ud835\udcdd[insert y s] x"}, {"tactic": "refine' mem_nhdsWithin.mpr \u27e8o \\ {y}, ho.sdiff isClosed_singleton, \u27e8hxo, hxy\u27e9, _\u27e9", "state_before": "case intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : T1Space \u03b1\nx y : \u03b1\ns : Set \u03b1\nhxy : x \u2260 y\nt : Set \u03b1\nht : t \u2208 \ud835\udcdd[s] x\no : Set \u03b1\nho : IsOpen o\nhxo : x \u2208 o\nhost : o \u2229 s \u2286 t\n\u22a2 t \u2208 \ud835\udcdd[insert y s] x", "state_after": "case intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : T1Space \u03b1\nx y : \u03b1\ns : Set \u03b1\nhxy : x \u2260 y\nt : Set \u03b1\nht : t \u2208 \ud835\udcdd[s] x\no : Set \u03b1\nho : IsOpen o\nhxo : x \u2208 o\nhost : o \u2229 s \u2286 t\n\u22a2 o \\ {y} \u2229 insert y s \u2286 t"}, {"tactic": "rw [inter_insert_of_not_mem <| not_mem_diff_of_mem (mem_singleton y)]", "state_before": "case intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : T1Space \u03b1\nx y : \u03b1\ns : Set \u03b1\nhxy : x \u2260 y\nt : Set \u03b1\nht : t \u2208 \ud835\udcdd[s] x\no : Set \u03b1\nho : IsOpen o\nhxo : x \u2208 o\nhost : o \u2229 s \u2286 t\n\u22a2 o \\ {y} \u2229 insert y s \u2286 t", "state_after": "case intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : T1Space \u03b1\nx y : \u03b1\ns : Set \u03b1\nhxy : x \u2260 y\nt : Set \u03b1\nht : t \u2208 \ud835\udcdd[s] x\no : Set \u03b1\nho : IsOpen o\nhxo : x \u2208 o\nhost : o \u2229 s \u2286 t\n\u22a2 o \\ {y} \u2229 s \u2286 t"}, {"tactic": "exact (inter_subset_inter (diff_subset _ _) Subset.rfl).trans host", "state_before": "case intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : T1Space \u03b1\nx y : \u03b1\ns : Set \u03b1\nhxy : x \u2260 y\nt : Set \u03b1\nht : t \u2208 \ud835\udcdd[s] x\no : Set \u03b1\nho : IsOpen o\nhxo : x \u2208 o\nhost : o \u2229 s \u2286 t\n\u22a2 o \\ {y} \u2229 s \u2286 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.ssubset_iff_of_subset", "start": [427, 1], "end": [428, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.exists_set_mem_of_union_eq_top", "start": [263, 1], "end": [267, 16], "traced_tactics": [{"tactic": "have p : x \u2208 \u22a4 := Set.mem_univ x", "state_before": "\u03b1 : Type ?u.55712\n\u03b2 : Type u_2\n\u03b3 : Type ?u.55718\n\u03b9\u271d : Sort ?u.55721\n\u03b9' : Sort ?u.55724\n\u03b9\u2082 : Sort ?u.55727\n\u03ba : \u03b9\u271d \u2192 Sort ?u.55732\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort ?u.55737\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort ?u.55742\n\u03ba' : \u03b9' \u2192 Sort ?u.55747\n\u03b9 : Type u_1\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b2\nw : (\u22c3 (i : \u03b9) (_ : i \u2208 t), s i) = \u22a4\nx : \u03b2\n\u22a2 \u2203 i, i \u2208 t \u2227 x \u2208 s i", "state_after": "\u03b1 : Type ?u.55712\n\u03b2 : Type u_2\n\u03b3 : Type ?u.55718\n\u03b9\u271d : Sort ?u.55721\n\u03b9' : Sort ?u.55724\n\u03b9\u2082 : Sort ?u.55727\n\u03ba : \u03b9\u271d \u2192 Sort ?u.55732\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort ?u.55737\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort ?u.55742\n\u03ba' : \u03b9' \u2192 Sort ?u.55747\n\u03b9 : Type u_1\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b2\nw : (\u22c3 (i : \u03b9) (_ : i \u2208 t), s i) = \u22a4\nx : \u03b2\np : x \u2208 \u22a4\n\u22a2 \u2203 i, i \u2208 t \u2227 x \u2208 s i"}, {"tactic": "rw [\u2190 w, Set.mem_iUnion] at p", "state_before": "\u03b1 : Type ?u.55712\n\u03b2 : Type u_2\n\u03b3 : Type ?u.55718\n\u03b9\u271d : Sort ?u.55721\n\u03b9' : Sort ?u.55724\n\u03b9\u2082 : Sort ?u.55727\n\u03ba : \u03b9\u271d \u2192 Sort ?u.55732\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort ?u.55737\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort ?u.55742\n\u03ba' : \u03b9' \u2192 Sort ?u.55747\n\u03b9 : Type u_1\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b2\nw : (\u22c3 (i : \u03b9) (_ : i \u2208 t), s i) = \u22a4\nx : \u03b2\np : x \u2208 \u22a4\n\u22a2 \u2203 i, i \u2208 t \u2227 x \u2208 s i", "state_after": "\u03b1 : Type ?u.55712\n\u03b2 : Type u_2\n\u03b3 : Type ?u.55718\n\u03b9\u271d : Sort ?u.55721\n\u03b9' : Sort ?u.55724\n\u03b9\u2082 : Sort ?u.55727\n\u03ba : \u03b9\u271d \u2192 Sort ?u.55732\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort ?u.55737\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort ?u.55742\n\u03ba' : \u03b9' \u2192 Sort ?u.55747\n\u03b9 : Type u_1\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b2\nw : (\u22c3 (i : \u03b9) (_ : i \u2208 t), s i) = \u22a4\nx : \u03b2\np : \u2203 i, x \u2208 \u22c3 (_ : i \u2208 t), s i\n\u22a2 \u2203 i, i \u2208 t \u2227 x \u2208 s i"}, {"tactic": "simpa using p", "state_before": "\u03b1 : Type ?u.55712\n\u03b2 : Type u_2\n\u03b3 : Type ?u.55718\n\u03b9\u271d : Sort ?u.55721\n\u03b9' : Sort ?u.55724\n\u03b9\u2082 : Sort ?u.55727\n\u03ba : \u03b9\u271d \u2192 Sort ?u.55732\n\u03ba\u2081 : \u03b9\u271d \u2192 Sort ?u.55737\n\u03ba\u2082 : \u03b9\u271d \u2192 Sort ?u.55742\n\u03ba' : \u03b9' \u2192 Sort ?u.55747\n\u03b9 : Type u_1\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b2\nw : (\u22c3 (i : \u03b9) (_ : i \u2208 t), s i) = \u22a4\nx : \u03b2\np : \u2203 i, x \u2208 \u22c3 (_ : i \u2208 t), s i\n\u22a2 \u2203 i, i \u2208 t \u2227 x \u2208 s i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/CauSeq.lean", "full_name": "CauSeq.coe_sup", "start": [839, 1], "end": [840, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/FunLike/Basic.lean", "full_name": "FunLike.exists_ne", "start": [195, 1], "end": [196, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Complex/Basic.lean", "full_name": "Complex.abs_I", "start": [987, 1], "end": [987, 59], "traced_tactics": [{"tactic": "simp [Complex.abs]", "state_before": "\u22a2 \u2191abs I = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/MinMax.lean", "full_name": "List.minimum_eq_none", "start": [319, 1], "end": [320, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Abelian/Exact.lean", "full_name": "CategoryTheory.Abelian.tfae_mono", "start": [257, 1], "end": [265, 14], "traced_tactics": [{"tactic": "tfae_have 3 \u2192 2", "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 TFAE [Mono f, kernel.\u03b9 f = 0, Exact 0 f]", "state_after": "case tfae_3_to_2\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 Exact 0 f \u2192 kernel.\u03b9 f = 0\n\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact 0 f \u2192 kernel.\u03b9 f = 0\n\u22a2 TFAE [Mono f, kernel.\u03b9 f = 0, Exact 0 f]"}, {"tactic": "tfae_have 1 \u2192 3", "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact 0 f \u2192 kernel.\u03b9 f = 0\n\u22a2 TFAE [Mono f, kernel.\u03b9 f = 0, Exact 0 f]", "state_after": "case tfae_1_to_3\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact 0 f \u2192 kernel.\u03b9 f = 0\n\u22a2 Mono f \u2192 Exact 0 f\n\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact 0 f \u2192 kernel.\u03b9 f = 0\ntfae_1_to_3 : Mono f \u2192 Exact 0 f\n\u22a2 TFAE [Mono f, kernel.\u03b9 f = 0, Exact 0 f]"}, {"tactic": "tfae_have 2 \u2192 1", "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact 0 f \u2192 kernel.\u03b9 f = 0\ntfae_1_to_3 : Mono f \u2192 Exact 0 f\n\u22a2 TFAE [Mono f, kernel.\u03b9 f = 0, Exact 0 f]", "state_after": "case tfae_2_to_1\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact 0 f \u2192 kernel.\u03b9 f = 0\ntfae_1_to_3 : Mono f \u2192 Exact 0 f\n\u22a2 kernel.\u03b9 f = 0 \u2192 Mono f\n\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact 0 f \u2192 kernel.\u03b9 f = 0\ntfae_1_to_3 : Mono f \u2192 Exact 0 f\ntfae_2_to_1 : kernel.\u03b9 f = 0 \u2192 Mono f\n\u22a2 TFAE [Mono f, kernel.\u03b9 f = 0, Exact 0 f]"}, {"tactic": "tfae_finish", "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact 0 f \u2192 kernel.\u03b9 f = 0\ntfae_1_to_3 : Mono f \u2192 Exact 0 f\ntfae_2_to_1 : kernel.\u03b9 f = 0 \u2192 Mono f\n\u22a2 TFAE [Mono f, kernel.\u03b9 f = 0, Exact 0 f]", "state_after": "no goals"}, {"tactic": "exact kernel_\u03b9_eq_zero_of_exact_zero_left Z", "state_before": "case tfae_3_to_2\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 Exact 0 f \u2192 kernel.\u03b9 f = 0", "state_after": "no goals"}, {"tactic": "intros", "state_before": "case tfae_1_to_3\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact 0 f \u2192 kernel.\u03b9 f = 0\n\u22a2 Mono f \u2192 Exact 0 f", "state_after": "case tfae_1_to_3\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact 0 f \u2192 kernel.\u03b9 f = 0\n\u271d : Mono f\n\u22a2 Exact 0 f"}, {"tactic": "exact exact_zero_left_of_mono Z", "state_before": "case tfae_1_to_3\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact 0 f \u2192 kernel.\u03b9 f = 0\n\u271d : Mono f\n\u22a2 Exact 0 f", "state_after": "no goals"}, {"tactic": "exact mono_of_kernel_\u03b9_eq_zero _", "state_before": "case tfae_2_to_1\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact 0 f \u2192 kernel.\u03b9 f = 0\ntfae_1_to_3 : Mono f \u2192 Exact 0 f\n\u22a2 kernel.\u03b9 f = 0 \u2192 Mono f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Basis.lean", "full_name": "Basis.mem_submodule_iff'", 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(toFinset s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Basic.lean", "full_name": "Set.Countable.exists_cycleOn", "start": [1888, 1], "end": [1906, 9], "traced_tactics": [{"tactic": "obtain hs' | hs' := s.finite_or_infinite", "state_before": "\u03b9 : Type ?u.3075067\n\u03b1 : Type u_1\n\u03b2 : Type ?u.3075073\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\ns : Set \u03b1\nhs : Set.Countable s\n\u22a2 \u2203 f, IsCycleOn f s \u2227 {x | \u2191f x \u2260 x} \u2286 s", "state_after": "case inl\n\u03b9 : Type ?u.3075067\n\u03b1 : Type u_1\n\u03b2 : Type ?u.3075073\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\ns : Set \u03b1\nhs : Set.Countable s\nhs' : Set.Finite s\n\u22a2 \u2203 f, IsCycleOn f s \u2227 {x | \u2191f x \u2260 x} \u2286 s\n\ncase inr\n\u03b9 : Type ?u.3075067\n\u03b1 : Type u_1\n\u03b2 : Type ?u.3075073\ninst\u271d : 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Type ?u.3075073\ninst\u271d : DecidableEq \u03b1\nf\u271d : Perm \u03b1\ns : Set \u03b1\nhs : Set.Countable s\nhs' : Set.Infinite s\nthis\u271d : Countable \u2191s\nthis : Infinite \u2191s\nf : \u2124 \u2243 \u2191s\n\u22a2 \u2203 f, IsCycleOn f s \u2227 {x | \u2191f x \u2260 x} \u2286 s"}, {"tactic": "refine'\n  \u27e8(Equiv.addRight 1).extendDomain f, _, fun x hx =>\n    of_not_not fun h => hx <| Perm.extendDomain_apply_not_subtype _ _ h\u27e9", "state_before": "case inr.intro\n\u03b9 : Type ?u.3075067\n\u03b1 : Type u_1\n\u03b2 : Type ?u.3075073\ninst\u271d : DecidableEq \u03b1\nf\u271d : Perm \u03b1\ns : Set \u03b1\nhs : Set.Countable s\nhs' : Set.Infinite s\nthis\u271d : Countable \u2191s\nthis : Infinite \u2191s\nf : \u2124 \u2243 \u2191s\n\u22a2 \u2203 f, IsCycleOn f s \u2227 {x | \u2191f x \u2260 x} \u2286 s", "state_after": "case inr.intro\n\u03b9 : Type ?u.3075067\n\u03b1 : Type u_1\n\u03b2 : Type ?u.3075073\ninst\u271d : DecidableEq \u03b1\nf\u271d : Perm \u03b1\ns : Set 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"https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Coprime/Basic.lean", "full_name": "IsCoprime.of_mul_add_left_right", "start": [214, 1], "end": [216, 32], "traced_tactics": [{"tactic": "rw [add_comm] at h", "state_before": "R : Type u\ninst\u271d : CommSemiring R\nx y z : R\nh : IsCoprime x (x * z + y)\n\u22a2 IsCoprime x y", "state_after": "R : Type u\ninst\u271d : CommSemiring R\nx y z : R\nh : IsCoprime x (y + x * z)\n\u22a2 IsCoprime x y"}, {"tactic": "exact h.of_add_mul_left_right", "state_before": "R : Type u\ninst\u271d : CommSemiring R\nx y z : R\nh : IsCoprime x (y + x * z)\n\u22a2 IsCoprime x y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Constructions/Pullbacks.lean", "full_name": 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Quotient.mk' (s \u00d7\u02e2 s) = Finset.sym2 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/NoncommProd.lean", "full_name": "Multiset.noncommProd_eq_prod", "start": [208, 1], "end": [211, 7], "traced_tactics": [{"tactic": "induction s using Quotient.inductionOn", "state_before": "F : Type ?u.135328\n\u03b9 : Type ?u.135331\n\u03b1\u271d : Type ?u.135334\n\u03b2 : Type ?u.135337\n\u03b3 : Type ?u.135340\nf : \u03b1\u271d \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1\u271d \u2192 \u03b1\u271d \u2192 \u03b1\u271d\ninst\u271d\u00b2 : Monoid \u03b1\u271d\ninst\u271d\u00b9 : Monoid \u03b2\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\ns : Multiset \u03b1\n\u22a2 noncommProd s (_ : \u2200 (x : \u03b1), x \u2208 {x | x \u2208 s} \u2192 \u2200 (x_2 : \u03b1), x_2 \u2208 {x | x \u2208 s} \u2192 x \u2260 x_2 \u2192 Commute x x_2) = prod s", "state_after": "case h\nF : Type ?u.135328\n\u03b9 : Type ?u.135331\n\u03b1\u271d : Type ?u.135334\n\u03b2 : Type ?u.135337\n\u03b3 : Type ?u.135340\nf : \u03b1\u271d \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1\u271d \u2192 \u03b1\u271d \u2192 \u03b1\u271d\ninst\u271d\u00b2 : Monoid \u03b1\u271d\ninst\u271d\u00b9 : Monoid \u03b2\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\na\u271d : List \u03b1\n\u22a2 noncommProd (Quotient.mk (List.isSetoid \u03b1) a\u271d)\n      (_ :\n        \u2200 (x : \u03b1),\n          x \u2208 {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} \u2192\n            \u2200 (x_2 : \u03b1), x_2 \u2208 {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} \u2192 x \u2260 x_2 \u2192 Commute x x_2) =\n    prod (Quotient.mk (List.isSetoid \u03b1) a\u271d)"}, {"tactic": "simp", "state_before": "case h\nF : Type ?u.135328\n\u03b9 : Type ?u.135331\n\u03b1\u271d : Type ?u.135334\n\u03b2 : Type ?u.135337\n\u03b3 : Type ?u.135340\nf : \u03b1\u271d \u2192 \u03b2 \u2192 \u03b2\nop : 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"traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Heyting/Basic.lean", "full_name": "sup_himp_self_left", "start": [427, 1], "end": [428, 47], "traced_tactics": [{"tactic": "rw [sup_himp_distrib, himp_self, top_inf_eq]", "state_before": "\u03b9 : Type ?u.72617\n\u03b1 : Type u_1\n\u03b2 : Type ?u.72623\ninst\u271d : GeneralizedHeytingAlgebra \u03b1\na\u271d b\u271d c d a b : \u03b1\n\u22a2 a \u2294 b \u21e8 a = b \u21e8 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "ZNum.of_nat_toZNumNeg", "start": [1560, 1], "end": [1560, 100], "traced_tactics": [{"tactic": "rw [\u2190 of_nat_toZNum, Num.zneg_toZNum]", "state_before": "\u03b1 : Type ?u.1066795\nn : \u2115\n\u22a2 Num.toZNumNeg \u2191n = -\u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/ContDiffDef.lean", "full_name": "Filter.EventuallyEq.iterated_fderiv_within'", "start": [879, 1], "end": [886, 68], "traced_tactics": [{"tactic": "induction' n with n ihn", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nht : t \u2286 s\nn : \u2115\n\u22a2 iteratedFDerivWithin \ud835\udd5c n f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c n f t", "state_after": "case zero\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nht : t \u2286 s\n\u22a2 iteratedFDerivWithin \ud835\udd5c Nat.zero f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c Nat.zero f t\n\ncase succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nht : t \u2286 s\nn : \u2115\nihn : iteratedFDerivWithin \ud835\udd5c n f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c n f t\n\u22a2 iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f t"}, {"tactic": "exact h.mono fun y hy => FunLike.ext _ _ fun _ => hy", "state_before": "case zero\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nht : t \u2286 s\n\u22a2 iteratedFDerivWithin \ud835\udd5c Nat.zero f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c Nat.zero f t", "state_after": "no goals"}, {"tactic": "have : fderivWithin \ud835\udd5c _ t =\u1da0[\ud835\udcdd[s] x] fderivWithin \ud835\udd5c _ t := ihn.fderiv_within' ht", "state_before": "case succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nht : t \u2286 s\nn : \u2115\nihn : iteratedFDerivWithin \ud835\udd5c n f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c n f t\n\u22a2 iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f t", "state_after": "case succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nht : t \u2286 s\nn : \u2115\nihn : iteratedFDerivWithin \ud835\udd5c n f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c n f t\nthis : fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f\u2081 t) t =\u1da0[\ud835\udcdd[s] x] fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f t) t\n\u22a2 iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f t"}, {"tactic": "apply this.mono", "state_before": "case succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nht : t \u2286 s\nn : \u2115\nihn : iteratedFDerivWithin \ud835\udd5c n f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c n f t\nthis : fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f\u2081 t) t =\u1da0[\ud835\udcdd[s] x] fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f t) t\n\u22a2 iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f t", "state_after": "case succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nht : t \u2286 s\nn : \u2115\nihn : iteratedFDerivWithin \ud835\udd5c n f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c n f t\nthis : fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f\u2081 t) t =\u1da0[\ud835\udcdd[s] x] fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f t) t\n\u22a2 \u2200 (x : E),\n    fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f\u2081 t) t x = fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f t) t x \u2192\n      iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f\u2081 t x = iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f t x"}, {"tactic": "intro y hy", "state_before": "case succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nht : t \u2286 s\nn : \u2115\nihn : iteratedFDerivWithin \ud835\udd5c n f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c n f t\nthis : fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f\u2081 t) t =\u1da0[\ud835\udcdd[s] x] fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f t) t\n\u22a2 \u2200 (x : E),\n    fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f\u2081 t) t x = fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f t) t x \u2192\n      iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f\u2081 t x = iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f t x", "state_after": "case succ\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nh : f\u2081 =\u1da0[\ud835\udcdd[s] x] f\nht : t \u2286 s\nn : \u2115\nihn : iteratedFDerivWithin \ud835\udd5c n f\u2081 t =\u1da0[\ud835\udcdd[s] x] iteratedFDerivWithin \ud835\udd5c n f t\nthis : fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f\u2081 t) t =\u1da0[\ud835\udcdd[s] x] fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f t) t\ny : E\nhy : fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f\u2081 t) t y = fderivWithin \ud835\udd5c (iteratedFDerivWithin \ud835\udd5c n f t) t y\n\u22a2 iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f\u2081 t y = iteratedFDerivWithin \ud835\udd5c (Nat.succ n) f t y"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Compactification/OnePoint.lean", "full_name": "OnePoint.denseRange_coe", "start": [391, 1], "end": [393, 32], "traced_tactics": [{"tactic": "rw [DenseRange, \u2190 compl_infty]", "state_before": "X : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d : NoncompactSpace X\n\u22a2 DenseRange some", "state_after": "X : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d : NoncompactSpace X\n\u22a2 Dense ({\u221e}\u1d9c)"}, {"tactic": "exact dense_compl_singleton _", "state_before": "X : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\ninst\u271d : NoncompactSpace X\n\u22a2 Dense ({\u221e}\u1d9c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/DirectLimit.lean", "full_name": "Module.DirectedSystem.map_self", "start": [75, 8], "end": [77, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Nat/Lemmas.lean", "full_name": "Nat.sum_cons", "start": [832, 9], "end": [832, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/MonoidLocalization.lean", "full_name": "AddSubmonoid.LocalizationMap.AwayMap.lift_comp", "start": [1333, 1], "end": [1334, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Count.lean", "full_name": "List.Sublist.countp_le", "start": [122, 1], "end": [123, 79], "traced_tactics": [{"tactic": "simpa only [countp_eq_length_filter] using length_le_of_sublist (s.filter p)", "state_before": "\u03b1 : Type u_1\nl : List \u03b1\np q : \u03b1 \u2192 Bool\nl\u2081 l\u2082 : List \u03b1\ns : l\u2081 <+ l\u2082\n\u22a2 countp p l\u2081 \u2264 countp p l\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/List/Lemmas.lean", "full_name": "List.join_nil", "start": [178, 1], "end": [178, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/ArithmeticFunction.lean", "full_name": "Nat.ArithmeticFunction.sum_eq_iff_sum_smul_moebius_eq", "start": [1086, 1], "end": [1121, 97], "traced_tactics": [{"tactic": "let f' : ArithmeticFunction R := \u27e8fun x => if x = 0 then 0 else f x, if_pos rfl\u27e9", "state_before": "R : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\n\u22a2 (\u2200 (n : \u2115), 0 < n \u2192 \u2211 i in divisors n, f i = g n) \u2194\n    \u2200 (n : \u2115), 0 < n \u2192 \u2211 x in divisorsAntidiagonal n, \u2191\u03bc x.fst \u2022 g x.snd = f n", "state_after": "R : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\n\u22a2 (\u2200 (n : \u2115), 0 < n \u2192 \u2211 i in divisors n, f i = g n) \u2194\n    \u2200 (n : \u2115), 0 < n \u2192 \u2211 x in divisorsAntidiagonal n, \u2191\u03bc x.fst \u2022 g x.snd = f n"}, {"tactic": "let g' : ArithmeticFunction R := \u27e8fun x => if x = 0 then 0 else g x, if_pos rfl\u27e9", "state_before": "R : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\n\u22a2 (\u2200 (n : \u2115), 0 < n \u2192 \u2211 i in divisors n, f i = g n) \u2194\n    \u2200 (n : \u2115), 0 < n \u2192 \u2211 x in divisorsAntidiagonal n, \u2191\u03bc x.fst \u2022 g x.snd = f n", "state_after": "R : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\n\u22a2 (\u2200 (n : \u2115), 0 < n \u2192 \u2211 i in divisors n, f i = g n) \u2194\n    \u2200 (n : \u2115), 0 < n \u2192 \u2211 x in divisorsAntidiagonal n, \u2191\u03bc x.fst \u2022 g x.snd = f n"}, {"tactic": "trans (\u03b6 : ArithmeticFunction \u2124) \u2022 f' = g'", "state_before": "R : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\n\u22a2 (\u2200 (n : \u2115), 0 < n \u2192 \u2211 i in divisors n, f i = g n) \u2194\n    \u2200 (n : \u2115), 0 < n \u2192 \u2211 x in divisorsAntidiagonal n, \u2191\u03bc x.fst \u2022 g x.snd = f n", "state_after": "R : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\n\u22a2 (\u2200 (n : \u2115), 0 < n \u2192 \u2211 i in divisors n, f i = g n) \u2194 \u2191\u03b6 \u2022 f' = g'\n\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\n\u22a2 \u2191\u03b6 \u2022 f' = g' \u2194 \u2200 (n : \u2115), 0 < n \u2192 \u2211 x in divisorsAntidiagonal n, \u2191\u03bc x.fst \u2022 g x.snd = f n"}, {"tactic": "trans \u03bc \u2022 g' = f'", "state_before": "R : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\n\u22a2 \u2191\u03b6 \u2022 f' = g' \u2194 \u2200 (n : \u2115), 0 < n \u2192 \u2211 x in divisorsAntidiagonal n, \u2191\u03bc x.fst \u2022 g x.snd = f n", "state_after": "R : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\n\u22a2 \u2191\u03b6 \u2022 f' = g' \u2194 \u03bc \u2022 g' = f'\n\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\n\u22a2 \u03bc \u2022 g' = f' \u2194 \u2200 (n : \u2115), 0 < n \u2192 \u2211 x in divisorsAntidiagonal n, \u2191\u03bc x.fst \u2022 g x.snd = f n"}, {"tactic": "rw [ext_iff]", "state_before": "R : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\n\u22a2 (\u2200 (n : \u2115), 0 < n \u2192 \u2211 i in divisors n, f i = g n) \u2194 \u2191\u03b6 \u2022 f' = g'", "state_after": "R : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\n\u22a2 (\u2200 (n : \u2115), 0 < n \u2192 \u2211 i in divisors n, f i = g n) \u2194 \u2200 (x : \u2115), \u2191(\u2191\u03b6 \u2022 f') x = \u2191g' x"}, {"tactic": "apply forall_congr'", "state_before": "R : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\n\u22a2 (\u2200 (n : \u2115), 0 < n \u2192 \u2211 i in divisors n, f i = g n) \u2194 \u2200 (x : \u2115), \u2191(\u2191\u03b6 \u2022 f') x = \u2191g' x", "state_after": "case h\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\n\u22a2 \u2200 (a : \u2115), 0 < a \u2192 \u2211 i in divisors a, f i = g a \u2194 \u2191(\u2191\u03b6 \u2022 f') a = \u2191g' a"}, {"tactic": "intro n", "state_before": "case h\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\n\u22a2 \u2200 (a : \u2115), 0 < a \u2192 \u2211 i in divisors a, f i = g a \u2194 \u2191(\u2191\u03b6 \u2022 f') a = \u2191g' a", "state_after": "case h\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn : \u2115\n\u22a2 0 < n \u2192 \u2211 i in divisors n, f i = g n \u2194 \u2191(\u2191\u03b6 \u2022 f') n = \u2191g' n"}, {"tactic": "cases n with\n| zero => simp\n| succ n =>\n  rw [coe_zeta_smul_apply]\n  simp only [n.succ_ne_zero, forall_prop_of_true, succ_pos', if_false, ZeroHom.coe_mk]\n  simp only [coe_mk, succ_ne_zero, ite_false]\n  rw [sum_congr rfl]  intros x hx         rw [if_neg (ne_of_gt (Nat.pos_of_mem_divisors ?_))]\n  exact n.succ; assumption", "state_before": "case h\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn : \u2115\n\u22a2 0 < n \u2192 \u2211 i in divisors n, f i = g n \u2194 \u2191(\u2191\u03b6 \u2022 f') n = \u2191g' n", "state_after": "no goals"}, {"tactic": "simp", "state_before": "case h.zero\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\n\u22a2 0 < Nat.zero \u2192 \u2211 i in divisors Nat.zero, f i = g Nat.zero \u2194 \u2191(\u2191\u03b6 \u2022 f') Nat.zero = \u2191g' Nat.zero", "state_after": "no goals"}, {"tactic": "rw [coe_zeta_smul_apply]", "state_before": "case h.succ\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn : \u2115\n\u22a2 0 < succ n \u2192 \u2211 i in divisors (succ n), f i = g (succ n) \u2194 \u2191(\u2191\u03b6 \u2022 f') (succ n) = \u2191g' (succ n)", "state_after": "case h.succ\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn : \u2115\n\u22a2 0 < succ n \u2192 \u2211 i in divisors (succ n), f i = g (succ n) \u2194 \u2211 i in divisors (succ n), \u2191f' i = \u2191g' (succ n)"}, {"tactic": "simp only [n.succ_ne_zero, forall_prop_of_true, succ_pos', if_false, ZeroHom.coe_mk]", "state_before": "case h.succ\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn : \u2115\n\u22a2 0 < succ n \u2192 \u2211 i in divisors (succ n), f i = g (succ n) \u2194 \u2211 i in divisors (succ n), \u2191f' i = \u2191g' (succ n)", "state_after": "case h.succ\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn : \u2115\n\u22a2 \u2211 i in divisors (succ n), f i = g (succ n) \u2194\n    \u2211 x in divisors (succ n),\n        \u2191{ toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) } x =\n      \u2191{ toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) } (succ n)"}, {"tactic": "simp only [coe_mk, succ_ne_zero, ite_false]", "state_before": "case h.succ\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn : \u2115\n\u22a2 \u2211 i in divisors (succ n), f i = g (succ n) \u2194\n    \u2211 x in divisors (succ n),\n        \u2191{ toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) } x =\n      \u2191{ toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) } (succ n)", "state_after": "case h.succ\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn : \u2115\n\u22a2 \u2211 i in divisors (succ n), f i = g (succ n) \u2194 (\u2211 x in divisors (succ n), if x = 0 then 0 else f x) = g (succ n)"}, {"tactic": "rw [sum_congr rfl]", "state_before": "case h.succ\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn : \u2115\n\u22a2 \u2211 i in divisors (succ n), f i = g (succ n) \u2194 (\u2211 x in divisors (succ n), if x = 0 then 0 else f x) = g (succ n)", "state_after": "case h.succ\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn : \u2115\n\u22a2 \u2200 (x : \u2115), x \u2208 divisors (succ n) \u2192 f x = if x = 0 then 0 else f x"}, {"tactic": "intros x hx", "state_before": "case h.succ\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn : \u2115\n\u22a2 \u2200 (x : \u2115), x \u2208 divisors (succ n) \u2192 f x = if x = 0 then 0 else f x", "state_after": "case h.succ\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn x : \u2115\nhx : x \u2208 divisors (succ n)\n\u22a2 f x = if x = 0 then 0 else f x"}, {"tactic": "rw [if_neg (ne_of_gt (Nat.pos_of_mem_divisors ?_))]", "state_before": "case h.succ\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn x : \u2115\nhx : x \u2208 divisors (succ n)\n\u22a2 f x = if x = 0 then 0 else f x", "state_after": "R : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn x : \u2115\nhx : x \u2208 divisors (succ n)\n\u22a2 \u2115\n\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn x : \u2115\nhx : x \u2208 divisors (succ n)\n\u22a2 x \u2208 divisors ?m.681674"}, {"tactic": "exact n.succ", "state_before": "R : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn x : \u2115\nhx : x \u2208 divisors (succ n)\n\u22a2 \u2115\n\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn x : \u2115\nhx : x \u2208 divisors (succ n)\n\u22a2 x \u2208 divisors ?m.681674", "state_after": "R : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn x : \u2115\nhx : x \u2208 divisors (succ n)\n\u22a2 x \u2208 divisors (succ n)"}, {"tactic": "assumption", "state_before": "R : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn x : \u2115\nhx : x \u2208 divisors (succ n)\n\u22a2 x \u2208 divisors (succ n)", "state_after": "no goals"}, {"tactic": "constructor <;> intro h", "state_before": "R : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\n\u22a2 \u2191\u03b6 \u2022 f' = g' \u2194 \u03bc \u2022 g' = f'", "state_after": "case mp\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nh : \u2191\u03b6 \u2022 f' = g'\n\u22a2 \u03bc \u2022 g' = f'\n\ncase mpr\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nh : \u03bc \u2022 g' = f'\n\u22a2 \u2191\u03b6 \u2022 f' = g'"}, {"tactic": "rw [\u2190 h, \u2190 mul_smul, moebius_mul_coe_zeta, one_smul]", "state_before": "case mp\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nh : \u2191\u03b6 \u2022 f' = g'\n\u22a2 \u03bc \u2022 g' = f'", "state_after": "no goals"}, {"tactic": "rw [\u2190 h, \u2190 mul_smul, coe_zeta_mul_moebius, one_smul]", "state_before": "case mpr\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nh : \u03bc \u2022 g' = f'\n\u22a2 \u2191\u03b6 \u2022 f' = g'", "state_after": "no goals"}, {"tactic": "rw [ext_iff]", "state_before": "R : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\n\u22a2 \u03bc \u2022 g' = f' \u2194 \u2200 (n : \u2115), 0 < n \u2192 \u2211 x in divisorsAntidiagonal n, \u2191\u03bc x.fst \u2022 g x.snd = f n", "state_after": "R : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\n\u22a2 (\u2200 (x : \u2115), \u2191(\u03bc \u2022 g') x = \u2191f' x) \u2194 \u2200 (n : \u2115), 0 < n \u2192 \u2211 x in divisorsAntidiagonal n, \u2191\u03bc x.fst \u2022 g x.snd = f n"}, {"tactic": "apply forall_congr'", "state_before": "R : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\n\u22a2 (\u2200 (x : \u2115), \u2191(\u03bc \u2022 g') x = \u2191f' x) \u2194 \u2200 (n : \u2115), 0 < n \u2192 \u2211 x in divisorsAntidiagonal n, \u2191\u03bc x.fst \u2022 g x.snd = f n", "state_after": "case h\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\n\u22a2 \u2200 (a : \u2115), \u2191(\u03bc \u2022 g') a = \u2191f' a \u2194 0 < a \u2192 \u2211 x in divisorsAntidiagonal a, \u2191\u03bc x.fst \u2022 g x.snd = f a"}, {"tactic": "intro n", "state_before": "case h\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\n\u22a2 \u2200 (a : \u2115), \u2191(\u03bc \u2022 g') a = \u2191f' a \u2194 0 < a \u2192 \u2211 x in divisorsAntidiagonal a, \u2191\u03bc x.fst \u2022 g x.snd = f a", "state_after": "case h\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn : \u2115\n\u22a2 \u2191(\u03bc \u2022 g') n = \u2191f' n \u2194 0 < n \u2192 \u2211 x in divisorsAntidiagonal n, \u2191\u03bc x.fst \u2022 g x.snd = f n"}, {"tactic": "cases n with\n| zero => simp\n| succ n =>\n  simp only [n.succ_ne_zero, forall_prop_of_true, succ_pos', smul_apply, if_false,\n    ZeroHom.coe_mk]\n  simp only [Nat.isUnit_iff, coe_mk, ZeroHom.toFun_eq_coe, succ_ne_zero, ite_false]\n  rw [sum_congr rfl]  intros x hx         rw [if_neg (ne_of_gt (Nat.pos_of_mem_divisors (snd_mem_divisors_of_mem_antidiagonal hx)))]", "state_before": "case h\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn : \u2115\n\u22a2 \u2191(\u03bc \u2022 g') n = \u2191f' n \u2194 0 < n \u2192 \u2211 x in divisorsAntidiagonal n, \u2191\u03bc x.fst \u2022 g x.snd = f n", "state_after": "no goals"}, {"tactic": "simp", "state_before": "case h.zero\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\n\u22a2 \u2191(\u03bc \u2022 g') Nat.zero = \u2191f' Nat.zero \u2194\n    0 < Nat.zero \u2192 \u2211 x in divisorsAntidiagonal Nat.zero, \u2191\u03bc x.fst \u2022 g x.snd = f Nat.zero", "state_after": "no goals"}, {"tactic": "simp only [n.succ_ne_zero, forall_prop_of_true, succ_pos', smul_apply, if_false,\n  ZeroHom.coe_mk]", "state_before": "case h.succ\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn : \u2115\n\u22a2 \u2191(\u03bc \u2022 g') (succ n) = \u2191f' (succ n) \u2194 0 < succ n \u2192 \u2211 x in divisorsAntidiagonal (succ n), \u2191\u03bc x.fst \u2022 g x.snd = f (succ n)", "state_after": "case h.succ\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn : \u2115\n\u22a2 \u2211 x in divisorsAntidiagonal (succ n),\n        \u2191\u03bc x.fst \u2022\n          \u2191{ toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) } x.snd =\n      \u2191{ toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) } (succ n) \u2194\n    \u2211 x in divisorsAntidiagonal (succ n), \u2191\u03bc x.fst \u2022 g x.snd = f (succ n)"}, {"tactic": "simp only [Nat.isUnit_iff, coe_mk, ZeroHom.toFun_eq_coe, succ_ne_zero, ite_false]", "state_before": "case h.succ\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn : \u2115\n\u22a2 \u2211 x in divisorsAntidiagonal (succ n),\n        \u2191\u03bc x.fst \u2022\n          \u2191{ toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) } x.snd =\n      \u2191{ toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) } (succ n) \u2194\n    \u2211 x in divisorsAntidiagonal (succ n), \u2191\u03bc x.fst \u2022 g x.snd = f (succ n)", "state_after": "case h.succ\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn : \u2115\n\u22a2 (\u2211 x in divisorsAntidiagonal (succ n), \u2191\u03bc x.fst \u2022 if x.snd = 0 then 0 else g x.snd) = f (succ n) \u2194\n    \u2211 x in divisorsAntidiagonal (succ n), \u2191\u03bc x.fst \u2022 g x.snd = f (succ n)"}, {"tactic": "rw [sum_congr rfl]", "state_before": "case h.succ\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn : \u2115\n\u22a2 (\u2211 x in divisorsAntidiagonal (succ n), \u2191\u03bc x.fst \u2022 if x.snd = 0 then 0 else g x.snd) = f (succ n) \u2194\n    \u2211 x in divisorsAntidiagonal (succ n), \u2191\u03bc x.fst \u2022 g x.snd = f (succ n)", "state_after": "case h.succ\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn : \u2115\n\u22a2 \u2200 (x : \u2115 \u00d7 \u2115), x \u2208 divisorsAntidiagonal (succ n) \u2192 (\u2191\u03bc x.fst \u2022 if x.snd = 0 then 0 else g x.snd) = \u2191\u03bc x.fst \u2022 g x.snd"}, {"tactic": "intros x hx", "state_before": "case h.succ\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn : \u2115\n\u22a2 \u2200 (x : \u2115 \u00d7 \u2115), x \u2208 divisorsAntidiagonal (succ n) \u2192 (\u2191\u03bc x.fst \u2022 if x.snd = 0 then 0 else g x.snd) = \u2191\u03bc x.fst \u2022 g x.snd", "state_after": "case h.succ\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn : \u2115\nx : \u2115 \u00d7 \u2115\nhx : x \u2208 divisorsAntidiagonal (succ n)\n\u22a2 (\u2191\u03bc x.fst \u2022 if x.snd = 0 then 0 else g x.snd) = \u2191\u03bc x.fst \u2022 g x.snd"}, {"tactic": "rw [if_neg (ne_of_gt (Nat.pos_of_mem_divisors (snd_mem_divisors_of_mem_antidiagonal hx)))]", "state_before": "case h.succ\nR : Type u_1\ninst\u271d : AddCommGroup R\nf g : \u2115 \u2192 R\nf' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else f x, map_zero' := (_ : (if 0 = 0 then 0 else f 0) = 0) }\ng' : ArithmeticFunction R :=\n  { toFun := fun x => if x = 0 then 0 else g x, map_zero' := (_ : (if 0 = 0 then 0 else g 0) = 0) }\nn : \u2115\nx : \u2115 \u00d7 \u2115\nhx : x \u2208 divisorsAntidiagonal (succ n)\n\u22a2 (\u2191\u03bc x.fst \u2022 if x.snd = 0 then 0 else g x.snd) = \u2191\u03bc x.fst \u2022 g x.snd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/Language.lean", "full_name": "Language.map_kstar", "start": [274, 1], "end": [277, 21], "traced_tactics": [{"tactic": "rw [kstar_eq_iSup_pow, kstar_eq_iSup_pow]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.100003\nl\u271d m : Language \u03b1\na b x : List \u03b1\nf : \u03b1 \u2192 \u03b2\nl : Language \u03b1\n\u22a2 \u2191(map f) l\u2217 = (\u2191(map f) l)\u2217", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.100003\nl\u271d m : Language \u03b1\na b x : List \u03b1\nf : \u03b1 \u2192 \u03b2\nl : Language \u03b1\n\u22a2 \u2191(map f) (\u2a06 (i : \u2115), l ^ i) = \u2a06 (i : \u2115), \u2191(map f) l ^ i"}, {"tactic": "simp_rw [\u2190 map_pow]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.100003\nl\u271d m : Language \u03b1\na b x : List \u03b1\nf : \u03b1 \u2192 \u03b2\nl : Language \u03b1\n\u22a2 \u2191(map f) (\u2a06 (i : \u2115), l ^ i) = \u2a06 (i : \u2115), \u2191(map f) l ^ i", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.100003\nl\u271d m : Language \u03b1\na b x : List \u03b1\nf : \u03b1 \u2192 \u03b2\nl : Language \u03b1\n\u22a2 \u2191(map f) (\u2a06 (i : \u2115), l ^ i) = \u2a06 (i : \u2115), \u2191(map f) (l ^ i)"}, {"tactic": "exact image_iUnion", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.100003\nl\u271d m : Language \u03b1\na b x : List \u03b1\nf : \u03b1 \u2192 \u03b2\nl : Language \u03b1\n\u22a2 \u2191(map f) (\u2a06 (i : \u2115), l ^ i) = \u2a06 (i : \u2115), \u2191(map f) (l ^ i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.restrict_piecewise_compl", "start": [115, 1], "end": [117, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean", "full_name": "BoxIntegral.TaggedPrepartition.IsSubordinate.biUnionPrepartition", "start": [265, 1], "end": [268, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Control/ForInStep/Lemmas.lean", "full_name": "ForInStep.bind_yield_bindList", "start": [31, 9], "end": [33, 85], "traced_tactics": [{"tactic": "cases s <;> simp", "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d : Monad m\nf : \u03b1 \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\ns : ForInStep \u03b2\nl : List \u03b1\n\u22a2 (ForInStep.bind s fun a => bindList f l (yield a)) = bindList f l s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SetFamily/HarrisKleitman.lean", "full_name": "IsLowerSet.le_card_inter_finset'", "start": [59, 1], "end": [94, 54], "traced_tactics": [{"tactic": "induction' s using Finset.induction with a s hs ih generalizing \ud835\udc9c \u212c", "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\ns : Finset \u03b1\na : \u03b1\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s\n\u22a2 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)", "state_after": "case empty\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns : Finset \u03b1\na : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 \u2205\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 \u2205\n\u22a2 card \ud835\udc9c * card \u212c \u2264 2 ^ card \u2205 * card (\ud835\udc9c \u2229 \u212c)\n\ncase insert\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\n\u22a2 card \ud835\udc9c * card \u212c \u2264 2 ^ card (insert a s) * card (\ud835\udc9c \u2229 \u212c)"}, {"tactic": "rw [card_insert_of_not_mem hs, \u2190 card_memberSubfamily_add_card_nonMemberSubfamily a \ud835\udc9c, \u2190\n  card_memberSubfamily_add_card_nonMemberSubfamily a \u212c, add_mul, mul_add, mul_add,\n  add_comm (_ * _), add_add_add_comm]", "state_before": "case insert\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\n\u22a2 card \ud835\udc9c * card \u212c \u2264 2 ^ card (insert a s) * card (\ud835\udc9c \u2229 \u212c)", "state_after": "case insert\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\n\u22a2 card (Finset.memberSubfamily a \ud835\udc9c) * card (Finset.nonMemberSubfamily a \u212c) +\n        card (Finset.nonMemberSubfamily a \ud835\udc9c) * card (Finset.memberSubfamily a \u212c) +\n      (card (Finset.memberSubfamily a \ud835\udc9c) * card (Finset.memberSubfamily a \u212c) +\n        card (Finset.nonMemberSubfamily a \ud835\udc9c) * card (Finset.nonMemberSubfamily a \u212c)) \u2264\n    2 ^ (card s + 1) * card (\ud835\udc9c \u2229 \u212c)"}, {"tactic": "refine'\n  (add_le_add_right\n        (mul_add_mul_le_mul_add_mul\n            (card_le_of_subset h\ud835\udc9c.memberSubfamily_subset_nonMemberSubfamily) <|\n          card_le_of_subset h\u212c.memberSubfamily_subset_nonMemberSubfamily)\n        _).trans\n    _", "state_before": "case insert\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\n\u22a2 card (Finset.memberSubfamily a \ud835\udc9c) * card (Finset.nonMemberSubfamily a \u212c) +\n        card (Finset.nonMemberSubfamily a \ud835\udc9c) * card (Finset.memberSubfamily a \u212c) +\n      (card (Finset.memberSubfamily a \ud835\udc9c) * card (Finset.memberSubfamily a \u212c) +\n        card (Finset.nonMemberSubfamily a \ud835\udc9c) * card (Finset.nonMemberSubfamily a \u212c)) \u2264\n    2 ^ (card s + 1) * card (\ud835\udc9c \u2229 \u212c)", "state_after": "case insert\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\n\u22a2 card (Finset.memberSubfamily a \ud835\udc9c) * card (Finset.memberSubfamily a \u212c) +\n        card (Finset.nonMemberSubfamily a \ud835\udc9c) * card (Finset.nonMemberSubfamily a \u212c) +\n      (card (Finset.memberSubfamily a \ud835\udc9c) * card (Finset.memberSubfamily a \u212c) +\n        card (Finset.nonMemberSubfamily a \ud835\udc9c) * card (Finset.nonMemberSubfamily a \u212c)) \u2264\n    2 ^ (card s + 1) * card (\ud835\udc9c \u2229 \u212c)"}, {"tactic": "rw [\u2190 two_mul, pow_succ, mul_assoc]", "state_before": "case insert\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\n\u22a2 card (Finset.memberSubfamily a \ud835\udc9c) * card (Finset.memberSubfamily a \u212c) +\n        card (Finset.nonMemberSubfamily a \ud835\udc9c) * card (Finset.nonMemberSubfamily a \u212c) +\n      (card (Finset.memberSubfamily a \ud835\udc9c) * card (Finset.memberSubfamily a \u212c) +\n        card (Finset.nonMemberSubfamily a \ud835\udc9c) * card (Finset.nonMemberSubfamily a \u212c)) \u2264\n    2 ^ (card s + 1) * card (\ud835\udc9c \u2229 \u212c)", "state_after": "case insert\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\n\u22a2 2 *\n      (card (Finset.memberSubfamily a \ud835\udc9c) * card (Finset.memberSubfamily a \u212c) +\n        card (Finset.nonMemberSubfamily a \ud835\udc9c) * card (Finset.nonMemberSubfamily a \u212c)) \u2264\n    2 * (2 ^ card s * card (\ud835\udc9c \u2229 \u212c))"}, {"tactic": "have h\u2080 : \u2200 \ud835\udc9e : Finset (Finset \u03b1), (\u2200 t \u2208 \ud835\udc9e, t \u2286 insert a s) \u2192\n    \u2200 t \u2208 \ud835\udc9e.nonMemberSubfamily a, t \u2286 s := by\n  rintro \ud835\udc9e h\ud835\udc9e t ht\n  rw [mem_nonMemberSubfamily] at ht\n  exact (subset_insert_iff_of_not_mem ht.2).1 (h\ud835\udc9e _ ht.1)", "state_before": "case insert\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\n\u22a2 2 *\n      (card (Finset.memberSubfamily a \ud835\udc9c) * card (Finset.memberSubfamily a \u212c) +\n        card (Finset.nonMemberSubfamily a \ud835\udc9c) * card (Finset.nonMemberSubfamily a \u212c)) \u2264\n    2 * (2 ^ card s * card (\ud835\udc9c \u2229 \u212c))", "state_after": "case insert\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\nh\u2080 :\n  \u2200 (\ud835\udc9e : Finset (Finset \u03b1)),\n    (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s) \u2192 \u2200 (t : Finset \u03b1), t \u2208 Finset.nonMemberSubfamily a \ud835\udc9e \u2192 t \u2286 s\n\u22a2 2 *\n      (card (Finset.memberSubfamily a \ud835\udc9c) * card (Finset.memberSubfamily a \u212c) +\n        card (Finset.nonMemberSubfamily a \ud835\udc9c) * card (Finset.nonMemberSubfamily a \u212c)) \u2264\n    2 * (2 ^ card s * card (\ud835\udc9c \u2229 \u212c))"}, {"tactic": "have h\u2081 : \u2200 \ud835\udc9e : Finset (Finset \u03b1), (\u2200 t \u2208 \ud835\udc9e, t \u2286 insert a s) \u2192\n    \u2200 t \u2208 \ud835\udc9e.memberSubfamily a, t \u2286 s := by\n  rintro \ud835\udc9e h\ud835\udc9e t ht\n  rw [mem_memberSubfamily] at ht\n  exact (subset_insert_iff_of_not_mem ht.2).1 ((subset_insert _ _).trans <| h\ud835\udc9e _ ht.1)", "state_before": "case insert\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\nh\u2080 :\n  \u2200 (\ud835\udc9e : Finset (Finset \u03b1)),\n    (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s) \u2192 \u2200 (t : Finset \u03b1), t \u2208 Finset.nonMemberSubfamily a \ud835\udc9e \u2192 t \u2286 s\n\u22a2 2 *\n      (card (Finset.memberSubfamily a \ud835\udc9c) * card (Finset.memberSubfamily a \u212c) +\n        card (Finset.nonMemberSubfamily a \ud835\udc9c) * card (Finset.nonMemberSubfamily a \u212c)) \u2264\n    2 * (2 ^ card s * card (\ud835\udc9c \u2229 \u212c))", "state_after": "case insert\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\nh\u2080 :\n  \u2200 (\ud835\udc9e : Finset (Finset \u03b1)),\n    (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s) \u2192 \u2200 (t : Finset \u03b1), t \u2208 Finset.nonMemberSubfamily a \ud835\udc9e \u2192 t \u2286 s\nh\u2081 :\n  \u2200 (\ud835\udc9e : Finset (Finset \u03b1)),\n    (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s) \u2192 \u2200 (t : Finset \u03b1), t \u2208 Finset.memberSubfamily a \ud835\udc9e \u2192 t \u2286 s\n\u22a2 2 *\n      (card (Finset.memberSubfamily a \ud835\udc9c) * card (Finset.memberSubfamily a \u212c) +\n        card (Finset.nonMemberSubfamily a \ud835\udc9c) * card (Finset.nonMemberSubfamily a \u212c)) \u2264\n    2 * (2 ^ card s * card (\ud835\udc9c \u2229 \u212c))"}, {"tactic": "refine' mul_le_mul_left' _ _", "state_before": "case insert\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\nh\u2080 :\n  \u2200 (\ud835\udc9e : Finset (Finset \u03b1)),\n    (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s) \u2192 \u2200 (t : Finset \u03b1), t \u2208 Finset.nonMemberSubfamily a \ud835\udc9e \u2192 t \u2286 s\nh\u2081 :\n  \u2200 (\ud835\udc9e : Finset (Finset \u03b1)),\n    (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s) \u2192 \u2200 (t : Finset \u03b1), t \u2208 Finset.memberSubfamily a \ud835\udc9e \u2192 t \u2286 s\n\u22a2 2 *\n      (card (Finset.memberSubfamily a \ud835\udc9c) * card (Finset.memberSubfamily a \u212c) +\n        card (Finset.nonMemberSubfamily a \ud835\udc9c) * card (Finset.nonMemberSubfamily a \u212c)) \u2264\n    2 * (2 ^ card s * card (\ud835\udc9c \u2229 \u212c))", "state_after": "case insert\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\nh\u2080 :\n  \u2200 (\ud835\udc9e : Finset (Finset \u03b1)),\n    (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s) \u2192 \u2200 (t : Finset \u03b1), t \u2208 Finset.nonMemberSubfamily a \ud835\udc9e \u2192 t \u2286 s\nh\u2081 :\n  \u2200 (\ud835\udc9e : Finset (Finset \u03b1)),\n    (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s) \u2192 \u2200 (t : Finset \u03b1), t \u2208 Finset.memberSubfamily a \ud835\udc9e \u2192 t \u2286 s\n\u22a2 card (Finset.memberSubfamily a \ud835\udc9c) * card (Finset.memberSubfamily a \u212c) +\n      card (Finset.nonMemberSubfamily a \ud835\udc9c) * card (Finset.nonMemberSubfamily a \u212c) \u2264\n    2 ^ card s * card (\ud835\udc9c \u2229 \u212c)"}, {"tactic": "refine' (add_le_add (ih h\ud835\udc9c.memberSubfamily h\u212c.memberSubfamily (h\u2081 _ h\ud835\udc9cs) <| h\u2081 _ h\u212cs) <|\n  ih h\ud835\udc9c.nonMemberSubfamily h\u212c.nonMemberSubfamily (h\u2080 _ h\ud835\udc9cs) <| h\u2080 _ h\u212cs).trans_eq _", "state_before": "case insert\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\nh\u2080 :\n  \u2200 (\ud835\udc9e : Finset (Finset \u03b1)),\n    (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s) \u2192 \u2200 (t : Finset \u03b1), t \u2208 Finset.nonMemberSubfamily a \ud835\udc9e \u2192 t \u2286 s\nh\u2081 :\n  \u2200 (\ud835\udc9e : Finset (Finset \u03b1)),\n    (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s) \u2192 \u2200 (t : Finset \u03b1), t \u2208 Finset.memberSubfamily a \ud835\udc9e \u2192 t \u2286 s\n\u22a2 card (Finset.memberSubfamily a \ud835\udc9c) * card (Finset.memberSubfamily a \u212c) +\n      card (Finset.nonMemberSubfamily a \ud835\udc9c) * card (Finset.nonMemberSubfamily a \u212c) \u2264\n    2 ^ card s * card (\ud835\udc9c \u2229 \u212c)", "state_after": "case insert\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\nh\u2080 :\n  \u2200 (\ud835\udc9e : Finset (Finset \u03b1)),\n    (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s) \u2192 \u2200 (t : Finset \u03b1), t \u2208 Finset.nonMemberSubfamily a \ud835\udc9e \u2192 t \u2286 s\nh\u2081 :\n  \u2200 (\ud835\udc9e : Finset (Finset \u03b1)),\n    (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s) \u2192 \u2200 (t : Finset \u03b1), t \u2208 Finset.memberSubfamily a \ud835\udc9e \u2192 t \u2286 s\n\u22a2 2 ^ card s * card (Finset.memberSubfamily a \ud835\udc9c \u2229 Finset.memberSubfamily a \u212c) +\n      2 ^ card s * card (Finset.nonMemberSubfamily a \ud835\udc9c \u2229 Finset.nonMemberSubfamily a \u212c) =\n    2 ^ card s * card (\ud835\udc9c \u2229 \u212c)"}, {"tactic": "rw [\u2190 mul_add, \u2190 memberSubfamily_inter, \u2190 nonMemberSubfamily_inter,\n  card_memberSubfamily_add_card_nonMemberSubfamily]", "state_before": "case insert\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\nh\u2080 :\n  \u2200 (\ud835\udc9e : Finset (Finset \u03b1)),\n    (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s) \u2192 \u2200 (t : Finset \u03b1), t \u2208 Finset.nonMemberSubfamily a \ud835\udc9e \u2192 t \u2286 s\nh\u2081 :\n  \u2200 (\ud835\udc9e : Finset (Finset \u03b1)),\n    (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s) \u2192 \u2200 (t : Finset \u03b1), t \u2208 Finset.memberSubfamily a \ud835\udc9e \u2192 t \u2286 s\n\u22a2 2 ^ card s * card (Finset.memberSubfamily a \ud835\udc9c \u2229 Finset.memberSubfamily a \u212c) +\n      2 ^ card s * card (Finset.nonMemberSubfamily a \ud835\udc9c \u2229 Finset.nonMemberSubfamily a \u212c) =\n    2 ^ card s * card (\ud835\udc9c \u2229 \u212c)", "state_after": "no goals"}, {"tactic": "simp_rw [subset_empty, \u2190 subset_singleton_iff', subset_singleton_iff] at h\ud835\udc9cs h\u212cs", "state_before": "case empty\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns : Finset \u03b1\na : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 \u2205\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 \u2205\n\u22a2 card \ud835\udc9c * card \u212c \u2264 2 ^ card \u2205 * card (\ud835\udc9c \u2229 \u212c)", "state_after": "case empty\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns : Finset \u03b1\na : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \ud835\udc9c = \u2205 \u2228 \ud835\udc9c = {\u2205}\nh\u212cs : \u212c = \u2205 \u2228 \u212c = {\u2205}\n\u22a2 card \ud835\udc9c * card \u212c \u2264 2 ^ card \u2205 * card (\ud835\udc9c \u2229 \u212c)"}, {"tactic": "obtain rfl | rfl := h\ud835\udc9cs", "state_before": "case empty\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns : Finset \u03b1\na : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \ud835\udc9c = \u2205 \u2228 \ud835\udc9c = {\u2205}\nh\u212cs : \u212c = \u2205 \u2228 \u212c = {\u2205}\n\u22a2 card \ud835\udc9c * card \u212c \u2264 2 ^ card \u2205 * card (\ud835\udc9c \u2229 \u212c)", "state_after": "case empty.inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c \u212c\u271d : Finset (Finset \u03b1)\ns : Finset \u03b1\na : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\n\u212c : Finset (Finset \u03b1)\nh\u212c : IsLowerSet \u2191\u212c\nh\u212cs : \u212c = \u2205 \u2228 \u212c = {\u2205}\nh\ud835\udc9c : IsLowerSet \u2191\u2205\n\u22a2 card \u2205 * card \u212c \u2264 2 ^ card \u2205 * card (\u2205 \u2229 \u212c)\n\ncase empty.inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c \u212c\u271d : Finset (Finset \u03b1)\ns : Finset \u03b1\na : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\n\u212c : Finset (Finset \u03b1)\nh\u212c : IsLowerSet \u2191\u212c\nh\u212cs : \u212c = \u2205 \u2228 \u212c = {\u2205}\nh\ud835\udc9c : IsLowerSet \u2191{\u2205}\n\u22a2 card {\u2205} * card \u212c \u2264 2 ^ card \u2205 * card ({\u2205} \u2229 \u212c)"}, {"tactic": "obtain rfl | rfl := h\u212cs", "state_before": "case empty.inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c \u212c\u271d : Finset (Finset \u03b1)\ns : Finset \u03b1\na : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\n\u212c : Finset (Finset \u03b1)\nh\u212c : IsLowerSet \u2191\u212c\nh\u212cs : \u212c = \u2205 \u2228 \u212c = {\u2205}\nh\ud835\udc9c : IsLowerSet \u2191{\u2205}\n\u22a2 card {\u2205} * card \u212c \u2264 2 ^ card \u2205 * card ({\u2205} \u2229 \u212c)", "state_after": "case empty.inr.inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\ns : Finset \u03b1\na : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\nh\u212c\u271d : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s\nh\ud835\udc9c : IsLowerSet \u2191{\u2205}\nh\u212c : IsLowerSet \u2191\u2205\n\u22a2 card {\u2205} * card \u2205 \u2264 2 ^ card \u2205 * card ({\u2205} \u2229 \u2205)\n\ncase empty.inr.inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\ns : Finset \u03b1\na : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\nh\u212c\u271d : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s\nh\ud835\udc9c h\u212c : IsLowerSet \u2191{\u2205}\n\u22a2 card {\u2205} * card {\u2205} \u2264 2 ^ card \u2205 * card ({\u2205} \u2229 {\u2205})"}, {"tactic": "simp only [card_empty, empty_inter, mul_zero, zero_mul]", "state_before": "case empty.inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c \u212c\u271d : Finset (Finset \u03b1)\ns : Finset \u03b1\na : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\n\u212c : Finset (Finset \u03b1)\nh\u212c : IsLowerSet \u2191\u212c\nh\u212cs : \u212c = \u2205 \u2228 \u212c = {\u2205}\nh\ud835\udc9c : IsLowerSet \u2191\u2205\n\u22a2 card \u2205 * card \u212c \u2264 2 ^ card \u2205 * card (\u2205 \u2229 \u212c)", "state_after": "no goals"}, {"tactic": "simp only [card_empty, inter_empty, mul_zero, zero_mul]", "state_before": "case empty.inr.inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\ns : Finset \u03b1\na : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\nh\u212c\u271d : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s\nh\ud835\udc9c : IsLowerSet \u2191{\u2205}\nh\u212c : IsLowerSet \u2191\u2205\n\u22a2 card {\u2205} * card \u2205 \u2264 2 ^ card \u2205 * card ({\u2205} \u2229 \u2205)", "state_after": "no goals"}, {"tactic": "simp only [card_empty, pow_zero, inter_singleton_of_mem, mem_singleton, card_singleton]", "state_before": "case empty.inr.inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\ns : Finset \u03b1\na : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\nh\u212c\u271d : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s\nh\ud835\udc9c h\u212c : IsLowerSet \u2191{\u2205}\n\u22a2 card {\u2205} * card {\u2205} \u2264 2 ^ card \u2205 * card ({\u2205} \u2229 {\u2205})", "state_after": "no goals"}, {"tactic": "rintro \ud835\udc9e h\ud835\udc9e t ht", "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\n\u22a2 \u2200 (\ud835\udc9e : Finset (Finset \u03b1)),\n    (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s) \u2192 \u2200 (t : Finset \u03b1), t \u2208 Finset.nonMemberSubfamily a \ud835\udc9e \u2192 t \u2286 s", "state_after": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\n\ud835\udc9e : Finset (Finset \u03b1)\nh\ud835\udc9e : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s\nt : Finset \u03b1\nht : t \u2208 Finset.nonMemberSubfamily a \ud835\udc9e\n\u22a2 t \u2286 s"}, {"tactic": "rw [mem_nonMemberSubfamily] at ht", "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\n\ud835\udc9e : Finset (Finset \u03b1)\nh\ud835\udc9e : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s\nt : Finset \u03b1\nht : t \u2208 Finset.nonMemberSubfamily a \ud835\udc9e\n\u22a2 t \u2286 s", "state_after": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\n\ud835\udc9e : Finset (Finset \u03b1)\nh\ud835\udc9e : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s\nt : Finset \u03b1\nht : t \u2208 \ud835\udc9e \u2227 \u00aca \u2208 t\n\u22a2 t \u2286 s"}, {"tactic": "exact (subset_insert_iff_of_not_mem ht.2).1 (h\ud835\udc9e _ ht.1)", "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\n\ud835\udc9e : Finset (Finset \u03b1)\nh\ud835\udc9e : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s\nt : Finset \u03b1\nht : t \u2208 \ud835\udc9e \u2227 \u00aca \u2208 t\n\u22a2 t \u2286 s", "state_after": "no goals"}, {"tactic": "rintro \ud835\udc9e h\ud835\udc9e t ht", "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\nh\u2080 :\n  \u2200 (\ud835\udc9e : Finset (Finset \u03b1)),\n    (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s) \u2192 \u2200 (t : Finset \u03b1), t \u2208 Finset.nonMemberSubfamily a \ud835\udc9e \u2192 t \u2286 s\n\u22a2 \u2200 (\ud835\udc9e : Finset (Finset \u03b1)),\n    (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s) \u2192 \u2200 (t : Finset \u03b1), t \u2208 Finset.memberSubfamily a \ud835\udc9e \u2192 t \u2286 s", "state_after": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\nh\u2080 :\n  \u2200 (\ud835\udc9e : Finset (Finset \u03b1)),\n    (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s) \u2192 \u2200 (t : Finset \u03b1), t \u2208 Finset.nonMemberSubfamily a \ud835\udc9e \u2192 t \u2286 s\n\ud835\udc9e : Finset (Finset \u03b1)\nh\ud835\udc9e : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s\nt : Finset \u03b1\nht : t \u2208 Finset.memberSubfamily a \ud835\udc9e\n\u22a2 t \u2286 s"}, {"tactic": "rw [mem_memberSubfamily] at ht", "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\nh\u2080 :\n  \u2200 (\ud835\udc9e : Finset (Finset \u03b1)),\n    (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s) \u2192 \u2200 (t : Finset \u03b1), t \u2208 Finset.nonMemberSubfamily a \ud835\udc9e \u2192 t \u2286 s\n\ud835\udc9e : Finset (Finset \u03b1)\nh\ud835\udc9e : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s\nt : Finset \u03b1\nht : t \u2208 Finset.memberSubfamily a \ud835\udc9e\n\u22a2 t \u2286 s", "state_after": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c \u2229 \u212c)\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsLowerSet \u2191\u212c\nh\ud835\udc9cs : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 insert a s\nh\u212cs : \u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 insert a s\nh\u2080 :\n  \u2200 (\ud835\udc9e : Finset (Finset \u03b1)),\n    (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s) \u2192 \u2200 (t : Finset \u03b1), t \u2208 Finset.nonMemberSubfamily a \ud835\udc9e \u2192 t \u2286 s\n\ud835\udc9e : Finset (Finset \u03b1)\nh\ud835\udc9e : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9e \u2192 t \u2286 insert a s\nt : Finset \u03b1\nht : insert a t \u2208 \ud835\udc9e \u2227 \u00aca \u2208 t\n\u22a2 t \u2286 s"}, {"tactic": "exact (subset_insert_iff_of_not_mem ht.2).1 ((subset_insert _ _).trans <| h\ud835\udc9e _ ht.1)", "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c\u271d : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\nh\ud835\udc9c\u271d : IsLowerSet \u2191\ud835\udc9c\u271d\nh\u212c\u271d : IsLowerSet \u2191\u212c\u271d\nh\ud835\udc9cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c\u271d \u2192 t \u2286 s\u271d\nh\u212cs\u271d : \u2200 (t : Finset \u03b1), t \u2208 \u212c\u271d \u2192 t \u2286 s\u271d\na : \u03b1\ns : Finset \u03b1\nhs : \u00aca \u2208 s\nih :\n  \u2200 {\ud835\udc9c \u212c : Finset (Finset \u03b1)},\n    IsLowerSet \u2191\ud835\udc9c \u2192\n      IsLowerSet \u2191\u212c \u2192\n        (\u2200 (t : Finset \u03b1), t \u2208 \ud835\udc9c \u2192 t \u2286 s) \u2192\n          (\u2200 (t : Finset \u03b1), t \u2208 \u212c \u2192 t \u2286 s) \u2192 card \ud835\udc9c * card \u212c \u2264 2 ^ card s * card (\ud835\udc9c 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"5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Eval.lean", "full_name": "Polynomial.coe_evalRingHom", "start": [1062, 1], "end": [1063, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/List/Lemmas.lean", "full_name": "List.disjoint_cons_right", "start": [1392, 9], "end": [1393, 74], "traced_tactics": [{"tactic": "rw [disjoint_cons_left]", "state_before": "\u03b1\u271d : Type u_1\nl\u2081 : List \u03b1\u271d\na : \u03b1\u271d\nl\u2082 : List \u03b1\u271d\n\u22a2 Disjoint (a :: l\u2082) l\u2081 \u2194 \u00aca \u2208 l\u2081 \u2227 Disjoint l\u2081 l\u2082", "state_after": "\u03b1\u271d : Type u_1\nl\u2081 : List \u03b1\u271d\na : \u03b1\u271d\nl\u2082 : List \u03b1\u271d\n\u22a2 \u00aca \u2208 l\u2081 \u2227 Disjoint l\u2082 l\u2081 \u2194 \u00aca \u2208 l\u2081 \u2227 Disjoint l\u2081 l\u2082"}, {"tactic": "simp [disjoint_comm]", "state_before": "\u03b1\u271d : Type u_1\nl\u2081 : List \u03b1\u271d\na : \u03b1\u271d\nl\u2082 : List \u03b1\u271d\n\u22a2 \u00aca \u2208 l\u2081 \u2227 Disjoint l\u2082 l\u2081 \u2194 \u00aca \u2208 l\u2081 \u2227 Disjoint l\u2081 l\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/ZeroObjects.lean", "full_name": "CategoryTheory.Limits.IsZero.eq_from", "start": [77, 1], "end": [78, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Int/Lemmas.lean", "full_name": "Int.natAbs_inj_of_nonneg_of_nonpos", "start": [74, 1], "end": [76, 96], "traced_tactics": [{"tactic": "simpa only [Int.natAbs_neg] using natAbs_inj_of_nonneg_of_nonneg ha (neg_nonneg_of_nonpos hb)", "state_before": "a\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\nha : 0 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\u211d\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u22a2 snorm' (fun x => b (f x) (g x)) p \u03bc \u2264 snorm' f q \u03bc * snorm' g r \u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016b (f a) (g a)\u2016\u208a ^ p \u2202\u03bc) ^ (1 / p) \u2264 snorm' f q \u03bc * snorm' g r \u03bc"}, {"tactic": "calc\n  (\u222b\u207b a : \u03b1, \u2191\u2016b (f a) (g a)\u2016\u208a ^ p \u2202\u03bc) ^ (1 / p) \u2264\n      (\u222b\u207b a : \u03b1, \u2191(\u2016f a\u2016\u208a * \u2016g a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) :=\n    (ENNReal.rpow_le_rpow_iff <| one_div_pos.mpr hp0_lt).mpr <|\n      lintegral_mono_ae <|\n        h.mono fun a ha => (ENNReal.rpow_le_rpow_iff hp0_lt).mpr <| ENNReal.coe_le_coe.mpr <| ha\n  _ \u2264 _ := ?_", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016b (f a) (g a)\u2016\u208a ^ p \u2202\u03bc) ^ (1 / p) \u2264 snorm' f q \u03bc * snorm' g r \u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191(\u2016f a\u2016\u208a * \u2016g a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 snorm' f q \u03bc * snorm' g r \u03bc"}, {"tactic": "simp_rw [snorm', ENNReal.coe_mul]", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191(\u2016f a\u2016\u208a * \u2016g a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 snorm' f q \u03bc * snorm' g r \u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u22a2 (\u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * \u2191\u2016g a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264\n    (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ r \u2202\u03bc) ^ (1 / r)"}, {"tactic": "exact ENNReal.lintegral_Lp_mul_le_Lq_mul_Lr hp0_lt hpq hpqr \u03bc hf.ennnorm hg.ennnorm", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq\u271d : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\np q r : \u211d\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 F\nhg : AEStronglyMeasurable g \u03bc\nb : E \u2192 F \u2192 G\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016b (f x) (g x)\u2016\u208a \u2264 \u2016f x\u2016\u208a * \u2016g x\u2016\u208a\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\n\u22a2 (\u222b\u207b (a : \u03b1), (\u2191\u2016f a\u2016\u208a * \u2191\u2016g a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264\n    (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ r \u2202\u03bc) ^ (1 / r)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/ContinuousOn.lean", "full_name": "ContinuousOn.congr_mono", "start": [822, 1], "end": [829, 49], "traced_tactics": [{"tactic": "intro x hx", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.330239\n\u03b4 : Type ?u.330242\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf g : \u03b1 \u2192 \u03b2\ns s\u2081 : Set \u03b1\nh : ContinuousOn f s\nh' : EqOn g f s\u2081\nh\u2081 : s\u2081 \u2286 s\n\u22a2 ContinuousOn g s\u2081", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.330239\n\u03b4 : Type ?u.330242\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf g : \u03b1 \u2192 \u03b2\ns s\u2081 : Set \u03b1\nh : ContinuousOn f s\nh' : EqOn g f s\u2081\nh\u2081 : s\u2081 \u2286 s\nx : \u03b1\nhx : x \u2208 s\u2081\n\u22a2 ContinuousWithinAt g s\u2081 x"}, {"tactic": "unfold ContinuousWithinAt", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.330239\n\u03b4 : Type ?u.330242\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf g : \u03b1 \u2192 \u03b2\ns s\u2081 : Set \u03b1\nh : ContinuousOn f s\nh' : EqOn g f s\u2081\nh\u2081 : s\u2081 \u2286 s\nx : \u03b1\nhx : x \u2208 s\u2081\n\u22a2 ContinuousWithinAt g s\u2081 x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.330239\n\u03b4 : Type ?u.330242\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf g : \u03b1 \u2192 \u03b2\ns s\u2081 : Set \u03b1\nh : ContinuousOn f s\nh' : EqOn g f s\u2081\nh\u2081 : s\u2081 \u2286 s\nx : \u03b1\nhx : x \u2208 s\u2081\n\u22a2 Tendsto g (\ud835\udcdd[s\u2081] x) (\ud835\udcdd (g x))"}, {"tactic": "have A := (h x (h\u2081 hx)).mono h\u2081", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.330239\n\u03b4 : Type ?u.330242\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf g : \u03b1 \u2192 \u03b2\ns s\u2081 : Set \u03b1\nh : ContinuousOn f s\nh' : EqOn g f s\u2081\nh\u2081 : s\u2081 \u2286 s\nx : \u03b1\nhx : x \u2208 s\u2081\n\u22a2 Tendsto g (\ud835\udcdd[s\u2081] x) (\ud835\udcdd (g x))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.330239\n\u03b4 : Type ?u.330242\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf g : \u03b1 \u2192 \u03b2\ns s\u2081 : Set \u03b1\nh : ContinuousOn f s\nh' : EqOn g f s\u2081\nh\u2081 : s\u2081 \u2286 s\nx : \u03b1\nhx : x \u2208 s\u2081\nA : ContinuousWithinAt f s\u2081 x\n\u22a2 Tendsto g (\ud835\udcdd[s\u2081] x) (\ud835\udcdd (g x))"}, {"tactic": "unfold ContinuousWithinAt at A", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.330239\n\u03b4 : Type ?u.330242\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf g : \u03b1 \u2192 \u03b2\ns s\u2081 : Set \u03b1\nh : ContinuousOn f s\nh' : EqOn g f s\u2081\nh\u2081 : s\u2081 \u2286 s\nx : \u03b1\nhx : x \u2208 s\u2081\nA : ContinuousWithinAt f s\u2081 x\n\u22a2 Tendsto g (\ud835\udcdd[s\u2081] x) (\ud835\udcdd (g x))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.330239\n\u03b4 : Type ?u.330242\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf g : \u03b1 \u2192 \u03b2\ns s\u2081 : Set \u03b1\nh : ContinuousOn f s\nh' : EqOn g f s\u2081\nh\u2081 : s\u2081 \u2286 s\nx : \u03b1\nhx : x \u2208 s\u2081\nA : Tendsto f (\ud835\udcdd[s\u2081] x) (\ud835\udcdd (f x))\n\u22a2 Tendsto g (\ud835\udcdd[s\u2081] x) (\ud835\udcdd (g x))"}, {"tactic": "rw [\u2190 h' hx] at A", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.330239\n\u03b4 : Type ?u.330242\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf g : \u03b1 \u2192 \u03b2\ns s\u2081 : Set \u03b1\nh : ContinuousOn f s\nh' : EqOn g f s\u2081\nh\u2081 : s\u2081 \u2286 s\nx : \u03b1\nhx : x \u2208 s\u2081\nA : Tendsto f (\ud835\udcdd[s\u2081] x) (\ud835\udcdd (f x))\n\u22a2 Tendsto g (\ud835\udcdd[s\u2081] x) (\ud835\udcdd (g x))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.330239\n\u03b4 : Type ?u.330242\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf g : \u03b1 \u2192 \u03b2\ns s\u2081 : Set \u03b1\nh : ContinuousOn f s\nh' : EqOn g f s\u2081\nh\u2081 : s\u2081 \u2286 s\nx : \u03b1\nhx : x \u2208 s\u2081\nA : Tendsto f (\ud835\udcdd[s\u2081] x) (\ud835\udcdd (g x))\n\u22a2 Tendsto g (\ud835\udcdd[s\u2081] x) (\ud835\udcdd (g x))"}, {"tactic": "exact A.congr' h'.eventuallyEq_nhdsWithin.symm", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.330239\n\u03b4 : Type ?u.330242\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf g : \u03b1 \u2192 \u03b2\ns s\u2081 : Set \u03b1\nh : ContinuousOn f s\nh' : EqOn g f s\u2081\nh\u2081 : s\u2081 \u2286 s\nx : \u03b1\nhx : x \u2208 s\u2081\nA : Tendsto f (\ud835\udcdd[s\u2081] x) (\ud835\udcdd (g x))\n\u22a2 Tendsto g (\ud835\udcdd[s\u2081] x) (\ud835\udcdd (g x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Hom/Lattice.lean", "full_name": "SupHom.coe_const", "start": [448, 1], "end": [449, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/Padics/PadicNumbers.lean", "full_name": "padicNormE.eq_of_norm_add_lt_left", "start": [955, 1], "end": [957, 80], "traced_tactics": [{"tactic": "rw [padicNormE.add_eq_max_of_ne hne]", "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nz1 z2 : \u211a_[p]\nh : \u2016z1 + z2\u2016 < \u2016z1\u2016\nhne : \u00ac\u2016z1\u2016 = \u2016z2\u2016\n\u22a2 \u2016z1 + z2\u2016 \u2265 \u2016z1\u2016", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nz1 z2 : \u211a_[p]\nh : \u2016z1 + z2\u2016 < \u2016z1\u2016\nhne : \u00ac\u2016z1\u2016 = \u2016z2\u2016\n\u22a2 max \u2016z1\u2016 \u2016z2\u2016 \u2265 \u2016z1\u2016"}, {"tactic": "apply le_max_left", "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nz1 z2 : \u211a_[p]\nh : \u2016z1 + z2\u2016 < \u2016z1\u2016\nhne : \u00ac\u2016z1\u2016 = \u2016z2\u2016\n\u22a2 max \u2016z1\u2016 \u2016z2\u2016 \u2265 \u2016z1\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicTopology/DoldKan/Normalized.lean", "full_name": "AlgebraicTopology.DoldKan.PInftyToNormalizedMooreComplex_comp_inclusionOfMooreComplexMap", "start": [78, 1], "end": [79, 93], "traced_tactics": [{"tactic": "aesop_cat", "state_before": "A : Type u_2\ninst\u271d\u00b9 : Category A\ninst\u271d : Abelian A\nX\u271d X : SimplicialObject A\n\u22a2 PInftyToNormalizedMooreComplex X \u226b inclusionOfMooreComplexMap X = PInfty", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean", "full_name": "Matrix.SpecialLinearGroup.coe_int_neg", "start": [278, 1], "end": [279, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Data/Char/Basic.lean", "full_name": "Char.isValidChar_zero", "start": [48, 1], "end": [49, 21], "traced_tactics": [{"tactic": "decide", "state_before": "\u22a2 0 < 55296", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.diff_union_self", "start": [2024, 1], "end": [2025, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/RingDivision.lean", "full_name": "Polynomial.mem_roots", "start": [580, 1], "end": [581, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Int/Bitwise.lean", "full_name": "Int.lor_bit", "start": [309, 1], "end": [310, 33], "traced_tactics": [{"tactic": "rw [\u2190 bitwise_or, bitwise_bit]", "state_before": "a : Bool\nm : \u2124\nb : Bool\nn : \u2124\n\u22a2 lor (bit a m) (bit b n) = bit (a || b) (lor m n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/OpenSubgroup.lean", "full_name": "OpenSubgroup.coe_top", "start": [149, 1], "end": [150, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/PiTensorProduct.lean", "full_name": "PiTensorProduct.lift_tprod", "start": [438, 1], "end": [439, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean", "full_name": "CircleDeg1Lift.dist_map_map_zero_lt", "start": [519, 1], "end": [521, 51], "traced_tactics": [{"tactic": "rw [dist_comm, Real.dist_eq, abs_lt, lt_sub_iff_add_lt', sub_lt_iff_lt_add', \u2190 sub_eq_add_neg]", "state_before": "f g : CircleDeg1Lift\n\u22a2 dist (\u2191f 0 + \u2191g 0) (\u2191f (\u2191g 0)) < 1", "state_after": "f g : CircleDeg1Lift\n\u22a2 \u2191f 0 + \u2191g 0 - 1 < \u2191f (\u2191g 0) \u2227 \u2191f (\u2191g 0) < \u2191f 0 + \u2191g 0 + 1"}, {"tactic": "exact \u27e8f.lt_map_map_zero g, f.map_map_zero_lt g\u27e9", "state_before": "f g : CircleDeg1Lift\n\u22a2 \u2191f 0 + \u2191g 0 - 1 < \u2191f (\u2191g 0) \u2227 \u2191f (\u2191g 0) < \u2191f 0 + \u2191g 0 + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Int/Parity.lean", "full_name": "Int.Odd.of_mul_right", "start": [153, 1], "end": [154, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Regular/SMul.lean", "full_name": "IsSMulRegular.not_zero", "start": [213, 1], "end": [214, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/ENNReal.lean", "full_name": "ENNReal.trichotomy\u2082", "start": [2282, 11], "end": [2295, 94], "traced_tactics": [{"tactic": "rcases eq_or_lt_of_le (bot_le : 0 \u2264 p) with ((rfl : 0 = p) | (hp : 0 < p))", "state_before": "\u03b1 : Type ?u.829236\n\u03b2 : Type ?u.829239\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\n\u22a2 p = 0 \u2227 q = 0 \u2228\n    p = 0 \u2227 q = \u22a4 \u2228\n      p = 0 \u2227 0 < ENNReal.toReal q \u2228\n        p = \u22a4 \u2227 q = \u22a4 \u2228\n          0 < ENNReal.toReal p \u2227 q = \u22a4 \u2228\n            0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q", "state_after": "case inl\n\u03b1 : Type ?u.829236\n\u03b2 : Type ?u.829239\na b c d : \u211d\u22650\u221e\nr p q\u271d : \u211d\u22650\nq : \u211d\u22650\u221e\nhpq : 0 \u2264 q\n\u22a2 0 = 0 \u2227 q = 0 \u2228\n    0 = 0 \u2227 q = \u22a4 \u2228\n      0 = 0 \u2227 0 < ENNReal.toReal q \u2228\n        0 = \u22a4 \u2227 q = \u22a4 \u2228\n          0 < ENNReal.toReal 0 \u2227 q = \u22a4 \u2228\n            0 < ENNReal.toReal 0 \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal 0 \u2264 ENNReal.toReal q\n\ncase inr\n\u03b1 : Type ?u.829236\n\u03b2 : Type ?u.829239\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\n\u22a2 p = 0 \u2227 q = 0 \u2228\n    p = 0 \u2227 q = \u22a4 \u2228\n      p = 0 \u2227 0 < ENNReal.toReal q \u2228\n        p = \u22a4 \u2227 q = \u22a4 \u2228\n          0 < ENNReal.toReal p \u2227 q = \u22a4 \u2228\n            0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q"}, {"tactic": "rcases eq_or_lt_of_le (le_top : q \u2264 \u221e) with (rfl | hq)", "state_before": "case inr\n\u03b1 : Type ?u.829236\n\u03b2 : Type ?u.829239\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\n\u22a2 p = 0 \u2227 q = 0 \u2228\n    p = 0 \u2227 q = \u22a4 \u2228\n      p = 0 \u2227 0 < ENNReal.toReal q \u2228\n        p = \u22a4 \u2227 q = \u22a4 \u2228\n          0 < ENNReal.toReal p \u2227 q = \u22a4 \u2228\n            0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q", "state_after": "case inr.inl\n\u03b1 : Type ?u.829236\n\u03b2 : Type ?u.829239\na b c d : \u211d\u22650\u221e\nr p\u271d q : \u211d\u22650\np : \u211d\u22650\u221e\nhp : 0 < p\nhpq : p \u2264 \u22a4\n\u22a2 p = 0 \u2227 \u22a4 = 0 \u2228\n    p = 0 \u2227 \u22a4 = \u22a4 \u2228\n      p = 0 \u2227 0 < ENNReal.toReal \u22a4 \u2228\n        p = \u22a4 \u2227 \u22a4 = \u22a4 \u2228\n          0 < ENNReal.toReal p \u2227 \u22a4 = \u22a4 \u2228\n            0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal \u22a4 \u2227 ENNReal.toReal p \u2264 ENNReal.toReal \u22a4\n\ncase inr.inr\n\u03b1 : Type ?u.829236\n\u03b2 : Type ?u.829239\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\n\u22a2 p = 0 \u2227 q = 0 \u2228\n    p = 0 \u2227 q = \u22a4 \u2228\n      p = 0 \u2227 0 < ENNReal.toReal q \u2228\n        p = \u22a4 \u2227 q = \u22a4 \u2228\n          0 < ENNReal.toReal p \u2227 q = \u22a4 \u2228\n            0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q"}, {"tactic": "repeat' right", "state_before": "case inr.inr\n\u03b1 : Type ?u.829236\n\u03b2 : Type ?u.829239\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\n\u22a2 p = 0 \u2227 q = 0 \u2228\n    p = 0 \u2227 q = \u22a4 \u2228\n      p = 0 \u2227 0 < ENNReal.toReal q \u2228\n        p = \u22a4 \u2227 q = \u22a4 \u2228\n          0 < ENNReal.toReal p \u2227 q = \u22a4 \u2228\n            0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q", "state_after": "case inr.inr.h.h.h.h.h\n\u03b1 : Type ?u.829236\n\u03b2 : Type ?u.829239\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\n\u22a2 0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q"}, {"tactic": "have hq' : 0 < q := lt_of_lt_of_le hp hpq", "state_before": "case inr.inr.h.h.h.h.h\n\u03b1 : Type ?u.829236\n\u03b2 : Type ?u.829239\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\n\u22a2 0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q", "state_after": "case inr.inr.h.h.h.h.h\n\u03b1 : Type ?u.829236\n\u03b2 : Type ?u.829239\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\nhq' : 0 < q\n\u22a2 0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q"}, {"tactic": "have hp' : p < \u221e := lt_of_le_of_lt hpq hq", "state_before": "case inr.inr.h.h.h.h.h\n\u03b1 : Type ?u.829236\n\u03b2 : Type ?u.829239\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\nhq' : 0 < q\n\u22a2 0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q", "state_after": "case inr.inr.h.h.h.h.h\n\u03b1 : Type ?u.829236\n\u03b2 : Type ?u.829239\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\nhq' : 0 < q\nhp' : p < \u22a4\n\u22a2 0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q"}, {"tactic": "simp [ENNReal.toReal_le_toReal hp'.ne hq.ne, ENNReal.toReal_pos_iff, hpq, hp, hp', hq', hq]", "state_before": "case inr.inr.h.h.h.h.h\n\u03b1 : Type ?u.829236\n\u03b2 : Type ?u.829239\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\nhq' : 0 < q\nhp' : p < \u22a4\n\u22a2 0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q", "state_after": "no goals"}, {"tactic": "simpa using q.trichotomy", "state_before": "case inl\n\u03b1 : Type ?u.829236\n\u03b2 : Type ?u.829239\na b c d : \u211d\u22650\u221e\nr p q\u271d : \u211d\u22650\nq : \u211d\u22650\u221e\nhpq : 0 \u2264 q\n\u22a2 0 = 0 \u2227 q = 0 \u2228\n    0 = 0 \u2227 q = \u22a4 \u2228\n      0 = 0 \u2227 0 < ENNReal.toReal q \u2228\n        0 = \u22a4 \u2227 q = \u22a4 \u2228\n          0 < ENNReal.toReal 0 \u2227 q = \u22a4 \u2228\n            0 < ENNReal.toReal 0 \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal 0 \u2264 ENNReal.toReal q", "state_after": "no goals"}, {"tactic": "simpa using p.trichotomy", "state_before": "case inr.inl\n\u03b1 : Type ?u.829236\n\u03b2 : Type ?u.829239\na b c d : \u211d\u22650\u221e\nr p\u271d q : \u211d\u22650\np : \u211d\u22650\u221e\nhp : 0 < p\nhpq : p \u2264 \u22a4\n\u22a2 p = 0 \u2227 \u22a4 = 0 \u2228\n    p = 0 \u2227 \u22a4 = \u22a4 \u2228\n      p = 0 \u2227 0 < ENNReal.toReal \u22a4 \u2228\n        p = \u22a4 \u2227 \u22a4 = \u22a4 \u2228\n          0 < ENNReal.toReal p \u2227 \u22a4 = \u22a4 \u2228\n            0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal \u22a4 \u2227 ENNReal.toReal p \u2264 ENNReal.toReal \u22a4", "state_after": "no goals"}, {"tactic": "right", "state_before": "case inr.inr.h.h.h.h.h\n\u03b1 : Type ?u.829236\n\u03b2 : Type ?u.829239\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\n\u22a2 0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q", "state_after": "case inr.inr.h.h.h.h.h\n\u03b1 : Type ?u.829236\n\u03b2 : Type ?u.829239\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\n\u22a2 0 < ENNReal.toReal p \u2227 0 < ENNReal.toReal q \u2227 ENNReal.toReal p \u2264 ENNReal.toReal q"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/Operations.lean", "full_name": "Ideal.map_sSup", "start": [1484, 1], "end": [1485, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/Padics/RingHoms.lean", "full_name": "PadicInt.zmodRepr_spec", "start": [203, 1], "end": [204, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/SymmDiff.lean", "full_name": "symmDiff_sdiff_right", "start": [429, 1], "end": [429, 95], "traced_tactics": [{"tactic": "rw [symmDiff_comm, symmDiff_sdiff_left]", "state_before": "\u03b9 : Type ?u.62639\n\u03b1 : Type u_1\n\u03b2 : Type ?u.62645\n\u03c0 : \u03b9 \u2192 Type ?u.62650\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\na b c d : \u03b1\n\u22a2 a \u2206 b \\ b = a \\ b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "full_name": "SimpleGraph.Walk.transfer_transfer", "start": [1738, 1], "end": [1744, 13], "traced_tactics": [{"tactic": "induction p with\n| nil => simp\n| cons _ _ ih =>\n  simp only [Walk.transfer, cons.injEq, heq_eq_eq, true_and]\n  apply ih", "state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : Walk G u v\nH : SimpleGraph V\nhp : \u2200 (e : Sym2 V), e \u2208 edges p \u2192 e \u2208 edgeSet H\nK : SimpleGraph V\nhp' : \u2200 (e : Sym2 V), e \u2208 edges (Walk.transfer p H hp) \u2192 e \u2208 edgeSet K\n\u22a2 Walk.transfer (Walk.transfer p H hp) K hp' = Walk.transfer p K (_ : \u2200 (e : Sym2 V), e \u2208 edges p \u2192 e \u2208 edgeSet K)", "state_after": "no goals"}, {"tactic": "simp", "state_before": "case nil\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : Walk G u v\nH K : SimpleGraph V\nu\u271d : V\nhp : \u2200 (e : Sym2 V), e \u2208 edges nil \u2192 e \u2208 edgeSet H\nhp' : \u2200 (e : Sym2 V), e \u2208 edges (Walk.transfer nil H hp) \u2192 e \u2208 edgeSet K\n\u22a2 Walk.transfer (Walk.transfer nil H hp) K hp' = Walk.transfer nil K (_ : \u2200 (e : Sym2 V), e \u2208 edges nil \u2192 e \u2208 edgeSet K)", "state_after": "no goals"}, {"tactic": "simp only [Walk.transfer, cons.injEq, heq_eq_eq, true_and]", "state_before": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : Walk G u v\nH K : SimpleGraph V\nu\u271d v\u271d w\u271d : V\nh\u271d : Adj G u\u271d v\u271d\np\u271d : Walk G v\u271d w\u271d\nih :\n  \u2200 (hp : \u2200 (e : Sym2 V), e \u2208 edges p\u271d \u2192 e \u2208 edgeSet H)\n    (hp' : \u2200 (e : Sym2 V), e \u2208 edges (Walk.transfer p\u271d H hp) \u2192 e \u2208 edgeSet K),\n    Walk.transfer (Walk.transfer p\u271d H hp) K hp' = Walk.transfer p\u271d K (_ : \u2200 (e : Sym2 V), e \u2208 edges p\u271d \u2192 e \u2208 edgeSet K)\nhp : \u2200 (e : Sym2 V), e \u2208 edges (cons h\u271d p\u271d) \u2192 e \u2208 edgeSet H\nhp' : \u2200 (e : Sym2 V), e \u2208 edges (Walk.transfer (cons h\u271d p\u271d) H hp) \u2192 e \u2208 edgeSet K\n\u22a2 Walk.transfer (Walk.transfer (cons h\u271d p\u271d) H hp) K hp' =\n    Walk.transfer (cons h\u271d p\u271d) K (_ : \u2200 (e : Sym2 V), e \u2208 edges (cons h\u271d p\u271d) \u2192 e \u2208 edgeSet K)", "state_after": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : Walk G u v\nH K : SimpleGraph V\nu\u271d v\u271d w\u271d : V\nh\u271d : Adj G u\u271d v\u271d\np\u271d : Walk G v\u271d w\u271d\nih :\n  \u2200 (hp : \u2200 (e : Sym2 V), e \u2208 edges p\u271d \u2192 e \u2208 edgeSet H)\n    (hp' : \u2200 (e : Sym2 V), e \u2208 edges (Walk.transfer p\u271d H hp) \u2192 e \u2208 edgeSet K),\n    Walk.transfer (Walk.transfer p\u271d H hp) K hp' = Walk.transfer p\u271d K (_ : \u2200 (e : Sym2 V), e \u2208 edges p\u271d \u2192 e \u2208 edgeSet K)\nhp : \u2200 (e : Sym2 V), e \u2208 edges (cons h\u271d p\u271d) \u2192 e \u2208 edgeSet H\nhp' : \u2200 (e : Sym2 V), e \u2208 edges (Walk.transfer (cons h\u271d p\u271d) H hp) \u2192 e \u2208 edgeSet K\n\u22a2 Walk.transfer (Walk.transfer p\u271d H (_ : \u2200 (e : Sym2 V), e \u2208 edges p\u271d \u2192 e \u2208 edgeSet H)) K\n      (_ :\n        \u2200 (e : Sym2 V),\n          e \u2208 edges (Walk.transfer p\u271d H (_ : \u2200 (e : Sym2 V), e \u2208 edges p\u271d \u2192 e \u2208 edgeSet H)) \u2192 e \u2208 edgeSet K) =\n    Walk.transfer p\u271d K (_ : \u2200 (e : Sym2 V), e \u2208 edges p\u271d \u2192 e \u2208 edgeSet K)"}, {"tactic": "apply ih", "state_before": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\np : Walk G u v\nH K : SimpleGraph V\nu\u271d v\u271d w\u271d : V\nh\u271d : Adj G u\u271d v\u271d\np\u271d : Walk G v\u271d w\u271d\nih :\n  \u2200 (hp : \u2200 (e : Sym2 V), e \u2208 edges p\u271d \u2192 e \u2208 edgeSet H)\n    (hp' : \u2200 (e : Sym2 V), e \u2208 edges (Walk.transfer p\u271d H hp) \u2192 e \u2208 edgeSet K),\n    Walk.transfer (Walk.transfer p\u271d H hp) K hp' = Walk.transfer p\u271d K (_ : \u2200 (e : Sym2 V), e \u2208 edges p\u271d \u2192 e \u2208 edgeSet K)\nhp : \u2200 (e : Sym2 V), e \u2208 edges (cons h\u271d p\u271d) \u2192 e \u2208 edgeSet H\nhp' : \u2200 (e : Sym2 V), e \u2208 edges (Walk.transfer (cons h\u271d p\u271d) H hp) \u2192 e \u2208 edgeSet K\n\u22a2 Walk.transfer (Walk.transfer p\u271d H (_ : \u2200 (e : Sym2 V), e \u2208 edges p\u271d \u2192 e \u2208 edgeSet H)) K\n      (_ :\n        \u2200 (e : Sym2 V),\n          e \u2208 edges (Walk.transfer p\u271d H (_ : \u2200 (e : Sym2 V), e \u2208 edges p\u271d \u2192 e \u2208 edgeSet H)) \u2192 e \u2208 edgeSet K) =\n    Walk.transfer p\u271d K (_ : \u2200 (e : Sym2 V), e \u2208 edges p\u271d \u2192 e \u2208 edgeSet K)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Transfer.lean", "full_name": "Subgroup.leftTransversals.diff_inv", "start": [77, 1], 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?u.290935\nR : Type u_3\nS : Type ?u.290941\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : Semiring S\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : AddCommMonoid N\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : AddCommMonoid P\ninst\u271d\u2075 : Module R P\n\u03b1' : Type ?u.291033\nM' : Type ?u.291036\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M'\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M\nv : \u03b1 \u2192 M\nv' : \u03b1' \u2192 M'\nc : R\na : \u03b1\n\u22a2 \u2191(Finsupp.total \u03b1 M R v) (single a c) = c \u2022 v a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Monotone/Basic.lean", "full_name": "strictAntiOn_toDual_comp_iff", "start": [216, 1], "end": [217, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Int/Lemmas.lean", "full_name": "Int.mul_nonneg_of_nonpos_of_nonpos", "start": [1210, 11], "end": [1213, 29], "traced_tactics": [{"tactic": "have : 0 * b \u2264 a * b := Int.mul_le_mul_of_nonpos_right ha hb", "state_before": "a b : Int\nha : a \u2264 0\nhb : b \u2264 0\n\u22a2 0 \u2264 a * b", "state_after": "a b : Int\nha : a \u2264 0\nhb : b \u2264 0\nthis : 0 * b \u2264 a * b\n\u22a2 0 \u2264 a * b"}, {"tactic": "rwa [Int.zero_mul] at this", "state_before": "a b : Int\nha : a \u2264 0\nhb : b \u2264 0\nthis : 0 * b \u2264 a * b\n\u22a2 0 \u2264 a * b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/FreeAbelianGroup.lean", "full_name": "FreeAbelianGroup.of_one", "start": [555, 1], "end": [556, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/Basic.lean", "full_name": "Real.iSup_nonneg", "start": [867, 11], "end": [868, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Complex/Basic.lean", "full_name": "Complex.int_cast_abs", "start": [1113, 1], "end": [1114, 37], "traced_tactics": [{"tactic": "rw [\u2190 ofReal_int_cast, abs_ofReal]", "state_before": "n : \u2124\n\u22a2 Abs.abs \u2191n = \u2191abs \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/IsROrC/Basic.lean", "full_name": "IsROrC.div_im", "start": [569, 1], "end": [571, 18], "traced_tactics": [{"tactic": "simp only [div_eq_mul_inv, mul_assoc, sub_eq_add_neg, add_comm, neg_mul, mul_neg, map_neg,\n  isROrC_simps]", "state_before": "K : Type u_1\nE : Type ?u.5746239\ninst\u271d : IsROrC K\nz w : K\n\u22a2 \u2191im (z / w) = \u2191im z * \u2191re w / \u2191normSq w - \u2191re z * \u2191im w / \u2191normSq w", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Prod.lean", "full_name": "HasFDerivAt.fst", "start": [161, 18], "end": [163, 8], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Normed/Group/Seminorm.lean", "full_name": "AddGroupSeminorm.smul_apply", "start": [487, 1], "end": [488, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/UpperLower/Basic.lean", "full_name": "LowerSet.coe_eq_empty", 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X \u27f6 Z\nc : PushoutCocone f g\n\u22a2 PullbackCone.fst (op c) = (inl c).op", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "topologicalGroup_iInf", "start": [1799, 1], "end": [1802, 60], "traced_tactics": [{"tactic": "rw [\u2190 sInf_range]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nG : Type w\nH : Type x\n\u03b9 : Sort u_1\ninst\u271d : Group G\nts' : \u03b9 \u2192 TopologicalSpace G\nh' : \u2200 (i : \u03b9), TopologicalGroup G\n\u22a2 TopologicalGroup G", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nG : Type w\nH : Type x\n\u03b9 : Sort u_1\ninst\u271d : Group G\nts' : \u03b9 \u2192 TopologicalSpace G\nh' : \u2200 (i : \u03b9), TopologicalGroup G\n\u22a2 TopologicalGroup G"}, {"tactic": "exact topologicalGroup_sInf (Set.forall_range_iff.mpr h')", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nG : Type w\nH : Type x\n\u03b9 : Sort u_1\ninst\u271d : Group G\nts' : \u03b9 \u2192 TopologicalSpace G\nh' : \u2200 (i : \u03b9), TopologicalGroup G\n\u22a2 TopologicalGroup G", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "full_name": "map_extChartAt_symm_nhdsWithin'", "start": [1181, 1], "end": [1184, 67], "traced_tactics": [{"tactic": "rwa [\u2190 extChartAt_source I]", "state_before": "\ud835\udd5c : Type u_3\nE : Type u_2\nM : Type u_1\nH : Type u_4\nE' : Type ?u.215272\nM' : Type ?u.215275\nH' : Type ?u.215278\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners \ud835\udd5c E H\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\nI' : ModelWithCorners \ud835\udd5c E' H'\nx : M\ns t : Set M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : ChartedSpace H' M'\ny : M\nhy : y \u2208 (extChartAt I x).source\n\u22a2 y \u2208 (chartAt H x).toLocalEquiv.source", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/LocalExtr.lean", "full_name": "IsLocalExtr.elim", "start": [86, 1], "end": [88, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.iIndepFun_iff_measure_inter_preimage_eq_mul", "start": [666, 1], "end": [694, 22], "traced_tactics": [{"tactic": "refine' \u27e8fun h S sets h_meas => h _ fun i hi_mem => \u27e8sets i, h_meas i hi_mem, rfl\u27e9, _\u27e9", "state_before": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\n\u22a2 iIndepFun m f \u2194\n    \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n      (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n        \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)", "state_after": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\n\u22a2 (\u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n      (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n        \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)) \u2192\n    iIndepFun m f"}, {"tactic": "intro h S sets\u03a9 h_meas", "state_before": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\n\u22a2 (\u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n      (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n        \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)) \u2192\n    iIndepFun m f", "state_after": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\n\u22a2 \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), sets\u03a9 i) = \u220f i in S, \u2191\u2191\u03bc (sets\u03a9 i)"}, {"tactic": "let sets\u03b2 : \u2200 i : \u03b9, Set (\u03b2 i) := fun i =>\n  dite (i \u2208 S) (fun hi_mem => (h_meas i hi_mem).choose) fun _ => Set.univ", "state_before": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\n\u22a2 \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), sets\u03a9 i) = \u220f i in S, \u2191\u2191\u03bc (sets\u03a9 i)", "state_after": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\n\u22a2 \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), sets\u03a9 i) = \u220f i in S, \u2191\u2191\u03bc (sets\u03a9 i)"}, {"tactic": "have h_meas\u03b2 : \u2200 i \u2208 S, MeasurableSet[m i] (sets\u03b2 i) := by\n  intro i hi_mem\n  simp_rw [dif_pos hi_mem]\n  exact (h_meas i hi_mem).choose_spec.1", "state_before": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\n\u22a2 \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), sets\u03a9 i) = \u220f i in S, \u2191\u2191\u03bc (sets\u03a9 i)", "state_after": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\n\u22a2 \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), sets\u03a9 i) = \u220f i in S, \u2191\u2191\u03bc (sets\u03a9 i)"}, {"tactic": "have h_preim : \u2200 i \u2208 S, sets\u03a9 i = f i \u207b\u00b9' sets\u03b2 i := by\n  intro i hi_mem\n  simp_rw [dif_pos hi_mem]\n  exact (h_meas i hi_mem).choose_spec.2.symm", "state_before": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\n\u22a2 \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), sets\u03a9 i) = \u220f i in S, \u2191\u2191\u03bc (sets\u03a9 i)", "state_after": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\nh_preim : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i = f i \u207b\u00b9' sets\u03b2 i\n\u22a2 \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), sets\u03a9 i) = \u220f i in S, \u2191\u2191\u03bc (sets\u03a9 i)"}, {"tactic": "have h_right_eq : (\u220f i in S, \u03bc (sets\u03a9 i)) = \u220f i in S, \u03bc (f i \u207b\u00b9' sets\u03b2 i) := by\n  refine' Finset.prod_congr rfl fun i hi_mem => _\n  rw [h_preim i hi_mem]", "state_before": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\nh_preim : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i = f i \u207b\u00b9' sets\u03b2 i\nh_left_eq : \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), sets\u03a9 i) = \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets\u03b2 i)\n\u22a2 \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), sets\u03a9 i) = \u220f i in S, \u2191\u2191\u03bc (sets\u03a9 i)", "state_after": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\nh_preim : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i = f i \u207b\u00b9' sets\u03b2 i\nh_left_eq : \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), sets\u03a9 i) = \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets\u03b2 i)\nh_right_eq : \u220f i in S, \u2191\u2191\u03bc (sets\u03a9 i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets\u03b2 i)\n\u22a2 \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), sets\u03a9 i) = \u220f i in S, \u2191\u2191\u03bc (sets\u03a9 i)"}, {"tactic": "rw [h_left_eq, h_right_eq]", "state_before": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\nh_preim : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i = f i \u207b\u00b9' sets\u03b2 i\nh_left_eq : \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), sets\u03a9 i) = \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets\u03b2 i)\nh_right_eq : \u220f i in S, \u2191\u2191\u03bc (sets\u03a9 i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets\u03b2 i)\n\u22a2 \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), sets\u03a9 i) = \u220f i in S, \u2191\u2191\u03bc (sets\u03a9 i)", "state_after": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\nh_preim : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i = f i \u207b\u00b9' sets\u03b2 i\nh_left_eq : \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), sets\u03a9 i) = \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets\u03b2 i)\nh_right_eq : \u220f i in S, \u2191\u2191\u03bc (sets\u03a9 i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets\u03b2 i)\n\u22a2 \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets\u03b2 i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets\u03b2 i)"}, {"tactic": "exact h S h_meas\u03b2", "state_before": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\nh_preim : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i = f i \u207b\u00b9' sets\u03b2 i\nh_left_eq : \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), sets\u03a9 i) = \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets\u03b2 i)\nh_right_eq : \u220f i in S, \u2191\u2191\u03bc (sets\u03a9 i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets\u03b2 i)\n\u22a2 \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets\u03b2 i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets\u03b2 i)", "state_after": "no goals"}, {"tactic": "intro i hi_mem", "state_before": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\n\u22a2 \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)", "state_after": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\ni : \u03b9\nhi_mem : i \u2208 S\n\u22a2 MeasurableSet (sets\u03b2 i)"}, {"tactic": "simp_rw [dif_pos hi_mem]", "state_before": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\ni : \u03b9\nhi_mem : i \u2208 S\n\u22a2 MeasurableSet (sets\u03b2 i)", "state_after": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\ni : \u03b9\nhi_mem : i \u2208 S\n\u22a2 MeasurableSet (Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i))"}, {"tactic": "exact (h_meas i hi_mem).choose_spec.1", "state_before": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\ni : \u03b9\nhi_mem : i \u2208 S\n\u22a2 MeasurableSet (Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i))", "state_after": "no goals"}, {"tactic": "intro i hi_mem", "state_before": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\n\u22a2 \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i = f i \u207b\u00b9' sets\u03b2 i", "state_after": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\ni : \u03b9\nhi_mem : i \u2208 S\n\u22a2 sets\u03a9 i = f i \u207b\u00b9' sets\u03b2 i"}, {"tactic": "simp_rw [dif_pos hi_mem]", "state_before": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\ni : \u03b9\nhi_mem : i \u2208 S\n\u22a2 sets\u03a9 i = f i \u207b\u00b9' sets\u03b2 i", "state_after": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\ni : \u03b9\nhi_mem : i \u2208 S\n\u22a2 sets\u03a9 i = f i \u207b\u00b9' Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i)"}, {"tactic": "exact (h_meas i hi_mem).choose_spec.2.symm", "state_before": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\ni : \u03b9\nhi_mem : i \u2208 S\n\u22a2 sets\u03a9 i = f i \u207b\u00b9' Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i)", "state_after": "no goals"}, {"tactic": "congr with x", "state_before": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\nh_preim : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i = f i \u207b\u00b9' sets\u03b2 i\n\u22a2 \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), sets\u03a9 i) = \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets\u03b2 i)", "state_after": "case e_a.h\n\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\nh_preim : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i = f i \u207b\u00b9' sets\u03b2 i\nx : \u03a9\n\u22a2 (x \u2208 \u22c2 (i : \u03b9) (_ : i \u2208 S), sets\u03a9 i) \u2194 x \u2208 \u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets\u03b2 i"}, {"tactic": "simp_rw [Set.mem_iInter]", "state_before": "case e_a.h\n\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\nh_preim : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i = f i \u207b\u00b9' sets\u03b2 i\nx : \u03a9\n\u22a2 (x \u2208 \u22c2 (i : \u03b9) (_ : i \u2208 S), sets\u03a9 i) \u2194 x \u2208 \u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets\u03b2 i", "state_after": "case e_a.h\n\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\nh_preim : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i = f i \u207b\u00b9' sets\u03b2 i\nx : \u03a9\n\u22a2 (\u2200 (i : \u03b9), i \u2208 S \u2192 x \u2208 sets\u03a9 i) \u2194\n    \u2200 (i : \u03b9),\n      i \u2208 S \u2192\n        x \u2208 f i \u207b\u00b9' if h : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ"}, {"tactic": "constructor <;> intro h i hi_mem <;> specialize h i hi_mem", "state_before": "case e_a.h\n\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\nh_preim : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i = f i \u207b\u00b9' sets\u03b2 i\nx : \u03a9\n\u22a2 (\u2200 (i : \u03b9), i \u2208 S \u2192 x \u2208 sets\u03a9 i) \u2194\n    \u2200 (i : \u03b9),\n      i \u2208 S \u2192\n        x \u2208 f i \u207b\u00b9' if h : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ", "state_after": "case e_a.h.mp\n\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh\u271d :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\nh_preim : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i = f i \u207b\u00b9' sets\u03b2 i\nx : \u03a9\ni : \u03b9\nhi_mem : i \u2208 S\nh : x \u2208 sets\u03a9 i\n\u22a2 x \u2208 f i \u207b\u00b9' if h : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\n\ncase e_a.h.mpr\n\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh\u271d :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\nh_preim : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i = f i \u207b\u00b9' sets\u03b2 i\nx : \u03a9\ni : \u03b9\nhi_mem : i \u2208 S\nh : x \u2208 f i \u207b\u00b9' if h : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\n\u22a2 x \u2208 sets\u03a9 i"}, {"tactic": "rwa [h_preim i hi_mem] at h", "state_before": "case e_a.h.mp\n\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh\u271d :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\nh_preim : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i = f i \u207b\u00b9' sets\u03b2 i\nx : \u03a9\ni : \u03b9\nhi_mem : i \u2208 S\nh : x \u2208 sets\u03a9 i\n\u22a2 x \u2208 f i \u207b\u00b9' if h : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ", "state_after": "no goals"}, {"tactic": "rwa [h_preim i hi_mem]", "state_before": "case e_a.h.mpr\n\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh\u271d :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\nh_preim : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i = f i \u207b\u00b9' sets\u03b2 i\nx : \u03a9\ni : \u03b9\nhi_mem : i \u2208 S\nh : x \u2208 f i \u207b\u00b9' if h : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\n\u22a2 x \u2208 sets\u03a9 i", "state_after": "no goals"}, {"tactic": "refine' Finset.prod_congr rfl fun i hi_mem => _", "state_before": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\nh_preim : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i = f i \u207b\u00b9' sets\u03b2 i\nh_left_eq : \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), sets\u03a9 i) = \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets\u03b2 i)\n\u22a2 \u220f i in S, \u2191\u2191\u03bc (sets\u03a9 i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets\u03b2 i)", "state_after": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\nh_preim : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i = f i \u207b\u00b9' sets\u03b2 i\nh_left_eq : \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), sets\u03a9 i) = \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets\u03b2 i)\ni : \u03b9\nhi_mem : i \u2208 S\n\u22a2 \u2191\u2191\u03bc (sets\u03a9 i) = \u2191\u2191\u03bc (f i \u207b\u00b9' sets\u03b2 i)"}, {"tactic": "rw [h_preim i hi_mem]", "state_before": "\u03a9 : Type u_3\n\u03b9\u271d : Type ?u.3089001\n\u03b2\u271d : Type ?u.3089004\n\u03b2' : Type ?u.3089007\n\u03b3 : Type ?u.3089010\n\u03b3' : Type ?u.3089013\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf\u271d : \u03a9 \u2192 \u03b2\u271d\ng : \u03a9 \u2192 \u03b2'\n\u03b9 : Type u_1\n\u03b2 : \u03b9 \u2192 Type u_2\nm : (x : \u03b9) \u2192 MeasurableSpace (\u03b2 x)\nf : (i : \u03b9) \u2192 \u03a9 \u2192 \u03b2 i\nh :\n  \u2200 (S : Finset \u03b9) {sets : (i : \u03b9) \u2192 Set (\u03b2 i)},\n    (\u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets i)) \u2192\n      \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets i) = \u220f i in S, \u2191\u2191\u03bc (f i \u207b\u00b9' sets i)\nS : Finset \u03b9\nsets\u03a9 : \u03b9 \u2192 Set \u03a9\nh_meas : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i\nsets\u03b2 : (i : \u03b9) \u2192 Set (\u03b2 i) :=\n  fun i => if hi_mem : i \u2208 S then Exists.choose (_ : sets\u03a9 i \u2208 (fun x => {s | MeasurableSet s}) i) else Set.univ\nh_meas\u03b2 : \u2200 (i : \u03b9), i \u2208 S \u2192 MeasurableSet (sets\u03b2 i)\nh_preim : \u2200 (i : \u03b9), i \u2208 S \u2192 sets\u03a9 i = f i \u207b\u00b9' sets\u03b2 i\nh_left_eq : \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), sets\u03a9 i) = \u2191\u2191\u03bc (\u22c2 (i : \u03b9) (_ : i \u2208 S), f i \u207b\u00b9' sets\u03b2 i)\ni : \u03b9\nhi_mem : i \u2208 S\n\u22a2 \u2191\u2191\u03bc (sets\u03a9 i) = \u2191\u2191\u03bc (f i \u207b\u00b9' sets\u03b2 i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Fold.lean", "full_name": "Multiset.fold_cons'_left", "start": [75, 1], "end": [76, 33], "traced_tactics": [{"tactic": "rw [fold_cons'_right, hc.comm]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.7230\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nhc : IsCommutative \u03b1 op\nha : IsAssociative \u03b1 op\nb a : \u03b1\ns : Multiset \u03b1\n\u22a2 fold op b (a ::\u2098 s) = fold op (op a b) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/ToIntervalMod.lean", "full_name": "toIocMod_add_left'", "start": [508, 1], "end": [509, 47], "traced_tactics": [{"tactic": "rw [add_comm, toIocMod_add_right', add_comm]", "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nh\u03b1 : Archimedean \u03b1\np : \u03b1\nhp : 0 < p\na\u271d b\u271d c : \u03b1\nn : \u2124\na b : \u03b1\n\u22a2 toIocMod hp (p + a) b = p + toIocMod hp a b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/QuasiSeparated.lean", "full_name": "isQuasiSeparated_univ_iff", "start": [58, 1], "end": [61, 26], "traced_tactics": [{"tactic": "rw [QuasiSeparatedSpace_iff]", "state_before": "\u03b1\u271d : Type ?u.585\n\u03b2 : Type ?u.588\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 IsQuasiSeparated Set.univ \u2194 QuasiSeparatedSpace \u03b1", "state_after": "\u03b1\u271d : Type ?u.585\n\u03b2 : Type ?u.588\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 IsQuasiSeparated Set.univ \u2194 \u2200 (U V : Set \u03b1), IsOpen U \u2192 IsCompact U \u2192 IsOpen V \u2192 IsCompact V \u2192 IsCompact (U \u2229 V)"}, {"tactic": "simp [IsQuasiSeparated]", "state_before": "\u03b1\u271d : Type ?u.585\n\u03b2 : Type ?u.588\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1\u271d \u2192 \u03b2\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u22a2 IsQuasiSeparated Set.univ \u2194 \u2200 (U V : Set \u03b1), IsOpen U \u2192 IsCompact U \u2192 IsOpen V \u2192 IsCompact V \u2192 IsCompact (U \u2229 V)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "MeasureTheory.Measurable.comp_nullMeasurable", "start": [430, 1], "end": [432, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.castAdd_castLT", "start": [1151, 1], "end": [1154, 7], "traced_tactics": [{"tactic": "ext", "state_before": "n m\u271d m : \u2115\ni : Fin (n + m)\nhi : \u2191i < n\n\u22a2 \u2191(castAdd m) (castLT i hi) = i", "state_after": "case h\nn m\u271d m : \u2115\ni : Fin (n + m)\nhi : \u2191i < n\n\u22a2 \u2191(\u2191(castAdd m) (castLT i hi)) = \u2191i"}, {"tactic": "simp", "state_before": "case h\nn m\u271d m : \u2115\ni : Fin (n + m)\nhi : \u2191i < n\n\u22a2 \u2191(\u2191(castAdd m) (castLT i hi)) = \u2191i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finsupp/Basic.lean", "full_name": "Finsupp.snd_sumFinsuppAddEquivProdFinsupp", "start": [1388, 1], "end": [1390, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Rotate.lean", "full_name": "List.isRotated_comm", "start": [454, 1], "end": [455, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Composition.lean", "full_name": "Composition.eq_single_iff_length", "start": [598, 1], "end": [609, 34], "traced_tactics": [{"tactic": "constructor", "state_before": "n\u271d : \u2115\nc\u271d : Composition n\u271d\nn : \u2115\nh : 0 < n\nc : Composition n\n\u22a2 c = single n h \u2194 length c = 1", "state_after": "case mp\nn\u271d : \u2115\nc\u271d : Composition n\u271d\nn : \u2115\nh : 0 < n\nc : Composition n\n\u22a2 c = single n h \u2192 length c = 1\n\ncase mpr\nn\u271d : \u2115\nc\u271d : Composition n\u271d\nn : \u2115\nh : 0 < n\nc : Composition n\n\u22a2 length c = 1 \u2192 c = single n h"}, {"tactic": "intro H", "state_before": "case mp\nn\u271d : \u2115\nc\u271d : Composition n\u271d\nn : \u2115\nh : 0 < n\nc : Composition n\n\u22a2 c = single n h \u2192 length c = 1", "state_after": "case mp\nn\u271d : \u2115\nc\u271d : Composition n\u271d\nn : \u2115\nh : 0 < n\nc : Composition n\nH : c = single n h\n\u22a2 length c = 1"}, {"tactic": "rw [H]", "state_before": "case mp\nn\u271d : \u2115\nc\u271d : Composition n\u271d\nn : \u2115\nh : 0 < n\nc : Composition n\nH : c = single n h\n\u22a2 length c = 1", "state_after": "case mp\nn\u271d : \u2115\nc\u271d : Composition n\u271d\nn : \u2115\nh : 0 < n\nc : Composition n\nH : c = single n h\n\u22a2 length (single n h) = 1"}, {"tactic": "exact single_length h", "state_before": "case mp\nn\u271d : \u2115\nc\u271d : Composition n\u271d\nn : \u2115\nh : 0 < n\nc : Composition n\nH : c = single n h\n\u22a2 length (single n h) = 1", "state_after": "no goals"}, {"tactic": "intro H", "state_before": "case mpr\nn\u271d : \u2115\nc\u271d : Composition n\u271d\nn : \u2115\nh : 0 < n\nc : Composition n\n\u22a2 length c = 1 \u2192 c = single n h", "state_after": "case mpr\nn\u271d : \u2115\nc\u271d : Composition n\u271d\nn : \u2115\nh : 0 < n\nc : Composition n\nH : length c = 1\n\u22a2 c = single n h"}, {"tactic": "ext1", "state_before": "case mpr\nn\u271d : \u2115\nc\u271d : Composition n\u271d\nn : \u2115\nh : 0 < n\nc : Composition n\nH : length c = 1\n\u22a2 c = single n h", "state_after": "case mpr.blocks\nn\u271d : \u2115\nc\u271d : Composition n\u271d\nn : \u2115\nh : 0 < n\nc : Composition n\nH : length c = 1\n\u22a2 c.blocks = (single n h).blocks"}, {"tactic": "have A : c.blocks.length = 1 := H \u25b8 c.blocks_length", "state_before": "case mpr.blocks\nn\u271d : \u2115\nc\u271d : Composition n\u271d\nn : \u2115\nh : 0 < n\nc : Composition n\nH : length c = 1\n\u22a2 c.blocks = (single n h).blocks", "state_after": "case mpr.blocks\nn\u271d : \u2115\nc\u271d : Composition n\u271d\nn : \u2115\nh : 0 < n\nc : Composition n\nH : length c = 1\nA : List.length c.blocks = 1\n\u22a2 c.blocks = (single n h).blocks"}, {"tactic": "have B : c.blocks.sum = n := c.blocks_sum", "state_before": "case mpr.blocks\nn\u271d : \u2115\nc\u271d : Composition n\u271d\nn : \u2115\nh : 0 < n\nc : Composition n\nH : length c = 1\nA : List.length c.blocks = 1\n\u22a2 c.blocks = (single n h).blocks", "state_after": "case mpr.blocks\nn\u271d : \u2115\nc\u271d : Composition n\u271d\nn : \u2115\nh : 0 < n\nc : Composition n\nH : length c = 1\nA : List.length c.blocks = 1\nB : sum c.blocks = n\n\u22a2 c.blocks = (single n h).blocks"}, {"tactic": "rw [eq_cons_of_length_one A] at B\u22a2", "state_before": "case mpr.blocks\nn\u271d : \u2115\nc\u271d : Composition n\u271d\nn : \u2115\nh : 0 < n\nc : Composition n\nH : length c = 1\nA : List.length c.blocks = 1\nB : sum c.blocks = n\n\u22a2 c.blocks = (single n h).blocks", "state_after": "case mpr.blocks\nn\u271d : \u2115\nc\u271d : Composition n\u271d\nn : \u2115\nh : 0 < n\nc : Composition n\nH : length c = 1\nA : List.length c.blocks = 1\nB : sum [nthLe c.blocks 0 (_ : 0 < List.length c.blocks)] = n\n\u22a2 [nthLe c.blocks 0 (_ : 0 < List.length c.blocks)] = (single n h).blocks"}, {"tactic": "simpa [single_blocks] using B", "state_before": "case mpr.blocks\nn\u271d : \u2115\nc\u271d : Composition n\u271d\nn : \u2115\nh : 0 < n\nc : Composition n\nH : length c = 1\nA : List.length c.blocks = 1\nB : sum [nthLe c.blocks 0 (_ : 0 < List.length c.blocks)] = n\n\u22a2 [nthLe c.blocks 0 (_ : 0 < List.length c.blocks)] = (single n h).blocks", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/ToIntervalMod.lean", "full_name": "toIxxMod_iff", "start": [853, 9], "end": [856, 33], "traced_tactics": [{"tactic": "rw [toIcoMod_eq_sub, toIocMod_eq_sub _ x\u2081, add_le_add_iff_right, \u2190 neg_sub x\u2081 x\u2083, toIocMod_neg,\n  neg_zero, le_sub_iff_add_le]", "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nh\u03b1 : Archimedean \u03b1\np : \u03b1\nhp : 0 < p\na b c : \u03b1\nn : \u2124\nx\u2081 x\u2082 x\u2083 : \u03b1\n\u22a2 toIcoMod hp x\u2081 x\u2082 \u2264 toIocMod hp x\u2081 x\u2083 \u2194 toIcoMod hp 0 (x\u2082 - x\u2081) + toIcoMod hp 0 (x\u2081 - x\u2083) \u2264 p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/QuadraticForm/Basic.lean", "full_name": "QuadraticForm.polar_add_right", "start": [292, 1], "end": [293, 70], "traced_tactics": [{"tactic": "rw [polar_comm Q x, polar_comm Q x, polar_comm Q x, polar_add_left]", "state_before": "S : Type ?u.163172\nR : Type u_1\nR\u2081 : Type ?u.163178\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : CommRing R\u2081\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx y y' : M\n\u22a2 polar (\u2191Q) x (y + y') = polar (\u2191Q) x y + polar (\u2191Q) x y'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Polynomial/Bernstein.lean", "full_name": "bernsteinPolynomial.iterate_derivative_at_0_eq_zero_of_lt", "start": [152, 1], "end": [167, 67], "traced_tactics": [{"tactic": "cases' \u03bd with \u03bd", "state_before": "R : Type u_1\ninst\u271d : CommRing R\nn \u03bd k : \u2115\n\u22a2 k < \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R n \u03bd)) = 0", "state_after": "case zero\nR : Type u_1\ninst\u271d : CommRing R\nn k : \u2115\n\u22a2 k < Nat.zero \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R n Nat.zero)) = 0\n\ncase succ\nR : Type u_1\ninst\u271d : CommRing R\nn k \u03bd : \u2115\n\u22a2 k < Nat.succ \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0"}, {"tactic": "rintro \u27e8\u27e9", "state_before": "case zero\nR : Type u_1\ninst\u271d : CommRing R\nn k : \u2115\n\u22a2 k < Nat.zero \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R n Nat.zero)) = 0", "state_after": "no goals"}, {"tactic": "rw [Nat.lt_succ_iff]", "state_before": "case succ\nR : Type u_1\ninst\u271d : CommRing R\nn k \u03bd : \u2115\n\u22a2 k < Nat.succ \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0", "state_after": "case succ\nR : Type u_1\ninst\u271d : CommRing R\nn k \u03bd : \u2115\n\u22a2 k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0"}, {"tactic": "induction' k with k ih generalizing n \u03bd", "state_before": "case succ\nR : Type u_1\ninst\u271d : CommRing R\nn k \u03bd : \u2115\n\u22a2 k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0", "state_after": "case succ.zero\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d n \u03bd : \u2115\n\u22a2 Nat.zero \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[Nat.zero]) (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\n\ncase succ.succ\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d k : \u2115\nih : \u2200 (n \u03bd : \u2115), k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\nn \u03bd : \u2115\n\u22a2 Nat.succ k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[Nat.succ k]) (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0"}, {"tactic": "simp [eval_at_0]", "state_before": "case succ.zero\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d n \u03bd : \u2115\n\u22a2 Nat.zero \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[Nat.zero]) (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0", "state_after": "no goals"}, {"tactic": "simp only [derivative_succ, Int.coe_nat_eq_zero, mul_eq_zero, Function.comp_apply,\n  Function.iterate_succ, Polynomial.iterate_derivative_sub,\n  Polynomial.iterate_derivative_nat_cast_mul, Polynomial.eval_mul, Polynomial.eval_nat_cast,\n  Polynomial.eval_sub]", "state_before": "case succ.succ\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d k : \u2115\nih : \u2200 (n \u03bd : \u2115), k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\nn \u03bd : \u2115\n\u22a2 Nat.succ k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[Nat.succ k]) (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0", "state_after": "case succ.succ\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d k : \u2115\nih : \u2200 (n \u03bd : \u2115), k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\nn \u03bd : \u2115\n\u22a2 Nat.succ k \u2264 \u03bd \u2192\n    \u2191n *\n        (Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R (n - 1) \u03bd)) -\n          Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R (n - 1) (\u03bd + 1)))) =\n      0"}, {"tactic": "intro h", "state_before": "case succ.succ\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d k : \u2115\nih : \u2200 (n \u03bd : \u2115), k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\nn \u03bd : \u2115\n\u22a2 Nat.succ k \u2264 \u03bd \u2192\n    \u2191n *\n        (Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R (n - 1) \u03bd)) -\n          Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R (n - 1) (\u03bd + 1)))) =\n      0", "state_after": "case succ.succ\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d k : \u2115\nih : \u2200 (n \u03bd : \u2115), k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\nn \u03bd : \u2115\nh : Nat.succ k \u2264 \u03bd\n\u22a2 \u2191n *\n      (Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R (n - 1) \u03bd)) -\n        Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R (n - 1) (\u03bd + 1)))) =\n    0"}, {"tactic": "apply mul_eq_zero_of_right", "state_before": "case succ.succ\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d k : \u2115\nih : \u2200 (n \u03bd : \u2115), k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\nn \u03bd : \u2115\nh : Nat.succ k \u2264 \u03bd\n\u22a2 \u2191n *\n      (Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R (n - 1) \u03bd)) -\n        Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R (n - 1) (\u03bd + 1)))) =\n    0", "state_after": "case succ.succ.h\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d k : \u2115\nih : \u2200 (n \u03bd : \u2115), k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\nn \u03bd : \u2115\nh : Nat.succ k \u2264 \u03bd\n\u22a2 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R (n - 1) \u03bd)) -\n      Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R (n - 1) (\u03bd + 1))) =\n    0"}, {"tactic": "rw [ih _ _ (Nat.le_of_succ_le h), sub_zero]", "state_before": "case succ.succ.h\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d k : \u2115\nih : \u2200 (n \u03bd : \u2115), k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\nn \u03bd : \u2115\nh : Nat.succ k \u2264 \u03bd\n\u22a2 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R (n - 1) \u03bd)) -\n      Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R (n - 1) (\u03bd + 1))) =\n    0", "state_after": "case succ.succ.h\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d k : \u2115\nih : \u2200 (n \u03bd : \u2115), k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\nn \u03bd : \u2115\nh : Nat.succ k \u2264 \u03bd\n\u22a2 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R (n - 1) \u03bd)) = 0"}, {"tactic": "convert ih _ _ (Nat.pred_le_pred h)", "state_before": "case succ.succ.h\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d k : \u2115\nih : \u2200 (n \u03bd : \u2115), k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\nn \u03bd : \u2115\nh : Nat.succ k \u2264 \u03bd\n\u22a2 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R (n - 1) \u03bd)) = 0", "state_after": "case h.e'_2.h.e'_4.h.h.e'_4.h.e'_4\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d k : \u2115\nih : \u2200 (n \u03bd : \u2115), k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\nn \u03bd : \u2115\nh : Nat.succ k \u2264 \u03bd\ne_2\u271d : Ring.toSemiring = CommSemiring.toSemiring\n\u22a2 \u03bd = Nat.succ (Nat.pred \u03bd)"}, {"tactic": "exact (Nat.succ_pred_eq_of_pos (k.succ_pos.trans_le h)).symm", "state_before": "case h.e'_2.h.e'_4.h.h.e'_4.h.e'_4\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d k : \u2115\nih : \u2200 (n \u03bd : \u2115), k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative^[k]) (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\nn \u03bd : \u2115\nh : Nat.succ k \u2264 \u03bd\ne_2\u271d : Ring.toSemiring = CommSemiring.toSemiring\n\u22a2 \u03bd = Nat.succ (Nat.pred \u03bd)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/GroupWithZero.lean", "full_name": "ContinuousWithinAt.div", "start": [180, 8], "end": [182, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Constructions.lean", "full_name": "MulOpposite.continuous_op", "start": [47, 1], "end": [48, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.le_add_left", "start": [654, 1], "end": [654, 100], "traced_tactics": [{"tactic": "simpa using add_le_add_right (zero_le t) s", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.42115\n\u03b3 : Type ?u.42118\ns t : Multiset \u03b1\n\u22a2 s \u2264 t + s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/SubsetProperties.lean", "full_name": "IsClosed.exists_minimal_nonempty_closed_subset", "start": [1224, 1], "end": [1260, 27], "traced_tactics": [{"tactic": "let opens := { U : Set \u03b1 | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 U\u1d9c.Nonempty }", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\n\u22a2 \u2203 V, V \u2286 S \u2227 Set.Nonempty V \u2227 IsClosed V \u2227 \u2200 (V' : Set \u03b1), V' \u2286 V \u2192 Set.Nonempty V' \u2192 IsClosed V' \u2192 V' = V", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\n\u22a2 \u2203 V, V \u2286 S \u2227 Set.Nonempty V \u2227 IsClosed V \u2227 \u2200 (V' : Set \u03b1), V' \u2286 V \u2192 Set.Nonempty V' \u2192 IsClosed V' \u2192 V' = V"}, {"tactic": "refine' \u27e8U\u1d9c, Set.compl_subset_comm.mp Uc, Ucne, Uo.isClosed_compl, _\u27e9", "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nU : Set \u03b1\nh : \u2200 (a : Set \u03b1), a \u2208 opens \u2192 U \u2286 a \u2192 a = U\nUc : S\u1d9c \u2286 U\nUo : IsOpen U\nUcne : Set.Nonempty (U\u1d9c)\n\u22a2 \u2203 V, V \u2286 S \u2227 Set.Nonempty V \u2227 IsClosed V \u2227 \u2200 (V' : Set \u03b1), V' \u2286 V \u2192 Set.Nonempty V' \u2192 IsClosed V' \u2192 V' = V", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nU : Set \u03b1\nh : \u2200 (a : Set \u03b1), a \u2208 opens \u2192 U \u2286 a \u2192 a = U\nUc : S\u1d9c \u2286 U\nUo : IsOpen U\nUcne : Set.Nonempty (U\u1d9c)\n\u22a2 \u2200 (V' : Set \u03b1), V' \u2286 U\u1d9c \u2192 Set.Nonempty V' \u2192 IsClosed V' \u2192 V' = U\u1d9c"}, {"tactic": "intro V' V'sub V'ne V'cls", "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nU : Set \u03b1\nh : \u2200 (a : Set \u03b1), a \u2208 opens \u2192 U \u2286 a \u2192 a = U\nUc : S\u1d9c \u2286 U\nUo : IsOpen U\nUcne : Set.Nonempty (U\u1d9c)\n\u22a2 \u2200 (V' : Set \u03b1), V' \u2286 U\u1d9c \u2192 Set.Nonempty V' \u2192 IsClosed V' \u2192 V' = U\u1d9c", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nU : Set \u03b1\nh : \u2200 (a : Set \u03b1), a \u2208 opens \u2192 U \u2286 a \u2192 a = U\nUc : S\u1d9c \u2286 U\nUo : IsOpen U\nUcne : Set.Nonempty (U\u1d9c)\nV' : Set \u03b1\nV'sub : V' \u2286 U\u1d9c\nV'ne : Set.Nonempty V'\nV'cls : IsClosed V'\n\u22a2 V' = U\u1d9c"}, {"tactic": "have : V'\u1d9c = U := by\n  refine' h (V'\u1d9c) \u27e8_, isOpen_compl_iff.mpr V'cls, _\u27e9 (Set.subset_compl_comm.mp V'sub)\n  exact Set.Subset.trans Uc (Set.subset_compl_comm.mp V'sub)\n  simp only [compl_compl, V'ne]", "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nU : Set \u03b1\nh : \u2200 (a : Set \u03b1), a \u2208 opens \u2192 U \u2286 a \u2192 a = U\nUc : S\u1d9c \u2286 U\nUo : IsOpen U\nUcne : Set.Nonempty (U\u1d9c)\nV' : Set \u03b1\nV'sub : V' \u2286 U\u1d9c\nV'ne : Set.Nonempty V'\nV'cls : IsClosed V'\n\u22a2 V' = U\u1d9c", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nU : Set \u03b1\nh : \u2200 (a : Set \u03b1), a \u2208 opens \u2192 U \u2286 a \u2192 a = U\nUc : S\u1d9c \u2286 U\nUo : IsOpen U\nUcne : Set.Nonempty (U\u1d9c)\nV' : Set \u03b1\nV'sub : V' \u2286 U\u1d9c\nV'ne : Set.Nonempty V'\nV'cls : IsClosed V'\nthis : V'\u1d9c = U\n\u22a2 V' = U\u1d9c"}, {"tactic": "rw [\u2190 this, compl_compl]", "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nU : Set \u03b1\nh : \u2200 (a : Set \u03b1), a \u2208 opens \u2192 U \u2286 a \u2192 a = U\nUc : S\u1d9c \u2286 U\nUo : IsOpen U\nUcne : Set.Nonempty (U\u1d9c)\nV' : Set \u03b1\nV'sub : V' \u2286 U\u1d9c\nV'ne : Set.Nonempty V'\nV'cls : IsClosed V'\nthis : V'\u1d9c = U\n\u22a2 V' = U\u1d9c", "state_after": "no goals"}, {"tactic": "by_cases hcne : c.Nonempty", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\n\u22a2 \u2203 ub, ub \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 ub", "state_after": "case pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nhcne : Set.Nonempty c\n\u22a2 \u2203 ub, ub \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 ub\n\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nhcne : \u00acSet.Nonempty c\n\u22a2 \u2203 ub, ub \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 ub"}, {"tactic": "obtain \u27e8U\u2080, hU\u2080\u27e9 := hcne", "state_before": "case pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nhcne : Set.Nonempty c\n\u22a2 \u2203 ub, ub \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 ub", "state_after": "case pos.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\n\u22a2 \u2203 ub, ub \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 ub"}, {"tactic": "haveI : Nonempty { U // U \u2208 c } := \u27e8\u27e8U\u2080, hU\u2080\u27e9\u27e9", "state_before": "case pos.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\n\u22a2 \u2203 ub, ub \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 ub", "state_after": "case pos.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\n\u22a2 \u2203 ub, ub \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 ub"}, {"tactic": "obtain \u27e8U\u2080compl, -, -\u27e9 := hc hU\u2080", "state_before": "case pos.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\n\u22a2 \u2203 ub, ub \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 ub", "state_after": "case pos.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 \u2203 ub, ub \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 ub"}, {"tactic": "use \u22c3\u2080 c", "state_before": "case pos.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 \u2203 ub, ub \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 ub", "state_after": "case pos.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 \u22c3\u2080 c \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 \u22c3\u2080 c"}, {"tactic": "refine' \u27e8\u27e8_, _, _\u27e9, fun U hU a ha => \u27e8U, hU, ha\u27e9\u27e9", "state_before": "case pos.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 \u22c3\u2080 c \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 \u22c3\u2080 c", "state_after": "case pos.intro.intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 S\u1d9c \u2286 \u22c3\u2080 c\n\ncase pos.intro.intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 IsOpen (\u22c3\u2080 c)\n\ncase pos.intro.intro.intro.refine'_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 Set.Nonempty ((\u22c3\u2080 c)\u1d9c)"}, {"tactic": "exact fun a ha => \u27e8U\u2080, hU\u2080, U\u2080compl ha\u27e9", "state_before": "case pos.intro.intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 S\u1d9c \u2286 \u22c3\u2080 c", "state_after": "no goals"}, {"tactic": "exact isOpen_sUnion fun _ h => (hc h).2.1", "state_before": "case pos.intro.intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 IsOpen (\u22c3\u2080 c)", "state_after": "no goals"}, {"tactic": "convert_to (\u22c2 U : { U // U \u2208 c }, U.1\u1d9c).Nonempty", "state_before": "case pos.intro.intro.intro.refine'_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 Set.Nonempty ((\u22c3\u2080 c)\u1d9c)", "state_after": "case h.e'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 (\u22c3\u2080 c)\u1d9c = \u22c2 (U : { U // U \u2208 c }), \u2191U\u1d9c\n\ncase pos.intro.intro.intro.refine'_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 Set.Nonempty (\u22c2 (U : { U // U \u2208 c }), \u2191U\u1d9c)"}, {"tactic": "apply IsCompact.nonempty_iInter_of_directed_nonempty_compact_closed", "state_before": "case pos.intro.intro.intro.refine'_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 Set.Nonempty (\u22c2 (U : { U // U \u2208 c }), \u2191U\u1d9c)", "state_after": "case pos.intro.intro.intro.refine'_3.hZd\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 Directed (fun x x_1 => x \u2287 x_1) fun i => \u2191i\u1d9c\n\ncase pos.intro.intro.intro.refine'_3.hZn\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 \u2200 (i : { U // U \u2208 c }), Set.Nonempty (\u2191i\u1d9c)\n\ncase pos.intro.intro.intro.refine'_3.hZc\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 \u2200 (i : { U // U \u2208 c }), IsCompact (\u2191i\u1d9c)\n\ncase pos.intro.intro.intro.refine'_3.hZcl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 \u2200 (i : { U // U \u2208 c }), IsClosed (\u2191i\u1d9c)"}, {"tactic": "ext", "state_before": "case h.e'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 (\u22c3\u2080 c)\u1d9c = \u22c2 (U : { U // U \u2208 c }), \u2191U\u1d9c", "state_after": "case h.e'_2.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 (\u22c3\u2080 c)\u1d9c \u2194 x\u271d \u2208 \u22c2 (U : { U // U \u2208 c }), \u2191U\u1d9c"}, {"tactic": "simp only [not_exists, exists_prop, not_and, Set.mem_iInter, Subtype.forall,\n  mem_setOf_eq, mem_compl_iff, mem_sUnion]", "state_before": "case h.e'_2.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 (\u22c3\u2080 c)\u1d9c \u2194 x\u271d \u2208 \u22c2 (U : { U // U \u2208 c }), \u2191U\u1d9c", "state_after": "no goals"}, {"tactic": "rintro \u27e8U, hU\u27e9 \u27e8U', hU'\u27e9", "state_before": "case pos.intro.intro.intro.refine'_3.hZd\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 Directed (fun x x_1 => x \u2287 x_1) fun i => \u2191i\u1d9c", "state_after": "case pos.intro.intro.intro.refine'_3.hZd.mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\nU : Set \u03b1\nhU : U \u2208 c\nU' : Set \u03b1\nhU' : U' \u2208 c\n\u22a2 \u2203 z,\n    (fun x x_1 => x \u2287 x_1) ((fun i => \u2191i\u1d9c) { val := U, property := hU }) ((fun i => \u2191i\u1d9c) z) \u2227\n      (fun x x_1 => x \u2287 x_1) ((fun i => \u2191i\u1d9c) { val := U', property := hU' }) ((fun i => \u2191i\u1d9c) z)"}, {"tactic": "obtain \u27e8V, hVc, hVU, hVU'\u27e9 := hz.directedOn U hU U' hU'", "state_before": "case pos.intro.intro.intro.refine'_3.hZd.mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\nU : Set \u03b1\nhU : U \u2208 c\nU' : Set \u03b1\nhU' : U' \u2208 c\n\u22a2 \u2203 z,\n    (fun x x_1 => x \u2287 x_1) ((fun i => \u2191i\u1d9c) { val := U, property := hU }) ((fun i => \u2191i\u1d9c) z) \u2227\n      (fun x x_1 => x \u2287 x_1) ((fun i => \u2191i\u1d9c) { val := U', property := hU' }) ((fun i => \u2191i\u1d9c) z)", "state_after": "case pos.intro.intro.intro.refine'_3.hZd.mk.mk.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\nU : Set \u03b1\nhU : U \u2208 c\nU' : Set \u03b1\nhU' : U' \u2208 c\nV : Set \u03b1\nhVc : V \u2208 c\nhVU : U \u2286 V\nhVU' : U' \u2286 V\n\u22a2 \u2203 z,\n    (fun x x_1 => x \u2287 x_1) ((fun i => \u2191i\u1d9c) { val := U, property := hU }) ((fun i => \u2191i\u1d9c) z) \u2227\n      (fun x x_1 => x \u2287 x_1) ((fun i => \u2191i\u1d9c) { val := U', property := hU' }) ((fun i => \u2191i\u1d9c) z)"}, {"tactic": "exact \u27e8\u27e8V, hVc\u27e9, Set.compl_subset_compl.mpr hVU, Set.compl_subset_compl.mpr hVU'\u27e9", "state_before": "case pos.intro.intro.intro.refine'_3.hZd.mk.mk.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\nU : Set \u03b1\nhU : U \u2208 c\nU' : Set \u03b1\nhU' : U' \u2208 c\nV : Set \u03b1\nhVc : V \u2208 c\nhVU : U \u2286 V\nhVU' : U' \u2286 V\n\u22a2 \u2203 z,\n    (fun x x_1 => x \u2287 x_1) ((fun i => \u2191i\u1d9c) { val := U, property := hU }) ((fun i => \u2191i\u1d9c) z) \u2227\n      (fun x x_1 => x \u2287 x_1) ((fun i => \u2191i\u1d9c) { val := U', property := hU' }) ((fun i => \u2191i\u1d9c) z)", "state_after": "no goals"}, {"tactic": "exact fun U => (hc U.2).2.2", "state_before": "case pos.intro.intro.intro.refine'_3.hZn\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 \u2200 (i : { U // U \u2208 c }), Set.Nonempty (\u2191i\u1d9c)", "state_after": "no goals"}, {"tactic": "exact fun U => (hc U.2).2.1.isClosed_compl.isCompact", "state_before": "case pos.intro.intro.intro.refine'_3.hZc\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 \u2200 (i : { U // U \u2208 c }), IsCompact (\u2191i\u1d9c)", "state_after": "no goals"}, {"tactic": "exact fun U => (hc U.2).2.1.isClosed_compl", "state_before": "case pos.intro.intro.intro.refine'_3.hZcl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nU\u2080 : Set \u03b1\nhU\u2080 : U\u2080 \u2208 c\nthis : Nonempty { U // U \u2208 c }\nU\u2080compl : S\u1d9c \u2286 U\u2080\n\u22a2 \u2200 (i : { U // U \u2208 c }), IsClosed (\u2191i\u1d9c)", "state_after": "no goals"}, {"tactic": "use S\u1d9c", "state_before": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nhcne : \u00acSet.Nonempty c\n\u22a2 \u2203 ub, ub \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 ub", "state_after": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nhcne : \u00acSet.Nonempty c\n\u22a2 S\u1d9c \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 S\u1d9c"}, {"tactic": "refine' \u27e8\u27e8Set.Subset.refl _, isOpen_compl_iff.mpr hS, _\u27e9, fun U Uc => (hcne \u27e8U, Uc\u27e9).elim\u27e9", "state_before": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nhcne : \u00acSet.Nonempty c\n\u22a2 S\u1d9c \u2208 opens \u2227 \u2200 (s : Set \u03b1), s \u2208 c \u2192 s \u2286 S\u1d9c", "state_after": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nhcne : \u00acSet.Nonempty c\n\u22a2 Set.Nonempty (S\u1d9c\u1d9c)"}, {"tactic": "rw [compl_compl]", "state_before": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nhcne : \u00acSet.Nonempty c\n\u22a2 Set.Nonempty (S\u1d9c\u1d9c)", "state_after": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nhcne : \u00acSet.Nonempty c\n\u22a2 Set.Nonempty S"}, {"tactic": "exact hne", "state_before": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nc : Set (Set \u03b1)\nhc : c \u2286 opens\nhz : IsChain (fun x x_1 => x \u2286 x_1) c\nhcne : \u00acSet.Nonempty c\n\u22a2 Set.Nonempty S", "state_after": "no goals"}, {"tactic": "refine' h (V'\u1d9c) \u27e8_, isOpen_compl_iff.mpr V'cls, _\u27e9 (Set.subset_compl_comm.mp V'sub)", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nU : Set \u03b1\nh : \u2200 (a : Set \u03b1), a \u2208 opens \u2192 U \u2286 a \u2192 a = U\nUc : S\u1d9c \u2286 U\nUo : IsOpen U\nUcne : Set.Nonempty (U\u1d9c)\nV' : Set \u03b1\nV'sub : V' \u2286 U\u1d9c\nV'ne : Set.Nonempty V'\nV'cls : IsClosed V'\n\u22a2 V'\u1d9c = U", "state_after": "case refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nU : Set \u03b1\nh : \u2200 (a : Set \u03b1), a \u2208 opens \u2192 U \u2286 a \u2192 a = U\nUc : S\u1d9c \u2286 U\nUo : IsOpen U\nUcne : Set.Nonempty (U\u1d9c)\nV' : Set \u03b1\nV'sub : V' \u2286 U\u1d9c\nV'ne : Set.Nonempty V'\nV'cls : IsClosed V'\n\u22a2 S\u1d9c \u2286 V'\u1d9c\n\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nU : Set \u03b1\nh : \u2200 (a : Set \u03b1), a \u2208 opens \u2192 U \u2286 a \u2192 a = U\nUc : S\u1d9c \u2286 U\nUo : IsOpen U\nUcne : Set.Nonempty (U\u1d9c)\nV' : Set \u03b1\nV'sub : V' \u2286 U\u1d9c\nV'ne : Set.Nonempty V'\nV'cls : IsClosed V'\n\u22a2 Set.Nonempty (V'\u1d9c\u1d9c)"}, {"tactic": "exact Set.Subset.trans Uc (Set.subset_compl_comm.mp V'sub)", "state_before": "case refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nU : Set \u03b1\nh : \u2200 (a : Set \u03b1), a \u2208 opens \u2192 U \u2286 a \u2192 a = U\nUc : S\u1d9c \u2286 U\nUo : IsOpen U\nUcne : Set.Nonempty (U\u1d9c)\nV' : Set \u03b1\nV'sub : V' \u2286 U\u1d9c\nV'ne : Set.Nonempty V'\nV'cls : IsClosed V'\n\u22a2 S\u1d9c \u2286 V'\u1d9c\n\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nU : Set \u03b1\nh : \u2200 (a : Set \u03b1), a \u2208 opens \u2192 U \u2286 a \u2192 a = U\nUc : S\u1d9c \u2286 U\nUo : IsOpen U\nUcne : Set.Nonempty (U\u1d9c)\nV' : Set \u03b1\nV'sub : V' \u2286 U\u1d9c\nV'ne : Set.Nonempty V'\nV'cls : IsClosed V'\n\u22a2 Set.Nonempty (V'\u1d9c\u1d9c)", "state_after": "case refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nU : Set \u03b1\nh : \u2200 (a : Set \u03b1), a \u2208 opens \u2192 U \u2286 a \u2192 a = U\nUc : S\u1d9c \u2286 U\nUo : IsOpen U\nUcne : Set.Nonempty (U\u1d9c)\nV' : Set \u03b1\nV'sub : V' \u2286 U\u1d9c\nV'ne : Set.Nonempty V'\nV'cls : IsClosed V'\n\u22a2 Set.Nonempty (V'\u1d9c\u1d9c)"}, {"tactic": "simp only [compl_compl, V'ne]", "state_before": "case refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.151556\n\u03c0 : \u03b9 \u2192 Type ?u.151561\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d : CompactSpace \u03b1\nS : Set \u03b1\nhS : IsClosed S\nhne : Set.Nonempty S\nopens : Set (Set \u03b1) := {U | S\u1d9c \u2286 U \u2227 IsOpen U \u2227 Set.Nonempty (U\u1d9c)}\nU : Set \u03b1\nh : \u2200 (a : Set \u03b1), a \u2208 opens \u2192 U \u2286 a \u2192 a = U\nUc : S\u1d9c \u2286 U\nUo : IsOpen U\nUcne : Set.Nonempty (U\u1d9c)\nV' : Set \u03b1\nV'sub : V' \u2286 U\u1d9c\nV'ne : Set.Nonempty V'\nV'cls : IsClosed V'\n\u22a2 Set.Nonempty (V'\u1d9c\u1d9c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Nat/Lemmas.lean", "full_name": "Nat.add_mod_right", "start": [490, 9], "end": [491, 63], "traced_tactics": [{"tactic": "rw [mod_eq_sub_mod (Nat.le_add_left ..), Nat.add_sub_cancel]", "state_before": "x z : Nat\n\u22a2 (x + z) % z = x % z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "iInf\u2082_eq_top", "start": [1075, 1], "end": [1076, 7], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.89702\n\u03b2\u2082 : Type ?u.89705\n\u03b3 : Type ?u.89708\n\u03b9 : Sort u_2\n\u03b9' : Sort ?u.89714\n\u03ba : \u03b9 \u2192 Sort u_3\n\u03ba' : \u03b9' \u2192 Sort ?u.89724\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 \u03b1\n\u22a2 (\u2a05 (i : \u03b9) (j : \u03ba i), f i j) = \u22a4 \u2194 \u2200 (i : \u03b9) (j : \u03ba i), f i j = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.Mem\u2112p.mono_measure", "start": [595, 1], "end": [596, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Function.lean", "full_name": "ConvexOn.convex_lt", "start": [546, 1], "end": [554, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Reverse.lean", "full_name": "Polynomial.reflect_support", "start": [107, 1], "end": [111, 94], "traced_tactics": [{"tactic": "rcases f with \u27e8\u27e9", "state_before": "R : Type u_1\ninst\u271d : Semiring R\nf\u271d : R[X]\nN : \u2115\nf : R[X]\n\u22a2 support (reflect N f) = image (\u2191(revAt N)) (support f)", "state_after": "case ofFinsupp\nR : Type u_1\ninst\u271d : Semiring R\nf : R[X]\nN : \u2115\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 support (reflect N { toFinsupp := toFinsupp\u271d }) = image (\u2191(revAt N)) (support { toFinsupp := toFinsupp\u271d })"}, {"tactic": "ext1", "state_before": "case ofFinsupp\nR : Type u_1\ninst\u271d : Semiring R\nf : R[X]\nN : \u2115\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 support (reflect N { toFinsupp := toFinsupp\u271d }) = image (\u2191(revAt N)) (support { toFinsupp := toFinsupp\u271d })", "state_after": "case ofFinsupp.a\nR : Type u_1\ninst\u271d : Semiring R\nf : R[X]\nN : \u2115\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\na\u271d : \u2115\n\u22a2 a\u271d \u2208 support (reflect N { toFinsupp := toFinsupp\u271d }) \u2194 a\u271d \u2208 image (\u2191(revAt N)) (support { toFinsupp := toFinsupp\u271d })"}, {"tactic": "simp only [reflect, support_ofFinsupp, support_embDomain, Finset.mem_map, Finset.mem_image]", "state_before": "case ofFinsupp.a\nR : Type u_1\ninst\u271d : Semiring R\nf : R[X]\nN : \u2115\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\na\u271d : \u2115\n\u22a2 a\u271d \u2208 support (reflect N { toFinsupp := toFinsupp\u271d }) \u2194 a\u271d \u2208 image (\u2191(revAt N)) 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"state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Lattice.lean", "full_name": "Multiset.sup_ndinsert", "start": [92, 1], "end": [93, 59], "traced_tactics": [{"tactic": "rw [\u2190 sup_dedup, dedup_ext.2, sup_dedup, sup_cons]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\n\u22a2 sup (ndinsert a s) = a \u2294 sup s", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\n\u22a2 \u2200 (a_1 : \u03b1), a_1 \u2208 ndinsert a s \u2194 a_1 \u2208 a ::\u2098 s"}, {"tactic": "simp", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : 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Module R P\n\u03b1 : Type u_2 := { N // FG N }\nH : WellFounded fun x x_1 => x > x_1\nN N\u2080 : Submodule R M\nh\u2081 : FG N\u2080\ne : N\u2080 \u2264 N\nh\u2082 : \u2200 (x : { N // FG N }), x \u2208 {N' | \u2191N' \u2264 N} \u2192 \u00acx > { val := N\u2080, property := h\u2081 }\n\u22a2 FG N"}, {"tactic": "convert h\u2081", "state_before": "case mpr.noetherian.intro.mk.intro\nR : Type u_1\nM : Type u_2\nP : Type ?u.113886\nN\u271d : Type w\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid N\u271d\ninst\u271d\u00b2 : Module R N\u271d\ninst\u271d\u00b9 : AddCommMonoid P\ninst\u271d : Module R P\n\u03b1 : Type u_2 := { N // FG N }\nH : WellFounded fun x x_1 => x > x_1\nN N\u2080 : Submodule R M\nh\u2081 : FG N\u2080\ne : N\u2080 \u2264 N\nh\u2082 : \u2200 (x : { N // FG N }), x \u2208 {N' | \u2191N' \u2264 N} \u2192 \u00acx > { val := N\u2080, property := h\u2081 }\n\u22a2 FG N", "state_after": "case h.e'_6\nR : Type u_1\nM : Type u_2\nP : Type ?u.113886\nN\u271d : Type w\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid N\u271d\ninst\u271d\u00b2 : Module R N\u271d\ninst\u271d\u00b9 : AddCommMonoid P\ninst\u271d : Module R P\n\u03b1 : Type u_2 := { N // FG N }\nH : WellFounded fun x x_1 => x > x_1\nN N\u2080 : Submodule R M\nh\u2081 : FG N\u2080\ne : N\u2080 \u2264 N\nh\u2082 : \u2200 (x : { N // FG N }), x \u2208 {N' | \u2191N' \u2264 N} \u2192 \u00acx > { val := N\u2080, property := h\u2081 }\n\u22a2 N = N\u2080"}, {"tactic": "refine' (e.antisymm _).symm", "state_before": "case h.e'_6\nR : Type u_1\nM : Type u_2\nP : Type ?u.113886\nN\u271d : Type w\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid N\u271d\ninst\u271d\u00b2 : Module R N\u271d\ninst\u271d\u00b9 : AddCommMonoid P\ninst\u271d : Module R P\n\u03b1 : Type u_2 := { N // FG N }\nH : WellFounded fun x x_1 => x > x_1\nN N\u2080 : Submodule R M\nh\u2081 : FG N\u2080\ne : N\u2080 \u2264 N\nh\u2082 : \u2200 (x : { N // FG N }), x \u2208 {N' | \u2191N' \u2264 N} \u2192 \u00acx > { val := N\u2080, property := h\u2081 }\n\u22a2 N = N\u2080", "state_after": "case h.e'_6\nR : Type u_1\nM : Type u_2\nP : Type ?u.113886\nN\u271d : Type w\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid N\u271d\ninst\u271d\u00b2 : Module R N\u271d\ninst\u271d\u00b9 : AddCommMonoid P\ninst\u271d : Module R P\n\u03b1 : Type u_2 := { N // FG N }\nH : WellFounded fun x x_1 => x > x_1\nN N\u2080 : Submodule R M\nh\u2081 : FG N\u2080\ne : N\u2080 \u2264 N\nh\u2082 : \u2200 (x : { N // FG N }), x \u2208 {N' | \u2191N' \u2264 N} \u2192 \u00acx > { val := N\u2080, property := h\u2081 }\n\u22a2 N \u2264 N\u2080"}, {"tactic": "by_contra h\u2083", "state_before": "case h.e'_6\nR : Type u_1\nM : Type u_2\nP : Type ?u.113886\nN\u271d : Type w\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid N\u271d\ninst\u271d\u00b2 : Module R N\u271d\ninst\u271d\u00b9 : AddCommMonoid P\ninst\u271d : Module R P\n\u03b1 : Type u_2 := { N // FG N }\nH : WellFounded fun x x_1 => x > x_1\nN N\u2080 : Submodule R M\nh\u2081 : FG N\u2080\ne : N\u2080 \u2264 N\nh\u2082 : \u2200 (x : { N // FG N }), x \u2208 {N' | \u2191N' \u2264 N} \u2192 \u00acx > { val := N\u2080, property := h\u2081 }\n\u22a2 N \u2264 N\u2080", "state_after": "case h.e'_6\nR : Type u_1\nM : Type u_2\nP : Type ?u.113886\nN\u271d : Type w\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid N\u271d\ninst\u271d\u00b2 : Module R N\u271d\ninst\u271d\u00b9 : AddCommMonoid P\ninst\u271d : Module R P\n\u03b1 : Type u_2 := { N // FG N }\nH : WellFounded fun x x_1 => x > x_1\nN N\u2080 : Submodule R M\nh\u2081 : FG N\u2080\ne : N\u2080 \u2264 N\nh\u2082 : \u2200 (x : { N // FG N }), x \u2208 {N' | \u2191N' \u2264 N} \u2192 \u00acx > { val := N\u2080, property := h\u2081 }\nh\u2083 : \u00acN \u2264 N\u2080\n\u22a2 False"}, {"tactic": "obtain \u27e8x, hx\u2081 : x \u2208 N, hx\u2082 : x \u2209 N\u2080\u27e9 := Set.not_subset.mp h\u2083", "state_before": "case h.e'_6\nR : Type u_1\nM : Type u_2\nP : Type ?u.113886\nN\u271d : Type w\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid N\u271d\ninst\u271d\u00b2 : Module R N\u271d\ninst\u271d\u00b9 : AddCommMonoid P\ninst\u271d : Module R P\n\u03b1 : Type u_2 := { N // FG N }\nH : WellFounded fun x x_1 => x > x_1\nN N\u2080 : Submodule R M\nh\u2081 : FG N\u2080\ne : N\u2080 \u2264 N\nh\u2082 : \u2200 (x : { N // FG N }), x \u2208 {N' | \u2191N' \u2264 N} \u2192 \u00acx > { val := N\u2080, property := h\u2081 }\nh\u2083 : \u00acN \u2264 N\u2080\n\u22a2 False", "state_after": "case h.e'_6.intro.intro\nR : Type u_1\nM : Type u_2\nP : Type ?u.113886\nN\u271d : Type w\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid N\u271d\ninst\u271d\u00b2 : Module R N\u271d\ninst\u271d\u00b9 : AddCommMonoid P\ninst\u271d : Module R P\n\u03b1 : Type u_2 := { N // FG N }\nH : WellFounded fun x x_1 => x > x_1\nN N\u2080 : Submodule R M\nh\u2081 : FG N\u2080\ne : N\u2080 \u2264 N\nh\u2082 : \u2200 (x : { N // FG N }), x \u2208 {N' | \u2191N' \u2264 N} \u2192 \u00acx > { val := N\u2080, property := h\u2081 }\nh\u2083 : \u00acN \u2264 N\u2080\nx : M\nhx\u2081 : x \u2208 N\nhx\u2082 : \u00acx \u2208 N\u2080\n\u22a2 False"}, {"tactic": "apply hx\u2082", "state_before": "case h.e'_6.intro.intro\nR : Type u_1\nM : Type u_2\nP : Type ?u.113886\nN\u271d : Type w\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid N\u271d\ninst\u271d\u00b2 : Module R N\u271d\ninst\u271d\u00b9 : AddCommMonoid P\ninst\u271d : Module R P\n\u03b1 : Type u_2 := { N // FG N }\nH : WellFounded fun x x_1 => x > x_1\nN N\u2080 : Submodule R M\nh\u2081 : FG N\u2080\ne : N\u2080 \u2264 N\nh\u2082 : \u2200 (x : { N // FG N }), x \u2208 {N' | \u2191N' \u2264 N} \u2192 \u00acx > { val := N\u2080, property := h\u2081 }\nh\u2083 : \u00acN \u2264 N\u2080\nx : M\nhx\u2081 : x \u2208 N\nhx\u2082 : \u00acx \u2208 N\u2080\n\u22a2 False", "state_after": "case h.e'_6.intro.intro\nR : Type u_1\nM : Type u_2\nP : Type ?u.113886\nN\u271d : Type w\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid N\u271d\ninst\u271d\u00b2 : Module R N\u271d\ninst\u271d\u00b9 : AddCommMonoid P\ninst\u271d : Module R P\n\u03b1 : Type u_2 := { N // FG N }\nH : WellFounded fun x x_1 => x > x_1\nN N\u2080 : Submodule R M\nh\u2081 : FG N\u2080\ne : N\u2080 \u2264 N\nh\u2082 : \u2200 (x : { N // FG N }), x \u2208 {N' | \u2191N' \u2264 N} \u2192 \u00acx > { val := N\u2080, property := h\u2081 }\nh\u2083 : \u00acN \u2264 N\u2080\nx : M\nhx\u2081 : x \u2208 N\nhx\u2082 : \u00acx \u2208 N\u2080\n\u22a2 x \u2208 N\u2080"}, {"tactic": "rw [eq_of_le_of_not_lt (le_sup_right : N\u2080 \u2264 _) (h\u2082\n  \u27e8_, Submodule.FG.sup \u27e8{x}, by rw [Finset.coe_singleton]\u27e9 h\u2081\u27e9 <|\n  sup_le ((Submodule.span_singleton_le_iff_mem _ _).mpr hx\u2081) e)]", "state_before": "case h.e'_6.intro.intro\nR : Type u_1\nM : Type u_2\nP : Type ?u.113886\nN\u271d : Type w\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid N\u271d\ninst\u271d\u00b2 : Module R N\u271d\ninst\u271d\u00b9 : AddCommMonoid P\ninst\u271d : Module R P\n\u03b1 : Type u_2 := { N // FG N }\nH : WellFounded fun x x_1 => x > x_1\nN N\u2080 : Submodule R M\nh\u2081 : FG N\u2080\ne : N\u2080 \u2264 N\nh\u2082 : \u2200 (x : { N // FG N }), x \u2208 {N' | \u2191N' \u2264 N} \u2192 \u00acx > { val := N\u2080, property := h\u2081 }\nh\u2083 : \u00acN \u2264 N\u2080\nx : M\nhx\u2081 : x \u2208 N\nhx\u2082 : \u00acx \u2208 N\u2080\n\u22a2 x \u2208 N\u2080", "state_after": "case h.e'_6.intro.intro\nR : Type u_1\nM : Type u_2\nP : Type ?u.113886\nN\u271d : Type w\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid N\u271d\ninst\u271d\u00b2 : Module R N\u271d\ninst\u271d\u00b9 : AddCommMonoid P\ninst\u271d : Module R P\n\u03b1 : Type u_2 := { N // FG N }\nH : WellFounded fun x x_1 => x > x_1\nN N\u2080 : Submodule R M\nh\u2081 : FG N\u2080\ne : N\u2080 \u2264 N\nh\u2082 : \u2200 (x : { N // FG N }), x \u2208 {N' | \u2191N' \u2264 N} \u2192 \u00acx > { val := N\u2080, property := h\u2081 }\nh\u2083 : \u00acN \u2264 N\u2080\nx : M\nhx\u2081 : x \u2208 N\nhx\u2082 : \u00acx \u2208 N\u2080\n\u22a2 x \u2208 span R {x} \u2294 N\u2080"}, {"tactic": "exact (le_sup_left : (R \u2219 x) \u2264 _) (Submodule.mem_span_singleton_self _)", "state_before": "case h.e'_6.intro.intro\nR : Type u_1\nM : Type u_2\nP : Type ?u.113886\nN\u271d : Type w\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid N\u271d\ninst\u271d\u00b2 : Module R N\u271d\ninst\u271d\u00b9 : AddCommMonoid P\ninst\u271d : Module R P\n\u03b1 : Type u_2 := { N // FG N }\nH : WellFounded fun x x_1 => x > x_1\nN N\u2080 : Submodule R M\nh\u2081 : FG N\u2080\ne : N\u2080 \u2264 N\nh\u2082 : \u2200 (x : { N // FG N }), x \u2208 {N' | \u2191N' \u2264 N} \u2192 \u00acx > { val := N\u2080, property := h\u2081 }\nh\u2083 : \u00acN \u2264 N\u2080\nx : M\nhx\u2081 : x \u2208 N\nhx\u2082 : \u00acx \u2208 N\u2080\n\u22a2 x \u2208 span R {x} \u2294 N\u2080", "state_after": "no goals"}, {"tactic": "rw [Finset.coe_singleton]", "state_before": "R : Type u_1\nM : Type u_2\nP : Type ?u.113886\nN\u271d : Type w\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid N\u271d\ninst\u271d\u00b2 : Module R N\u271d\ninst\u271d\u00b9 : AddCommMonoid P\ninst\u271d : Module R P\n\u03b1 : Type u_2 := { N // FG N }\nH : WellFounded fun x x_1 => x > x_1\nN N\u2080 : Submodule R M\nh\u2081 : FG N\u2080\ne : N\u2080 \u2264 N\nh\u2082 : \u2200 (x : { N // FG N }), x \u2208 {N' | \u2191N' \u2264 N} \u2192 \u00acx > { val := N\u2080, property := h\u2081 }\nh\u2083 : \u00acN \u2264 N\u2080\nx : M\nhx\u2081 : x \u2208 N\nhx\u2082 : \u00acx \u2208 N\u2080\n\u22a2 span R \u2191{x} = span R {x}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Pow.lean", "full_name": "Nat.one_lt_pow_iff", "start": [82, 1], "end": [93, 70], "traced_tactics": [{"tactic": "rcases n with (rfl | n)", "state_before": "k n : \u2115\nh : 0 \u2260 k\n\u22a2 1 < n ^ k \u2194 1 < n", "state_after": "case zero\nk : \u2115\nh : 0 \u2260 k\n\u22a2 1 < zero ^ k \u2194 1 < zero\n\ncase succ\nk : \u2115\nh : 0 \u2260 k\nn : \u2115\n\u22a2 1 < succ n ^ k \u2194 1 < succ n"}, {"tactic": "rcases n with (rfl | n)", "state_before": "case succ\nk : \u2115\nh : 0 \u2260 k\nn : \u2115\n\u22a2 1 < succ n ^ k \u2194 1 < succ n", "state_after": "case succ.zero\nk : \u2115\nh : 0 \u2260 k\n\u22a2 1 < succ zero ^ k \u2194 1 < succ zero\n\ncase succ.succ\nk : \u2115\nh : 0 \u2260 k\nn : \u2115\n\u22a2 1 < succ (succ n) ^ k \u2194 1 < succ (succ n)"}, {"tactic": "refine' \u27e8fun _ => one_lt_succ_succ n, fun _ => _\u27e9", "state_before": "case succ.succ\nk : \u2115\nh : 0 \u2260 k\nn : \u2115\n\u22a2 1 < succ (succ n) ^ k \u2194 1 < succ (succ n)", "state_after": "case succ.succ\nk : \u2115\nh : 0 \u2260 k\nn : \u2115\nx\u271d : 1 < succ (succ n)\n\u22a2 1 < succ (succ n) ^ k"}, {"tactic": "induction' k with k hk", "state_before": "case succ.succ\nk : \u2115\nh : 0 \u2260 k\nn : \u2115\nx\u271d : 1 < succ (succ n)\n\u22a2 1 < succ (succ n) ^ k", "state_after": "case succ.succ.zero\nk : \u2115\nh\u271d : 0 \u2260 k\nn : \u2115\nx\u271d : 1 < succ (succ n)\nh : 0 \u2260 zero\n\u22a2 1 < succ (succ n) ^ zero\n\ncase succ.succ.succ\nk\u271d : \u2115\nh\u271d : 0 \u2260 k\u271d\nn : \u2115\nx\u271d : 1 < succ (succ n)\nk : \u2115\nhk : 0 \u2260 k \u2192 1 < succ (succ n) ^ k\nh : 0 \u2260 succ k\n\u22a2 1 < succ (succ n) ^ succ k"}, {"tactic": "rcases k with (rfl | k)", "state_before": "case succ.succ.succ\nk\u271d : \u2115\nh\u271d : 0 \u2260 k\u271d\nn : \u2115\nx\u271d : 1 < succ (succ n)\nk : \u2115\nhk : 0 \u2260 k \u2192 1 < succ (succ n) ^ k\nh : 0 \u2260 succ k\n\u22a2 1 < succ (succ n) ^ succ k", "state_after": "case succ.succ.succ.zero\nk : \u2115\nh\u271d : 0 \u2260 k\nn : \u2115\nx\u271d : 1 < succ (succ n)\nhk : 0 \u2260 zero \u2192 1 < succ (succ n) ^ zero\nh : 0 \u2260 succ zero\n\u22a2 1 < succ (succ n) ^ succ zero\n\ncase succ.succ.succ.succ\nk\u271d : \u2115\nh\u271d : 0 \u2260 k\u271d\nn : \u2115\nx\u271d : 1 < succ (succ n)\nk : \u2115\nhk : 0 \u2260 succ k \u2192 1 < succ (succ n) ^ succ k\nh : 0 \u2260 succ (succ k)\n\u22a2 1 < succ (succ n) ^ succ (succ k)"}, {"tactic": "rw [pow_succ']", "state_before": "case succ.succ.succ.succ\nk\u271d : \u2115\nh\u271d : 0 \u2260 k\u271d\nn : \u2115\nx\u271d : 1 < succ (succ n)\nk : \u2115\nhk : 0 \u2260 succ k \u2192 1 < succ (succ n) ^ succ k\nh : 0 \u2260 succ (succ k)\n\u22a2 1 < succ (succ n) ^ succ (succ k)", "state_after": "case succ.succ.succ.succ\nk\u271d : \u2115\nh\u271d : 0 \u2260 k\u271d\nn : \u2115\nx\u271d : 1 < succ (succ n)\nk : \u2115\nhk : 0 \u2260 succ k \u2192 1 < succ (succ n) ^ succ k\nh : 0 \u2260 succ (succ k)\n\u22a2 1 < succ (succ n) * succ (succ n) ^ (k + 1)"}, {"tactic": "exact one_lt_mul (one_lt_succ_succ _).le (hk (succ_ne_zero k).symm)", "state_before": "case succ.succ.succ.succ\nk\u271d : \u2115\nh\u271d : 0 \u2260 k\u271d\nn : \u2115\nx\u271d : 1 < succ (succ n)\nk : \u2115\nhk : 0 \u2260 succ k \u2192 1 < succ (succ n) ^ succ k\nh : 0 \u2260 succ (succ k)\n\u22a2 1 < succ (succ n) * succ (succ n) ^ (k + 1)", "state_after": "no goals"}, {"tactic": "cases k <;> simp [zero_pow_eq]", "state_before": "case zero\nk : \u2115\nh : 0 \u2260 k\n\u22a2 1 < zero ^ k \u2194 1 < zero", "state_after": "no goals"}, {"tactic": "rw [\u2190 Nat.one_eq_succ_zero, one_pow]", "state_before": "case succ.zero\nk : \u2115\nh : 0 \u2260 k\n\u22a2 1 < succ zero ^ k \u2194 1 < succ zero", "state_after": "no goals"}, {"tactic": "exact absurd rfl h", "state_before": "case succ.succ.zero\nk : \u2115\nh\u271d : 0 \u2260 k\nn : \u2115\nx\u271d : 1 < succ (succ n)\nh : 0 \u2260 zero\n\u22a2 1 < succ (succ n) ^ zero", "state_after": "no goals"}, {"tactic": "simp [\u2190 Nat.one_eq_succ_zero]", "state_before": "case succ.succ.succ.zero\nk : \u2115\nh\u271d : 0 \u2260 k\nn : \u2115\nx\u271d : 1 < succ (succ n)\nhk : 0 \u2260 zero \u2192 1 < succ (succ n) ^ zero\nh : 0 \u2260 succ zero\n\u22a2 1 < succ (succ n) ^ succ zero", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/NaturalOps.lean", "full_name": "Ordinal.le_of_nadd_le_nadd_right", "start": [414, 1], "end": [415, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "full_name": "MeasureTheory.Measure.AbsolutelyContinuous.isOpenPosMeasure", "start": [75, 11], "end": [76, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Intervals.lean", "full_name": "Finset.sum_Ico_reflect", "start": [167, 1], "end": [169, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/NNReal.lean", "full_name": "NNReal.iSup_mul_iSup_le", "start": [989, 1], "end": [991, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.SimpleFunc.integral_eq_lintegral", "start": [408, 1], "end": [413, 87], "traced_tactics": [{"tactic": "have : f =\u1d50[\u03bc] f.map (ENNReal.toReal \u2218 ENNReal.ofReal) :=\n  h_pos.mono fun a h => (ENNReal.toReal_ofReal h).symm", "state_before": "\u03b1 : Type u_1\nE : Type ?u.155996\nF : Type ?u.155999\n\ud835\udd5c : Type ?u.156002\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type ?u.156104\nF' : Type ?u.156107\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192\u209b \u211d\nhf : Integrable \u2191f\nh_pos : 0 \u2264\u1d50[\u03bc] \u2191f\n\u22a2 integral \u03bc f = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2191f a) \u2202\u03bc)", "state_after": "\u03b1 : Type u_1\nE : Type ?u.155996\nF : Type ?u.155999\n\ud835\udd5c : Type ?u.156002\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type ?u.156104\nF' : Type ?u.156107\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192\u209b \u211d\nhf : Integrable \u2191f\nh_pos : 0 \u2264\u1d50[\u03bc] \u2191f\nthis : \u2191f =\u1d50[\u03bc] \u2191(map (ENNReal.toReal \u2218 ENNReal.ofReal) f)\n\u22a2 integral \u03bc f = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2191f a) \u2202\u03bc)"}, {"tactic": "rw [\u2190 integral_eq_lintegral' hf]", "state_before": "\u03b1 : Type u_1\nE : Type ?u.155996\nF : Type ?u.155999\n\ud835\udd5c : Type ?u.156002\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type ?u.156104\nF' : Type ?u.156107\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192\u209b \u211d\nhf : Integrable \u2191f\nh_pos : 0 \u2264\u1d50[\u03bc] \u2191f\nthis : \u2191f =\u1d50[\u03bc] \u2191(map (ENNReal.toReal \u2218 ENNReal.ofReal) f)\n\u22a2 integral \u03bc f = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2191f a) \u2202\u03bc)", "state_after": "\u03b1 : Type u_1\nE : Type ?u.155996\nF : Type ?u.155999\n\ud835\udd5c : Type ?u.156002\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type ?u.156104\nF' : Type ?u.156107\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192\u209b \u211d\nhf : Integrable \u2191f\nh_pos : 0 \u2264\u1d50[\u03bc] \u2191f\nthis : \u2191f =\u1d50[\u03bc] \u2191(map (ENNReal.toReal \u2218 ENNReal.ofReal) f)\n\u22a2 integral \u03bc f = integral \u03bc (map (ENNReal.toReal \u2218 ENNReal.ofReal) f)\n\ncase hg0\n\u03b1 : Type u_1\nE : Type ?u.155996\nF : Type ?u.155999\n\ud835\udd5c : Type ?u.156002\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type ?u.156104\nF' : Type ?u.156107\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192\u209b \u211d\nhf : Integrable \u2191f\nh_pos : 0 \u2264\u1d50[\u03bc] \u2191f\nthis : \u2191f =\u1d50[\u03bc] \u2191(map (ENNReal.toReal \u2218 ENNReal.ofReal) f)\n\u22a2 ENNReal.ofReal 0 = 0\n\ncase ht\n\u03b1 : Type u_1\nE : Type ?u.155996\nF : Type ?u.155999\n\ud835\udd5c : Type ?u.156002\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type ?u.156104\nF' : Type ?u.156107\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192\u209b \u211d\nhf : Integrable \u2191f\nh_pos : 0 \u2264\u1d50[\u03bc] \u2191f\nthis : \u2191f =\u1d50[\u03bc] \u2191(map (ENNReal.toReal \u2218 ENNReal.ofReal) f)\n\u22a2 \u2200 (b : \u211d), ENNReal.ofReal b \u2260 \u22a4"}, {"tactic": "exacts [integral_congr hf this, ENNReal.ofReal_zero, fun b => ENNReal.ofReal_ne_top]", "state_before": "\u03b1 : Type u_1\nE : Type ?u.155996\nF : Type ?u.155999\n\ud835\udd5c : Type ?u.156002\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type ?u.156104\nF' : Type ?u.156107\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192\u209b \u211d\nhf : Integrable \u2191f\nh_pos : 0 \u2264\u1d50[\u03bc] \u2191f\nthis : \u2191f =\u1d50[\u03bc] \u2191(map (ENNReal.toReal \u2218 ENNReal.ofReal) f)\n\u22a2 integral \u03bc f = integral \u03bc (map (ENNReal.toReal \u2218 ENNReal.ofReal) f)\n\ncase hg0\n\u03b1 : Type u_1\nE : Type ?u.155996\nF : Type ?u.155999\n\ud835\udd5c : Type ?u.156002\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type ?u.156104\nF' : Type ?u.156107\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192\u209b \u211d\nhf : Integrable \u2191f\nh_pos : 0 \u2264\u1d50[\u03bc] \u2191f\nthis : \u2191f =\u1d50[\u03bc] \u2191(map (ENNReal.toReal \u2218 ENNReal.ofReal) f)\n\u22a2 ENNReal.ofReal 0 = 0\n\ncase ht\n\u03b1 : Type u_1\nE : Type ?u.155996\nF : Type ?u.155999\n\ud835\udd5c : Type ?u.156002\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\np : \u211d\u22650\u221e\nG : Type ?u.156104\nF' : Type ?u.156107\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : NormedSpace \u211d F'\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192\u209b \u211d\nhf : Integrable \u2191f\nh_pos : 0 \u2264\u1d50[\u03bc] \u2191f\nthis : \u2191f =\u1d50[\u03bc] \u2191(map (ENNReal.toReal \u2218 ENNReal.ofReal) f)\n\u22a2 \u2200 (b : \u211d), ENNReal.ofReal b \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "ContinuousAt.integral_sub_linear_isLittleO_ae", "start": [1016, 1], "end": [1022, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finsupp/Defs.lean", "full_name": "Finsupp.embDomain_inj", "start": [876, 1], "end": [877, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Matrix/Block.lean", "full_name": "Matrix.blockDiagonal_apply", "start": [356, 1], "end": [360, 6], "traced_tactics": [{"tactic": "cases ik", "state_before": "l : Type ?u.128706\nm : Type u_1\nn : Type u_2\no : Type u_4\np : Type ?u.128718\nq : Type ?u.128721\nm' : o \u2192 Type ?u.128726\nn' : o \u2192 Type ?u.128731\np' : o \u2192 Type ?u.128736\nR : Type ?u.128739\nS : Type ?u.128742\n\u03b1 : Type u_3\n\u03b2 : Type ?u.128748\ninst\u271d\u00b2 : DecidableEq o\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : Zero \u03b2\nM : o \u2192 Matrix m n \u03b1\nik : m \u00d7 o\njk : n \u00d7 o\n\u22a2 blockDiagonal M ik jk = if ik.snd = jk.snd then M ik.snd ik.fst jk.fst else 0", "state_after": "case mk\nl : Type ?u.128706\nm : Type u_1\nn : Type u_2\no : Type u_4\np : Type ?u.128718\nq : Type ?u.128721\nm' : o \u2192 Type ?u.128726\nn' : o \u2192 Type ?u.128731\np' : o \u2192 Type ?u.128736\nR : Type ?u.128739\nS : Type ?u.128742\n\u03b1 : Type u_3\n\u03b2 : Type ?u.128748\ninst\u271d\u00b2 : DecidableEq o\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : Zero \u03b2\nM : o \u2192 Matrix m n \u03b1\njk : n \u00d7 o\nfst\u271d : m\nsnd\u271d : o\n\u22a2 blockDiagonal M (fst\u271d, snd\u271d) jk = if (fst\u271d, snd\u271d).snd = jk.snd then M (fst\u271d, snd\u271d).snd (fst\u271d, snd\u271d).fst jk.fst else 0"}, {"tactic": "cases jk", "state_before": "case mk\nl : Type ?u.128706\nm : Type u_1\nn : Type u_2\no : Type u_4\np : Type ?u.128718\nq : Type ?u.128721\nm' : o \u2192 Type ?u.128726\nn' : o \u2192 Type ?u.128731\np' : o \u2192 Type ?u.128736\nR : Type ?u.128739\nS : Type ?u.128742\n\u03b1 : Type u_3\n\u03b2 : Type ?u.128748\ninst\u271d\u00b2 : DecidableEq o\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : Zero \u03b2\nM : o \u2192 Matrix m n \u03b1\njk : n \u00d7 o\nfst\u271d : m\nsnd\u271d : o\n\u22a2 blockDiagonal M (fst\u271d, snd\u271d) jk = if (fst\u271d, snd\u271d).snd = jk.snd then M (fst\u271d, snd\u271d).snd (fst\u271d, snd\u271d).fst jk.fst else 0", "state_after": "case mk.mk\nl : Type ?u.128706\nm : Type u_1\nn : Type u_2\no : Type u_4\np : Type ?u.128718\nq : Type ?u.128721\nm' : o \u2192 Type ?u.128726\nn' : o \u2192 Type ?u.128731\np' : o \u2192 Type ?u.128736\nR : Type ?u.128739\nS : Type ?u.128742\n\u03b1 : Type u_3\n\u03b2 : Type ?u.128748\ninst\u271d\u00b2 : DecidableEq o\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : Zero \u03b2\nM : o \u2192 Matrix m n \u03b1\nfst\u271d\u00b9 : m\nsnd\u271d\u00b9 : o\nfst\u271d : n\nsnd\u271d : o\n\u22a2 blockDiagonal M (fst\u271d\u00b9, snd\u271d\u00b9) (fst\u271d, snd\u271d) =\n    if (fst\u271d\u00b9, snd\u271d\u00b9).snd = (fst\u271d, snd\u271d).snd then M (fst\u271d\u00b9, snd\u271d\u00b9).snd (fst\u271d\u00b9, snd\u271d\u00b9).fst (fst\u271d, snd\u271d).fst else 0"}, {"tactic": "rfl", "state_before": "case mk.mk\nl : Type ?u.128706\nm : Type u_1\nn : Type u_2\no : Type u_4\np : Type ?u.128718\nq : Type ?u.128721\nm' : o \u2192 Type ?u.128726\nn' : o \u2192 Type ?u.128731\np' : o \u2192 Type ?u.128736\nR : Type ?u.128739\nS : Type ?u.128742\n\u03b1 : Type u_3\n\u03b2 : Type ?u.128748\ninst\u271d\u00b2 : DecidableEq o\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : Zero \u03b2\nM : o \u2192 Matrix m n \u03b1\nfst\u271d\u00b9 : m\nsnd\u271d\u00b9 : o\nfst\u271d : n\nsnd\u271d : o\n\u22a2 blockDiagonal M (fst\u271d\u00b9, snd\u271d\u00b9) (fst\u271d, snd\u271d) =\n    if (fst\u271d\u00b9, snd\u271d\u00b9).snd = (fst\u271d, snd\u271d).snd then M (fst\u271d\u00b9, snd\u271d\u00b9).snd (fst\u271d\u00b9, snd\u271d\u00b9).fst (fst\u271d, snd\u271d).fst else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Bounds/Basic.lean", "full_name": "bddBelow_Ici", "start": [538, 1], "end": [539, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/UniformMulAction.lean", "full_name": "UniformContinuous.const_smul", "start": [99, 1], "end": [101, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/Operations.lean", "full_name": "Ideal.basisSpanSingleton_apply", "start": [1946, 1], "end": [1950, 86], "traced_tactics": [{"tactic": "simp only [basisSpanSingleton, Basis.map_apply, LinearEquiv.trans_apply,\n  Submodule.restrictScalarsEquiv_apply, LinearEquiv.ofInjective_apply, LinearEquiv.coe_ofEq_apply,\n  LinearEquiv.restrictScalars_apply, Algebra.coe_lmul_eq_mul, LinearMap.mul_apply']", "state_before": "\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nb : Basis \u03b9 R S\nx : S\nhx : x \u2260 0\ni : \u03b9\n\u22a2 \u2191(\u2191(basisSpanSingleton b hx) i) = x * \u2191b i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/OmegaCompletePartialOrder.lean", "full_name": "OmegaCompletePartialOrder.ContinuousHom.seq_continuous'", "start": [667, 1], "end": [672, 33], "traced_tactics": [{"tactic": "simp only [seq_eq_bind_map]", "state_before": "\u03b1 : Type u\n\u03b1' : Type ?u.60695\n\u03b2\u271d : Type v\n\u03b2' : Type ?u.60700\n\u03b3\u271d : Type ?u.60703\n\u03c6 : Type ?u.60706\ninst\u271d\u2075 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u2074 : OmegaCompletePartialOrder \u03b2\u271d\ninst\u271d\u00b3 : OmegaCompletePartialOrder \u03b3\u271d\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03c6\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1'\ninst\u271d : OmegaCompletePartialOrder \u03b2'\n\u03b2 \u03b3 : Type v\nf : \u03b1 \u2192 Part (\u03b2 \u2192 \u03b3)\ng : \u03b1 \u2192 Part \u03b2\nhf : Continuous' f\nhg : Continuous' g\n\u22a2 Continuous' fun x => Seq.seq (f x) fun x_1 => g x", "state_after": "\u03b1 : Type u\n\u03b1' : Type ?u.60695\n\u03b2\u271d : Type 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\u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nh1 : c \u2264 a\nh2 : c \u2264 b\n\u22a2 a - c = b - c \u2194 a = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean", "full_name": "CategoryTheory.Limits.inv_prodComparison_map_snd", "start": [1294, 1], "end": [1295, 90], "traced_tactics": [{"tactic": "simp [IsIso.inv_comp_eq]", "state_before": "C : Type u\ninst\u271d\u2076 : Category C\nX Y : C\nD : Type u\u2082\ninst\u271d\u2075 : Category D\nF : C \u2964 D\nA A' B B' : C\ninst\u271d\u2074 : HasBinaryProduct A B\ninst\u271d\u00b3 : HasBinaryProduct A' B'\ninst\u271d\u00b2 : HasBinaryProduct (F.obj A) (F.obj B)\ninst\u271d\u00b9 : HasBinaryProduct (F.obj A') (F.obj B')\ninst\u271d : IsIso (prodComparison F A B)\n\u22a2 inv (prodComparison F A B) \u226b F.map prod.snd = prod.snd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Int/DivMod.lean", "full_name": "Int.add_emod_emod", "start": [424, 9], "end": [425, 49], "traced_tactics": [{"tactic": "rw [Int.add_comm, emod_add_emod, Int.add_comm]", "state_before": "m n k : Int\n\u22a2 (m + n % k) % k = (m + n) % k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/CofilteredSystem.lean", "full_name": "nonempty_sections_of_finite_inverse_system", "start": [115, 1], "end": [121, 58], "traced_tactics": [{"tactic": "cases isEmpty_or_nonempty J", "state_before": "J : Type u\ninst\u271d\u00b3 : Preorder J\ninst\u271d\u00b2 : IsDirected J fun x x_1 => x \u2264 x_1\nF : J\u1d52\u1d56 \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J\u1d52\u1d56), Finite (F.obj j)\ninst\u271d : \u2200 (j : J\u1d52\u1d56), _root_.Nonempty (F.obj j)\n\u22a2 Set.Nonempty (Functor.sections F)", "state_after": "case inl\nJ : Type u\ninst\u271d\u00b3 : Preorder J\ninst\u271d\u00b2 : IsDirected J fun x x_1 => x \u2264 x_1\nF : J\u1d52\u1d56 \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J\u1d52\u1d56), Finite (F.obj j)\ninst\u271d : \u2200 (j : J\u1d52\u1d56), _root_.Nonempty (F.obj j)\nh\u271d : IsEmpty J\n\u22a2 Set.Nonempty (Functor.sections F)\n\ncase inr\nJ : Type u\ninst\u271d\u00b3 : Preorder J\ninst\u271d\u00b2 : IsDirected J fun x x_1 => x \u2264 x_1\nF : J\u1d52\u1d56 \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J\u1d52\u1d56), Finite (F.obj j)\ninst\u271d : \u2200 (j : J\u1d52\u1d56), _root_.Nonempty (F.obj j)\nh\u271d : _root_.Nonempty J\n\u22a2 Set.Nonempty (Functor.sections F)"}, {"tactic": "haveI : IsEmpty J\u1d52\u1d56 := \u27e8fun j => isEmptyElim j.unop\u27e9", "state_before": "case inl\nJ : Type u\ninst\u271d\u00b3 : Preorder J\ninst\u271d\u00b2 : IsDirected J fun x x_1 => x \u2264 x_1\nF : J\u1d52\u1d56 \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J\u1d52\u1d56), Finite (F.obj j)\ninst\u271d : \u2200 (j : J\u1d52\u1d56), _root_.Nonempty (F.obj j)\nh\u271d : IsEmpty J\n\u22a2 Set.Nonempty (Functor.sections F)", "state_after": "case inl\nJ : Type u\ninst\u271d\u00b3 : Preorder J\ninst\u271d\u00b2 : IsDirected J fun x x_1 => x \u2264 x_1\nF : J\u1d52\u1d56 \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J\u1d52\u1d56), Finite (F.obj j)\ninst\u271d : \u2200 (j : J\u1d52\u1d56), _root_.Nonempty (F.obj j)\nh\u271d : IsEmpty J\nthis : IsEmpty J\u1d52\u1d56\n\u22a2 Set.Nonempty (Functor.sections F)"}, {"tactic": "exact \u27e8isEmptyElim, by apply isEmptyElim\u27e9", "state_before": "case inl\nJ : Type u\ninst\u271d\u00b3 : Preorder J\ninst\u271d\u00b2 : IsDirected J fun x x_1 => x \u2264 x_1\nF : J\u1d52\u1d56 \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J\u1d52\u1d56), Finite (F.obj j)\ninst\u271d : \u2200 (j : J\u1d52\u1d56), _root_.Nonempty (F.obj j)\nh\u271d : IsEmpty J\nthis : IsEmpty J\u1d52\u1d56\n\u22a2 Set.Nonempty (Functor.sections F)", "state_after": "no goals"}, {"tactic": "apply isEmptyElim", "state_before": "J : Type u\ninst\u271d\u00b3 : Preorder J\ninst\u271d\u00b2 : IsDirected J fun x x_1 => x \u2264 x_1\nF : J\u1d52\u1d56 \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J\u1d52\u1d56), Finite (F.obj j)\ninst\u271d : \u2200 (j : J\u1d52\u1d56), _root_.Nonempty (F.obj j)\nh\u271d : IsEmpty J\nthis : IsEmpty J\u1d52\u1d56\n\u22a2 (fun a => isEmptyElim a) \u2208 Functor.sections F", "state_after": "no goals"}, {"tactic": "exact nonempty_sections_of_finite_cofiltered_system _", "state_before": "case inr\nJ : Type u\ninst\u271d\u00b3 : Preorder J\ninst\u271d\u00b2 : IsDirected J fun x x_1 => x \u2264 x_1\nF : J\u1d52\u1d56 \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J\u1d52\u1d56), Finite (F.obj j)\ninst\u271d : \u2200 (j : J\u1d52\u1d56), _root_.Nonempty (F.obj j)\nh\u271d : _root_.Nonempty J\n\u22a2 Set.Nonempty (Functor.sections F)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/Multilinear.lean", "full_name": "ContinuousMultilinearMap.compContinuousLinearMapEquivL_symm", "start": [1229, 1], "end": [1232, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/OreLocalization/Basic.lean", "full_name": "OreLocalization.right_distrib", "start": [734, 1], "end": [742, 66], "traced_tactics": [{"tactic": "induction' x using OreLocalization.ind with r\u2081 s\u2081", "state_before": "R : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nx y z : OreLocalization R S\n\u22a2 (x + y) * z = x * z + y * z", "state_after": "case c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\ny z : OreLocalization R S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\n\u22a2 (r\u2081 /\u2092 s\u2081 + y) * z = r\u2081 /\u2092 s\u2081 * z + y * z"}, {"tactic": "induction' y using OreLocalization.ind with r\u2082 s\u2082", "state_before": "case c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\ny z : OreLocalization R S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\n\u22a2 (r\u2081 /\u2092 s\u2081 + y) * z = r\u2081 /\u2092 s\u2081 * z + y * z", "state_after": "case c.c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nz : OreLocalization R S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\n\u22a2 (r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082) * z = r\u2081 /\u2092 s\u2081 * z + r\u2082 /\u2092 s\u2082 * z"}, {"tactic": "induction' z using OreLocalization.ind with r\u2083 s\u2083", "state_before": "case c.c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nz : OreLocalization R S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\n\u22a2 (r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082) * z = r\u2081 /\u2092 s\u2081 * z + r\u2082 /\u2092 s\u2082 * z", "state_after": "case c.c.c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\n\u22a2 (r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083) + r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083)"}, {"tactic": "rcases oreDivAddChar' r\u2081 r\u2082 s\u2081 s\u2082 with \u27e8ra, sa, ha, ha'\u27e9", "state_before": "case c.c.c\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\n\u22a2 (r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083) + r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083)", "state_after": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nha' : r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 = (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa)\n\u22a2 (r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083) + r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083)"}, {"tactic": "rw [ha']", "state_before": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nha' : r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 = (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa)\n\u22a2 (r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083) + r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083)", "state_after": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nha' : r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 = (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa)\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083) + r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083)"}, {"tactic": "clear ha'", "state_before": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\nha' : r\u2081 /\u2092 s\u2081 + r\u2082 /\u2092 s\u2082 = (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa)\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083) + r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083)", "state_after": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083) + r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083)"}, {"tactic": "norm_cast  at ha", "state_before": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191s\u2081 * \u2191sa = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083) + r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083)", "state_after": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083) + r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083)"}, {"tactic": "rw [OreLocalization.expand' r\u2081 s\u2081 sa]", "state_before": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) = r\u2081 /\u2092 s\u2081 * (r\u2083 /\u2092 s\u2083) + r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083)", "state_after": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) = r\u2081 * \u2191sa /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) + r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083)"}, {"tactic": "rw [OreLocalization.expand r\u2082 s\u2082 ra (by rw [\u2190 ha]; apply SetLike.coe_mem)]", "state_before": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) = r\u2081 * \u2191sa /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) + r\u2082 /\u2092 s\u2082 * (r\u2083 /\u2092 s\u2083)", "state_after": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) =\n    r\u2081 * \u2191sa /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) + r\u2082 * ra /\u2092 { val := \u2191s\u2082 * ra, property := (_ : \u2191s\u2082 * ra \u2208 S) } * (r\u2083 /\u2092 s\u2083)"}, {"tactic": "rw [\u2190 Subtype.coe_eq_of_eq_mk ha]", "state_before": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) =\n    r\u2081 * \u2191sa /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) + r\u2082 * ra /\u2092 { val := \u2191s\u2082 * ra, property := (_ : \u2191s\u2082 * ra \u2208 S) } * (r\u2083 /\u2092 s\u2083)", "state_after": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) =\n    r\u2081 * \u2191sa /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) + r\u2082 * ra /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083)"}, {"tactic": "repeat' rw [oreDiv_mul_oreDiv]; simp only [add_mul, add_oreDiv]", "state_before": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa + r\u2082 * ra) /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) =\n    r\u2081 * \u2191sa /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083) + r\u2082 * ra /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083)", "state_after": "no goals"}, {"tactic": "rw [\u2190 ha]", "state_before": "R : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 \u2191s\u2082 * ra \u2208 S", "state_after": "R : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 \u2191(s\u2081 * sa) \u2208 S"}, {"tactic": "apply SetLike.coe_mem", "state_before": "R : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 \u2191(s\u2081 * sa) \u2208 S", "state_after": "no goals"}, {"tactic": "rw [oreDiv_mul_oreDiv]", "state_before": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa * oreNum r\u2083 (s\u2081 * sa) + r\u2082 * ra * oreNum r\u2083 (s\u2081 * sa)) /\u2092 (s\u2083 * oreDenom r\u2083 (s\u2081 * sa)) =\n    r\u2081 * \u2191sa * oreNum r\u2083 (s\u2081 * sa) /\u2092 (s\u2083 * oreDenom r\u2083 (s\u2081 * sa)) + r\u2082 * ra /\u2092 (s\u2081 * sa) * (r\u2083 /\u2092 s\u2083)", "state_after": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa * oreNum r\u2083 (s\u2081 * sa) + r\u2082 * ra * oreNum r\u2083 (s\u2081 * sa)) /\u2092 (s\u2083 * oreDenom r\u2083 (s\u2081 * sa)) =\n    r\u2081 * \u2191sa * oreNum r\u2083 (s\u2081 * sa) /\u2092 (s\u2083 * oreDenom r\u2083 (s\u2081 * sa)) +\n      r\u2082 * ra * oreNum r\u2083 (s\u2081 * sa) /\u2092 (s\u2083 * oreDenom r\u2083 (s\u2081 * sa))"}, {"tactic": "simp only [add_mul, add_oreDiv]", "state_before": "case c.c.c.mk.mk.intro\nR : Type u_1\ninst\u271d\u00b9 : Semiring R\nS : Submonoid R\ninst\u271d : OreSet S\nr\u2081 : R\ns\u2081 : { x // x \u2208 S }\nr\u2082 : R\ns\u2082 : { x // x \u2208 S }\nr\u2083 : R\ns\u2083 : { x // x \u2208 S }\nra : R\nsa : { x // x \u2208 S }\nha : \u2191(s\u2081 * sa) = \u2191s\u2082 * ra\n\u22a2 (r\u2081 * \u2191sa * oreNum r\u2083 (s\u2081 * sa) + r\u2082 * ra * oreNum r\u2083 (s\u2081 * sa)) /\u2092 (s\u2083 * oreDenom r\u2083 (s\u2081 * sa)) =\n    r\u2081 * \u2191sa * oreNum r\u2083 (s\u2081 * sa) /\u2092 (s\u2083 * oreDenom r\u2083 (s\u2081 * sa)) +\n      r\u2082 * ra * oreNum r\u2083 (s\u2081 * sa) /\u2092 (s\u2083 * oreDenom r\u2083 (s\u2081 * sa))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Localization/Basic.lean", "full_name": "IsLocalization.map_units", "start": [123, 1], "end": [124, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "full_name": "CategoryTheory.MonoidalCategory.comp_tensor_id", "start": [227, 1], "end": [229, 7], "traced_tactics": [{"tactic": "rw [\u2190 tensor_comp]", "state_before": "C\u271d : Type u\n\ud835\udc9e : Category C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\nf : W \u27f6 X\ng : X \u27f6 Y\n\u22a2 f \u226b g \u2297 \ud835\udfd9 Z = (f \u2297 \ud835\udfd9 Z) \u226b (g \u2297 \ud835\udfd9 Z)", "state_after": "C\u271d : Type u\n\ud835\udc9e : Category C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\nf : W \u27f6 X\ng : X \u27f6 Y\n\u22a2 f \u226b g \u2297 \ud835\udfd9 Z = f \u226b g \u2297 \ud835\udfd9 Z \u226b \ud835\udfd9 Z"}, {"tactic": "simp", "state_before": "C\u271d : Type u\n\ud835\udc9e : Category C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\nf : W \u27f6 X\ng : X \u27f6 Y\n\u22a2 f \u226b g \u2297 \ud835\udfd9 Z = f \u226b 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"Filter.Subsingleton.atBot_eq", "start": [299, 1], "end": [300, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Polish.lean", "full_name": "IsOpen.polishSpace", "start": [354, 1], "end": [359, 53], "traced_tactics": [{"tactic": "letI := upgradePolishSpace \u03b1", "state_before": "\u03b1\u271d : Type ?u.34143\n\u03b2 : Type ?u.34146\ninst\u271d\u00b2 : MetricSpace \u03b1\u271d\ns\u271d : Opens \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsOpen s\n\u22a2 PolishSpace \u2191s", "state_after": "\u03b1\u271d : Type ?u.34143\n\u03b2 : Type ?u.34146\ninst\u271d\u00b2 : MetricSpace \u03b1\u271d\ns\u271d : Opens \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsOpen s\nthis : UpgradedPolishSpace \u03b1 := upgradePolishSpace \u03b1\n\u22a2 PolishSpace \u2191s"}, {"tactic": "lift s to Opens \u03b1 using hs", "state_before": "\u03b1\u271d : Type ?u.34143\n\u03b2 : Type ?u.34146\ninst\u271d\u00b2 : MetricSpace \u03b1\u271d\ns\u271d : Opens \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsOpen s\nthis : UpgradedPolishSpace \u03b1 := upgradePolishSpace \u03b1\n\u22a2 PolishSpace \u2191s", "state_after": "case intro\n\u03b1\u271d : Type ?u.34143\n\u03b2 : Type ?u.34146\ninst\u271d\u00b2 : MetricSpace \u03b1\u271d\ns\u271d : Opens \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : PolishSpace \u03b1\nthis : UpgradedPolishSpace \u03b1 := upgradePolishSpace \u03b1\ns : Opens \u03b1\n\u22a2 PolishSpace \u2191\u2191s"}, {"tactic": "have : SecondCountableTopology s.CompleteCopy := inferInstanceAs (SecondCountableTopology s)", "state_before": "case 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"full_name": "Set.not_mem_Ioi_self", "start": [732, 1], "end": [732, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Prod.lean", "full_name": "Submodule.ker_inl", "start": [589, 1], "end": [589, 82], "traced_tactics": [{"tactic": "rw [ker, \u2190 prod_bot, prod_comap_inl]", "state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type ?u.315515\nM\u2086 : Type ?u.315518\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 ker (inl R M M\u2082) = \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Maps.lean", "full_name": "Inducing.continuousAt_iff", "start": [129, 1], "end": [131, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/MeanInequalitiesPow.lean", "full_name": "ENNReal.rpow_add_le_add_rpow", "start": [341, 1], "end": [348, 50], "traced_tactics": [{"tactic": "rcases hp.eq_or_lt with (rfl | hp_pos)", "state_before": "\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp : 0 \u2264 p\nhp1 : p \u2264 1\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p", "state_after": "case inl\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\u221e\nhp : 0 \u2264 0\nhp1 : 0 \u2264 1\n\u22a2 (a + b) ^ 0 \u2264 a ^ 0 + b ^ 0\n\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p"}, {"tactic": "have h := rpow_add_rpow_le a b hp_pos hp1", "state_before": "case inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p", "state_after": "case inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\nh : (a ^ 1 + b ^ 1) ^ (1 / 1) \u2264 (a ^ p + b ^ p) ^ (1 / p)\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p"}, {"tactic": "rw [one_div_one] at h", "state_before": "case inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\nh : (a ^ 1 + b ^ 1) ^ (1 / 1) \u2264 (a ^ p + b ^ p) ^ (1 / p)\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p", "state_after": "case inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\nh : (a ^ 1 + b ^ 1) ^ 1 \u2264 (a ^ p + b ^ p) ^ (1 / p)\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p"}, {"tactic": "repeat' rw [ENNReal.rpow_one] at h", "state_before": "case inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\nh : (a ^ 1 + b ^ 1) ^ 1 \u2264 (a ^ p + b ^ p) ^ (1 / p)\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p", "state_after": "case inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\nh : a + b \u2264 (a ^ p + b ^ p) ^ (1 / p)\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p"}, {"tactic": "exact (ENNReal.le_rpow_one_div_iff hp_pos).mp h", "state_before": "case inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\nh : a + b \u2264 (a ^ p + b ^ p) ^ (1 / p)\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p", "state_after": "no goals"}, {"tactic": "simp", "state_before": "case inl\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\u221e\nhp : 0 \u2264 0\nhp1 : 0 \u2264 1\n\u22a2 (a + b) ^ 0 \u2264 a ^ 0 + b ^ 0", "state_after": "no goals"}, {"tactic": "rw [ENNReal.rpow_one] at h", "state_before": "case inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\nh : a + b \u2264 (a ^ p + b ^ p) ^ (1 / p)\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p", "state_after": "case inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\nh : a + b \u2264 (a ^ p + b ^ p) ^ (1 / p)\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Derivative.lean", "full_name": "Polynomial.degree_derivative_lt", "start": [205, 1], "end": [208, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "full_name": "HasSum.add_disjoint", "start": [335, 1], "end": [339, 18], "traced_tactics": [{"tactic": "rw [hasSum_subtype_iff_indicator] at *", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.248888\n\u03b4 : Type ?u.248891\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\ns t : Set \u03b2\nhs : Disjoint s t\nha : HasSum (f \u2218 Subtype.val) a\nhb : HasSum (f \u2218 Subtype.val) b\n\u22a2 HasSum (f \u2218 Subtype.val) (a + b)", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.248888\n\u03b4 : Type ?u.248891\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\ns t : Set \u03b2\nhs : Disjoint s t\nha : HasSum (Set.indicator s f) a\nhb : HasSum (Set.indicator t f) b\n\u22a2 HasSum (Set.indicator (s \u222a t) f) (a + b)"}, {"tactic": "rw [Set.indicator_union_of_disjoint hs]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.248888\n\u03b4 : Type ?u.248891\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\ns t : Set \u03b2\nhs : Disjoint s t\nha : HasSum (Set.indicator s f) a\nhb : HasSum (Set.indicator t f) b\n\u22a2 HasSum (Set.indicator (s \u222a t) f) (a + b)", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.248888\n\u03b4 : Type ?u.248891\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\ns t : Set \u03b2\nhs : Disjoint s t\nha : HasSum (Set.indicator s f) a\nhb : HasSum (Set.indicator t f) b\n\u22a2 HasSum (fun a => Set.indicator s f a + Set.indicator t f a) (a + b)"}, {"tactic": "exact ha.add hb", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.248888\n\u03b4 : Type ?u.248891\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na b : \u03b1\ns\u271d : Finset \u03b2\ninst\u271d : ContinuousAdd \u03b1\ns t : Set \u03b2\nhs : Disjoint s t\nha : HasSum (Set.indicator s f) a\nhb : HasSum (Set.indicator t f) b\n\u22a2 HasSum (fun a => Set.indicator s f a + Set.indicator t f a) (a + b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/FreeModule/PID.lean", "full_name": "Ideal.exists_smith_normal_form", "start": [555, 1], "end": [567, 77], "traced_tactics": [{"tactic": "cases nonempty_fintype \u03b9", "state_before": "\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type ?u.301599\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nb\u271d : \u03b9 \u2192 M\nS : Type u_3\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R S\nI : Ideal S\nhI : I \u2260 \u22a5\n\u22a2 \u2203 b' a ab', \u2200 (i : \u03b9), \u2191(\u2191ab' i) = a i \u2022 \u2191b' i", "state_after": "case intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type ?u.301599\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nb\u271d : \u03b9 \u2192 M\nS : Type u_3\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R S\nI : Ideal S\nhI : I \u2260 \u22a5\nval\u271d : Fintype \u03b9\n\u22a2 \u2203 b' a ab', \u2200 (i : \u03b9), \u2191(\u2191ab' i) = a i \u2022 \u2191b' i"}, {"tactic": "let \u27e8bS, bI, f, a, snf\u27e9 := I.smithNormalForm b hI", "state_before": "case intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type ?u.301599\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nb\u271d : \u03b9 \u2192 M\nS : Type u_3\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R S\nI : Ideal S\nhI : I \u2260 \u22a5\nval\u271d : Fintype \u03b9\n\u22a2 \u2203 b' a ab', \u2200 (i : \u03b9), \u2191(\u2191ab' i) = a i \u2022 \u2191b' i", "state_after": "case intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type ?u.301599\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nb\u271d : \u03b9 \u2192 M\nS : Type u_3\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R S\nI : Ideal S\nhI : I \u2260 \u22a5\nval\u271d : Fintype \u03b9\nbS : Basis \u03b9 R S\nbI : Basis (Fin (Fintype.card \u03b9)) R { x // x \u2208 restrictScalars R I }\nf : Fin (Fintype.card \u03b9) \u21aa \u03b9\na : Fin (Fintype.card \u03b9) \u2192 R\nsnf : \u2200 (i : Fin (Fintype.card \u03b9)), \u2191(\u2191bI i) = a i \u2022 \u2191bS (\u2191f i)\n\u22a2 \u2203 b' a ab', \u2200 (i : \u03b9), \u2191(\u2191ab' i) = a i \u2022 \u2191b' i"}, {"tactic": "let e : Fin (Fintype.card \u03b9) \u2243 \u03b9 :=\n  Equiv.ofBijective f\n    ((Fintype.bijective_iff_injective_and_card f).mpr \u27e8f.injective, Fintype.card_fin _\u27e9)", "state_before": "case intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type ?u.301599\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nb\u271d : \u03b9 \u2192 M\nS : Type u_3\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R S\nI : Ideal S\nhI : I \u2260 \u22a5\nval\u271d : Fintype \u03b9\nbS : Basis \u03b9 R S\nbI : Basis (Fin (Fintype.card \u03b9)) R { x // x \u2208 restrictScalars R I }\nf : Fin (Fintype.card \u03b9) \u21aa \u03b9\na : Fin (Fintype.card \u03b9) \u2192 R\nsnf : \u2200 (i : Fin (Fintype.card \u03b9)), \u2191(\u2191bI i) = a i \u2022 \u2191bS (\u2191f i)\n\u22a2 \u2203 b' a ab', \u2200 (i : \u03b9), \u2191(\u2191ab' i) = a i \u2022 \u2191b' i", "state_after": "case intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type ?u.301599\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nb\u271d : \u03b9 \u2192 M\nS : Type u_3\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R S\nI : Ideal S\nhI : I \u2260 \u22a5\nval\u271d : Fintype \u03b9\nbS : Basis \u03b9 R S\nbI : Basis (Fin (Fintype.card \u03b9)) R { x // x \u2208 restrictScalars R I }\nf : Fin (Fintype.card \u03b9) \u21aa \u03b9\na : Fin (Fintype.card \u03b9) \u2192 R\nsnf : \u2200 (i : Fin (Fintype.card \u03b9)), \u2191(\u2191bI i) = a i \u2022 \u2191bS (\u2191f i)\ne : Fin (Fintype.card \u03b9) \u2243 \u03b9 := Equiv.ofBijective \u2191f (_ : Function.Bijective \u2191f)\n\u22a2 \u2203 b' a ab', \u2200 (i : \u03b9), \u2191(\u2191ab' i) = a i \u2022 \u2191b' i"}, {"tactic": "have fe : \u2200 i, f (e.symm i) = i := e.apply_symm_apply", "state_before": "case intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type ?u.301599\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nb\u271d : \u03b9 \u2192 M\nS : Type u_3\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R S\nI : Ideal S\nhI : I \u2260 \u22a5\nval\u271d : Fintype \u03b9\nbS : Basis \u03b9 R S\nbI : Basis (Fin (Fintype.card \u03b9)) R { x // x \u2208 restrictScalars R I }\nf : Fin (Fintype.card \u03b9) \u21aa \u03b9\na : Fin (Fintype.card \u03b9) \u2192 R\nsnf : \u2200 (i : Fin (Fintype.card \u03b9)), \u2191(\u2191bI i) = a i \u2022 \u2191bS (\u2191f i)\ne : Fin (Fintype.card \u03b9) \u2243 \u03b9 := Equiv.ofBijective \u2191f (_ : Function.Bijective \u2191f)\n\u22a2 \u2203 b' a ab', \u2200 (i : \u03b9), \u2191(\u2191ab' i) = a i \u2022 \u2191b' i", "state_after": "case intro\n\u03b9 : Type u_1\nR : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\nM : Type ?u.301599\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nb\u271d : \u03b9 \u2192 M\nS : Type u_3\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 R S\nI : Ideal S\nhI : I \u2260 \u22a5\nval\u271d : Fintype \u03b9\nbS : Basis \u03b9 R S\nbI : Basis (Fin (Fintype.card \u03b9)) R { x // x \u2208 restrictScalars R I }\nf : Fin (Fintype.card \u03b9) \u21aa \u03b9\na : Fin (Fintype.card \u03b9) \u2192 R\nsnf : \u2200 (i : Fin (Fintype.card \u03b9)), \u2191(\u2191bI i) = a i \u2022 \u2191bS (\u2191f i)\ne : Fin (Fintype.card \u03b9) \u2243 \u03b9 := Equiv.ofBijective \u2191f (_ : Function.Bijective \u2191f)\nfe : \u2200 (i : \u03b9), \u2191f (\u2191e.symm i) = i\n\u22a2 \u2203 b' a ab', \u2200 (i : \u03b9), \u2191(\u2191ab' i) = a i \u2022 \u2191b' i"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/GroupAction/Opposite.lean", "full_name": "MulOpposite.op_smul_eq_op_smul_op", "start": [64, 1], "end": [66, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Bits.lean", "full_name": "Nat.bits_append_bit", "start": [217, 1], "end": [220, 8], "traced_tactics": [{"tactic": "rw [Nat.bits, binaryRec_eq']", "state_before": "n\u271d n : \u2115\nb : Bool\nhn : n = 0 \u2192 b = true\n\u22a2 bits (bit b n) = b :: bits n", "state_after": "case h\nn\u271d n : \u2115\nb : Bool\nhn : n = 0 \u2192 b = true\n\u22a2 [false] = [] \u2228 (n = 0 \u2192 b = true)"}, {"tactic": "simpa", "state_before": "case h\nn\u271d n : \u2115\nb : Bool\nhn : n = 0 \u2192 b = true\n\u22a2 [false] = [] \u2228 (n = 0 \u2192 b = true)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/String/Lemmas.lean", "full_name": "String.Pos.sub_eq", "start": [103, 1], "end": [103, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.int_cast_cast", "start": [264, 1], "end": [265, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBNode.balLeft_toList", "start": [427, 9], "end": [429, 49], "traced_tactics": [{"tactic": "unfold balLeft", "state_before": "\u03b1 : Type u_1\nl : RBNode \u03b1\nv : \u03b1\nr : RBNode \u03b1\n\u22a2 toList (balLeft l v r) = toList l ++ v :: toList r", "state_after": "\u03b1 : Type u_1\nl : RBNode \u03b1\nv : \u03b1\nr : RBNode \u03b1\n\u22a2 toList\n      (match l with\n      | node red a x b => node red (node black a x b) v r\n      | l =>\n        match r with\n        | node black a y b => balance2 l v (node red a y b)\n        | node red (node black a y b) z c => node red (node black l v a) y (balance2 b z (setRed c))\n        | r => node red l v r) =\n    toList l ++ v :: toList r"}, {"tactic": "split <;> simp", "state_before": "\u03b1 : Type u_1\nl : RBNode \u03b1\nv : \u03b1\nr : RBNode \u03b1\n\u22a2 toList\n      (match l with\n      | node red a x b => node red (node black a x b) v r\n      | l =>\n        match r with\n        | node black a y b => balance2 l v (node red a y b)\n        | node red (node black a y b) z c => node red (node black l v a) y (balance2 b z (setRed c))\n        | r => node red l v r) =\n    toList l ++ v :: toList r", "state_after": "case h_2\n\u03b1 : Type u_1\nl : RBNode \u03b1\nv : \u03b1\nr l\u271d : RBNode \u03b1\nx\u271d : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), l = node red a x b \u2192 False\n\u22a2 toList\n      (match r with\n      | node black a y b => balance2 l v (node red a y b)\n      | node red (node black a y b) z c => node red (node black l v a) y (balance2 b z (setRed c))\n      | r => node red l v r) =\n    toList l ++ v :: toList r"}, {"tactic": "split <;> simp", "state_before": "case h_2\n\u03b1 : Type u_1\nl : RBNode \u03b1\nv : \u03b1\nr l\u271d : RBNode \u03b1\nx\u271d : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), l = node red a x b \u2192 False\n\u22a2 toList\n      (match r with\n      | node black a y b => balance2 l v (node red a y b)\n      | node red (node black a y b) z c => node red (node black l v a) y (balance2 b z (setRed c))\n      | r => node red l v r) =\n    toList l ++ v :: toList r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/Irrational.lean", "full_name": "Irrational.ne_nat", "start": [171, 1], "end": [172, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Data/Nat/Basic.lean", "full_name": "Nat.sub_le_of_le_add", "start": [642, 1], "end": [651, 13], "traced_tactics": [{"tactic": "match le.dest h, Nat.le_total a b with\n| _, Or.inl hle =>\n  rw [Nat.sub_eq_zero_of_le hle]\n  apply Nat.zero_le\n| \u27e8d, hd\u27e9, Or.inr hge =>\n  apply @le.intro _ _ d\n  have hd := Nat.sub_eq_of_eq_add hd\n  rw [Nat.add_comm, \u2190 Nat.add_sub_assoc hge, Nat.add_comm]\n  exact hd", "state_before": "a b c : Nat\nh : a \u2264 c + b\n\u22a2 a - b \u2264 c", "state_after": "no goals"}, {"tactic": "rw [Nat.sub_eq_zero_of_le hle]", "state_before": "a b c : Nat\nh : a \u2264 c + b\nx\u271d : Exists fun k => a + k = c + b\nhle : a \u2264 b\n\u22a2 a - b \u2264 c", "state_after": "a b c : Nat\nh : a \u2264 c + b\nx\u271d : Exists fun k => a + k = c + b\nhle : a \u2264 b\n\u22a2 0 \u2264 c"}, {"tactic": "apply Nat.zero_le", "state_before": "a b c : Nat\nh : a \u2264 c + b\nx\u271d : Exists fun k => a + k = c + b\nhle : a \u2264 b\n\u22a2 0 \u2264 c", "state_after": "no goals"}, {"tactic": "apply @le.intro _ _ d", "state_before": "a b c : Nat\nh : a \u2264 c + b\nd : Nat\nhd : a + d = c + b\nhge : b \u2264 a\n\u22a2 a - b \u2264 c", "state_after": "a b c : Nat\nh : a \u2264 c + b\nd : Nat\nhd : a + d = c + b\nhge : b \u2264 a\n\u22a2 a - b + d = c"}, {"tactic": "have hd := Nat.sub_eq_of_eq_add hd", "state_before": "a b c : Nat\nh : a \u2264 c + b\nd : Nat\nhd : a + d = c + b\nhge : b \u2264 a\n\u22a2 a - b + d = c", "state_after": "a b c : Nat\nh : a \u2264 c + b\nd : Nat\nhd\u271d : a + d = c + b\nhge : b \u2264 a\nhd : a + d - b = c\n\u22a2 a - b + d = c"}, {"tactic": "rw [Nat.add_comm, \u2190 Nat.add_sub_assoc hge, Nat.add_comm]", "state_before": "a b c : Nat\nh : a \u2264 c + b\nd : Nat\nhd\u271d : a + d = c + b\nhge : b \u2264 a\nhd : a + d - b = c\n\u22a2 a - b + d = c", "state_after": "a b c : Nat\nh : a \u2264 c + b\nd : Nat\nhd\u271d : a + d = c + b\nhge : b \u2264 a\nhd : a + d - b = c\n\u22a2 a + d - b = c"}, {"tactic": "exact hd", "state_before": "a b c : Nat\nh : a \u2264 c + b\nd : Nat\nhd\u271d : a + d = c + b\nhge : b \u2264 a\nhd : a + d - b = c\n\u22a2 a + d - b = c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Setoid/Basic.lean", "full_name": "Setoid.ext_iff", "start": [64, 1], "end": [65, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "full_name": "direction_affineSpan", "start": [558, 1], "end": [566, 52], "traced_tactics": [{"tactic": "apply le_antisymm", "state_before": "k : Type u_3\nV : Type u_2\nP : Type u_1\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\n\u22a2 AffineSubspace.direction (affineSpan k s) = vectorSpan k s", "state_after": "case a\nk : Type u_3\nV : Type u_2\nP : Type u_1\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\n\u22a2 AffineSubspace.direction (affineSpan k s) \u2264 vectorSpan k s\n\ncase a\nk : Type u_3\nV : Type u_2\nP : Type u_1\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\n\u22a2 vectorSpan k s \u2264 AffineSubspace.direction (affineSpan k s)"}, {"tactic": "refine' Submodule.span_le.2 _", "state_before": "case a\nk : Type u_3\nV : Type u_2\nP : Type u_1\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\n\u22a2 AffineSubspace.direction (affineSpan k s) \u2264 vectorSpan k s", "state_after": "case a\nk : Type u_3\nV : Type u_2\nP : Type u_1\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\n\u22a2 \u2191(affineSpan k s) -\u1d65 \u2191(affineSpan k s) \u2286 \u2191(vectorSpan k s)"}, {"tactic": "rintro v \u27e8p1, p3, \u27e8p2, hp2, v1, hv1, hp1\u27e9, \u27e8p4, hp4, v2, hv2, hp3\u27e9, rfl\u27e9", "state_before": "case a\nk : Type u_3\nV : Type u_2\nP : Type u_1\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\n\u22a2 \u2191(affineSpan k s) -\u1d65 \u2191(affineSpan k s) \u2286 \u2191(vectorSpan k s)", "state_after": "case a.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_1\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np1 p3 p2 : P\nhp2 : p2 \u2208 s\nv1 : V\nhv1 : v1 \u2208 vectorSpan k s\nhp1 : p1 = v1 +\u1d65 p2\np4 : P\nhp4 : p4 \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhp3 : p3 = v2 +\u1d65 p4\n\u22a2 (fun x x_1 => x -\u1d65 x_1) p1 p3 \u2208 \u2191(vectorSpan k s)"}, {"tactic": "simp only [SetLike.mem_coe]", "state_before": "case a.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_1\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np1 p3 p2 : P\nhp2 : p2 \u2208 s\nv1 : V\nhv1 : v1 \u2208 vectorSpan k s\nhp1 : p1 = v1 +\u1d65 p2\np4 : P\nhp4 : p4 \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhp3 : p3 = v2 +\u1d65 p4\n\u22a2 (fun x x_1 => x -\u1d65 x_1) p1 p3 \u2208 \u2191(vectorSpan k s)", "state_after": "case a.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_1\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np1 p3 p2 : P\nhp2 : p2 \u2208 s\nv1 : V\nhv1 : v1 \u2208 vectorSpan k s\nhp1 : p1 = v1 +\u1d65 p2\np4 : P\nhp4 : p4 \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhp3 : p3 = v2 +\u1d65 p4\n\u22a2 p1 -\u1d65 p3 \u2208 vectorSpan k s"}, {"tactic": "rw [hp1, hp3, vsub_vadd_eq_vsub_sub, vadd_vsub_assoc]", "state_before": "case a.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_1\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np1 p3 p2 : P\nhp2 : p2 \u2208 s\nv1 : V\nhv1 : v1 \u2208 vectorSpan k s\nhp1 : p1 = v1 +\u1d65 p2\np4 : P\nhp4 : p4 \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhp3 : p3 = v2 +\u1d65 p4\n\u22a2 p1 -\u1d65 p3 \u2208 vectorSpan k s", "state_after": "case a.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_1\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np1 p3 p2 : P\nhp2 : p2 \u2208 s\nv1 : V\nhv1 : v1 \u2208 vectorSpan k s\nhp1 : p1 = v1 +\u1d65 p2\np4 : P\nhp4 : p4 \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhp3 : p3 = v2 +\u1d65 p4\n\u22a2 v1 + (p2 -\u1d65 p4) - v2 \u2208 vectorSpan k s"}, {"tactic": "exact\n  (vectorSpan k s).sub_mem ((vectorSpan k s).add_mem hv1 (vsub_mem_vectorSpan k hp2 hp4)) hv2", "state_before": "case a.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_1\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np1 p3 p2 : P\nhp2 : p2 \u2208 s\nv1 : V\nhv1 : v1 \u2208 vectorSpan k s\nhp1 : p1 = v1 +\u1d65 p2\np4 : P\nhp4 : p4 \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhp3 : p3 = v2 +\u1d65 p4\n\u22a2 v1 + (p2 -\u1d65 p4) - v2 \u2208 vectorSpan k s", "state_after": "no goals"}, {"tactic": "exact vectorSpan_mono k (subset_spanPoints k s)", "state_before": "case a\nk : Type u_3\nV : Type u_2\nP : Type u_1\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\n\u22a2 vectorSpan k s \u2264 AffineSubspace.direction (affineSpan k s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.diff_empty", "start": [1906, 1], "end": [1907, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/Inverse.lean", "full_name": "ApproximatesLinearOn.toLocalHomeomorph_coe", "start": [499, 1], "end": [502, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/NaturalOps.lean", "full_name": "NatOrdinal.toOrdinal_one", "start": [109, 1], "end": [110, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Bounded.lean", "full_name": "Set.bounded_self", "start": [171, 1], "end": [172, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/CauSeqCompletion.lean", "full_name": "CauSeq.Completion.ofRat_natCast", "start": [138, 1], "end": [139, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Digits.lean", "full_name": "Nat.digits_add", "start": [138, 1], "end": [148, 23], "traced_tactics": [{"tactic": "rcases exists_eq_add_of_le' h with \u27e8b, rfl : _ = _ + 2\u27e9", "state_before": "n b : \u2115\nh : 1 < b\nx y : \u2115\nhxb : x < b\nhxy : x \u2260 0 \u2228 y \u2260 0\n\u22a2 digits b (x + b * y) = x :: digits b y", "state_after": "case intro\nn x y : \u2115\nhxy : x \u2260 0 \u2228 y \u2260 0\nb : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\n\u22a2 digits (b + 2) (x + (b + 2) * y) = x :: digits (b + 2) y"}, {"tactic": "cases y", "state_before": "case intro\nn x y : \u2115\nhxy : x \u2260 0 \u2228 y \u2260 0\nb : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\n\u22a2 digits (b + 2) (x + (b + 2) * y) = x :: digits (b + 2) y", "state_after": "case intro.zero\nn x b : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\nhxy : x \u2260 0 \u2228 zero \u2260 0\n\u22a2 digits (b + 2) (x + (b + 2) * zero) = x :: digits (b + 2) zero\n\ncase intro.succ\nn x b : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\nn\u271d : \u2115\nhxy : x \u2260 0 \u2228 succ n\u271d \u2260 0\n\u22a2 digits (b + 2) (x + (b + 2) * succ n\u271d) = x :: digits (b + 2) (succ n\u271d)"}, {"tactic": "dsimp [digits]", "state_before": "case intro.succ\nn x b : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\nn\u271d : \u2115\nhxy : x \u2260 0 \u2228 succ n\u271d \u2260 0\n\u22a2 digits (b + 2) (x + (b + 2) * succ n\u271d) = x :: digits (b + 2) (succ n\u271d)", "state_after": "case intro.succ\nn x b : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\nn\u271d : \u2115\nhxy : x \u2260 0 \u2228 succ n\u271d \u2260 0\n\u22a2 digitsAux (b + 0 + 2) (_ : 2 \u2264 b + 0 + 2) (x + (b + 2) * succ n\u271d) =\n    x :: digitsAux (b + 0 + 2) (_ : 2 \u2264 b + 0 + 2) (succ n\u271d)"}, {"tactic": "rw [digitsAux_def]", "state_before": "case intro.succ\nn x b : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\nn\u271d : \u2115\nhxy : x \u2260 0 \u2228 succ n\u271d \u2260 0\n\u22a2 digitsAux (b + 0 + 2) (_ : 2 \u2264 b + 0 + 2) (x + (b + 2) * succ n\u271d) =\n    x :: digitsAux (b + 0 + 2) (_ : 2 \u2264 b + 0 + 2) (succ n\u271d)", "state_after": "case intro.succ\nn x b : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\nn\u271d : \u2115\nhxy : x \u2260 0 \u2228 succ n\u271d \u2260 0\n\u22a2 (x + (b + 2) * succ n\u271d) % (b + 0 + 2) ::\n      digitsAux (b + 0 + 2) (_ : 2 \u2264 b + 0 + 2) ((x + (b + 2) * succ n\u271d) / (b + 0 + 2)) =\n    x :: digitsAux (b + 0 + 2) (_ : 2 \u2264 b + 0 + 2) (succ n\u271d)\n\ncase intro.succ.w\nn x b : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\nn\u271d : \u2115\nhxy : x \u2260 0 \u2228 succ n\u271d \u2260 0\n\u22a2 0 < x + (b + 2) * succ n\u271d"}, {"tactic": "simp [hxb, hxy.resolve_right (absurd rfl)]", "state_before": "case intro.zero\nn x b : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\nhxy : x \u2260 0 \u2228 zero \u2260 0\n\u22a2 digits (b + 2) (x + (b + 2) * zero) = x :: digits (b + 2) zero", "state_after": "no goals"}, {"tactic": "congr", "state_before": "case intro.succ\nn x b : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\nn\u271d : \u2115\nhxy : x \u2260 0 \u2228 succ n\u271d \u2260 0\n\u22a2 (x + (b + 2) * succ n\u271d) % (b + 0 + 2) ::\n      digitsAux (b + 0 + 2) (_ : 2 \u2264 b + 0 + 2) ((x + (b + 2) * succ n\u271d) / (b + 0 + 2)) =\n    x :: digitsAux (b + 0 + 2) (_ : 2 \u2264 b + 0 + 2) (succ n\u271d)", "state_after": "case intro.succ.e_head\nn x b : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\nn\u271d : \u2115\nhxy : x \u2260 0 \u2228 succ n\u271d \u2260 0\n\u22a2 (x + (b + 2) * succ n\u271d) % (b + 0 + 2) = x\n\ncase intro.succ.e_tail.e_a\nn x b : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\nn\u271d : \u2115\nhxy : x \u2260 0 \u2228 succ n\u271d \u2260 0\n\u22a2 (x + (b + 2) * succ n\u271d) / (b + 0 + 2) = succ n\u271d"}, {"tactic": "simp [Nat.add_mod, mod_eq_of_lt hxb]", "state_before": "case intro.succ.e_head\nn x b : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\nn\u271d : \u2115\nhxy : x \u2260 0 \u2228 succ n\u271d \u2260 0\n\u22a2 (x + (b + 2) * succ n\u271d) % (b + 0 + 2) = x", "state_after": "no goals"}, {"tactic": "simp [add_mul_div_left, div_eq_of_lt hxb]", "state_before": "case intro.succ.e_tail.e_a\nn x b : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\nn\u271d : \u2115\nhxy : x \u2260 0 \u2228 succ n\u271d \u2260 0\n\u22a2 (x + (b + 2) * succ n\u271d) / (b + 0 + 2) = succ n\u271d", "state_after": "no goals"}, {"tactic": "apply Nat.succ_pos", "state_before": "case intro.succ.w\nn x b : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\nn\u271d : \u2115\nhxy : x \u2260 0 \u2228 succ n\u271d \u2260 0\n\u22a2 0 < x + (b + 2) * succ n\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Directed.lean", "full_name": "Directed.extend_bot", "start": [132, 1], "end": [144, 47], "traced_tactics": [{"tactic": "intro a b", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\nr r' s : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderBot \u03b1\ne : \u03b9 \u2192 \u03b2\nf : \u03b9 \u2192 \u03b1\nhf : Directed (fun x x_1 => x \u2264 x_1) f\nhe : Injective e\n\u22a2 Directed (fun x x_1 => x \u2264 x_1) (extend e f \u22a5)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\nr r' s : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderBot \u03b1\ne : \u03b9 \u2192 \u03b2\nf : \u03b9 \u2192 \u03b1\nhf : Directed (fun x x_1 => x \u2264 x_1) f\nhe : Injective e\na b : \u03b2\n\u22a2 \u2203 z,\n    (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 a) (extend e f \u22a5 z) \u2227 (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 b) (extend e f \u22a5 z)"}, {"tactic": "rcases(em (\u2203 i, e i = a)).symm with (ha | \u27e8i, rfl\u27e9)", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\nr r' s : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderBot \u03b1\ne : \u03b9 \u2192 \u03b2\nf : \u03b9 \u2192 \u03b1\nhf : Directed (fun x x_1 => x \u2264 x_1) f\nhe : Injective e\na b : \u03b2\n\u22a2 \u2203 z,\n    (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 a) (extend e f \u22a5 z) \u2227 (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 b) (extend e f \u22a5 z)", "state_after": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\nr r' s : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderBot \u03b1\ne : \u03b9 \u2192 \u03b2\nf : \u03b9 \u2192 \u03b1\nhf : Directed (fun x x_1 => x \u2264 x_1) f\nhe : Injective e\na b : \u03b2\nha : \u00ac\u2203 i, e i = a\n\u22a2 \u2203 z,\n    (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 a) (extend e f \u22a5 z) \u2227 (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 b) (extend e f \u22a5 z)\n\ncase inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\nr r' s : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderBot \u03b1\ne : \u03b9 \u2192 \u03b2\nf : \u03b9 \u2192 \u03b1\nhf : Directed (fun x x_1 => x \u2264 x_1) f\nhe : Injective e\nb : \u03b2\ni : \u03b9\n\u22a2 \u2203 z,\n    (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 (e i)) (extend e f \u22a5 z) \u2227\n      (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 b) (extend e f \u22a5 z)"}, {"tactic": "rcases(em (\u2203 i, e i = b)).symm with (hb | \u27e8j, rfl\u27e9)", "state_before": "case inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\nr r' s : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderBot \u03b1\ne : \u03b9 \u2192 \u03b2\nf : \u03b9 \u2192 \u03b1\nhf : Directed (fun x x_1 => x \u2264 x_1) f\nhe : Injective e\nb : \u03b2\ni : \u03b9\n\u22a2 \u2203 z,\n    (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 (e i)) (extend e f \u22a5 z) \u2227\n      (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 b) (extend e f \u22a5 z)", "state_after": "case inr.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\nr r' s : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderBot \u03b1\ne : \u03b9 \u2192 \u03b2\nf : \u03b9 \u2192 \u03b1\nhf : Directed (fun x x_1 => x \u2264 x_1) f\nhe : Injective e\nb : \u03b2\ni : \u03b9\nhb : \u00ac\u2203 i, e i = b\n\u22a2 \u2203 z,\n    (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 (e i)) (extend e f \u22a5 z) \u2227\n      (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 b) (extend e f \u22a5 z)\n\ncase inr.intro.inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\nr r' s : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderBot \u03b1\ne : \u03b9 \u2192 \u03b2\nf : \u03b9 \u2192 \u03b1\nhf : Directed (fun x x_1 => x \u2264 x_1) f\nhe : Injective e\ni j : \u03b9\n\u22a2 \u2203 z,\n    (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 (e i)) (extend e f \u22a5 z) \u2227\n      (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 (e j)) (extend e f \u22a5 z)"}, {"tactic": "rcases hf i j with \u27e8k, hi, hj\u27e9", "state_before": "case inr.intro.inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\nr r' s : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderBot \u03b1\ne : \u03b9 \u2192 \u03b2\nf : \u03b9 \u2192 \u03b1\nhf : Directed (fun x x_1 => x \u2264 x_1) f\nhe : Injective e\ni j : \u03b9\n\u22a2 \u2203 z,\n    (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 (e i)) (extend e f \u22a5 z) \u2227\n      (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 (e j)) (extend e f \u22a5 z)", "state_after": "case inr.intro.inr.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\nr r' s : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderBot \u03b1\ne : \u03b9 \u2192 \u03b2\nf : \u03b9 \u2192 \u03b1\nhf : Directed (fun x x_1 => x \u2264 x_1) f\nhe : Injective e\ni j k : \u03b9\nhi : f i \u2264 f k\nhj : f j \u2264 f k\n\u22a2 \u2203 z,\n    (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 (e i)) (extend e f \u22a5 z) \u2227\n      (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 (e j)) (extend e f \u22a5 z)"}, {"tactic": "use e k", "state_before": "case inr.intro.inr.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\nr r' s : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderBot \u03b1\ne : \u03b9 \u2192 \u03b2\nf : \u03b9 \u2192 \u03b1\nhf : Directed (fun x x_1 => x \u2264 x_1) f\nhe : Injective e\ni j k : \u03b9\nhi : f i \u2264 f k\nhj : f j \u2264 f k\n\u22a2 \u2203 z,\n    (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 (e i)) (extend e f \u22a5 z) \u2227\n      (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 (e j)) (extend e f \u22a5 z)", "state_after": "case inr.intro.inr.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\nr r' s : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderBot \u03b1\ne : \u03b9 \u2192 \u03b2\nf : \u03b9 \u2192 \u03b1\nhf : Directed (fun x x_1 => x \u2264 x_1) f\nhe : Injective e\ni j k : \u03b9\nhi : f i \u2264 f k\nhj : f j \u2264 f k\n\u22a2 (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 (e i)) (extend e f \u22a5 (e k)) \u2227\n    (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 (e j)) (extend e f \u22a5 (e k))"}, {"tactic": "simp only [he.extend_apply, *, true_and_iff]", "state_before": "case inr.intro.inr.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\nr r' s : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderBot \u03b1\ne : \u03b9 \u2192 \u03b2\nf : \u03b9 \u2192 \u03b1\nhf : Directed (fun x x_1 => x \u2264 x_1) f\nhe : Injective e\ni j k : \u03b9\nhi : f i \u2264 f k\nhj : f j \u2264 f k\n\u22a2 (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 (e i)) (extend e f \u22a5 (e k)) \u2227\n    (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 (e j)) (extend e f \u22a5 (e k))", "state_after": "no goals"}, {"tactic": "use b", "state_before": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\nr r' s : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderBot \u03b1\ne : \u03b9 \u2192 \u03b2\nf : \u03b9 \u2192 \u03b1\nhf : Directed (fun x x_1 => x \u2264 x_1) f\nhe : Injective e\na b : \u03b2\nha : \u00ac\u2203 i, e i = a\n\u22a2 \u2203 z,\n    (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 a) (extend e f \u22a5 z) \u2227 (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 b) (extend e f \u22a5 z)", "state_after": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\nr r' s : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderBot \u03b1\ne : \u03b9 \u2192 \u03b2\nf : \u03b9 \u2192 \u03b1\nhf : Directed (fun x x_1 => x \u2264 x_1) f\nhe : Injective e\na b : \u03b2\nha : \u00ac\u2203 i, e i = a\n\u22a2 (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 a) (extend e f \u22a5 b) \u2227 (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 b) (extend e f \u22a5 b)"}, {"tactic": "simp [Function.extend_apply' _ _ _ ha]", "state_before": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\nr r' s : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderBot \u03b1\ne : \u03b9 \u2192 \u03b2\nf : \u03b9 \u2192 \u03b1\nhf : Directed (fun x x_1 => x \u2264 x_1) f\nhe : Injective e\na b : \u03b2\nha : \u00ac\u2203 i, e i = a\n\u22a2 (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 a) (extend e f \u22a5 b) \u2227 (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 b) (extend e f \u22a5 b)", "state_after": "no goals"}, {"tactic": "use e i", "state_before": "case inr.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\nr r' s : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderBot \u03b1\ne : \u03b9 \u2192 \u03b2\nf : \u03b9 \u2192 \u03b1\nhf : Directed (fun x x_1 => x \u2264 x_1) f\nhe : Injective e\nb : \u03b2\ni : \u03b9\nhb : \u00ac\u2203 i, e i = b\n\u22a2 \u2203 z,\n    (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 (e i)) (extend e f \u22a5 z) \u2227\n      (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 b) (extend e f \u22a5 z)", "state_after": "case inr.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\nr r' s : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderBot \u03b1\ne : \u03b9 \u2192 \u03b2\nf : \u03b9 \u2192 \u03b1\nhf : Directed (fun x x_1 => x \u2264 x_1) f\nhe : Injective e\nb : \u03b2\ni : \u03b9\nhb : \u00ac\u2203 i, e i = b\n\u22a2 (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 (e i)) (extend e f \u22a5 (e i)) \u2227\n    (fun x x_1 => x \u2264 x_1) (extend e f \u22a5 b) (extend e f \u22a5 (e i))"}, {"tactic": "simp [Function.extend_apply' _ _ _ hb]", "state_before": "case inr.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\nr r' s : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderBot \u03b1\ne : \u03b9 \u2192 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"Mathlib/Topology/MetricSpace/Holder.lean", "full_name": "HolderOnWith.ediam_image_le_of_subset", "start": [167, 1], "end": [169, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Log/Monotone.lean", "full_name": "Real.log_div_sqrt_antitoneOn", "start": [89, 1], "end": [92, 11], "traced_tactics": [{"tactic": "simp_rw [sqrt_eq_rpow]", "state_before": "x y : \u211d\n\u22a2 AntitoneOn (fun x => log x / sqrt x) {x | exp 2 \u2264 x}", "state_after": "x y : \u211d\n\u22a2 AntitoneOn (fun x => log x / x ^ (1 / 2)) {x | exp 2 \u2264 x}"}, {"tactic": "convert @log_div_self_rpow_antitoneOn (1 / 2) (by norm_num)", "state_before": "x y : \u211d\n\u22a2 AntitoneOn (fun x => log x / x ^ (1 / 2)) {x | exp 2 \u2264 x}", "state_after": "case h.e'_6.h.e'_2.h.h.e'_3.h.e'_1\nx y x\u271d : \u211d\n\u22a2 2 = 1 / (1 / 2)"}, {"tactic": "norm_num", "state_before": "case h.e'_6.h.e'_2.h.h.e'_3.h.e'_1\nx y x\u271d : \u211d\n\u22a2 2 = 1 / (1 / 2)", "state_after": "no goals"}, {"tactic": "norm_num", "state_before": "x y : \u211d\n\u22a2 0 < 1 / 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Sites/Subsheaf.lean", "full_name": "CategoryTheory.GrothendieckTopology.Subpresheaf.sheafify_le", "start": [326, 1], "end": [337, 6], "traced_tactics": [{"tactic": "intro U x hx", "state_before": "C : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\n\u22a2 sheafify J G \u2264 G'", "state_after": "C : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\nU : C\u1d52\u1d56\nx : F.obj U\nhx : x \u2208 obj (sheafify J G) U\n\u22a2 x \u2208 obj G' U"}, {"tactic": "convert((G.sheafifyLift (Subpresheaf.homOfLe h) hG').app U \u27e8x, hx\u27e9).2", "state_before": "C : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\nU : C\u1d52\u1d56\nx : F.obj U\nhx : x \u2208 obj (sheafify J G) U\n\u22a2 x \u2208 obj G' U", "state_after": "case h.e'_4\nC : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\nU : C\u1d52\u1d56\nx : F.obj U\nhx : x \u2208 obj (sheafify J G) U\n\u22a2 x = \u2191((sheafifyLift G (homOfLe h) hG').app U { val := x, property := hx })"}, {"tactic": "apply (hF _ hx).isSeparatedFor.ext", "state_before": "case h.e'_4\nC : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\nU : C\u1d52\u1d56\nx : F.obj U\nhx : x \u2208 obj (sheafify J G) U\n\u22a2 x = \u2191((sheafifyLift G (homOfLe h) hG').app U { val := x, property := hx })", "state_after": "case h.e'_4\nC : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\nU : C\u1d52\u1d56\nx : F.obj U\nhx : x \u2208 obj (sheafify J G) U\n\u22a2 \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 U.unop\u2984,\n    (sieveOfSection G x).arrows f \u2192\n      F.map f.op x = F.map f.op \u2191((sheafifyLift G (homOfLe h) hG').app U { val := x, property := hx })"}, {"tactic": "intro V i hi", "state_before": "case h.e'_4\nC : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\nU : C\u1d52\u1d56\nx : F.obj U\nhx : x \u2208 obj (sheafify J G) U\n\u22a2 \u2200 \u2983Y : C\u2984 \u2983f : Y \u27f6 U.unop\u2984,\n    (sieveOfSection G x).arrows f \u2192\n      F.map f.op x = F.map f.op \u2191((sheafifyLift G (homOfLe h) hG').app U { val := x, property := hx })", "state_after": "case h.e'_4\nC : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\nU : C\u1d52\u1d56\nx : F.obj U\nhx : x \u2208 obj (sheafify J G) U\nV : C\ni : V \u27f6 U.unop\nhi : (sieveOfSection G x).arrows i\n\u22a2 F.map i.op x = F.map i.op \u2191((sheafifyLift G (homOfLe h) hG').app U { val := x, property := hx })"}, {"tactic": "have :=\n  congr_arg (fun f : G.toPresheaf \u27f6 G'.toPresheaf => (NatTrans.app f (op V) \u27e8_, hi\u27e9).1)\n    (G.to_sheafifyLift (Subpresheaf.homOfLe h) hG')", "state_before": "case h.e'_4\nC : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\nU : C\u1d52\u1d56\nx : F.obj U\nhx : x \u2208 obj (sheafify J G) U\nV : C\ni : V \u27f6 U.unop\nhi : (sieveOfSection G x).arrows i\n\u22a2 F.map i.op x = F.map i.op \u2191((sheafifyLift G (homOfLe h) hG').app U { val := x, property := hx })", "state_after": "case h.e'_4\nC : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\nU : C\u1d52\u1d56\nx : F.obj U\nhx : x \u2208 obj (sheafify J G) U\nV : C\ni : V \u27f6 U.unop\nhi : (sieveOfSection G x).arrows i\nthis :\n  (fun f => \u2191(f.app V.op { val := F.map i.op x, property := hi }))\n      (homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G (homOfLe h) hG') =\n    (fun f => \u2191(f.app V.op { val := F.map i.op x, property := hi })) (homOfLe h)\n\u22a2 F.map i.op x = F.map i.op \u2191((sheafifyLift G (homOfLe h) hG').app U { val := x, property := hx })"}, {"tactic": "convert this.symm", "state_before": "case h.e'_4\nC : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\nU : C\u1d52\u1d56\nx : F.obj U\nhx : x \u2208 obj (sheafify J G) U\nV : C\ni : V \u27f6 U.unop\nhi : (sieveOfSection G x).arrows i\nthis :\n  (fun f => \u2191(f.app V.op { val := F.map i.op x, property := hi }))\n      (homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G (homOfLe h) hG') =\n    (fun f => \u2191(f.app V.op { val := F.map i.op x, property := hi })) (homOfLe h)\n\u22a2 F.map i.op x = F.map i.op \u2191((sheafifyLift G (homOfLe h) hG').app U { val := x, property := hx })", "state_after": "case h.e'_3\nC : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\nU : C\u1d52\u1d56\nx : F.obj U\nhx : x \u2208 obj (sheafify J G) U\nV : C\ni : V \u27f6 U.unop\nhi : (sieveOfSection G x).arrows i\nthis :\n  (fun f => \u2191(f.app V.op { val := F.map i.op x, property := hi }))\n      (homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G (homOfLe h) hG') =\n    (fun f => \u2191(f.app V.op { val := F.map i.op x, property := hi })) (homOfLe h)\n\u22a2 F.map i.op \u2191((sheafifyLift G (homOfLe h) hG').app U { val := x, property := hx }) =\n    (fun f => \u2191(f.app V.op { val := F.map i.op x, property := hi }))\n      (homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G (homOfLe h) hG')"}, {"tactic": "erw [\u2190 Subpresheaf.nat_trans_naturality]", "state_before": "case h.e'_3\nC : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\nU : C\u1d52\u1d56\nx : F.obj U\nhx : x \u2208 obj (sheafify J G) U\nV : C\ni : V \u27f6 U.unop\nhi : (sieveOfSection G x).arrows i\nthis :\n  (fun f => \u2191(f.app V.op { val := F.map i.op x, property := hi }))\n      (homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G (homOfLe h) hG') =\n    (fun f => \u2191(f.app V.op { val := F.map i.op x, property := hi })) (homOfLe h)\n\u22a2 F.map i.op \u2191((sheafifyLift G (homOfLe h) hG').app U { val := x, property := hx }) =\n    (fun f => \u2191(f.app V.op { val := F.map i.op x, property := hi }))\n      (homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G (homOfLe h) hG')", "state_after": "case h.e'_3\nC : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\nU : C\u1d52\u1d56\nx : F.obj U\nhx : x \u2208 obj (sheafify J G) U\nV : C\ni : V \u27f6 U.unop\nhi : (sieveOfSection G x).arrows i\nthis :\n  (fun f => \u2191(f.app V.op { val := F.map i.op x, property := hi }))\n      (homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G (homOfLe h) hG') =\n    (fun f => \u2191(f.app V.op { val := F.map i.op x, property := hi })) (homOfLe h)\n\u22a2 \u2191((sheafifyLift G (homOfLe h) hG').app V.op ((toPresheaf (sheafify J G)).map i.op { val := x, property := hx })) =\n    (fun f => \u2191(f.app V.op { val := F.map i.op x, property := hi }))\n      (homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G (homOfLe h) hG')"}, {"tactic": "rfl", "state_before": "case h.e'_3\nC : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nh : G \u2264 G'\nhF : Presieve.IsSheaf J F\nhG' : Presieve.IsSheaf J (toPresheaf G')\nU : C\u1d52\u1d56\nx : F.obj U\nhx : x \u2208 obj (sheafify J G) U\nV : C\ni : V \u27f6 U.unop\nhi : (sieveOfSection G x).arrows i\nthis :\n  (fun f => \u2191(f.app V.op { val := F.map i.op x, property := hi }))\n      (homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G (homOfLe h) hG') =\n    (fun f => \u2191(f.app V.op { val := F.map i.op x, property := hi })) (homOfLe h)\n\u22a2 \u2191((sheafifyLift G (homOfLe h) hG').app V.op ((toPresheaf (sheafify J G)).map i.op { val := x, property := hx })) =\n    (fun f => \u2191(f.app V.op { val := F.map i.op x, property := hi }))\n      (homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G (homOfLe h) hG')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Hom/Equiv/Units/Basic.lean", "full_name": "Equiv.mulLeft_symm_apply", "start": [137, 1], "end": [138, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "iInf_option_elim", "start": [1589, 1], "end": [1590, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "full_name": "Subalgebra.coe_sub", "start": [404, 11], "end": [405, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Perm/Basic.lean", "full_name": "Equiv.Perm.ofSubtype_subtypePerm", "start": [437, 1], "end": [446, 54], "traced_tactics": [{"tactic": "by_cases hx : p x", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d : Perm \u03b1\ninst\u271d : DecidablePred p\na : \u03b1\nf : Perm \u03b1\nh\u2081 : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nh\u2082 : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 p x\nx : \u03b1\n\u22a2 \u2191(\u2191ofSubtype (subtypePerm f h\u2081)) x = \u2191f x", "state_after": "case pos\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d : Perm \u03b1\ninst\u271d : DecidablePred p\na : \u03b1\nf : Perm \u03b1\nh\u2081 : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nh\u2082 : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 p x\nx : \u03b1\nhx : p x\n\u22a2 \u2191(\u2191ofSubtype (subtypePerm f h\u2081)) x = \u2191f x\n\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d : Perm \u03b1\ninst\u271d : DecidablePred p\na : \u03b1\nf : Perm \u03b1\nh\u2081 : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nh\u2082 : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 p x\nx : \u03b1\nhx : \u00acp x\n\u22a2 \u2191(\u2191ofSubtype (subtypePerm f h\u2081)) x = \u2191f x"}, {"tactic": "exact (subtypePerm f h\u2081).extendDomain_apply_subtype _ hx", "state_before": "case pos\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d : Perm \u03b1\ninst\u271d : DecidablePred p\na : \u03b1\nf : Perm \u03b1\nh\u2081 : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nh\u2082 : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 p x\nx : \u03b1\nhx : p x\n\u22a2 \u2191(\u2191ofSubtype (subtypePerm f h\u2081)) x = \u2191f x", "state_after": "no goals"}, {"tactic": "rw [ofSubtype, MonoidHom.coe_mk]", "state_before": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d : Perm \u03b1\ninst\u271d : DecidablePred p\na : \u03b1\nf : Perm \u03b1\nh\u2081 : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nh\u2082 : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 p x\nx : \u03b1\nhx : \u00acp x\n\u22a2 \u2191(\u2191ofSubtype (subtypePerm f h\u2081)) x = \u2191f x", "state_after": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d : Perm \u03b1\ninst\u271d : DecidablePred p\na : \u03b1\nf : Perm \u03b1\nh\u2081 : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nh\u2082 : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 p x\nx : \u03b1\nhx : \u00acp x\n\u22a2 \u2191(\u2191{ toFun := fun f => extendDomain f (Equiv.refl { x // p x }),\n              map_one' := (_ : extendDomain 1 (Equiv.refl { x // p x }) = 1) }\n          (subtypePerm f h\u2081))\n      x =\n    \u2191f x"}, {"tactic": "dsimp only [OneHom.coe_mk]", "state_before": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d : Perm \u03b1\ninst\u271d : DecidablePred p\na : \u03b1\nf : Perm \u03b1\nh\u2081 : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nh\u2082 : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 p x\nx : \u03b1\nhx : \u00acp x\n\u22a2 \u2191(\u2191{ toFun := fun f => extendDomain f (Equiv.refl { x // p x }),\n              map_one' := (_ : extendDomain 1 (Equiv.refl { x // p x }) = 1) }\n          (subtypePerm f h\u2081))\n      x =\n    \u2191f x", "state_after": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d : Perm \u03b1\ninst\u271d : DecidablePred p\na : \u03b1\nf : Perm \u03b1\nh\u2081 : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nh\u2082 : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 p x\nx : \u03b1\nhx : \u00acp x\n\u22a2 \u2191(extendDomain (subtypePerm f h\u2081) (Equiv.refl { x // p x })) x = \u2191f x"}, {"tactic": "rw [Equiv.Perm.extendDomain_apply_not_subtype _ _ hx]", "state_before": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d : Perm \u03b1\ninst\u271d : DecidablePred p\na : \u03b1\nf : Perm \u03b1\nh\u2081 : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nh\u2082 : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 p x\nx : \u03b1\nhx : \u00acp x\n\u22a2 \u2191(extendDomain (subtypePerm f h\u2081) (Equiv.refl { x // p x })) x = \u2191f x", "state_after": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d : Perm \u03b1\ninst\u271d : DecidablePred p\na : \u03b1\nf : Perm \u03b1\nh\u2081 : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nh\u2082 : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 p x\nx : \u03b1\nhx : \u00acp x\n\u22a2 x = \u2191f x"}, {"tactic": "exact not_not.mp fun h => hx (h\u2082 x (Ne.symm h))", "state_before": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d : Perm \u03b1\ninst\u271d : DecidablePred p\na : \u03b1\nf : Perm \u03b1\nh\u2081 : \u2200 (x : \u03b1), p x \u2194 p (\u2191f x)\nh\u2082 : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 p x\nx : \u03b1\nhx : \u00acp x\n\u22a2 x = \u2191f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Quiver/Cast.lean", "full_name": "Quiver.Path.cast_eq_iff_heq", "start": [121, 1], "end": [124, 31], "traced_tactics": [{"tactic": "rw [Path.cast_eq_cast]", "state_before": "U : Type u_1\ninst\u271d : Quiver U\nu v u' v' : U\nhu : u = u'\nhv : v = v'\np : Path u v\np' : Path u' v'\n\u22a2 cast hu hv p = p' \u2194 HEq p p'", "state_after": "U : Type u_1\ninst\u271d : Quiver U\nu v u' v' : U\nhu : u = u'\nhv : v = v'\np : Path u v\np' : Path u' v'\n\u22a2 _root_.cast (_ : Path u v = Path u' v') p = p' \u2194 HEq p p'"}, {"tactic": "exact _root_.cast_eq_iff_heq", "state_before": "U : Type u_1\ninst\u271d : Quiver U\nu v u' v' : U\nhu : u = u'\nhv : v = v'\np : Path u v\np' : Path u' v'\n\u22a2 _root_.cast (_ : Path u v = Path u' v') p = p' \u2194 HEq p p'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Divisibility/Basic.lean", "full_name": "dvd_rfl", "start": [123, 1], "end": [123, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "full_name": "Ring.inverse_mul_cancel_left", "start": [128, 1], "end": [129, 52], "traced_tactics": [{"tactic": "rw [\u2190 mul_assoc, inverse_mul_cancel x h, one_mul]", "state_before": "\u03b1 : Type ?u.7902\nM\u2080 : Type u_1\nG\u2080 : Type ?u.7908\nM\u2080' : Type ?u.7911\nG\u2080' : Type ?u.7914\nF : Type ?u.7917\nF' : Type ?u.7920\ninst\u271d : MonoidWithZero M\u2080\nx y : M\u2080\nh : IsUnit x\n\u22a2 inverse x * (x * y) = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/NAry.lean", "full_name": "Set.image2_eq_empty_iff", "start": [175, 1], "end": [177, 35], "traced_tactics": [{"tactic": "rw [\u2190 not_nonempty_iff_eq_empty, image2_nonempty_iff, not_and_or]", "state_before": "\u03b1 : Type u_2\n\u03b1' : Type ?u.21222\n\u03b2 : Type u_3\n\u03b2' : Type ?u.21228\n\u03b3 : Type u_1\n\u03b3' : Type ?u.21234\n\u03b4 : Type ?u.21237\n\u03b4' : Type ?u.21240\n\u03b5 : Type ?u.21243\n\u03b5' : Type ?u.21246\n\u03b6 : Type ?u.21249\n\u03b6' : Type ?u.21252\n\u03bd : Type ?u.21255\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\n\u22a2 image2 f s t = \u2205 \u2194 s = \u2205 \u2228 t = \u2205", "state_after": "\u03b1 : Type u_2\n\u03b1' : Type ?u.21222\n\u03b2 : Type u_3\n\u03b2' : Type ?u.21228\n\u03b3 : Type u_1\n\u03b3' : Type ?u.21234\n\u03b4 : Type ?u.21237\n\u03b4' : Type ?u.21240\n\u03b5 : Type ?u.21243\n\u03b5' : Type ?u.21246\n\u03b6 : Type ?u.21249\n\u03b6' : Type ?u.21252\n\u03bd : Type ?u.21255\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\n\u22a2 \u00acSet.Nonempty s \u2228 \u00acSet.Nonempty t \u2194 s = \u2205 \u2228 t = \u2205"}, {"tactic": "simp [not_nonempty_iff_eq_empty]", "state_before": "\u03b1 : Type u_2\n\u03b1' : Type ?u.21222\n\u03b2 : Type u_3\n\u03b2' : Type ?u.21228\n\u03b3 : Type u_1\n\u03b3' : Type ?u.21234\n\u03b4 : Type ?u.21237\n\u03b4' : Type ?u.21240\n\u03b5 : Type ?u.21243\n\u03b5' : Type ?u.21246\n\u03b6 : Type ?u.21249\n\u03b6' : Type ?u.21252\n\u03bd : Type ?u.21255\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\n\u22a2 \u00acSet.Nonempty s \u2228 \u00acSet.Nonempty t \u2194 s = \u2205 \u2228 t = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "full_name": "NNReal.rpow_eq_rpow_iff", "start": [246, 1], "end": [247, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Constructions.lean", "full_name": "closedEmbedding_inr", "start": [947, 1], "end": [948, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Fib.lean", "full_name": "Nat.fib_two", "start": [88, 1], "end": [89, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Clique.lean", "full_name": "SimpleGraph.cliqueFree_of_card_lt", "start": [227, 1], "end": [231, 86], "traced_tactics": [{"tactic": "by_contra h", "state_before": "\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\ninst\u271d : Fintype \u03b1\nhc : Fintype.card \u03b1 < n\n\u22a2 CliqueFree G n", "state_after": "\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\ninst\u271d : Fintype \u03b1\nhc : Fintype.card \u03b1 < n\nh : \u00acCliqueFree G n\n\u22a2 False"}, {"tactic": "refine' Nat.lt_le_antisymm hc _", "state_before": "\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\ninst\u271d : Fintype \u03b1\nhc : Fintype.card \u03b1 < n\nh : \u00acCliqueFree G n\n\u22a2 False", "state_after": "\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\ninst\u271d : Fintype \u03b1\nhc : Fintype.card \u03b1 < n\nh : \u00acCliqueFree G n\n\u22a2 n \u2264 Fintype.card \u03b1"}, {"tactic": "rw [cliqueFree_iff, not_isEmpty_iff] at h", "state_before": "\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\ninst\u271d : Fintype \u03b1\nhc : Fintype.card \u03b1 < n\nh : \u00acCliqueFree G n\n\u22a2 n \u2264 Fintype.card \u03b1", "state_after": "\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\ninst\u271d : Fintype \u03b1\nhc : Fintype.card \u03b1 < n\nh : Nonempty (\u22a4 \u21aag G)\n\u22a2 n \u2264 Fintype.card \u03b1"}, {"tactic": "simpa only [Fintype.card_fin] using Fintype.card_le_of_embedding h.some.toEmbedding", "state_before": "\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\ninst\u271d : Fintype \u03b1\nhc : Fintype.card \u03b1 < n\nh : Nonempty (\u22a4 \u21aag G)\n\u22a2 n \u2264 Fintype.card \u03b1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Setoid/Basic.lean", "full_name": "Quot.subsingleton_iff", "start": [471, 1], "end": [477, 45], "traced_tactics": [{"tactic": "simp only [_root_.subsingleton_iff, _root_.eq_top_iff, Pi.le_def, Pi.top_apply, forall_const]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.23424\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 Subsingleton (Quot r) \u2194 EqvGen r = \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.23424\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 (\u2200 (x y : Quot r), x = y) \u2194 \u2200 (i i_1 : \u03b1), \u22a4 \u2264 EqvGen r i i_1"}, {"tactic": "refine' (surjective_quot_mk _).forall.trans (forall_congr' fun a => _)", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.23424\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 (\u2200 (x y : Quot r), x = y) \u2194 \u2200 (i i_1 : \u03b1), \u22a4 \u2264 EqvGen r i i_1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.23424\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\n\u22a2 (\u2200 (y : Quot r), mk r a = y) \u2194 \u2200 (i : \u03b1), \u22a4 \u2264 EqvGen r a i"}, {"tactic": "refine' (surjective_quot_mk _).forall.trans (forall_congr' fun b => _)", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.23424\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\n\u22a2 (\u2200 (y : Quot r), mk r a = y) \u2194 \u2200 (i : \u03b1), \u22a4 \u2264 EqvGen r a i", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.23424\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\n\u22a2 mk r a = mk r b \u2194 \u22a4 \u2264 EqvGen r a b"}, {"tactic": "rw [Quot.eq]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.23424\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\n\u22a2 mk r a = mk r b \u2194 \u22a4 \u2264 EqvGen r a b", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.23424\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\n\u22a2 EqvGen r a b \u2194 \u22a4 \u2264 EqvGen r a b"}, {"tactic": "simp only [forall_const, le_Prop_eq, OrderTop.toTop, Pi.orderTop, Pi.top_apply,\n           Prop.top_eq_true, true_implies]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.23424\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\n\u22a2 EqvGen r a b \u2194 \u22a4 \u2264 EqvGen r a b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/PNat/Basic.lean", "full_name": "PNat.lt_add_right", "start": [256, 1], "end": [257, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.tendsto_iInf_iInf", "start": [3015, 1], "end": [3017, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/BoxIntegral/Partition/SubboxInduction.lean", "full_name": "BoxIntegral.Box.exists_taggedPartition_isHenstock_isSubordinate_homothetic", "start": [109, 1], "end": [137, 9], "traced_tactics": [{"tactic": "refine' subbox_induction_on I (fun J _ hJ => _) fun z _ => _", "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\n\u22a2 \u2203 \u03c0,\n    TaggedPrepartition.IsPartition \u03c0 \u2227\n      IsHenstock \u03c0 \u2227\n        IsSubordinate \u03c0 r \u2227\n          (\u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 \u2203 m, \u2200 (i : \u03b9), upper J i - lower J i = (upper I i - lower I i) / 2 ^ m) \u2227\n            TaggedPrepartition.distortion \u03c0 = distortion I", "state_after": "case refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\nhJ :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2203 \u03c0,\n        TaggedPrepartition.IsPartition \u03c0 \u2227\n          IsHenstock \u03c0 \u2227\n            IsSubordinate \u03c0 r \u2227\n              (\u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 \u2203 m, \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ m) \u2227\n                TaggedPrepartition.distortion \u03c0 = distortion J'\n\u22a2 \u2203 \u03c0,\n    TaggedPrepartition.IsPartition \u03c0 \u2227\n      IsHenstock \u03c0 \u2227\n        IsSubordinate \u03c0 r \u2227\n          (\u2200 (J_1 : Box \u03b9), J_1 \u2208 \u03c0 \u2192 \u2203 m, \u2200 (i : \u03b9), upper J_1 i - lower J_1 i = (upper J i - lower J i) / 2 ^ m) \u2227\n            TaggedPrepartition.distortion \u03c0 = distortion J\n\ncase refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nz : \u03b9 \u2192 \u211d\nx\u271d : z \u2208 \u2191Box.Icc I\n\u22a2 \u2203 U,\n    U \u2208 \ud835\udcdd[\u2191Box.Icc I] z \u2227\n      \u2200 (J : Box \u03b9),\n        J \u2264 I \u2192\n          \u2200 (m : \u2115),\n            z \u2208 \u2191Box.Icc J \u2192\n              \u2191Box.Icc J \u2286 U \u2192\n                (\u2200 (i : \u03b9), upper J i - lower J i = (upper I i - lower I i) / 2 ^ m) \u2192\n                  \u2203 \u03c0,\n                    TaggedPrepartition.IsPartition \u03c0 \u2227\n                      IsHenstock \u03c0 \u2227\n                        IsSubordinate \u03c0 r \u2227\n                          (\u2200 (J_1 : Box \u03b9),\n                              J_1 \u2208 \u03c0 \u2192 \u2203 m, \u2200 (i : \u03b9), upper J_1 i - lower J_1 i = (upper J i - lower J i) / 2 ^ m) \u2227\n                            TaggedPrepartition.distortion \u03c0 = distortion J"}, {"tactic": "choose! \u03c0i hP hHen hr Hn _ using hJ", "state_before": "case refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\nhJ :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2203 \u03c0,\n        TaggedPrepartition.IsPartition \u03c0 \u2227\n          IsHenstock \u03c0 \u2227\n            IsSubordinate \u03c0 r \u2227\n              (\u2200 (J : Box \u03b9), J \u2208 \u03c0 \u2192 \u2203 m, \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ m) \u2227\n                TaggedPrepartition.distortion \u03c0 = distortion J'\n\u22a2 \u2203 \u03c0,\n    TaggedPrepartition.IsPartition \u03c0 \u2227\n      IsHenstock \u03c0 \u2227\n        IsSubordinate \u03c0 r \u2227\n          (\u2200 (J_1 : Box \u03b9), J_1 \u2208 \u03c0 \u2192 \u2203 m, \u2200 (i : \u03b9), upper J_1 i - lower J_1 i = (upper J i - lower J i) / 2 ^ m) \u2227\n            TaggedPrepartition.distortion \u03c0 = distortion J", "state_after": "case refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\n\u03c0i : (J' : Box \u03b9) \u2192 TaggedPrepartition J'\nhP : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.IsPartition (\u03c0i J')\nhHen : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsHenstock (\u03c0i J')\nhr : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsSubordinate (\u03c0i J') r\nHn :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2200 (J : Box \u03b9), J \u2208 \u03c0i J' \u2192 \u2203 m, \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ m\n\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.distortion (\u03c0i J') = distortion J'\n\u22a2 \u2203 \u03c0,\n    TaggedPrepartition.IsPartition \u03c0 \u2227\n      IsHenstock \u03c0 \u2227\n        IsSubordinate \u03c0 r \u2227\n          (\u2200 (J_1 : Box \u03b9), J_1 \u2208 \u03c0 \u2192 \u2203 m, \u2200 (i : \u03b9), upper J_1 i - lower J_1 i = (upper J i - lower J i) / 2 ^ m) \u2227\n            TaggedPrepartition.distortion \u03c0 = distortion J"}, {"tactic": "choose! n hn using Hn", "state_before": "case refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\n\u03c0i : (J' : Box \u03b9) \u2192 TaggedPrepartition J'\nhP : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.IsPartition (\u03c0i J')\nhHen : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsHenstock (\u03c0i J')\nhr : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsSubordinate (\u03c0i J') r\nHn :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2200 (J : Box \u03b9), J \u2208 \u03c0i J' \u2192 \u2203 m, \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ m\n\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.distortion (\u03c0i J') = distortion J'\n\u22a2 \u2203 \u03c0,\n    TaggedPrepartition.IsPartition \u03c0 \u2227\n      IsHenstock \u03c0 \u2227\n        IsSubordinate \u03c0 r \u2227\n          (\u2200 (J_1 : Box \u03b9), J_1 \u2208 \u03c0 \u2192 \u2203 m, \u2200 (i : \u03b9), upper J_1 i - lower J_1 i = (upper J i - lower J i) / 2 ^ m) \u2227\n            TaggedPrepartition.distortion \u03c0 = distortion J", "state_after": "case refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\n\u03c0i : (J' : Box \u03b9) \u2192 TaggedPrepartition J'\nhP : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.IsPartition (\u03c0i J')\nhHen : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsHenstock (\u03c0i J')\nhr : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsSubordinate (\u03c0i J') r\n\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.distortion (\u03c0i J') = distortion J'\nn : Box \u03b9 \u2192 Box \u03b9 \u2192 \u2115\nhn :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2200 (J : Box \u03b9), J \u2208 \u03c0i J' \u2192 \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ n J' J\n\u22a2 \u2203 \u03c0,\n    TaggedPrepartition.IsPartition \u03c0 \u2227\n      IsHenstock \u03c0 \u2227\n        IsSubordinate \u03c0 r \u2227\n          (\u2200 (J_1 : Box \u03b9), J_1 \u2208 \u03c0 \u2192 \u2203 m, \u2200 (i : \u03b9), upper J_1 i - lower J_1 i = (upper J i - lower J i) / 2 ^ m) \u2227\n            TaggedPrepartition.distortion \u03c0 = distortion J"}, {"tactic": "have hP : ((splitCenter J).biUnionTagged \u03c0i).IsPartition :=\n  (isPartition_splitCenter _).biUnionTagged hP", "state_before": "case refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\n\u03c0i : (J' : Box \u03b9) \u2192 TaggedPrepartition J'\nhP : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.IsPartition (\u03c0i J')\nhHen : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsHenstock (\u03c0i J')\nhr : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsSubordinate (\u03c0i J') r\n\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.distortion (\u03c0i J') = distortion J'\nn : Box \u03b9 \u2192 Box \u03b9 \u2192 \u2115\nhn :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2200 (J : Box \u03b9), J \u2208 \u03c0i J' \u2192 \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ n J' J\n\u22a2 \u2203 \u03c0,\n    TaggedPrepartition.IsPartition \u03c0 \u2227\n      IsHenstock \u03c0 \u2227\n        IsSubordinate \u03c0 r \u2227\n          (\u2200 (J_1 : Box \u03b9), J_1 \u2208 \u03c0 \u2192 \u2203 m, \u2200 (i : \u03b9), upper J_1 i - lower J_1 i = (upper J i - lower J i) / 2 ^ m) \u2227\n            TaggedPrepartition.distortion \u03c0 = distortion J", "state_after": "case refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\n\u03c0i : (J' : Box \u03b9) \u2192 TaggedPrepartition J'\nhP\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.IsPartition (\u03c0i J')\nhHen : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsHenstock (\u03c0i J')\nhr : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsSubordinate (\u03c0i J') r\n\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.distortion (\u03c0i J') = distortion J'\nn : Box \u03b9 \u2192 Box \u03b9 \u2192 \u2115\nhn :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2200 (J : Box \u03b9), J \u2208 \u03c0i J' \u2192 \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ n J' J\nhP : TaggedPrepartition.IsPartition (biUnionTagged (splitCenter J) \u03c0i)\n\u22a2 \u2203 \u03c0,\n    TaggedPrepartition.IsPartition \u03c0 \u2227\n      IsHenstock \u03c0 \u2227\n        IsSubordinate \u03c0 r \u2227\n          (\u2200 (J_1 : Box \u03b9), J_1 \u2208 \u03c0 \u2192 \u2203 m, \u2200 (i : \u03b9), upper J_1 i - lower J_1 i = (upper J i - lower J i) / 2 ^ m) \u2227\n            TaggedPrepartition.distortion \u03c0 = distortion J"}, {"tactic": "have hsub : \u2200 J' \u2208 (splitCenter J).biUnionTagged \u03c0i, \u2203 n : \u2115, \u2200 i,\n    (J' : _).upper i - J'.lower i = (J.upper i - J.lower i) / 2 ^ n := by\n  intro J' hJ'\n  rcases (splitCenter J).mem_biUnionTagged.1 hJ' with \u27e8J\u2081, h\u2081, h\u2082\u27e9\n  refine' \u27e8n J\u2081 J' + 1, fun i => _\u27e9\n  simp only [hn J\u2081 h\u2081 J' h\u2082, upper_sub_lower_of_mem_splitCenter h\u2081, pow_succ, div_div]", "state_before": "case refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\n\u03c0i : (J' : Box \u03b9) \u2192 TaggedPrepartition J'\nhP\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.IsPartition (\u03c0i J')\nhHen : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsHenstock (\u03c0i J')\nhr : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsSubordinate (\u03c0i J') r\n\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.distortion (\u03c0i J') = distortion J'\nn : Box \u03b9 \u2192 Box \u03b9 \u2192 \u2115\nhn :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2200 (J : Box \u03b9), J \u2208 \u03c0i J' \u2192 \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ n J' J\nhP : TaggedPrepartition.IsPartition (biUnionTagged (splitCenter J) \u03c0i)\n\u22a2 \u2203 \u03c0,\n    TaggedPrepartition.IsPartition \u03c0 \u2227\n      IsHenstock \u03c0 \u2227\n        IsSubordinate \u03c0 r \u2227\n          (\u2200 (J_1 : Box \u03b9), J_1 \u2208 \u03c0 \u2192 \u2203 m, \u2200 (i : \u03b9), upper J_1 i - lower J_1 i = (upper J i - lower J i) / 2 ^ m) \u2227\n            TaggedPrepartition.distortion \u03c0 = distortion J", "state_after": "case refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\n\u03c0i : (J' : Box \u03b9) \u2192 TaggedPrepartition J'\nhP\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.IsPartition (\u03c0i J')\nhHen : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsHenstock (\u03c0i J')\nhr : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsSubordinate (\u03c0i J') r\n\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.distortion (\u03c0i J') = distortion J'\nn : Box \u03b9 \u2192 Box \u03b9 \u2192 \u2115\nhn :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2200 (J : Box \u03b9), J \u2208 \u03c0i J' \u2192 \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ n J' J\nhP : TaggedPrepartition.IsPartition (biUnionTagged (splitCenter J) \u03c0i)\nhsub :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 biUnionTagged (splitCenter J) \u03c0i \u2192 \u2203 n, \u2200 (i : \u03b9), upper J' i - lower J' i = (upper J i - lower J i) / 2 ^ n\n\u22a2 \u2203 \u03c0,\n    TaggedPrepartition.IsPartition \u03c0 \u2227\n      IsHenstock \u03c0 \u2227\n        IsSubordinate \u03c0 r \u2227\n          (\u2200 (J_1 : Box \u03b9), J_1 \u2208 \u03c0 \u2192 \u2203 m, \u2200 (i : \u03b9), upper J_1 i - lower J_1 i = (upper J i - lower J i) / 2 ^ m) \u2227\n            TaggedPrepartition.distortion \u03c0 = distortion J"}, {"tactic": "refine' \u27e8_, hP, isHenstock_biUnionTagged.2 hHen, isSubordinate_biUnionTagged.2 hr, hsub, _\u27e9", "state_before": "case refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\n\u03c0i : (J' : Box \u03b9) \u2192 TaggedPrepartition J'\nhP\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.IsPartition (\u03c0i J')\nhHen : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsHenstock (\u03c0i J')\nhr : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsSubordinate (\u03c0i J') r\n\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.distortion (\u03c0i J') = distortion J'\nn : Box \u03b9 \u2192 Box \u03b9 \u2192 \u2115\nhn :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2200 (J : Box \u03b9), J \u2208 \u03c0i J' \u2192 \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ n J' J\nhP : TaggedPrepartition.IsPartition (biUnionTagged (splitCenter J) \u03c0i)\nhsub :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 biUnionTagged (splitCenter J) \u03c0i \u2192 \u2203 n, \u2200 (i : \u03b9), upper J' i - lower J' i = (upper J i - lower J i) / 2 ^ n\n\u22a2 \u2203 \u03c0,\n    TaggedPrepartition.IsPartition \u03c0 \u2227\n      IsHenstock \u03c0 \u2227\n        IsSubordinate \u03c0 r \u2227\n          (\u2200 (J_1 : Box \u03b9), J_1 \u2208 \u03c0 \u2192 \u2203 m, \u2200 (i : \u03b9), upper J_1 i - lower J_1 i = (upper J i - lower J i) / 2 ^ m) \u2227\n            TaggedPrepartition.distortion \u03c0 = distortion J", "state_after": "case refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\n\u03c0i : (J' : Box \u03b9) \u2192 TaggedPrepartition J'\nhP\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.IsPartition (\u03c0i J')\nhHen : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsHenstock (\u03c0i J')\nhr : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsSubordinate (\u03c0i J') r\n\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.distortion (\u03c0i J') = distortion J'\nn : Box \u03b9 \u2192 Box \u03b9 \u2192 \u2115\nhn :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2200 (J : Box \u03b9), J \u2208 \u03c0i J' \u2192 \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ n J' J\nhP : TaggedPrepartition.IsPartition (biUnionTagged (splitCenter J) \u03c0i)\nhsub :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 biUnionTagged (splitCenter J) \u03c0i \u2192 \u2203 n, \u2200 (i : \u03b9), upper J' i - lower J' i = (upper J i - lower J i) / 2 ^ n\n\u22a2 TaggedPrepartition.distortion (biUnionTagged (splitCenter J) \u03c0i) = distortion J"}, {"tactic": "refine' TaggedPrepartition.distortion_of_const _ hP.nonempty_boxes fun J' h' => _", "state_before": "case refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\n\u03c0i : (J' : Box \u03b9) \u2192 TaggedPrepartition J'\nhP\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.IsPartition (\u03c0i J')\nhHen : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsHenstock (\u03c0i J')\nhr : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsSubordinate (\u03c0i J') r\n\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.distortion (\u03c0i J') = distortion J'\nn : Box \u03b9 \u2192 Box \u03b9 \u2192 \u2115\nhn :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2200 (J : Box \u03b9), J \u2208 \u03c0i J' \u2192 \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ n J' J\nhP : TaggedPrepartition.IsPartition (biUnionTagged (splitCenter J) \u03c0i)\nhsub :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 biUnionTagged (splitCenter J) \u03c0i \u2192 \u2203 n, \u2200 (i : \u03b9), upper J' i - lower J' i = (upper J i - lower J i) / 2 ^ n\n\u22a2 TaggedPrepartition.distortion (biUnionTagged (splitCenter J) \u03c0i) = distortion J", "state_after": "case refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\n\u03c0i : (J' : Box \u03b9) \u2192 TaggedPrepartition J'\nhP\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.IsPartition (\u03c0i J')\nhHen : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsHenstock (\u03c0i J')\nhr : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsSubordinate (\u03c0i J') r\n\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.distortion (\u03c0i J') = distortion J'\nn : Box \u03b9 \u2192 Box \u03b9 \u2192 \u2115\nhn :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2200 (J : Box \u03b9), J \u2208 \u03c0i J' \u2192 \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ n J' J\nhP : TaggedPrepartition.IsPartition (biUnionTagged (splitCenter J) \u03c0i)\nhsub :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 biUnionTagged (splitCenter J) \u03c0i \u2192 \u2203 n, \u2200 (i : \u03b9), upper J' i - lower J' i = (upper J i - lower J i) / 2 ^ n\nJ' : Box \u03b9\nh' : J' \u2208 biUnionTagged (splitCenter J) \u03c0i\n\u22a2 distortion J' = distortion J"}, {"tactic": "rcases hsub J' h' with \u27e8n, hn\u27e9", "state_before": "case refine'_1\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\n\u03c0i : (J' : Box \u03b9) \u2192 TaggedPrepartition J'\nhP\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.IsPartition (\u03c0i J')\nhHen : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsHenstock (\u03c0i J')\nhr : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsSubordinate (\u03c0i J') r\n\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.distortion (\u03c0i J') = distortion J'\nn : Box \u03b9 \u2192 Box \u03b9 \u2192 \u2115\nhn :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2200 (J : Box \u03b9), J \u2208 \u03c0i J' \u2192 \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ n J' J\nhP : TaggedPrepartition.IsPartition (biUnionTagged (splitCenter J) \u03c0i)\nhsub :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 biUnionTagged (splitCenter J) \u03c0i \u2192 \u2203 n, \u2200 (i : \u03b9), upper J' i - lower J' i = (upper J i - lower J i) / 2 ^ n\nJ' : Box \u03b9\nh' : J' \u2208 biUnionTagged (splitCenter J) \u03c0i\n\u22a2 distortion J' = distortion J", "state_after": "case refine'_1.intro\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\n\u03c0i : (J' : Box \u03b9) \u2192 TaggedPrepartition J'\nhP\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.IsPartition (\u03c0i J')\nhHen : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsHenstock (\u03c0i J')\nhr : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsSubordinate (\u03c0i J') r\n\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.distortion (\u03c0i J') = distortion J'\nn\u271d : Box \u03b9 \u2192 Box \u03b9 \u2192 \u2115\nhn\u271d :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2200 (J : Box \u03b9), J \u2208 \u03c0i J' \u2192 \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ n\u271d J' J\nhP : TaggedPrepartition.IsPartition (biUnionTagged (splitCenter J) \u03c0i)\nhsub :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 biUnionTagged (splitCenter J) \u03c0i \u2192 \u2203 n, \u2200 (i : \u03b9), upper J' i - lower J' i = (upper J i - lower J i) / 2 ^ n\nJ' : Box \u03b9\nh' : J' \u2208 biUnionTagged (splitCenter J) \u03c0i\nn : \u2115\nhn : \u2200 (i : \u03b9), upper J' i - lower J' i = (upper J i - lower J i) / 2 ^ n\n\u22a2 distortion J' = distortion J"}, {"tactic": "exact Box.distortion_eq_of_sub_eq_div hn", "state_before": "case refine'_1.intro\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\n\u03c0i : (J' : Box \u03b9) \u2192 TaggedPrepartition J'\nhP\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.IsPartition (\u03c0i J')\nhHen : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsHenstock (\u03c0i J')\nhr : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsSubordinate (\u03c0i J') r\n\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.distortion (\u03c0i J') = distortion J'\nn\u271d : Box \u03b9 \u2192 Box \u03b9 \u2192 \u2115\nhn\u271d :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2200 (J : Box \u03b9), J \u2208 \u03c0i J' \u2192 \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ n\u271d J' J\nhP : TaggedPrepartition.IsPartition (biUnionTagged (splitCenter J) \u03c0i)\nhsub :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 biUnionTagged (splitCenter J) \u03c0i \u2192 \u2203 n, \u2200 (i : \u03b9), upper J' i - lower J' i = (upper J i - lower J i) / 2 ^ n\nJ' : Box \u03b9\nh' : J' \u2208 biUnionTagged (splitCenter J) \u03c0i\nn : \u2115\nhn : \u2200 (i : \u03b9), upper J' i - lower J' i = (upper J i - lower J i) / 2 ^ n\n\u22a2 distortion J' = distortion J", "state_after": "no goals"}, {"tactic": "intro J' hJ'", "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\n\u03c0i : (J' : Box \u03b9) \u2192 TaggedPrepartition J'\nhP\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.IsPartition (\u03c0i J')\nhHen : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsHenstock (\u03c0i J')\nhr : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsSubordinate (\u03c0i J') r\n\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.distortion (\u03c0i J') = distortion J'\nn : Box \u03b9 \u2192 Box \u03b9 \u2192 \u2115\nhn :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2200 (J : Box \u03b9), J \u2208 \u03c0i J' \u2192 \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ n J' J\nhP : TaggedPrepartition.IsPartition (biUnionTagged (splitCenter J) \u03c0i)\n\u22a2 \u2200 (J' : Box \u03b9),\n    J' \u2208 biUnionTagged (splitCenter J) \u03c0i \u2192 \u2203 n, \u2200 (i : \u03b9), upper J' i - lower J' i = (upper J i - lower J i) / 2 ^ n", "state_after": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\n\u03c0i : (J' : Box \u03b9) \u2192 TaggedPrepartition J'\nhP\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.IsPartition (\u03c0i J')\nhHen : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsHenstock (\u03c0i J')\nhr : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsSubordinate (\u03c0i J') r\n\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.distortion (\u03c0i J') = distortion J'\nn : Box \u03b9 \u2192 Box \u03b9 \u2192 \u2115\nhn :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2200 (J : Box \u03b9), J \u2208 \u03c0i J' \u2192 \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ n J' J\nhP : TaggedPrepartition.IsPartition (biUnionTagged (splitCenter J) \u03c0i)\nJ' : Box \u03b9\nhJ' : J' \u2208 biUnionTagged (splitCenter J) \u03c0i\n\u22a2 \u2203 n, \u2200 (i : \u03b9), upper J' i - lower J' i = (upper J i - lower J i) / 2 ^ n"}, {"tactic": "rcases (splitCenter J).mem_biUnionTagged.1 hJ' with \u27e8J\u2081, h\u2081, h\u2082\u27e9", "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\n\u03c0i : (J' : Box \u03b9) \u2192 TaggedPrepartition J'\nhP\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.IsPartition (\u03c0i J')\nhHen : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsHenstock (\u03c0i J')\nhr : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsSubordinate (\u03c0i J') r\n\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.distortion (\u03c0i J') = distortion J'\nn : Box \u03b9 \u2192 Box \u03b9 \u2192 \u2115\nhn :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2200 (J : Box \u03b9), J \u2208 \u03c0i J' \u2192 \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ n J' J\nhP : TaggedPrepartition.IsPartition (biUnionTagged (splitCenter J) \u03c0i)\nJ' : Box \u03b9\nhJ' : J' \u2208 biUnionTagged (splitCenter J) \u03c0i\n\u22a2 \u2203 n, \u2200 (i : \u03b9), upper J' i - lower J' i = (upper J i - lower J i) / 2 ^ n", "state_after": "case intro.intro\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\n\u03c0i : (J' : Box \u03b9) \u2192 TaggedPrepartition J'\nhP\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.IsPartition (\u03c0i J')\nhHen : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsHenstock (\u03c0i J')\nhr : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsSubordinate (\u03c0i J') r\n\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.distortion (\u03c0i J') = distortion J'\nn : Box \u03b9 \u2192 Box \u03b9 \u2192 \u2115\nhn :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2200 (J : Box \u03b9), J \u2208 \u03c0i J' \u2192 \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ n J' J\nhP : TaggedPrepartition.IsPartition (biUnionTagged (splitCenter J) \u03c0i)\nJ' : Box \u03b9\nhJ' : J' \u2208 biUnionTagged (splitCenter J) \u03c0i\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 splitCenter J\nh\u2082 : J' \u2208 \u03c0i J\u2081\n\u22a2 \u2203 n, \u2200 (i : \u03b9), upper J' i - lower J' i = (upper J i - lower J i) / 2 ^ n"}, {"tactic": "refine' \u27e8n J\u2081 J' + 1, fun i => _\u27e9", "state_before": "case intro.intro\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\n\u03c0i : (J' : Box \u03b9) \u2192 TaggedPrepartition J'\nhP\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.IsPartition (\u03c0i J')\nhHen : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsHenstock (\u03c0i J')\nhr : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsSubordinate (\u03c0i J') r\n\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.distortion (\u03c0i J') = distortion J'\nn : Box \u03b9 \u2192 Box \u03b9 \u2192 \u2115\nhn :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2200 (J : Box \u03b9), J \u2208 \u03c0i J' \u2192 \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ n J' J\nhP : TaggedPrepartition.IsPartition (biUnionTagged (splitCenter J) \u03c0i)\nJ' : Box \u03b9\nhJ' : J' \u2208 biUnionTagged (splitCenter J) \u03c0i\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 splitCenter J\nh\u2082 : J' \u2208 \u03c0i J\u2081\n\u22a2 \u2203 n, \u2200 (i : \u03b9), upper J' i - lower J' i = (upper J i - lower J i) / 2 ^ n", "state_after": "case intro.intro\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\n\u03c0i : (J' : Box \u03b9) \u2192 TaggedPrepartition J'\nhP\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.IsPartition (\u03c0i J')\nhHen : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsHenstock (\u03c0i J')\nhr : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsSubordinate (\u03c0i J') r\n\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.distortion (\u03c0i J') = distortion J'\nn : Box \u03b9 \u2192 Box \u03b9 \u2192 \u2115\nhn :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2200 (J : Box \u03b9), J \u2208 \u03c0i J' \u2192 \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ n J' J\nhP : TaggedPrepartition.IsPartition (biUnionTagged (splitCenter J) \u03c0i)\nJ' : Box \u03b9\nhJ' : J' \u2208 biUnionTagged (splitCenter J) \u03c0i\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 splitCenter J\nh\u2082 : J' \u2208 \u03c0i J\u2081\ni : \u03b9\n\u22a2 upper J' i - lower J' i = (upper J i - lower J i) / 2 ^ (n J\u2081 J' + 1)"}, {"tactic": "simp only [hn J\u2081 h\u2081 J' h\u2082, upper_sub_lower_of_mem_splitCenter h\u2081, pow_succ, div_div]", "state_before": "case intro.intro\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nJ : Box \u03b9\nx\u271d : J \u2264 I\n\u03c0i : (J' : Box \u03b9) \u2192 TaggedPrepartition J'\nhP\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.IsPartition (\u03c0i J')\nhHen : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsHenstock (\u03c0i J')\nhr : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 IsSubordinate (\u03c0i J') r\n\u271d : \u2200 (J' : Box \u03b9), J' \u2208 splitCenter J \u2192 TaggedPrepartition.distortion (\u03c0i J') = distortion J'\nn : Box \u03b9 \u2192 Box \u03b9 \u2192 \u2115\nhn :\n  \u2200 (J' : Box \u03b9),\n    J' \u2208 splitCenter J \u2192\n      \u2200 (J : Box \u03b9), J \u2208 \u03c0i J' \u2192 \u2200 (i : \u03b9), upper J i - lower J i = (upper J' i - lower J' i) / 2 ^ n J' J\nhP : TaggedPrepartition.IsPartition (biUnionTagged (splitCenter J) \u03c0i)\nJ' : Box \u03b9\nhJ' : J' \u2208 biUnionTagged (splitCenter J) \u03c0i\nJ\u2081 : Box \u03b9\nh\u2081 : J\u2081 \u2208 splitCenter J\nh\u2082 : J' \u2208 \u03c0i J\u2081\ni : \u03b9\n\u22a2 upper J' i - lower J' i = (upper J i - lower J i) / 2 ^ (n J\u2081 J' + 1)", "state_after": "no goals"}, {"tactic": "refine' \u27e8Box.Icc I \u2229 closedBall z (r z),\n  inter_mem_nhdsWithin _ (closedBall_mem_nhds _ (r z).coe_prop), _\u27e9", "state_before": "case refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nz : \u03b9 \u2192 \u211d\nx\u271d : z \u2208 \u2191Box.Icc I\n\u22a2 \u2203 U,\n    U \u2208 \ud835\udcdd[\u2191Box.Icc I] z \u2227\n      \u2200 (J : Box \u03b9),\n        J \u2264 I \u2192\n          \u2200 (m : \u2115),\n            z \u2208 \u2191Box.Icc J \u2192\n              \u2191Box.Icc J \u2286 U \u2192\n                (\u2200 (i : \u03b9), upper J i - lower J i = (upper I i - lower I i) / 2 ^ m) \u2192\n                  \u2203 \u03c0,\n                    TaggedPrepartition.IsPartition \u03c0 \u2227\n                      IsHenstock \u03c0 \u2227\n                        IsSubordinate \u03c0 r \u2227\n                          (\u2200 (J_1 : Box \u03b9),\n                              J_1 \u2208 \u03c0 \u2192 \u2203 m, \u2200 (i : \u03b9), upper J_1 i - lower J_1 i = (upper J i - lower J i) / 2 ^ m) \u2227\n                            TaggedPrepartition.distortion \u03c0 = distortion J", "state_after": "case refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nz : \u03b9 \u2192 \u211d\nx\u271d : z \u2208 \u2191Box.Icc I\n\u22a2 \u2200 (J : Box \u03b9),\n    J \u2264 I \u2192\n      \u2200 (m : \u2115),\n        z \u2208 \u2191Box.Icc J \u2192\n          \u2191Box.Icc J \u2286 \u2191Box.Icc I \u2229 closedBall z \u2191(r z) \u2192\n            (\u2200 (i : \u03b9), upper J i - lower J i = (upper I i - lower I i) / 2 ^ m) \u2192\n              \u2203 \u03c0,\n                TaggedPrepartition.IsPartition \u03c0 \u2227\n                  IsHenstock \u03c0 \u2227\n                    IsSubordinate \u03c0 r \u2227\n                      (\u2200 (J_1 : Box \u03b9),\n                          J_1 \u2208 \u03c0 \u2192 \u2203 m, \u2200 (i : \u03b9), upper J_1 i - lower J_1 i = (upper J i - lower J i) / 2 ^ m) \u2227\n                        TaggedPrepartition.distortion \u03c0 = distortion J"}, {"tactic": "intro J _ n Hmem HIcc Hsub", "state_before": "case refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nz : \u03b9 \u2192 \u211d\nx\u271d : z \u2208 \u2191Box.Icc I\n\u22a2 \u2200 (J : Box \u03b9),\n    J \u2264 I \u2192\n      \u2200 (m : \u2115),\n        z \u2208 \u2191Box.Icc J \u2192\n          \u2191Box.Icc J \u2286 \u2191Box.Icc I \u2229 closedBall z \u2191(r z) \u2192\n            (\u2200 (i : \u03b9), upper J i - lower J i = (upper I i - lower I i) / 2 ^ m) \u2192\n              \u2203 \u03c0,\n                TaggedPrepartition.IsPartition \u03c0 \u2227\n                  IsHenstock \u03c0 \u2227\n                    IsSubordinate \u03c0 r \u2227\n                      (\u2200 (J_1 : Box \u03b9),\n                          J_1 \u2208 \u03c0 \u2192 \u2203 m, \u2200 (i : \u03b9), upper J_1 i - lower J_1 i = (upper J i - lower J i) / 2 ^ m) \u2227\n                        TaggedPrepartition.distortion \u03c0 = distortion J", "state_after": "case refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nz : \u03b9 \u2192 \u211d\nx\u271d : z \u2208 \u2191Box.Icc I\nJ : Box \u03b9\na\u271d : J \u2264 I\nn : \u2115\nHmem : z \u2208 \u2191Box.Icc J\nHIcc : \u2191Box.Icc J \u2286 \u2191Box.Icc I \u2229 closedBall z \u2191(r z)\nHsub : \u2200 (i : \u03b9), upper J i - lower J i = (upper I i - lower I i) / 2 ^ n\n\u22a2 \u2203 \u03c0,\n    TaggedPrepartition.IsPartition \u03c0 \u2227\n      IsHenstock \u03c0 \u2227\n        IsSubordinate \u03c0 r \u2227\n          (\u2200 (J_1 : Box \u03b9), J_1 \u2208 \u03c0 \u2192 \u2203 m, \u2200 (i : \u03b9), upper J_1 i - lower J_1 i = (upper J i - lower J i) / 2 ^ m) \u2227\n            TaggedPrepartition.distortion \u03c0 = distortion J"}, {"tactic": "rw [Set.subset_inter_iff] at HIcc", "state_before": "case refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nz : \u03b9 \u2192 \u211d\nx\u271d : z \u2208 \u2191Box.Icc I\nJ : Box \u03b9\na\u271d : J \u2264 I\nn : \u2115\nHmem : z \u2208 \u2191Box.Icc J\nHIcc : \u2191Box.Icc J \u2286 \u2191Box.Icc I \u2229 closedBall z \u2191(r z)\nHsub : \u2200 (i : \u03b9), upper J i - lower J i = (upper I i - lower I i) / 2 ^ n\n\u22a2 \u2203 \u03c0,\n    TaggedPrepartition.IsPartition \u03c0 \u2227\n      IsHenstock \u03c0 \u2227\n        IsSubordinate \u03c0 r \u2227\n          (\u2200 (J_1 : Box \u03b9), J_1 \u2208 \u03c0 \u2192 \u2203 m, \u2200 (i : \u03b9), upper J_1 i - lower J_1 i = (upper J i - lower J i) / 2 ^ m) \u2227\n            TaggedPrepartition.distortion \u03c0 = distortion J", "state_after": "case refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nz : \u03b9 \u2192 \u211d\nx\u271d : z \u2208 \u2191Box.Icc I\nJ : Box \u03b9\na\u271d : J \u2264 I\nn : \u2115\nHmem : z \u2208 \u2191Box.Icc J\nHIcc : \u2191Box.Icc J \u2286 \u2191Box.Icc I \u2227 \u2191Box.Icc J \u2286 closedBall z \u2191(r z)\nHsub : \u2200 (i : \u03b9), upper J i - lower J i = (upper I i - lower I i) / 2 ^ n\n\u22a2 \u2203 \u03c0,\n    TaggedPrepartition.IsPartition \u03c0 \u2227\n      IsHenstock \u03c0 \u2227\n        IsSubordinate \u03c0 r \u2227\n          (\u2200 (J_1 : Box \u03b9), J_1 \u2208 \u03c0 \u2192 \u2203 m, \u2200 (i : \u03b9), upper J_1 i - lower J_1 i = (upper J i - lower J i) / 2 ^ m) \u2227\n            TaggedPrepartition.distortion \u03c0 = distortion J"}, {"tactic": "refine' \u27e8single _ _ le_rfl _ Hmem, isPartition_single _, isHenstock_single _,\n  (isSubordinate_single _ _).2 HIcc.2, _, distortion_single _ _\u27e9", "state_before": "case refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nz : \u03b9 \u2192 \u211d\nx\u271d : z \u2208 \u2191Box.Icc I\nJ : Box \u03b9\na\u271d : J \u2264 I\nn : \u2115\nHmem : z \u2208 \u2191Box.Icc J\nHIcc : \u2191Box.Icc J \u2286 \u2191Box.Icc I \u2227 \u2191Box.Icc J \u2286 closedBall z \u2191(r z)\nHsub : \u2200 (i : \u03b9), upper J i - lower J i = (upper I i - lower I i) / 2 ^ n\n\u22a2 \u2203 \u03c0,\n    TaggedPrepartition.IsPartition \u03c0 \u2227\n      IsHenstock \u03c0 \u2227\n        IsSubordinate \u03c0 r \u2227\n          (\u2200 (J_1 : Box \u03b9), J_1 \u2208 \u03c0 \u2192 \u2203 m, \u2200 (i : \u03b9), upper J_1 i - lower J_1 i = (upper J i - lower J i) / 2 ^ m) \u2227\n            TaggedPrepartition.distortion \u03c0 = distortion J", "state_after": "case refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nz : \u03b9 \u2192 \u211d\nx\u271d : z \u2208 \u2191Box.Icc I\nJ : Box \u03b9\na\u271d : J \u2264 I\nn : \u2115\nHmem : z \u2208 \u2191Box.Icc J\nHIcc : \u2191Box.Icc J \u2286 \u2191Box.Icc I \u2227 \u2191Box.Icc J \u2286 closedBall z \u2191(r z)\nHsub : \u2200 (i : \u03b9), upper J i - lower J i = (upper I i - lower I i) / 2 ^ n\n\u22a2 \u2200 (J_1 : Box \u03b9),\n    J_1 \u2208 TaggedPrepartition.single J J (_ : J \u2264 J) z Hmem \u2192\n      \u2203 m, \u2200 (i : \u03b9), upper J_1 i - lower J_1 i = (upper J i - lower J i) / 2 ^ m"}, {"tactic": "simp only [TaggedPrepartition.mem_single, forall_eq]", "state_before": "case refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nz : \u03b9 \u2192 \u211d\nx\u271d : z \u2208 \u2191Box.Icc I\nJ : Box \u03b9\na\u271d : J \u2264 I\nn : \u2115\nHmem : z \u2208 \u2191Box.Icc J\nHIcc : \u2191Box.Icc J \u2286 \u2191Box.Icc I \u2227 \u2191Box.Icc J \u2286 closedBall z \u2191(r z)\nHsub : \u2200 (i : \u03b9), upper J i - lower J i = (upper I i - lower I i) / 2 ^ n\n\u22a2 \u2200 (J_1 : Box \u03b9),\n    J_1 \u2208 TaggedPrepartition.single J J (_ : J \u2264 J) z Hmem \u2192\n      \u2203 m, \u2200 (i : \u03b9), upper J_1 i - lower J_1 i = (upper J i - lower J i) / 2 ^ m", "state_after": "case refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nz : \u03b9 \u2192 \u211d\nx\u271d : z \u2208 \u2191Box.Icc I\nJ : Box \u03b9\na\u271d : J \u2264 I\nn : \u2115\nHmem : z \u2208 \u2191Box.Icc J\nHIcc : \u2191Box.Icc J \u2286 \u2191Box.Icc I \u2227 \u2191Box.Icc J \u2286 closedBall z \u2191(r z)\nHsub : \u2200 (i : \u03b9), upper J i - lower J i = (upper I i - lower I i) / 2 ^ n\n\u22a2 \u2203 m, \u2200 (i : \u03b9), upper J i - lower J i = (upper J i - lower J i) / 2 ^ m"}, {"tactic": "refine' \u27e80, fun i => _\u27e9", "state_before": "case refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nz : \u03b9 \u2192 \u211d\nx\u271d : z \u2208 \u2191Box.Icc I\nJ : Box \u03b9\na\u271d : J \u2264 I\nn : \u2115\nHmem : z \u2208 \u2191Box.Icc J\nHIcc : \u2191Box.Icc J \u2286 \u2191Box.Icc I \u2227 \u2191Box.Icc J \u2286 closedBall z \u2191(r z)\nHsub : \u2200 (i : \u03b9), upper J i - lower J i = (upper I i - lower I i) / 2 ^ n\n\u22a2 \u2203 m, \u2200 (i : \u03b9), upper J i - lower J i = (upper J i - lower J i) / 2 ^ m", "state_after": "case refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nz : \u03b9 \u2192 \u211d\nx\u271d : z \u2208 \u2191Box.Icc I\nJ : Box \u03b9\na\u271d : J \u2264 I\nn : \u2115\nHmem : z \u2208 \u2191Box.Icc J\nHIcc : \u2191Box.Icc J \u2286 \u2191Box.Icc I \u2227 \u2191Box.Icc J \u2286 closedBall z \u2191(r z)\nHsub : \u2200 (i : \u03b9), upper J i - lower J i = (upper I i - lower I i) / 2 ^ n\ni : \u03b9\n\u22a2 upper J i - lower J i = (upper J i - lower J i) / 2 ^ 0"}, {"tactic": "simp", "state_before": "case refine'_2\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J\u271d I : Box \u03b9\nr : (\u03b9 \u2192 \u211d) \u2192 \u2191(Ioi 0)\nz : \u03b9 \u2192 \u211d\nx\u271d : z \u2208 \u2191Box.Icc I\nJ : Box \u03b9\na\u271d : J \u2264 I\nn : \u2115\nHmem : z \u2208 \u2191Box.Icc J\nHIcc : \u2191Box.Icc J \u2286 \u2191Box.Icc I \u2227 \u2191Box.Icc J \u2286 closedBall z \u2191(r z)\nHsub : \u2200 (i : \u03b9), upper J i - lower J i = (upper I i - lower I i) / 2 ^ n\ni : \u03b9\n\u22a2 upper J i - lower J i = (upper J i - lower J i) / 2 ^ 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.generate_union", "start": [542, 1], "end": [544, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Int/Lemmas.lean", "full_name": "Int.negOfNat_mul_negSucc", "start": [391, 1], "end": [392, 56], "traced_tactics": [{"tactic": "rw [Int.mul_comm, negSucc_mul_negOfNat, Nat.mul_comm]", "state_before": "m n : Nat\n\u22a2 negOfNat n * -[m+1] = ofNat (n * succ m)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Dfinsupp/NeLocus.lean", "full_name": "Dfinsupp.neLocus_sub_left", "start": [153, 1], "end": [154, 75], "traced_tactics": [{"tactic": "simp only [sub_eq_add_neg, @neLocus_add_left \u03b1 N _ _ _, neLocus_neg_neg]", "state_before": "\u03b1 : Type u_1\nN : \u03b1 \u2192 Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : (a : \u03b1) \u2192 DecidableEq (N a)\ninst\u271d : (a : \u03b1) \u2192 AddGroup (N a)\nf f\u2081 f\u2082 g g\u2081 g\u2082 : \u03a0\u2080 (a : \u03b1), N a\n\u22a2 neLocus (f - g\u2081) (f - g\u2082) = neLocus g\u2081 g\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.union_congr_left", "start": [1485, 1], "end": [1486, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.pair_eq_singleton", "start": [1381, 1], "end": [1382, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Basic.lean", "full_name": "Submodule.comap_smul'", "start": [1085, 1], "end": [1087, 56], "traced_tactics": [{"tactic": "classical by_cases h : a = 0 <;> simp [h, comap_smul]", "state_before": "R : Type ?u.964606\nR\u2081 : Type ?u.964609\nR\u2082 : Type ?u.964612\nR\u2083 : Type ?u.964615\nR\u2084 : Type ?u.964618\nS : Type ?u.964621\nK : Type u_1\nK\u2082 : Type ?u.964627\nM : Type ?u.964630\nM' : Type ?u.964633\nM\u2081 : Type ?u.964636\nM\u2082 : Type ?u.964639\nM\u2083 : Type ?u.964642\nM\u2084 : Type ?u.964645\nN : Type ?u.964648\nN\u2082 : Type ?u.964651\n\u03b9 : Type ?u.964654\nV : Type u_2\nV\u2082 : Type u_3\ninst\u271d\u2074 : Semifield K\ninst\u271d\u00b3 : AddCommMonoid V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : AddCommMonoid V\u2082\ninst\u271d : Module K V\u2082\nf : V \u2192\u2097[K] V\u2082\np : Submodule K V\u2082\na : K\n\u22a2 comap (a \u2022 f) p = \u2a05 (_ : a \u2260 0), comap f p", "state_after": "no goals"}, {"tactic": "by_cases h : a = 0 <;> simp [h, comap_smul]", "state_before": "R : Type ?u.964606\nR\u2081 : Type ?u.964609\nR\u2082 : Type ?u.964612\nR\u2083 : Type ?u.964615\nR\u2084 : Type ?u.964618\nS : Type ?u.964621\nK : Type u_1\nK\u2082 : Type ?u.964627\nM : Type ?u.964630\nM' : Type ?u.964633\nM\u2081 : Type ?u.964636\nM\u2082 : Type ?u.964639\nM\u2083 : Type ?u.964642\nM\u2084 : Type ?u.964645\nN : Type ?u.964648\nN\u2082 : Type ?u.964651\n\u03b9 : Type ?u.964654\nV : Type u_2\nV\u2082 : Type u_3\ninst\u271d\u2074 : Semifield K\ninst\u271d\u00b3 : AddCommMonoid V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : AddCommMonoid V\u2082\ninst\u271d : Module K V\u2082\nf : V \u2192\u2097[K] V\u2082\np : Submodule K V\u2082\na : K\n\u22a2 comap (a \u2022 f) p = \u2a05 (_ : a \u2260 0), comap f p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "full_name": "norm_sub_eq_norm_add", "start": [1526, 1], "end": [1530, 14], "traced_tactics": [{"tactic": "rw [\u2190 mul_self_inj_of_nonneg (norm_nonneg _) (norm_nonneg _)]", "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.3120837\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nv w : E\nh : inner v w = 0\n\u22a2 \u2016w - v\u2016 = \u2016w + v\u2016", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.3120837\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nv w : E\nh : inner v w = 0\n\u22a2 \u2016w - v\u2016 * \u2016w - v\u2016 = \u2016w + v\u2016 * \u2016w + v\u2016"}, {"tactic": "simp only [h, \u2190 @inner_self_eq_norm_mul_norm \ud835\udd5c, sub_neg_eq_add, sub_zero, map_sub, zero_re',\n  zero_sub, add_zero, map_add, inner_add_right, inner_sub_left, inner_sub_right, inner_re_symm,\n  zero_add]", "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.3120837\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nv w : E\nh : inner v w = 0\n\u22a2 \u2016w - v\u2016 * \u2016w - v\u2016 = \u2016w + v\u2016 * \u2016w + v\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Bases.lean", "full_name": "TopologicalSpace.IsSeparable.mono", "start": [385, 1], "end": [387, 34], "traced_tactics": [{"tactic": "rcases hs with \u27e8c, c_count, hs\u27e9", "state_before": "\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns u : Set \u03b1\nhs : IsSeparable s\nhu : u \u2286 s\n\u22a2 IsSeparable u", "state_after": "case intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns u : Set \u03b1\nhu : u \u2286 s\nc : Set \u03b1\nc_count : Set.Countable c\nhs : s \u2286 closure c\n\u22a2 IsSeparable u"}, {"tactic": "exact \u27e8c, c_count, hu.trans hs\u27e9", "state_before": "case intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\ns u : Set \u03b1\nhu : u \u2286 s\nc : Set \u03b1\nc_count : Set.Countable c\nhs : s \u2286 closure c\n\u22a2 IsSeparable u", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.hasFiniteIntegral_add_measure", "start": [212, 1], "end": [214, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.subset_iInter\u2082", "start": [299, 1], "end": [301, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Sylow.lean", "full_name": "Sylow.normal_of_normalizerCondition", "start": [754, 1], "end": [757, 97], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Bases.lean", "full_name": "TopologicalSpace.IsTopologicalBasis.dense_iff", "start": [220, 1], "end": [223, 79], "traced_tactics": [{"tactic": "simp only [Dense, hb.mem_closure_iff]", "state_before": "\u03b1 : Type u\nt : TopologicalSpace \u03b1\nb : Set (Set \u03b1)\nhb : IsTopologicalBasis b\ns : Set \u03b1\n\u22a2 Dense s \u2194 \u2200 (o : Set \u03b1), o \u2208 b \u2192 Set.Nonempty o \u2192 Set.Nonempty (o \u2229 s)", "state_after": "\u03b1 : Type u\nt : TopologicalSpace \u03b1\nb : Set (Set \u03b1)\nhb : IsTopologicalBasis b\ns : Set \u03b1\n\u22a2 (\u2200 (x : \u03b1) (o : Set \u03b1), o \u2208 b \u2192 x \u2208 o \u2192 Set.Nonempty (o \u2229 s)) \u2194\n    \u2200 (o : Set \u03b1), o \u2208 b \u2192 Set.Nonempty o \u2192 Set.Nonempty (o \u2229 s)"}, {"tactic": "exact \u27e8fun h o hb \u27e8a, ha\u27e9 => h a o hb ha, fun h a o hb ha => h o hb \u27e8a, ha\u27e9\u27e9", "state_before": "\u03b1 : Type u\nt : TopologicalSpace \u03b1\nb : Set (Set \u03b1)\nhb : IsTopologicalBasis b\ns : Set \u03b1\n\u22a2 (\u2200 (x : \u03b1) (o : Set \u03b1), o \u2208 b \u2192 x \u2208 o \u2192 Set.Nonempty (o \u2229 s)) \u2194\n    \u2200 (o : Set \u03b1), o \u2208 b \u2192 Set.Nonempty o \u2192 Set.Nonempty (o \u2229 s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Finprod.lean", "full_name": "finprod_mem_coe_finset", "start": [527, 1], "end": [529, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Composition.lean", "full_name": "Composition.lt_sizeUpTo_index_succ", "start": [339, 1], "end": [340, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/GaloisConnection.lean", "full_name": "sInf_image2_eq_sInf_sSup", "start": [408, 1], "end": [411, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/MeasurableSpace.lean", "full_name": "MeasurableSet.measurableSet_limsup", "start": [1905, 1], "end": [1907, 74], "traced_tactics": [{"tactic": "simpa only [\u2190 blimsup_true] using measurableSet_blimsup fun n _ => hs n", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.912025\n\u03b3 : Type ?u.912028\n\u03b4 : Type ?u.912031\n\u03b4' : Type ?u.912034\n\u03b9 : Sort u\u03b9\ns\u271d t u : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\ns : \u2115 \u2192 Set \u03b1\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\n\u22a2 MeasurableSet (limsup s atTop)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Archimedean.lean", "full_name": "existsUnique_add_zsmul_mem_Ioc", "start": [98, 1], "end": [103, 48], "traced_tactics": [{"tactic": "simpa only [add_zsmul, sub_lt_iff_lt_add', le_sub_iff_add_le', \u2190 add_assoc, and_comm, mem_Ioc,\n  Equiv.coe_addRight, one_zsmul, add_le_add_iff_right] using\n  existsUnique_zsmul_near_of_pos ha (c - b)", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup \u03b1\ninst\u271d : Archimedean \u03b1\na : \u03b1\nha : 0 < a\nb c : \u03b1\n\u22a2 \u2203! x, b + \u2191(Equiv.addRight 1) x \u2022 a \u2208 Ioc c (c + a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Algebraic.lean", "full_name": "Algebra.isAlgebraic_trans", "start": [217, 1], "end": [220, 37], "traced_tactics": [{"tactic": "simp only [IsAlgebraic, isAlgebraic_iff_isIntegral] at L_alg A_alg\u22a2", "state_before": "K : Type u_1\nL : Type u_2\nR : Type ?u.254369\nS : Type ?u.254372\nA : Type u_3\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra L A\ninst\u271d\u2075 : Algebra K A\ninst\u271d\u2074 : IsScalarTower K L A\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R S A\nL_alg : IsAlgebraic K L\nA_alg : IsAlgebraic L A\n\u22a2 IsAlgebraic K A", "state_after": "K : Type u_1\nL : Type u_2\nR : Type ?u.254369\nS : Type ?u.254372\nA : Type u_3\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra L A\ninst\u271d\u2075 : Algebra K A\ninst\u271d\u2074 : IsScalarTower K L A\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R S A\nL_alg : \u2200 (x : L), IsIntegral K x\nA_alg : \u2200 (x : A), IsIntegral L x\n\u22a2 \u2200 (x : A), IsIntegral K x"}, {"tactic": "exact isIntegral_trans L_alg A_alg", "state_before": "K : Type u_1\nL : Type u_2\nR : Type ?u.254369\nS : Type ?u.254372\nA : Type u_3\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra L A\ninst\u271d\u2075 : Algebra K A\ninst\u271d\u2074 : IsScalarTower K L A\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : IsScalarTower R S A\nL_alg : \u2200 (x : L), IsIntegral K x\nA_alg : \u2200 (x : A), IsIntegral L x\n\u22a2 \u2200 (x : A), IsIntegral K x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean", "full_name": "gramSchmidt_orthonormal", "start": [296, 1], "end": [305, 41], "traced_tactics": [{"tactic": "unfold Orthonormal", "state_before": "\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nh\u2080 : LinearIndependent \ud835\udd5c f\n\u22a2 Orthonormal \ud835\udd5c (gramSchmidtNormed \ud835\udd5c f)", "state_after": "\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nh\u2080 : LinearIndependent \ud835\udd5c f\n\u22a2 (\u2200 (i : \u03b9), \u2016gramSchmidtNormed \ud835\udd5c f i\u2016 = 1) \u2227\n    \u2200 {i j : \u03b9}, i \u2260 j \u2192 inner (gramSchmidtNormed \ud835\udd5c f i) (gramSchmidtNormed \ud835\udd5c f j) = 0"}, {"tactic": "constructor", "state_before": "\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nh\u2080 : LinearIndependent \ud835\udd5c f\n\u22a2 (\u2200 (i : \u03b9), \u2016gramSchmidtNormed \ud835\udd5c f i\u2016 = 1) \u2227\n    \u2200 {i j : \u03b9}, i \u2260 j \u2192 inner (gramSchmidtNormed \ud835\udd5c f i) (gramSchmidtNormed \ud835\udd5c f j) = 0", "state_after": "case left\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nh\u2080 : LinearIndependent \ud835\udd5c f\n\u22a2 \u2200 (i : \u03b9), \u2016gramSchmidtNormed \ud835\udd5c f i\u2016 = 1\n\ncase right\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nh\u2080 : LinearIndependent \ud835\udd5c f\n\u22a2 \u2200 {i j : \u03b9}, i \u2260 j \u2192 inner (gramSchmidtNormed \ud835\udd5c f i) (gramSchmidtNormed \ud835\udd5c f j) = 0"}, {"tactic": "simp only [gramSchmidtNormed_unit_length, h\u2080, eq_self_iff_true, imp_true_iff]", "state_before": "case left\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nh\u2080 : LinearIndependent \ud835\udd5c f\n\u22a2 \u2200 (i : \u03b9), \u2016gramSchmidtNormed \ud835\udd5c f i\u2016 = 1", "state_after": "no goals"}, {"tactic": "intro i j hij", "state_before": "case right\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nh\u2080 : LinearIndependent \ud835\udd5c f\n\u22a2 \u2200 {i j : \u03b9}, i \u2260 j \u2192 inner (gramSchmidtNormed \ud835\udd5c f i) (gramSchmidtNormed \ud835\udd5c f j) = 0", "state_after": "case right\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nh\u2080 : LinearIndependent \ud835\udd5c f\ni j : \u03b9\nhij : i \u2260 j\n\u22a2 inner (gramSchmidtNormed \ud835\udd5c f i) (gramSchmidtNormed \ud835\udd5c f j) = 0"}, {"tactic": "simp only [gramSchmidtNormed, inner_smul_left, inner_smul_right, IsROrC.conj_inv,\n  IsROrC.conj_ofReal, mul_eq_zero, inv_eq_zero, IsROrC.ofReal_eq_zero, norm_eq_zero]", "state_before": "case right\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nh\u2080 : LinearIndependent \ud835\udd5c f\ni j : \u03b9\nhij : i \u2260 j\n\u22a2 inner (gramSchmidtNormed \ud835\udd5c f i) (gramSchmidtNormed \ud835\udd5c f j) = 0", "state_after": "case right\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nh\u2080 : LinearIndependent \ud835\udd5c f\ni j : \u03b9\nhij : i \u2260 j\n\u22a2 gramSchmidt \ud835\udd5c f j = 0 \u2228 gramSchmidt \ud835\udd5c f i = 0 \u2228 inner (gramSchmidt \ud835\udd5c f i) (gramSchmidt \ud835\udd5c f j) = 0"}, {"tactic": "repeat' right", "state_before": "case right\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nh\u2080 : LinearIndependent \ud835\udd5c f\ni j : \u03b9\nhij : i \u2260 j\n\u22a2 gramSchmidt \ud835\udd5c f j = 0 \u2228 gramSchmidt \ud835\udd5c f i = 0 \u2228 inner (gramSchmidt \ud835\udd5c f i) (gramSchmidt \ud835\udd5c f j) = 0", "state_after": "case right.h.h\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nh\u2080 : LinearIndependent \ud835\udd5c f\ni j : \u03b9\nhij : i \u2260 j\n\u22a2 inner (gramSchmidt \ud835\udd5c f i) (gramSchmidt \ud835\udd5c f j) = 0"}, {"tactic": "exact gramSchmidt_orthogonal \ud835\udd5c f hij", "state_before": "case right.h.h\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nh\u2080 : LinearIndependent \ud835\udd5c f\ni j : \u03b9\nhij : i \u2260 j\n\u22a2 inner (gramSchmidt \ud835\udd5c f i) (gramSchmidt \ud835\udd5c f j) = 0", "state_after": "no goals"}, {"tactic": "right", "state_before": "case right.h.h\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nh\u2080 : LinearIndependent \ud835\udd5c f\ni j : \u03b9\nhij : i \u2260 j\n\u22a2 inner (gramSchmidt \ud835\udd5c f i) (gramSchmidt \ud835\udd5c f j) = 0", "state_after": "case right.h.h\n\ud835\udd5c : Type u_2\nE : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nh\u2080 : LinearIndependent \ud835\udd5c f\ni j : \u03b9\nhij : i \u2260 j\n\u22a2 inner (gramSchmidt \ud835\udd5c f i) (gramSchmidt \ud835\udd5c f j) = 0"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "mul_singleton_mem_nhds_of_nhds_one", "start": [1323, 1], "end": [1324, 56], "traced_tactics": [{"tactic": "simpa only [one_mul] using mul_singleton_mem_nhds a h", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nG : Type w\nH : Type x\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : ContinuousConstSMul \u03b1\u1d50\u1d52\u1d56 \u03b1\ns t : Set \u03b1\na : \u03b1\nh : s \u2208 \ud835\udcdd 1\n\u22a2 s * {a} \u2208 \ud835\udcdd a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.testAgainstNN_add", "start": [383, 1], "end": [387, 94], "traced_tactics": [{"tactic": "simp only [\u2190 ENNReal.coe_eq_coe, BoundedContinuousFunction.coe_add, ENNReal.coe_add, Pi.add_apply,\n  testAgainstNN_coe_eq]", "state_before": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type ?u.70395\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 testAgainstNN \u03bc (f\u2081 + f\u2082) = testAgainstNN \u03bc f\u2081 + testAgainstNN \u03bc f\u2082", "state_after": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type ?u.70395\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f\u2081 \u03c9) + \u2191(\u2191f\u2082 \u03c9) \u2202\u2191\u03bc) = (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f\u2081 \u03c9) \u2202\u2191\u03bc) + \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f\u2082 \u03c9) \u2202\u2191\u03bc"}, {"tactic": "exact lintegral_add_left (BoundedContinuousFunction.NNReal.coe_ennreal_comp_measurable _) _", "state_before": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type ?u.70395\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f\u2081 \u03c9) + \u2191(\u2191f\u2082 \u03c9) \u2202\u2191\u03bc) = (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f\u2081 \u03c9) \u2202\u2191\u03bc) + \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f\u2082 \u03c9) \u2202\u2191\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/ENNReal.lean", "full_name": "ENNReal.cancel_of_ne", "start": [1064, 1], "end": [1065, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fintype/Basic.lean", "full_name": "Function.Injective.invOfMemRange_surjective", "start": [511, 1], "end": [512, 41], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.62835\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\na : \u03b1\n\u22a2 invOfMemRange hf { val := f a, property := (_ : f a \u2208 Set.range f) } = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/LucasLehmer.lean", "full_name": "LucasLehmer.X.neg_fst", "start": [223, 1], "end": [224, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Interval.lean", "full_name": "NonemptyInterval.inv_pure", "start": [512, 1], "end": [513, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Dynamics/Flow.lean", "full_name": "Flow.ext", "start": [112, 1], "end": [116, 16], "traced_tactics": [{"tactic": "congr", "state_before": "\u03c4 : Type u_1\ninst\u271d\u00b3 : AddMonoid \u03c4\ninst\u271d\u00b2 : TopologicalSpace \u03c4\ninst\u271d\u00b9 : ContinuousAdd \u03c4\n\u03b1 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\n\u03d5 : Flow \u03c4 \u03b1\nf\u2081 : \u03c4 \u2192 \u03b1 \u2192 \u03b1\ncont'\u271d\u00b9 : Continuous (uncurry f\u2081)\nmap_add'\u271d\u00b9 : \u2200 (t\u2081 t\u2082 : \u03c4) (x : \u03b1), f\u2081 (t\u2081 + t\u2082) x = f\u2081 t\u2081 (f\u2081 t\u2082 x)\nmap_zero'\u271d\u00b9 : \u2200 (x : \u03b1), f\u2081 0 x = x\nf\u2082 : \u03c4 \u2192 \u03b1 \u2192 \u03b1\ncont'\u271d : Continuous (uncurry f\u2082)\nmap_add'\u271d : \u2200 (t\u2081 t\u2082 : \u03c4) (x : \u03b1), f\u2082 (t\u2081 + t\u2082) x = f\u2082 t\u2081 (f\u2082 t\u2082 x)\nmap_zero'\u271d : \u2200 (x : \u03b1), f\u2082 0 x = x\nh :\n  \u2200 (t : \u03c4) (x : \u03b1),\n    toFun { toFun := f\u2081, cont' := cont'\u271d\u00b9, map_add' := map_add'\u271d\u00b9, map_zero' := map_zero'\u271d\u00b9 } t x =\n      toFun { toFun := f\u2082, cont' := cont'\u271d, map_add' := map_add'\u271d, map_zero' := map_zero'\u271d } t x\n\u22a2 { toFun := f\u2081, cont' := cont'\u271d\u00b9, map_add' := map_add'\u271d\u00b9, map_zero' := map_zero'\u271d\u00b9 } =\n    { toFun := f\u2082, cont' := cont'\u271d, map_add' := map_add'\u271d, map_zero' := map_zero'\u271d }", "state_after": "case e_toFun\n\u03c4 : Type u_1\ninst\u271d\u00b3 : AddMonoid \u03c4\ninst\u271d\u00b2 : TopologicalSpace \u03c4\ninst\u271d\u00b9 : ContinuousAdd \u03c4\n\u03b1 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\n\u03d5 : Flow \u03c4 \u03b1\nf\u2081 : \u03c4 \u2192 \u03b1 \u2192 \u03b1\ncont'\u271d\u00b9 : Continuous (uncurry f\u2081)\nmap_add'\u271d\u00b9 : \u2200 (t\u2081 t\u2082 : \u03c4) (x : \u03b1), f\u2081 (t\u2081 + t\u2082) x = f\u2081 t\u2081 (f\u2081 t\u2082 x)\nmap_zero'\u271d\u00b9 : \u2200 (x : \u03b1), f\u2081 0 x = x\nf\u2082 : \u03c4 \u2192 \u03b1 \u2192 \u03b1\ncont'\u271d : Continuous (uncurry f\u2082)\nmap_add'\u271d : \u2200 (t\u2081 t\u2082 : \u03c4) (x : \u03b1), f\u2082 (t\u2081 + t\u2082) x = f\u2082 t\u2081 (f\u2082 t\u2082 x)\nmap_zero'\u271d : \u2200 (x : \u03b1), f\u2082 0 x = x\nh :\n  \u2200 (t : \u03c4) (x : \u03b1),\n    toFun { toFun := f\u2081, cont' := cont'\u271d\u00b9, map_add' := map_add'\u271d\u00b9, map_zero' := map_zero'\u271d\u00b9 } t x =\n      toFun { toFun := f\u2082, cont' := cont'\u271d, map_add' := map_add'\u271d, map_zero' := map_zero'\u271d } t x\n\u22a2 f\u2081 = f\u2082"}, {"tactic": "funext", "state_before": "case e_toFun\n\u03c4 : Type u_1\ninst\u271d\u00b3 : AddMonoid \u03c4\ninst\u271d\u00b2 : TopologicalSpace \u03c4\ninst\u271d\u00b9 : ContinuousAdd \u03c4\n\u03b1 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\n\u03d5 : Flow \u03c4 \u03b1\nf\u2081 : \u03c4 \u2192 \u03b1 \u2192 \u03b1\ncont'\u271d\u00b9 : Continuous (uncurry f\u2081)\nmap_add'\u271d\u00b9 : \u2200 (t\u2081 t\u2082 : \u03c4) (x : \u03b1), f\u2081 (t\u2081 + t\u2082) x = f\u2081 t\u2081 (f\u2081 t\u2082 x)\nmap_zero'\u271d\u00b9 : \u2200 (x : \u03b1), f\u2081 0 x = x\nf\u2082 : \u03c4 \u2192 \u03b1 \u2192 \u03b1\ncont'\u271d : Continuous (uncurry f\u2082)\nmap_add'\u271d : \u2200 (t\u2081 t\u2082 : \u03c4) (x : \u03b1), f\u2082 (t\u2081 + t\u2082) x = f\u2082 t\u2081 (f\u2082 t\u2082 x)\nmap_zero'\u271d : \u2200 (x : \u03b1), f\u2082 0 x = x\nh :\n  \u2200 (t : \u03c4) (x : \u03b1),\n    toFun { toFun := f\u2081, cont' := cont'\u271d\u00b9, map_add' := map_add'\u271d\u00b9, map_zero' := map_zero'\u271d\u00b9 } t x =\n      toFun { toFun := f\u2082, cont' := cont'\u271d, map_add' := map_add'\u271d, map_zero' := map_zero'\u271d } t x\n\u22a2 f\u2081 = f\u2082", "state_after": "case e_toFun.h.h\n\u03c4 : Type u_1\ninst\u271d\u00b3 : AddMonoid \u03c4\ninst\u271d\u00b2 : TopologicalSpace \u03c4\ninst\u271d\u00b9 : ContinuousAdd \u03c4\n\u03b1 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\n\u03d5 : Flow \u03c4 \u03b1\nf\u2081 : \u03c4 \u2192 \u03b1 \u2192 \u03b1\ncont'\u271d\u00b9 : Continuous (uncurry f\u2081)\nmap_add'\u271d\u00b9 : \u2200 (t\u2081 t\u2082 : \u03c4) (x : \u03b1), f\u2081 (t\u2081 + t\u2082) x = f\u2081 t\u2081 (f\u2081 t\u2082 x)\nmap_zero'\u271d\u00b9 : \u2200 (x : \u03b1), f\u2081 0 x = x\nf\u2082 : \u03c4 \u2192 \u03b1 \u2192 \u03b1\ncont'\u271d : Continuous (uncurry f\u2082)\nmap_add'\u271d : \u2200 (t\u2081 t\u2082 : \u03c4) (x : \u03b1), f\u2082 (t\u2081 + t\u2082) x = f\u2082 t\u2081 (f\u2082 t\u2082 x)\nmap_zero'\u271d : \u2200 (x : \u03b1), f\u2082 0 x = x\nh :\n  \u2200 (t : \u03c4) (x : \u03b1),\n    toFun { toFun := f\u2081, cont' := cont'\u271d\u00b9, map_add' := map_add'\u271d\u00b9, map_zero' := map_zero'\u271d\u00b9 } t x =\n      toFun { toFun := f\u2082, cont' := cont'\u271d, map_add' := map_add'\u271d, map_zero' := map_zero'\u271d } t x\nx\u271d\u00b9 : \u03c4\nx\u271d : \u03b1\n\u22a2 f\u2081 x\u271d\u00b9 x\u271d = f\u2082 x\u271d\u00b9 x\u271d"}, {"tactic": "exact h _ _", "state_before": "case e_toFun.h.h\n\u03c4 : Type u_1\ninst\u271d\u00b3 : AddMonoid \u03c4\ninst\u271d\u00b2 : TopologicalSpace \u03c4\ninst\u271d\u00b9 : ContinuousAdd \u03c4\n\u03b1 : Type u_2\ninst\u271d : TopologicalSpace \u03b1\n\u03d5 : Flow \u03c4 \u03b1\nf\u2081 : \u03c4 \u2192 \u03b1 \u2192 \u03b1\ncont'\u271d\u00b9 : Continuous (uncurry f\u2081)\nmap_add'\u271d\u00b9 : \u2200 (t\u2081 t\u2082 : \u03c4) (x : \u03b1), f\u2081 (t\u2081 + t\u2082) x = f\u2081 t\u2081 (f\u2081 t\u2082 x)\nmap_zero'\u271d\u00b9 : \u2200 (x : \u03b1), f\u2081 0 x = x\nf\u2082 : \u03c4 \u2192 \u03b1 \u2192 \u03b1\ncont'\u271d : Continuous (uncurry f\u2082)\nmap_add'\u271d : \u2200 (t\u2081 t\u2082 : \u03c4) (x : \u03b1), f\u2082 (t\u2081 + t\u2082) x = f\u2082 t\u2081 (f\u2082 t\u2082 x)\nmap_zero'\u271d : \u2200 (x : \u03b1), f\u2082 0 x = x\nh :\n  \u2200 (t : \u03c4) (x : \u03b1),\n    toFun { toFun := f\u2081, cont' := cont'\u271d\u00b9, map_add' := map_add'\u271d\u00b9, map_zero' := map_zero'\u271d\u00b9 } t x =\n      toFun { toFun := f\u2082, cont' := cont'\u271d, map_add' := map_add'\u271d, map_zero' := map_zero'\u271d } t x\nx\u271d\u00b9 : \u03c4\nx\u271d : \u03b1\n\u22a2 f\u2081 x\u271d\u00b9 x\u271d = f\u2082 x\u271d\u00b9 x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Gluing.lean", "full_name": "Metric.isometry_inr", "start": [287, 1], "end": [288, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm.lean", "full_name": "ContinuousLinearMap.op_norm_comp_linearIsometryEquiv", "start": [1811, 1], "end": [1824, 62], "traced_tactics": [{"tactic": "cases subsingleton_or_nontrivial F'", "state_before": "\ud835\udd5c : Type ?u.3597667\n\ud835\udd5c\u2082 : Type u_1\n\ud835\udd5c\u2083 : Type u_2\nE : Type ?u.3597676\nE\u2097 : Type ?u.3597679\nF : Type u_3\nF\u2097 : Type ?u.3597685\nG : Type u_4\nG\u2097 : Type ?u.3597691\n\ud835\udcd5 : Type ?u.3597694\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\ninst\u271d\u00b9\u2079 : NormedAddCommGroup G\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\ud835\udd5c\u2082' : Type u_5\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082'\nF' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082' F'\n\u03c3\u2082' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082'' : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2082'\n\u03c3\u2082\u2083' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2077 : RingHomInvPair \u03c3\u2082' \u03c3\u2082''\ninst\u271d\u2076 : RingHomInvPair \u03c3\u2082'' \u03c3\u2082'\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2082' \u03c3\u2082\u2083 \u03c3\u2082\u2083'\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082'' \u03c3\u2082\u2083' \u03c3\u2082\u2083\ninst\u271d\u00b3 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2082'\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082''\ninst\u271d : RingHomIsometric \u03c3\u2082\u2083'\nf : F \u2192SL[\u03c3\u2082\u2083] G\ng : F' \u2243\u209b\u2097\u1d62[\u03c3\u2082'] F\n\u22a2 \u2016comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g))\u2016 = \u2016f\u2016", "state_after": "case inl\n\ud835\udd5c : Type ?u.3597667\n\ud835\udd5c\u2082 : Type u_1\n\ud835\udd5c\u2083 : Type u_2\nE : Type ?u.3597676\nE\u2097 : Type ?u.3597679\nF : Type u_3\nF\u2097 : Type ?u.3597685\nG : Type u_4\nG\u2097 : Type ?u.3597691\n\ud835\udcd5 : Type ?u.3597694\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\ninst\u271d\u00b9\u2079 : NormedAddCommGroup G\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\ud835\udd5c\u2082' : Type u_5\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082'\nF' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082' F'\n\u03c3\u2082' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082'' : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2082'\n\u03c3\u2082\u2083' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2077 : RingHomInvPair \u03c3\u2082' \u03c3\u2082''\ninst\u271d\u2076 : RingHomInvPair \u03c3\u2082'' \u03c3\u2082'\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2082' \u03c3\u2082\u2083 \u03c3\u2082\u2083'\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082'' \u03c3\u2082\u2083' \u03c3\u2082\u2083\ninst\u271d\u00b3 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2082'\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082''\ninst\u271d : RingHomIsometric \u03c3\u2082\u2083'\nf : F \u2192SL[\u03c3\u2082\u2083] G\ng : F' \u2243\u209b\u2097\u1d62[\u03c3\u2082'] F\nh\u271d : Subsingleton F'\n\u22a2 \u2016comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g))\u2016 = \u2016f\u2016\n\ncase inr\n\ud835\udd5c : Type ?u.3597667\n\ud835\udd5c\u2082 : Type u_1\n\ud835\udd5c\u2083 : Type u_2\nE : Type ?u.3597676\nE\u2097 : Type ?u.3597679\nF : Type u_3\nF\u2097 : Type ?u.3597685\nG : Type u_4\nG\u2097 : Type ?u.3597691\n\ud835\udcd5 : Type ?u.3597694\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\ninst\u271d\u00b9\u2079 : NormedAddCommGroup G\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\ud835\udd5c\u2082' : Type u_5\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082'\nF' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082' F'\n\u03c3\u2082' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082'' : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2082'\n\u03c3\u2082\u2083' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2077 : RingHomInvPair \u03c3\u2082' \u03c3\u2082''\ninst\u271d\u2076 : RingHomInvPair \u03c3\u2082'' \u03c3\u2082'\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2082' \u03c3\u2082\u2083 \u03c3\u2082\u2083'\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082'' \u03c3\u2082\u2083' \u03c3\u2082\u2083\ninst\u271d\u00b3 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2082'\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082''\ninst\u271d : RingHomIsometric \u03c3\u2082\u2083'\nf : F \u2192SL[\u03c3\u2082\u2083] G\ng : F' \u2243\u209b\u2097\u1d62[\u03c3\u2082'] F\nh\u271d : Nontrivial F'\n\u22a2 \u2016comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g))\u2016 = \u2016f\u2016"}, {"tactic": "refine' le_antisymm _ _", "state_before": "case inr\n\ud835\udd5c : Type ?u.3597667\n\ud835\udd5c\u2082 : Type u_1\n\ud835\udd5c\u2083 : Type u_2\nE : Type ?u.3597676\nE\u2097 : Type ?u.3597679\nF : Type u_3\nF\u2097 : Type ?u.3597685\nG : Type u_4\nG\u2097 : Type ?u.3597691\n\ud835\udcd5 : Type ?u.3597694\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\ninst\u271d\u00b9\u2079 : NormedAddCommGroup G\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\ud835\udd5c\u2082' : Type u_5\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082'\nF' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082' F'\n\u03c3\u2082' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082'' : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2082'\n\u03c3\u2082\u2083' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2077 : RingHomInvPair \u03c3\u2082' \u03c3\u2082''\ninst\u271d\u2076 : RingHomInvPair \u03c3\u2082'' \u03c3\u2082'\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2082' \u03c3\u2082\u2083 \u03c3\u2082\u2083'\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082'' \u03c3\u2082\u2083' \u03c3\u2082\u2083\ninst\u271d\u00b3 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2082'\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082''\ninst\u271d : RingHomIsometric \u03c3\u2082\u2083'\nf : F \u2192SL[\u03c3\u2082\u2083] G\ng : F' \u2243\u209b\u2097\u1d62[\u03c3\u2082'] F\nh\u271d : Nontrivial F'\n\u22a2 \u2016comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g))\u2016 = \u2016f\u2016", "state_after": "case inr.refine'_1\n\ud835\udd5c : Type ?u.3597667\n\ud835\udd5c\u2082 : Type u_1\n\ud835\udd5c\u2083 : Type u_2\nE : Type ?u.3597676\nE\u2097 : Type ?u.3597679\nF : Type u_3\nF\u2097 : Type ?u.3597685\nG : Type u_4\nG\u2097 : Type ?u.3597691\n\ud835\udcd5 : Type ?u.3597694\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\ninst\u271d\u00b9\u2079 : NormedAddCommGroup G\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\ud835\udd5c\u2082' : Type u_5\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082'\nF' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082' F'\n\u03c3\u2082' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082'' : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2082'\n\u03c3\u2082\u2083' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2077 : RingHomInvPair \u03c3\u2082' \u03c3\u2082''\ninst\u271d\u2076 : RingHomInvPair \u03c3\u2082'' \u03c3\u2082'\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2082' \u03c3\u2082\u2083 \u03c3\u2082\u2083'\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082'' \u03c3\u2082\u2083' \u03c3\u2082\u2083\ninst\u271d\u00b3 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2082'\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082''\ninst\u271d : RingHomIsometric \u03c3\u2082\u2083'\nf : F \u2192SL[\u03c3\u2082\u2083] G\ng : F' \u2243\u209b\u2097\u1d62[\u03c3\u2082'] F\nh\u271d : Nontrivial F'\n\u22a2 \u2016comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g))\u2016 \u2264 \u2016f\u2016\n\ncase inr.refine'_2\n\ud835\udd5c : Type ?u.3597667\n\ud835\udd5c\u2082 : Type u_1\n\ud835\udd5c\u2083 : Type u_2\nE : Type ?u.3597676\nE\u2097 : Type ?u.3597679\nF : Type u_3\nF\u2097 : Type ?u.3597685\nG : Type u_4\nG\u2097 : Type ?u.3597691\n\ud835\udcd5 : Type ?u.3597694\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\ninst\u271d\u00b9\u2079 : NormedAddCommGroup G\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\ud835\udd5c\u2082' : Type u_5\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082'\nF' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082' F'\n\u03c3\u2082' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082'' : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2082'\n\u03c3\u2082\u2083' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2077 : RingHomInvPair \u03c3\u2082' \u03c3\u2082''\ninst\u271d\u2076 : RingHomInvPair \u03c3\u2082'' \u03c3\u2082'\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2082' \u03c3\u2082\u2083 \u03c3\u2082\u2083'\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082'' \u03c3\u2082\u2083' \u03c3\u2082\u2083\ninst\u271d\u00b3 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2082'\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082''\ninst\u271d : RingHomIsometric \u03c3\u2082\u2083'\nf : F \u2192SL[\u03c3\u2082\u2083] G\ng : F' \u2243\u209b\u2097\u1d62[\u03c3\u2082'] F\nh\u271d : Nontrivial F'\n\u22a2 \u2016f\u2016 \u2264 \u2016comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g))\u2016"}, {"tactic": "haveI := g.symm.toLinearEquiv.toEquiv.subsingleton", "state_before": "case inl\n\ud835\udd5c : Type ?u.3597667\n\ud835\udd5c\u2082 : Type u_1\n\ud835\udd5c\u2083 : Type u_2\nE : Type ?u.3597676\nE\u2097 : Type ?u.3597679\nF : Type u_3\nF\u2097 : Type ?u.3597685\nG : Type u_4\nG\u2097 : Type ?u.3597691\n\ud835\udcd5 : Type ?u.3597694\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\ninst\u271d\u00b9\u2079 : NormedAddCommGroup G\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\ud835\udd5c\u2082' : Type u_5\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082'\nF' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082' F'\n\u03c3\u2082' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082'' : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2082'\n\u03c3\u2082\u2083' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2077 : RingHomInvPair \u03c3\u2082' \u03c3\u2082''\ninst\u271d\u2076 : RingHomInvPair \u03c3\u2082'' \u03c3\u2082'\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2082' \u03c3\u2082\u2083 \u03c3\u2082\u2083'\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082'' \u03c3\u2082\u2083' \u03c3\u2082\u2083\ninst\u271d\u00b3 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2082'\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082''\ninst\u271d : RingHomIsometric \u03c3\u2082\u2083'\nf : F \u2192SL[\u03c3\u2082\u2083] G\ng : F' \u2243\u209b\u2097\u1d62[\u03c3\u2082'] F\nh\u271d : Subsingleton F'\n\u22a2 \u2016comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g))\u2016 = \u2016f\u2016", "state_after": "case inl\n\ud835\udd5c : Type ?u.3597667\n\ud835\udd5c\u2082 : Type u_1\n\ud835\udd5c\u2083 : Type u_2\nE : Type ?u.3597676\nE\u2097 : Type ?u.3597679\nF : Type u_3\nF\u2097 : Type ?u.3597685\nG : Type u_4\nG\u2097 : Type ?u.3597691\n\ud835\udcd5 : Type ?u.3597694\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\ninst\u271d\u00b9\u2079 : NormedAddCommGroup G\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\ud835\udd5c\u2082' : Type u_5\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082'\nF' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082' F'\n\u03c3\u2082' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082'' : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2082'\n\u03c3\u2082\u2083' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2077 : RingHomInvPair \u03c3\u2082' \u03c3\u2082''\ninst\u271d\u2076 : RingHomInvPair \u03c3\u2082'' \u03c3\u2082'\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2082' \u03c3\u2082\u2083 \u03c3\u2082\u2083'\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082'' \u03c3\u2082\u2083' \u03c3\u2082\u2083\ninst\u271d\u00b3 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2082'\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082''\ninst\u271d : RingHomIsometric \u03c3\u2082\u2083'\nf : F \u2192SL[\u03c3\u2082\u2083] G\ng : F' \u2243\u209b\u2097\u1d62[\u03c3\u2082'] F\nh\u271d : Subsingleton F'\nthis : Subsingleton F\n\u22a2 \u2016comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g))\u2016 = \u2016f\u2016"}, {"tactic": "simp", "state_before": "case inl\n\ud835\udd5c : Type ?u.3597667\n\ud835\udd5c\u2082 : Type u_1\n\ud835\udd5c\u2083 : Type u_2\nE : Type ?u.3597676\nE\u2097 : Type ?u.3597679\nF : Type u_3\nF\u2097 : Type ?u.3597685\nG : Type u_4\nG\u2097 : Type ?u.3597691\n\ud835\udcd5 : Type ?u.3597694\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\ninst\u271d\u00b9\u2079 : NormedAddCommGroup G\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\ud835\udd5c\u2082' : Type u_5\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082'\nF' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082' F'\n\u03c3\u2082' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082'' : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2082'\n\u03c3\u2082\u2083' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2077 : RingHomInvPair \u03c3\u2082' \u03c3\u2082''\ninst\u271d\u2076 : RingHomInvPair \u03c3\u2082'' \u03c3\u2082'\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2082' \u03c3\u2082\u2083 \u03c3\u2082\u2083'\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082'' \u03c3\u2082\u2083' \u03c3\u2082\u2083\ninst\u271d\u00b3 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2082'\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082''\ninst\u271d : RingHomIsometric \u03c3\u2082\u2083'\nf : F \u2192SL[\u03c3\u2082\u2083] G\ng : F' \u2243\u209b\u2097\u1d62[\u03c3\u2082'] F\nh\u271d : Subsingleton F'\nthis : Subsingleton F\n\u22a2 \u2016comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g))\u2016 = \u2016f\u2016", "state_after": "no goals"}, {"tactic": "convert f.op_norm_comp_le g.toLinearIsometry.toContinuousLinearMap", "state_before": "case inr.refine'_1\n\ud835\udd5c : Type ?u.3597667\n\ud835\udd5c\u2082 : Type u_1\n\ud835\udd5c\u2083 : Type u_2\nE : Type ?u.3597676\nE\u2097 : Type ?u.3597679\nF : Type u_3\nF\u2097 : Type ?u.3597685\nG : Type u_4\nG\u2097 : Type ?u.3597691\n\ud835\udcd5 : Type ?u.3597694\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\ninst\u271d\u00b9\u2079 : NormedAddCommGroup G\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\ud835\udd5c\u2082' : Type u_5\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082'\nF' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082' F'\n\u03c3\u2082' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082'' : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2082'\n\u03c3\u2082\u2083' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2077 : RingHomInvPair \u03c3\u2082' \u03c3\u2082''\ninst\u271d\u2076 : RingHomInvPair \u03c3\u2082'' \u03c3\u2082'\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2082' \u03c3\u2082\u2083 \u03c3\u2082\u2083'\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082'' \u03c3\u2082\u2083' \u03c3\u2082\u2083\ninst\u271d\u00b3 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2082'\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082''\ninst\u271d : RingHomIsometric \u03c3\u2082\u2083'\nf : F \u2192SL[\u03c3\u2082\u2083] G\ng : F' \u2243\u209b\u2097\u1d62[\u03c3\u2082'] F\nh\u271d : Nontrivial F'\n\u22a2 \u2016comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g))\u2016 \u2264 \u2016f\u2016", "state_after": "case h.e'_4\n\ud835\udd5c : Type ?u.3597667\n\ud835\udd5c\u2082 : Type u_1\n\ud835\udd5c\u2083 : Type u_2\nE : Type ?u.3597676\nE\u2097 : Type ?u.3597679\nF : Type u_3\nF\u2097 : Type ?u.3597685\nG : Type u_4\nG\u2097 : Type ?u.3597691\n\ud835\udcd5 : Type ?u.3597694\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\ninst\u271d\u00b9\u2079 : NormedAddCommGroup G\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\ud835\udd5c\u2082' : Type u_5\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082'\nF' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082' F'\n\u03c3\u2082' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082'' : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2082'\n\u03c3\u2082\u2083' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2077 : RingHomInvPair \u03c3\u2082' \u03c3\u2082''\ninst\u271d\u2076 : RingHomInvPair \u03c3\u2082'' \u03c3\u2082'\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2082' \u03c3\u2082\u2083 \u03c3\u2082\u2083'\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082'' \u03c3\u2082\u2083' \u03c3\u2082\u2083\ninst\u271d\u00b3 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2082'\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082''\ninst\u271d : RingHomIsometric \u03c3\u2082\u2083'\nf : F \u2192SL[\u03c3\u2082\u2083] G\ng : F' \u2243\u209b\u2097\u1d62[\u03c3\u2082'] F\nh\u271d : Nontrivial F'\n\u22a2 \u2016f\u2016 = \u2016f\u2016 * \u2016LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g)\u2016"}, {"tactic": "simp [g.toLinearIsometry.norm_toContinuousLinearMap]", "state_before": "case h.e'_4\n\ud835\udd5c : Type ?u.3597667\n\ud835\udd5c\u2082 : Type u_1\n\ud835\udd5c\u2083 : Type u_2\nE : Type ?u.3597676\nE\u2097 : Type ?u.3597679\nF : Type u_3\nF\u2097 : Type ?u.3597685\nG : Type u_4\nG\u2097 : Type ?u.3597691\n\ud835\udcd5 : Type ?u.3597694\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\ninst\u271d\u00b9\u2079 : NormedAddCommGroup G\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\ud835\udd5c\u2082' : Type u_5\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082'\nF' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082' F'\n\u03c3\u2082' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082'' : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2082'\n\u03c3\u2082\u2083' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2077 : RingHomInvPair \u03c3\u2082' \u03c3\u2082''\ninst\u271d\u2076 : RingHomInvPair \u03c3\u2082'' \u03c3\u2082'\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2082' \u03c3\u2082\u2083 \u03c3\u2082\u2083'\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082'' \u03c3\u2082\u2083' \u03c3\u2082\u2083\ninst\u271d\u00b3 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2082'\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082''\ninst\u271d : RingHomIsometric \u03c3\u2082\u2083'\nf : F \u2192SL[\u03c3\u2082\u2083] G\ng : F' \u2243\u209b\u2097\u1d62[\u03c3\u2082'] F\nh\u271d : Nontrivial F'\n\u22a2 \u2016f\u2016 = \u2016f\u2016 * \u2016LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g)\u2016", "state_after": "no goals"}, {"tactic": "convert(f.comp g.toLinearIsometry.toContinuousLinearMap).op_norm_comp_le\n    g.symm.toLinearIsometry.toContinuousLinearMap", "state_before": "case inr.refine'_2\n\ud835\udd5c : Type ?u.3597667\n\ud835\udd5c\u2082 : Type u_1\n\ud835\udd5c\u2083 : Type u_2\nE : Type ?u.3597676\nE\u2097 : Type ?u.3597679\nF : Type u_3\nF\u2097 : Type ?u.3597685\nG : Type u_4\nG\u2097 : Type ?u.3597691\n\ud835\udcd5 : Type ?u.3597694\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\ninst\u271d\u00b9\u2079 : NormedAddCommGroup G\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\ud835\udd5c\u2082' : Type u_5\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082'\nF' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082' F'\n\u03c3\u2082' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082'' : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2082'\n\u03c3\u2082\u2083' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2077 : RingHomInvPair \u03c3\u2082' \u03c3\u2082''\ninst\u271d\u2076 : RingHomInvPair \u03c3\u2082'' \u03c3\u2082'\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2082' \u03c3\u2082\u2083 \u03c3\u2082\u2083'\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082'' \u03c3\u2082\u2083' \u03c3\u2082\u2083\ninst\u271d\u00b3 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2082'\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082''\ninst\u271d : RingHomIsometric \u03c3\u2082\u2083'\nf : F \u2192SL[\u03c3\u2082\u2083] G\ng : F' \u2243\u209b\u2097\u1d62[\u03c3\u2082'] F\nh\u271d : Nontrivial F'\n\u22a2 \u2016f\u2016 \u2264 \u2016comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g))\u2016", "state_after": "case h.e'_3.h.e'_3\n\ud835\udd5c : Type ?u.3597667\n\ud835\udd5c\u2082 : Type u_1\n\ud835\udd5c\u2083 : Type u_2\nE : Type ?u.3597676\nE\u2097 : Type ?u.3597679\nF : Type u_3\nF\u2097 : Type ?u.3597685\nG : Type u_4\nG\u2097 : Type ?u.3597691\n\ud835\udcd5 : Type ?u.3597694\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\ninst\u271d\u00b9\u2079 : NormedAddCommGroup G\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\ud835\udd5c\u2082' : Type u_5\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082'\nF' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082' F'\n\u03c3\u2082' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082'' : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2082'\n\u03c3\u2082\u2083' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2077 : RingHomInvPair \u03c3\u2082' \u03c3\u2082''\ninst\u271d\u2076 : RingHomInvPair \u03c3\u2082'' \u03c3\u2082'\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2082' \u03c3\u2082\u2083 \u03c3\u2082\u2083'\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082'' \u03c3\u2082\u2083' \u03c3\u2082\u2083\ninst\u271d\u00b3 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2082'\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082''\ninst\u271d : RingHomIsometric \u03c3\u2082\u2083'\nf : F \u2192SL[\u03c3\u2082\u2083] G\ng : F' \u2243\u209b\u2097\u1d62[\u03c3\u2082'] F\nh\u271d : Nontrivial F'\n\u22a2 f =\n    comp (comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g)))\n      (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry (LinearIsometryEquiv.symm g)))\n\ncase h.e'_4\n\ud835\udd5c : Type ?u.3597667\n\ud835\udd5c\u2082 : Type u_1\n\ud835\udd5c\u2083 : Type u_2\nE : Type ?u.3597676\nE\u2097 : Type ?u.3597679\nF : Type u_3\nF\u2097 : Type ?u.3597685\nG : Type u_4\nG\u2097 : Type ?u.3597691\n\ud835\udcd5 : Type ?u.3597694\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\ninst\u271d\u00b9\u2079 : NormedAddCommGroup G\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\ud835\udd5c\u2082' : Type u_5\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082'\nF' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082' F'\n\u03c3\u2082' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082'' : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2082'\n\u03c3\u2082\u2083' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2077 : RingHomInvPair \u03c3\u2082' \u03c3\u2082''\ninst\u271d\u2076 : RingHomInvPair \u03c3\u2082'' \u03c3\u2082'\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2082' \u03c3\u2082\u2083 \u03c3\u2082\u2083'\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082'' \u03c3\u2082\u2083' \u03c3\u2082\u2083\ninst\u271d\u00b3 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2082'\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082''\ninst\u271d : RingHomIsometric \u03c3\u2082\u2083'\nf : F \u2192SL[\u03c3\u2082\u2083] G\ng : F' \u2243\u209b\u2097\u1d62[\u03c3\u2082'] F\nh\u271d : Nontrivial F'\n\u22a2 \u2016comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g))\u2016 =\n    \u2016comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g))\u2016 *\n      \u2016LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry (LinearIsometryEquiv.symm g))\u2016"}, {"tactic": "haveI := g.symm.surjective.nontrivial", "state_before": "case h.e'_4\n\ud835\udd5c : Type ?u.3597667\n\ud835\udd5c\u2082 : Type u_1\n\ud835\udd5c\u2083 : Type u_2\nE : Type ?u.3597676\nE\u2097 : Type ?u.3597679\nF : Type u_3\nF\u2097 : Type ?u.3597685\nG : Type u_4\nG\u2097 : Type ?u.3597691\n\ud835\udcd5 : Type ?u.3597694\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\ninst\u271d\u00b9\u2079 : NormedAddCommGroup G\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\ud835\udd5c\u2082' : Type u_5\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082'\nF' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082' F'\n\u03c3\u2082' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082'' : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2082'\n\u03c3\u2082\u2083' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2077 : RingHomInvPair \u03c3\u2082' \u03c3\u2082''\ninst\u271d\u2076 : RingHomInvPair \u03c3\u2082'' \u03c3\u2082'\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2082' \u03c3\u2082\u2083 \u03c3\u2082\u2083'\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082'' \u03c3\u2082\u2083' \u03c3\u2082\u2083\ninst\u271d\u00b3 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2082'\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082''\ninst\u271d : RingHomIsometric \u03c3\u2082\u2083'\nf : F \u2192SL[\u03c3\u2082\u2083] G\ng : F' \u2243\u209b\u2097\u1d62[\u03c3\u2082'] F\nh\u271d : Nontrivial F'\n\u22a2 \u2016comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g))\u2016 =\n    \u2016comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g))\u2016 *\n      \u2016LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry (LinearIsometryEquiv.symm g))\u2016", "state_after": "case h.e'_4\n\ud835\udd5c : Type ?u.3597667\n\ud835\udd5c\u2082 : Type u_1\n\ud835\udd5c\u2083 : Type u_2\nE : Type ?u.3597676\nE\u2097 : Type ?u.3597679\nF : Type u_3\nF\u2097 : Type ?u.3597685\nG : Type u_4\nG\u2097 : Type ?u.3597691\n\ud835\udcd5 : Type ?u.3597694\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\ninst\u271d\u00b9\u2079 : NormedAddCommGroup G\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\ud835\udd5c\u2082' : Type u_5\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082'\nF' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082' F'\n\u03c3\u2082' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082'' : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2082'\n\u03c3\u2082\u2083' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2077 : RingHomInvPair \u03c3\u2082' \u03c3\u2082''\ninst\u271d\u2076 : RingHomInvPair \u03c3\u2082'' \u03c3\u2082'\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2082' \u03c3\u2082\u2083 \u03c3\u2082\u2083'\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082'' \u03c3\u2082\u2083' \u03c3\u2082\u2083\ninst\u271d\u00b3 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2082'\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082''\ninst\u271d : RingHomIsometric \u03c3\u2082\u2083'\nf : F \u2192SL[\u03c3\u2082\u2083] G\ng : F' \u2243\u209b\u2097\u1d62[\u03c3\u2082'] F\nh\u271d : Nontrivial F'\nthis : Nontrivial F\n\u22a2 \u2016comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g))\u2016 =\n    \u2016comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g))\u2016 *\n      \u2016LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry (LinearIsometryEquiv.symm g))\u2016"}, {"tactic": "simp [g.symm.toLinearIsometry.norm_toContinuousLinearMap]", "state_before": "case h.e'_4\n\ud835\udd5c : Type ?u.3597667\n\ud835\udd5c\u2082 : Type u_1\n\ud835\udd5c\u2083 : Type u_2\nE : Type ?u.3597676\nE\u2097 : Type ?u.3597679\nF : Type u_3\nF\u2097 : Type ?u.3597685\nG : Type u_4\nG\u2097 : Type ?u.3597691\n\ud835\udcd5 : Type ?u.3597694\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\ninst\u271d\u00b9\u2079 : NormedAddCommGroup G\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\ud835\udd5c\u2082' : Type u_5\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082'\nF' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082' F'\n\u03c3\u2082' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082'' : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2082'\n\u03c3\u2082\u2083' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2077 : RingHomInvPair \u03c3\u2082' \u03c3\u2082''\ninst\u271d\u2076 : RingHomInvPair \u03c3\u2082'' \u03c3\u2082'\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2082' \u03c3\u2082\u2083 \u03c3\u2082\u2083'\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082'' \u03c3\u2082\u2083' \u03c3\u2082\u2083\ninst\u271d\u00b3 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2082'\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082''\ninst\u271d : RingHomIsometric \u03c3\u2082\u2083'\nf : F \u2192SL[\u03c3\u2082\u2083] G\ng : F' \u2243\u209b\u2097\u1d62[\u03c3\u2082'] F\nh\u271d : Nontrivial F'\nthis : Nontrivial F\n\u22a2 \u2016comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g))\u2016 =\n    \u2016comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g))\u2016 *\n      \u2016LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry (LinearIsometryEquiv.symm g))\u2016", "state_after": "no goals"}, {"tactic": "ext", "state_before": "case h.e'_3.h.e'_3\n\ud835\udd5c : Type ?u.3597667\n\ud835\udd5c\u2082 : Type u_1\n\ud835\udd5c\u2083 : Type u_2\nE : Type ?u.3597676\nE\u2097 : Type ?u.3597679\nF : Type u_3\nF\u2097 : Type ?u.3597685\nG : Type u_4\nG\u2097 : Type ?u.3597691\n\ud835\udcd5 : Type ?u.3597694\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\ninst\u271d\u00b9\u2079 : NormedAddCommGroup G\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\ud835\udd5c\u2082' : Type u_5\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082'\nF' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082' F'\n\u03c3\u2082' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082'' : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2082'\n\u03c3\u2082\u2083' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2077 : RingHomInvPair \u03c3\u2082' \u03c3\u2082''\ninst\u271d\u2076 : RingHomInvPair \u03c3\u2082'' \u03c3\u2082'\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2082' \u03c3\u2082\u2083 \u03c3\u2082\u2083'\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082'' \u03c3\u2082\u2083' \u03c3\u2082\u2083\ninst\u271d\u00b3 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2082'\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082''\ninst\u271d : RingHomIsometric \u03c3\u2082\u2083'\nf : F \u2192SL[\u03c3\u2082\u2083] G\ng : F' \u2243\u209b\u2097\u1d62[\u03c3\u2082'] F\nh\u271d : Nontrivial F'\n\u22a2 f =\n    comp (comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g)))\n      (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry (LinearIsometryEquiv.symm g)))", "state_after": "case h.e'_3.h.e'_3.h\n\ud835\udd5c : Type ?u.3597667\n\ud835\udd5c\u2082 : Type u_1\n\ud835\udd5c\u2083 : Type u_2\nE : Type ?u.3597676\nE\u2097 : Type ?u.3597679\nF : Type u_3\nF\u2097 : Type ?u.3597685\nG : Type u_4\nG\u2097 : Type ?u.3597691\n\ud835\udcd5 : Type ?u.3597694\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\ninst\u271d\u00b9\u2079 : NormedAddCommGroup G\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\ud835\udd5c\u2082' : Type u_5\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082'\nF' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082' F'\n\u03c3\u2082' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082'' : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2082'\n\u03c3\u2082\u2083' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2077 : RingHomInvPair \u03c3\u2082' \u03c3\u2082''\ninst\u271d\u2076 : RingHomInvPair \u03c3\u2082'' \u03c3\u2082'\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2082' \u03c3\u2082\u2083 \u03c3\u2082\u2083'\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082'' \u03c3\u2082\u2083' \u03c3\u2082\u2083\ninst\u271d\u00b3 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2082'\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082''\ninst\u271d : RingHomIsometric \u03c3\u2082\u2083'\nf : F \u2192SL[\u03c3\u2082\u2083] G\ng : F' \u2243\u209b\u2097\u1d62[\u03c3\u2082'] F\nh\u271d : Nontrivial F'\nx\u271d : F\n\u22a2 \u2191f x\u271d =\n    \u2191(comp (comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g)))\n          (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry (LinearIsometryEquiv.symm g))))\n      x\u271d"}, {"tactic": "simp", "state_before": "case h.e'_3.h.e'_3.h\n\ud835\udd5c : Type ?u.3597667\n\ud835\udd5c\u2082 : Type u_1\n\ud835\udd5c\u2083 : Type u_2\nE : Type ?u.3597676\nE\u2097 : Type ?u.3597679\nF : Type u_3\nF\u2097 : Type ?u.3597685\nG : Type u_4\nG\u2097 : Type ?u.3597691\n\ud835\udcd5 : Type ?u.3597694\ninst\u271d\u00b2\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b2\u2070 : NormedAddCommGroup F\ninst\u271d\u00b9\u2079 : NormedAddCommGroup G\ninst\u271d\u00b9\u2078 : NormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\ud835\udd5c\u2082' : Type u_5\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082'\nF' : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082' F'\n\u03c3\u2082' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082'' : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2082'\n\u03c3\u2082\u2083' : \ud835\udd5c\u2082' \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2077 : RingHomInvPair \u03c3\u2082' \u03c3\u2082''\ninst\u271d\u2076 : RingHomInvPair \u03c3\u2082'' \u03c3\u2082'\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2082' \u03c3\u2082\u2083 \u03c3\u2082\u2083'\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082'' \u03c3\u2082\u2083' \u03c3\u2082\u2083\ninst\u271d\u00b3 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2082'\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082''\ninst\u271d : RingHomIsometric \u03c3\u2082\u2083'\nf : F \u2192SL[\u03c3\u2082\u2083] G\ng : F' \u2243\u209b\u2097\u1d62[\u03c3\u2082'] F\nh\u271d : Nontrivial F'\nx\u271d : F\n\u22a2 \u2191f x\u271d =\n    \u2191(comp (comp f (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry g)))\n          (LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry (LinearIsometryEquiv.symm g))))\n      x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "full_name": "CategoryTheory.MonoidalCategory.id_tensor_comp_tensor_id", "start": [239, 1], "end": [241, 7], "traced_tactics": [{"tactic": "rw [\u2190 tensor_comp]", "state_before": "C\u271d : Type u\n\ud835\udc9e : Category C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\nf : W \u27f6 X\ng : Y \u27f6 Z\n\u22a2 (\ud835\udfd9 Y \u2297 f) \u226b (g \u2297 \ud835\udfd9 X) = g \u2297 f", "state_after": "C\u271d : Type u\n\ud835\udc9e : Category C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\nf : W \u27f6 X\ng : Y \u27f6 Z\n\u22a2 \ud835\udfd9 Y \u226b g \u2297 f \u226b \ud835\udfd9 X = g \u2297 f"}, {"tactic": "simp", "state_before": "C\u271d : Type u\n\ud835\udc9e : Category C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\nf : W \u27f6 X\ng : Y \u27f6 Z\n\u22a2 \ud835\udfd9 Y \u226b g \u2297 f \u226b \ud835\udfd9 X = g \u2297 f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.average_eq_integral", "start": [100, 1], "end": [101, 48], "traced_tactics": [{"tactic": "rw [average, measure_univ, inv_one, one_smul]", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.188331\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\n\u03bc : Measure \u03b1\ns : Set E\ninst\u271d : IsProbabilityMeasure \u03bc\nf : \u03b1 \u2192 E\n\u22a2 (\u2a0d (x : \u03b1), f x \u2202\u03bc) = \u222b (x : \u03b1), f x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.Disjoint.symm", "start": [2896, 1], "end": [2897, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Core.lean", "full_name": "Ne.intro", "start": [569, 1], "end": [569, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "Real.cosh_pos", "start": [1561, 1], "end": [1562, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean", "full_name": "Real.exists_extension_norm_eq", "start": [45, 1], "end": [60, 26], "traced_tactics": [{"tactic": "rcases exists_extension_of_le_sublinear \u27e8p, f\u27e9 (fun x => \u2016f\u2016 * \u2016x\u2016)\n    (fun c hc x => by simp only [norm_smul c x, Real.norm_eq_abs, abs_of_pos hc, mul_left_comm])\n    (fun x y => by rw [\u2190 left_distrib]\n      exact mul_le_mul_of_nonneg_left (norm_add_le x y) (@norm_nonneg _ _ f))\n    fun x => le_trans (le_abs_self _) (f.le_op_norm _) with \u27e8g, g_eq, g_le\u27e9", "state_before": "E : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\n\u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\n\u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016"}, {"tactic": "set g' :=\n  g.mkContinuous \u2016f\u2016 fun x => abs_le.2 \u27e8neg_le.1 <| g.map_neg x \u25b8 norm_neg x \u25b8 g_le (-x), g_le x\u27e9", "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\n\u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), abs (\u2191g x) \u2264 \u2016f\u2016 * \u2016x\u2016)\n\u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016"}, {"tactic": "simp only [norm_smul c x, Real.norm_eq_abs, abs_of_pos hc, mul_left_comm]", "state_before": "E : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\nc : \u211d\nhc : 0 < c\nx : E\n\u22a2 (fun x => \u2016f\u2016 * \u2016x\u2016) (c \u2022 x) = c * (fun x => \u2016f\u2016 * \u2016x\u2016) x", "state_after": "no goals"}, {"tactic": "rw [\u2190 left_distrib]", "state_before": "E : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\nx y : E\n\u22a2 (fun x => \u2016f\u2016 * \u2016x\u2016) (x + y) \u2264 (fun x => \u2016f\u2016 * \u2016x\u2016) x + (fun x => \u2016f\u2016 * \u2016x\u2016) y", "state_after": "E : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\nx y : E\n\u22a2 (fun x => \u2016f\u2016 * \u2016x\u2016) (x + y) \u2264 \u2016f\u2016 * (\u2016x\u2016 + \u2016y\u2016)"}, {"tactic": "exact mul_le_mul_of_nonneg_left (norm_add_le x y) (@norm_nonneg _ _ f)", "state_before": "E : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\nx y : E\n\u22a2 (fun x => \u2016f\u2016 * \u2016x\u2016) (x + y) \u2264 \u2016f\u2016 * (\u2016x\u2016 + \u2016y\u2016)", "state_after": "no goals"}, {"tactic": "refine' \u27e8g', g_eq, _\u27e9", "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), abs (\u2191g x) \u2264 \u2016f\u2016 * \u2016x\u2016)\n\u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), abs (\u2191g x) \u2264 \u2016f\u2016 * \u2016x\u2016)\n\u22a2 \u2016g'\u2016 = \u2016f\u2016"}, {"tactic": "apply le_antisymm (g.mkContinuous_norm_le (norm_nonneg f) _)", "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), abs (\u2191g x) \u2264 \u2016f\u2016 * \u2016x\u2016)\n\u22a2 \u2016g'\u2016 = \u2016f\u2016", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), abs (\u2191g x) \u2264 \u2016f\u2016 * \u2016x\u2016)\n\u22a2 \u2016f\u2016 \u2264 \u2016LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), abs (\u2191g x) \u2264 \u2016f\u2016 * \u2016x\u2016)\u2016"}, {"tactic": "refine' f.op_norm_le_bound (norm_nonneg _) fun x => _", "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), abs (\u2191g x) \u2264 \u2016f\u2016 * \u2016x\u2016)\n\u22a2 \u2016f\u2016 \u2264 \u2016LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), abs (\u2191g x) \u2264 \u2016f\u2016 * \u2016x\u2016)\u2016", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), abs (\u2191g x) \u2264 \u2016f\u2016 * \u2016x\u2016)\nx : { x // x \u2208 p }\n\u22a2 \u2016\u2191f x\u2016 \u2264 \u2016LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), abs (\u2191g x) \u2264 \u2016f\u2016 * \u2016x\u2016)\u2016 * \u2016x\u2016"}, {"tactic": "dsimp at g_eq", "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), abs (\u2191g x) \u2264 \u2016f\u2016 * \u2016x\u2016)\nx : { x // x \u2208 p }\n\u22a2 \u2016\u2191f x\u2016 \u2264 \u2016LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), abs (\u2191g x) \u2264 \u2016f\u2016 * \u2016x\u2016)\u2016 * \u2016x\u2016", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), abs (\u2191g x) \u2264 \u2016f\u2016 * \u2016x\u2016)\nx : { x // x \u2208 p }\n\u22a2 \u2016\u2191f x\u2016 \u2264 \u2016LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), abs (\u2191g x) \u2264 \u2016f\u2016 * \u2016x\u2016)\u2016 * \u2016x\u2016"}, {"tactic": "rw [\u2190 g_eq]", "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), abs (\u2191g x) \u2264 \u2016f\u2016 * \u2016x\u2016)\nx : { x // x \u2208 p }\n\u22a2 \u2016\u2191f x\u2016 \u2264 \u2016LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), abs (\u2191g x) \u2264 \u2016f\u2016 * \u2016x\u2016)\u2016 * \u2016x\u2016", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), abs (\u2191g x) \u2264 \u2016f\u2016 * \u2016x\u2016)\nx : { x // x \u2208 p }\n\u22a2 \u2016\u2191g \u2191x\u2016 \u2264 \u2016LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), abs (\u2191g x) \u2264 \u2016f\u2016 * \u2016x\u2016)\u2016 * \u2016x\u2016"}, {"tactic": "apply g'.le_op_norm", "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), abs (\u2191g x) \u2264 \u2016f\u2016 * \u2016x\u2016)\nx : { x // x \u2208 p }\n\u22a2 \u2016\u2191g \u2191x\u2016 \u2264 \u2016LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), abs (\u2191g x) \u2264 \u2016f\u2016 * \u2016x\u2016)\u2016 * \u2016x\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "full_name": "WithTop.iSup_coe_lt_top", "start": [1544, 1], "end": [1546, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.Measure.tprod_tprod", "start": [255, 1], "end": [259, 65], "traced_tactics": [{"tactic": "induction' l with i l ih", "state_before": "\u03b9 : Type ?u.2388864\n\u03b9' : Type ?u.2388867\n\u03b1 : \u03b9 \u2192 Type ?u.2388872\ninst\u271d\u00b3 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1 i)\n\u03b4 : Type u_1\n\u03c0 : \u03b4 \u2192 Type u_2\ninst\u271d\u00b9 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\nl : List \u03b4\n\u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b4), SigmaFinite (\u03bc i)\ns : (i : \u03b4) \u2192 Set (\u03c0 i)\n\u22a2 \u2191\u2191(Measure.tprod l \u03bc) (Set.tprod l s) = List.prod (List.map (fun i => \u2191\u2191(\u03bc i) (s i)) l)", "state_after": "case nil\n\u03b9 : Type ?u.2388864\n\u03b9' : Type ?u.2388867\n\u03b1 : \u03b9 \u2192 Type ?u.2388872\ninst\u271d\u00b3 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1 i)\n\u03b4 : Type u_1\n\u03c0 : \u03b4 \u2192 Type u_2\ninst\u271d\u00b9 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\n\u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b4), SigmaFinite (\u03bc i)\ns : (i : \u03b4) \u2192 Set (\u03c0 i)\n\u22a2 \u2191\u2191(Measure.tprod [] \u03bc) (Set.tprod [] s) = List.prod (List.map (fun i => \u2191\u2191(\u03bc i) (s i)) [])\n\ncase cons\n\u03b9 : Type ?u.2388864\n\u03b9' : Type ?u.2388867\n\u03b1 : \u03b9 \u2192 Type ?u.2388872\ninst\u271d\u00b3 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1 i)\n\u03b4 : Type u_1\n\u03c0 : \u03b4 \u2192 Type u_2\ninst\u271d\u00b9 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\n\u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b4), SigmaFinite (\u03bc i)\ns : (i : \u03b4) \u2192 Set (\u03c0 i)\ni : \u03b4\nl : List \u03b4\nih : \u2191\u2191(Measure.tprod l \u03bc) (Set.tprod l s) = List.prod (List.map (fun i => \u2191\u2191(\u03bc i) (s i)) l)\n\u22a2 \u2191\u2191(Measure.tprod (i :: l) \u03bc) (Set.tprod (i :: l) s) = List.prod (List.map (fun i => \u2191\u2191(\u03bc i) (s i)) (i :: l))"}, {"tactic": "rw [tprod_cons, Set.tprod, prod_prod, map_cons, prod_cons, ih]", "state_before": "case cons\n\u03b9 : Type ?u.2388864\n\u03b9' : Type ?u.2388867\n\u03b1 : \u03b9 \u2192 Type ?u.2388872\ninst\u271d\u00b3 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1 i)\n\u03b4 : Type u_1\n\u03c0 : \u03b4 \u2192 Type u_2\ninst\u271d\u00b9 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\n\u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b4), SigmaFinite (\u03bc i)\ns : (i : \u03b4) \u2192 Set (\u03c0 i)\ni : \u03b4\nl : List \u03b4\nih : \u2191\u2191(Measure.tprod l \u03bc) (Set.tprod l s) = List.prod (List.map (fun i => \u2191\u2191(\u03bc i) (s i)) l)\n\u22a2 \u2191\u2191(Measure.tprod (i :: l) \u03bc) (Set.tprod (i :: l) s) = List.prod (List.map (fun i => \u2191\u2191(\u03bc i) (s i)) (i :: l))", "state_after": "no goals"}, {"tactic": "simp", "state_before": "case nil\n\u03b9 : Type ?u.2388864\n\u03b9' : Type ?u.2388867\n\u03b1 : \u03b9 \u2192 Type ?u.2388872\ninst\u271d\u00b3 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (\u03b1 i)\n\u03b4 : Type u_1\n\u03c0 : \u03b4 \u2192 Type u_2\ninst\u271d\u00b9 : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\n\u03bc : (i : \u03b4) \u2192 Measure (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b4), SigmaFinite (\u03bc i)\ns : (i : \u03b4) \u2192 Set (\u03c0 i)\n\u22a2 \u2191\u2191(Measure.tprod [] \u03bc) (Set.tprod [] s) = List.prod (List.map (fun i => \u2191\u2191(\u03bc i) (s i)) [])", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Strict.lean", "full_name": "StrictConvex.is_linear_preimage", "start": [152, 1], "end": [155, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/IntervalAverage.lean", "full_name": "interval_average_eq", "start": [46, 1], "end": [52, 91], "traced_tactics": [{"tactic": "cases' le_or_lt a b with h h", "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\na b : \u211d\n\u22a2 (\u2a0d (x : \u211d) in \u0399 a b, f x) = (b - a)\u207b\u00b9 \u2022 \u222b (x : \u211d) in a..b, f x", "state_after": "case inl\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\na b : \u211d\nh : a \u2264 b\n\u22a2 (\u2a0d (x : \u211d) in \u0399 a b, f x) = (b - a)\u207b\u00b9 \u2022 \u222b (x : \u211d) in a..b, f x\n\ncase inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\na b : \u211d\nh : b < a\n\u22a2 (\u2a0d (x : \u211d) in \u0399 a b, f x) = (b - a)\u207b\u00b9 \u2022 \u222b (x : \u211d) in a..b, f x"}, {"tactic": "rw [set_average_eq, uIoc_of_le h, Real.volume_Ioc, intervalIntegral.integral_of_le h,\n  ENNReal.toReal_ofReal (sub_nonneg.2 h)]", "state_before": "case inl\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\na b : \u211d\nh : a \u2264 b\n\u22a2 (\u2a0d (x : \u211d) in \u0399 a b, f x) = (b - a)\u207b\u00b9 \u2022 \u222b (x : \u211d) in a..b, f x", "state_after": "no goals"}, {"tactic": "rw [set_average_eq, uIoc_of_lt h, Real.volume_Ioc, intervalIntegral.integral_of_ge h.le,\n  ENNReal.toReal_ofReal (sub_nonneg.2 h.le), smul_neg, \u2190 neg_smul, \u2190 inv_neg, neg_sub]", "state_before": "case inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u211d \u2192 E\na b : \u211d\nh : b < a\n\u22a2 (\u2a0d (x : \u211d) in \u0399 a b, f x) = (b - a)\u207b\u00b9 \u2022 \u222b (x : \u211d) in a..b, f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Int/Lemmas.lean", "full_name": "Int.negOfNat_eq_subNatNat_zero", "start": [410, 1], "end": [410, 90], "traced_tactics": [{"tactic": "cases n <;> rfl", "state_before": "n : Nat\n\u22a2 negOfNat n = subNatNat 0 n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Germ.lean", "full_name": "Filter.Germ.map\u2082_const", "start": [274, 1], "end": [276, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Basic.lean", "full_name": "comp_mul_right", "start": [46, 1], "end": [48, 19], "traced_tactics": [{"tactic": "ext z", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.797\nG : Type ?u.800\ninst\u271d : Semigroup \u03b1\nx y : \u03b1\n\u22a2 ((fun x_1 => x_1 * x) \u2218 fun x => x * y) = fun x_1 => x_1 * (y * x)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.797\nG : Type ?u.800\ninst\u271d : Semigroup \u03b1\nx y z : \u03b1\n\u22a2 ((fun x_1 => x_1 * x) \u2218 fun x => x * y) z = z * (y * x)"}, {"tactic": "simp [mul_assoc]", "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.797\nG : Type ?u.800\ninst\u271d : Semigroup \u03b1\nx y z : \u03b1\n\u22a2 ((fun x_1 => x_1 * x) \u2218 fun x => x * y) z = z * (y * x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Game/PGame.lean", "full_name": "PGame.add_right_neg_le_zero", "start": [1689, 1], "end": [1690, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Lipschitz.lean", "full_name": "LipschitzWith.of_le_add_mul", "start": [333, 11], "end": [334, 98], "traced_tactics": [{"tactic": "simpa only [Real.toNNReal_coe] using LipschitzWith.of_le_add_mul' K h", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : PseudoMetricSpace \u03b3\nK\u271d : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\nf : \u03b1 \u2192 \u211d\nK : \u211d\u22650\nh : \u2200 (x y : \u03b1), f x \u2264 f y + \u2191K * dist x y\n\u22a2 LipschitzWith K f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/FractionalIdeal.lean", "full_name": "FractionalIdeal.mul_mem_mul", "start": [578, 1], "end": [581, 36], "traced_tactics": [{"tactic": "simp only [mul_def]", "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Submonoid R\nP : Type u_2\ninst\u271d\u00b9 : CommRing P\ninst\u271d : Algebra R P\nloc : IsLocalization S P\nI J : FractionalIdeal S P\ni j : P\nhi : i \u2208 I\nhj : j \u2208 J\n\u22a2 i * j \u2208 I * J", "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Submonoid R\nP : Type u_2\ninst\u271d\u00b9 : CommRing P\ninst\u271d : Algebra R P\nloc : IsLocalization S P\nI J : FractionalIdeal S P\ni j : P\nhi : i \u2208 I\nhj : j \u2208 J\n\u22a2 i * j \u2208 { val := \u2191I * \u2191J, property := (_ : IsFractional S (\u2191I * \u2191J)) }"}, {"tactic": "exact Submodule.mul_mem_mul hi hj", "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\nS : Submonoid R\nP : Type u_2\ninst\u271d\u00b9 : CommRing P\ninst\u271d : Algebra R P\nloc : IsLocalization S P\nI J : FractionalIdeal S P\ni j : P\nhi : i \u2208 I\nhj : j \u2208 J\n\u22a2 i * j \u2208 { val := \u2191I * \u2191J, property := (_ : IsFractional S (\u2191I * \u2191J)) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finsupp/Basic.lean", "full_name": "Finsupp.mem_splitSupport_iff_nonzero", "start": [1805, 1], "end": [1810, 26], "traced_tactics": [{"tactic": "rw [splitSupport, @mem_image _ _ (Classical.decEq _), Ne.def, \u2190 support_eq_empty, \u2190 Ne.def, \u2190\n  Finset.nonempty_iff_ne_empty, split, comapDomain, Finset.Nonempty]", "state_before": "\u03b1 : Type ?u.738687\n\u03b2 : Type ?u.738690\n\u03b3 : Type ?u.738693\n\u03b9 : Type u_1\nM : Type u_2\nM' : Type ?u.738702\nN : Type ?u.738705\nP : Type ?u.738708\nG : Type ?u.738711\nH : Type ?u.738714\nR : Type ?u.738717\nS : Type ?u.738720\n\u03b1s : \u03b9 \u2192 Type u_3\ninst\u271d : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\ni : \u03b9\n\u22a2 i \u2208 splitSupport l \u2194 split l i \u2260 0", "state_after": "\u03b1 : Type ?u.738687\n\u03b2 : Type ?u.738690\n\u03b3 : Type ?u.738693\n\u03b9 : Type u_1\nM : Type u_2\nM' : Type ?u.738702\nN : Type ?u.738705\nP : Type ?u.738708\nG : Type ?u.738711\nH : Type ?u.738714\nR : Type ?u.738717\nS : Type ?u.738720\n\u03b1s : \u03b9 \u2192 Type u_3\ninst\u271d : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\ni : \u03b9\n\u22a2 (\u2203 a, a \u2208 l.support \u2227 a.fst = i) \u2194\n    \u2203 x,\n      x \u2208\n        {\n            support :=\n              preimage l.support (Sigma.mk i)\n                (_ :\n                  \u2200 (x : \u03b1s i),\n                    x \u2208 Sigma.mk i \u207b\u00b9' \u2191l.support \u2192\n                      \u2200 (x_2 : \u03b1s i),\n                        x_2 \u2208 Sigma.mk i \u207b\u00b9' \u2191l.support \u2192 { fst := i, snd := x } = { fst := i, snd := x_2 } \u2192 x = x_2),\n            toFun := fun a => \u2191l { fst := i, snd := a },\n            mem_support_toFun :=\n              (_ :\n                \u2200 (a : \u03b1s i),\n                  a \u2208\n                      preimage l.support (Sigma.mk i)\n                        (_ :\n                          \u2200 (x : \u03b1s i),\n                            x \u2208 Sigma.mk i \u207b\u00b9' \u2191l.support \u2192\n                              \u2200 (x_2 : \u03b1s i),\n                                x_2 \u2208 Sigma.mk i \u207b\u00b9' \u2191l.support \u2192\n                                  { fst := i, snd := x } = { fst := i, snd := x_2 } \u2192 x = x_2) \u2194\n                    (fun a => \u2191l { fst := i, snd := a }) a \u2260 0) }.support"}, {"tactic": "simp only [exists_prop, Finset.mem_preimage, exists_and_right, exists_eq_right, mem_support_iff,\n  Sigma.exists, Ne.def]", "state_before": "\u03b1 : Type ?u.738687\n\u03b2 : Type ?u.738690\n\u03b3 : Type ?u.738693\n\u03b9 : Type u_1\nM : Type u_2\nM' : Type ?u.738702\nN : Type ?u.738705\nP : Type ?u.738708\nG : Type ?u.738711\nH : Type ?u.738714\nR : Type ?u.738717\nS : Type ?u.738720\n\u03b1s : \u03b9 \u2192 Type u_3\ninst\u271d : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\ni : \u03b9\n\u22a2 (\u2203 a, a \u2208 l.support \u2227 a.fst = i) \u2194\n    \u2203 x,\n      x \u2208\n        {\n            support :=\n              preimage l.support (Sigma.mk i)\n                (_ :\n                  \u2200 (x : \u03b1s i),\n                    x \u2208 Sigma.mk i \u207b\u00b9' \u2191l.support \u2192\n                      \u2200 (x_2 : \u03b1s i),\n                        x_2 \u2208 Sigma.mk i \u207b\u00b9' \u2191l.support \u2192 { fst := i, snd := x } = { fst := i, snd := x_2 } \u2192 x = x_2),\n            toFun := fun a => \u2191l { fst := i, snd := a },\n            mem_support_toFun :=\n              (_ :\n                \u2200 (a : \u03b1s i),\n                  a \u2208\n                      preimage l.support (Sigma.mk i)\n                        (_ :\n                          \u2200 (x : \u03b1s i),\n                            x \u2208 Sigma.mk i \u207b\u00b9' \u2191l.support \u2192\n                              \u2200 (x_2 : \u03b1s i),\n                                x_2 \u2208 Sigma.mk i \u207b\u00b9' \u2191l.support \u2192\n                                  { fst := i, snd := x } = { fst := i, snd := x_2 } \u2192 x = x_2) \u2194\n                    (fun a => \u2191l { fst := i, snd := a }) a \u2260 0) }.support", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "full_name": "MeasureTheory.SignedMeasure.singularPart_mutuallySingular", "start": [849, 1], "end": [864, 40], "traced_tactics": [{"tactic": "by_cases hl : s.HaveLebesgueDecomposition \u03bc", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhl : HaveLebesgueDecomposition s \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhl : \u00acHaveLebesgueDecomposition s \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc"}, {"tactic": "haveI := hl", "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhl : HaveLebesgueDecomposition s \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhl this : HaveLebesgueDecomposition s \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc"}, {"tactic": "obtain \u27e8i, hi, hpos, hneg\u27e9 := s.toJordanDecomposition.mutuallySingular", "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhl this : HaveLebesgueDecomposition s \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "case pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhl this : HaveLebesgueDecomposition s \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\nhpos : \u2191\u2191(toJordanDecomposition s).posPart i = 0\nhneg : \u2191\u2191(toJordanDecomposition s).negPart (i\u1d9c) = 0\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc"}, {"tactic": "rw [s.toJordanDecomposition.posPart.haveLebesgueDecomposition_add \u03bc] at hpos", "state_before": "case pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhl this : HaveLebesgueDecomposition s \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\nhpos : \u2191\u2191(toJordanDecomposition s).posPart i = 0\nhneg : \u2191\u2191(toJordanDecomposition s).negPart (i\u1d9c) = 0\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "case pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhl this : HaveLebesgueDecomposition s \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\nhpos :\n  \u2191\u2191(Measure.singularPart (toJordanDecomposition s).posPart \u03bc +\n            withDensity \u03bc (rnDeriv (toJordanDecomposition s).posPart \u03bc))\n      i =\n    0\nhneg : \u2191\u2191(toJordanDecomposition s).negPart (i\u1d9c) = 0\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc"}, {"tactic": "rw [s.toJordanDecomposition.negPart.haveLebesgueDecomposition_add \u03bc] at hneg", "state_before": "case pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhl this : HaveLebesgueDecomposition s \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\nhpos :\n  \u2191\u2191(Measure.singularPart (toJordanDecomposition s).posPart \u03bc +\n            withDensity \u03bc (rnDeriv (toJordanDecomposition s).posPart \u03bc))\n      i =\n    0\nhneg : \u2191\u2191(toJordanDecomposition s).negPart (i\u1d9c) = 0\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "case pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhl this : HaveLebesgueDecomposition s \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\nhpos :\n  \u2191\u2191(Measure.singularPart (toJordanDecomposition s).posPart \u03bc +\n            withDensity \u03bc (rnDeriv (toJordanDecomposition s).posPart \u03bc))\n      i =\n    0\nhneg :\n  \u2191\u2191(Measure.singularPart (toJordanDecomposition s).negPart \u03bc +\n            withDensity \u03bc (rnDeriv (toJordanDecomposition s).negPart \u03bc))\n      (i\u1d9c) =\n    0\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc"}, {"tactic": "rw [add_apply, add_eq_zero_iff] at hpos hneg", "state_before": "case pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhl this : HaveLebesgueDecomposition s \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\nhpos :\n  \u2191\u2191(Measure.singularPart (toJordanDecomposition s).posPart \u03bc +\n            withDensity \u03bc (rnDeriv (toJordanDecomposition s).posPart \u03bc))\n      i =\n    0\nhneg :\n  \u2191\u2191(Measure.singularPart (toJordanDecomposition s).negPart \u03bc +\n            withDensity \u03bc (rnDeriv (toJordanDecomposition s).negPart \u03bc))\n      (i\u1d9c) =\n    0\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "case pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhl this : HaveLebesgueDecomposition s \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\nhpos :\n  \u2191\u2191(Measure.singularPart (toJordanDecomposition s).posPart \u03bc) i = 0 \u2227\n    \u2191\u2191(withDensity \u03bc (rnDeriv (toJordanDecomposition s).posPart \u03bc)) i = 0\nhneg :\n  \u2191\u2191(Measure.singularPart (toJordanDecomposition s).negPart \u03bc) (i\u1d9c) = 0 \u2227\n    \u2191\u2191(withDensity \u03bc (rnDeriv (toJordanDecomposition s).negPart \u03bc)) (i\u1d9c) = 0\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc"}, {"tactic": "exact \u27e8i, hi, hpos.1, hneg.1\u27e9", "state_before": "case pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhl this : HaveLebesgueDecomposition s \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\nhpos :\n  \u2191\u2191(Measure.singularPart (toJordanDecomposition s).posPart \u03bc) i = 0 \u2227\n    \u2191\u2191(withDensity \u03bc (rnDeriv (toJordanDecomposition s).posPart \u03bc)) i = 0\nhneg :\n  \u2191\u2191(Measure.singularPart (toJordanDecomposition s).negPart \u03bc) (i\u1d9c) = 0 \u2227\n    \u2191\u2191(withDensity \u03bc (rnDeriv (toJordanDecomposition s).negPart \u03bc)) (i\u1d9c) = 0\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "no goals"}, {"tactic": "rw [not_haveLebesgueDecomposition_iff] at hl", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhl : \u00acHaveLebesgueDecomposition s \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhl :\n  \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart \u03bc \u2228\n    \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).negPart \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc"}, {"tactic": "cases' hl with hp hn", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhl :\n  \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart \u03bc \u2228\n    \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).negPart \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhp : \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc\n\ncase neg.inr\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhn : \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).negPart \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc"}, {"tactic": "rw [Measure.singularPart, dif_neg hp]", "state_before": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhp : \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhp : \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart \u03bc\n\u22a2 0 \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc"}, {"tactic": "exact MutuallySingular.zero_left", "state_before": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhp : \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart \u03bc\n\u22a2 0 \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "no goals"}, {"tactic": "rw [Measure.singularPart, Measure.singularPart, dif_neg hn]", "state_before": "case neg.inr\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhn : \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).negPart \u03bc\n\u22a2 Measure.singularPart (toJordanDecomposition s).posPart \u03bc \u27c2\u2098 Measure.singularPart (toJordanDecomposition s).negPart \u03bc", "state_after": "case neg.inr\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhn : \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).negPart \u03bc\n\u22a2 (if h : Measure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart \u03bc then\n      (Classical.choose\n          (_ :\n            \u2203 p, Measurable p.snd \u2227 p.fst \u27c2\u2098 \u03bc \u2227 (toJordanDecomposition s).posPart = p.fst + withDensity \u03bc p.snd)).fst\n    else 0) \u27c2\u2098\n    0"}, {"tactic": "exact MutuallySingular.zero_right", "state_before": "case neg.inr\n\u03b1 : Type u_1\n\u03b2 : Type ?u.300315\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ns : SignedMeasure \u03b1\n\u03bc : Measure \u03b1\nhn : \u00acMeasure.HaveLebesgueDecomposition (toJordanDecomposition s).negPart \u03bc\n\u22a2 (if h : Measure.HaveLebesgueDecomposition (toJordanDecomposition s).posPart \u03bc then\n      (Classical.choose\n          (_ :\n            \u2203 p, Measurable p.snd \u2227 p.fst \u27c2\u2098 \u03bc \u2227 (toJordanDecomposition s).posPart = p.fst + withDensity \u03bc p.snd)).fst\n    else 0) \u27c2\u2098\n    0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "full_name": "Subgroup.normalClosure_le_normal", "start": [2484, 1], "end": [2490, 22], "traced_tactics": [{"tactic": "intro a w", "state_before": "G : Type u_1\nG' : Type ?u.441144\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group G'\nA : Type ?u.441153\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN : Subgroup G\ninst\u271d : Normal N\nh : s \u2286 \u2191N\n\u22a2 normalClosure s \u2264 N", "state_after": "G : Type u_1\nG' : Type ?u.441144\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group G'\nA : Type ?u.441153\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN : Subgroup G\ninst\u271d : Normal N\nh : s \u2286 \u2191N\na : G\nw : a \u2208 normalClosure s\n\u22a2 a \u2208 N"}, {"tactic": "refine' closure_induction w (fun x hx => _) _ (fun x y ihx ihy => _) fun x ihx => _", "state_before": "G : Type u_1\nG' : Type ?u.441144\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group G'\nA : Type ?u.441153\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN : Subgroup G\ninst\u271d : Normal N\nh : s \u2286 \u2191N\na : G\nw : a \u2208 normalClosure s\n\u22a2 a \u2208 N", "state_after": "case refine'_1\nG : Type u_1\nG' : Type ?u.441144\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group G'\nA : Type ?u.441153\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN : Subgroup G\ninst\u271d : Normal N\nh : s \u2286 \u2191N\na : G\nw : a \u2208 normalClosure s\nx : G\nhx : x \u2208 conjugatesOfSet s\n\u22a2 x \u2208 N\n\ncase refine'_2\nG : Type u_1\nG' : Type ?u.441144\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group G'\nA : Type ?u.441153\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN : Subgroup G\ninst\u271d : Normal N\nh : s \u2286 \u2191N\na : G\nw : a \u2208 normalClosure s\n\u22a2 1 \u2208 N\n\ncase refine'_3\nG : Type u_1\nG' : Type ?u.441144\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group G'\nA : Type ?u.441153\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN : Subgroup G\ninst\u271d : Normal N\nh : s \u2286 \u2191N\na : G\nw : a \u2208 normalClosure s\nx y : G\nihx : x \u2208 N\nihy : y \u2208 N\n\u22a2 x * y \u2208 N\n\ncase refine'_4\nG : Type u_1\nG' : Type ?u.441144\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group G'\nA : Type ?u.441153\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN : Subgroup G\ninst\u271d : Normal N\nh : s \u2286 \u2191N\na : G\nw : a \u2208 normalClosure s\nx : G\nihx : x \u2208 N\n\u22a2 x\u207b\u00b9 \u2208 N"}, {"tactic": "exact conjugatesOfSet_subset h hx", "state_before": "case refine'_1\nG : Type u_1\nG' : Type ?u.441144\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group G'\nA : Type ?u.441153\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN : Subgroup G\ninst\u271d : Normal N\nh : s \u2286 \u2191N\na : G\nw : a \u2208 normalClosure s\nx : G\nhx : x \u2208 conjugatesOfSet s\n\u22a2 x \u2208 N", "state_after": "no goals"}, {"tactic": "exact one_mem _", "state_before": "case refine'_2\nG : Type u_1\nG' : Type ?u.441144\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group G'\nA : Type ?u.441153\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN : Subgroup G\ninst\u271d : Normal N\nh : s \u2286 \u2191N\na : G\nw : a \u2208 normalClosure s\n\u22a2 1 \u2208 N", "state_after": "no goals"}, {"tactic": "exact mul_mem ihx ihy", "state_before": "case refine'_3\nG : Type u_1\nG' : Type ?u.441144\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group G'\nA : Type ?u.441153\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN : Subgroup G\ninst\u271d : Normal N\nh : s \u2286 \u2191N\na : G\nw : a \u2208 normalClosure s\nx y : G\nihx : x \u2208 N\nihy : y \u2208 N\n\u22a2 x * y \u2208 N", "state_after": "no goals"}, {"tactic": "exact inv_mem ihx", "state_before": "case refine'_4\nG : Type u_1\nG' : Type ?u.441144\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group G'\nA : Type ?u.441153\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN : Subgroup G\ninst\u271d : Normal N\nh : s \u2286 \u2191N\na : G\nw : a \u2208 normalClosure s\nx : G\nihx : x \u2208 N\n\u22a2 x\u207b\u00b9 \u2208 N", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Rat/Lemmas.lean", "full_name": "Rat.substr_num_den'", "start": [170, 1], "end": [173, 25], "traced_tactics": [{"tactic": "rw [sub_eq_add_neg, sub_eq_add_neg, \u2190 neg_mul, \u2190 num_neg_eq_neg_num, \u2190 den_neg_eq_den r,\n  add_num_den' q (-r)]", "state_before": "q r : \u211a\n\u22a2 (q - r).num * \u2191q.den * \u2191r.den = (q.num * \u2191r.den - r.num * \u2191q.den) * \u2191(q - r).den", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "full_name": "WithBot.add_lt_add_of_lt_of_le", "start": [720, 11], "end": [723, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/Hyperreal.lean", "full_name": "Hyperreal.IsSt.isSt_st", "start": [345, 1], "end": [346, 18], "traced_tactics": [{"tactic": "rwa [hxr.st_eq]", "state_before": "x : \u211d*\nr : \u211d\nhxr : IsSt x r\n\u22a2 IsSt x (st x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "full_name": "Left.mul_eq_mul_iff_eq_and_eq", "start": [1242, 1], "end": [1253, 46], "traced_tactics": [{"tactic": "rcases hac.eq_or_lt with (rfl | hac)", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.62108\ninst\u271d\u2075 : Semigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c d : \u03b1\nhac : a \u2264 c\nhbd : b \u2264 d\nh : a * b = c * d\n\u22a2 a = c \u2227 b = d", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type ?u.62108\ninst\u271d\u2075 : Semigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b d : \u03b1\nhbd : b \u2264 d\nhac : a \u2264 a\nh : a * b = a * d\n\u22a2 a = a \u2227 b = d\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type ?u.62108\ninst\u271d\u2075 : Semigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c d : \u03b1\nhac\u271d : a \u2264 c\nhbd : b \u2264 d\nh : a * b = c * d\nhac : a < c\n\u22a2 a = c \u2227 b = d"}, {"tactic": "rcases eq_or_lt_of_le hbd with (rfl | hbd)", "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type ?u.62108\ninst\u271d\u2075 : Semigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c d : \u03b1\nhac\u271d : a \u2264 c\nhbd : b \u2264 d\nh : a * b = c * d\nhac : a < c\n\u22a2 a = c \u2227 b = d", "state_after": "case inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type ?u.62108\ninst\u271d\u2075 : Semigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\nhac\u271d : a \u2264 c\nhac : a < c\nhbd : b \u2264 b\nh : a * b = c * b\n\u22a2 a = c \u2227 b = b\n\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type ?u.62108\ninst\u271d\u2075 : Semigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c d : \u03b1\nhac\u271d : a \u2264 c\nhbd\u271d : b \u2264 d\nh : a * b = c * d\nhac : a < c\nhbd : b < d\n\u22a2 a = c \u2227 b = d"}, {"tactic": "exact ((Left.mul_lt_mul hac hbd).ne h).elim", "state_before": "case inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type ?u.62108\ninst\u271d\u2075 : Semigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c d : \u03b1\nhac\u271d : a \u2264 c\nhbd\u271d : b \u2264 d\nh : a * b = c * d\nhac : a < c\nhbd : b < d\n\u22a2 a = c \u2227 b = d", "state_after": "no goals"}, {"tactic": "exact \u27e8rfl, mul_left_cancel'' h\u27e9", "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type ?u.62108\ninst\u271d\u2075 : Semigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b d : \u03b1\nhbd : b \u2264 d\nhac : a \u2264 a\nh : a * b = a * d\n\u22a2 a = a \u2227 b = d", "state_after": "no goals"}, {"tactic": "exact \u27e8mul_right_cancel'' h, rfl\u27e9", "state_before": "case inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type ?u.62108\ninst\u271d\u2075 : Semigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\nhac\u271d : a \u2264 c\nhac : a < c\nhbd : b \u2264 b\nh : a * b = c * b\n\u22a2 a = c \u2227 b = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/Calculus.lean", "full_name": "contDiff_euclidean", "start": [354, 1], "end": [356, 6], "traced_tactics": [{"tactic": "rw [\u2190 (EuclideanSpace.equiv \u03b9 \ud835\udd5c).comp_contDiff_iff, contDiff_pi]", "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_3\nH : Type u_2\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\nn : \u2115\u221e\n\u22a2 ContDiff \ud835\udd5c n f \u2194 \u2200 (i : \u03b9), ContDiff \ud835\udd5c n fun x => f x i", "state_after": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_3\nH : Type u_2\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\nn : \u2115\u221e\n\u22a2 (\u2200 (i : \u03b9), ContDiff \ud835\udd5c n fun x => (\u2191(EuclideanSpace.equiv \u03b9 \ud835\udd5c) \u2218 f) x i) \u2194 \u2200 (i : \u03b9), ContDiff \ud835\udd5c n fun x => f x i"}, {"tactic": "rfl", "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_3\nH : Type u_2\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\nn : \u2115\u221e\n\u22a2 (\u2200 (i : \u03b9), ContDiff \ud835\udd5c n fun x => (\u2191(EuclideanSpace.equiv \u03b9 \ud835\udd5c) \u2218 f) x i) \u2194 \u2200 (i : \u03b9), ContDiff \ud835\udd5c n fun x => f x i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Maps.lean", "full_name": "OpenEmbedding.tendsto_nhds_iff", "start": [568, 1], "end": [570, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Int.le_floor_add", "start": [734, 1], "end": [736, 45], "traced_tactics": [{"tactic": "rw [le_floor, Int.cast_add]", "state_before": "F : Type ?u.131188\n\u03b1 : Type u_1\n\u03b2 : Type ?u.131194\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 \u230aa\u230b + \u230ab\u230b \u2264 \u230aa + b\u230b", "state_after": "F : Type ?u.131188\n\u03b1 : Type u_1\n\u03b2 : Type ?u.131194\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 \u2191\u230aa\u230b + \u2191\u230ab\u230b \u2264 a + b"}, {"tactic": "exact add_le_add (floor_le _) (floor_le _)", "state_before": "F : Type ?u.131188\n\u03b1 : Type u_1\n\u03b2 : Type ?u.131194\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 \u2191\u230aa\u230b + \u2191\u230ab\u230b \u2264 a + b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "full_name": "Filter.HasBasis.cauchySeq_iff", "start": [285, 1], "end": [291, 81], "traced_tactics": [{"tactic": "rw [cauchySeq_iff_tendsto, \u2190 prod_atTop_atTop_eq]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b3 : Sort u_1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\np : \u03b3 \u2192 Prop\ns : \u03b3 \u2192 Set (\u03b1 \u00d7 \u03b1)\nh : HasBasis (\ud835\udce4 \u03b1) p s\n\u22a2 CauchySeq u \u2194 \u2200 (i : \u03b3), p i \u2192 \u2203 N, \u2200 (m : \u03b2), N \u2264 m \u2192 \u2200 (n : \u03b2), N \u2264 n \u2192 (u m, u n) \u2208 s i", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b3 : Sort u_1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\np : \u03b3 \u2192 Prop\ns : \u03b3 \u2192 Set (\u03b1 \u00d7 \u03b1)\nh : HasBasis (\ud835\udce4 \u03b1) p s\n\u22a2 Tendsto (Prod.map u u) (atTop \u00d7\u02e2 atTop) (\ud835\udce4 \u03b1) \u2194\n    \u2200 (i : \u03b3), p i \u2192 \u2203 N, \u2200 (m : \u03b2), N \u2264 m \u2192 \u2200 (n : \u03b2), N \u2264 n \u2192 (u m, u n) \u2208 s i"}, {"tactic": "refine' (atTop_basis.prod_self.tendsto_iff h).trans _", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b3 : Sort u_1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\np : \u03b3 \u2192 Prop\ns : \u03b3 \u2192 Set (\u03b1 \u00d7 \u03b1)\nh : HasBasis (\ud835\udce4 \u03b1) p s\n\u22a2 Tendsto (Prod.map u u) (atTop \u00d7\u02e2 atTop) (\ud835\udce4 \u03b1) \u2194\n    \u2200 (i : \u03b3), p i \u2192 \u2203 N, \u2200 (m : \u03b2), N \u2264 m \u2192 \u2200 (n : \u03b2), N \u2264 n \u2192 (u m, u n) \u2208 s i", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b3 : Sort u_1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\np : \u03b3 \u2192 Prop\ns : \u03b3 \u2192 Set (\u03b1 \u00d7 \u03b1)\nh : HasBasis (\ud835\udce4 \u03b1) p s\n\u22a2 (\u2200 (ib : \u03b3), p ib \u2192 \u2203 ia, True \u2227 \u2200 (x : \u03b2 \u00d7 \u03b2), x \u2208 Ici ia \u00d7\u02e2 Ici ia \u2192 Prod.map u u x \u2208 s ib) \u2194\n    \u2200 (i : \u03b3), p i \u2192 \u2203 N, \u2200 (m : \u03b2), N \u2264 m \u2192 \u2200 (n : \u03b2), N \u2264 n \u2192 (u m, u n) \u2208 s i"}, {"tactic": "simp only [exists_prop, true_and_iff, MapsTo, preimage, subset_def, Prod.forall, mem_prod_eq,\n  mem_setOf_eq, mem_Ici, and_imp, Prod.map, ge_iff_le, @forall_swap (_ \u2264 _) \u03b2]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b3 : Sort u_1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\np : \u03b3 \u2192 Prop\ns : \u03b3 \u2192 Set (\u03b1 \u00d7 \u03b1)\nh : HasBasis (\ud835\udce4 \u03b1) p s\n\u22a2 (\u2200 (ib : \u03b3), p ib \u2192 \u2203 ia, True \u2227 \u2200 (x : \u03b2 \u00d7 \u03b2), x \u2208 Ici ia \u00d7\u02e2 Ici ia \u2192 Prod.map u u x \u2208 s ib) \u2194\n    \u2200 (i : \u03b3), p i \u2192 \u2203 N, \u2200 (m : \u03b2), N \u2264 m \u2192 \u2200 (n : \u03b2), N \u2264 n \u2192 (u m, u n) \u2208 s i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/TrivSqZeroExt.lean", "full_name": "TrivSqZeroExt.inl_sum", "start": [339, 1], "end": [341, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/BilinearForm.lean", "full_name": "BilinForm.linMulLin_compLeft", "start": [741, 1], "end": [743, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subsemigroup/Basic.lean", "full_name": "Subsemigroup.coe_top", "start": [204, 1], "end": [205, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Int/Lemmas.lean", "full_name": "Int.zero_ne_one", "start": [483, 11], "end": [483, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "full_name": "antitone_mul_left", "start": [417, 1], "end": [418, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/LocalHomeomorph.lean", "full_name": "LocalHomeomorph.source_inter_preimage_inv_preimage", "start": [295, 1], "end": [297, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/MvPolynomial/Variables.lean", "full_name": "MvPolynomial.totalDegree_zero", "start": [627, 1], "end": [628, 42], "traced_tactics": [{"tactic": "rw [\u2190 C_0]", "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type ?u.394806\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\n\u22a2 totalDegree 0 = 0", "state_after": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type ?u.394806\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\n\u22a2 totalDegree (\u2191C 0) = 0"}, {"tactic": "exact totalDegree_C (0 : R)", "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type ?u.394806\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\n\u22a2 totalDegree (\u2191C 0) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "full_name": "Real.cos_eq_sqrt_one_sub_sin_sq", "start": [492, 1], "end": [494, 88], "traced_tactics": [{"tactic": "rw [\u2190 abs_cos_eq_sqrt_one_sub_sin_sq, abs_of_nonneg (cos_nonneg_of_mem_Icc \u27e8hl, hu\u27e9)]", "state_before": "x : \u211d\nhl : -(\u03c0 / 2) \u2264 x\nhu : x \u2264 \u03c0 / 2\n\u22a2 cos x = sqrt (1 - sin x ^ 2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Basic.lean", "full_name": "Metric.tendsto_atTop", "start": [1098, 1], "end": [1101, 30], "traced_tactics": [{"tactic": "simp only [true_and]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.140991\n\u03b9 : Type ?u.140994\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 (\u2200 (ib : \u211d), 0 < ib \u2192 \u2203 ia, True \u2227 \u2200 (x : \u03b2), x \u2208 Ici ia \u2192 u x \u2208 ball a ib) \u2194\n    \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b2), n \u2265 N \u2192 dist (u n) a < \u03b5", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.140991\n\u03b9 : Type ?u.140994\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 (\u2200 (ib : \u211d), 0 < ib \u2192 \u2203 ia, \u2200 (x : \u03b2), x \u2208 Ici ia \u2192 u x \u2208 ball a ib) \u2194\n    \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b2), n \u2265 N \u2192 dist (u n) a < \u03b5"}, {"tactic": "rfl", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.140991\n\u03b9 : Type ?u.140994\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 (\u2200 (ib : \u211d), 0 < ib \u2192 \u2203 ia, \u2200 (x : \u03b2), x \u2208 Ici ia \u2192 u x \u2208 ball a ib) \u2194\n    \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b2), n \u2265 N \u2192 dist (u n) a < \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.eq_singleton_iff_unique_mem", "start": [738, 1], "end": [744, 48], "traced_tactics": [{"tactic": "constructor <;> intro t", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.30589\n\u03b3 : Type ?u.30592\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\n\u22a2 s = {a} \u2194 a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type ?u.30589\n\u03b3 : Type ?u.30592\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : s = {a}\n\u22a2 a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\n\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type ?u.30589\n\u03b3 : Type ?u.30592\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\n\u22a2 s = {a}"}, {"tactic": "rw [t]", "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type ?u.30589\n\u03b3 : Type ?u.30592\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : s = {a}\n\u22a2 a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type ?u.30589\n\u03b3 : Type ?u.30592\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : s = {a}\n\u22a2 a \u2208 {a} \u2227 \u2200 (x : \u03b1), x \u2208 {a} \u2192 x = a"}, {"tactic": "exact \u27e8Finset.mem_singleton_self _, fun _ => Finset.mem_singleton.1\u27e9", "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type ?u.30589\n\u03b3 : Type ?u.30592\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : s = {a}\n\u22a2 a \u2208 {a} \u2227 \u2200 (x : \u03b1), x \u2208 {a} \u2192 x = a", "state_after": "no goals"}, {"tactic": "ext", "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type ?u.30589\n\u03b3 : Type ?u.30592\ns\u271d : Finset \u03b1\na\u271d b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\n\u22a2 s = {a}", "state_after": "case mpr.a\n\u03b1 : Type u_1\n\u03b2 : Type ?u.30589\n\u03b3 : Type ?u.30592\ns\u271d : Finset \u03b1\na\u271d\u00b9 b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 s \u2194 a\u271d \u2208 {a}"}, {"tactic": "rw [Finset.mem_singleton]", "state_before": "case mpr.a\n\u03b1 : Type u_1\n\u03b2 : Type ?u.30589\n\u03b3 : Type ?u.30592\ns\u271d : Finset \u03b1\na\u271d\u00b9 b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 s \u2194 a\u271d \u2208 {a}", "state_after": "case mpr.a\n\u03b1 : Type u_1\n\u03b2 : Type ?u.30589\n\u03b3 : Type ?u.30592\ns\u271d : Finset \u03b1\na\u271d\u00b9 b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 s \u2194 a\u271d = a"}, {"tactic": "exact \u27e8t.right _, fun r => r.symm \u25b8 t.left\u27e9", "state_before": "case mpr.a\n\u03b1 : Type u_1\n\u03b2 : Type ?u.30589\n\u03b3 : Type ?u.30592\ns\u271d : Finset \u03b1\na\u271d\u00b9 b : \u03b1\ns : Finset \u03b1\na : \u03b1\nt : a \u2208 s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = a\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 s \u2194 a\u271d = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/SuccPred/LinearLocallyFinite.lean", "full_name": "LinearLocallyFiniteOrder.isGLB_Ioc_of_isGLB_Ioi", "start": [81, 1], "end": [88, 51], "traced_tactics": [{"tactic": "simp_rw [IsGLB, IsGreatest, mem_upperBounds, mem_lowerBounds] at h\u22a2", "state_before": "\u03b9 : Type u_1\ninst\u271d : LinearOrder \u03b9\ni j k : \u03b9\nhij_lt : i < j\nh : IsGLB (Set.Ioi i) k\n\u22a2 IsGLB (Set.Ioc i j) k", "state_after": "\u03b9 : Type u_1\ninst\u271d : LinearOrder \u03b9\ni j k : \u03b9\nhij_lt : i < j\nh : (\u2200 (x : \u03b9), x \u2208 Set.Ioi i \u2192 k \u2264 x) \u2227 \u2200 (x : \u03b9), (\u2200 (x_1 : \u03b9), x_1 \u2208 Set.Ioi i \u2192 x \u2264 x_1) \u2192 x \u2264 k\n\u22a2 (\u2200 (x : \u03b9), x \u2208 Set.Ioc i j \u2192 k \u2264 x) \u2227 \u2200 (x : \u03b9), (\u2200 (x_1 : \u03b9), x_1 \u2208 Set.Ioc i j \u2192 x \u2264 x_1) \u2192 x \u2264 k"}, {"tactic": "refine' \u27e8fun x hx \u21a6 h.1 x hx.1, fun x hx \u21a6 h.2 x _\u27e9", "state_before": "\u03b9 : Type u_1\ninst\u271d : LinearOrder \u03b9\ni j k : \u03b9\nhij_lt : i < j\nh : (\u2200 (x : \u03b9), x \u2208 Set.Ioi i \u2192 k \u2264 x) \u2227 \u2200 (x : \u03b9), (\u2200 (x_1 : \u03b9), x_1 \u2208 Set.Ioi i \u2192 x \u2264 x_1) \u2192 x \u2264 k\n\u22a2 (\u2200 (x : \u03b9), x \u2208 Set.Ioc i j \u2192 k \u2264 x) \u2227 \u2200 (x : \u03b9), (\u2200 (x_1 : \u03b9), x_1 \u2208 Set.Ioc i j \u2192 x \u2264 x_1) \u2192 x \u2264 k", "state_after": "\u03b9 : Type u_1\ninst\u271d : LinearOrder \u03b9\ni j k : \u03b9\nhij_lt : i < j\nh : (\u2200 (x : \u03b9), x \u2208 Set.Ioi i \u2192 k \u2264 x) \u2227 \u2200 (x : \u03b9), (\u2200 (x_1 : \u03b9), x_1 \u2208 Set.Ioi i \u2192 x \u2264 x_1) \u2192 x \u2264 k\nx : \u03b9\nhx : \u2200 (x_1 : \u03b9), x_1 \u2208 Set.Ioc i j \u2192 x \u2264 x_1\n\u22a2 \u2200 (x_1 : \u03b9), x_1 \u2208 Set.Ioi i \u2192 x \u2264 x_1"}, {"tactic": "intro y hy", "state_before": "\u03b9 : Type u_1\ninst\u271d : LinearOrder \u03b9\ni j k : \u03b9\nhij_lt : i < j\nh : (\u2200 (x : \u03b9), x \u2208 Set.Ioi i \u2192 k \u2264 x) \u2227 \u2200 (x : \u03b9), (\u2200 (x_1 : \u03b9), x_1 \u2208 Set.Ioi i \u2192 x \u2264 x_1) \u2192 x \u2264 k\nx : \u03b9\nhx : \u2200 (x_1 : \u03b9), x_1 \u2208 Set.Ioc i j \u2192 x \u2264 x_1\n\u22a2 \u2200 (x_1 : \u03b9), x_1 \u2208 Set.Ioi i \u2192 x \u2264 x_1", "state_after": "\u03b9 : Type u_1\ninst\u271d : LinearOrder \u03b9\ni j k : \u03b9\nhij_lt : i < j\nh : (\u2200 (x : \u03b9), x \u2208 Set.Ioi i \u2192 k \u2264 x) \u2227 \u2200 (x : \u03b9), (\u2200 (x_1 : \u03b9), x_1 \u2208 Set.Ioi i \u2192 x \u2264 x_1) \u2192 x \u2264 k\nx : \u03b9\nhx : \u2200 (x_1 : \u03b9), x_1 \u2208 Set.Ioc i j \u2192 x \u2264 x_1\ny : \u03b9\nhy : y \u2208 Set.Ioi i\n\u22a2 x \u2264 y"}, {"tactic": "cases' le_or_lt y j with h_le h_lt", "state_before": "\u03b9 : Type u_1\ninst\u271d : LinearOrder \u03b9\ni j k : \u03b9\nhij_lt : i < j\nh : (\u2200 (x : \u03b9), x \u2208 Set.Ioi i \u2192 k \u2264 x) \u2227 \u2200 (x : \u03b9), (\u2200 (x_1 : \u03b9), x_1 \u2208 Set.Ioi i \u2192 x \u2264 x_1) \u2192 x \u2264 k\nx : \u03b9\nhx : \u2200 (x_1 : \u03b9), x_1 \u2208 Set.Ioc i j \u2192 x \u2264 x_1\ny : \u03b9\nhy : y \u2208 Set.Ioi i\n\u22a2 x \u2264 y", "state_after": "case inl\n\u03b9 : Type u_1\ninst\u271d : LinearOrder \u03b9\ni j k : \u03b9\nhij_lt : i < j\nh : (\u2200 (x : \u03b9), x \u2208 Set.Ioi i \u2192 k \u2264 x) \u2227 \u2200 (x : \u03b9), (\u2200 (x_1 : \u03b9), x_1 \u2208 Set.Ioi i \u2192 x \u2264 x_1) \u2192 x \u2264 k\nx : \u03b9\nhx : \u2200 (x_1 : \u03b9), x_1 \u2208 Set.Ioc i j \u2192 x \u2264 x_1\ny : \u03b9\nhy : y \u2208 Set.Ioi i\nh_le : y \u2264 j\n\u22a2 x \u2264 y\n\ncase inr\n\u03b9 : Type u_1\ninst\u271d : LinearOrder \u03b9\ni j k : \u03b9\nhij_lt : i < j\nh : (\u2200 (x : \u03b9), x \u2208 Set.Ioi i \u2192 k \u2264 x) \u2227 \u2200 (x : \u03b9), (\u2200 (x_1 : \u03b9), x_1 \u2208 Set.Ioi i \u2192 x \u2264 x_1) \u2192 x \u2264 k\nx : \u03b9\nhx : \u2200 (x_1 : \u03b9), x_1 \u2208 Set.Ioc i j \u2192 x \u2264 x_1\ny : \u03b9\nhy : y \u2208 Set.Ioi i\nh_lt : j < y\n\u22a2 x \u2264 y"}, {"tactic": "exact hx y \u27e8hy, h_le\u27e9", "state_before": "case inl\n\u03b9 : Type u_1\ninst\u271d : LinearOrder \u03b9\ni j k : \u03b9\nhij_lt : i < j\nh : (\u2200 (x : \u03b9), x \u2208 Set.Ioi i \u2192 k \u2264 x) \u2227 \u2200 (x : \u03b9), (\u2200 (x_1 : \u03b9), x_1 \u2208 Set.Ioi i \u2192 x \u2264 x_1) \u2192 x \u2264 k\nx : \u03b9\nhx : \u2200 (x_1 : \u03b9), x_1 \u2208 Set.Ioc i j \u2192 x \u2264 x_1\ny : \u03b9\nhy : y \u2208 Set.Ioi i\nh_le : y \u2264 j\n\u22a2 x \u2264 y", "state_after": "no goals"}, {"tactic": "exact le_trans (hx j \u27e8hij_lt, le_rfl\u27e9) h_lt.le", "state_before": "case inr\n\u03b9 : Type u_1\ninst\u271d : LinearOrder \u03b9\ni j k : \u03b9\nhij_lt : i < j\nh : (\u2200 (x : \u03b9), x \u2208 Set.Ioi i \u2192 k \u2264 x) \u2227 \u2200 (x : \u03b9), (\u2200 (x_1 : \u03b9), x_1 \u2208 Set.Ioi i \u2192 x \u2264 x_1) \u2192 x \u2264 k\nx : \u03b9\nhx : \u2200 (x_1 : \u03b9), x_1 \u2208 Set.Ioc i j \u2192 x \u2264 x_1\ny : \u03b9\nhy : y \u2208 Set.Ioi i\nh_lt : j < y\n\u22a2 x \u2264 y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Control/Functor/Multivariate.lean", "full_name": "MvFunctor.LiftP_def", "start": [141, 1], "end": [142, 89], "traced_tactics": [{"tactic": "simp [MvFunctor.map_map]", "state_before": "n : \u2115\n\u03b1 \u03b2 \u03b3 : TypeVec n\nF : TypeVec n \u2192 Type v\ninst\u271d\u00b9 : MvFunctor F\nP : \u03b1 \u27f9 TypeVec.repeat n Prop\nR : \u03b1 \u2297 \u03b1 \u27f9 TypeVec.repeat n Prop\ninst\u271d : LawfulMvFunctor F\nx : F \u03b1\n\u22a2 \u2200 (u : F fun i => Subtype ((fun i x => ofRepeat (P i x)) i)),\n    (fun i => Subtype.val) <$$> u = x \u2194 subtypeVal P <$$> toSubtype P <$$> u = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Basic.lean", "full_name": "Submodule.map_equiv_eq_comap_symm", "start": [2506, 1], "end": [2508, 79], "traced_tactics": [{"tactic": "rw [mem_map_equiv, mem_comap, LinearEquiv.coe_coe]", "state_before": "R : Type u_1\nR\u2081 : Type ?u.2273908\nR\u2082 : Type u_2\nR\u2083 : Type ?u.2273914\nR\u2084 : Type ?u.2273917\nS : Type ?u.2273920\nK\u271d : Type ?u.2273923\nK\u2082 : Type ?u.2273926\nM : Type u_3\nM' : Type ?u.2273932\nM\u2081 : Type ?u.2273935\nM\u2082 : Type u_4\nM\u2083 : Type ?u.2273941\nM\u2084 : Type ?u.2273944\nN : Type ?u.2273947\nN\u2082 : Type ?u.2273950\n\u03b9 : Type ?u.2273953\nV : Type ?u.2273956\nV\u2082 : Type ?u.2273959\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\ne : M \u2243\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nK : Submodule R M\nx\u271d : M\u2082\n\u22a2 x\u271d \u2208 map (\u2191e) K \u2194 x\u271d \u2208 comap (\u2191(LinearEquiv.symm e)) K", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Lists.lean", "full_name": "Lists'.mem_of_subset'", "start": [196, 1], "end": [203, 31], "traced_tactics": [{"tactic": "cases h", "state_before": "\u03b1 : Type u_1\na : Lists \u03b1\nx\u271d : Lists' \u03b1 true\nh : a \u2208 toList nil\n\u22a2 a \u2208 x\u271d", "state_after": "no goals"}, {"tactic": "cases' s with _ _ _ _ _ e m s", "state_before": "\u03b1 : Type u_1\na : Lists \u03b1\nb\u271d : Bool\na0 : Lists' \u03b1 b\u271d\nl0 l\u2082 : Lists' \u03b1 true\ns : cons' a0 l0 \u2286 l\u2082\nh : a \u2208 toList (cons' a0 l0)\n\u22a2 a \u2208 l\u2082", "state_after": "case cons\n\u03b1 : Type u_1\na : Lists \u03b1\nl0 l\u2082 : Lists' \u03b1 true\na\u271d a'\u271d : Lists \u03b1\ne : a\u271d ~ a'\u271d\nh : a \u2208 toList (cons' a\u271d.snd l0)\nm : a'\u271d \u2208 toList l\u2082\ns : Subset l0 l\u2082\n\u22a2 a \u2208 l\u2082"}, {"tactic": "simp only [toList, Sigma.eta, List.find?, List.mem_cons] at h", "state_before": "case cons\n\u03b1 : Type u_1\na : Lists \u03b1\nl0 l\u2082 : Lists' \u03b1 true\na\u271d a'\u271d : Lists \u03b1\ne : a\u271d ~ a'\u271d\nh : a \u2208 toList (cons' a\u271d.snd l0)\nm : a'\u271d \u2208 toList l\u2082\ns : Subset l0 l\u2082\n\u22a2 a \u2208 l\u2082", "state_after": "case cons\n\u03b1 : Type u_1\na : Lists \u03b1\nl0 l\u2082 : Lists' \u03b1 true\na\u271d a'\u271d : Lists \u03b1\ne : a\u271d ~ a'\u271d\nm : a'\u271d \u2208 toList l\u2082\ns : Subset l0 l\u2082\nh : a = a\u271d \u2228 a \u2208 toList l0\n\u22a2 a \u2208 l\u2082"}, {"tactic": "rcases h with (rfl | h)", "state_before": "case cons\n\u03b1 : Type u_1\na : Lists \u03b1\nl0 l\u2082 : Lists' \u03b1 true\na\u271d a'\u271d : Lists \u03b1\ne : a\u271d ~ a'\u271d\nm : a'\u271d \u2208 toList l\u2082\ns : Subset l0 l\u2082\nh : a = a\u271d \u2228 a \u2208 toList l0\n\u22a2 a \u2208 l\u2082", "state_after": "case cons.inl\n\u03b1 : Type u_1\na : Lists \u03b1\nl0 l\u2082 : Lists' \u03b1 true\na'\u271d : Lists \u03b1\nm : a'\u271d \u2208 toList l\u2082\ns : Subset l0 l\u2082\ne : a ~ a'\u271d\n\u22a2 a \u2208 l\u2082\n\ncase cons.inr\n\u03b1 : Type u_1\na : Lists \u03b1\nl0 l\u2082 : Lists' \u03b1 true\na\u271d a'\u271d : Lists \u03b1\ne : a\u271d ~ a'\u271d\nm : a'\u271d \u2208 toList l\u2082\ns : Subset l0 l\u2082\nh : a \u2208 toList l0\n\u22a2 a \u2208 l\u2082"}, {"tactic": "exact \u27e8_, m, e\u27e9", "state_before": "case cons.inl\n\u03b1 : Type u_1\na : Lists \u03b1\nl0 l\u2082 : Lists' \u03b1 true\na'\u271d : Lists \u03b1\nm : a'\u271d \u2208 toList l\u2082\ns : Subset l0 l\u2082\ne : a ~ a'\u271d\n\u22a2 a \u2208 l\u2082", "state_after": "no goals"}, {"tactic": "exact mem_of_subset' s h", "state_before": "case cons.inr\n\u03b1 : Type u_1\na : Lists \u03b1\nl0 l\u2082 : Lists' \u03b1 true\na\u271d a'\u271d : Lists \u03b1\ne : a\u271d ~ a'\u271d\nm : a'\u271d \u2208 toList l\u2082\ns : Subset l0 l\u2082\nh : a \u2208 toList l0\n\u22a2 a \u2208 l\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/Ordered.lean", "full_name": "right_lt_lineMap_iff_lt", "start": [103, 1], "end": [104, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Determinant.lean", "full_name": "LinearEquiv.det_refl", "start": [389, 1], "end": [390, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Compactification/OnePoint.lean", "full_name": "OnePoint.not_mem_range_coe_iff", "start": [155, 1], "end": [156, 59], "traced_tactics": [{"tactic": "rw [\u2190 mem_compl_iff, compl_range_coe, mem_singleton_iff]", "state_before": "X : Type u_1\nx : OnePoint X\n\u22a2 \u00acx \u2208 range some \u2194 x = \u221e", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean", "full_name": "AffineMap.coe_comp", "start": [418, 1], "end": [419, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Function.lean", "full_name": "convexOn_const", "start": [209, 1], "end": [210, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "full_name": "Finsupp.sum_apply", "start": [312, 1], "end": [314, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Young/SemistandardTableau.lean", "full_name": "Ssyt.ext", "start": [87, 1], "end": [90, 12], "traced_tactics": [{"tactic": "funext", "state_before": "\u03bc : YoungDiagram\nT T' : Ssyt \u03bc\nh : \u2200 (i j : \u2115), \u2191T i j = \u2191T' i j\nx\u271d : \u2115\n\u22a2 \u2191T x\u271d = \u2191T' x\u271d", "state_after": "case h\n\u03bc : YoungDiagram\nT T' : Ssyt \u03bc\nh : \u2200 (i j : \u2115), \u2191T i j = \u2191T' i j\nx\u271d\u00b9 x\u271d : \u2115\n\u22a2 \u2191T x\u271d\u00b9 x\u271d = \u2191T' x\u271d\u00b9 x\u271d"}, {"tactic": "apply h", "state_before": "case h\n\u03bc : YoungDiagram\nT T' : Ssyt \u03bc\nh : \u2200 (i j : \u2115), \u2191T i j = \u2191T' i j\nx\u271d\u00b9 x\u271d : \u2115\n\u22a2 \u2191T x\u271d\u00b9 x\u271d = \u2191T' x\u271d\u00b9 x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Basic.lean", "full_name": "Set.Ici_prod_eq", "start": [1891, 1], "end": [1892, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "full_name": "Complex.cos_int_mul_two_pi", "start": [1235, 1], "end": [1236, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "full_name": "Metric.cthickening_eq_biUnion_closedBall", "start": [1422, 1], "end": [1434, 26], "traced_tactics": [{"tactic": "rcases eq_empty_or_nonempty E with (rfl | hne)", "state_before": "\u03b9 : Sort ?u.148509\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nE : Set \u03b1\nh\u03b4 : 0 \u2264 \u03b4\n\u22a2 cthickening \u03b4 E = \u22c3 (x : \u03b1) (_ : x \u2208 closure E), closedBall x \u03b4", "state_after": "case inl\n\u03b9 : Sort ?u.148509\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nh\u03b4 : 0 \u2264 \u03b4\n\u22a2 cthickening \u03b4 \u2205 = \u22c3 (x : \u03b1) (_ : x \u2208 closure \u2205), closedBall x \u03b4\n\ncase inr\n\u03b9 : Sort ?u.148509\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nE : Set \u03b1\nh\u03b4 : 0 \u2264 \u03b4\nhne : Set.Nonempty E\n\u22a2 cthickening \u03b4 E = \u22c3 (x : \u03b1) (_ : x \u2208 closure E), closedBall x \u03b4"}, {"tactic": "rw [\u2190 cthickening_closure]", "state_before": "case inr\n\u03b9 : Sort ?u.148509\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nE : Set \u03b1\nh\u03b4 : 0 \u2264 \u03b4\nhne : Set.Nonempty E\n\u22a2 cthickening \u03b4 E = \u22c3 (x : \u03b1) (_ : x \u2208 closure E), closedBall x \u03b4", "state_after": "case inr\n\u03b9 : Sort ?u.148509\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nE : Set \u03b1\nh\u03b4 : 0 \u2264 \u03b4\nhne : Set.Nonempty E\n\u22a2 cthickening \u03b4 (closure E) = \u22c3 (x : \u03b1) (_ : x \u2208 closure E), closedBall x \u03b4"}, {"tactic": "refine Subset.antisymm (fun x hx \u21a6 ?_)\n  (iUnion\u2082_subset fun x hx \u21a6 closedBall_subset_cthickening hx _)", "state_before": "case inr\n\u03b9 : Sort ?u.148509\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nE : Set \u03b1\nh\u03b4 : 0 \u2264 \u03b4\nhne : Set.Nonempty E\n\u22a2 cthickening \u03b4 (closure E) = \u22c3 (x : \u03b1) (_ : x \u2208 closure E), closedBall x \u03b4", "state_after": "case inr\n\u03b9 : Sort ?u.148509\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nE : Set \u03b1\nh\u03b4 : 0 \u2264 \u03b4\nhne : Set.Nonempty E\nx : \u03b1\nhx : x \u2208 cthickening \u03b4 (closure E)\n\u22a2 x \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 closure E), closedBall x \u03b4"}, {"tactic": "obtain \u27e8y, yE, hy\u27e9 : \u2203 y \u2208 closure E, infDist x (closure E) = dist x y :=\n  isClosed_closure.exists_infDist_eq_dist (closure_nonempty_iff.mpr hne) x", "state_before": "case inr\n\u03b9 : Sort ?u.148509\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nE : Set \u03b1\nh\u03b4 : 0 \u2264 \u03b4\nhne : Set.Nonempty E\nx : \u03b1\nhx : x \u2208 cthickening \u03b4 (closure E)\n\u22a2 x \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 closure E), closedBall x \u03b4", "state_after": "case inr.intro.intro\n\u03b9 : Sort ?u.148509\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nE : Set \u03b1\nh\u03b4 : 0 \u2264 \u03b4\nhne : Set.Nonempty E\nx : \u03b1\nhx : x \u2208 cthickening \u03b4 (closure E)\ny : \u03b1\nyE : y \u2208 closure E\nhy : infDist x (closure E) = dist x y\n\u22a2 x \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 closure E), closedBall x \u03b4"}, {"tactic": "replace hy : dist x y \u2264 \u03b4 :=\n  (ENNReal.ofReal_le_ofReal_iff h\u03b4).mp\n    (((congr_arg ENNReal.ofReal hy.symm).le.trans ENNReal.ofReal_toReal_le).trans hx)", "state_before": "case inr.intro.intro\n\u03b9 : Sort ?u.148509\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nE : Set \u03b1\nh\u03b4 : 0 \u2264 \u03b4\nhne : Set.Nonempty E\nx : \u03b1\nhx : x \u2208 cthickening \u03b4 (closure E)\ny : \u03b1\nyE : y \u2208 closure E\nhy : infDist x (closure E) = dist x y\n\u22a2 x \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 closure E), closedBall x \u03b4", "state_after": "case inr.intro.intro\n\u03b9 : Sort ?u.148509\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nE : Set \u03b1\nh\u03b4 : 0 \u2264 \u03b4\nhne : Set.Nonempty E\nx : \u03b1\nhx : x \u2208 cthickening \u03b4 (closure E)\ny : \u03b1\nyE : y \u2208 closure E\nhy : dist x y \u2264 \u03b4\n\u22a2 x \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 closure E), closedBall x \u03b4"}, {"tactic": "exact mem_biUnion yE hy", "state_before": "case inr.intro.intro\n\u03b9 : Sort ?u.148509\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx\u271d : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nE : Set \u03b1\nh\u03b4 : 0 \u2264 \u03b4\nhne : Set.Nonempty E\nx : \u03b1\nhx : x \u2208 cthickening \u03b4 (closure E)\ny : \u03b1\nyE : y \u2208 closure E\nhy : dist x y \u2264 \u03b4\n\u22a2 x \u2208 \u22c3 (x : \u03b1) (_ : x \u2208 closure E), closedBall x \u03b4", "state_after": "no goals"}, {"tactic": "simp only [cthickening_empty, biUnion_empty, closure_empty]", "state_before": "case inl\n\u03b9 : Sort ?u.148509\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\u271d\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nh\u03b4 : 0 \u2264 \u03b4\n\u22a2 cthickening \u03b4 \u2205 = \u22c3 (x : \u03b1) (_ : x \u2208 closure \u2205), closedBall x \u03b4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/WithBot.lean", "full_name": "WithBot.not_lt_none", "start": [279, 1], "end": [280, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Adjoin/FG.lean", "full_name": "Subalgebra.fg_adjoin_finset", "start": [100, 1], "end": [101, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Subring/Basic.lean", "full_name": "Subring.coe_inf", "start": [712, 1], "end": [713, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.not_mem_mono", "start": [366, 1], "end": [367, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "full_name": "ContinuousLinearMap.coe_zero'", "start": [640, 1], "end": [641, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/EReal.lean", "full_name": "EReal.coe_neg", "start": [750, 20], "end": [750, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "full_name": "mem_maximalAtlas_iff", "start": [926, 1], "end": [928, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Sum/Basic.lean", "full_name": "Sum.map_injective", "start": [594, 1], "end": [599, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Data/Nat/Linear.lean", "full_name": "Nat.Linear.PolyCnstr.eq_false_of_isUnsat", "start": [639, 1], "end": [650, 9], "traced_tactics": [{"tactic": "cases c", "state_before": "ctx : Context\nc : PolyCnstr\n\u22a2 isUnsat c = true \u2192 denote ctx c = False", "state_after": "case mk\nctx : Context\neq\u271d : Bool\nlhs\u271d rhs\u271d : Poly\n\u22a2 isUnsat { eq := eq\u271d, lhs := lhs\u271d, rhs := rhs\u271d } = true \u2192 denote ctx { eq := eq\u271d, lhs := lhs\u271d, rhs := rhs\u271d } = False"}, {"tactic": "rename_i eq lhs rhs", "state_before": "case mk\nctx : Context\neq\u271d : Bool\nlhs\u271d rhs\u271d : Poly\n\u22a2 isUnsat { eq := eq\u271d, lhs := lhs\u271d, rhs := rhs\u271d } = true \u2192 denote ctx { eq := eq\u271d, lhs := lhs\u271d, rhs := rhs\u271d } = False", "state_after": "case mk\nctx : Context\neq : Bool\nlhs rhs : Poly\n\u22a2 isUnsat { eq := eq, lhs := lhs, rhs := rhs } = true \u2192 denote ctx { eq := eq, lhs := lhs, rhs := rhs } = False"}, {"tactic": "simp [isUnsat]", "state_before": "case mk\nctx : Context\neq : Bool\nlhs rhs : Poly\n\u22a2 isUnsat { eq := eq, lhs := lhs, rhs := rhs } = true \u2192 denote ctx { eq := eq, lhs := lhs, rhs := rhs } = False", "state_after": "case mk\nctx : Context\neq : Bool\nlhs rhs : Poly\n\u22a2 (bif eq then Poly.isZero lhs && Poly.isNonZero rhs || Poly.isNonZero lhs && Poly.isZero rhs\n      else Poly.isNonZero lhs && Poly.isZero rhs) =\n      true \u2192\n    denote ctx { eq := eq, lhs := lhs, rhs := rhs } = False"}, {"tactic": "by_cases he : eq = true <;> simp [he, denote, Poly.denote_eq, Poly.denote_le]", "state_before": "case mk\nctx : Context\neq : Bool\nlhs rhs : Poly\n\u22a2 (bif eq then Poly.isZero lhs && Poly.isNonZero rhs || Poly.isNonZero lhs && Poly.isZero rhs\n      else Poly.isNonZero lhs && Poly.isZero rhs) =\n      true \u2192\n    denote ctx { eq := eq, lhs := lhs, rhs := rhs } = False", "state_after": "case mk.inl\nctx : Context\neq : Bool\nlhs rhs : Poly\nhe : eq = true\n\u22a2 Poly.isZero lhs = true \u2227 Poly.isNonZero rhs = true \u2228 Poly.isNonZero lhs = true \u2227 Poly.isZero rhs = true \u2192\n    (Poly.denote ctx lhs = Poly.denote ctx rhs) = False\n\ncase mk.inr\nctx : Context\neq : Bool\nlhs rhs : Poly\nhe : \u00aceq = true\n\u22a2 Poly.isNonZero lhs = true \u2227 Poly.isZero rhs = true \u2192 (Poly.denote ctx lhs \u2264 Poly.denote ctx rhs) = False"}, {"tactic": "intro\n  | Or.inl \u27e8h\u2081, h\u2082\u27e9 => simp [Poly.of_isZero, h\u2081]; have := Nat.not_eq_zero_of_lt (Poly.of_isNonZero ctx h\u2082); simp [this.symm]\n  | Or.inr \u27e8h\u2081, h\u2082\u27e9 => simp [Poly.of_isZero, h\u2082]; have := Nat.not_eq_zero_of_lt (Poly.of_isNonZero ctx h\u2081); simp [this]", "state_before": "case mk.inl\nctx : Context\neq : Bool\nlhs rhs : Poly\nhe : eq = true\n\u22a2 Poly.isZero lhs = true \u2227 Poly.isNonZero rhs = true \u2228 Poly.isNonZero lhs = true \u2227 Poly.isZero rhs = true \u2192\n    (Poly.denote ctx lhs = Poly.denote ctx rhs) = False", "state_after": "no goals"}, {"tactic": "simp [Poly.of_isZero, h\u2081]", "state_before": "ctx : Context\neq : Bool\nlhs rhs : Poly\nhe : eq = true\nx\u271d : Poly.isZero lhs = true \u2227 Poly.isNonZero rhs = true \u2228 Poly.isNonZero lhs = true \u2227 Poly.isZero rhs = true\nh\u2081 : Poly.isZero lhs = true\nh\u2082 : Poly.isNonZero rhs = true\n\u22a2 (Poly.denote ctx lhs = Poly.denote ctx rhs) = False", "state_after": "ctx : Context\neq : Bool\nlhs rhs : Poly\nhe : eq = true\nx\u271d : Poly.isZero lhs = true \u2227 Poly.isNonZero rhs = true \u2228 Poly.isNonZero lhs = true \u2227 Poly.isZero rhs = true\nh\u2081 : Poly.isZero lhs = true\nh\u2082 : Poly.isNonZero rhs = true\n\u22a2 (0 = Poly.denote ctx rhs) = False"}, {"tactic": "have := Nat.not_eq_zero_of_lt (Poly.of_isNonZero ctx h\u2082)", "state_before": "ctx : Context\neq : Bool\nlhs rhs : Poly\nhe : eq = true\nx\u271d : Poly.isZero lhs = true \u2227 Poly.isNonZero rhs = true \u2228 Poly.isNonZero lhs = true \u2227 Poly.isZero rhs = true\nh\u2081 : Poly.isZero lhs = true\nh\u2082 : Poly.isNonZero rhs = true\n\u22a2 (0 = Poly.denote ctx rhs) = False", "state_after": "ctx : Context\neq : Bool\nlhs rhs : Poly\nhe : eq = true\nx\u271d : Poly.isZero lhs = true \u2227 Poly.isNonZero rhs = true \u2228 Poly.isNonZero lhs = true \u2227 Poly.isZero rhs = true\nh\u2081 : Poly.isZero lhs = true\nh\u2082 : Poly.isNonZero rhs = true\nthis : Poly.denote ctx rhs \u2260 0\n\u22a2 (0 = Poly.denote ctx rhs) = False"}, {"tactic": "simp [this.symm]", "state_before": "ctx : Context\neq : Bool\nlhs rhs : Poly\nhe : eq = true\nx\u271d : Poly.isZero lhs = true \u2227 Poly.isNonZero rhs = true \u2228 Poly.isNonZero lhs = true \u2227 Poly.isZero rhs = true\nh\u2081 : Poly.isZero lhs = true\nh\u2082 : Poly.isNonZero rhs = true\nthis : Poly.denote ctx rhs \u2260 0\n\u22a2 (0 = Poly.denote ctx rhs) = False", "state_after": "no goals"}, {"tactic": "simp [Poly.of_isZero, h\u2082]", "state_before": "ctx : Context\neq : Bool\nlhs rhs : Poly\nhe : eq = true\nx\u271d : Poly.isZero lhs = true \u2227 Poly.isNonZero rhs = true \u2228 Poly.isNonZero lhs = true \u2227 Poly.isZero rhs = true\nh\u2081 : Poly.isNonZero lhs = true\nh\u2082 : Poly.isZero rhs = true\n\u22a2 (Poly.denote ctx lhs = Poly.denote ctx rhs) = False", "state_after": "ctx : Context\neq : Bool\nlhs rhs : Poly\nhe : eq = true\nx\u271d : Poly.isZero lhs = true \u2227 Poly.isNonZero rhs = true \u2228 Poly.isNonZero lhs = true \u2227 Poly.isZero rhs = true\nh\u2081 : Poly.isNonZero lhs = true\nh\u2082 : Poly.isZero rhs = true\n\u22a2 (Poly.denote ctx lhs = 0) = False"}, {"tactic": "have := Nat.not_eq_zero_of_lt (Poly.of_isNonZero ctx h\u2081)", "state_before": "ctx : Context\neq : Bool\nlhs rhs : Poly\nhe : eq = true\nx\u271d : Poly.isZero lhs = true \u2227 Poly.isNonZero rhs = true \u2228 Poly.isNonZero lhs = true \u2227 Poly.isZero rhs = true\nh\u2081 : Poly.isNonZero lhs = true\nh\u2082 : Poly.isZero rhs = true\n\u22a2 (Poly.denote ctx lhs = 0) = False", "state_after": "ctx : Context\neq : Bool\nlhs rhs : Poly\nhe : eq = true\nx\u271d : Poly.isZero lhs = true \u2227 Poly.isNonZero rhs = true \u2228 Poly.isNonZero lhs = true \u2227 Poly.isZero rhs = true\nh\u2081 : Poly.isNonZero lhs = true\nh\u2082 : Poly.isZero rhs = true\nthis : Poly.denote ctx lhs \u2260 0\n\u22a2 (Poly.denote ctx lhs = 0) = False"}, {"tactic": "simp [this]", "state_before": "ctx : Context\neq : Bool\nlhs rhs : Poly\nhe : eq = true\nx\u271d : Poly.isZero lhs = true \u2227 Poly.isNonZero rhs = true \u2228 Poly.isNonZero lhs = true \u2227 Poly.isZero rhs = true\nh\u2081 : Poly.isNonZero lhs = true\nh\u2082 : Poly.isZero rhs = true\nthis : Poly.denote ctx lhs \u2260 0\n\u22a2 (Poly.denote ctx lhs = 0) = False", "state_after": "no goals"}, {"tactic": "intro \u27e8h\u2081, h\u2082\u27e9", "state_before": "case mk.inr\nctx : Context\neq : Bool\nlhs rhs : Poly\nhe : \u00aceq = true\n\u22a2 Poly.isNonZero lhs = true \u2227 Poly.isZero rhs = true \u2192 (Poly.denote ctx lhs \u2264 Poly.denote ctx rhs) = False", "state_after": "case mk.inr\nctx : Context\neq : Bool\nlhs rhs : Poly\nhe : \u00aceq = true\nh\u2081 : Poly.isNonZero lhs = true\nh\u2082 : Poly.isZero rhs = true\n\u22a2 (Poly.denote ctx lhs \u2264 Poly.denote ctx rhs) = False"}, {"tactic": "simp [Poly.of_isZero, h\u2082]", "state_before": "case mk.inr\nctx : Context\neq : Bool\nlhs rhs : Poly\nhe : \u00aceq = true\nh\u2081 : Poly.isNonZero lhs = true\nh\u2082 : Poly.isZero rhs = true\n\u22a2 (Poly.denote ctx lhs \u2264 Poly.denote ctx rhs) = False", "state_after": "case mk.inr\nctx : Context\neq : Bool\nlhs rhs : Poly\nhe : \u00aceq = true\nh\u2081 : Poly.isNonZero lhs = true\nh\u2082 : Poly.isZero rhs = true\n\u22a2 (Poly.denote ctx lhs = 0) = False"}, {"tactic": "have := Nat.not_eq_zero_of_lt (Poly.of_isNonZero ctx h\u2081)", "state_before": "case mk.inr\nctx : Context\neq : Bool\nlhs rhs : Poly\nhe : \u00aceq = true\nh\u2081 : Poly.isNonZero lhs = true\nh\u2082 : Poly.isZero rhs = true\n\u22a2 (Poly.denote ctx lhs = 0) = False", "state_after": "case mk.inr\nctx : Context\neq : Bool\nlhs rhs : Poly\nhe : \u00aceq = true\nh\u2081 : Poly.isNonZero lhs = true\nh\u2082 : Poly.isZero rhs = true\nthis : Poly.denote ctx lhs \u2260 0\n\u22a2 (Poly.denote ctx lhs = 0) = False"}, {"tactic": "simp [this]", "state_before": "case mk.inr\nctx : Context\neq : Bool\nlhs rhs : Poly\nhe : \u00aceq = true\nh\u2081 : Poly.isNonZero lhs = true\nh\u2082 : Poly.isZero rhs = true\nthis : Poly.denote ctx lhs \u2260 0\n\u22a2 (Poly.denote ctx lhs = 0) = False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean", "full_name": "Matrix.nonsing_inv_mul_cancel_right", "start": [335, 1], "end": [336, 47], "traced_tactics": [{"tactic": "simp [Matrix.mul_assoc, nonsing_inv_mul A h]", "state_before": "l : Type ?u.212026\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : CommRing \u03b1\nA B\u271d : Matrix n n \u03b1\nB : Matrix m n \u03b1\nh : IsUnit (det A)\n\u22a2 B \u2b1d A\u207b\u00b9 \u2b1d A = B", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/Partrec.lean", "full_name": "Computable.sum_casesOn", "start": [715, 1], "end": [723, 52], "traced_tactics": [{"tactic": "cases' f a with b c <;> simp [Nat.div2_val]", "state_before": "\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\ng : \u03b1 \u2192 \u03b2 \u2192 \u03c3\nh : \u03b1 \u2192 \u03b3 \u2192 \u03c3\nhf : Computable f\nhg : Computable\u2082 g\nhh : Computable\u2082 h\na : \u03b1\n\u22a2 (bif Nat.bodd (encode (f a)) then Option.map (h a) (decode (Nat.div2 (encode (f a))))\n    else Option.map (g a) (decode (Nat.div2 (encode (f a))))) =\n    Option.some (Sum.casesOn (f a) (g a) (h a))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/IntermediateField.lean", "full_name": "IntermediateField.coe_add", "start": [251, 11], "end": [252, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/NatAntidiagonal.lean", "full_name": "Finset.Nat.sum_antidiagonal_succ'", "start": [51, 1], "end": [53, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.scanl_nil", "start": [2606, 1], "end": [2607, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/Padics/PadicIntegers.lean", "full_name": "PadicInt.coe_nat_cast", "start": [159, 1], "end": [159, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Submonoid/Inverses.lean", "full_name": "Submonoid.fromLeftInv_eq_iff", "start": [137, 1], "end": [138, 90], "traced_tactics": [{"tactic": "rw [\u2190 IsUnit.mul_right_inj (leftInv_le_isUnit _ a.prop), S.mul_fromLeftInv, eq_comm]", "state_before": "M : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\na : { x // x \u2208 leftInv S }\nb : M\n\u22a2 \u2191(fromLeftInv S a) = b \u2194 \u2191a * b = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Init/Algebra/Order.lean", "full_name": "le_of_not_le", "start": [307, 1], "end": [308, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "tendsto_inv_nhdsWithin_Ioi_inv", "start": [572, 1], "end": [573, 69], "traced_tactics": [{"tactic": "simpa only [inv_inv] using @tendsto_inv_nhdsWithin_Ioi _ _ _ _ a\u207b\u00b9", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nG : Type w\nH : Type x\ninst\u271d\u2076 : TopologicalSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalGroup G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf : \u03b1 \u2192 G\ns : Set \u03b1\nx : \u03b1\ninst\u271d\u00b2 : TopologicalSpace H\ninst\u271d\u00b9 : OrderedCommGroup H\ninst\u271d : ContinuousInv H\na : H\n\u22a2 Tendsto Inv.inv (\ud835\udcdd[Ioi a\u207b\u00b9] a\u207b\u00b9) (\ud835\udcdd[Iio a] a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finsupp/Basic.lean", "full_name": "Finsupp.mem_support_multiset_sum", "start": [1180, 1], "end": [1190, 53], "traced_tactics": [{"tactic": "simp at h", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.521672\n\u03b3 : Type ?u.521675\n\u03b9 : Type ?u.521678\nM : Type u_1\nM' : Type ?u.521684\nN : Type ?u.521687\nP : Type ?u.521690\nG : Type ?u.521693\nH : Type ?u.521696\nR : Type ?u.521699\nS : Type ?u.521702\ninst\u271d : AddCommMonoid M\ns : Multiset (\u03b1 \u2192\u2080 M)\na : \u03b1\nh : a \u2208 (Multiset.sum 0).support\n\u22a2 False", "state_after": "no goals"}, {"tactic": "intro f s ih ha", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.521672\n\u03b3 : Type ?u.521675\n\u03b9 : Type ?u.521678\nM : Type u_1\nM' : Type ?u.521684\nN : Type ?u.521687\nP : Type ?u.521690\nG : Type ?u.521693\nH : Type ?u.521696\nR : Type ?u.521699\nS : Type ?u.521702\ninst\u271d : AddCommMonoid M\ns : Multiset (\u03b1 \u2192\u2080 M)\na : \u03b1\n\u22a2 \u2200 \u2983a_1 : \u03b1 \u2192\u2080 M\u2984 {s : Multiset (\u03b1 \u2192\u2080 M)},\n    (a \u2208 (Multiset.sum s).support \u2192 \u2203 f, f \u2208 s \u2227 a \u2208 f.support) \u2192\n      a \u2208 (Multiset.sum (a_1 ::\u2098 s)).support \u2192 \u2203 f, f \u2208 a_1 ::\u2098 s \u2227 a \u2208 f.support", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type ?u.521672\n\u03b3 : Type ?u.521675\n\u03b9 : Type ?u.521678\nM : Type u_1\nM' : Type ?u.521684\nN : Type ?u.521687\nP : Type ?u.521690\nG : Type ?u.521693\nH : Type ?u.521696\nR : Type ?u.521699\nS : Type ?u.521702\ninst\u271d : AddCommMonoid M\ns\u271d : Multiset (\u03b1 \u2192\u2080 M)\na : \u03b1\nf : \u03b1 \u2192\u2080 M\ns : Multiset (\u03b1 \u2192\u2080 M)\nih : a \u2208 (Multiset.sum s).support \u2192 \u2203 f, f \u2208 s \u2227 a \u2208 f.support\nha : a \u2208 (Multiset.sum (f ::\u2098 s)).support\n\u22a2 \u2203 f_1, f_1 \u2208 f ::\u2098 s \u2227 a \u2208 f_1.support"}, {"tactic": "by_cases h : a \u2208 f.support", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.521672\n\u03b3 : Type ?u.521675\n\u03b9 : Type ?u.521678\nM : Type u_1\nM' : Type ?u.521684\nN : Type ?u.521687\nP : Type ?u.521690\nG : Type ?u.521693\nH : Type ?u.521696\nR : Type ?u.521699\nS : Type ?u.521702\ninst\u271d : AddCommMonoid M\ns\u271d : Multiset (\u03b1 \u2192\u2080 M)\na : \u03b1\nf : \u03b1 \u2192\u2080 M\ns : Multiset (\u03b1 \u2192\u2080 M)\nih : a \u2208 (Multiset.sum s).support \u2192 \u2203 f, f \u2208 s \u2227 a \u2208 f.support\nha : a \u2208 (Multiset.sum (f ::\u2098 s)).support\n\u22a2 \u2203 f_1, f_1 \u2208 f ::\u2098 s \u2227 a \u2208 f_1.support", "state_after": "case pos\n\u03b1 : Type u_2\n\u03b2 : Type ?u.521672\n\u03b3 : Type ?u.521675\n\u03b9 : Type ?u.521678\nM : Type u_1\nM' : Type ?u.521684\nN : Type ?u.521687\nP : Type ?u.521690\nG : Type ?u.521693\nH : Type ?u.521696\nR : Type ?u.521699\nS : Type ?u.521702\ninst\u271d : AddCommMonoid M\ns\u271d : Multiset (\u03b1 \u2192\u2080 M)\na : \u03b1\nf : \u03b1 \u2192\u2080 M\ns : Multiset (\u03b1 \u2192\u2080 M)\nih : a \u2208 (Multiset.sum s).support \u2192 \u2203 f, f \u2208 s \u2227 a \u2208 f.support\nha : a \u2208 (Multiset.sum (f ::\u2098 s)).support\nh : a \u2208 f.support\n\u22a2 \u2203 f_1, f_1 \u2208 f ::\u2098 s \u2227 a \u2208 f_1.support\n\ncase neg\n\u03b1 : Type u_2\n\u03b2 : Type ?u.521672\n\u03b3 : Type ?u.521675\n\u03b9 : Type ?u.521678\nM : Type u_1\nM' : Type ?u.521684\nN : Type ?u.521687\nP : Type ?u.521690\nG : Type ?u.521693\nH : Type ?u.521696\nR : Type ?u.521699\nS : Type ?u.521702\ninst\u271d : AddCommMonoid M\ns\u271d : Multiset (\u03b1 \u2192\u2080 M)\na : \u03b1\nf : \u03b1 \u2192\u2080 M\ns : Multiset (\u03b1 \u2192\u2080 M)\nih : a \u2208 (Multiset.sum s).support \u2192 \u2203 f, f \u2208 s \u2227 a \u2208 f.support\nha : a \u2208 (Multiset.sum (f ::\u2098 s)).support\nh : \u00aca \u2208 f.support\n\u22a2 \u2203 f_1, f_1 \u2208 f ::\u2098 s \u2227 a \u2208 f_1.support"}, {"tactic": "exact \u27e8f, Multiset.mem_cons_self _ _, h\u27e9", "state_before": "case pos\n\u03b1 : Type u_2\n\u03b2 : Type ?u.521672\n\u03b3 : Type ?u.521675\n\u03b9 : Type ?u.521678\nM : Type u_1\nM' : Type ?u.521684\nN : Type ?u.521687\nP : Type ?u.521690\nG : Type ?u.521693\nH : Type ?u.521696\nR : Type ?u.521699\nS : Type ?u.521702\ninst\u271d : AddCommMonoid M\ns\u271d : Multiset (\u03b1 \u2192\u2080 M)\na : \u03b1\nf : \u03b1 \u2192\u2080 M\ns : Multiset (\u03b1 \u2192\u2080 M)\nih : a \u2208 (Multiset.sum s).support \u2192 \u2203 f, f \u2208 s \u2227 a \u2208 f.support\nha : a \u2208 (Multiset.sum (f ::\u2098 s)).support\nh : a \u2208 f.support\n\u22a2 \u2203 f_1, f_1 \u2208 f ::\u2098 s \u2227 a \u2208 f_1.support", "state_after": "no goals"}, {"tactic": "simp only [Multiset.sum_cons, mem_support_iff, add_apply, not_mem_support_iff.1 h,\n  zero_add] at ha", "state_before": "case neg\n\u03b1 : Type u_2\n\u03b2 : Type ?u.521672\n\u03b3 : Type ?u.521675\n\u03b9 : Type ?u.521678\nM : Type u_1\nM' : Type ?u.521684\nN : Type ?u.521687\nP : Type ?u.521690\nG : Type ?u.521693\nH : Type ?u.521696\nR : Type ?u.521699\nS : Type ?u.521702\ninst\u271d : AddCommMonoid M\ns\u271d : Multiset (\u03b1 \u2192\u2080 M)\na : \u03b1\nf : \u03b1 \u2192\u2080 M\ns : Multiset (\u03b1 \u2192\u2080 M)\nih : a \u2208 (Multiset.sum s).support \u2192 \u2203 f, f \u2208 s \u2227 a \u2208 f.support\nha : a \u2208 (Multiset.sum (f ::\u2098 s)).support\nh : \u00aca \u2208 f.support\n\u22a2 \u2203 f_1, f_1 \u2208 f ::\u2098 s \u2227 a \u2208 f_1.support", "state_after": "case neg\n\u03b1 : Type u_2\n\u03b2 : Type ?u.521672\n\u03b3 : Type ?u.521675\n\u03b9 : Type ?u.521678\nM : Type u_1\nM' : Type ?u.521684\nN : Type ?u.521687\nP : Type ?u.521690\nG : Type ?u.521693\nH : Type ?u.521696\nR : Type ?u.521699\nS : Type ?u.521702\ninst\u271d : AddCommMonoid M\ns\u271d : Multiset (\u03b1 \u2192\u2080 M)\na : \u03b1\nf : \u03b1 \u2192\u2080 M\ns : Multiset (\u03b1 \u2192\u2080 M)\nih : a \u2208 (Multiset.sum s).support \u2192 \u2203 f, f \u2208 s \u2227 a \u2208 f.support\nh : \u00aca \u2208 f.support\nha : \u2191(Multiset.sum s) a \u2260 0\n\u22a2 \u2203 f_1, f_1 \u2208 f ::\u2098 s \u2227 a \u2208 f_1.support"}, {"tactic": "rcases ih (mem_support_iff.2 ha) with \u27e8f', h\u2080, h\u2081\u27e9", "state_before": "case neg\n\u03b1 : Type u_2\n\u03b2 : Type ?u.521672\n\u03b3 : Type ?u.521675\n\u03b9 : Type ?u.521678\nM : Type u_1\nM' : Type ?u.521684\nN : Type ?u.521687\nP : Type ?u.521690\nG : Type ?u.521693\nH : Type ?u.521696\nR : Type ?u.521699\nS : Type ?u.521702\ninst\u271d : AddCommMonoid M\ns\u271d : Multiset (\u03b1 \u2192\u2080 M)\na : \u03b1\nf : \u03b1 \u2192\u2080 M\ns : Multiset (\u03b1 \u2192\u2080 M)\nih : a \u2208 (Multiset.sum s).support \u2192 \u2203 f, f \u2208 s \u2227 a \u2208 f.support\nh : \u00aca \u2208 f.support\nha : \u2191(Multiset.sum s) a \u2260 0\n\u22a2 \u2203 f_1, f_1 \u2208 f ::\u2098 s \u2227 a \u2208 f_1.support", "state_after": "case neg.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.521672\n\u03b3 : Type ?u.521675\n\u03b9 : Type ?u.521678\nM : Type u_1\nM' : Type ?u.521684\nN : Type ?u.521687\nP : Type ?u.521690\nG : Type ?u.521693\nH : Type ?u.521696\nR : Type ?u.521699\nS : Type ?u.521702\ninst\u271d : AddCommMonoid M\ns\u271d : Multiset (\u03b1 \u2192\u2080 M)\na : \u03b1\nf : \u03b1 \u2192\u2080 M\ns : Multiset (\u03b1 \u2192\u2080 M)\nih : a \u2208 (Multiset.sum s).support \u2192 \u2203 f, f \u2208 s \u2227 a \u2208 f.support\nh : \u00aca \u2208 f.support\nha : \u2191(Multiset.sum s) a \u2260 0\nf' : \u03b1 \u2192\u2080 M\nh\u2080 : f' \u2208 s\nh\u2081 : a \u2208 f'.support\n\u22a2 \u2203 f_1, f_1 \u2208 f ::\u2098 s \u2227 a \u2208 f_1.support"}, {"tactic": "exact \u27e8f', Multiset.mem_cons_of_mem h\u2080, h\u2081\u27e9", "state_before": "case neg.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.521672\n\u03b3 : Type ?u.521675\n\u03b9 : Type ?u.521678\nM : Type u_1\nM' : Type ?u.521684\nN : Type ?u.521687\nP : Type ?u.521690\nG : Type ?u.521693\nH : Type ?u.521696\nR : Type ?u.521699\nS : Type ?u.521702\ninst\u271d : AddCommMonoid M\ns\u271d : Multiset (\u03b1 \u2192\u2080 M)\na : \u03b1\nf : \u03b1 \u2192\u2080 M\ns : Multiset (\u03b1 \u2192\u2080 M)\nih : a \u2208 (Multiset.sum s).support \u2192 \u2203 f, f \u2208 s \u2227 a \u2208 f.support\nh : \u00aca \u2208 f.support\nha : \u2191(Multiset.sum s) a \u2260 0\nf' : \u03b1 \u2192\u2080 M\nh\u2080 : f' \u2208 s\nh\u2081 : a \u2208 f'.support\n\u22a2 \u2203 f_1, f_1 \u2208 f ::\u2098 s \u2227 a \u2208 f_1.support", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Module/LocalizedModule.lean", "full_name": "LocalizedModule.lift_comp", "start": [664, 1], "end": [667, 64], "traced_tactics": [{"tactic": "ext x", "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type ?u.554447\nM'' : Type u_3\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\nf : M \u2192\u2097[R] M'\ng\u271d g : M \u2192\u2097[R] M''\nh : \u2200 (x : { x // x \u2208 S }), IsUnit (\u2191(algebraMap R (Module.End R M'')) \u2191x)\n\u22a2 LinearMap.comp (lift S g h) (mkLinearMap S M) = g", "state_after": "case h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type ?u.554447\nM'' : Type u_3\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\nf : M \u2192\u2097[R] M'\ng\u271d g : M \u2192\u2097[R] M''\nh : \u2200 (x : { x // x \u2208 S }), IsUnit (\u2191(algebraMap R (Module.End R M'')) \u2191x)\nx : M\n\u22a2 \u2191(LinearMap.comp (lift S g h) (mkLinearMap S M)) x = \u2191g x"}, {"tactic": "dsimp", "state_before": "case h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type ?u.554447\nM'' : Type u_3\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\nf : M \u2192\u2097[R] M'\ng\u271d g : M \u2192\u2097[R] M''\nh : \u2200 (x : { x // x \u2208 S }), IsUnit (\u2191(algebraMap R (Module.End R M'')) \u2191x)\nx : M\n\u22a2 \u2191(LinearMap.comp (lift S g h) (mkLinearMap S M)) x = \u2191g x", "state_after": "case h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type ?u.554447\nM'' : Type u_3\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\nf : M \u2192\u2097[R] M'\ng\u271d g : M \u2192\u2097[R] M''\nh : \u2200 (x : { x // x \u2208 S }), IsUnit (\u2191(algebraMap R (Module.End R M'')) \u2191x)\nx : M\n\u22a2 \u2191(lift S g h) (mk x 1) = \u2191g x"}, {"tactic": "rw [LocalizedModule.lift_mk]", "state_before": "case h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type ?u.554447\nM'' : Type u_3\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\nf : M \u2192\u2097[R] M'\ng\u271d g : M 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\u2192 Prop\nn : \u2115\n\u22a2 card \u2191n = \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/List/Lemmas.lean", "full_name": "List.nil_union", "start": [1431, 9], "end": [1431, 88], "traced_tactics": [{"tactic": "simp [List.union, foldr]", "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\n\u22a2 List.union [] l = l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/MeasurableSpace.lean", "full_name": "Measurable.piecewise", "start": [323, 11], "end": [327, 31], "traced_tactics": [{"tactic": "intro t ht", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.43295\n\u03b4 : Type ?u.43298\n\u03b4' : Type ?u.43301\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nf g : \u03b1 \u2192 \u03b2\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nx\u271d : DecidablePred fun x => x \u2208 s\nhs : MeasurableSet s\nhf : Measurable f\nhg : Measurable g\n\u22a2 Measurable (piecewise s f g)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.43295\n\u03b4 : Type ?u.43298\n\u03b4' : Type ?u.43301\n\u03b9 : Sort u\u03b9\ns t\u271d u : Set \u03b1\nf g : \u03b1 \u2192 \u03b2\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nx\u271d : DecidablePred fun x => x \u2208 s\nhs : MeasurableSet s\nhf : Measurable f\nhg : Measurable g\nt : Set \u03b2\nht : MeasurableSet t\n\u22a2 MeasurableSet (piecewise s f g \u207b\u00b9' t)"}, {"tactic": "rw [piecewise_preimage]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.43295\n\u03b4 : Type ?u.43298\n\u03b4' : Type ?u.43301\n\u03b9 : Sort u\u03b9\ns t\u271d u : Set \u03b1\nf g : \u03b1 \u2192 \u03b2\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nx\u271d : DecidablePred fun x => x \u2208 s\nhs : MeasurableSet s\nhf : Measurable f\nhg : Measurable g\nt : Set \u03b2\nht : MeasurableSet t\n\u22a2 MeasurableSet (piecewise s f g \u207b\u00b9' t)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.43295\n\u03b4 : Type ?u.43298\n\u03b4' : Type ?u.43301\n\u03b9 : Sort u\u03b9\ns t\u271d u : Set \u03b1\nf g : \u03b1 \u2192 \u03b2\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nx\u271d : DecidablePred fun x => x \u2208 s\nhs : MeasurableSet s\nhf : Measurable f\nhg : Measurable g\nt : Set \u03b2\nht : MeasurableSet t\n\u22a2 MeasurableSet (Set.ite s (f \u207b\u00b9' t) (g \u207b\u00b9' t))"}, {"tactic": "exact hs.ite (hf ht) (hg ht)", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.43295\n\u03b4 : Type ?u.43298\n\u03b4' : Type ?u.43301\n\u03b9 : Sort u\u03b9\ns t\u271d u : Set \u03b1\nf g : \u03b1 \u2192 \u03b2\nm : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nx\u271d : DecidablePred fun x => x \u2208 s\nhs : MeasurableSet s\nhf : Measurable f\nhg : Measurable g\nt : Set \u03b2\nht : MeasurableSet t\n\u22a2 MeasurableSet (Set.ite s (f \u207b\u00b9' t) (g \u207b\u00b9' t))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Range.lean", "full_name": "Multiset.coe_range", "start": [28, 1], "end": [29, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "full_name": "Subgroup.mem_center_iff", "start": [2082, 1], "end": [2083, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "full_name": "ContinuousLinearMap.coe_id'", "start": [686, 1], "end": [687, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Polynomial/Content.lean", "full_name": "Polynomial.isPrimitive_primPart", "start": [267, 1], "end": [272, 90], "traced_tactics": [{"tactic": "by_cases h : p = 0", "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np : R[X]\n\u22a2 IsPrimitive (primPart p)", "state_after": "case pos\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np : R[X]\nh : p = 0\n\u22a2 IsPrimitive (primPart p)\n\ncase neg\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np : R[X]\nh : \u00acp = 0\n\u22a2 IsPrimitive (primPart p)"}, {"tactic": "rw [\u2190 content_eq_zero_iff] at h", "state_before": "case neg\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np : R[X]\nh : \u00acp = 0\n\u22a2 IsPrimitive (primPart p)", "state_after": "case neg\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np : R[X]\nh\u271d : \u00acp = 0\nh : \u00accontent p = 0\n\u22a2 IsPrimitive (primPart p)"}, {"tactic": "rw [isPrimitive_iff_content_eq_one]", "state_before": "case neg\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np : R[X]\nh\u271d : \u00acp = 0\nh : \u00accontent p = 0\n\u22a2 IsPrimitive (primPart p)", "state_after": "case neg\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np : R[X]\nh\u271d : \u00acp = 0\nh : \u00accontent p = 0\n\u22a2 content (primPart p) = 1"}, {"tactic": "apply mul_left_cancel\u2080 h", "state_before": "case neg\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np : R[X]\nh\u271d : \u00acp = 0\nh : \u00accontent p = 0\n\u22a2 content (primPart p) = 1", "state_after": "case neg\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np : R[X]\nh\u271d : \u00acp = 0\nh : \u00accontent p = 0\n\u22a2 content p * content (primPart p) = content p * 1"}, {"tactic": "conv_rhs => rw [p.eq_C_content_mul_primPart, mul_one, content_C_mul, normalize_content]", "state_before": "case neg\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np : R[X]\nh\u271d : \u00acp = 0\nh : \u00accontent p = 0\n\u22a2 content p * content (primPart p) = content p * 1", "state_after": "no goals"}, {"tactic": "simp [h]", "state_before": "case pos\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np : R[X]\nh : p = 0\n\u22a2 IsPrimitive (primPart p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/LocalHomeomorph.lean", "full_name": "LocalHomeomorph.image_source_inter_eq'", "start": [276, 1], "end": [277, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Subsemiring/Basic.lean", "full_name": "Subsemiring.map_le_iff_le_comap", "start": [568, 1], "end": [570, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Lagrange.lean", "full_name": "Lagrange.eval_basisDivisor_right", "start": [170, 1], "end": [171, 96], "traced_tactics": [{"tactic": "simp only [basisDivisor, eval_mul, eval_C, eval_sub, eval_X, sub_self, MulZeroClass.mul_zero]", "state_before": "F : Type u_1\ninst\u271d : Field F\nx y : F\n\u22a2 eval y (basisDivisor x y) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/SMul.lean", "full_name": "upperBounds_smul_of_pos", "start": [285, 1], "end": [286, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "full_name": "EMetric.le_infEdist", "start": [64, 1], "end": [65, 36], "traced_tactics": [{"tactic": "simp only [infEdist, le_iInf_iff]", "state_before": "\u03b9 : Sort ?u.793\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nd : \u211d\u22650\u221e\n\u22a2 d \u2264 infEdist x s \u2194 \u2200 (y : \u03b1), y \u2208 s \u2192 d \u2264 edist x y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Data/Nat/Gcd.lean", "full_name": "Nat.gcd_self", "start": [38, 9], "end": [39, 30], "traced_tactics": [{"tactic": "cases n <;> simp [gcd_succ]", "state_before": "n : Nat\n\u22a2 gcd n n = n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Factorization/Basic.lean", "full_name": "Nat.ord_proj_mul_ord_compl_eq_self", "start": [402, 1], "end": [403, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.EventuallyLE.sup", "start": [1775, 1], "end": [1777, 66], "traced_tactics": [{"tactic": "filter_upwards [hf, hg] with x hfx hgx using sup_le_sup hfx hgx", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.216914\n\u03b9 : Sort x\ninst\u271d : SemilatticeSup \u03b2\nl : Filter \u03b1\nf\u2081 f\u2082 g\u2081 g\u2082 : \u03b1 \u2192 \u03b2\nhf : f\u2081 \u2264\u1da0[l] f\u2082\nhg : g\u2081 \u2264\u1da0[l] g\u2082\n\u22a2 f\u2081 \u2294 g\u2081 \u2264\u1da0[l] f\u2082 \u2294 g\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.to_ofOption", "start": [380, 1], "end": [380, 85], "traced_tactics": [{"tactic": "cases o <;> rfl", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.11305\n\u03b3 : Type ?u.11308\no : Option \u03b1\n\u22a2 toOption \u2191o = o", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.Tape.map_mk\u2081", "start": [732, 1], "end": [734, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.ssubset_iff_exists_subset_erase", "start": [1961, 1], "end": [1964, 42], "traced_tactics": [{"tactic": "refine' \u27e8fun h => _, fun \u27e8a, ha, h\u27e9 => ssubset_of_subset_of_ssubset h <| erase_ssubset ha\u27e9", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.214645\n\u03b3 : Type ?u.214648\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\n\u22a2 s \u2282 t \u2194 \u2203 a, a \u2208 t \u2227 s \u2286 erase t a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.214645\n\u03b3 : Type ?u.214648\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\nh : s \u2282 t\n\u22a2 \u2203 a, a \u2208 t \u2227 s \u2286 erase t a"}, {"tactic": "obtain \u27e8a, ht, hs\u27e9 := not_subset.1 h.2", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.214645\n\u03b3 : Type ?u.214648\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u v : Finset \u03b1\na b : \u03b1\ns t : Finset \u03b1\nh : s \u2282 t\n\u22a2 \u2203 a, a \u2208 t \u2227 s \u2286 erase t a", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.214645\n\u03b3 : Type ?u.214648\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u v : Finset \u03b1\na\u271d b : \u03b1\ns t : Finset \u03b1\nh : s \u2282 t\na : \u03b1\nht : a \u2208 t\nhs : \u00aca \u2208 s\n\u22a2 \u2203 a, a \u2208 t \u2227 s \u2286 erase t a"}, {"tactic": "exact \u27e8a, ht, subset_erase.2 \u27e8h.1, hs\u27e9\u27e9", "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.214645\n\u03b3 : Type ?u.214648\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u v : Finset \u03b1\na\u271d b : \u03b1\ns t : Finset \u03b1\nh : s \u2282 t\na : \u03b1\nht : a \u2208 t\nhs : \u00aca \u2208 s\n\u22a2 \u2203 a, a \u2208 t \u2227 s \u2286 erase t a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Instances/ENNReal.lean", "full_name": "ENNReal.tendsto_mul", "start": [334, 11], "end": [353, 95], "traced_tactics": [{"tactic": "have ht : \u2200 b : \u211d\u22650\u221e, b \u2260 0 \u2192\n    Tendsto (fun p : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e => p.1 * p.2) (\ud835\udcdd ((\u22a4 : \u211d\u22650\u221e), b)) (\ud835\udcdd \u22a4) := fun b hb => by\n  refine' tendsto_nhds_top_iff_nnreal.2 fun n => _\n  rcases lt_iff_exists_nnreal_btwn.1 (pos_iff_ne_zero.2 hb) with \u27e8\u03b5, h\u03b5, h\u03b5b\u27e9\n  have : \u2200\u1da0 c : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e in \ud835\udcdd (\u221e, b), \u2191n / \u2191\u03b5 < c.1 \u2227 \u2191\u03b5 < c.2 :=\n    (lt_mem_nhds <| div_lt_top coe_ne_top h\u03b5.ne').prod_nhds (lt_mem_nhds h\u03b5b)\n  refine' this.mono fun c hc => _\n  exact (ENNReal.div_mul_cancel h\u03b5.ne' coe_ne_top).symm.trans_lt (mul_lt_mul hc.1 hc.2)", "state_before": "\u03b1 : Type ?u.89302\n\u03b2 : Type ?u.89305\n\u03b3 : Type ?u.89308\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nha : a \u2260 0 \u2228 b \u2260 \u22a4\nhb : b \u2260 0 \u2228 a \u2260 \u22a4\n\u22a2 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (a, b)) (\ud835\udcdd (a * b))", "state_after": "\u03b1 : Type ?u.89302\n\u03b2 : Type ?u.89305\n\u03b3 : Type ?u.89308\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nha : a \u2260 0 \u2228 b \u2260 \u22a4\nhb : b \u2260 0 \u2228 a \u2260 \u22a4\nht : \u2200 (b : \u211d\u22650\u221e), b \u2260 0 \u2192 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (\u22a4, b)) (\ud835\udcdd \u22a4)\n\u22a2 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (a, b)) (\ud835\udcdd (a * b))"}, {"tactic": "refine' tendsto_nhds_top_iff_nnreal.2 fun n => _", "state_before": "\u03b1 : Type ?u.89302\n\u03b2 : Type ?u.89305\n\u03b3 : Type ?u.89308\na b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nha : a \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d : b\u271d \u2260 0 \u2228 a \u2260 \u22a4\nb : \u211d\u22650\u221e\nhb : b \u2260 0\n\u22a2 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (\u22a4, b)) (\ud835\udcdd \u22a4)", "state_after": "\u03b1 : Type ?u.89302\n\u03b2 : Type ?u.89305\n\u03b3 : Type ?u.89308\na b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nha : a \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d : b\u271d \u2260 0 \u2228 a \u2260 \u22a4\nb : \u211d\u22650\u221e\nhb : b \u2260 0\nn : \u211d\u22650\n\u22a2 \u2200\u1da0 (a : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e) in \ud835\udcdd (\u22a4, b), \u2191n < a.fst * a.snd"}, {"tactic": "rcases lt_iff_exists_nnreal_btwn.1 (pos_iff_ne_zero.2 hb) with \u27e8\u03b5, h\u03b5, h\u03b5b\u27e9", "state_before": "\u03b1 : Type ?u.89302\n\u03b2 : Type ?u.89305\n\u03b3 : Type ?u.89308\na b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nha : a \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d : b\u271d \u2260 0 \u2228 a \u2260 \u22a4\nb : \u211d\u22650\u221e\nhb : b \u2260 0\nn : \u211d\u22650\n\u22a2 \u2200\u1da0 (a : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e) in \ud835\udcdd (\u22a4, b), \u2191n < a.fst * a.snd", "state_after": "case intro.intro\n\u03b1 : Type ?u.89302\n\u03b2 : Type ?u.89305\n\u03b3 : Type ?u.89308\na b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nha : a \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d : b\u271d \u2260 0 \u2228 a \u2260 \u22a4\nb : \u211d\u22650\u221e\nhb : b \u2260 0\nn \u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u2191\u03b5\nh\u03b5b : \u2191\u03b5 < b\n\u22a2 \u2200\u1da0 (a : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e) in \ud835\udcdd (\u22a4, b), \u2191n < a.fst * a.snd"}, {"tactic": "have : \u2200\u1da0 c : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e in \ud835\udcdd (\u221e, b), \u2191n / \u2191\u03b5 < c.1 \u2227 \u2191\u03b5 < c.2 :=\n  (lt_mem_nhds <| div_lt_top coe_ne_top h\u03b5.ne').prod_nhds (lt_mem_nhds h\u03b5b)", "state_before": "case intro.intro\n\u03b1 : Type ?u.89302\n\u03b2 : Type ?u.89305\n\u03b3 : Type ?u.89308\na b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nha : a \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d : b\u271d \u2260 0 \u2228 a \u2260 \u22a4\nb : \u211d\u22650\u221e\nhb : b \u2260 0\nn \u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u2191\u03b5\nh\u03b5b : \u2191\u03b5 < b\n\u22a2 \u2200\u1da0 (a : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e) in \ud835\udcdd (\u22a4, b), \u2191n < a.fst * a.snd", "state_after": "case intro.intro\n\u03b1 : Type ?u.89302\n\u03b2 : Type ?u.89305\n\u03b3 : Type ?u.89308\na b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nha : a \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d : b\u271d \u2260 0 \u2228 a \u2260 \u22a4\nb : \u211d\u22650\u221e\nhb : b \u2260 0\nn \u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u2191\u03b5\nh\u03b5b : \u2191\u03b5 < b\nthis : \u2200\u1da0 (c : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e) in \ud835\udcdd (\u22a4, b), \u2191n / \u2191\u03b5 < c.fst \u2227 \u2191\u03b5 < c.snd\n\u22a2 \u2200\u1da0 (a : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e) in \ud835\udcdd (\u22a4, b), \u2191n < a.fst * a.snd"}, {"tactic": "refine' this.mono fun c hc => _", "state_before": "case intro.intro\n\u03b1 : Type ?u.89302\n\u03b2 : Type ?u.89305\n\u03b3 : Type ?u.89308\na b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nha : a \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d : b\u271d \u2260 0 \u2228 a \u2260 \u22a4\nb : \u211d\u22650\u221e\nhb : b \u2260 0\nn \u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u2191\u03b5\nh\u03b5b : \u2191\u03b5 < b\nthis : \u2200\u1da0 (c : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e) in \ud835\udcdd (\u22a4, b), \u2191n / \u2191\u03b5 < c.fst \u2227 \u2191\u03b5 < c.snd\n\u22a2 \u2200\u1da0 (a : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e) in \ud835\udcdd (\u22a4, b), \u2191n < a.fst * a.snd", "state_after": "case intro.intro\n\u03b1 : Type ?u.89302\n\u03b2 : Type ?u.89305\n\u03b3 : Type ?u.89308\na b\u271d c\u271d d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nha : a \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d : b\u271d \u2260 0 \u2228 a \u2260 \u22a4\nb : \u211d\u22650\u221e\nhb : b \u2260 0\nn \u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u2191\u03b5\nh\u03b5b : \u2191\u03b5 < b\nthis : \u2200\u1da0 (c : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e) in \ud835\udcdd (\u22a4, b), \u2191n / \u2191\u03b5 < c.fst \u2227 \u2191\u03b5 < c.snd\nc : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e\nhc : \u2191n / \u2191\u03b5 < c.fst \u2227 \u2191\u03b5 < c.snd\n\u22a2 \u2191n < c.fst * c.snd"}, {"tactic": "exact (ENNReal.div_mul_cancel h\u03b5.ne' coe_ne_top).symm.trans_lt (mul_lt_mul hc.1 hc.2)", "state_before": "case intro.intro\n\u03b1 : Type ?u.89302\n\u03b2 : Type ?u.89305\n\u03b3 : Type ?u.89308\na b\u271d c\u271d d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nha : a \u2260 0 \u2228 b\u271d \u2260 \u22a4\nhb\u271d : b\u271d \u2260 0 \u2228 a \u2260 \u22a4\nb : \u211d\u22650\u221e\nhb : b \u2260 0\nn \u03b5 : \u211d\u22650\nh\u03b5 : 0 < \u2191\u03b5\nh\u03b5b : \u2191\u03b5 < b\nthis : \u2200\u1da0 (c : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e) in \ud835\udcdd (\u22a4, b), \u2191n / \u2191\u03b5 < c.fst \u2227 \u2191\u03b5 < c.snd\nc : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e\nhc : \u2191n / \u2191\u03b5 < c.fst \u2227 \u2191\u03b5 < c.snd\n\u22a2 \u2191n < c.fst * c.snd", "state_after": "no goals"}, {"tactic": "simp only [ne_eq, or_false] at hb", "state_before": "case top\n\u03b1 : Type ?u.89302\n\u03b2 : Type ?u.89305\n\u03b3 : Type ?u.89308\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nht : \u2200 (b : \u211d\u22650\u221e), b \u2260 0 \u2192 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (\u22a4, b)) (\ud835\udcdd \u22a4)\nha : \u22a4 \u2260 0 \u2228 b \u2260 \u22a4\nhb : b \u2260 0 \u2228 \u22a4 \u2260 \u22a4\n\u22a2 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (\u22a4, b)) (\ud835\udcdd (\u22a4 * b))", "state_after": "case top\n\u03b1 : Type ?u.89302\n\u03b2 : Type ?u.89305\n\u03b3 : Type ?u.89308\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nht : \u2200 (b : \u211d\u22650\u221e), b \u2260 0 \u2192 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (\u22a4, b)) (\ud835\udcdd \u22a4)\nha : \u22a4 \u2260 0 \u2228 b \u2260 \u22a4\nhb : \u00acb = 0\n\u22a2 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (\u22a4, b)) (\ud835\udcdd (\u22a4 * b))"}, {"tactic": "simp [ht b hb, top_mul hb]", "state_before": "case top\n\u03b1 : Type ?u.89302\n\u03b2 : Type ?u.89305\n\u03b3 : Type ?u.89308\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nht : \u2200 (b : \u211d\u22650\u221e), b \u2260 0 \u2192 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (\u22a4, b)) (\ud835\udcdd \u22a4)\nha : \u22a4 \u2260 0 \u2228 b \u2260 \u22a4\nhb : \u00acb = 0\n\u22a2 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (\u22a4, b)) (\ud835\udcdd (\u22a4 * b))", "state_after": "no goals"}, {"tactic": "simp only [ne_eq, or_false] at ha", "state_before": "case coe.top\n\u03b1 : Type ?u.89302\n\u03b2 : Type ?u.89305\n\u03b3 : Type ?u.89308\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nht : \u2200 (b : \u211d\u22650\u221e), b \u2260 0 \u2192 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (\u22a4, b)) (\ud835\udcdd \u22a4)\na : \u211d\u22650\nha : \u2191a \u2260 0 \u2228 \u22a4 \u2260 \u22a4\nhb : \u22a4 \u2260 0 \u2228 \u2191a \u2260 \u22a4\n\u22a2 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (\u2191a, \u22a4)) (\ud835\udcdd (\u2191a * \u22a4))", "state_after": "case coe.top\n\u03b1 : Type ?u.89302\n\u03b2 : Type ?u.89305\n\u03b3 : Type ?u.89308\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nht : \u2200 (b : \u211d\u22650\u221e), b \u2260 0 \u2192 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (\u22a4, b)) (\ud835\udcdd \u22a4)\na : \u211d\u22650\nhb : \u22a4 \u2260 0 \u2228 \u2191a \u2260 \u22a4\nha : \u00ac\u2191a = 0\n\u22a2 Tendsto (fun p => p.fst * p.snd) (\ud835\udcdd (\u2191a, \u22a4)) (\ud835\udcdd (\u2191a * \u22a4))"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Images.lean", "full_name": "CategoryTheory.Limits.image.as_\u03b9", "start": [321, 1], "end": [321, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Covering/OneDim.lean", "full_name": "Real.Icc_mem_vitaliFamily_at_left", "start": [47, 1], "end": [51, 48], "traced_tactics": [{"tactic": "rw [Icc_eq_closedBall]", "state_before": "x y : \u211d\nhxy : x < y\n\u22a2 Icc x y \u2208 VitaliFamily.setsAt (vitaliFamily volume 1) y", "state_after": "x y : \u211d\nhxy : x < y\n\u22a2 Metric.closedBall ((x + y) / 2) ((y - x) / 2) \u2208 VitaliFamily.setsAt (vitaliFamily volume 1) y"}, {"tactic": "refine' closedBall_mem_vitaliFamily_of_dist_le_mul _ _ (by linarith)", "state_before": "x y : \u211d\nhxy : x < y\n\u22a2 Metric.closedBall ((x + y) / 2) ((y - x) / 2) \u2208 VitaliFamily.setsAt (vitaliFamily volume 1) y", "state_after": "x y : \u211d\nhxy : x < y\n\u22a2 dist y ((x + y) / 2) \u2264 1 * ((y - x) / 2)"}, {"tactic": "rw [Real.dist_eq, abs_of_nonneg] <;> linarith", "state_before": "x y : \u211d\nhxy : x < y\n\u22a2 dist y ((x + y) / 2) \u2264 1 * ((y - x) / 2)", "state_after": "no goals"}, {"tactic": "linarith", "state_before": "x y : \u211d\nhxy : x < y\n\u22a2 0 < (y - x) / 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "ContinuousMap.coeFn_toLp", "start": [1718, 1], "end": [1720, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean", "full_name": "TopCat.pullback_fst_range", "start": [218, 1], "end": [227, 9], "traced_tactics": [{"tactic": "ext x", "state_before": "J : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\n\u22a2 Set.range ((forget TopCat).map pullback.fst) = {x | \u2203 y, (forget TopCat).map f x = (forget TopCat).map g y}", "state_after": "case h\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\nx : (forget TopCat).obj X\n\u22a2 x \u2208 Set.range ((forget TopCat).map pullback.fst) \u2194 x \u2208 {x | \u2203 y, (forget TopCat).map f x = (forget TopCat).map g y}"}, {"tactic": "constructor", "state_before": "case h\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\nx : (forget TopCat).obj X\n\u22a2 x \u2208 Set.range ((forget TopCat).map pullback.fst) \u2194 x \u2208 {x | \u2203 y, (forget TopCat).map f x = (forget TopCat).map g y}", "state_after": "case h.mp\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\nx : (forget TopCat).obj X\n\u22a2 x \u2208 Set.range ((forget TopCat).map pullback.fst) \u2192 x \u2208 {x | \u2203 y, (forget TopCat).map f x = (forget TopCat).map g y}\n\ncase h.mpr\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\nx : (forget TopCat).obj X\n\u22a2 x \u2208 {x | \u2203 y, (forget TopCat).map f x = (forget TopCat).map g y} \u2192 x \u2208 Set.range ((forget TopCat).map pullback.fst)"}, {"tactic": "rintro \u27e8y, rfl\u27e9", "state_before": "case h.mp\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\nx : (forget TopCat).obj X\n\u22a2 x \u2208 Set.range ((forget TopCat).map pullback.fst) \u2192 x \u2208 {x | \u2203 y, (forget TopCat).map f x = (forget TopCat).map g y}", "state_after": "case h.mp.intro\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\ny : (forget TopCat).obj (pullback f g)\n\u22a2 (forget TopCat).map pullback.fst y \u2208 {x | \u2203 y, (forget TopCat).map f x = (forget TopCat).map g y}"}, {"tactic": "use (pullback.snd : pullback f g \u27f6 _) y", "state_before": "case h.mp.intro\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\ny : (forget TopCat).obj (pullback f g)\n\u22a2 (forget TopCat).map pullback.fst y \u2208 {x | \u2203 y, (forget TopCat).map f x = (forget TopCat).map g y}", "state_after": "case h.mp.intro\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\ny : (forget TopCat).obj (pullback f g)\n\u22a2 (forget TopCat).map f ((forget TopCat).map pullback.fst y) =\n    (forget TopCat).map g ((forget TopCat).map pullback.snd y)"}, {"tactic": "exact ConcreteCategory.congr_hom pullback.condition y", "state_before": "case h.mp.intro\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\ny : (forget TopCat).obj (pullback f g)\n\u22a2 (forget TopCat).map f ((forget TopCat).map pullback.fst y) =\n    (forget TopCat).map g ((forget TopCat).map pullback.snd y)", "state_after": "no goals"}, {"tactic": "rintro \u27e8y, eq\u27e9", "state_before": "case h.mpr\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\nx : (forget TopCat).obj X\n\u22a2 x \u2208 {x | \u2203 y, (forget TopCat).map f x = (forget TopCat).map g y} \u2192 x \u2208 Set.range ((forget TopCat).map pullback.fst)", "state_after": "case h.mpr.intro\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\nx : (forget TopCat).obj X\ny : \u2191Y\neq : (forget TopCat).map f x = (forget TopCat).map g y\n\u22a2 x \u2208 Set.range ((forget TopCat).map pullback.fst)"}, {"tactic": "use (TopCat.pullbackIsoProdSubtype f g).inv \u27e8\u27e8x, y\u27e9, eq\u27e9", "state_before": "case h.mpr.intro\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\nx : (forget TopCat).obj X\ny : \u2191Y\neq : (forget TopCat).map f x = (forget TopCat).map g y\n\u22a2 x \u2208 Set.range ((forget TopCat).map pullback.fst)", "state_after": "case h.mpr.intro\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\nx : (forget TopCat).obj X\ny : \u2191Y\neq : (forget TopCat).map f x = (forget TopCat).map g y\n\u22a2 (forget TopCat).map pullback.fst\n      ((forget TopCat).map (pullbackIsoProdSubtype f g).inv { val := (x, y), property := eq }) =\n    x"}, {"tactic": "simp", "state_before": "case h.mpr.intro\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\ng : Y \u27f6 S\nx : (forget TopCat).obj X\ny : \u2191Y\neq : (forget TopCat).map f x = (forget TopCat).map g y\n\u22a2 (forget TopCat).map pullback.fst\n      ((forget TopCat).map (pullbackIsoProdSubtype f g).inv { val := (x, y), property := eq }) =\n    x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.eq_of_forall_apply_eq", "start": [198, 1], "end": [201, 91], "traced_tactics": [{"tactic": "ext1 s s_mble", "state_before": "\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc \u03bd : FiniteMeasure \u03a9\nh : \u2200 (s : Set \u03a9), MeasurableSet s \u2192 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s = (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bd s)) s\n\u22a2 \u03bc = \u03bd", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc \u03bd : FiniteMeasure \u03a9\nh : \u2200 (s : Set \u03a9), MeasurableSet s \u2192 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s = (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bd s)) s\ns : Set \u03a9\ns_mble : MeasurableSet s\n\u22a2 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s"}, {"tactic": "simpa [ennreal_coeFn_eq_coeFn_toMeasure] using congr_arg ((\u2191) : \u211d\u22650 \u2192 \u211d\u22650\u221e) (h s s_mble)", "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc \u03bd : FiniteMeasure \u03a9\nh : \u2200 (s : Set \u03a9), MeasurableSet s \u2192 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s = (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bd s)) s\ns : Set \u03a9\ns_mble : MeasurableSet s\n\u22a2 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "singleton_mul_mem_nhds_of_nhds_one", "start": [1285, 1], "end": [1286, 56], "traced_tactics": [{"tactic": "simpa only [mul_one] using singleton_mul_mem_nhds a h", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nG : Type w\nH : Type x\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : ContinuousConstSMul \u03b1 \u03b1\ns t : Set \u03b1\na : \u03b1\nh : s \u2208 \ud835\udcdd 1\n\u22a2 {a} * s \u2208 \ud835\udcdd a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Matrix/Block.lean", "full_name": "Matrix.blockDiagonal'_apply_ne", "start": [663, 1], "end": [665, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/IsEmpty.lean", "full_name": "isEmpty_prod", "start": [170, 1], "end": [171, 60], "traced_tactics": [{"tactic": "simp only [\u2190 not_nonempty_iff, nonempty_prod, not_and_or]", "state_before": "\u03b1\u271d : Sort ?u.3022\n\u03b2\u271d : Sort ?u.3025\n\u03b3 : Sort ?u.3028\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u22a2 IsEmpty (\u03b1 \u00d7 \u03b2) \u2194 IsEmpty \u03b1 \u2228 IsEmpty \u03b2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "full_name": "Zsqrtd.norm_eq_mul_conj", "start": [559, 1], "end": [561, 58], "traced_tactics": [{"tactic": "cases n", "state_before": "d : \u2124\nn : \u2124\u221ad\n\u22a2 \u2191(norm n) = n * star n", "state_after": "case mk\nd re\u271d im\u271d : \u2124\n\u22a2 \u2191(norm { re := re\u271d, im := im\u271d }) = { re := re\u271d, im := im\u271d } * star { re := re\u271d, im := im\u271d }"}, {"tactic": "simp [norm, star, Zsqrtd.ext, mul_comm, sub_eq_add_neg]", "state_before": "case mk\nd re\u271d im\u271d : \u2124\n\u22a2 \u2191(norm { re := re\u271d, im := im\u271d }) = { re := re\u271d, im := im\u271d } * star { re := re\u271d, im := im\u271d }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/Basic.lean", "full_name": "inf_uniformity", "start": [1235, 1], "end": [1236, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/ModEq.lean", "full_name": "AddCommGroup.ModEq.add_left", "start": [230, 11], "end": [231, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.succ_injective", "start": [897, 1], "end": [898, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Reverse.lean", "full_name": "Polynomial.eval\u2082_reverse_eq_zero_iff", "start": [358, 1], "end": [360, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Analysis/Filter.lean", "full_name": "Filter.Realizer.principal_\u03c3", "start": [167, 1], "end": [168, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Control/Fold.lean", "full_name": "Traversable.foldMap_hom_free", "start": [293, 1], "end": [295, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "Antitone.map_csSup_of_continuousAt", "start": [2825, 1], "end": [2828, 9], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/MStructure.lean", "full_name": "IsLprojection.Lcomplement", "start": [98, 1], "end": [101, 21], "traced_tactics": [{"tactic": "rw [add_comm, sub_sub_cancel]", "state_before": "X : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b9 : Ring M\ninst\u271d : Module M X\nP : M\nh : IsLprojection X P\nx : X\n\u22a2 \u2016x\u2016 = \u2016(1 - P) \u2022 x\u2016 + \u2016(1 - (1 - P)) \u2022 x\u2016", "state_after": "X : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b9 : Ring M\ninst\u271d : Module M X\nP : M\nh : IsLprojection X P\nx : X\n\u22a2 \u2016x\u2016 = \u2016P \u2022 x\u2016 + \u2016(1 - P) \u2022 x\u2016"}, {"tactic": "exact h.Lnorm x", "state_before": "X : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup X\nM : Type u_2\ninst\u271d\u00b9 : Ring M\ninst\u271d : Module M X\nP : M\nh : IsLprojection X P\nx : X\n\u22a2 \u2016x\u2016 = \u2016P \u2022 x\u2016 + \u2016(1 - P) \u2022 x\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Sheaves/SheafCondition/Sites.lean", "full_name": "TopCat.Presheaf.covering_presieve_eq_self", "start": [93, 1], "end": [97, 94], "traced_tactics": [{"tactic": "funext Z", "state_before": "X : TopCat\nY : Opens \u2191X\nR : Presieve Y\n\u22a2 presieveOfCoveringAux (coveringOfPresieve Y R) Y = R", "state_after": "case h\nX : TopCat\nY : Opens \u2191X\nR : Presieve Y\nZ : Opens \u2191X\n\u22a2 presieveOfCoveringAux (coveringOfPresieve Y R) Y = R"}, {"tactic": "ext f", "state_before": "case h\nX : TopCat\nY : Opens \u2191X\nR : Presieve Y\nZ : Opens \u2191X\n\u22a2 presieveOfCoveringAux (coveringOfPresieve Y R) Y = R", "state_after": "case h.h\nX : TopCat\nY : Opens \u2191X\nR : Presieve Y\nZ : Opens \u2191X\nf : Z \u27f6 Y\n\u22a2 f \u2208 presieveOfCoveringAux (coveringOfPresieve Y R) Y \u2194 f \u2208 R"}, {"tactic": "exact \u27e8fun \u27e8\u27e8_, f', h\u27e9, rfl\u27e9 => by rwa [Subsingleton.elim f f'], fun h => \u27e8\u27e8Z, f, h\u27e9, rfl\u27e9\u27e9", "state_before": "case h.h\nX : TopCat\nY : Opens \u2191X\nR : Presieve Y\nZ : Opens \u2191X\nf : Z \u27f6 Y\n\u22a2 f \u2208 presieveOfCoveringAux (coveringOfPresieve Y R) Y \u2194 f \u2208 R", "state_after": "no goals"}, {"tactic": "rwa [Subsingleton.elim f f']", "state_before": "X : TopCat\nY : Opens \u2191X\nR : Presieve Y\nZ : Opens \u2191X\nf : Z \u27f6 Y\nx\u271d : f \u2208 presieveOfCoveringAux (coveringOfPresieve Y R) Y\nf' : Z \u27f6 Y\nh : R f'\n\u22a2 f \u2208 R", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/PFunctor/Univariate/M.lean", "full_name": "PFunctor.M.bisim'", "start": [746, 1], "end": [758, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.range_unique", "start": [1107, 1], "end": [1114, 37], "traced_tactics": [{"tactic": "ext x", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.95746\n\u03b3 : Type ?u.95749\n\u03b9 : Sort u_1\n\u03b9' : Sort ?u.95755\nf : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\nh : Unique \u03b9\n\u22a2 range f = {f default}", "state_after": "case h\n\u03b1 : Type u_2\n\u03b2 : Type ?u.95746\n\u03b3 : Type ?u.95749\n\u03b9 : Sort u_1\n\u03b9' : Sort ?u.95755\nf : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\nh : Unique \u03b9\nx : \u03b1\n\u22a2 x \u2208 range f \u2194 x \u2208 {f default}"}, {"tactic": "rw [mem_range]", "state_before": "case h\n\u03b1 : Type u_2\n\u03b2 : Type ?u.95746\n\u03b3 : Type ?u.95749\n\u03b9 : Sort u_1\n\u03b9' : Sort ?u.95755\nf : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\nh : Unique \u03b9\nx : \u03b1\n\u22a2 x \u2208 range f \u2194 x \u2208 {f default}", "state_after": "case h\n\u03b1 : Type u_2\n\u03b2 : Type ?u.95746\n\u03b3 : Type ?u.95749\n\u03b9 : Sort u_1\n\u03b9' : Sort ?u.95755\nf : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\nh : Unique \u03b9\nx : \u03b1\n\u22a2 (\u2203 y, f y = x) \u2194 x \u2208 {f default}"}, {"tactic": "constructor", "state_before": "case h\n\u03b1 : Type u_2\n\u03b2 : Type ?u.95746\n\u03b3 : Type ?u.95749\n\u03b9 : Sort u_1\n\u03b9' : Sort ?u.95755\nf : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\nh : Unique \u03b9\nx : \u03b1\n\u22a2 (\u2203 y, f y = x) \u2194 x \u2208 {f default}", "state_after": "case h.mp\n\u03b1 : Type u_2\n\u03b2 : Type ?u.95746\n\u03b3 : Type ?u.95749\n\u03b9 : Sort u_1\n\u03b9' : Sort ?u.95755\nf : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\nh : Unique \u03b9\nx : \u03b1\n\u22a2 (\u2203 y, f y = x) \u2192 x \u2208 {f default}\n\ncase h.mpr\n\u03b1 : Type u_2\n\u03b2 : Type ?u.95746\n\u03b3 : Type ?u.95749\n\u03b9 : Sort u_1\n\u03b9' : Sort ?u.95755\nf : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\nh : Unique \u03b9\nx : \u03b1\n\u22a2 x \u2208 {f default} \u2192 \u2203 y, f y = x"}, {"tactic": "rintro \u27e8i, hi\u27e9", "state_before": "case h.mp\n\u03b1 : Type u_2\n\u03b2 : Type ?u.95746\n\u03b3 : Type ?u.95749\n\u03b9 : Sort u_1\n\u03b9' : Sort ?u.95755\nf : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\nh : Unique \u03b9\nx : \u03b1\n\u22a2 (\u2203 y, f y = x) \u2192 x \u2208 {f default}", "state_after": "case h.mp.intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.95746\n\u03b3 : Type ?u.95749\n\u03b9 : Sort u_1\n\u03b9' : Sort ?u.95755\nf : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\nh : Unique \u03b9\nx : \u03b1\ni : \u03b9\nhi : f i = x\n\u22a2 x \u2208 {f default}"}, {"tactic": "rw [h.uniq i] at hi", "state_before": "case h.mp.intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.95746\n\u03b3 : Type ?u.95749\n\u03b9 : Sort u_1\n\u03b9' : Sort ?u.95755\nf : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\nh : Unique \u03b9\nx : \u03b1\ni : \u03b9\nhi : f i = x\n\u22a2 x \u2208 {f default}", "state_after": "case h.mp.intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.95746\n\u03b3 : Type ?u.95749\n\u03b9 : Sort u_1\n\u03b9' : Sort ?u.95755\nf : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\nh : Unique \u03b9\nx : \u03b1\ni : \u03b9\nhi : f default = x\n\u22a2 x \u2208 {f default}"}, {"tactic": "exact hi \u25b8 mem_singleton _", "state_before": "case h.mp.intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.95746\n\u03b3 : Type ?u.95749\n\u03b9 : Sort u_1\n\u03b9' : Sort ?u.95755\nf : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\nh : Unique \u03b9\nx : \u03b1\ni : \u03b9\nhi : f default = x\n\u22a2 x \u2208 {f default}", "state_after": "no goals"}, {"tactic": "exact fun h => \u27e8default, h.symm\u27e9", "state_before": "case h.mpr\n\u03b1 : Type u_2\n\u03b2 : Type ?u.95746\n\u03b3 : Type ?u.95749\n\u03b9 : Sort u_1\n\u03b9' : Sort ?u.95755\nf : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\nh : Unique \u03b9\nx : \u03b1\n\u22a2 x \u2208 {f default} \u2192 \u2203 y, f y = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Basic.lean", "full_name": "lt_or_lt_iff_ne", "start": [485, 1], "end": [486, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.bind_some_eq_map", "start": [528, 1], "end": [529, 27], "traced_tactics": [{"tactic": "simp [eq_comm]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.22801\nf : \u03b1 \u2192 \u03b2\nx : Part \u03b1\n\u22a2 \u2200 (a : \u03b2), a \u2208 Part.bind x (some \u2218 f) \u2194 a \u2208 map f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Integrals.lean", "full_name": "integral_sin_pow_antitone", "start": [695, 1], "end": [696, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Add.lean", "full_name": "HasStrictFDerivAt.sub_const", "start": [527, 1], "end": [529, 54], "traced_tactics": [{"tactic": "simpa only [sub_eq_add_neg] using hf.add_const (-c)", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type ?u.479400\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type ?u.479495\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhf : HasStrictFDerivAt f f' x\nc : F\n\u22a2 HasStrictFDerivAt (fun x => f x - c) f' x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "Real.cos_sq", "start": [1294, 8], "end": [1295, 39], "traced_tactics": [{"tactic": "simp [cos_sq]", "state_before": "x y : \u211d\n\u22a2 \u2191(cos x ^ 2) = \u2191(1 / 2 + cos (2 * x) / 2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Instances.lean", "full_name": "Set.Ioo.coe_mul", "start": [359, 1], "end": [360, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Quaternion.lean", "full_name": "Quaternion.hasSum_coe", "start": [242, 1], "end": [244, 85], "traced_tactics": [{"tactic": "simpa only using h.map (show \u210d \u2192\u2097[\u211d] \u211d from QuaternionAlgebra.re\u2097 _ _) continuous_re", "state_before": "\u03b1 : Type u_1\nf : \u03b1 \u2192 \u211d\nr : \u211d\nh : HasSum (fun a => \u2191(f a)) \u2191r\n\u22a2 HasSum f r", "state_after": "no goals"}, {"tactic": "simpa only using h.map (algebraMap \u211d \u210d) (continuous_algebraMap _ _)", "state_before": "\u03b1 : Type u_1\nf : \u03b1 \u2192 \u211d\nr : \u211d\nh : HasSum f r\n\u22a2 HasSum (fun a => \u2191(f a)) \u2191r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Alternating.lean", "full_name": "AlternatingMap.map_update_zero", "start": [224, 1], "end": [225, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.map_one", "start": [121, 11], "end": [122, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/ZFC/Basic.lean", "full_name": "ZFSet.singleton_injective", "start": [1112, 1], "end": [1114, 51], "traced_tactics": [{"tactic": "let this := congr_arg sUnion H", "state_before": "x y : ZFSet\nH : {x} = {y}\n\u22a2 x = y", "state_after": "x y : ZFSet\nH : {x} = {y}\nthis : \u22c3\u2080 {x} = \u22c3\u2080 {y} := congr_arg sUnion H\n\u22a2 x = y"}, {"tactic": "rwa [sUnion_singleton, sUnion_singleton] at this", "state_before": "x y : ZFSet\nH : {x} = {y}\nthis : \u22c3\u2080 {x} = \u22c3\u2080 {y} := congr_arg sUnion H\n\u22a2 x = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Matrix/Basic.lean", "full_name": "Matrix.submatrix_zero", "start": [2428, 1], "end": [2430, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Constructions.lean", "full_name": "continuous_mulSingle", "start": [1257, 1], "end": [1259, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/ModelTheory/LanguageMap.lean", "full_name": "FirstOrder.Language.LHom.funMap_sumInl", "start": [299, 1], "end": [301, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Closed/Cartesian.lean", "full_name": "CategoryTheory.CartesianClosed.curry_natural_right", "start": [186, 1], "end": [188, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.hom_eq_hom", "start": [472, 1], "end": [474, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/SuccPred/Limit.lean", "full_name": "Order.IsSuccLimit.isMin_of_noMax", "start": [102, 1], "end": [106, 42], "traced_tactics": [{"tactic": "rcases hb.exists_succ_iterate with \u27e8_ | n, rfl\u27e9", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : Preorder \u03b1\na : \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : NoMaxOrder \u03b1\nh : IsSuccLimit a\nb : \u03b1\nhb : b \u2264 a\n\u22a2 a \u2264 b", "state_after": "case intro.zero\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : NoMaxOrder \u03b1\nb : \u03b1\nh : IsSuccLimit ((succ^[Nat.zero]) b)\nhb : b \u2264 (succ^[Nat.zero]) b\n\u22a2 (succ^[Nat.zero]) b \u2264 b\n\ncase intro.succ\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : NoMaxOrder \u03b1\nb : \u03b1\nn : \u2115\nh : IsSuccLimit ((succ^[Nat.succ n]) b)\nhb : b \u2264 (succ^[Nat.succ n]) b\n\u22a2 (succ^[Nat.succ n]) b \u2264 b"}, {"tactic": "exact le_rfl", "state_before": "case intro.zero\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : NoMaxOrder \u03b1\nb : \u03b1\nh : IsSuccLimit ((succ^[Nat.zero]) b)\nhb : b \u2264 (succ^[Nat.zero]) b\n\u22a2 (succ^[Nat.zero]) b \u2264 b", "state_after": "no goals"}, {"tactic": "rw [iterate_succ_apply'] at h", "state_before": "case intro.succ\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : NoMaxOrder \u03b1\nb : \u03b1\nn : \u2115\nh : IsSuccLimit ((succ^[Nat.succ n]) b)\nhb : b \u2264 (succ^[Nat.succ n]) b\n\u22a2 (succ^[Nat.succ n]) b \u2264 b", "state_after": "case intro.succ\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : NoMaxOrder \u03b1\nb : \u03b1\nn : \u2115\nh : IsSuccLimit (succ ((succ^[n]) b))\nhb : b \u2264 (succ^[Nat.succ n]) b\n\u22a2 (succ^[Nat.succ n]) b \u2264 b"}, {"tactic": "exact (not_isSuccLimit_succ _ h).elim", "state_before": "case intro.succ\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\ninst\u271d\u00b9 : IsSuccArchimedean \u03b1\ninst\u271d : NoMaxOrder \u03b1\nb : \u03b1\nn : \u2115\nh : IsSuccLimit (succ ((succ^[n]) b))\nhb : b \u2264 (succ^[Nat.succ n]) b\n\u22a2 (succ^[Nat.succ n]) b \u2264 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Fold.lean", "full_name": "Multiset.fold_eq_foldl", "start": [52, 1], "end": [54, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/ContDiff.lean", "full_name": "ContDiff.inv", "start": [1714, 1], "end": [1716, 71], "traced_tactics": [{"tactic": "rw [contDiff_iff_contDiffAt]", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2074 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\nX : Type ?u.2664072\ninst\u271d\u2076 : NormedAddCommGroup X\ninst\u271d\u2075 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf\u271d f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nR : Type ?u.2667550\ninst\u271d\u2074 : NormedRing R\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd5c R\n\ud835\udd5c' : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : CompleteSpace \ud835\udd5c'\nf : E \u2192 \ud835\udd5c'\nn : \u2115\u221e\nhf : ContDiff \ud835\udd5c n f\nh : \u2200 (x : E), f x \u2260 0\n\u22a2 ContDiff \ud835\udd5c n fun x => (f x)\u207b\u00b9", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2074 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\nX : Type ?u.2664072\ninst\u271d\u2076 : NormedAddCommGroup X\ninst\u271d\u2075 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf\u271d f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nR : Type ?u.2667550\ninst\u271d\u2074 : NormedRing R\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd5c R\n\ud835\udd5c' : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : CompleteSpace \ud835\udd5c'\nf : E \u2192 \ud835\udd5c'\nn : \u2115\u221e\nhf : ContDiff \ud835\udd5c n f\nh : \u2200 (x : E), f x \u2260 0\n\u22a2 \u2200 (x : E), ContDiffAt \ud835\udd5c n (fun x => (f x)\u207b\u00b9) x"}, {"tactic": "exact fun x => hf.contDiffAt.inv (h x)", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2074 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\nX : Type ?u.2664072\ninst\u271d\u2076 : NormedAddCommGroup X\ninst\u271d\u2075 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf\u271d f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nR : Type ?u.2667550\ninst\u271d\u2074 : NormedRing R\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd5c R\n\ud835\udd5c' : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d : CompleteSpace \ud835\udd5c'\nf : E \u2192 \ud835\udd5c'\nn : \u2115\u221e\nhf : ContDiff \ud835\udd5c n f\nh : \u2200 (x : E), f x \u2260 0\n\u22a2 \u2200 (x : E), ContDiffAt \ud835\udd5c n (fun x => (f x)\u207b\u00b9) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/RingDivision.lean", "full_name": "Polynomial.bUnion_roots_finite", "start": [920, 1], "end": [933, 37], "traced_tactics": [{"tactic": "let \u03c0 : R[X] \u2192 Fin (d + 1) \u2192 R := fun f i => f.coeff i", "state_before": "R\u271d : Type u\nS\u271d : Type v\nT : Type w\na b : R\u271d\nn : \u2115\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\np q : R\u271d[X]\ninst\u271d\u00b3 : CommRing T\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : IsDomain S\nm : R \u2192+* S\nd : \u2115\nU : Set R\nh : Set.Finite U\n\u22a2 Set.Finite fun f => natDegree f \u2264 d \u2227 \u2200 (i : \u2115), coeff f i \u2208 U", "state_after": "R\u271d : Type u\nS\u271d : Type v\nT : Type w\na b : R\u271d\nn : \u2115\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\np q : R\u271d[X]\ninst\u271d\u00b3 : CommRing T\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : IsDomain S\nm : R \u2192+* S\nd : \u2115\nU : Set R\nh : Set.Finite U\n\u03c0 : R[X] \u2192 Fin (d + 1) \u2192 R := fun f i => coeff f \u2191i\n\u22a2 Set.Finite fun f => natDegree f \u2264 d \u2227 \u2200 (i : \u2115), coeff f i \u2208 U"}, {"tactic": "refine' ((Set.Finite.pi fun _ => h).subset <| _).of_finite_image (_ : Set.InjOn \u03c0 _)", "state_before": "R\u271d : Type u\nS\u271d : Type v\nT : Type w\na b : R\u271d\nn : \u2115\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\np q : R\u271d[X]\ninst\u271d\u00b3 : CommRing T\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : IsDomain S\nm : R \u2192+* S\nd : \u2115\nU : Set R\nh : Set.Finite U\n\u03c0 : R[X] \u2192 Fin (d + 1) \u2192 R := fun f i => coeff f \u2191i\n\u22a2 Set.Finite fun f => natDegree f \u2264 d \u2227 \u2200 (i : \u2115), coeff f i \u2208 U", "state_after": "case refine'_1\nR\u271d : Type u\nS\u271d : Type v\nT : Type w\na b : R\u271d\nn : \u2115\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\np q : R\u271d[X]\ninst\u271d\u00b3 : CommRing T\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : IsDomain S\nm : R \u2192+* S\nd : \u2115\nU : Set R\nh : Set.Finite U\n\u03c0 : R[X] \u2192 Fin (d + 1) \u2192 R := fun f i => coeff f \u2191i\n\u22a2 (\u03c0 '' fun f => natDegree f \u2264 d \u2227 \u2200 (i : \u2115), coeff f i \u2208 U) \u2286 Set.pi Set.univ fun x => U\n\ncase refine'_2\nR\u271d : Type u\nS\u271d : Type v\nT : Type w\na b : R\u271d\nn : \u2115\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\np q : R\u271d[X]\ninst\u271d\u00b3 : CommRing T\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : IsDomain S\nm : R \u2192+* S\nd : \u2115\nU : Set R\nh : Set.Finite U\n\u03c0 : R[X] \u2192 Fin (d + 1) \u2192 R := fun f i => coeff f \u2191i\n\u22a2 Set.InjOn \u03c0 fun f => natDegree f \u2264 d \u2227 \u2200 (i : \u2115), coeff f i \u2208 U"}, {"tactic": "exact Set.image_subset_iff.2 fun f hf i _ => hf.2 i", "state_before": "case refine'_1\nR\u271d : Type u\nS\u271d : Type v\nT : Type w\na b : R\u271d\nn : \u2115\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\np q : R\u271d[X]\ninst\u271d\u00b3 : CommRing T\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : IsDomain S\nm : R \u2192+* S\nd : \u2115\nU : Set R\nh : Set.Finite U\n\u03c0 : R[X] \u2192 Fin (d + 1) \u2192 R := fun f i => coeff f \u2191i\n\u22a2 (\u03c0 '' fun f => natDegree f \u2264 d \u2227 \u2200 (i : \u2115), coeff f i \u2208 U) \u2286 Set.pi Set.univ fun x => U", "state_after": "no goals"}, {"tactic": "refine' fun x hx y hy hxy => (ext_iff_natDegree_le hx.1 hy.1).2 fun i hi => _", "state_before": "case refine'_2\nR\u271d : Type u\nS\u271d : Type v\nT : Type w\na b : R\u271d\nn : \u2115\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\np q : R\u271d[X]\ninst\u271d\u00b3 : CommRing T\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : IsDomain S\nm : R \u2192+* S\nd : \u2115\nU : Set R\nh : Set.Finite U\n\u03c0 : R[X] \u2192 Fin (d + 1) \u2192 R := fun f i => coeff f \u2191i\n\u22a2 Set.InjOn \u03c0 fun f => natDegree f \u2264 d \u2227 \u2200 (i : \u2115), coeff f i \u2208 U", "state_after": "case refine'_2\nR\u271d : Type u\nS\u271d : Type v\nT : Type w\na b : R\u271d\nn : \u2115\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\np q : R\u271d[X]\ninst\u271d\u00b3 : CommRing T\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : IsDomain S\nm : R \u2192+* S\nd : \u2115\nU : Set R\nh : Set.Finite U\n\u03c0 : R[X] \u2192 Fin (d + 1) \u2192 R := fun f i => coeff f \u2191i\nx : R[X]\nhx : x \u2208 fun f => natDegree f \u2264 d \u2227 \u2200 (i : \u2115), coeff f i \u2208 U\ny : R[X]\nhy : y \u2208 fun f => natDegree f \u2264 d \u2227 \u2200 (i : \u2115), coeff f i \u2208 U\nhxy : \u03c0 x = \u03c0 y\ni : \u2115\nhi : i \u2264 d\n\u22a2 coeff x i = coeff y i"}, {"tactic": "exact id congr_fun hxy \u27e8i, Nat.lt_succ_of_le hi\u27e9", "state_before": "case refine'_2\nR\u271d : Type u\nS\u271d : Type v\nT : Type w\na b : R\u271d\nn : \u2115\ninst\u271d\u2075 : CommRing R\u271d\ninst\u271d\u2074 : IsDomain R\u271d\np q : R\u271d[X]\ninst\u271d\u00b3 : CommRing T\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : IsDomain S\nm : R \u2192+* S\nd : \u2115\nU : Set R\nh : Set.Finite U\n\u03c0 : R[X] \u2192 Fin (d + 1) \u2192 R := fun f i => coeff f \u2191i\nx : R[X]\nhx : x \u2208 fun f => natDegree f \u2264 d \u2227 \u2200 (i : \u2115), coeff f i \u2208 U\ny : R[X]\nhy : y \u2208 fun f => natDegree f \u2264 d \u2227 \u2200 (i : \u2115), coeff f i \u2208 U\nhxy : \u03c0 x = \u03c0 y\ni : \u2115\nhi : i \u2264 d\n\u22a2 coeff x i = coeff y i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/FiberBundle/Trivialization.lean", "full_name": "Trivialization.coordChangeHomeomorph_coe", "start": [721, 1], "end": [723, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Maps.lean", "full_name": "OpenEmbedding.isOpenMap_iff", "start": [618, 1], "end": [620, 99], "traced_tactics": [{"tactic": "simp_rw [isOpenMap_iff_nhds_le, \u2190 map_map, comp, \u2190 hg.map_nhds_eq, Filter.map_le_map_iff hg.inj]", "state_before": "\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.259259\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nhg : OpenEmbedding g\n\u22a2 IsOpenMap f \u2194 IsOpenMap (g \u2218 f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Multiplicity.lean", "full_name": "multiplicity_eq_zero_of_coprime", "start": [652, 1], "end": [660, 16], "traced_tactics": [{"tactic": "rw [multiplicity_le_multiplicity_iff] at hle", "state_before": "\u03b1 : Type ?u.2262511\np a b : \u2115\nhp : p \u2260 1\nhle : multiplicity p a \u2264 multiplicity p b\nhab : coprime a b\n\u22a2 multiplicity p a = 0", "state_after": "\u03b1 : Type ?u.2262511\np a b : \u2115\nhp : p \u2260 1\nhle : \u2200 (n : \u2115), p ^ n \u2223 a \u2192 p ^ n \u2223 b\nhab : coprime a b\n\u22a2 multiplicity p a = 0"}, {"tactic": "rw [\u2190 nonpos_iff_eq_zero, \u2190 not_lt, PartENat.pos_iff_one_le, \u2190 Nat.cast_one, \u2190\n  pow_dvd_iff_le_multiplicity]", "state_before": "\u03b1 : Type ?u.2262511\np a b : \u2115\nhp : p \u2260 1\nhle : \u2200 (n : \u2115), p ^ n \u2223 a \u2192 p ^ n \u2223 b\nhab : coprime a b\n\u22a2 multiplicity p a = 0", "state_after": "\u03b1 : Type ?u.2262511\np a b : \u2115\nhp : p \u2260 1\nhle : \u2200 (n : \u2115), p ^ n \u2223 a \u2192 p ^ n \u2223 b\nhab : coprime a b\n\u22a2 \u00acp ^ 1 \u2223 a"}, {"tactic": "intro h", "state_before": "\u03b1 : Type ?u.2262511\np a b : \u2115\nhp : p \u2260 1\nhle : \u2200 (n : \u2115), p ^ n \u2223 a \u2192 p ^ n \u2223 b\nhab : coprime a b\n\u22a2 \u00acp ^ 1 \u2223 a", "state_after": "\u03b1 : Type ?u.2262511\np a b : \u2115\nhp : p \u2260 1\nhle : \u2200 (n : \u2115), p ^ n \u2223 a \u2192 p ^ n \u2223 b\nhab : coprime a b\nh : p ^ 1 \u2223 a\n\u22a2 False"}, {"tactic": "have := Nat.dvd_gcd h (hle _ h)", "state_before": "\u03b1 : Type ?u.2262511\np a b : \u2115\nhp : p \u2260 1\nhle : \u2200 (n : \u2115), p ^ n \u2223 a \u2192 p ^ n \u2223 b\nhab : coprime a b\nh : p ^ 1 \u2223 a\n\u22a2 False", "state_after": "\u03b1 : Type ?u.2262511\np a b : \u2115\nhp : p \u2260 1\nhle : \u2200 (n : \u2115), p ^ n \u2223 a \u2192 p ^ n \u2223 b\nhab : coprime a b\nh : p ^ 1 \u2223 a\nthis : p ^ 1 \u2223 Nat.gcd a b\n\u22a2 False"}, {"tactic": "rw [coprime.gcd_eq_one hab, Nat.dvd_one, pow_one] at this", "state_before": "\u03b1 : Type ?u.2262511\np a b : \u2115\nhp : p \u2260 1\nhle : \u2200 (n : \u2115), p ^ n \u2223 a \u2192 p ^ n \u2223 b\nhab : coprime a b\nh : p ^ 1 \u2223 a\nthis : p ^ 1 \u2223 Nat.gcd a b\n\u22a2 False", "state_after": "\u03b1 : Type ?u.2262511\np a b : \u2115\nhp : p \u2260 1\nhle : \u2200 (n : \u2115), p ^ n \u2223 a \u2192 p ^ n \u2223 b\nhab : coprime a b\nh : p ^ 1 \u2223 a\nthis : p = 1\n\u22a2 False"}, {"tactic": "exact hp this", "state_before": "\u03b1 : Type ?u.2262511\np a b : \u2115\nhp : p \u2260 1\nhle : \u2200 (n : \u2115), p ^ n \u2223 a \u2192 p ^ n \u2223 b\nhab : coprime a b\nh : p ^ 1 \u2223 a\nthis : p = 1\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Abelian/NonPreadditive.lean", "full_name": "CategoryTheory.NonPreadditiveAbelian.add_neg_self", "start": [403, 1], "end": [403, 85], "traced_tactics": [{"tactic": "rw [add_neg, sub_self]", "state_before": "C : Type u\ninst\u271d\u00b9 : Category C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n\u22a2 a + -a = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "full_name": "CategoryTheory.Limits.hasColimitsOfShape_of_equivalence", "start": [1185, 1], "end": [1189, 41], "traced_tactics": [{"tactic": "constructor", "state_before": "J : Type u\u2081\ninst\u271d\u2074 : Category J\nK : Type u\u2082\ninst\u271d\u00b3 : Category K\nC : Type u\ninst\u271d\u00b2 : Category C\nF : J \u2964 C\nJ' : Type u\u2082\ninst\u271d\u00b9 : Category J'\ne : J \u224c J'\ninst\u271d : HasColimitsOfShape J C\n\u22a2 HasColimitsOfShape J' C", "state_after": "case has_colimit\nJ : Type u\u2081\ninst\u271d\u2074 : Category J\nK : Type u\u2082\ninst\u271d\u00b3 : Category K\nC : Type u\ninst\u271d\u00b2 : Category C\nF : J \u2964 C\nJ' : Type u\u2082\ninst\u271d\u00b9 : Category J'\ne : J \u224c J'\ninst\u271d : HasColimitsOfShape J C\n\u22a2 autoParam (\u2200 (F : J' \u2964 C), HasColimit F) _auto\u271d"}, {"tactic": "intro F", "state_before": "case has_colimit\nJ : Type u\u2081\ninst\u271d\u2074 : Category J\nK : Type u\u2082\ninst\u271d\u00b3 : Category K\nC : Type u\ninst\u271d\u00b2 : Category C\nF : J \u2964 C\nJ' : Type u\u2082\ninst\u271d\u00b9 : Category J'\ne : J \u224c J'\ninst\u271d : HasColimitsOfShape J C\n\u22a2 autoParam (\u2200 (F : J' \u2964 C), HasColimit F) _auto\u271d", "state_after": "case has_colimit\nJ : Type u\u2081\ninst\u271d\u2074 : Category J\nK : Type u\u2082\ninst\u271d\u00b3 : Category K\nC : Type u\ninst\u271d\u00b2 : Category C\nF\u271d : J \u2964 C\nJ' : Type u\u2082\ninst\u271d\u00b9 : Category J'\ne : J \u224c J'\ninst\u271d : HasColimitsOfShape J C\nF : J' \u2964 C\n\u22a2 HasColimit F"}, {"tactic": "apply hasColimit_of_equivalence_comp e", "state_before": "case has_colimit\nJ : Type u\u2081\ninst\u271d\u2074 : Category J\nK : Type u\u2082\ninst\u271d\u00b3 : Category K\nC : Type u\ninst\u271d\u00b2 : Category C\nF\u271d : J \u2964 C\nJ' : Type u\u2082\ninst\u271d\u00b9 : Category J'\ne : J \u224c J'\ninst\u271d : HasColimitsOfShape J C\nF : J' \u2964 C\n\u22a2 HasColimit F", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Sym.lean", "full_name": "Finset.sym2_univ", "start": [91, 1], "end": [93, 51], "traced_tactics": [{"tactic": "ext", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nm : Sym2 \u03b1\ninst\u271d : Fintype \u03b1\n\u22a2 Finset.sym2 univ = univ", "state_after": "case a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nm : Sym2 \u03b1\ninst\u271d : Fintype \u03b1\na\u271d : Sym2 \u03b1\n\u22a2 a\u271d \u2208 Finset.sym2 univ \u2194 a\u271d \u2208 univ"}, {"tactic": "simp only [mem_sym2_iff, mem_univ, implies_true]", "state_before": "case a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ns t : Finset \u03b1\na b : \u03b1\nm : Sym2 \u03b1\ninst\u271d : Fintype \u03b1\na\u271d : Sym2 \u03b1\n\u22a2 a\u271d \u2208 Finset.sym2 univ \u2194 a\u271d \u2208 univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Order.lean", "full_name": "continuous_coinduced_dom", "start": [719, 1], "end": [721, 61], "traced_tactics": [{"tactic": "simp only [continuous_iff_coinduced_le, coinduced_compose]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2\n\u03b9 : Sort ?u.37354\ng : \u03b2 \u2192 \u03b3\nt\u2081 : TopologicalSpace \u03b1\nt\u2082 : TopologicalSpace \u03b3\n\u22a2 Continuous g \u2194 Continuous (g \u2218 f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Prod.lean", "full_name": "hasStrictFDerivAt_fst", "start": [138, 1], "end": [139, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/ZFC/Basic.lean", "full_name": "Class.sInter_empty", "start": [1732, 1], "end": [1733, 57], "traced_tactics": [{"tactic": "rw [sInter, classToCong_empty, Set.sInter_empty, univ]", "state_before": "\u22a2 \u22c2\u2080 \u2205 = univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.mem_unique", "start": [146, 1], "end": [147, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/List/Init/Lemmas.lean", "full_name": "List.cons_bind", "start": [102, 9], "end": [103, 77], "traced_tactics": [{"tactic": "simp [join, List.bind]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nx : \u03b1\nxs : List \u03b1\nf : \u03b1 \u2192 List \u03b2\n\u22a2 List.bind (x :: xs) f = f x ++ List.bind xs f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "full_name": "abs_dvd_abs", "start": [141, 1], "end": [142, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Chain.lean", "full_name": "List.chain'_map", "start": [227, 1], "end": [229, 39], "traced_tactics": [{"tactic": "cases l <;> [rfl; exact chain_map _]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nf : \u03b2 \u2192 \u03b1\nl : List \u03b2\n\u22a2 Chain' R (map f l) \u2194 Chain' (fun a b => R (f a) (f b)) l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/StoneCech.lean", "full_name": "continuous_ultrafilter_extend", "start": [184, 1], "end": [192, 47], "traced_tactics": [{"tactic": "have h : \u2200 b : Ultrafilter \u03b1, \u2203 c, Tendsto f (comap pure (\ud835\udcdd b)) (\ud835\udcdd c) := fun b =>\n  let \u27e8c, _, h'\u27e9 :=\n    isCompact_univ.ultrafilter_le_nhds (b.map f) (by rw [le_principal_iff]; exact univ_mem)\n  \u27e8c, le_trans (map_mono (ultrafilter_comap_pure_nhds _)) h'\u27e9", "state_before": "\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\n\u22a2 Continuous (Ultrafilter.extend f)", "state_after": "\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nh : \u2200 (b : Ultrafilter \u03b1), \u2203 c, Tendsto f (comap pure (\ud835\udcdd b)) (\ud835\udcdd c)\n\u22a2 Continuous (Ultrafilter.extend f)"}, {"tactic": "letI : TopologicalSpace \u03b1 := \u22a5", "state_before": "\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nh : \u2200 (b : Ultrafilter \u03b1), \u2203 c, Tendsto f (comap pure (\ud835\udcdd b)) (\ud835\udcdd c)\n\u22a2 Continuous (Ultrafilter.extend f)", "state_after": "\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nh : \u2200 (b : Ultrafilter \u03b1), \u2203 c, Tendsto f (comap pure (\ud835\udcdd b)) (\ud835\udcdd c)\nthis : TopologicalSpace \u03b1 := \u22a5\n\u22a2 Continuous (Ultrafilter.extend f)"}, {"tactic": "haveI : NormalSpace \u03b3 := normalOfCompactT2", "state_before": "\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nh : \u2200 (b : Ultrafilter \u03b1), \u2203 c, Tendsto f (comap pure (\ud835\udcdd b)) (\ud835\udcdd c)\nthis : TopologicalSpace \u03b1 := \u22a5\n\u22a2 Continuous (Ultrafilter.extend f)", "state_after": "\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nh : \u2200 (b : Ultrafilter \u03b1), \u2203 c, Tendsto f (comap pure (\ud835\udcdd b)) (\ud835\udcdd c)\nthis\u271d : TopologicalSpace \u03b1 := \u22a5\nthis : NormalSpace \u03b3\n\u22a2 Continuous (Ultrafilter.extend f)"}, {"tactic": "exact denseInducing_pure.continuous_extend h", "state_before": "\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nh : \u2200 (b : Ultrafilter \u03b1), \u2203 c, Tendsto f (comap pure (\ud835\udcdd b)) (\ud835\udcdd c)\nthis\u271d : TopologicalSpace \u03b1 := \u22a5\nthis : NormalSpace \u03b3\n\u22a2 Continuous (Ultrafilter.extend f)", "state_after": "no goals"}, {"tactic": "rw [le_principal_iff]", "state_before": "\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nb : Ultrafilter \u03b1\n\u22a2 \u2191(Ultrafilter.map f b) \u2264 \ud835\udcdf univ", "state_after": "\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nb : Ultrafilter \u03b1\n\u22a2 univ \u2208 \u2191(Ultrafilter.map f b)"}, {"tactic": "exact univ_mem", "state_before": "\u03b1 : Type u\n\u03b3 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : T2Space \u03b3\ninst\u271d : CompactSpace \u03b3\nf : \u03b1 \u2192 \u03b3\nb : Ultrafilter \u03b1\n\u22a2 univ \u2208 \u2191(Ultrafilter.map f b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/Exponential.lean", "full_name": "Commute.exp", "start": [159, 1], "end": [160, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/LocalExtr.lean", "full_name": "IsLocalMinOn.comp_antitone", "start": [261, 8], "end": [263, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "full_name": "NNReal.one_lt_rpow", "start": [216, 1], "end": [217, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Probability/Integration.lean", "full_name": "ProbabilityTheory.IndepFun.integrable_mul", "start": [140, 1], "end": [155, 43], "traced_tactics": [{"tactic": "let nX : \u03a9 \u2192 ENNReal := fun a => \u2016X a\u2016\u208a", "state_before": "\u03a9 : Type u_2\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX\u271d Y\u271d : \u03a9 \u2192 \u211d\n\u03b2 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nX Y : \u03a9 \u2192 \u03b2\ninst\u271d\u00b9 : NormedDivisionRing \u03b2\ninst\u271d : BorelSpace \u03b2\nhXY : IndepFun X Y\nhX : Integrable X\nhY : Integrable Y\n\u22a2 Integrable (X * Y)", "state_after": "\u03a9 : Type u_2\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX\u271d Y\u271d : \u03a9 \u2192 \u211d\n\u03b2 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nX Y : \u03a9 \u2192 \u03b2\ninst\u271d\u00b9 : NormedDivisionRing \u03b2\ninst\u271d : BorelSpace \u03b2\nhXY : IndepFun X Y\nhX : Integrable X\nhY : Integrable Y\nnX : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016X a\u2016\u208a\n\u22a2 Integrable (X * Y)"}, {"tactic": "let nY : \u03a9 \u2192 ENNReal := fun a => \u2016Y a\u2016\u208a", "state_before": "\u03a9 : Type u_2\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX\u271d Y\u271d : \u03a9 \u2192 \u211d\n\u03b2 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nX Y : \u03a9 \u2192 \u03b2\ninst\u271d\u00b9 : NormedDivisionRing \u03b2\ninst\u271d : BorelSpace \u03b2\nhXY : IndepFun X Y\nhX : Integrable X\nhY : Integrable Y\nnX : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016X a\u2016\u208a\n\u22a2 Integrable (X * Y)", "state_after": "\u03a9 : Type u_2\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX\u271d Y\u271d : \u03a9 \u2192 \u211d\n\u03b2 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nX Y : \u03a9 \u2192 \u03b2\ninst\u271d\u00b9 : NormedDivisionRing \u03b2\ninst\u271d : BorelSpace \u03b2\nhXY : IndepFun X Y\nhX : Integrable X\nhY : Integrable Y\nnX : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016X a\u2016\u208a\nnY : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016Y a\u2016\u208a\n\u22a2 Integrable (X * Y)"}, {"tactic": "have hXY' : IndepFun (fun a => \u2016X a\u2016\u208a) (fun a => \u2016Y a\u2016\u208a) \u03bc :=\n  hXY.comp measurable_nnnorm measurable_nnnorm", "state_before": "\u03a9 : Type u_2\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX\u271d Y\u271d : \u03a9 \u2192 \u211d\n\u03b2 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nX Y : \u03a9 \u2192 \u03b2\ninst\u271d\u00b9 : NormedDivisionRing \u03b2\ninst\u271d : BorelSpace \u03b2\nhXY : IndepFun X Y\nhX : Integrable X\nhY : Integrable Y\nnX : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016X a\u2016\u208a\nnY : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016Y a\u2016\u208a\n\u22a2 Integrable (X * Y)", "state_after": "\u03a9 : Type u_2\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX\u271d Y\u271d : \u03a9 \u2192 \u211d\n\u03b2 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nX Y : \u03a9 \u2192 \u03b2\ninst\u271d\u00b9 : NormedDivisionRing \u03b2\ninst\u271d : BorelSpace \u03b2\nhXY : IndepFun X Y\nhX : Integrable X\nhY : Integrable Y\nnX : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016X a\u2016\u208a\nnY : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016Y a\u2016\u208a\nhXY' : IndepFun (fun a => \u2016X a\u2016\u208a) fun a => \u2016Y a\u2016\u208a\n\u22a2 Integrable (X * Y)"}, {"tactic": "have hXY'' : IndepFun nX nY \u03bc :=\n  hXY'.comp measurable_coe_nnreal_ennreal measurable_coe_nnreal_ennreal", "state_before": "\u03a9 : Type u_2\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX\u271d Y\u271d : \u03a9 \u2192 \u211d\n\u03b2 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nX Y : \u03a9 \u2192 \u03b2\ninst\u271d\u00b9 : NormedDivisionRing \u03b2\ninst\u271d : BorelSpace \u03b2\nhXY : IndepFun X Y\nhX : Integrable X\nhY : Integrable Y\nnX : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016X a\u2016\u208a\nnY : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016Y a\u2016\u208a\nhXY' : IndepFun (fun a => \u2016X a\u2016\u208a) fun a => \u2016Y a\u2016\u208a\n\u22a2 Integrable (X * Y)", "state_after": "\u03a9 : Type u_2\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX\u271d Y\u271d : \u03a9 \u2192 \u211d\n\u03b2 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nX Y : \u03a9 \u2192 \u03b2\ninst\u271d\u00b9 : NormedDivisionRing \u03b2\ninst\u271d : BorelSpace \u03b2\nhXY : IndepFun X Y\nhX : Integrable X\nhY : Integrable Y\nnX : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016X a\u2016\u208a\nnY : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016Y a\u2016\u208a\nhXY' : IndepFun (fun a => \u2016X a\u2016\u208a) fun a => \u2016Y a\u2016\u208a\nhXY'' : IndepFun nX nY\n\u22a2 Integrable (X * Y)"}, {"tactic": "have hnX : AEMeasurable nX \u03bc := hX.1.aemeasurable.nnnorm.coe_nnreal_ennreal", "state_before": "\u03a9 : Type u_2\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX\u271d Y\u271d : \u03a9 \u2192 \u211d\n\u03b2 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nX Y : \u03a9 \u2192 \u03b2\ninst\u271d\u00b9 : NormedDivisionRing \u03b2\ninst\u271d : BorelSpace \u03b2\nhXY : IndepFun X Y\nhX : Integrable X\nhY : Integrable Y\nnX : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016X a\u2016\u208a\nnY : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016Y a\u2016\u208a\nhXY' : IndepFun (fun a => \u2016X a\u2016\u208a) fun a => \u2016Y a\u2016\u208a\nhXY'' : IndepFun nX nY\n\u22a2 Integrable (X * Y)", "state_after": "\u03a9 : Type u_2\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX\u271d Y\u271d : \u03a9 \u2192 \u211d\n\u03b2 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nX Y : \u03a9 \u2192 \u03b2\ninst\u271d\u00b9 : NormedDivisionRing \u03b2\ninst\u271d : BorelSpace \u03b2\nhXY : IndepFun X Y\nhX : Integrable X\nhY : Integrable Y\nnX : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016X a\u2016\u208a\nnY : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016Y a\u2016\u208a\nhXY' : IndepFun (fun a => \u2016X a\u2016\u208a) fun a => \u2016Y a\u2016\u208a\nhXY'' : IndepFun nX nY\nhnX : AEMeasurable nX\n\u22a2 Integrable (X * Y)"}, {"tactic": "have hnY : AEMeasurable nY \u03bc := hY.1.aemeasurable.nnnorm.coe_nnreal_ennreal", "state_before": "\u03a9 : Type u_2\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX\u271d Y\u271d : \u03a9 \u2192 \u211d\n\u03b2 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nX Y : \u03a9 \u2192 \u03b2\ninst\u271d\u00b9 : NormedDivisionRing \u03b2\ninst\u271d : BorelSpace \u03b2\nhXY : IndepFun X Y\nhX : Integrable X\nhY : Integrable Y\nnX : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016X a\u2016\u208a\nnY : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016Y a\u2016\u208a\nhXY' : IndepFun (fun a => \u2016X a\u2016\u208a) fun a => \u2016Y a\u2016\u208a\nhXY'' : IndepFun nX nY\nhnX : AEMeasurable nX\n\u22a2 Integrable (X * Y)", "state_after": "\u03a9 : Type u_2\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX\u271d Y\u271d : \u03a9 \u2192 \u211d\n\u03b2 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nX Y : \u03a9 \u2192 \u03b2\ninst\u271d\u00b9 : NormedDivisionRing \u03b2\ninst\u271d : BorelSpace \u03b2\nhXY : IndepFun X Y\nhX : Integrable X\nhY : Integrable Y\nnX : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016X a\u2016\u208a\nnY : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016Y a\u2016\u208a\nhXY' : IndepFun (fun a => \u2016X a\u2016\u208a) fun a => \u2016Y a\u2016\u208a\nhXY'' : IndepFun nX nY\nhnX : AEMeasurable nX\nhnY : AEMeasurable nY\n\u22a2 Integrable (X * Y)"}, {"tactic": "have hmul : (\u222b\u207b a, nX a * nY a \u2202\u03bc) = (\u222b\u207b a, nX a \u2202\u03bc) * \u222b\u207b a, nY a \u2202\u03bc :=\n  lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun' hnX hnY hXY''", "state_before": "\u03a9 : Type u_2\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX\u271d Y\u271d : \u03a9 \u2192 \u211d\n\u03b2 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nX Y : \u03a9 \u2192 \u03b2\ninst\u271d\u00b9 : NormedDivisionRing \u03b2\ninst\u271d : BorelSpace \u03b2\nhXY : IndepFun X Y\nhX : Integrable X\nhY : Integrable Y\nnX : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016X a\u2016\u208a\nnY : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016Y a\u2016\u208a\nhXY' : IndepFun (fun a => \u2016X a\u2016\u208a) fun a => \u2016Y a\u2016\u208a\nhXY'' : IndepFun nX nY\nhnX : AEMeasurable nX\nhnY : AEMeasurable nY\n\u22a2 Integrable (X * Y)", "state_after": "\u03a9 : Type u_2\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX\u271d Y\u271d : \u03a9 \u2192 \u211d\n\u03b2 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nX Y : \u03a9 \u2192 \u03b2\ninst\u271d\u00b9 : NormedDivisionRing \u03b2\ninst\u271d : BorelSpace \u03b2\nhXY : IndepFun X Y\nhX : Integrable X\nhY : Integrable Y\nnX : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016X a\u2016\u208a\nnY : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016Y a\u2016\u208a\nhXY' : IndepFun (fun a => \u2016X a\u2016\u208a) fun a => \u2016Y a\u2016\u208a\nhXY'' : IndepFun nX nY\nhnX : AEMeasurable nX\nhnY : AEMeasurable nY\nhmul : (\u222b\u207b (a : \u03a9), nX a * nY a \u2202\u03bc) = (\u222b\u207b (a : \u03a9), nX a \u2202\u03bc) * \u222b\u207b (a : \u03a9), nY a \u2202\u03bc\n\u22a2 Integrable (X * Y)"}, {"tactic": "refine' \u27e8hX.1.mul hY.1, _\u27e9", "state_before": "\u03a9 : Type u_2\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX\u271d Y\u271d : \u03a9 \u2192 \u211d\n\u03b2 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nX Y : \u03a9 \u2192 \u03b2\ninst\u271d\u00b9 : NormedDivisionRing \u03b2\ninst\u271d : BorelSpace \u03b2\nhXY : IndepFun X Y\nhX : Integrable X\nhY : Integrable Y\nnX : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016X a\u2016\u208a\nnY : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016Y a\u2016\u208a\nhXY' : IndepFun (fun a => \u2016X a\u2016\u208a) fun a => \u2016Y a\u2016\u208a\nhXY'' : IndepFun nX nY\nhnX : AEMeasurable nX\nhnY : AEMeasurable nY\nhmul : (\u222b\u207b (a : \u03a9), nX a * nY a \u2202\u03bc) = (\u222b\u207b (a : \u03a9), nX a \u2202\u03bc) * \u222b\u207b (a : \u03a9), nY a \u2202\u03bc\n\u22a2 Integrable (X * Y)", "state_after": "\u03a9 : Type u_2\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX\u271d Y\u271d : \u03a9 \u2192 \u211d\n\u03b2 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nX Y : \u03a9 \u2192 \u03b2\ninst\u271d\u00b9 : NormedDivisionRing \u03b2\ninst\u271d : BorelSpace \u03b2\nhXY : IndepFun X Y\nhX : Integrable X\nhY : Integrable Y\nnX : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016X a\u2016\u208a\nnY : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016Y a\u2016\u208a\nhXY' : IndepFun (fun a => \u2016X a\u2016\u208a) fun a => \u2016Y a\u2016\u208a\nhXY'' : IndepFun nX nY\nhnX : AEMeasurable nX\nhnY : AEMeasurable nY\nhmul : (\u222b\u207b (a : \u03a9), nX a * nY a \u2202\u03bc) = (\u222b\u207b (a : \u03a9), nX a \u2202\u03bc) * \u222b\u207b (a : \u03a9), nY a \u2202\u03bc\n\u22a2 HasFiniteIntegral (X * Y)"}, {"tactic": "simp_rw [HasFiniteIntegral, Pi.mul_apply, nnnorm_mul, ENNReal.coe_mul, hmul]", "state_before": "\u03a9 : Type u_2\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX\u271d Y\u271d : \u03a9 \u2192 \u211d\n\u03b2 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nX Y : \u03a9 \u2192 \u03b2\ninst\u271d\u00b9 : NormedDivisionRing \u03b2\ninst\u271d : BorelSpace \u03b2\nhXY : IndepFun X Y\nhX : Integrable X\nhY : Integrable Y\nnX : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016X a\u2016\u208a\nnY : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016Y a\u2016\u208a\nhXY' : IndepFun (fun a => \u2016X a\u2016\u208a) fun a => \u2016Y a\u2016\u208a\nhXY'' : IndepFun nX nY\nhnX : AEMeasurable nX\nhnY : AEMeasurable nY\nhmul : (\u222b\u207b (a : \u03a9), nX a * nY a \u2202\u03bc) = (\u222b\u207b (a : \u03a9), nX a \u2202\u03bc) * \u222b\u207b (a : \u03a9), nY a \u2202\u03bc\n\u22a2 HasFiniteIntegral (X * Y)", "state_after": "\u03a9 : Type u_2\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX\u271d Y\u271d : \u03a9 \u2192 \u211d\n\u03b2 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nX Y : \u03a9 \u2192 \u03b2\ninst\u271d\u00b9 : NormedDivisionRing \u03b2\ninst\u271d : BorelSpace \u03b2\nhXY : IndepFun X Y\nhX : Integrable X\nhY : Integrable Y\nnX : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016X a\u2016\u208a\nnY : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016Y a\u2016\u208a\nhXY' : IndepFun (fun a => \u2016X a\u2016\u208a) fun a => \u2016Y a\u2016\u208a\nhXY'' : IndepFun nX nY\nhnX : AEMeasurable nX\nhnY : AEMeasurable nY\nhmul : (\u222b\u207b (a : \u03a9), nX a * nY a \u2202\u03bc) = (\u222b\u207b (a : \u03a9), nX a \u2202\u03bc) * \u222b\u207b (a : \u03a9), nY a \u2202\u03bc\n\u22a2 ((\u222b\u207b (a : \u03a9), \u2191\u2016X a\u2016\u208a \u2202\u03bc) * \u222b\u207b (a : \u03a9), \u2191\u2016Y a\u2016\u208a \u2202\u03bc) < \u22a4"}, {"tactic": "exact ENNReal.mul_lt_top hX.2.ne hY.2.ne", "state_before": "\u03a9 : Type u_2\nm\u03a9 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\nf g : \u03a9 \u2192 \u211d\u22650\u221e\nX\u271d Y\u271d : \u03a9 \u2192 \u211d\n\u03b2 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nX Y : \u03a9 \u2192 \u03b2\ninst\u271d\u00b9 : NormedDivisionRing \u03b2\ninst\u271d : BorelSpace \u03b2\nhXY : IndepFun X Y\nhX : Integrable X\nhY : Integrable Y\nnX : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016X a\u2016\u208a\nnY : \u03a9 \u2192 \u211d\u22650\u221e := fun a => \u2191\u2016Y a\u2016\u208a\nhXY' : IndepFun (fun a => \u2016X a\u2016\u208a) fun a => \u2016Y a\u2016\u208a\nhXY'' : IndepFun nX nY\nhnX : AEMeasurable nX\nhnY : AEMeasurable nY\nhmul : (\u222b\u207b (a : \u03a9), nX a * nY a \u2202\u03bc) = (\u222b\u207b (a : \u03a9), nX a \u2202\u03bc) * \u222b\u207b (a : \u03a9), nY a \u2202\u03bc\n\u22a2 ((\u222b\u207b (a : \u03a9), \u2191\u2016X a\u2016\u208a \u2202\u03bc) * \u222b\u207b (a : \u03a9), \u2191\u2016Y a\u2016\u208a \u2202\u03bc) < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/W/Basic.lean", "full_name": "WType.depth_lt_depth_mk", "start": [138, 1], "end": [139, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.sum_of_empty", "start": [2111, 1], "end": [2112, 74], "traced_tactics": [{"tactic": "rw [\u2190 measure_univ_eq_zero, sum_apply _ MeasurableSet.univ, tsum_empty]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.345266\n\u03b3 : Type ?u.345269\n\u03b4 : Type ?u.345272\n\u03b9 : Type u_1\nR : Type ?u.345278\nR' : Type ?u.345281\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : IsEmpty \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 sum \u03bc = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Valuation/ValuationRing.lean", "full_name": "ValuationRing.of_integers", "start": [413, 1], "end": [420, 32], "traced_tactics": [{"tactic": "constructor", "state_before": "\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\n\u22a2 ValuationRing \ud835\udcaa", "state_after": "case cond'\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\n\u22a2 \u2200 (a b : \ud835\udcaa), \u2203 c, a * c = b \u2228 b * c = a"}, {"tactic": "intro a b", "state_before": "case cond'\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\n\u22a2 \u2200 (a b : \ud835\udcaa), \u2203 c, a * c = b \u2228 b * c = a", "state_after": "case cond'\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\na b : \ud835\udcaa\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a"}, {"tactic": "cases' le_total (v (algebraMap \ud835\udcaa K a)) (v (algebraMap \ud835\udcaa K b)) with h h", "state_before": "case cond'\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\na b : \ud835\udcaa\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a", "state_after": "case cond'.inl\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\na b : \ud835\udcaa\nh : \u2191v (\u2191(algebraMap \ud835\udcaa K) a) \u2264 \u2191v (\u2191(algebraMap \ud835\udcaa K) b)\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a\n\ncase cond'.inr\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\na b : \ud835\udcaa\nh : \u2191v (\u2191(algebraMap \ud835\udcaa K) b) \u2264 \u2191v (\u2191(algebraMap \ud835\udcaa K) a)\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a"}, {"tactic": "obtain \u27e8c, hc\u27e9 := Valuation.Integers.dvd_of_le hh h", "state_before": "case cond'.inl\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\na b : \ud835\udcaa\nh : \u2191v (\u2191(algebraMap \ud835\udcaa K) a) \u2264 \u2191v (\u2191(algebraMap \ud835\udcaa K) b)\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a", "state_after": "case cond'.inl.intro\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\na b : \ud835\udcaa\nh : \u2191v (\u2191(algebraMap \ud835\udcaa K) a) \u2264 \u2191v (\u2191(algebraMap \ud835\udcaa K) b)\nc : \ud835\udcaa\nhc : a = b * c\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a"}, {"tactic": "use c", "state_before": "case cond'.inl.intro\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\na b : \ud835\udcaa\nh : \u2191v (\u2191(algebraMap \ud835\udcaa K) a) \u2264 \u2191v (\u2191(algebraMap \ud835\udcaa K) b)\nc : \ud835\udcaa\nhc : a = b * c\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a", "state_after": "case cond'.inl.intro\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\na b : \ud835\udcaa\nh : \u2191v (\u2191(algebraMap \ud835\udcaa K) a) \u2264 \u2191v (\u2191(algebraMap \ud835\udcaa K) b)\nc : \ud835\udcaa\nhc : a = b * c\n\u22a2 a * c = b \u2228 b * c = a"}, {"tactic": "exact Or.inr hc.symm", "state_before": "case cond'.inl.intro\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\na b : \ud835\udcaa\nh : \u2191v (\u2191(algebraMap \ud835\udcaa K) a) \u2264 \u2191v (\u2191(algebraMap \ud835\udcaa K) b)\nc : \ud835\udcaa\nhc : a = b * c\n\u22a2 a * c = b \u2228 b * c = a", "state_after": "no goals"}, {"tactic": "obtain \u27e8c, hc\u27e9 := Valuation.Integers.dvd_of_le hh h", "state_before": "case cond'.inr\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\na b : \ud835\udcaa\nh : \u2191v (\u2191(algebraMap \ud835\udcaa K) b) \u2264 \u2191v (\u2191(algebraMap \ud835\udcaa K) a)\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a", "state_after": "case cond'.inr.intro\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\na b : \ud835\udcaa\nh : \u2191v (\u2191(algebraMap \ud835\udcaa K) b) \u2264 \u2191v (\u2191(algebraMap \ud835\udcaa K) a)\nc : \ud835\udcaa\nhc : b = a * c\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a"}, {"tactic": "use c", "state_before": "case cond'.inr.intro\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\na b : \ud835\udcaa\nh : \u2191v (\u2191(algebraMap \ud835\udcaa K) b) \u2264 \u2191v (\u2191(algebraMap \ud835\udcaa K) a)\nc : \ud835\udcaa\nhc : b = a * c\n\u22a2 \u2203 c, a * c = b \u2228 b * c = a", "state_after": "case cond'.inr.intro\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\na b : \ud835\udcaa\nh : \u2191v (\u2191(algebraMap \ud835\udcaa K) b) \u2264 \u2191v (\u2191(algebraMap \ud835\udcaa K) a)\nc : \ud835\udcaa\nhc : b = a * c\n\u22a2 a * c = b \u2228 b * c = a"}, {"tactic": "exact Or.inl hc.symm", "state_before": "case cond'.inr.intro\n\ud835\udcaa : Type u\nK : Type v\n\u0393 : Type w\ninst\u271d\u2074 : CommRing \ud835\udcaa\ninst\u271d\u00b3 : IsDomain \ud835\udcaa\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra \ud835\udcaa K\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\nv : Valuation K \u0393\nhh : Valuation.Integers v \ud835\udcaa\na b : \ud835\udcaa\nh : \u2191v (\u2191(algebraMap \ud835\udcaa K) b) \u2264 \u2191v (\u2191(algebraMap \ud835\udcaa K) a)\nc : \ud835\udcaa\nhc : b = a * c\n\u22a2 a * c = b \u2228 b * c = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Exp.lean", "full_name": "Real.tendsto_exp_neg_atTop_nhds_0", "start": [185, 1], "end": [186, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/LocalExtr.lean", "full_name": "IsLocalMinOn.add", "start": [377, 8], "end": [379, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Chebyshev.lean", "full_name": "AntivaryOn.card_mul_sum_le_sum_mul_sum", "start": [121, 1], "end": [124, 42], "traced_tactics": [{"tactic": "rw [\u2190 nsmul_eq_mul]", "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type ?u.18351\ninst\u271d : LinearOrderedRing \u03b1\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf g : \u03b9 \u2192 \u03b1\nhfg : AntivaryOn f g \u2191s\n\u22a2 \u2191(card s) * \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i) * \u2211 i in s, g i", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type ?u.18351\ninst\u271d : LinearOrderedRing \u03b1\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf g : \u03b9 \u2192 \u03b1\nhfg : AntivaryOn f g \u2191s\n\u22a2 card s \u2022 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i) * \u2211 i in s, g i"}, {"tactic": "exact hfg.card_smul_sum_le_sum_smul_sum", "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type ?u.18351\ninst\u271d : LinearOrderedRing \u03b1\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf g : \u03b9 \u2192 \u03b1\nhfg : AntivaryOn f g \u2191s\n\u22a2 card s \u2022 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i) * \u2211 i in s, g i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fin/Interval.lean", "full_name": "Fin.map_valEmbedding_Ico", "start": [64, 1], "end": [65, 59], "traced_tactics": [{"tactic": "simp [Ico_eq_finset_subtype, Finset.fin, Finset.map_map]", "state_before": "n : \u2115\na b : Fin n\n\u22a2 map valEmbedding (Ico a b) = Ico \u2191a \u2191b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean", "full_name": "integral_Ioi_rpow_of_lt", "start": [79, 1], "end": [89, 42], "traced_tactics": [{"tactic": "have hd : \u2200 (x : \u211d) (_ : x \u2208 Ici c), HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x := by\n  intro x hx\n  convert (hasDerivAt_rpow_const (p := a + 1) (Or.inl (hc.trans_le hx).ne')).div_const _ using 1\n  field_simp [show a + 1 \u2260 0 from ne_of_lt (by linarith), mul_comm]", "state_before": "a : \u211d\nha : a < -1\nc : \u211d\nhc : 0 < c\n\u22a2 (\u222b (t : \u211d) in Ioi c, t ^ a) = -c ^ (a + 1) / (a + 1)", "state_after": "a : \u211d\nha : a < -1\nc : \u211d\nhc : 0 < c\nhd : \u2200 (x : \u211d), x \u2208 Ici c \u2192 HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x\n\u22a2 (\u222b (t : \u211d) in Ioi c, t ^ a) = -c ^ (a + 1) / (a + 1)"}, {"tactic": "have ht : Tendsto (fun t => t ^ (a + 1) / (a + 1)) atTop (\ud835\udcdd (0 / (a + 1))) := by\n  apply Tendsto.div_const\n  simpa only [neg_neg] using tendsto_rpow_neg_atTop (by linarith : 0 < -(a + 1))", "state_before": "a : \u211d\nha : a < -1\nc : \u211d\nhc : 0 < c\nhd : \u2200 (x : \u211d), x \u2208 Ici c \u2192 HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x\n\u22a2 (\u222b (t : \u211d) in Ioi c, t ^ a) = -c ^ (a + 1) / (a + 1)", "state_after": "a : \u211d\nha : a < -1\nc : \u211d\nhc : 0 < c\nhd : \u2200 (x : \u211d), x \u2208 Ici c \u2192 HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x\nht : Tendsto (fun t => t ^ (a + 1) / (a + 1)) atTop (\ud835\udcdd (0 / (a + 1)))\n\u22a2 (\u222b (t : \u211d) in Ioi c, t ^ a) = -c ^ (a + 1) / (a + 1)"}, {"tactic": "convert integral_Ioi_of_hasDerivAt_of_tendsto' hd (integrableOn_Ioi_rpow_of_lt ha hc) ht using 1", "state_before": "a : \u211d\nha : a < -1\nc : \u211d\nhc : 0 < c\nhd : \u2200 (x : \u211d), x \u2208 Ici c \u2192 HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x\nht : Tendsto (fun t => t ^ (a + 1) / (a + 1)) atTop (\ud835\udcdd (0 / (a + 1)))\n\u22a2 (\u222b (t : \u211d) in Ioi c, t ^ a) = -c ^ (a + 1) / (a + 1)", "state_after": "case h.e'_3\na : \u211d\nha : a < -1\nc : \u211d\nhc : 0 < c\nhd : \u2200 (x : \u211d), x \u2208 Ici c \u2192 HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x\nht : Tendsto (fun t => t ^ (a + 1) / (a + 1)) atTop (\ud835\udcdd (0 / (a + 1)))\n\u22a2 -c ^ (a + 1) / (a + 1) = 0 / (a + 1) - c ^ (a + 1) / (a + 1)"}, {"tactic": "simp only [neg_div, zero_div, zero_sub]", "state_before": "case h.e'_3\na : \u211d\nha : a < -1\nc : \u211d\nhc : 0 < c\nhd : \u2200 (x : \u211d), x \u2208 Ici c \u2192 HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x\nht : Tendsto (fun t => t ^ (a + 1) / (a + 1)) atTop (\ud835\udcdd (0 / (a + 1)))\n\u22a2 -c ^ (a + 1) / (a + 1) = 0 / (a + 1) - c ^ (a + 1) / (a + 1)", "state_after": "no goals"}, {"tactic": "intro x hx", "state_before": "a : \u211d\nha : a < -1\nc : \u211d\nhc : 0 < c\n\u22a2 \u2200 (x : \u211d), x \u2208 Ici c \u2192 HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x", "state_after": "a : \u211d\nha : a < -1\nc : \u211d\nhc : 0 < c\nx : \u211d\nhx : x \u2208 Ici c\n\u22a2 HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x"}, {"tactic": "convert (hasDerivAt_rpow_const (p := a + 1) (Or.inl (hc.trans_le hx).ne')).div_const _ using 1", "state_before": "a : \u211d\nha : a < -1\nc : \u211d\nhc : 0 < c\nx : \u211d\nhx : x \u2208 Ici c\n\u22a2 HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x", "state_after": "case h.e'_7\na : \u211d\nha : a < -1\nc : \u211d\nhc : 0 < c\nx : \u211d\nhx : x \u2208 Ici c\n\u22a2 x ^ a = (a + 1) * x ^ (a + 1 - 1) / (a + 1)"}, {"tactic": "field_simp [show a + 1 \u2260 0 from ne_of_lt (by linarith), mul_comm]", "state_before": "case h.e'_7\na : \u211d\nha : a < -1\nc : \u211d\nhc : 0 < c\nx : \u211d\nhx : x \u2208 Ici c\n\u22a2 x ^ a = (a + 1) * x ^ (a + 1 - 1) / (a + 1)", "state_after": "no goals"}, {"tactic": "linarith", "state_before": "a : \u211d\nha : a < -1\nc : \u211d\nhc : 0 < c\nx : \u211d\nhx : x \u2208 Ici c\n\u22a2 a + 1 < 0", "state_after": "no goals"}, {"tactic": "apply Tendsto.div_const", "state_before": "a : \u211d\nha : a < -1\nc : \u211d\nhc : 0 < c\nhd : \u2200 (x : \u211d), x \u2208 Ici c \u2192 HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x\n\u22a2 Tendsto (fun t => t ^ (a + 1) / (a + 1)) atTop (\ud835\udcdd (0 / (a + 1)))", "state_after": "case hf\na : \u211d\nha : a < -1\nc : \u211d\nhc : 0 < c\nhd : \u2200 (x : \u211d), x \u2208 Ici c \u2192 HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x\n\u22a2 Tendsto (fun a_1 => a_1 ^ (a + 1)) atTop (\ud835\udcdd 0)"}, {"tactic": "simpa only [neg_neg] using tendsto_rpow_neg_atTop (by linarith : 0 < -(a + 1))", "state_before": "case hf\na : \u211d\nha : a < -1\nc : \u211d\nhc : 0 < c\nhd : \u2200 (x : \u211d), x \u2208 Ici c \u2192 HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x\n\u22a2 Tendsto (fun a_1 => a_1 ^ (a + 1)) atTop (\ud835\udcdd 0)", "state_after": "no goals"}, {"tactic": "linarith", "state_before": "a : \u211d\nha : a < -1\nc : \u211d\nhc : 0 < c\nhd : \u2200 (x : \u211d), x \u2208 Ici c \u2192 HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x\n\u22a2 0 < -(a + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Nat/Gcd.lean", "full_name": "Nat.gcd_dvd_gcd_mul_right", "start": [134, 1], "end": [135, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Inseparable.lean", "full_name": "specializes_iff_closure_subset", "start": [146, 1], "end": [147, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Card.lean", "full_name": "Finset.card_disjoint_union", "start": [427, 1], "end": [428, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean", "full_name": "MeasureTheory.tendstoInMeasure_of_tendsto_snorm_top", "start": [326, 1], "end": [345, 36], "traced_tactics": [{"tactic": "intro \u03b4 h\u03b4", "state_before": "\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nhfg : Tendsto (fun n => snorm (f n - g) \u22a4 \u03bc) l (\ud835\udcdd 0)\n\u22a2 TendstoInMeasure \u03bc f l g", "state_after": "\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nhfg : Tendsto (fun n => snorm (f n - g) \u22a4 \u03bc) l (\ud835\udcdd 0)\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u22a2 Tendsto (fun i => \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f i x) (g x)}) l (\ud835\udcdd 0)"}, {"tactic": "simp only [snorm_exponent_top, snormEssSup] at hfg", "state_before": "\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nhfg : Tendsto (fun n => snorm (f n - g) \u22a4 \u03bc) l (\ud835\udcdd 0)\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u22a2 Tendsto (fun i => \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f i x) (g x)}) l (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nhfg : Tendsto (fun n => essSup (fun x => \u2191\u2016(f n - g) x\u2016\u208a) \u03bc) l (\ud835\udcdd 0)\n\u22a2 Tendsto (fun i => \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f i x) (g x)}) l (\ud835\udcdd 0)"}, {"tactic": "rw [ENNReal.tendsto_nhds_zero] at hfg\u22a2", "state_before": "\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nhfg : Tendsto (fun n => essSup (fun x => \u2191\u2016(f n - g) x\u2016\u208a) \u03bc) l (\ud835\udcdd 0)\n\u22a2 Tendsto (fun i => \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f i x) (g x)}) l (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nhfg : \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016(f x - g) x_1\u2016\u208a) \u03bc \u2264 \u03b5\n\u22a2 \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, \u2191\u2191\u03bc {x_1 | \u03b4 \u2264 dist (f x x_1) (g x_1)} \u2264 \u03b5"}, {"tactic": "intro \u03b5 h\u03b5", "state_before": "\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nhfg : \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016(f x - g) x_1\u2016\u208a) \u03bc \u2264 \u03b5\n\u22a2 \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, \u2191\u2191\u03bc {x_1 | \u03b4 \u2264 dist (f x x_1) (g x_1)} \u2264 \u03b5", "state_after": "\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nhfg : \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016(f x - g) x_1\u2016\u208a) \u03bc \u2264 \u03b5\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, \u2191\u2191\u03bc {x_1 | \u03b4 \u2264 dist (f x x_1) (g x_1)} \u2264 \u03b5"}, {"tactic": "specialize hfg (ENNReal.ofReal \u03b4 / 2)\n    (ENNReal.div_pos_iff.2 \u27e8(ENNReal.ofReal_pos.2 h\u03b4).ne.symm, ENNReal.two_ne_top\u27e9)", "state_before": "\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nhfg : \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016(f x - g) x_1\u2016\u208a) \u03bc \u2264 \u03b5\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, \u2191\u2191\u03bc {x_1 | \u03b4 \u2264 dist (f x x_1) (g x_1)} \u2264 \u03b5", "state_after": "\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016(f x - g) x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, \u2191\u2191\u03bc {x_1 | \u03b4 \u2264 dist (f x x_1) (g x_1)} \u2264 \u03b5"}, {"tactic": "refine' hfg.mono fun n hn => _", "state_before": "\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016(f x - g) x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, \u2191\u2191\u03bc {x_1 | \u03b4 \u2264 dist (f x x_1) (g x_1)} \u2264 \u03b5", "state_after": "\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016(f x - g) x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016(f n - g) x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\n\u22a2 \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f n x) (g x)} \u2264 \u03b5"}, {"tactic": "simp only [true_and_iff, gt_iff_lt, ge_iff_le, zero_tsub, zero_le, zero_add, Set.mem_Icc,\n  Pi.sub_apply] at *", "state_before": "\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016(f x - g) x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016(f n - g) x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\n\u22a2 \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f n x) (g x)} \u2264 \u03b5", "state_after": "\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f n x) (g x)} \u2264 \u03b5"}, {"tactic": "have : essSup (fun x : \u03b1 => (\u2016f n x - g x\u2016\u208a : \u211d\u22650\u221e)) \u03bc < ENNReal.ofReal \u03b4 :=\n  lt_of_le_of_lt hn\n    (ENNReal.half_lt_self (ENNReal.ofReal_pos.2 h\u03b4).ne.symm ENNReal.ofReal_lt_top.ne)", "state_before": "\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f n x) (g x)} \u2264 \u03b5", "state_after": "\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nh\u03b5 : 0 < \u03b5\nthis : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc < ENNReal.ofReal \u03b4\n\u22a2 \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f n x) (g x)} \u2264 \u03b5"}, {"tactic": "refine' ((le_of_eq _).trans (ae_lt_of_essSup_lt this).le).trans h\u03b5.le", "state_before": "\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nh\u03b5 : 0 < \u03b5\nthis : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc < ENNReal.ofReal \u03b4\n\u22a2 \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f n x) (g x)} \u2264 \u03b5", "state_after": "\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nh\u03b5 : 0 < \u03b5\nthis : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc < ENNReal.ofReal \u03b4\n\u22a2 \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f n x) (g x)} = \u2191\u2191\u03bc ({x | (fun y => \u2191\u2016f n y - g y\u2016\u208a < ENNReal.ofReal \u03b4) x}\u1d9c)"}, {"tactic": "congr with x", "state_before": "\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nh\u03b5 : 0 < \u03b5\nthis : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc < ENNReal.ofReal \u03b4\n\u22a2 \u2191\u2191\u03bc {x | \u03b4 \u2264 dist (f n x) (g x)} = \u2191\u2191\u03bc ({x | (fun y => \u2191\u2016f n y - g y\u2016\u208a < ENNReal.ofReal \u03b4) x}\u1d9c)", "state_after": "case e_a.h\n\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nh\u03b5 : 0 < \u03b5\nthis : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc < ENNReal.ofReal \u03b4\nx : \u03b1\n\u22a2 x \u2208 {x | \u03b4 \u2264 dist (f n x) (g x)} \u2194 x \u2208 {x | (fun y => \u2191\u2016f n y - g y\u2016\u208a < ENNReal.ofReal \u03b4) x}\u1d9c"}, {"tactic": "simp only [ENNReal.ofReal_le_iff_le_toReal ENNReal.coe_lt_top.ne, ENNReal.coe_toReal, not_lt,\n  coe_nnnorm, Set.mem_setOf_eq, Set.mem_compl_iff]", "state_before": "case e_a.h\n\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nh\u03b5 : 0 < \u03b5\nthis : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc < ENNReal.ofReal \u03b4\nx : \u03b1\n\u22a2 x \u2208 {x | \u03b4 \u2264 dist (f n x) (g x)} \u2194 x \u2208 {x | (fun y => \u2191\u2016f n y - g y\u2016\u208a < ENNReal.ofReal \u03b4) x}\u1d9c", "state_after": "case e_a.h\n\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nh\u03b5 : 0 < \u03b5\nthis : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc < ENNReal.ofReal \u03b4\nx : \u03b1\n\u22a2 \u03b4 \u2264 dist (f n x) (g x) \u2194 \u03b4 \u2264 \u2016f n x - g x\u2016"}, {"tactic": "rw [\u2190 dist_eq_norm (f n x) (g x)]", "state_before": "case e_a.h\n\u03b1 : Type u_3\n\u03b9 : Type u_2\nE\u271d : Type ?u.53469\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\u271d\np : \u211d\u22650\u221e\nf\u271d : \u03b9 \u2192 \u03b1 \u2192 E\u271d\ng\u271d : \u03b1 \u2192 E\u271d\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf : \u03b9 \u2192 \u03b1 \u2192 E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\n\u03b5 : \u211d\u22650\u221e\nhfg : \u2200\u1da0 (x : \u03b9) in l, essSup (fun x_1 => \u2191\u2016f x x_1 - g x_1\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nn : \u03b9\nhn : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc \u2264 ENNReal.ofReal \u03b4 / 2\nh\u03b5 : 0 < \u03b5\nthis : essSup (fun x => \u2191\u2016f n x - g x\u2016\u208a) \u03bc < ENNReal.ofReal \u03b4\nx : \u03b1\n\u22a2 \u03b4 \u2264 dist (f n x) (g x) \u2194 \u03b4 \u2264 \u2016f n x - g x\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.ae_eq_comp'", "start": [2876, 1], "end": [2878, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subgroup/Pointwise.lean", "full_name": "Subgroup.closure_induction_left", "start": [72, 1], "end": [76, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/EReal.lean", "full_name": "EReal.coe_eq_zero", "start": [369, 1], "end": [370, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/WittVector/IsPoly.lean", "full_name": "WittVector.poly_eq_of_wittPolynomial_bind_eq'", "start": [186, 1], "end": [194, 61], "traced_tactics": [{"tactic": "ext1 n", "state_before": "p : \u2115\nR S : Type u\n\u03c3 : Type ?u.9710\nidx : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Fact (Nat.Prime p)\nf g : \u2115 \u2192 MvPolynomial (idx \u00d7 \u2115) \u2124\nh : \u2200 (n : \u2115), \u2191(bind\u2081 f) (wittPolynomial p \u2124 n) = \u2191(bind\u2081 g) (wittPolynomial p \u2124 n)\n\u22a2 f = g", "state_after": "case h\np : \u2115\nR S : Type u\n\u03c3 : Type ?u.9710\nidx : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Fact (Nat.Prime p)\nf g : \u2115 \u2192 MvPolynomial (idx \u00d7 \u2115) \u2124\nh : \u2200 (n : \u2115), \u2191(bind\u2081 f) (wittPolynomial p \u2124 n) = \u2191(bind\u2081 g) (wittPolynomial p \u2124 n)\nn : \u2115\n\u22a2 f n = g n"}, {"tactic": "apply MvPolynomial.map_injective (Int.castRingHom \u211a) Int.cast_injective", "state_before": "case h\np : \u2115\nR S : Type u\n\u03c3 : Type ?u.9710\nidx : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Fact (Nat.Prime p)\nf g : \u2115 \u2192 MvPolynomial (idx \u00d7 \u2115) \u2124\nh : \u2200 (n : \u2115), \u2191(bind\u2081 f) (wittPolynomial p \u2124 n) = \u2191(bind\u2081 g) (wittPolynomial p \u2124 n)\nn : \u2115\n\u22a2 f n = g n", "state_after": "case h.a\np : \u2115\nR S : Type u\n\u03c3 : Type ?u.9710\nidx : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Fact (Nat.Prime p)\nf g : \u2115 \u2192 MvPolynomial (idx \u00d7 \u2115) \u2124\nh : \u2200 (n : \u2115), \u2191(bind\u2081 f) (wittPolynomial p \u2124 n) = \u2191(bind\u2081 g) (wittPolynomial p \u2124 n)\nn : \u2115\n\u22a2 \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (f n) = \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (g n)"}, {"tactic": "rw [\u2190 Function.funext_iff] at h", "state_before": "case h.a\np : \u2115\nR S : Type u\n\u03c3 : Type ?u.9710\nidx : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Fact (Nat.Prime p)\nf g : \u2115 \u2192 MvPolynomial (idx \u00d7 \u2115) \u2124\nh : \u2200 (n : \u2115), \u2191(bind\u2081 f) (wittPolynomial p \u2124 n) = \u2191(bind\u2081 g) (wittPolynomial p \u2124 n)\nn : \u2115\n\u22a2 \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (f n) = \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (g n)", "state_after": "case h.a\np : \u2115\nR S : Type u\n\u03c3 : Type ?u.9710\nidx : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Fact (Nat.Prime p)\nf g : \u2115 \u2192 MvPolynomial (idx \u00d7 \u2115) \u2124\nh : (fun n => \u2191(bind\u2081 f) (wittPolynomial p \u2124 n)) = fun n => \u2191(bind\u2081 g) (wittPolynomial p \u2124 n)\nn : \u2115\n\u22a2 \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (f n) = \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (g n)"}, {"tactic": "replace h :=\n  congr_arg (fun fam => bind\u2081 (MvPolynomial.map (Int.castRingHom \u211a) \u2218 fam) (xInTermsOfW p \u211a n)) h", "state_before": "case h.a\np : \u2115\nR S : Type u\n\u03c3 : Type ?u.9710\nidx : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Fact (Nat.Prime p)\nf g : \u2115 \u2192 MvPolynomial (idx \u00d7 \u2115) \u2124\nh : (fun n => \u2191(bind\u2081 f) (wittPolynomial p \u2124 n)) = fun n => \u2191(bind\u2081 g) (wittPolynomial p \u2124 n)\nn : \u2115\n\u22a2 \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (f n) = \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (g n)", "state_after": "case h.a\np : \u2115\nR S : Type u\n\u03c3 : Type ?u.9710\nidx : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Fact (Nat.Prime p)\nf g : \u2115 \u2192 MvPolynomial (idx \u00d7 \u2115) \u2124\nn : \u2115\nh :\n  ((fun fam => \u2191(bind\u2081 (\u2191(MvPolynomial.map (Int.castRingHom \u211a)) \u2218 fam)) (xInTermsOfW p \u211a n)) fun n =>\n      \u2191(bind\u2081 f) (wittPolynomial p \u2124 n)) =\n    (fun fam => \u2191(bind\u2081 (\u2191(MvPolynomial.map (Int.castRingHom \u211a)) \u2218 fam)) (xInTermsOfW p \u211a n)) fun n =>\n      \u2191(bind\u2081 g) (wittPolynomial p \u2124 n)\n\u22a2 \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (f n) = \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (g n)"}, {"tactic": "simpa only [Function.comp, map_bind\u2081, map_wittPolynomial, \u2190 bind\u2081_bind\u2081,\n  bind\u2081_wittPolynomial_xInTermsOfW, bind\u2081_X_right] using h", "state_before": "case h.a\np : \u2115\nR S : Type u\n\u03c3 : Type ?u.9710\nidx : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Fact (Nat.Prime p)\nf g : \u2115 \u2192 MvPolynomial (idx \u00d7 \u2115) \u2124\nn : \u2115\nh :\n  ((fun fam => \u2191(bind\u2081 (\u2191(MvPolynomial.map (Int.castRingHom \u211a)) \u2218 fam)) (xInTermsOfW p \u211a n)) fun n =>\n      \u2191(bind\u2081 f) (wittPolynomial p \u2124 n)) =\n    (fun fam => \u2191(bind\u2081 (\u2191(MvPolynomial.map (Int.castRingHom \u211a)) \u2218 fam)) (xInTermsOfW p \u211a n)) fun n =>\n      \u2191(bind\u2081 g) (wittPolynomial p \u2124 n)\n\u22a2 \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (f n) = \u2191(MvPolynomial.map (Int.castRingHom \u211a)) (g n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Monotone.lean", "full_name": "Antitone.Ioi", "start": [80, 11], "end": [81, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Bases.lean", "full_name": "Filter.HasBasis.hasBasis_self_subset", "start": [448, 1], "end": [450, 52], "traced_tactics": [{"tactic": "simpa only [and_assoc] using h.restrict_subset hV", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.24375\n\u03b3 : Type ?u.24378\n\u03b9 : Sort ?u.24381\n\u03b9' : Sort ?u.24384\nl l' : Filter \u03b1\np\u271d : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\np : Set \u03b1 \u2192 Prop\nh : HasBasis l (fun s => s \u2208 l \u2227 p s) id\nV : Set \u03b1\nhV : V \u2208 l\n\u22a2 HasBasis l (fun s => s \u2208 l \u2227 p s \u2227 s \u2286 V) id", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Interval.lean", "full_name": "NonemptyInterval.fst_pow", "start": [260, 1], "end": [261, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/UpperLower/Basic.lean", "full_name": "UpperSet.sup_prod", "start": [1576, 1], "end": [1577, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/BilinearForm.lean", "full_name": "mem_pairSelfAdjointMatricesSubmodule'", "start": [507, 1], "end": [509, 51], "traced_tactics": [{"tactic": "simp only [mem_pairSelfAdjointMatricesSubmodule]", "state_before": "R : Type ?u.2690088\nM : Type ?u.2690091\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type ?u.2690127\nM\u2081 : Type ?u.2690130\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type ?u.2690739\nM\u2082 : Type ?u.2690742\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_1\nM\u2083 : Type ?u.2690932\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type ?u.2691520\nK : Type ?u.2691523\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_2\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\n\u22a2 A \u2208 pairSelfAdjointMatricesSubmodule J J\u2083 \u2194 IsAdjointPair J J\u2083 A A", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Group/Defs.lean", "full_name": "Right.self_lt_inv", "start": [509, 1], "end": [510, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/CauSeq.lean", "full_name": "CauSeq.coe_mul", "start": [298, 1], "end": [299, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.measure_toMeasurable_inter_of_sigmaFinite", "start": [3712, 1], "end": [3720, 45], "traced_tactics": [{"tactic": "have : t \u2286 \u22c3 n, spanningSets \u03bc n := by\n  rw [iUnion_spanningSets]\n  exact subset_univ _", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.754890\n\u03b3 : Type ?u.754893\n\u03b4 : Type ?u.754896\n\u03b9 : Type ?u.754899\nR : Type ?u.754902\nR' : Type ?u.754905\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc t \u2229 s) = \u2191\u2191\u03bc (t \u2229 s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.754890\n\u03b3 : Type ?u.754893\n\u03b4 : Type ?u.754896\n\u03b9 : Type ?u.754899\nR : Type ?u.754902\nR' : Type ?u.754905\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nthis : t \u2286 \u22c3 (n : \u2115), spanningSets \u03bc n\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc t \u2229 s) = \u2191\u2191\u03bc (t \u2229 s)"}, {"tactic": "refine measure_toMeasurable_inter_of_cover hs this fun n => ne_of_lt ?_", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.754890\n\u03b3 : Type ?u.754893\n\u03b4 : Type ?u.754896\n\u03b9 : Type ?u.754899\nR : Type ?u.754902\nR' : Type ?u.754905\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nthis : t \u2286 \u22c3 (n : \u2115), spanningSets \u03bc n\n\u22a2 \u2191\u2191\u03bc (toMeasurable \u03bc t \u2229 s) = \u2191\u2191\u03bc (t \u2229 s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.754890\n\u03b3 : Type ?u.754893\n\u03b4 : Type ?u.754896\n\u03b9 : Type ?u.754899\nR : Type ?u.754902\nR' : Type ?u.754905\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nthis : t \u2286 \u22c3 (n : \u2115), spanningSets \u03bc n\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (t \u2229 spanningSets \u03bc n) < \u22a4"}, {"tactic": "calc\n  \u03bc (t \u2229 spanningSets \u03bc n) \u2264 \u03bc (spanningSets \u03bc n) := measure_mono (inter_subset_right _ _)\n  _ < \u221e := measure_spanningSets_lt_top \u03bc n", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.754890\n\u03b3 : Type ?u.754893\n\u03b4 : Type ?u.754896\n\u03b9 : Type ?u.754899\nR : Type ?u.754902\nR' : Type ?u.754905\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\nthis : t \u2286 \u22c3 (n : \u2115), spanningSets \u03bc n\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (t \u2229 spanningSets \u03bc n) < \u22a4", "state_after": "no goals"}, {"tactic": "rw [iUnion_spanningSets]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.754890\n\u03b3 : Type ?u.754893\n\u03b4 : Type ?u.754896\n\u03b9 : Type ?u.754899\nR : Type ?u.754902\nR' : Type ?u.754905\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 t \u2286 \u22c3 (n : \u2115), spanningSets \u03bc n", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.754890\n\u03b3 : Type ?u.754893\n\u03b4 : Type ?u.754896\n\u03b9 : Type ?u.754899\nR : Type ?u.754902\nR' : Type ?u.754905\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 t \u2286 univ"}, {"tactic": "exact subset_univ _", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.754890\n\u03b3 : Type ?u.754893\n\u03b4 : Type ?u.754896\n\u03b9 : Type ?u.754899\nR : Type ?u.754902\nR' : Type ?u.754905\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\ns : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 t \u2286 univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean", "full_name": "HasFDerivWithinAt.rpow", "start": [414, 1], "end": [417, 97], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Duplicate.lean", "full_name": "List.not_duplicate_singleton", "start": [81, 1], "end": [81, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/TwoPointing.lean", "full_name": "TwoPointing.nonempty_two_pointing_iff", "start": [83, 1], "end": [84, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Basic.lean", "full_name": "LinearEquiv.map_eq_comap", "start": [1861, 1], "end": [1863, 57], "traced_tactics": [{"tactic": "simp [e.image_eq_preimage]", "state_before": "R : Type u_1\nR\u2081 : Type ?u.1656611\nR\u2082 : Type u_4\nR\u2083 : Type ?u.1656617\nR\u2084 : Type ?u.1656620\nS : Type ?u.1656623\nK : Type ?u.1656626\nK\u2082 : Type ?u.1656629\nM : Type u_2\nM' : Type ?u.1656635\nM\u2081 : Type ?u.1656638\nM\u2082 : Type u_3\nM\u2083 : Type ?u.1656644\nM\u2084 : Type ?u.1656647\nN : Type ?u.1656650\nN\u2082 : Type ?u.1656653\n\u03b9 : Type ?u.1656656\nV : Type ?u.1656659\nV\u2082 : Type ?u.1656662\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : Semiring R\u2082\ninst\u271d\u2075 : Semiring R\u2083\ninst\u271d\u2074 : Semiring R\u2084\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : AddCommMonoid M\u2084\nmodule_M : Module R M\nmodule_M\u2082 : Module R\u2082 M\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ne e' : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\np : Submodule R M\n\u22a2 \u2191(Submodule.map (\u2191e) p) = \u2191(Submodule.comap (\u2191(symm e)) p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.ext_of_generateFrom_of_iUnion", "start": [1954, 1], "end": [1961, 14], "traced_tactics": [{"tactic": "refine' ext_of_generateFrom_of_cover_subset hA hC _ (countable_range B) h1B _ h_eq", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.323239\n\u03b3 : Type ?u.323242\n\u03b4 : Type ?u.323245\n\u03b9 : Type ?u.323248\nR : Type ?u.323251\nR' : Type ?u.323254\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nC : Set (Set \u03b1)\nB : \u2115 \u2192 Set \u03b1\nhA : m0 = generateFrom C\nhC : IsPiSystem C\nh1B : (\u22c3 (i : \u2115), B i) = univ\nh2B : \u2200 (i : \u2115), B i \u2208 C\nh\u03bcB : \u2200 (i : \u2115), \u2191\u2191\u03bc (B i) \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\n\u22a2 \u03bc = \u03bd", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type ?u.323239\n\u03b3 : Type ?u.323242\n\u03b4 : Type ?u.323245\n\u03b9 : Type ?u.323248\nR : Type ?u.323251\nR' : Type ?u.323254\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nC : Set (Set \u03b1)\nB : \u2115 \u2192 Set \u03b1\nhA : m0 = generateFrom C\nhC : IsPiSystem C\nh1B : (\u22c3 (i : \u2115), B i) = univ\nh2B : \u2200 (i : \u2115), B i \u2208 C\nh\u03bcB : \u2200 (i : \u2115), \u2191\u2191\u03bc (B i) \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\n\u22a2 range B \u2286 C\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type ?u.323239\n\u03b3 : Type ?u.323242\n\u03b4 : Type ?u.323245\n\u03b9 : Type ?u.323248\nR : Type ?u.323251\nR' : Type ?u.323254\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nC : Set (Set \u03b1)\nB : \u2115 \u2192 Set \u03b1\nhA : m0 = generateFrom C\nhC : IsPiSystem C\nh1B : (\u22c3 (i : \u2115), B i) = univ\nh2B : \u2200 (i : \u2115), B i \u2208 C\nh\u03bcB : \u2200 (i : \u2115), \u2191\u2191\u03bc (B i) \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\n\u22a2 \u2200 (s : Set \u03b1), s \u2208 range B \u2192 \u2191\u2191\u03bc s \u2260 \u22a4"}, {"tactic": "rintro _ \u27e8i, rfl\u27e9", "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type ?u.323239\n\u03b3 : Type ?u.323242\n\u03b4 : Type ?u.323245\n\u03b9 : Type ?u.323248\nR : Type ?u.323251\nR' : Type ?u.323254\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nC : Set (Set \u03b1)\nB : \u2115 \u2192 Set \u03b1\nhA : m0 = generateFrom C\nhC : IsPiSystem C\nh1B : (\u22c3 (i : \u2115), B i) = univ\nh2B : \u2200 (i : \u2115), B i \u2208 C\nh\u03bcB : \u2200 (i : \u2115), \u2191\u2191\u03bc (B i) \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\n\u22a2 range B \u2286 C", "state_after": "case refine'_1.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.323239\n\u03b3 : Type ?u.323242\n\u03b4 : Type ?u.323245\n\u03b9 : Type ?u.323248\nR : Type ?u.323251\nR' : Type ?u.323254\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nC : Set (Set \u03b1)\nB : \u2115 \u2192 Set \u03b1\nhA : m0 = generateFrom C\nhC : IsPiSystem C\nh1B : (\u22c3 (i : \u2115), B i) = univ\nh2B : \u2200 (i : \u2115), B i \u2208 C\nh\u03bcB : \u2200 (i : \u2115), \u2191\u2191\u03bc (B i) \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\ni : \u2115\n\u22a2 B i \u2208 C"}, {"tactic": "apply h2B", "state_before": "case refine'_1.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.323239\n\u03b3 : Type ?u.323242\n\u03b4 : Type ?u.323245\n\u03b9 : Type ?u.323248\nR : Type ?u.323251\nR' : Type ?u.323254\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nC : Set (Set \u03b1)\nB : \u2115 \u2192 Set \u03b1\nhA : m0 = generateFrom C\nhC : IsPiSystem C\nh1B : (\u22c3 (i : \u2115), B i) = univ\nh2B : \u2200 (i : \u2115), B i \u2208 C\nh\u03bcB : \u2200 (i : \u2115), \u2191\u2191\u03bc (B i) \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\ni : \u2115\n\u22a2 B i \u2208 C", "state_after": "no goals"}, {"tactic": "rintro _ \u27e8i, rfl\u27e9", "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type ?u.323239\n\u03b3 : Type ?u.323242\n\u03b4 : Type ?u.323245\n\u03b9 : Type ?u.323248\nR : Type ?u.323251\nR' : Type ?u.323254\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nC : Set (Set \u03b1)\nB : \u2115 \u2192 Set \u03b1\nhA : m0 = generateFrom C\nhC : IsPiSystem C\nh1B : (\u22c3 (i : \u2115), B i) = univ\nh2B : \u2200 (i : \u2115), B i \u2208 C\nh\u03bcB : \u2200 (i : \u2115), \u2191\u2191\u03bc (B i) \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\n\u22a2 \u2200 (s : Set \u03b1), s \u2208 range B \u2192 \u2191\u2191\u03bc s \u2260 \u22a4", "state_after": "case refine'_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.323239\n\u03b3 : Type ?u.323242\n\u03b4 : Type ?u.323245\n\u03b9 : Type ?u.323248\nR : Type ?u.323251\nR' : Type ?u.323254\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nC : Set (Set \u03b1)\nB : \u2115 \u2192 Set \u03b1\nhA : m0 = generateFrom C\nhC : IsPiSystem C\nh1B : (\u22c3 (i : \u2115), B i) = univ\nh2B : \u2200 (i : \u2115), B i \u2208 C\nh\u03bcB : \u2200 (i : \u2115), \u2191\u2191\u03bc (B i) \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\ni : \u2115\n\u22a2 \u2191\u2191\u03bc (B i) \u2260 \u22a4"}, {"tactic": "apply h\u03bcB", "state_before": "case refine'_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.323239\n\u03b3 : Type ?u.323242\n\u03b4 : Type ?u.323245\n\u03b9 : Type ?u.323248\nR : Type ?u.323251\nR' : Type ?u.323254\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nC : Set (Set \u03b1)\nB : \u2115 \u2192 Set \u03b1\nhA : m0 = generateFrom C\nhC : IsPiSystem C\nh1B : (\u22c3 (i : \u2115), B i) = univ\nh2B : \u2200 (i : \u2115), B i \u2208 C\nh\u03bcB : \u2200 (i : \u2115), \u2191\u2191\u03bc (B i) \u2260 \u22a4\nh_eq : \u2200 (s : Set \u03b1), s \u2208 C \u2192 \u2191\u2191\u03bc s = \u2191\u2191\u03bd s\ni : \u2115\n\u22a2 \u2191\u2191\u03bc (B i) \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean", "full_name": "CategoryTheory.Limits.PreservesPullback.iso_inv_fst", "start": [133, 1], "end": [135, 48], "traced_tactics": [{"tactic": "simp [PreservesPullback.iso, Iso.inv_comp_eq]", "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category C\nD : Type u\u2082\ninst\u271d\u00b3 : Category D\nG : C \u2964 D\nW X Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\nh : W \u27f6 X\nk : W \u27f6 Y\ncomm : h \u226b f = k \u226b g\ninst\u271d\u00b2 : PreservesLimit (cospan f g) G\ninst\u271d\u00b9 : HasPullback f g\ninst\u271d : HasPullback (G.map f) (G.map g)\n\u22a2 (iso G f g).inv \u226b G.map pullback.fst = pullback.fst", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Types.lean", "full_name": "CategoryTheory.Limits.Types.type_equalizer_iff_unique", "start": [416, 1], "end": [419, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Index.lean", "full_name": "Subgroup.mul_self_mem_of_index_two", "start": [210, 1], "end": [211, 34], "traced_tactics": [{"tactic": "rw [mul_mem_iff_of_index_two h]", "state_before": "G : Type u_1\ninst\u271d : Group G\nH K L : Subgroup G\nh : index H = 2\na : G\n\u22a2 a * a \u2208 H", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Interval.lean", "full_name": "Interval.mul_eq_one_iff", "start": [583, 11], "end": [590, 42], "traced_tactics": [{"tactic": "cases s", "state_before": "\u03b9 : Type ?u.351009\n\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\ns t : Interval \u03b1\n\u22a2 s * t = 1 \u2194 \u2203 a b, s = pure a \u2227 t = pure b \u2227 a * b = 1", "state_after": "case none\n\u03b9 : Type ?u.351009\n\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\nt : Interval \u03b1\n\u22a2 none * t = 1 \u2194 \u2203 a b, none = pure a \u2227 t = pure b \u2227 a * b = 1\n\ncase some\n\u03b9 : Type ?u.351009\n\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\nt : Interval \u03b1\nval\u271d : NonemptyInterval \u03b1\n\u22a2 some val\u271d * t = 1 \u2194 \u2203 a b, some val\u271d = pure a \u2227 t = pure b \u2227 a * b = 1"}, {"tactic": "cases t", "state_before": "case some\n\u03b9 : Type ?u.351009\n\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\nt : Interval \u03b1\nval\u271d : NonemptyInterval \u03b1\n\u22a2 some val\u271d * t = 1 \u2194 \u2203 a b, some val\u271d = pure a \u2227 t = pure b \u2227 a * b = 1", "state_after": "case some.none\n\u03b9 : Type ?u.351009\n\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\nval\u271d : NonemptyInterval \u03b1\n\u22a2 some val\u271d * none = 1 \u2194 \u2203 a b, some val\u271d = pure a \u2227 none = pure b \u2227 a * b = 1\n\ncase some.some\n\u03b9 : Type ?u.351009\n\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\nval\u271d\u00b9 val\u271d : NonemptyInterval \u03b1\n\u22a2 some val\u271d\u00b9 * some val\u271d = 1 \u2194 \u2203 a b, some val\u271d\u00b9 = pure a \u2227 some val\u271d = pure b \u2227 a * b = 1"}, {"tactic": "simp [WithBot.none_eq_bot]", "state_before": "case none\n\u03b9 : Type ?u.351009\n\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\nt : Interval \u03b1\n\u22a2 none * t = 1 \u2194 \u2203 a b, none = pure a \u2227 t = pure b \u2227 a * b = 1", "state_after": "no goals"}, {"tactic": "simp [WithBot.none_eq_bot]", "state_before": "case some.none\n\u03b9 : Type ?u.351009\n\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\nval\u271d : NonemptyInterval \u03b1\n\u22a2 some val\u271d * none = 1 \u2194 \u2203 a b, some val\u271d = pure a \u2227 none = pure b \u2227 a * b = 1", "state_after": "no goals"}, {"tactic": "simp_rw [WithBot.some_eq_coe, \u2190 NonemptyInterval.coe_mul_interval,\n  \u2190 NonemptyInterval.coe_one_interval, WithBot.coe_inj, NonemptyInterval.coe_eq_pure]", "state_before": "case some.some\n\u03b9 : Type ?u.351009\n\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\nval\u271d\u00b9 val\u271d : NonemptyInterval \u03b1\n\u22a2 some val\u271d\u00b9 * some val\u271d = 1 \u2194 \u2203 a b, some val\u271d\u00b9 = pure a \u2227 some val\u271d = pure b \u2227 a * b = 1", "state_after": "case some.some\n\u03b9 : Type ?u.351009\n\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\nval\u271d\u00b9 val\u271d : NonemptyInterval \u03b1\n\u22a2 val\u271d\u00b9 * val\u271d = 1 \u2194 \u2203 a b, val\u271d\u00b9 = NonemptyInterval.pure a \u2227 val\u271d = NonemptyInterval.pure b \u2227 a * b = 1"}, {"tactic": "exact NonemptyInterval.mul_eq_one_iff", "state_before": "case some.some\n\u03b9 : Type ?u.351009\n\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\nval\u271d\u00b9 val\u271d : NonemptyInterval \u03b1\n\u22a2 val\u271d\u00b9 * val\u271d = 1 \u2194 \u2203 a b, val\u271d\u00b9 = NonemptyInterval.pure a \u2227 val\u271d = NonemptyInterval.pure b \u2227 a * b = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "full_name": "bit1_mul", "start": [570, 1], "end": [572, 34], "traced_tactics": [{"tactic": "dsimp [bit1]", "state_before": "\u03b1 : Type ?u.270117\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonAssocRing R\nn r : R\n\u22a2 bit1 n * r = 2 \u2022 (n * r) + r", "state_after": "\u03b1 : Type ?u.270117\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonAssocRing R\nn r : R\n\u22a2 (bit0 n + 1) * r = 2 \u2022 (n * r) + r"}, {"tactic": "rw [add_mul, bit0_mul, one_mul]", "state_before": "\u03b1 : Type ?u.270117\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : NonAssocRing R\nn r : R\n\u22a2 (bit0 n + 1) * r = 2 \u2022 (n * r) + r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Isometry.lean", "full_name": "IsometryEquiv.range_eq_univ", "start": [451, 1], "end": [452, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.sup_eq_zero_iff", "start": [1293, 1], "end": [1299, 19], "traced_tactics": [{"tactic": "refine'\n  \u27e8fun h i => _, fun h =>\n    le_antisymm (sup_le fun i => Ordinal.le_zero.2 (h i)) (Ordinal.zero_le _)\u27e9", "state_before": "\u03b1 : Type ?u.284060\n\u03b2 : Type ?u.284063\n\u03b3 : Type ?u.284066\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b9 : Type u\nf : \u03b9 \u2192 Ordinal\n\u22a2 sup f = 0 \u2194 \u2200 (i : \u03b9), f i = 0", "state_after": "\u03b1 : Type ?u.284060\n\u03b2 : Type ?u.284063\n\u03b3 : Type ?u.284066\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b9 : Type u\nf : \u03b9 \u2192 Ordinal\nh : sup f = 0\ni : \u03b9\n\u22a2 f i = 0"}, {"tactic": "rw [\u2190 Ordinal.le_zero, \u2190 h]", "state_before": "\u03b1 : Type ?u.284060\n\u03b2 : Type ?u.284063\n\u03b3 : Type ?u.284066\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b9 : Type u\nf : \u03b9 \u2192 Ordinal\nh : sup f = 0\ni : \u03b9\n\u22a2 f i = 0", "state_after": "\u03b1 : Type ?u.284060\n\u03b2 : Type ?u.284063\n\u03b3 : Type ?u.284066\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b9 : Type u\nf : \u03b9 \u2192 Ordinal\nh : sup f = 0\ni : \u03b9\n\u22a2 f i \u2264 sup f"}, {"tactic": "exact le_sup f i", "state_before": "\u03b1 : Type ?u.284060\n\u03b2 : Type ?u.284063\n\u03b3 : Type ?u.284066\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b9 : Type u\nf : \u03b9 \u2192 Ordinal\nh : sup f = 0\ni : \u03b9\n\u22a2 f i \u2264 sup f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "full_name": "measurableSet_eq_fun_of_countable", "start": [399, 1], "end": [408, 41], "traced_tactics": [{"tactic": "have : { x | f x = g x } = \u22c3 j, { x | f x = j } \u2229 { x | g x = j } := by\n  ext1 x\n  simp only [Set.mem_setOf_eq, Set.mem_iUnion, Set.mem_inter_iff, exists_eq_right']", "state_before": "G : Type ?u.2100312\n\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : MeasureTheory.Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : Countable E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\n\u22a2 MeasurableSet {x | f x = g x}", "state_after": "G : Type ?u.2100312\n\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : MeasureTheory.Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : Countable E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\nthis : {x | f x = g x} = \u22c3 (j : E), {x | f x = j} \u2229 {x | g x = j}\n\u22a2 MeasurableSet {x | f x = g x}"}, {"tactic": "rw [this]", "state_before": "G : Type ?u.2100312\n\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : MeasureTheory.Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : Countable E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\nthis : {x | f x = g x} = \u22c3 (j : E), {x | f x = j} \u2229 {x | g x = j}\n\u22a2 MeasurableSet {x | f x = g x}", "state_after": "G : Type ?u.2100312\n\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : MeasureTheory.Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : Countable E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\nthis : {x | f x = g x} = \u22c3 (j : E), {x | f x = j} \u2229 {x | g x = j}\n\u22a2 MeasurableSet (\u22c3 (j : E), {x | f x = j} \u2229 {x | g x = j})"}, {"tactic": "refine' MeasurableSet.iUnion fun j => MeasurableSet.inter _ _", "state_before": "G : Type ?u.2100312\n\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : MeasureTheory.Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : Countable E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\nthis : {x | f x = g x} = \u22c3 (j : E), {x | f x = j} \u2229 {x | g x = j}\n\u22a2 MeasurableSet (\u22c3 (j : E), {x | f x = j} \u2229 {x | g x = j})", "state_after": "case refine'_1\nG : Type ?u.2100312\n\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : MeasureTheory.Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : Countable E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\nthis : {x | f x = g x} = \u22c3 (j : E), {x | f x = j} \u2229 {x | g x = j}\nj : E\n\u22a2 MeasurableSet {x | f x = j}\n\ncase refine'_2\nG : Type ?u.2100312\n\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : MeasureTheory.Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : Countable E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\nthis : {x | f x = g x} = \u22c3 (j : E), {x | f x = j} \u2229 {x | g x = j}\nj : E\n\u22a2 MeasurableSet {x | g x = j}"}, {"tactic": "ext1 x", "state_before": "G : Type ?u.2100312\n\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : MeasureTheory.Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : Countable E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\n\u22a2 {x | f x = g x} = \u22c3 (j : E), {x | f x = j} \u2229 {x | g x = j}", "state_after": "case h\nG : Type ?u.2100312\n\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : MeasureTheory.Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : Countable E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\nx : \u03b1\n\u22a2 x \u2208 {x | f x = g x} \u2194 x \u2208 \u22c3 (j : E), {x | f x = j} \u2229 {x | g x = j}"}, {"tactic": "simp only [Set.mem_setOf_eq, Set.mem_iUnion, Set.mem_inter_iff, exists_eq_right']", "state_before": "case h\nG : Type ?u.2100312\n\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : MeasureTheory.Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : Countable E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\nx : \u03b1\n\u22a2 x \u2208 {x | f x = g x} \u2194 x \u2208 \u22c3 (j : E), {x | f x = j} \u2229 {x | g x = j}", "state_after": "no goals"}, {"tactic": "exact hf (measurableSet_singleton j)", "state_before": "case refine'_1\nG : Type ?u.2100312\n\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : MeasureTheory.Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : Countable E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\nthis : {x | f x = g x} = \u22c3 (j : E), {x | f x = j} \u2229 {x | g x = j}\nj : E\n\u22a2 MeasurableSet {x | f x = j}", "state_after": "no goals"}, {"tactic": "exact hg (measurableSet_singleton j)", "state_before": "case refine'_2\nG : Type ?u.2100312\n\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace G\ninst\u271d\u00b3 : Div G\nm\u271d : MeasurableSpace \u03b1\nf\u271d g\u271d : \u03b1 \u2192 G\n\u03bc : MeasureTheory.Measure \u03b1\nm : MeasurableSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : MeasurableSingletonClass E\ninst\u271d : Countable E\nf g : \u03b1 \u2192 E\nhf : Measurable f\nhg : Measurable g\nthis : {x | f x = g x} = \u22c3 (j : E), {x | f x = j} \u2229 {x | g x = j}\nj : E\n\u22a2 MeasurableSet {x | g x = j}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.image_subset_image_iff", "start": [561, 1], "end": [564, 24], "traced_tactics": [{"tactic": "refine' Iff.symm <| (Iff.intro (image_subset f)) fun h => _", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.55016\n\u03b9 : Sort ?u.55019\n\u03b9' : Sort ?u.55022\nf\u271d : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\n\u22a2 f '' s \u2286 f '' t \u2194 s \u2286 t", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.55016\n\u03b9 : Sort ?u.55019\n\u03b9' : Sort ?u.55022\nf\u271d : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nh : f '' s \u2286 f '' t\n\u22a2 s \u2286 t"}, {"tactic": "rw [\u2190 preimage_image_eq s hf, \u2190 preimage_image_eq t hf]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.55016\n\u03b9 : Sort ?u.55019\n\u03b9' : Sort ?u.55022\nf\u271d : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nh : f '' s \u2286 f '' t\n\u22a2 s \u2286 t", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.55016\n\u03b9 : Sort ?u.55019\n\u03b9' : Sort ?u.55022\nf\u271d : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nh : f '' s \u2286 f '' t\n\u22a2 f \u207b\u00b9' (f '' s) \u2286 f \u207b\u00b9' (f '' t)"}, {"tactic": "exact preimage_mono h", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.55016\n\u03b9 : Sort ?u.55019\n\u03b9' : Sort ?u.55022\nf\u271d : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nh : f '' s \u2286 f '' t\n\u22a2 f \u207b\u00b9' (f '' s) \u2286 f \u207b\u00b9' (f '' t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SetFamily/Intersecting.lean", "full_name": "Set.Intersecting.not_compl_mem", "start": [150, 1], "end": [151, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Extr.lean", "full_name": "IsExtrOn.elim", "start": [130, 1], "end": [131, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/NaturalOps.lean", "full_name": "NatOrdinal.toOrdinal_zero", "start": [104, 1], "end": [105, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/ENNReal.lean", "full_name": "ENNReal.toNNReal_coe", "start": [173, 1], "end": [173, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Basic.lean", "full_name": "LinearOrder.ext", "start": [662, 1], "end": [665, 14], "traced_tactics": [{"tactic": "ext x y", "state_before": "\u03b9 : Type ?u.20121\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type ?u.20132\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nA B : LinearOrder \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\n\u22a2 A = B", "state_after": "case a.a.a.le.h.h.a\n\u03b9 : Type ?u.20121\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type ?u.20132\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nA B : LinearOrder \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\nx y : \u03b1\n\u22a2 x \u2264 y \u2194 x \u2264 y"}, {"tactic": "exact H x y", "state_before": "case a.a.a.le.h.h.a\n\u03b9 : Type ?u.20121\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type ?u.20132\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nA B : LinearOrder \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\nx y : \u03b1\n\u22a2 x \u2264 y \u2194 x \u2264 y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Noetherian.lean", "full_name": "Module.Finite.of_injective", "start": [173, 1], "end": [175, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Maps.lean", "full_name": "IsOpenMap.of_inverse", "start": [396, 1], "end": [398, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean", "full_name": "Matrix.toLinearMap\u2082'_toMatrix'", "start": [251, 1], "end": [253, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "full_name": "Polynomial.unique_int_coeff_of_cycl", "start": [240, 1], "end": [246, 16], "traced_tactics": [{"tactic": "obtain \u27e8P, hP\u27e9 := int_coeff_of_cyclotomic' h", "state_before": "K\u271d : Type ?u.290364\ninst\u271d\u00b3 : Field K\u271d\nK : Type u_1\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : IsDomain K\ninst\u271d : CharZero K\n\u03b6 : K\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 \u2203! P, map (Int.castRingHom K) P = cyclotomic' (\u2191n) K", "state_after": "case intro\nK\u271d : Type ?u.290364\ninst\u271d\u00b3 : Field K\u271d\nK : Type u_1\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : IsDomain K\ninst\u271d : CharZero K\n\u03b6 : K\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nP : \u2124[X]\nhP : map (Int.castRingHom K) P = cyclotomic' (\u2191n) K \u2227 degree P = degree (cyclotomic' (\u2191n) K) \u2227 Monic P\n\u22a2 \u2203! P, map (Int.castRingHom K) P = cyclotomic' (\u2191n) K"}, {"tactic": "refine' \u27e8P, hP.1, fun Q hQ => _\u27e9", "state_before": "case intro\nK\u271d : Type ?u.290364\ninst\u271d\u00b3 : Field K\u271d\nK : Type u_1\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : IsDomain K\ninst\u271d : CharZero K\n\u03b6 : K\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nP : \u2124[X]\nhP : map (Int.castRingHom K) P = cyclotomic' (\u2191n) K \u2227 degree P = degree (cyclotomic' (\u2191n) K) \u2227 Monic P\n\u22a2 \u2203! P, map (Int.castRingHom K) P = cyclotomic' (\u2191n) K", "state_after": "case intro\nK\u271d : Type ?u.290364\ninst\u271d\u00b3 : Field K\u271d\nK : Type u_1\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : IsDomain K\ninst\u271d : CharZero K\n\u03b6 : K\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nP : \u2124[X]\nhP : map (Int.castRingHom K) P = cyclotomic' (\u2191n) K \u2227 degree P = degree (cyclotomic' (\u2191n) K) \u2227 Monic P\nQ : \u2124[X]\nhQ : (fun P => map (Int.castRingHom K) P = cyclotomic' (\u2191n) K) Q\n\u22a2 Q = P"}, {"tactic": "apply map_injective (Int.castRingHom K) Int.cast_injective", "state_before": "case intro\nK\u271d : Type ?u.290364\ninst\u271d\u00b3 : Field K\u271d\nK : Type u_1\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : IsDomain K\ninst\u271d : CharZero K\n\u03b6 : K\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\nP : \u2124[X]\nhP : map (Int.castRingHom K) P = cyclotomic' (\u2191n) K \u2227 degree P = degree (cyclotomic' (\u2191n) K) 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(\u2191n) K \u2227 degree P = degree (cyclotomic' (\u2191n) K) \u2227 Monic P\nQ : \u2124[X]\nhQ : (fun P => map (Int.castRingHom K) P = cyclotomic' (\u2191n) K) Q\n\u22a2 map (Int.castRingHom K) Q = map (Int.castRingHom K) P", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/FiberBundle/Trivialization.lean", "full_name": "Trivialization.mem_target", "start": [386, 1], "end": [387, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Clique.lean", "full_name": "SimpleGraph.not_cliqueFree_iff", "start": [196, 1], "end": [197, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Pi.lean", "full_name": "Set.disjoint_pi_univ_Ioc_update_left_right", "start": [115, 1], "end": [121, 65], "traced_tactics": [{"tactic": "rw [disjoint_left]", "state_before": "\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx\u271d y\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nx y : (i : \u03b9) \u2192 \u03b1 i\ni\u2080 : \u03b9\nm : \u03b1 i\u2080\n\u22a2 Disjoint (pi univ fun i => Ioc (x i) (update y i\u2080 m i)) (pi univ fun i => Ioc (update x i\u2080 m i) (y i))", "state_after": "\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx\u271d y\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nx y : (i : \u03b9) \u2192 \u03b1 i\ni\u2080 : \u03b9\nm : \u03b1 i\u2080\n\u22a2 \u2200 \u2983a : (i : \u03b9) \u2192 \u03b1 i\u2984,\n    (a \u2208 pi univ fun i => Ioc (x i) (update y i\u2080 m i)) \u2192 \u00aca \u2208 pi univ fun i => Ioc (update x i\u2080 m i) (y i)"}, {"tactic": "rintro z h\u2081 h\u2082", "state_before": "\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx\u271d y\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nx y : (i : \u03b9) \u2192 \u03b1 i\ni\u2080 : \u03b9\nm : \u03b1 i\u2080\n\u22a2 \u2200 \u2983a : (i : \u03b9) \u2192 \u03b1 i\u2984,\n    (a \u2208 pi univ fun i => Ioc (x i) (update y i\u2080 m i)) \u2192 \u00aca \u2208 pi univ fun i => Ioc (update x i\u2080 m i) (y i)", "state_after": "\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx\u271d y\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nx y : (i : \u03b9) \u2192 \u03b1 i\ni\u2080 : \u03b9\nm : \u03b1 i\u2080\nz : (i : \u03b9) \u2192 \u03b1 i\nh\u2081 : z \u2208 pi univ fun i => Ioc (x i) (update y i\u2080 m i)\nh\u2082 : z \u2208 pi univ fun i => Ioc (update x i\u2080 m i) (y i)\n\u22a2 False"}, {"tactic": "refine' (h\u2081 i\u2080 (mem_univ _)).2.not_lt _", "state_before": "\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx\u271d y\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nx y : (i : \u03b9) \u2192 \u03b1 i\ni\u2080 : \u03b9\nm : \u03b1 i\u2080\nz : (i : \u03b9) \u2192 \u03b1 i\nh\u2081 : z \u2208 pi univ fun i => Ioc (x i) (update y i\u2080 m i)\nh\u2082 : z \u2208 pi univ fun i => Ioc (update x i\u2080 m i) (y i)\n\u22a2 False", "state_after": "\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx\u271d y\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nx y : (i : \u03b9) \u2192 \u03b1 i\ni\u2080 : \u03b9\nm : \u03b1 i\u2080\nz : (i : \u03b9) \u2192 \u03b1 i\nh\u2081 : z \u2208 pi univ fun i => Ioc (x i) (update y i\u2080 m i)\nh\u2082 : z \u2208 pi univ fun i => Ioc (update x i\u2080 m i) (y i)\n\u22a2 update y i\u2080 m i\u2080 < z i\u2080"}, {"tactic": "simpa only [Function.update_same] using (h\u2082 i\u2080 (mem_univ _)).1", "state_before": "\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx\u271d y\u271d : (i : \u03b9) \u2192 \u03b1 i\ninst\u271d : DecidableEq \u03b9\nx y : (i : \u03b9) \u2192 \u03b1 i\ni\u2080 : \u03b9\nm : \u03b1 i\u2080\nz : (i : \u03b9) \u2192 \u03b1 i\nh\u2081 : z \u2208 pi univ fun i => Ioc (x i) (update y i\u2080 m i)\nh\u2082 : z \u2208 pi univ fun i => Ioc (update x i\u2080 m i) (y i)\n\u22a2 update y i\u2080 m i\u2080 < z i\u2080", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Basic.lean", "full_name": "Metric.tendsto_nhdsWithin_nhdsWithin", "start": [1033, 1], "end": [1037, 76], "traced_tactics": [{"tactic": "simp only [inter_comm _ s, inter_comm _ t, mem_inter_iff, and_imp]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.98106\n\u03b9 : Type ?u.98109\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 (\u2200 (ib : \u211d), 0 < ib \u2192 \u2203 ia, 0 < ia \u2227 \u2200 (x : \u03b1), x \u2208 ball a ia \u2229 s \u2192 f x \u2208 ball b ib \u2229 t) \u2194\n    \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 {x : \u03b1}, x \u2208 s \u2192 dist x a < \u03b4 \u2192 f x \u2208 t \u2227 dist (f x) b < \u03b5", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.98106\n\u03b9 : Type ?u.98109\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 (\u2200 (ib : \u211d), 0 < ib \u2192 \u2203 ia, 0 < ia \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 ball a ia \u2192 f x \u2208 t \u2227 f x \u2208 ball b ib) \u2194\n    \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 {x : \u03b1}, x \u2208 s \u2192 dist x a < \u03b4 \u2192 f x \u2208 t \u2227 dist (f x) b < \u03b5"}, {"tactic": "rfl", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.98106\n\u03b9 : Type ?u.98109\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 (\u2200 (ib : \u211d), 0 < ib \u2192 \u2203 ia, 0 < ia \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x \u2208 ball a ia \u2192 f x \u2208 t \u2227 f x \u2208 ball b ib) \u2194\n    \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 {x : \u03b1}, x \u2208 s \u2192 dist x a < \u03b4 \u2192 f x \u2208 t \u2227 dist (f x) b < \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/BinomialHeap.lean", "full_name": "Std.BinomialHeapImp.Heap.WellFormed.rankGT", "start": [357, 1], "end": [360, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Ring/Divisibility.lean", "full_name": "dvd_iff_dvd_of_dvd_sub", "start": [118, 1], "end": [119, 45], "traced_tactics": [{"tactic": "rw [\u2190 sub_add_cancel b c, dvd_add_right h]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.11507\ninst\u271d : NonUnitalRing \u03b1\na b c : \u03b1\nh : a \u2223 b - c\n\u22a2 a \u2223 b \u2194 a \u2223 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/LocallyFinite.lean", "full_name": "Multiset.mem_Ioc", "start": [562, 1], "end": [563, 45], "traced_tactics": [{"tactic": "rw [Ioc, \u2190 Finset.mem_def, Finset.mem_Ioc]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.32714\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b x : \u03b1\n\u22a2 x \u2208 Ioc a b \u2194 a < x \u2227 x \u2264 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "full_name": "AffineSubspace.Parallel.symm", "start": [1722, 1], "end": [1726, 35], "traced_tactics": [{"tactic": "rcases h with \u27e8v, rfl\u27e9", "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : AffineSubspace k P\nh : s\u2081 \u2225 s\u2082\n\u22a2 s\u2082 \u2225 s\u2081", "state_after": "case intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 : AffineSubspace k P\nv : V\n\u22a2 map (\u2191(constVAdd k P v)) s\u2081 \u2225 s\u2081"}, {"tactic": "refine' \u27e8-v, _\u27e9", "state_before": "case intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 : AffineSubspace k P\nv : V\n\u22a2 map (\u2191(constVAdd k P v)) s\u2081 \u2225 s\u2081", "state_after": "case intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 : AffineSubspace k P\nv : V\n\u22a2 s\u2081 = map (\u2191(constVAdd k P (-v))) (map (\u2191(constVAdd k P v)) s\u2081)"}, {"tactic": "rw [map_map, \u2190 coe_trans_to_affineMap, \u2190 constVAdd_add, neg_add_self, constVAdd_zero,\n  coe_refl_to_affineMap, map_id]", "state_before": "case intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 : AffineSubspace k P\nv : V\n\u22a2 s\u2081 = map (\u2191(constVAdd k P (-v))) (map (\u2191(constVAdd k P v)) s\u2081)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Abelian/NonPreadditive.lean", "full_name": "CategoryTheory.NonPreadditiveAbelian.neg_add_self", "start": [406, 1], "end": [406, 90], "traced_tactics": [{"tactic": "rw [add_comm, add_neg_self]", "state_before": "C : Type u\ninst\u271d\u00b9 : Category C\ninst\u271d : NonPreadditiveAbelian C\nX Y : C\na : X \u27f6 Y\n\u22a2 -a + a = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "full_name": "EMetric.hausdorffEdist_le_ediam", "start": [346, 1], "end": [354, 95], "traced_tactics": [{"tactic": "rcases hs with \u27e8x, xs\u27e9", "state_before": "\u03b9 : Sort ?u.46050\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nhs : Set.Nonempty s\nht : Set.Nonempty t\n\u22a2 hausdorffEdist s t \u2264 diam (s \u222a t)", "state_after": "case intro\n\u03b9 : Sort ?u.46050\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nht : Set.Nonempty t\nx : \u03b1\nxs : x \u2208 s\n\u22a2 hausdorffEdist s t \u2264 diam (s \u222a t)"}, {"tactic": "rcases ht with \u27e8y, yt\u27e9", "state_before": "case intro\n\u03b9 : Sort ?u.46050\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nht : Set.Nonempty t\nx : \u03b1\nxs : x \u2208 s\n\u22a2 hausdorffEdist s t \u2264 diam (s \u222a t)", "state_after": "case intro.intro\n\u03b9 : Sort ?u.46050\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y\u271d : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\n\u22a2 hausdorffEdist s t \u2264 diam (s \u222a t)"}, {"tactic": "refine' hausdorffEdist_le_of_mem_edist _ _", "state_before": "case intro.intro\n\u03b9 : Sort ?u.46050\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y\u271d : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\n\u22a2 hausdorffEdist s t \u2264 diam (s \u222a t)", "state_after": "case intro.intro.refine'_1\n\u03b9 : Sort ?u.46050\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y\u271d : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 edist x y \u2264 diam (s \u222a t)\n\ncase intro.intro.refine'_2\n\u03b9 : Sort ?u.46050\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y\u271d : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\n\u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 edist x y \u2264 diam (s \u222a t)"}, {"tactic": "intro z hz", "state_before": "case intro.intro.refine'_1\n\u03b9 : Sort ?u.46050\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y\u271d : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 edist x y \u2264 diam (s \u222a t)", "state_after": "case intro.intro.refine'_1\n\u03b9 : Sort ?u.46050\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y\u271d : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\nz : \u03b1\nhz : z \u2208 s\n\u22a2 \u2203 y, y \u2208 t \u2227 edist z y \u2264 diam (s \u222a t)"}, {"tactic": "exact \u27e8y, yt, edist_le_diam_of_mem (subset_union_left _ _ hz) (subset_union_right _ _ yt)\u27e9", "state_before": "case intro.intro.refine'_1\n\u03b9 : Sort ?u.46050\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y\u271d : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\nz : \u03b1\nhz : z \u2208 s\n\u22a2 \u2203 y, y \u2208 t \u2227 edist z y \u2264 diam (s \u222a t)", "state_after": "no goals"}, {"tactic": "intro z hz", "state_before": "case intro.intro.refine'_2\n\u03b9 : Sort ?u.46050\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y\u271d : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\n\u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 edist x y \u2264 diam (s \u222a t)", "state_after": "case intro.intro.refine'_2\n\u03b9 : Sort ?u.46050\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y\u271d : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\nz : \u03b1\nhz : z \u2208 t\n\u22a2 \u2203 y, y \u2208 s \u2227 edist z y \u2264 diam (s \u222a t)"}, {"tactic": "exact \u27e8x, xs, edist_le_diam_of_mem (subset_union_right _ _ hz) (subset_union_left _ _ xs)\u27e9", "state_before": "case intro.intro.refine'_2\n\u03b9 : Sort ?u.46050\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx\u271d y\u271d : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\nz : \u03b1\nhz : z \u2208 t\n\u22a2 \u2203 y, y \u2208 s \u2227 edist z y \u2264 diam (s \u222a t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/BoundedOrder.lean", "full_name": "StrictMono.maximal_preimage_top", "start": [216, 1], "end": [222, 6], "traced_tactics": [{"tactic": "rw [h_top]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type ?u.17940\n\u03b4 : Type ?u.17943\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : OrderTop \u03b2\nf : \u03b1 \u2192 \u03b2\nH : StrictMono f\na : \u03b1\nh_top : f a = \u22a4\nx : \u03b1\np : \u03b2\n\u22a2 p \u2264 f a", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type ?u.17940\n\u03b4 : Type ?u.17943\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : OrderTop \u03b2\nf : \u03b1 \u2192 \u03b2\nH : StrictMono f\na : \u03b1\nh_top : f a = \u22a4\nx : \u03b1\np : \u03b2\n\u22a2 p \u2264 \u22a4"}, {"tactic": "exact le_top", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type ?u.17940\n\u03b4 : Type ?u.17943\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : OrderTop \u03b2\nf : \u03b1 \u2192 \u03b2\nH : StrictMono f\na : \u03b1\nh_top : f a = \u22a4\nx : \u03b1\np : \u03b2\n\u22a2 p \u2264 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.sInter_comap_sets", "start": [2468, 1], "end": [2478, 27], "traced_tactics": [{"tactic": "ext x", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.278063\n\u03b9 : Sort x\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nm : \u03b1 \u2192 \u03b2\nm' : \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nF : Filter \u03b2\n\u22a2 \u22c2\u2080 (comap f F).sets = \u22c2 (U : Set \u03b2) (_ : U \u2208 F), f \u207b\u00b9' U", "state_after": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.278063\n\u03b9 : Sort x\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nm : \u03b1 \u2192 \u03b2\nm' : \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nF : Filter \u03b2\nx : \u03b1\n\u22a2 x \u2208 \u22c2\u2080 (comap f F).sets \u2194 x \u2208 \u22c2 (U : Set \u03b2) (_ : U \u2208 F), f \u207b\u00b9' U"}, {"tactic": "suffices (\u2200 (A : Set \u03b1) (B : Set \u03b2), B \u2208 F \u2192 f \u207b\u00b9' B \u2286 A \u2192 x \u2208 A) \u2194\n    \u2200 B : Set \u03b2, B \u2208 F \u2192 f x \u2208 B by\n  simp only [mem_sInter, mem_iInter, Filter.mem_sets, mem_comap, this, and_imp, exists_prop,\n    mem_preimage, exists_imp]", "state_before": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.278063\n\u03b9 : Sort x\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nm : \u03b1 \u2192 \u03b2\nm' : \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nF : Filter \u03b2\nx : \u03b1\n\u22a2 x \u2208 \u22c2\u2080 (comap f F).sets \u2194 x \u2208 \u22c2 (U : Set \u03b2) (_ : U \u2208 F), f \u207b\u00b9' U", "state_after": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.278063\n\u03b9 : Sort x\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nm : \u03b1 \u2192 \u03b2\nm' : \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nF : Filter \u03b2\nx : \u03b1\n\u22a2 (\u2200 (A : Set \u03b1) (B : Set \u03b2), B \u2208 F \u2192 f \u207b\u00b9' B \u2286 A \u2192 x \u2208 A) \u2194 \u2200 (B : Set \u03b2), B \u2208 F \u2192 f x \u2208 B"}, {"tactic": "constructor", "state_before": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.278063\n\u03b9 : Sort x\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nm : \u03b1 \u2192 \u03b2\nm' : \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nF : Filter \u03b2\nx : \u03b1\n\u22a2 (\u2200 (A : Set \u03b1) (B : Set \u03b2), B \u2208 F \u2192 f \u207b\u00b9' B \u2286 A \u2192 x \u2208 A) \u2194 \u2200 (B : Set \u03b2), B \u2208 F \u2192 f x \u2208 B", "state_after": "case h.mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.278063\n\u03b9 : Sort x\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nm : \u03b1 \u2192 \u03b2\nm' : \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nF : Filter \u03b2\nx : \u03b1\n\u22a2 (\u2200 (A : Set \u03b1) (B : Set \u03b2), B \u2208 F \u2192 f \u207b\u00b9' B \u2286 A \u2192 x \u2208 A) \u2192 \u2200 (B : Set \u03b2), B \u2208 F \u2192 f x \u2208 B\n\ncase h.mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.278063\n\u03b9 : Sort x\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nm : \u03b1 \u2192 \u03b2\nm' : \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nF : Filter \u03b2\nx : \u03b1\n\u22a2 (\u2200 (B : Set \u03b2), B \u2208 F \u2192 f x \u2208 B) \u2192 \u2200 (A : Set \u03b1) (B : Set \u03b2), B \u2208 F \u2192 f \u207b\u00b9' B \u2286 A \u2192 x \u2208 A"}, {"tactic": "simp only [mem_sInter, mem_iInter, Filter.mem_sets, mem_comap, this, and_imp, exists_prop,\n  mem_preimage, exists_imp]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.278063\n\u03b9 : Sort x\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nm : \u03b1 \u2192 \u03b2\nm' : \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nF : Filter \u03b2\nx : \u03b1\nthis : (\u2200 (A : Set \u03b1) (B : Set \u03b2), B \u2208 F \u2192 f \u207b\u00b9' B \u2286 A \u2192 x \u2208 A) \u2194 \u2200 (B : Set \u03b2), B \u2208 F \u2192 f x \u2208 B\n\u22a2 x \u2208 \u22c2\u2080 (comap f F).sets \u2194 x \u2208 \u22c2 (U : Set \u03b2) (_ : U \u2208 F), f \u207b\u00b9' U", "state_after": "no goals"}, {"tactic": "intro h U U_in", "state_before": "case h.mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.278063\n\u03b9 : Sort x\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nm : \u03b1 \u2192 \u03b2\nm' : \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nF : Filter \u03b2\nx : \u03b1\n\u22a2 (\u2200 (A : Set \u03b1) (B : Set \u03b2), B \u2208 F \u2192 f \u207b\u00b9' B \u2286 A \u2192 x \u2208 A) \u2192 \u2200 (B : Set \u03b2), B \u2208 F \u2192 f x \u2208 B", "state_after": "case h.mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.278063\n\u03b9 : Sort x\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nm : \u03b1 \u2192 \u03b2\nm' : \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nF : Filter \u03b2\nx : \u03b1\nh : \u2200 (A : Set \u03b1) (B : Set \u03b2), B \u2208 F \u2192 f \u207b\u00b9' B \u2286 A \u2192 x \u2208 A\nU : Set \u03b2\nU_in : U \u2208 F\n\u22a2 f x \u2208 U"}, {"tactic": "simpa only [Subset.rfl, forall_prop_of_true, mem_preimage] using h (f \u207b\u00b9' U) U U_in", "state_before": "case h.mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.278063\n\u03b9 : Sort x\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nm : \u03b1 \u2192 \u03b2\nm' : \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nF : Filter \u03b2\nx : \u03b1\nh : \u2200 (A : Set \u03b1) (B : Set \u03b2), B \u2208 F \u2192 f \u207b\u00b9' B \u2286 A \u2192 x \u2208 A\nU : Set \u03b2\nU_in : U \u2208 F\n\u22a2 f x \u2208 U", "state_after": "no goals"}, {"tactic": "intro h V U U_in f_U_V", "state_before": "case h.mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.278063\n\u03b9 : Sort x\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nm : \u03b1 \u2192 \u03b2\nm' : \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nF : Filter \u03b2\nx : \u03b1\n\u22a2 (\u2200 (B : Set \u03b2), B \u2208 F \u2192 f x \u2208 B) \u2192 \u2200 (A : Set \u03b1) (B : Set \u03b2), B \u2208 F \u2192 f \u207b\u00b9' B \u2286 A \u2192 x \u2208 A", "state_after": "case h.mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.278063\n\u03b9 : Sort x\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nm : \u03b1 \u2192 \u03b2\nm' : \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nF : Filter \u03b2\nx : \u03b1\nh : \u2200 (B : Set \u03b2), B \u2208 F \u2192 f x \u2208 B\nV : Set \u03b1\nU : Set \u03b2\nU_in : U \u2208 F\nf_U_V : f \u207b\u00b9' U \u2286 V\n\u22a2 x \u2208 V"}, {"tactic": "exact f_U_V (h U U_in)", "state_before": "case h.mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.278063\n\u03b9 : Sort x\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nm : \u03b1 \u2192 \u03b2\nm' : \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nf : \u03b1 \u2192 \u03b2\nF : Filter \u03b2\nx : \u03b1\nh : \u2200 (B : Set \u03b2), B \u2208 F \u2192 f x \u2208 B\nV : Set \u03b1\nU : Set \u03b2\nU_in : U \u2208 F\nf_U_V : f \u207b\u00b9' U \u2286 V\n\u22a2 x \u2208 V", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/Partrec.lean", "full_name": "Partrec.option_some_iff", "start": [768, 1], "end": [774, 49], "traced_tactics": [{"tactic": "simp [Part.bind_assoc, \u2190 Function.comp_apply (f := Part.some) (g := encode), bind_some_eq_map,\n  -Function.comp_apply]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.211200\n\u03b3 : Type ?u.211203\n\u03c3 : Type u_2\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192. \u03c3\nh : Partrec fun a => Part.map Option.some (f a)\nn : \u2115\n\u22a2 (do\n      let n \u2190 Part.bind \u2191(decode n) fun a => Part.map encode ((fun a => Part.map Option.some (f a)) a)\n      \u2191(Nat.ppred n)) =\n    Part.bind \u2191(decode n) fun a => Part.map encode (f a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean", "full_name": "span_gramSchmidt", "start": [177, 1], "end": [181, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Rat/Lemmas.lean", "full_name": "Rat.ofInt_den", "start": [163, 9], "end": [163, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicTopology/SimplexCategory.lean", "full_name": "SimplexCategory.\u03b4_comp_\u03c3_self'", "start": [287, 1], "end": [290, 21], "traced_tactics": [{"tactic": "subst H", "state_before": "n : \u2115\nj : Fin (n + 2)\ni : Fin (n + 1)\nH : j = \u2191Fin.castSucc i\n\u22a2 \u03b4 j \u226b \u03c3 i = \ud835\udfd9 [n]", "state_after": "n : \u2115\ni : Fin (n + 1)\n\u22a2 \u03b4 (\u2191Fin.castSucc i) \u226b \u03c3 i = \ud835\udfd9 [n]"}, {"tactic": "rw [\u03b4_comp_\u03c3_self]", "state_before": "n : \u2115\ni : Fin (n + 1)\n\u22a2 \u03b4 (\u2191Fin.castSucc i) \u226b \u03c3 i = \ud835\udfd9 [n]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "full_name": "Real.tendsto_log_nat_add_one_sub_log", "start": [424, 1], "end": [425, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/LocalExtr.lean", "full_name": "isLocalMinOn_const", "start": [190, 1], "end": [191, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Int/Lemmas.lean", "full_name": "Int.sub_self", "start": [342, 11], "end": [343, 45], "traced_tactics": [{"tactic": "rw [Int.sub_eq_add_neg, Int.add_right_neg]", "state_before": "a : Int\n\u22a2 a - a = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/DedekindDomain/Ideal.lean", "full_name": "idealFactorsFunOfQuotHom_comp", "start": [1086, 1], "end": [1094, 37], "traced_tactics": [{"tactic": "refine OrderHom.ext _ _ (funext fun x => ?_)", "state_before": "R : Type u_1\nA : Type u_2\nK : Type ?u.981852\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : IsDedekindDomain A\nI : Ideal R\nJ : Ideal A\nB : Type u_3\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : IsDomain B\ninst\u271d : IsDedekindDomain B\nL : Ideal B\nf : R \u29f8 I \u2192+* A \u29f8 J\ng : A \u29f8 J \u2192+* B \u29f8 L\nhf : Function.Surjective \u2191f\nhg : Function.Surjective \u2191g\n\u22a2 OrderHom.comp (idealFactorsFunOfQuotHom hg) (idealFactorsFunOfQuotHom hf) =\n    idealFactorsFunOfQuotHom (_ : Function.Surjective \u2191(RingHom.comp g f))", "state_after": "R : Type u_1\nA : Type u_2\nK : Type ?u.981852\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : IsDedekindDomain A\nI : Ideal R\nJ : Ideal A\nB : Type u_3\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : IsDomain B\ninst\u271d : IsDedekindDomain B\nL : Ideal B\nf : R \u29f8 I \u2192+* A \u29f8 J\ng : A \u29f8 J \u2192+* B \u29f8 L\nhf : Function.Surjective \u2191f\nhg : Function.Surjective \u2191g\nx : \u2191{p | p \u2223 I}\n\u22a2 \u2191(OrderHom.comp (idealFactorsFunOfQuotHom hg) (idealFactorsFunOfQuotHom hf)) x =\n    \u2191(idealFactorsFunOfQuotHom (_ : Function.Surjective \u2191(RingHom.comp g f))) x"}, {"tactic": "rw [idealFactorsFunOfQuotHom, idealFactorsFunOfQuotHom, OrderHom.comp_coe, OrderHom.coe_fun_mk,\n  OrderHom.coe_fun_mk, Function.comp_apply, idealFactorsFunOfQuotHom, OrderHom.coe_fun_mk,\n  Subtype.mk_eq_mk, Subtype.coe_mk, map_comap_of_surjective (Ideal.Quotient.mk J)\n  Quotient.mk_surjective, map_map]", "state_before": "R : Type u_1\nA : Type u_2\nK : Type ?u.981852\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : IsDedekindDomain A\nI : Ideal R\nJ : Ideal A\nB : Type u_3\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : IsDomain B\ninst\u271d : IsDedekindDomain B\nL : Ideal B\nf : R \u29f8 I \u2192+* A \u29f8 J\ng : A \u29f8 J \u2192+* B \u29f8 L\nhf : Function.Surjective \u2191f\nhg : Function.Surjective \u2191g\nx : \u2191{p | p \u2223 I}\n\u22a2 \u2191(OrderHom.comp (idealFactorsFunOfQuotHom hg) (idealFactorsFunOfQuotHom hf)) x =\n    \u2191(idealFactorsFunOfQuotHom (_ : Function.Surjective \u2191(RingHom.comp g f))) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Sigma/Basic.lean", "full_name": "PSigma.exists", "start": [249, 1], "end": [250, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "full_name": "Asymptotics.isLittleO_iff_exists_eq_mul", "start": [1969, 1], "end": [1976, 75], "traced_tactics": [{"tactic": "constructor", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.634360\nE : Type ?u.634363\nF : Type ?u.634366\nG : Type ?u.634369\nE' : Type ?u.634372\nF' : Type ?u.634375\nG' : Type ?u.634378\nE'' : Type ?u.634381\nF'' : Type ?u.634384\nG'' : Type ?u.634387\nR : Type ?u.634390\nR' : Type ?u.634393\n\ud835\udd5c : Type u_2\n\ud835\udd5c' : Type ?u.634399\ninst\u271d\u00b9\u00b2 : Norm E\ninst\u271d\u00b9\u00b9 : Norm F\ninst\u271d\u00b9\u2070 : Norm G\ninst\u271d\u2079 : SeminormedAddCommGroup E'\ninst\u271d\u2078 : SeminormedAddCommGroup F'\ninst\u271d\u2077 : SeminormedAddCommGroup G'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F''\ninst\u271d\u2074 : NormedAddCommGroup G''\ninst\u271d\u00b3 : SeminormedRing R\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\n\u22a2 u =o[l] v \u2194 \u2203 \u03c6 _h\u03c6, u =\u1da0[l] \u03c6 * v", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type ?u.634360\nE : Type ?u.634363\nF : Type ?u.634366\nG : Type ?u.634369\nE' : Type ?u.634372\nF' : Type ?u.634375\nG' : Type ?u.634378\nE'' : Type ?u.634381\nF'' : Type ?u.634384\nG'' : Type ?u.634387\nR : Type ?u.634390\nR' : Type ?u.634393\n\ud835\udd5c : Type u_2\n\ud835\udd5c' : Type ?u.634399\ninst\u271d\u00b9\u00b2 : Norm E\ninst\u271d\u00b9\u00b9 : Norm F\ninst\u271d\u00b9\u2070 : Norm G\ninst\u271d\u2079 : SeminormedAddCommGroup E'\ninst\u271d\u2078 : SeminormedAddCommGroup F'\ninst\u271d\u2077 : SeminormedAddCommGroup G'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F''\ninst\u271d\u2074 : NormedAddCommGroup G''\ninst\u271d\u00b3 : SeminormedRing R\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\n\u22a2 u =o[l] v \u2192 \u2203 \u03c6 _h\u03c6, u =\u1da0[l] \u03c6 * v\n\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type ?u.634360\nE : Type ?u.634363\nF : Type ?u.634366\nG : Type ?u.634369\nE' : Type ?u.634372\nF' : Type ?u.634375\nG' : Type ?u.634378\nE'' : Type ?u.634381\nF'' : Type ?u.634384\nG'' : Type ?u.634387\nR : Type ?u.634390\nR' : Type ?u.634393\n\ud835\udd5c : Type u_2\n\ud835\udd5c' : Type ?u.634399\ninst\u271d\u00b9\u00b2 : Norm E\ninst\u271d\u00b9\u00b9 : Norm F\ninst\u271d\u00b9\u2070 : Norm G\ninst\u271d\u2079 : SeminormedAddCommGroup E'\ninst\u271d\u2078 : SeminormedAddCommGroup F'\ninst\u271d\u2077 : SeminormedAddCommGroup G'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F''\ninst\u271d\u2074 : NormedAddCommGroup G''\ninst\u271d\u00b3 : SeminormedRing R\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\n\u22a2 (\u2203 \u03c6 _h\u03c6, u =\u1da0[l] \u03c6 * v) \u2192 u =o[l] v"}, {"tactic": "exact fun h => \u27e8fun x => u x / v x, h.tendsto_div_nhds_zero, h.eventually_mul_div_cancel.symm\u27e9", "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type ?u.634360\nE : Type ?u.634363\nF : Type ?u.634366\nG : Type ?u.634369\nE' : Type ?u.634372\nF' : Type ?u.634375\nG' : Type ?u.634378\nE'' : Type ?u.634381\nF'' : Type ?u.634384\nG'' : Type ?u.634387\nR : Type ?u.634390\nR' : Type ?u.634393\n\ud835\udd5c : Type u_2\n\ud835\udd5c' : Type ?u.634399\ninst\u271d\u00b9\u00b2 : Norm E\ninst\u271d\u00b9\u00b9 : Norm F\ninst\u271d\u00b9\u2070 : Norm G\ninst\u271d\u2079 : SeminormedAddCommGroup E'\ninst\u271d\u2078 : SeminormedAddCommGroup F'\ninst\u271d\u2077 : SeminormedAddCommGroup G'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F''\ninst\u271d\u2074 : NormedAddCommGroup G''\ninst\u271d\u00b3 : SeminormedRing R\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\n\u22a2 u =o[l] v \u2192 \u2203 \u03c6 _h\u03c6, u =\u1da0[l] \u03c6 * v", "state_after": "no goals"}, {"tactic": "simp only [IsLittleO_def]", "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type ?u.634360\nE : Type ?u.634363\nF : Type ?u.634366\nG : Type ?u.634369\nE' : Type ?u.634372\nF' : Type ?u.634375\nG' : Type ?u.634378\nE'' : Type ?u.634381\nF'' : Type ?u.634384\nG'' : Type ?u.634387\nR : Type ?u.634390\nR' : Type ?u.634393\n\ud835\udd5c : Type u_2\n\ud835\udd5c' : Type ?u.634399\ninst\u271d\u00b9\u00b2 : Norm E\ninst\u271d\u00b9\u00b9 : Norm F\ninst\u271d\u00b9\u2070 : Norm G\ninst\u271d\u2079 : SeminormedAddCommGroup E'\ninst\u271d\u2078 : SeminormedAddCommGroup F'\ninst\u271d\u2077 : SeminormedAddCommGroup G'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F''\ninst\u271d\u2074 : NormedAddCommGroup G''\ninst\u271d\u00b3 : SeminormedRing R\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\n\u22a2 (\u2203 \u03c6 _h\u03c6, u =\u1da0[l] \u03c6 * v) \u2192 u =o[l] v", "state_after": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type ?u.634360\nE : Type ?u.634363\nF : Type ?u.634366\nG : Type ?u.634369\nE' : Type ?u.634372\nF' : Type ?u.634375\nG' : Type ?u.634378\nE'' : Type ?u.634381\nF'' : Type ?u.634384\nG'' : Type ?u.634387\nR : Type ?u.634390\nR' : Type ?u.634393\n\ud835\udd5c : Type u_2\n\ud835\udd5c' : Type ?u.634399\ninst\u271d\u00b9\u00b2 : Norm E\ninst\u271d\u00b9\u00b9 : Norm F\ninst\u271d\u00b9\u2070 : Norm G\ninst\u271d\u2079 : SeminormedAddCommGroup E'\ninst\u271d\u2078 : SeminormedAddCommGroup F'\ninst\u271d\u2077 : SeminormedAddCommGroup G'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F''\ninst\u271d\u2074 : NormedAddCommGroup G''\ninst\u271d\u00b3 : SeminormedRing R\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\n\u22a2 (\u2203 \u03c6 _h\u03c6, u =\u1da0[l] \u03c6 * v) \u2192 \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l u v"}, {"tactic": "rintro \u27e8\u03c6, h\u03c6, huv\u03c6\u27e9 c hpos", "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type ?u.634360\nE : Type ?u.634363\nF : Type ?u.634366\nG : Type ?u.634369\nE' : Type ?u.634372\nF' : Type ?u.634375\nG' : Type ?u.634378\nE'' : Type ?u.634381\nF'' : Type ?u.634384\nG'' : Type ?u.634387\nR : Type ?u.634390\nR' : Type ?u.634393\n\ud835\udd5c : Type u_2\n\ud835\udd5c' : Type ?u.634399\ninst\u271d\u00b9\u00b2 : Norm E\ninst\u271d\u00b9\u00b9 : Norm F\ninst\u271d\u00b9\u2070 : Norm G\ninst\u271d\u2079 : SeminormedAddCommGroup E'\ninst\u271d\u2078 : SeminormedAddCommGroup F'\ninst\u271d\u2077 : SeminormedAddCommGroup G'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F''\ninst\u271d\u2074 : NormedAddCommGroup G''\ninst\u271d\u00b3 : SeminormedRing R\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v : \u03b1 \u2192 \ud835\udd5c\n\u22a2 (\u2203 \u03c6 _h\u03c6, u =\u1da0[l] \u03c6 * v) \u2192 \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l u v", "state_after": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.634360\nE : Type ?u.634363\nF : Type ?u.634366\nG : Type ?u.634369\nE' : Type ?u.634372\nF' : Type ?u.634375\nG' : Type ?u.634378\nE'' : Type ?u.634381\nF'' : Type ?u.634384\nG'' : Type ?u.634387\nR : Type ?u.634390\nR' : Type ?u.634393\n\ud835\udd5c : Type u_2\n\ud835\udd5c' : Type ?u.634399\ninst\u271d\u00b9\u00b2 : Norm E\ninst\u271d\u00b9\u00b9 : Norm F\ninst\u271d\u00b9\u2070 : Norm G\ninst\u271d\u2079 : SeminormedAddCommGroup E'\ninst\u271d\u2078 : SeminormedAddCommGroup F'\ninst\u271d\u2077 : SeminormedAddCommGroup G'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F''\ninst\u271d\u2074 : NormedAddCommGroup G''\ninst\u271d\u00b3 : SeminormedRing R\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v \u03c6 : \u03b1 \u2192 \ud835\udd5c\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 0)\nhuv\u03c6 : u =\u1da0[l] \u03c6 * v\nc : \u211d\nhpos : 0 < c\n\u22a2 IsBigOWith c l u v"}, {"tactic": "rw [NormedAddCommGroup.tendsto_nhds_zero] at h\u03c6", "state_before": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.634360\nE : Type ?u.634363\nF : Type ?u.634366\nG : Type ?u.634369\nE' : Type ?u.634372\nF' : Type ?u.634375\nG' : Type ?u.634378\nE'' : Type ?u.634381\nF'' : Type ?u.634384\nG'' : Type ?u.634387\nR : Type ?u.634390\nR' : Type ?u.634393\n\ud835\udd5c : Type u_2\n\ud835\udd5c' : Type ?u.634399\ninst\u271d\u00b9\u00b2 : Norm E\ninst\u271d\u00b9\u00b9 : Norm F\ninst\u271d\u00b9\u2070 : Norm G\ninst\u271d\u2079 : SeminormedAddCommGroup E'\ninst\u271d\u2078 : SeminormedAddCommGroup F'\ninst\u271d\u2077 : SeminormedAddCommGroup G'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F''\ninst\u271d\u2074 : NormedAddCommGroup G''\ninst\u271d\u00b3 : SeminormedRing R\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v \u03c6 : \u03b1 \u2192 \ud835\udd5c\nh\u03c6 : Tendsto \u03c6 l (\ud835\udcdd 0)\nhuv\u03c6 : u =\u1da0[l] \u03c6 * v\nc : \u211d\nhpos : 0 < c\n\u22a2 IsBigOWith c l u v", "state_after": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.634360\nE : Type ?u.634363\nF : Type ?u.634366\nG : Type ?u.634369\nE' : Type ?u.634372\nF' : Type ?u.634375\nG' : Type ?u.634378\nE'' : Type ?u.634381\nF'' : Type ?u.634384\nG'' : Type ?u.634387\nR : Type ?u.634390\nR' : Type ?u.634393\n\ud835\udd5c : Type u_2\n\ud835\udd5c' : Type ?u.634399\ninst\u271d\u00b9\u00b2 : Norm E\ninst\u271d\u00b9\u00b9 : Norm F\ninst\u271d\u00b9\u2070 : Norm G\ninst\u271d\u2079 : SeminormedAddCommGroup E'\ninst\u271d\u2078 : SeminormedAddCommGroup F'\ninst\u271d\u2077 : SeminormedAddCommGroup G'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F''\ninst\u271d\u2074 : NormedAddCommGroup G''\ninst\u271d\u00b3 : SeminormedRing R\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v \u03c6 : \u03b1 \u2192 \ud835\udd5c\nh\u03c6 : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x\u2016 < \u03b5\nhuv\u03c6 : u =\u1da0[l] \u03c6 * v\nc : \u211d\nhpos : 0 < c\n\u22a2 IsBigOWith c l u v"}, {"tactic": "exact isBigOWith_of_eq_mul _ ((h\u03c6 c hpos).mono fun x => le_of_lt) huv\u03c6", "state_before": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.634360\nE : Type ?u.634363\nF : Type ?u.634366\nG : Type ?u.634369\nE' : Type ?u.634372\nF' : Type ?u.634375\nG' : Type ?u.634378\nE'' : Type ?u.634381\nF'' : Type ?u.634384\nG'' : Type ?u.634387\nR : Type ?u.634390\nR' : Type ?u.634393\n\ud835\udd5c : Type u_2\n\ud835\udd5c' : Type ?u.634399\ninst\u271d\u00b9\u00b2 : Norm E\ninst\u271d\u00b9\u00b9 : Norm F\ninst\u271d\u00b9\u2070 : Norm G\ninst\u271d\u2079 : SeminormedAddCommGroup E'\ninst\u271d\u2078 : SeminormedAddCommGroup F'\ninst\u271d\u2077 : SeminormedAddCommGroup G'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F''\ninst\u271d\u2074 : NormedAddCommGroup G''\ninst\u271d\u00b3 : SeminormedRing R\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nu v \u03c6 : \u03b1 \u2192 \ud835\udd5c\nh\u03c6 : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016\u03c6 x\u2016 < \u03b5\nhuv\u03c6 : u =\u1da0[l] \u03c6 * v\nc : \u211d\nhpos : 0 < c\n\u22a2 IsBigOWith c l u v", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Finprod.lean", "full_name": "finprod_mem_union", "start": [808, 1], "end": [810, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.length_concat'", "start": [575, 1], "end": [576, 54], "traced_tactics": [{"tactic": "simp only [concat_eq_append, length_append, length]", "state_before": "\u03b9 : Type ?u.25204\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 length (concat l a) = succ (length l)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "full_name": "UniqueFactorizationMonoid.normalizedFactors_zero", "start": [650, 1], "end": [651, 36], "traced_tactics": [{"tactic": "simp [normalizedFactors, factors]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\n\u22a2 normalizedFactors 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/MonoidLocalization.lean", "full_name": "Submonoid.LocalizationMap.exists_of_sec_mk'", "start": [818, 1], "end": [820, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Extr.lean", "full_name": "IsExtrOn.comp_mono", "start": [351, 1], "end": [353, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/LocallyConstant/Basic.lean", "full_name": "LocallyConstant.continuous", "start": [302, 11], "end": [303, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/PartENat.lean", "full_name": "PartENat.lt_find", "start": [789, 1], "end": [798, 15], "traced_tactics": [{"tactic": "rw [coe_lt_iff]", "state_before": "P : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\n\u22a2 \u2191n < find P", "state_after": "P : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\n\u22a2 \u2200 (h : (find P).Dom), n < Part.get (find P) h"}, {"tactic": "intro h\u2081", "state_before": "P : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\n\u22a2 \u2200 (h : (find P).Dom), n < Part.get (find P) h", "state_after": "P : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\nh\u2081 : (find P).Dom\n\u22a2 n < Part.get (find P) h\u2081"}, {"tactic": "rw [find_get]", "state_before": "P : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\nh\u2081 : (find P).Dom\n\u22a2 n < Part.get (find P) h\u2081", "state_after": "P : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\nh\u2081 : (find P).Dom\n\u22a2 n < Nat.find h\u2081"}, {"tactic": "have h\u2082 := @Nat.find_spec P _ h\u2081", "state_before": "P : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\nh\u2081 : (find P).Dom\n\u22a2 n < Nat.find h\u2081", "state_after": "P : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\nh\u2081 : (find P).Dom\nh\u2082 : P (Nat.find h\u2081)\n\u22a2 n < Nat.find h\u2081"}, {"tactic": "revert h\u2082", "state_before": "P : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\nh\u2081 : (find P).Dom\nh\u2082 : P (Nat.find h\u2081)\n\u22a2 n < Nat.find h\u2081", "state_after": "P : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\nh\u2081 : (find P).Dom\n\u22a2 P (Nat.find h\u2081) \u2192 n < Nat.find h\u2081"}, {"tactic": "contrapose", "state_before": "P : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\nh\u2081 : (find P).Dom\n\u22a2 P (Nat.find h\u2081) \u2192 n < Nat.find h\u2081", "state_after": "P : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\nh\u2081 : (find P).Dom\n\u22a2 \u00acn < Nat.find h\u2081 \u2192 \u00acP (Nat.find h\u2081)"}, {"tactic": "intro h\u2082", "state_before": "P : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\nh\u2081 : (find P).Dom\n\u22a2 \u00acn < Nat.find h\u2081 \u2192 \u00acP (Nat.find h\u2081)", "state_after": "P : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\nh\u2081 : (find P).Dom\nh\u2082 : \u00acn < Nat.find h\u2081\n\u22a2 \u00acP (Nat.find h\u2081)"}, {"tactic": "rw [not_lt] at h\u2082", "state_before": "P : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\nh\u2081 : (find P).Dom\nh\u2082 : \u00acn < Nat.find h\u2081\n\u22a2 \u00acP (Nat.find h\u2081)", "state_after": "P : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\nh\u2081 : (find P).Dom\nh\u2082 : Nat.find h\u2081 \u2264 n\n\u22a2 \u00acP (Nat.find h\u2081)"}, {"tactic": "exact h _ h\u2082", "state_before": "P : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : \u2200 (m : \u2115), m \u2264 n \u2192 \u00acP m\nh\u2081 : (find P).Dom\nh\u2082 : Nat.find h\u2081 \u2264 n\n\u22a2 \u00acP (Nat.find h\u2081)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/Operations.lean", "full_name": "RingHom.sub_mem_ker_iff", "start": [2057, 1], "end": [2057, 99], "traced_tactics": [{"tactic": "rw [mem_ker, map_sub, sub_eq_zero]", "state_before": "R : Type u\nS : Type v\nT : Type w\nF : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nx y : R\n\u22a2 x - y \u2208 ker f \u2194 \u2191f x = \u2191f y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Subobject/Limits.lean", "full_name": "CategoryTheory.Limits.kernelSubobject_comp_mono", "start": [235, 1], "end": [237, 100], "traced_tactics": [{"tactic": "simp", "state_before": "C : Type u\ninst\u271d\u2074 : Category C\nX Y Z\u271d : C\ninst\u271d\u00b3 : HasZeroMorphisms C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b2 : HasKernel f\u271d\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasKernel f\nZ : C\nh : Y \u27f6 Z\ninst\u271d : Mono h\n\u22a2 (arrow (kernelSubobject (f \u226b h)) \u226b f) \u226b h = 0 \u226b h", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/ENNReal.lean", "full_name": "ENNReal.toNNReal_le_toNNReal", "start": [2033, 1], "end": [2034, 89], "traced_tactics": [{"tactic": "rwa [\u2190 coe_toNNReal ha, \u2190 coe_toNNReal hb, coe_le_coe]", "state_before": "\u03b1 : Type ?u.799347\n\u03b2 : Type ?u.799350\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 \u22a4\nhb : b \u2260 \u22a4\nh : ENNReal.toNNReal a \u2264 ENNReal.toNNReal b\n\u22a2 a \u2264 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Derivative.lean", "full_name": "Polynomial.dvd_iterate_derivative_pow", "start": [494, 1], "end": [499, 26], "traced_tactics": [{"tactic": "obtain \u27e8m, rfl\u27e9 := Nat.exists_eq_succ_of_ne_zero hm", "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : CommSemiring R\nf : R[X]\nn m : \u2115\nc : R\nhm : m \u2260 0\n\u22a2 \u2191n \u2223 eval c ((\u2191derivative^[m]) (f ^ n))", "state_after": "case intro\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : CommSemiring R\nf : R[X]\nn : \u2115\nc : R\nm : \u2115\nhm : Nat.succ m \u2260 0\n\u22a2 \u2191n \u2223 eval c ((\u2191derivative^[Nat.succ m]) (f ^ n))"}, {"tactic": "rw [Function.iterate_succ_apply, derivative_pow, mul_assoc, C_eq_nat_cast,\n  iterate_derivative_nat_cast_mul, eval_mul, eval_nat_cast]", "state_before": "case intro\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : CommSemiring R\nf : R[X]\nn : \u2115\nc : R\nm : \u2115\nhm : Nat.succ m \u2260 0\n\u22a2 \u2191n \u2223 eval c ((\u2191derivative^[Nat.succ m]) (f ^ n))", "state_after": "case intro\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : CommSemiring R\nf : R[X]\nn : \u2115\nc : R\nm : \u2115\nhm : Nat.succ m \u2260 0\n\u22a2 \u2191n \u2223 \u2191n * eval c ((\u2191derivative^[m]) (f ^ (n - 1) * \u2191derivative f))"}, {"tactic": "exact dvd_mul_right _ _", "state_before": "case intro\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : CommSemiring R\nf : R[X]\nn : \u2115\nc : R\nm : \u2115\nhm : Nat.succ m \u2260 0\n\u22a2 \u2191n \u2223 \u2191n * eval c ((\u2191derivative^[m]) (f ^ (n - 1) * \u2191derivative f))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.succAbove_pos", "start": [2125, 1], "end": [2128, 48], "traced_tactics": [{"tactic": "by_cases H : castSucc i < p", "state_before": "n m : \u2115\ninst\u271d : NeZero n\np : Fin (n + 1)\ni : Fin n\nh : 0 < i\n\u22a2 0 < \u2191(succAbove p) i", "state_after": "case pos\nn m : \u2115\ninst\u271d : NeZero n\np : Fin (n + 1)\ni : Fin n\nh : 0 < i\nH : \u2191castSucc i < p\n\u22a2 0 < \u2191(succAbove p) i\n\ncase neg\nn m : \u2115\ninst\u271d : NeZero n\np : Fin (n + 1)\ni : Fin n\nh : 0 < i\nH : \u00ac\u2191castSucc i < p\n\u22a2 0 < \u2191(succAbove p) i"}, {"tactic": "simpa [succAbove_below _ _ H] using castSucc_pos h", "state_before": "case pos\nn m : \u2115\ninst\u271d : NeZero n\np : Fin (n + 1)\ni : Fin n\nh : 0 < i\nH : \u2191castSucc i < p\n\u22a2 0 < \u2191(succAbove p) i", "state_after": "no goals"}, {"tactic": "simp [succAbove_above _ _ (le_of_not_lt H)]", "state_before": "case neg\nn m : \u2115\ninst\u271d : NeZero n\np : Fin (n + 1)\ni : Fin n\nh : 0 < i\nH : \u00ac\u2191castSucc i < p\n\u22a2 0 < \u2191(succAbove p) i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Splits.lean", "full_name": "Polynomial.splits_of_exists_multiset", "start": [394, 1], "end": [407, 31], "traced_tactics": [{"tactic": "rw [irreducible_iff_prime] at hp", "state_before": "F : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Irreducible p\nhdp : p \u2223 map i f\n\u22a2 degree p = 1", "state_after": "F : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 map i f\n\u22a2 degree p = 1"}, {"tactic": "rw [hs, \u2190 Multiset.prod_toList] at hdp", "state_before": "F : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 map i f\n\u22a2 degree p = 1", "state_after": "F : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 \u2191C (\u2191i (leadingCoeff f)) * List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\n\u22a2 degree p = 1"}, {"tactic": "obtain hd | hd := hp.2.2 _ _ hdp", "state_before": "F : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 \u2191C (\u2191i (leadingCoeff f)) * List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\n\u22a2 degree p = 1", "state_after": "case inl\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 \u2191C (\u2191i (leadingCoeff f)) * List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\nhd : p \u2223 \u2191C (\u2191i (leadingCoeff f))\n\u22a2 degree p = 1\n\ncase inr\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 \u2191C (\u2191i (leadingCoeff f)) * List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\nhd : p \u2223 List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\n\u22a2 degree p = 1"}, {"tactic": "refine' (hp.2.1 <| isUnit_of_dvd_unit hd _).elim", "state_before": "case inl\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 \u2191C (\u2191i (leadingCoeff f)) * List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\nhd : p \u2223 \u2191C (\u2191i (leadingCoeff f))\n\u22a2 degree p = 1", "state_after": "case inl\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 \u2191C (\u2191i (leadingCoeff f)) * List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\nhd : p \u2223 \u2191C (\u2191i (leadingCoeff f))\n\u22a2 IsUnit (\u2191C (\u2191i (leadingCoeff f)))"}, {"tactic": "exact isUnit_C.2 ((leadingCoeff_ne_zero.2 hf0).isUnit.map i)", "state_before": "case inl\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 \u2191C (\u2191i (leadingCoeff f)) * List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\nhd : p \u2223 \u2191C (\u2191i (leadingCoeff f))\n\u22a2 IsUnit (\u2191C (\u2191i (leadingCoeff f)))", "state_after": "no goals"}, {"tactic": "obtain \u27e8q, hq, hd\u27e9 := hp.dvd_prod_iff.1 hd", "state_before": "case inr\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 \u2191C (\u2191i (leadingCoeff f)) * List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\nhd : p \u2223 List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\n\u22a2 degree p = 1", "state_after": "case inr.intro.intro\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 \u2191C (\u2191i (leadingCoeff f)) * List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\nhd\u271d : p \u2223 List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\nq : L[X]\nhq : q \u2208 Multiset.toList (Multiset.map (fun a => X - \u2191C a) s)\nhd : p \u2223 q\n\u22a2 degree p = 1"}, {"tactic": "obtain \u27e8a, _, rfl\u27e9 := Multiset.mem_map.1 (Multiset.mem_toList.1 hq)", "state_before": "case inr.intro.intro\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 \u2191C (\u2191i (leadingCoeff f)) * List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\nhd\u271d : p \u2223 List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\nq : L[X]\nhq : q \u2208 Multiset.toList (Multiset.map (fun a => X - \u2191C a) s)\nhd : p \u2223 q\n\u22a2 degree p = 1", "state_after": "case inr.intro.intro.intro.intro\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 \u2191C (\u2191i (leadingCoeff f)) * List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\nhd\u271d : p \u2223 List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\na : L\nleft\u271d : a \u2208 s\nhq : X - \u2191C a \u2208 Multiset.toList (Multiset.map (fun a => X - \u2191C a) s)\nhd : p \u2223 X - \u2191C a\n\u22a2 degree p = 1"}, {"tactic": "rw [degree_eq_degree_of_associated ((hp.dvd_prime_iff_associated <| prime_X_sub_C a).1 hd)]", "state_before": "case inr.intro.intro.intro.intro\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 \u2191C (\u2191i (leadingCoeff f)) * List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\nhd\u271d : p \u2223 List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\na : L\nleft\u271d : a \u2208 s\nhq : X - \u2191C a \u2208 Multiset.toList (Multiset.map (fun a => X - \u2191C a) s)\nhd : p \u2223 X - \u2191C a\n\u22a2 degree p = 1", "state_after": "case inr.intro.intro.intro.intro\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 \u2191C (\u2191i (leadingCoeff f)) * List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\nhd\u271d : p \u2223 List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\na : L\nleft\u271d : a \u2208 s\nhq : X - \u2191C a \u2208 Multiset.toList (Multiset.map (fun a => X - \u2191C a) s)\nhd : p \u2223 X - \u2191C a\n\u22a2 degree (X - \u2191C a) = 1"}, {"tactic": "exact degree_X_sub_C a", "state_before": "case inr.intro.intro.intro.intro\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nf : K[X]\ns : Multiset L\nhs : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)\nhf0 : \u00acf = 0\np : L[X]\nhp : Prime p\nhdp : p \u2223 \u2191C (\u2191i (leadingCoeff f)) * List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\nhd\u271d : p \u2223 List.prod (Multiset.toList (Multiset.map (fun a => X - \u2191C a) s))\na : L\nleft\u271d : a \u2208 s\nhq : X - \u2191C a \u2208 Multiset.toList (Multiset.map (fun a => X - \u2191C a) s)\nhd : p \u2223 X - \u2191C a\n\u22a2 degree (X - \u2191C a) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/LocalExtr.lean", "full_name": "IsLocalMin.neg", "start": [396, 8], "end": [397, 9], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Monad/Basic.lean", "full_name": "CategoryTheory.Comonad.left_counit", "start": [167, 1], "end": [169, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Small/Basic.lean", "full_name": "Small.mk'", "start": [37, 1], "end": [38, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Localization/Basic.lean", "full_name": "Localization.mk_nat_cast", "start": [1043, 1], "end": [1044, 43], "traced_tactics": [{"tactic": "simpa using @mk_algebraMap R _ M \u2115 _ _ m", "state_before": "R : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type ?u.2909039\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type ?u.2909206\ninst\u271d : CommSemiring P\nm : \u2115\n\u22a2 mk (\u2191m) 1 = \u2191m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/Portmanteau.lean", "full_name": "MeasureTheory.ProbabilityMeasure.limsup_measure_closed_le_of_tendsto", "start": [380, 1], "end": [385, 94], "traced_tactics": [{"tactic": "apply FiniteMeasure.limsup_measure_closed_le_of_tendsto\n  ((ProbabilityMeasure.tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds L).mp \u03bcs_lim) F_closed", "state_before": "\u03a9\u271d : Type ?u.34617\ninst\u271d\u00b3 : MeasurableSpace \u03a9\u271d\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nL : Filter \u03b9\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : PseudoEMetricSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : ProbabilityMeasure \u03a9\n\u03bcs : \u03b9 \u2192 ProbabilityMeasure \u03a9\n\u03bcs_lim : Tendsto \u03bcs L (\ud835\udcdd \u03bc)\nF : Set \u03a9\nF_closed : IsClosed F\n\u22a2 limsup (fun i => \u2191\u2191\u2191(\u03bcs i) F) L \u2264 \u2191\u2191\u2191\u03bc F", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Basic.lean", "full_name": "AntitoneOn.convex_lt", "start": [365, 1], "end": [367, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Instances/TrivSqZeroExt.lean", "full_name": "TrivSqZeroExt.embedding_inr", "start": [81, 1], "end": [82, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Div.lean", "full_name": "Polynomial.rootMultiplicity_pos", "start": [577, 1], "end": [579, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean", "full_name": "HasDerivWithinAt.log", "start": [102, 1], "end": [105, 55], "traced_tactics": [{"tactic": "rw [div_eq_inv_mul]", "state_before": "f : \u211d \u2192 \u211d\nx f' : \u211d\ns : Set \u211d\nhf : HasDerivWithinAt f f' s x\nhx : f x \u2260 0\n\u22a2 HasDerivWithinAt (fun y => Real.log (f y)) (f' / f x) s x", "state_after": "f : \u211d \u2192 \u211d\nx f' : \u211d\ns : Set \u211d\nhf : HasDerivWithinAt f f' s x\nhx : f x \u2260 0\n\u22a2 HasDerivWithinAt (fun y => Real.log (f y)) ((f x)\u207b\u00b9 * f') s x"}, {"tactic": "exact (hasDerivAt_log hx).comp_hasDerivWithinAt x hf", "state_before": "f : \u211d \u2192 \u211d\nx f' : \u211d\ns : Set \u211d\nhf : HasDerivWithinAt f f' s x\nhx : f x \u2260 0\n\u22a2 HasDerivWithinAt (fun y => Real.log (f y)) ((f x)\u207b\u00b9 * f') s x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "full_name": "MeasureTheory.inter_ae_eq_left_of_ae_eq_univ", "start": [536, 1], "end": [538, 18], "traced_tactics": [{"tactic": "convert ae_eq_set_inter (ae_eq_refl s) h", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.125038\n\u03b3 : Type ?u.125041\n\u03b4 : Type ?u.125044\n\u03b9 : Type ?u.125047\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh : t =\u1d50[\u03bc] univ\n\u22a2 s \u2229 t =\u1d50[\u03bc] s", "state_after": "case h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type ?u.125038\n\u03b3 : Type ?u.125041\n\u03b4 : Type ?u.125044\n\u03b9 : Type ?u.125047\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh : t =\u1d50[\u03bc] univ\n\u22a2 s = s \u2229 univ"}, {"tactic": "rw [inter_univ]", "state_before": "case h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type ?u.125038\n\u03b3 : Type ?u.125041\n\u03b4 : Type ?u.125044\n\u03b9 : Type ?u.125047\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\nh : t =\u1d50[\u03bc] univ\n\u22a2 s = s \u2229 univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/AList.lean", "full_name": "AList.lookup_erase", "start": [245, 1], "end": [246, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "BoundedContinuousFunction.toLp_injective", "start": [1682, 1], "end": [1684, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicTopology/SimplexCategory.lean", "full_name": "SimplexCategory.\u03b4_comp_\u03c3_of_gt'", "start": [327, 1], "end": [333, 12], "traced_tactics": [{"tactic": "simp only [Fin.not_lt_zero, hi] at H", "state_before": "n : \u2115\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : Fin.succ j < i\nhi : i = 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rw [\u2190 \u03b4_comp_\u03c3_of_gt]", "state_before": "n : \u2115\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : Fin.succ j < i\n\u22a2 \u03b4 i \u226b \u03c3 j = \u03c3 (Fin.castLT j (_ : \u2191j < n + 1)) \u226b \u03b4 (Fin.pred i (_ : i = 0 \u2192 False))", "state_after": "n : \u2115\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : Fin.succ j < i\n\u22a2 \u03b4 i \u226b \u03c3 j = \u03b4 (Fin.succ (Fin.pred i (_ : i = 0 \u2192 False))) \u226b \u03c3 (\u2191Fin.castSucc (Fin.castLT j (_ : \u2191j < n + 1)))\n\nn : \u2115\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : Fin.succ j < i\n\u22a2 \u2191Fin.castSucc (Fin.castLT j (_ : \u2191j < n + 1)) < Fin.pred i (_ : i = 0 \u2192 False)"}, {"tactic": "simp", "state_before": "n : \u2115\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : Fin.succ j < i\n\u22a2 \u03b4 i \u226b \u03c3 j = \u03b4 (Fin.succ (Fin.pred i (_ : i = 0 \u2192 False))) \u226b \u03c3 (\u2191Fin.castSucc (Fin.castLT j (_ : \u2191j < n + 1)))", "state_after": "no goals"}, {"tactic": "rw [Fin.castSucc_castLT, \u2190 Fin.succ_lt_succ_iff, Fin.succ_pred]", "state_before": "n : \u2115\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : Fin.succ j < i\n\u22a2 \u2191Fin.castSucc (Fin.castLT j (_ : \u2191j < n + 1)) < Fin.pred i (_ : i = 0 \u2192 False)", "state_after": "n : \u2115\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : Fin.succ j < i\n\u22a2 Fin.succ j < i"}, {"tactic": "exact H", "state_before": "n : \u2115\ni : Fin (n + 3)\nj : Fin (n + 2)\nH : Fin.succ j < i\n\u22a2 Fin.succ j < i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/Finite/Basic.lean", "full_name": "ZMod.pow_card_sub_one_eq_one", "start": [433, 1], "end": [436, 27], "traced_tactics": [{"tactic": "have h := FiniteField.pow_card_sub_one_eq_one a ha", "state_before": "K : Type ?u.1085344\nR : Type ?u.1085347\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : ZMod p\nha : a \u2260 0\n\u22a2 a ^ (p - 1) = 1", "state_after": "K : Type ?u.1085344\nR : Type ?u.1085347\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : ZMod p\nha : a \u2260 0\nh : a ^ (Fintype.card (ZMod p) - 1) = 1\n\u22a2 a ^ (p - 1) = 1"}, {"tactic": "rwa [ZMod.card p] at h", "state_before": "K : Type ?u.1085344\nR : Type ?u.1085347\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : ZMod p\nha : a \u2260 0\nh : a ^ (Fintype.card (ZMod p) - 1) = 1\n\u22a2 a ^ (p - 1) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/MeasurableSpace.lean", "full_name": "MeasurableEquiv.symm_mk", "start": [1240, 1], "end": [1242, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.erase_eq", "start": [2226, 1], "end": [2227, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean", "full_name": "AlgebraicTopology.DoldKan.identity_N\u2082_objectwise", "start": [227, 1], "end": [246, 72], "traced_tactics": [{"tactic": "ext n", "state_before": "C : Type u_1\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\n\u22a2 N\u2082\u0393\u2082.inv.app (N\u2082.obj P) \u226b N\u2082.map (\u0393\u2082N\u2082.natTrans.app P) = \ud835\udfd9 (N\u2082.obj P)", "state_after": "case h.h\nC : Type u_1\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f (N\u2082\u0393\u2082.inv.app (N\u2082.obj P) \u226b N\u2082.map (\u0393\u2082N\u2082.natTrans.app P)).f n =\n    HomologicalComplex.Hom.f (\ud835\udfd9 (N\u2082.obj P)).f n"}, {"tactic": "have eq\u2081 : (N\u2082\u0393\u2082.inv.app (N\u2082.obj P)).f.f n = PInfty.f n \u226b P.p.app (op [n]) \u226b\n    (\u0393\u2080.splitting (N\u2082.obj P).X).\u03b9Summand (Splitting.IndexSet.id (op [n])) := by\n  simp only [N\u2082\u0393\u2082_inv_app_f_f, N\u2082_obj_p_f, assoc]", "state_before": "case h.h\nC : Type u_1\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f (N\u2082\u0393\u2082.inv.app (N\u2082.obj P) \u226b N\u2082.map (\u0393\u2082N\u2082.natTrans.app P)).f n =\n    HomologicalComplex.Hom.f (\ud835\udfd9 (N\u2082.obj P)).f n", "state_after": "case h.h\nC : Type u_1\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\nn : \u2115\neq\u2081 :\n  HomologicalComplex.Hom.f (N\u2082\u0393\u2082.inv.app (N\u2082.obj P)).f n =\n    HomologicalComplex.Hom.f PInfty n \u226b\n      P.p.app [n].op \u226b Splitting.\u03b9Summand (\u0393\u2080.splitting (N\u2082.obj P).X) (Splitting.IndexSet.id [n].op)\n\u22a2 HomologicalComplex.Hom.f (N\u2082\u0393\u2082.inv.app (N\u2082.obj P) \u226b N\u2082.map (\u0393\u2082N\u2082.natTrans.app P)).f n =\n    HomologicalComplex.Hom.f (\ud835\udfd9 (N\u2082.obj P)).f n"}, {"tactic": "have eq\u2082 : (\u0393\u2080.splitting (N\u2082.obj P).X).\u03b9Summand (Splitting.IndexSet.id (op [n])) \u226b\n    (N\u2082.map (\u0393\u2082N\u2082.natTrans.app P)).f.f n = PInfty.f n \u226b P.p.app (op [n]) := by\n  dsimp\n  simp only [assoc, \u0393\u2082N\u2082.natTrans_app_f_app, Functor.comp_map, NatTrans.comp_app,\n    Karoubi.comp_f, compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082_hom_app, eqToHom_refl, Karoubi.eqToHom_f,\n    PInfty_on_\u0393\u2080_splitting_summand_eq_self_assoc, Functor.comp_obj]\n  dsimp [N\u2082]\n  simp only [Splitting.\u03b9_desc_assoc, assoc, id_comp, unop_op,\n    Splitting.IndexSet.id_fst, len_mk, Splitting.IndexSet.e,\n    Splitting.IndexSet.id_snd_coe, op_id, P.X.map_id, id_comp,\n    PInfty_f_naturality_assoc, PInfty_f_idem_assoc, app_idem]", "state_before": "case h.h\nC : Type u_1\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\nn : \u2115\neq\u2081 :\n  HomologicalComplex.Hom.f (N\u2082\u0393\u2082.inv.app (N\u2082.obj P)).f n =\n    HomologicalComplex.Hom.f PInfty n \u226b\n      P.p.app [n].op \u226b Splitting.\u03b9Summand (\u0393\u2080.splitting (N\u2082.obj P).X) (Splitting.IndexSet.id [n].op)\n\u22a2 HomologicalComplex.Hom.f (N\u2082\u0393\u2082.inv.app (N\u2082.obj P) \u226b N\u2082.map (\u0393\u2082N\u2082.natTrans.app P)).f n =\n    HomologicalComplex.Hom.f (\ud835\udfd9 (N\u2082.obj P)).f n", "state_after": "case h.h\nC : Type u_1\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\nn : \u2115\neq\u2081 :\n  HomologicalComplex.Hom.f (N\u2082\u0393\u2082.inv.app (N\u2082.obj P)).f n =\n    HomologicalComplex.Hom.f PInfty n \u226b\n      P.p.app [n].op \u226b Splitting.\u03b9Summand (\u0393\u2080.splitting (N\u2082.obj P).X) (Splitting.IndexSet.id [n].op)\neq\u2082 :\n  Splitting.\u03b9Summand (\u0393\u2080.splitting (N\u2082.obj P).X) (Splitting.IndexSet.id [n].op) \u226b\n      HomologicalComplex.Hom.f (N\u2082.map (\u0393\u2082N\u2082.natTrans.app P)).f n =\n    HomologicalComplex.Hom.f PInfty n \u226b P.p.app [n].op\n\u22a2 HomologicalComplex.Hom.f (N\u2082\u0393\u2082.inv.app (N\u2082.obj P) \u226b N\u2082.map (\u0393\u2082N\u2082.natTrans.app P)).f n =\n    HomologicalComplex.Hom.f (\ud835\udfd9 (N\u2082.obj P)).f n"}, {"tactic": "simp only [Karoubi.comp_f, HomologicalComplex.comp_f, Karoubi.id_eq, N\u2082_obj_p_f, assoc,\n  eq\u2081, eq\u2082, PInfty_f_naturality_assoc, app_idem, PInfty_f_idem_assoc]", "state_before": "case h.h\nC : Type u_1\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\nn : \u2115\neq\u2081 :\n  HomologicalComplex.Hom.f (N\u2082\u0393\u2082.inv.app (N\u2082.obj P)).f n =\n    HomologicalComplex.Hom.f PInfty n \u226b\n      P.p.app [n].op \u226b Splitting.\u03b9Summand (\u0393\u2080.splitting (N\u2082.obj P).X) (Splitting.IndexSet.id [n].op)\neq\u2082 :\n  Splitting.\u03b9Summand (\u0393\u2080.splitting (N\u2082.obj P).X) (Splitting.IndexSet.id [n].op) \u226b\n      HomologicalComplex.Hom.f (N\u2082.map (\u0393\u2082N\u2082.natTrans.app P)).f n =\n    HomologicalComplex.Hom.f PInfty n \u226b P.p.app [n].op\n\u22a2 HomologicalComplex.Hom.f (N\u2082\u0393\u2082.inv.app (N\u2082.obj P) \u226b N\u2082.map (\u0393\u2082N\u2082.natTrans.app P)).f n =\n    HomologicalComplex.Hom.f (\ud835\udfd9 (N\u2082.obj P)).f n", "state_after": "no goals"}, {"tactic": "simp only [N\u2082\u0393\u2082_inv_app_f_f, N\u2082_obj_p_f, assoc]", "state_before": "C : Type u_1\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f (N\u2082\u0393\u2082.inv.app (N\u2082.obj P)).f n =\n    HomologicalComplex.Hom.f PInfty n \u226b\n      P.p.app [n].op \u226b Splitting.\u03b9Summand (\u0393\u2080.splitting (N\u2082.obj P).X) (Splitting.IndexSet.id [n].op)", "state_after": "no goals"}, {"tactic": "dsimp", "state_before": "C : Type u_1\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\nn : \u2115\neq\u2081 :\n  HomologicalComplex.Hom.f (N\u2082\u0393\u2082.inv.app (N\u2082.obj P)).f n =\n    HomologicalComplex.Hom.f PInfty n \u226b\n      P.p.app [n].op \u226b Splitting.\u03b9Summand (\u0393\u2080.splitting (N\u2082.obj P).X) (Splitting.IndexSet.id [n].op)\n\u22a2 Splitting.\u03b9Summand (\u0393\u2080.splitting (N\u2082.obj P).X) (Splitting.IndexSet.id [n].op) \u226b\n      HomologicalComplex.Hom.f (N\u2082.map (\u0393\u2082N\u2082.natTrans.app P)).f n =\n    HomologicalComplex.Hom.f PInfty n \u226b P.p.app [n].op", "state_after": "C : Type u_1\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\nn : \u2115\neq\u2081 :\n  HomologicalComplex.Hom.f (N\u2082\u0393\u2082.inv.app (N\u2082.obj P)).f n =\n    HomologicalComplex.Hom.f PInfty n \u226b\n      P.p.app [n].op \u226b Splitting.\u03b9Summand (\u0393\u2080.splitting (N\u2082.obj P).X) (Splitting.IndexSet.id [n].op)\n\u22a2 Splitting.\u03b9Summand (\u0393\u2080.splitting (N\u2082.obj P).X) (Splitting.IndexSet.id [n].op) \u226b\n      HomologicalComplex.Hom.f PInfty n \u226b (\u0393\u2082N\u2082.natTrans.app P).f.app [n].op =\n    HomologicalComplex.Hom.f PInfty n \u226b P.p.app [n].op"}, {"tactic": "simp only [assoc, \u0393\u2082N\u2082.natTrans_app_f_app, Functor.comp_map, NatTrans.comp_app,\n  Karoubi.comp_f, compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082_hom_app, eqToHom_refl, Karoubi.eqToHom_f,\n  PInfty_on_\u0393\u2080_splitting_summand_eq_self_assoc, Functor.comp_obj]", "state_before": "C : Type u_1\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\nn : \u2115\neq\u2081 :\n  HomologicalComplex.Hom.f (N\u2082\u0393\u2082.inv.app (N\u2082.obj P)).f n =\n    HomologicalComplex.Hom.f PInfty n \u226b\n      P.p.app [n].op \u226b Splitting.\u03b9Summand (\u0393\u2080.splitting (N\u2082.obj P).X) (Splitting.IndexSet.id [n].op)\n\u22a2 Splitting.\u03b9Summand (\u0393\u2080.splitting (N\u2082.obj P).X) (Splitting.IndexSet.id [n].op) \u226b\n      HomologicalComplex.Hom.f PInfty n \u226b (\u0393\u2082N\u2082.natTrans.app P).f.app [n].op =\n    HomologicalComplex.Hom.f PInfty n \u226b P.p.app [n].op", "state_after": "C : Type u_1\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\nn : \u2115\neq\u2081 :\n  HomologicalComplex.Hom.f (N\u2082\u0393\u2082.inv.app (N\u2082.obj P)).f n =\n    HomologicalComplex.Hom.f PInfty n \u226b\n      P.p.app [n].op \u226b Splitting.\u03b9Summand (\u0393\u2080.splitting (N\u2082.obj P).X) (Splitting.IndexSet.id [n].op)\n\u22a2 Splitting.\u03b9Summand (\u0393\u2080.splitting (N\u2082.obj P).X) (Splitting.IndexSet.id [n].op) \u226b\n      ((\u0393\u2082.map (N\u2082.map (Karoubi.decompId_i P))).f \u226b\n            (\u0393\u2082.obj (N\u2082.obj (Karoubi.mk P.X (\ud835\udfd9 P.X)))).p \u226b\n              \ud835\udfd9 (\u0393\u2082.obj (N\u2082.obj (Karoubi.mk P.X (\ud835\udfd9 P.X)))).X \u226b (\u0393\u2082N\u2081.natTrans.app P.X).f \u226b (Karoubi.decompId_p P).f).app\n        [n].op =\n    HomologicalComplex.Hom.f PInfty n \u226b P.p.app [n].op"}, {"tactic": "dsimp [N\u2082]", "state_before": "C : Type u_1\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\nn : \u2115\neq\u2081 :\n  HomologicalComplex.Hom.f (N\u2082\u0393\u2082.inv.app (N\u2082.obj P)).f n =\n    HomologicalComplex.Hom.f PInfty n \u226b\n      P.p.app [n].op \u226b Splitting.\u03b9Summand (\u0393\u2080.splitting (N\u2082.obj P).X) (Splitting.IndexSet.id [n].op)\n\u22a2 Splitting.\u03b9Summand (\u0393\u2080.splitting (N\u2082.obj P).X) (Splitting.IndexSet.id [n].op) \u226b\n      ((\u0393\u2082.map (N\u2082.map (Karoubi.decompId_i P))).f \u226b\n            (\u0393\u2082.obj (N\u2082.obj (Karoubi.mk P.X (\ud835\udfd9 P.X)))).p \u226b\n              \ud835\udfd9 (\u0393\u2082.obj (N\u2082.obj (Karoubi.mk P.X (\ud835\udfd9 P.X)))).X \u226b (\u0393\u2082N\u2081.natTrans.app P.X).f \u226b (Karoubi.decompId_p P).f).app\n        [n].op =\n    HomologicalComplex.Hom.f PInfty n \u226b P.p.app [n].op", "state_after": "C : Type u_1\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\nn : \u2115\neq\u2081 :\n  HomologicalComplex.Hom.f (N\u2082\u0393\u2082.inv.app (N\u2082.obj P)).f n =\n    HomologicalComplex.Hom.f PInfty n \u226b\n      P.p.app [n].op \u226b Splitting.\u03b9Summand (\u0393\u2080.splitting (N\u2082.obj P).X) (Splitting.IndexSet.id [n].op)\n\u22a2 Splitting.\u03b9Summand (\u0393\u2080.splitting K[P.X]) (Splitting.IndexSet.id [n].op) \u226b\n      (Splitting.desc (\u0393\u2080.splitting K[P.X]) [n].op fun A =>\n          (HomologicalComplex.Hom.f PInfty (len A.fst.unop) \u226b P.p.app A.fst) \u226b\n            Splitting.\u03b9Summand (\u0393\u2080.splitting K[P.X]) A) \u226b\n        (Splitting.desc (\u0393\u2080.splitting K[P.X]) [n].op fun A =>\n            (HomologicalComplex.Hom.f PInfty (len A.fst.unop) \u226b \ud835\udfd9 (P.X.obj A.fst)) \u226b\n              Splitting.\u03b9Summand (\u0393\u2080.splitting K[P.X]) A) \u226b\n          \ud835\udfd9 (\u0393\u2080.Obj.obj\u2082 K[P.X] [n].op) \u226b\n            (Splitting.desc (\u0393\u2080.splitting K[P.X]) [n].op fun A =>\n                HomologicalComplex.Hom.f PInfty (len A.fst.unop) \u226b P.X.map (Splitting.IndexSet.e A).op) \u226b\n              P.p.app [n].op =\n    HomologicalComplex.Hom.f PInfty n \u226b P.p.app [n].op"}, {"tactic": "simp only [Splitting.\u03b9_desc_assoc, assoc, id_comp, unop_op,\n  Splitting.IndexSet.id_fst, len_mk, Splitting.IndexSet.e,\n  Splitting.IndexSet.id_snd_coe, op_id, P.X.map_id, id_comp,\n  PInfty_f_naturality_assoc, PInfty_f_idem_assoc, app_idem]", "state_before": "C : Type u_1\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\nn : \u2115\neq\u2081 :\n  HomologicalComplex.Hom.f (N\u2082\u0393\u2082.inv.app (N\u2082.obj P)).f n =\n    HomologicalComplex.Hom.f PInfty n \u226b\n      P.p.app [n].op \u226b Splitting.\u03b9Summand (\u0393\u2080.splitting (N\u2082.obj P).X) (Splitting.IndexSet.id [n].op)\n\u22a2 Splitting.\u03b9Summand (\u0393\u2080.splitting K[P.X]) (Splitting.IndexSet.id [n].op) \u226b\n      (Splitting.desc (\u0393\u2080.splitting K[P.X]) [n].op fun A =>\n          (HomologicalComplex.Hom.f PInfty (len A.fst.unop) \u226b P.p.app A.fst) \u226b\n            Splitting.\u03b9Summand (\u0393\u2080.splitting K[P.X]) A) \u226b\n        (Splitting.desc (\u0393\u2080.splitting K[P.X]) [n].op fun A =>\n            (HomologicalComplex.Hom.f PInfty (len A.fst.unop) \u226b \ud835\udfd9 (P.X.obj A.fst)) \u226b\n              Splitting.\u03b9Summand (\u0393\u2080.splitting K[P.X]) A) \u226b\n          \ud835\udfd9 (\u0393\u2080.Obj.obj\u2082 K[P.X] [n].op) \u226b\n            (Splitting.desc (\u0393\u2080.splitting K[P.X]) [n].op fun A =>\n                HomologicalComplex.Hom.f PInfty (len A.fst.unop) \u226b P.X.map (Splitting.IndexSet.e A).op) \u226b\n              P.p.app [n].op =\n    HomologicalComplex.Hom.f PInfty n \u226b P.p.app [n].op", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/FreeGroup.lean", "full_name": "FreeGroup.reduce.idem", "start": [1205, 1], "end": [1206, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Sylow.lean", "full_name": "Sylow.smul_subtype", "start": [257, 1], "end": [259, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/TensorProduct.lean", "full_name": "TensorProduct.lift.tmul'", "start": [485, 1], "end": [486, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/Adjoin.lean", "full_name": "IntermediateField.minpoly.natDegree_le", "start": [878, 1], "end": [880, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/RelIso/Basic.lean", "full_name": "RelEmbedding.coe_trans", "start": [318, 1], "end": [319, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Star/Pointwise.lean", "full_name": "Set.star_add", "start": [128, 11], "end": [130, 13], "traced_tactics": [{"tactic": "simp_rw [\u2190 image_star, \u2190 image2_add, image_image2, image2_image_left, image2_image_right,\n  star_add]", "state_before": "\u03b1 : Type u_1\ns\u271d t\u271d : Set \u03b1\na : \u03b1\ninst\u271d\u00b9 : AddMonoid \u03b1\ninst\u271d : StarAddMonoid \u03b1\ns t : Set \u03b1\n\u22a2 (s + t)\u22c6 = s\u22c6 + t\u22c6", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.cast_id", "start": [227, 1], "end": [229, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean", "full_name": "BoxIntegral.TaggedPrepartition.IsHenstock.card_filter_tag_eq_le", "start": [238, 1], "end": [246, 72], "traced_tactics": [{"tactic": "refine' Finset.card_le_of_subset fun J hJ => _", "state_before": "\u03b9 : Type u_1\nI J : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nx\u271d : \u03b9 \u2192 \u211d\ninst\u271d : Fintype \u03b9\nh : IsHenstock \u03c0\nx : \u03b9 \u2192 \u211d\n\u22a2 Finset.card (Finset.filter (fun J => tag \u03c0 J = x) \u03c0.boxes) \u2264\n    Finset.card (Finset.filter (fun J => x \u2208 \u2191Box.Icc J) \u03c0.boxes)", "state_after": "\u03b9 : Type u_1\nI J\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nx\u271d : \u03b9 \u2192 \u211d\ninst\u271d : Fintype \u03b9\nh : IsHenstock \u03c0\nx : \u03b9 \u2192 \u211d\nJ : Box \u03b9\nhJ : J \u2208 Finset.filter (fun J => tag \u03c0 J = x) \u03c0.boxes\n\u22a2 J \u2208 Finset.filter (fun J => x \u2208 \u2191Box.Icc J) \u03c0.boxes"}, {"tactic": "rw [Finset.mem_filter] at hJ\u22a2", "state_before": "\u03b9 : Type u_1\nI J\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nx\u271d : \u03b9 \u2192 \u211d\ninst\u271d : Fintype \u03b9\nh : IsHenstock \u03c0\nx : \u03b9 \u2192 \u211d\nJ : Box \u03b9\nhJ : J \u2208 Finset.filter (fun J => tag \u03c0 J = x) \u03c0.boxes\n\u22a2 J \u2208 Finset.filter (fun J => x \u2208 \u2191Box.Icc J) \u03c0.boxes", "state_after": "\u03b9 : Type u_1\nI J\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nx\u271d : \u03b9 \u2192 \u211d\ninst\u271d : Fintype \u03b9\nh : IsHenstock \u03c0\nx : \u03b9 \u2192 \u211d\nJ : Box \u03b9\nhJ : J \u2208 \u03c0.boxes \u2227 tag \u03c0 J = x\n\u22a2 J \u2208 \u03c0.boxes \u2227 x \u2208 \u2191Box.Icc J"}, {"tactic": "rcases hJ with \u27e8hJ, rfl\u27e9", "state_before": "\u03b9 : Type u_1\nI J\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nx\u271d : \u03b9 \u2192 \u211d\ninst\u271d : Fintype \u03b9\nh : IsHenstock \u03c0\nx : \u03b9 \u2192 \u211d\nJ : Box \u03b9\nhJ : J \u2208 \u03c0.boxes \u2227 tag \u03c0 J = x\n\u22a2 J \u2208 \u03c0.boxes \u2227 x \u2208 \u2191Box.Icc J", "state_after": "case intro\n\u03b9 : Type u_1\nI J\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nx : \u03b9 \u2192 \u211d\ninst\u271d : Fintype \u03b9\nh : IsHenstock \u03c0\nJ : Box \u03b9\nhJ : J \u2208 \u03c0.boxes\n\u22a2 J \u2208 \u03c0.boxes \u2227 tag \u03c0 J \u2208 \u2191Box.Icc J"}, {"tactic": "exact \u27e8hJ, h J hJ\u27e9", "state_before": "case intro\n\u03b9 : Type u_1\nI J\u271d : Box \u03b9\n\u03c0 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I\nx : \u03b9 \u2192 \u211d\ninst\u271d : Fintype \u03b9\nh : IsHenstock \u03c0\nJ : Box \u03b9\nhJ : J \u2208 \u03c0.boxes\n\u22a2 J \u2208 \u03c0.boxes \u2227 tag \u03c0 J \u2208 \u2191Box.Icc J", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean", "full_name": "NonUnitalSubsemiring.comap_top", "start": [765, 1], "end": [766, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Hom/CompleteLattice.lean", "full_name": "sSupHom.symm_dual_id", "start": [811, 1], "end": [812, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/WellFoundedSet.lean", "full_name": "Set.wellFoundedOn_singleton", "start": [498, 1], "end": [499, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Group/Prod.lean", "full_name": "MeasureTheory.lintegral_lintegral_mul_inv", "start": [198, 1], "end": [210, 21], "traced_tactics": [{"tactic": "have h : Measurable fun z : G \u00d7 G => (z.2 * z.1, z.1\u207b\u00b9) :=\n  (measurable_snd.mul measurable_fst).prod_mk measurable_fst.inv", "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nf : G \u2192 G \u2192 \u211d\u22650\u221e\nhf : AEMeasurable (uncurry f)\n\u22a2 (\u222b\u207b (x : G), \u222b\u207b (y : G), f (y * x) x\u207b\u00b9 \u2202\u03bd \u2202\u03bc) = \u222b\u207b (x : G), \u222b\u207b (y : G), f x y \u2202\u03bd \u2202\u03bc", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nf : G \u2192 G \u2192 \u211d\u22650\u221e\nhf : AEMeasurable (uncurry f)\nh : Measurable fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\n\u22a2 (\u222b\u207b (x : G), \u222b\u207b (y : G), f (y * x) x\u207b\u00b9 \u2202\u03bd \u2202\u03bc) = \u222b\u207b (x : G), \u222b\u207b (y : G), f x y \u2202\u03bd \u2202\u03bc"}, {"tactic": "have h2f : AEMeasurable (uncurry fun x y => f (y * x) x\u207b\u00b9) (\u03bc.prod \u03bd) :=\n  hf.comp_quasiMeasurePreserving (measurePreserving_mul_prod_inv \u03bc \u03bd).quasiMeasurePreserving", "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nf : G \u2192 G \u2192 \u211d\u22650\u221e\nhf : AEMeasurable (uncurry f)\nh : Measurable fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\n\u22a2 (\u222b\u207b (x : G), \u222b\u207b (y : G), f (y * x) x\u207b\u00b9 \u2202\u03bd \u2202\u03bc) = \u222b\u207b (x : G), \u222b\u207b (y : G), f x y \u2202\u03bd \u2202\u03bc", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nf : G \u2192 G \u2192 \u211d\u22650\u221e\nhf : AEMeasurable (uncurry f)\nh : Measurable fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x\u207b\u00b9)\n\u22a2 (\u222b\u207b (x : G), \u222b\u207b (y : G), f (y * x) x\u207b\u00b9 \u2202\u03bd \u2202\u03bc) = \u222b\u207b (x : G), \u222b\u207b (y : G), f x y \u2202\u03bd \u2202\u03bc"}, {"tactic": "simp_rw [lintegral_lintegral h2f, lintegral_lintegral hf]", "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nf : G \u2192 G \u2192 \u211d\u22650\u221e\nhf : AEMeasurable (uncurry f)\nh : Measurable fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x\u207b\u00b9)\n\u22a2 (\u222b\u207b (x : G), \u222b\u207b (y : G), f (y * x) x\u207b\u00b9 \u2202\u03bd \u2202\u03bc) = \u222b\u207b (x : G), \u222b\u207b (y : G), f x y \u2202\u03bd \u2202\u03bc", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nf : G \u2192 G \u2192 \u211d\u22650\u221e\nhf : AEMeasurable (uncurry f)\nh : Measurable fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x\u207b\u00b9)\n\u22a2 (\u222b\u207b (z : G \u00d7 G), f (z.snd * z.fst) z.fst\u207b\u00b9 \u2202Measure.prod \u03bc \u03bd) = \u222b\u207b (z : G \u00d7 G), f z.fst z.snd \u2202Measure.prod \u03bc \u03bd"}, {"tactic": "conv_rhs => rw [\u2190 (measurePreserving_mul_prod_inv \u03bc \u03bd).map_eq]", "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nf : G \u2192 G \u2192 \u211d\u22650\u221e\nhf : AEMeasurable (uncurry f)\nh : Measurable fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x\u207b\u00b9)\n\u22a2 (\u222b\u207b (z : G \u00d7 G), f (z.snd * z.fst) z.fst\u207b\u00b9 \u2202Measure.prod \u03bc \u03bd) = \u222b\u207b (z : G \u00d7 G), f z.fst z.snd \u2202Measure.prod \u03bc \u03bd", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nf : G \u2192 G \u2192 \u211d\u22650\u221e\nhf : AEMeasurable (uncurry f)\nh : Measurable fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x\u207b\u00b9)\n\u22a2 (\u222b\u207b (z : G \u00d7 G), f (z.snd * z.fst) z.fst\u207b\u00b9 \u2202Measure.prod \u03bc \u03bd) =\n    \u222b\u207b (z : G \u00d7 G), f z.fst z.snd \u2202map (fun z => (z.snd * z.fst, z.fst\u207b\u00b9)) (Measure.prod \u03bc \u03bd)"}, {"tactic": "symm", "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nf : G \u2192 G \u2192 \u211d\u22650\u221e\nhf : AEMeasurable (uncurry f)\nh : Measurable fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x\u207b\u00b9)\n\u22a2 (\u222b\u207b (z : G \u00d7 G), f (z.snd * z.fst) z.fst\u207b\u00b9 \u2202Measure.prod \u03bc \u03bd) =\n    \u222b\u207b (z : G \u00d7 G), f z.fst z.snd \u2202map (fun z => (z.snd * z.fst, z.fst\u207b\u00b9)) (Measure.prod \u03bc \u03bd)", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nf : G \u2192 G \u2192 \u211d\u22650\u221e\nhf : AEMeasurable (uncurry f)\nh : Measurable fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x\u207b\u00b9)\n\u22a2 (\u222b\u207b (z : G \u00d7 G), f z.fst z.snd \u2202map (fun z => (z.snd * z.fst, z.fst\u207b\u00b9)) (Measure.prod \u03bc \u03bd)) =\n    \u222b\u207b (z : G \u00d7 G), f (z.snd * z.fst) z.fst\u207b\u00b9 \u2202Measure.prod \u03bc \u03bd"}, {"tactic": "exact\n  lintegral_map' (hf.mono' (measurePreserving_mul_prod_inv \u03bc \u03bd).map_eq.absolutelyContinuous)\n    h.aemeasurable", "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nf : G \u2192 G \u2192 \u211d\u22650\u221e\nhf : AEMeasurable (uncurry f)\nh : Measurable fun z => (z.snd * z.fst, z.fst\u207b\u00b9)\nh2f : AEMeasurable (uncurry fun x y => f (y * x) x\u207b\u00b9)\n\u22a2 (\u222b\u207b (z : G \u00d7 G), f z.fst z.snd \u2202map (fun z => (z.snd * z.fst, z.fst\u207b\u00b9)) (Measure.prod \u03bc \u03bd)) =\n    \u222b\u207b (z : G \u00d7 G), f (z.snd * z.fst) z.fst\u207b\u00b9 \u2202Measure.prod \u03bc \u03bd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean", "full_name": "MeasureTheory.Integrable.comp_smul", "start": [155, 1], "end": [158, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/GaloisConnection.lean", "full_name": "GaloisConnection.exists_eq_l", "start": [222, 1], "end": [223, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/ToIntervalMod.lean", "full_name": "toIcoMod_mem_Ico'", "start": [89, 1], "end": [91, 26], "traced_tactics": [{"tactic": "convert toIcoMod_mem_Ico hp 0 b", "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nh\u03b1 : Archimedean \u03b1\np : \u03b1\nhp : 0 < p\na b\u271d c : \u03b1\nn : \u2124\nb : \u03b1\n\u22a2 toIcoMod hp 0 b \u2208 Set.Ico 0 p", "state_after": "case h.e'_5.h.e'_4\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nh\u03b1 : Archimedean \u03b1\np : \u03b1\nhp : 0 < p\na b\u271d c : \u03b1\nn : \u2124\nb : \u03b1\n\u22a2 p = 0 + p"}, {"tactic": "exact (zero_add p).symm", "state_before": "case h.e'_5.h.e'_4\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nh\u03b1 : Archimedean \u03b1\np : \u03b1\nhp : 0 < p\na b\u271d c : \u03b1\nn : \u2124\nb : \u03b1\n\u22a2 p = 0 + p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Rel.lean", "full_name": "Rel.mem_core", "start": [215, 1], "end": [216, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Holder.lean", "full_name": "HolderWith.continuous", "start": [220, 11], "end": [221, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "full_name": "Metric.hausdorffDist_le_of_mem_dist", "start": [769, 1], "end": [777, 50], "traced_tactics": [{"tactic": "apply hausdorffDist_le_of_infDist hr", "state_before": "\u03b9 : Sort ?u.77239\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 dist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 dist x y \u2264 r\n\u22a2 hausdorffDist s t \u2264 r", "state_after": "case H1\n\u03b9 : Sort ?u.77239\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 dist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 dist x y \u2264 r\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r\n\ncase H2\n\u03b9 : Sort ?u.77239\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 dist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 dist x y \u2264 r\n\u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r"}, {"tactic": "intro x xs", "state_before": "case H1\n\u03b9 : Sort ?u.77239\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 dist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 dist x y \u2264 r\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r", "state_after": "case H1\n\u03b9 : Sort ?u.77239\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 dist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 dist x y \u2264 r\nx : \u03b1\nxs : x \u2208 s\n\u22a2 infDist x t \u2264 r"}, {"tactic": "rcases H1 x xs with \u27e8y, yt, hy\u27e9", "state_before": "case H1\n\u03b9 : Sort ?u.77239\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 dist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 dist x y \u2264 r\nx : \u03b1\nxs : x \u2208 s\n\u22a2 infDist x t \u2264 r", "state_after": "case H1.intro.intro\n\u03b9 : Sort ?u.77239\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y\u271d : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 dist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 dist x y \u2264 r\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\nhy : dist x y \u2264 r\n\u22a2 infDist x t \u2264 r"}, {"tactic": "exact le_trans (infDist_le_dist_of_mem yt) hy", "state_before": "case H1.intro.intro\n\u03b9 : Sort ?u.77239\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y\u271d : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 dist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 dist x y \u2264 r\nx : \u03b1\nxs : x \u2208 s\ny : \u03b1\nyt : y \u2208 t\nhy : dist x y \u2264 r\n\u22a2 infDist x t \u2264 r", "state_after": "no goals"}, {"tactic": "intro x xt", "state_before": "case H2\n\u03b9 : Sort ?u.77239\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 dist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 dist x y \u2264 r\n\u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r", "state_after": "case H2\n\u03b9 : Sort ?u.77239\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 dist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 dist x y \u2264 r\nx : \u03b1\nxt : x \u2208 t\n\u22a2 infDist x s \u2264 r"}, {"tactic": "rcases H2 x xt with \u27e8y, ys, hy\u27e9", "state_before": "case H2\n\u03b9 : Sort ?u.77239\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 dist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 dist x y \u2264 r\nx : \u03b1\nxt : x \u2208 t\n\u22a2 infDist x s \u2264 r", "state_after": "case H2.intro.intro\n\u03b9 : Sort ?u.77239\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y\u271d : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 dist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 dist x y \u2264 r\nx : \u03b1\nxt : x \u2208 t\ny : \u03b1\nys : y \u2208 s\nhy : dist x y \u2264 r\n\u22a2 infDist x s \u2264 r"}, {"tactic": "exact le_trans (infDist_le_dist_of_mem ys) hy", "state_before": "case H2.intro.intro\n\u03b9 : Sort ?u.77239\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\ns t u : Set \u03b1\nx\u271d y\u271d : \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\nr : \u211d\nhr : 0 \u2264 r\nH1 : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 dist x y \u2264 r\nH2 : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 y, y \u2208 s \u2227 dist x y \u2264 r\nx : \u03b1\nxt : x \u2208 t\ny : \u03b1\nys : y \u2208 s\nhy : dist x y \u2264 r\n\u22a2 infDist x s \u2264 r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/CauSeq.lean", "full_name": "CauSeq.pow_apply", "start": [397, 1], "end": [398, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Deprecated/Subgroup.lean", "full_name": "Group.mem_closure_union_iff", "start": [651, 1], "end": [660, 48], "traced_tactics": [{"tactic": "simp only [closure_eq_mclosure, Monoid.mem_closure_union_iff, exists_prop, preimage_union]", "state_before": "G\u271d : Type ?u.136865\nH : Type ?u.136868\nA : Type ?u.136871\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_1\ninst\u271d : CommGroup G\ns t : Set G\nx : G\n\u22a2 x \u2208 closure (s \u222a t) \u2194 \u2203 y, y \u2208 closure s \u2227 \u2203 z, z \u2208 closure t \u2227 y * z = x", "state_after": "G\u271d : Type ?u.136865\nH : Type ?u.136868\nA : Type ?u.136871\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_1\ninst\u271d : CommGroup G\ns t : Set G\nx : G\n\u22a2 (\u2203 y,\n      (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure t \u2227 y_1 * z = y) \u2227\n        \u2203 z,\n          (\u2203 y, y \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227\n            y * z = x) \u2194\n    \u2203 y,\n      (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 y_1 * z = y) \u2227\n        \u2203 z, (\u2203 y, y \u2208 Monoid.Closure t \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227 y * z = x"}, {"tactic": "constructor", "state_before": "G\u271d : Type ?u.136865\nH : Type ?u.136868\nA : Type ?u.136871\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_1\ninst\u271d : CommGroup G\ns t : Set G\nx : G\n\u22a2 (\u2203 y,\n      (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure t \u2227 y_1 * z = y) \u2227\n        \u2203 z,\n          (\u2203 y, y \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227\n            y * z = x) \u2194\n    \u2203 y,\n      (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 y_1 * z = y) \u2227\n        \u2203 z, (\u2203 y, y \u2208 Monoid.Closure t \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227 y * z = x", "state_after": "case mp\nG\u271d : Type ?u.136865\nH : Type ?u.136868\nA : Type ?u.136871\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_1\ninst\u271d : CommGroup G\ns t : Set G\nx : G\n\u22a2 (\u2203 y,\n      (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure t \u2227 y_1 * z = y) \u2227\n        \u2203 z,\n          (\u2203 y, y \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227\n            y * z = x) \u2192\n    \u2203 y,\n      (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 y_1 * z = y) \u2227\n        \u2203 z, (\u2203 y, y \u2208 Monoid.Closure t \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227 y * z = x\n\ncase mpr\nG\u271d : Type ?u.136865\nH : Type ?u.136868\nA : Type ?u.136871\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_1\ninst\u271d : CommGroup G\ns t : Set G\nx : G\n\u22a2 (\u2203 y,\n      (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 y_1 * z = y) \u2227\n        \u2203 z, (\u2203 y, y \u2208 Monoid.Closure t \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227 y * z = x) \u2192\n    \u2203 y,\n      (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure t \u2227 y_1 * z = y) \u2227\n        \u2203 z,\n          (\u2203 y, y \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227\n            y * z = x"}, {"tactic": "rintro \u27e8_, \u27e8ys, hys, yt, hyt, rfl\u27e9, _, \u27e8zs, hzs, zt, hzt, rfl\u27e9, rfl\u27e9", "state_before": "case mp\nG\u271d : Type ?u.136865\nH : Type ?u.136868\nA : Type ?u.136871\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_1\ninst\u271d : CommGroup G\ns t : Set G\nx : G\n\u22a2 (\u2203 y,\n      (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure t \u2227 y_1 * z = y) \u2227\n        \u2203 z,\n          (\u2203 y, y \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227\n            y * z = x) \u2192\n    \u2203 y,\n      (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 y_1 * z = y) \u2227\n        \u2203 z, (\u2203 y, y \u2208 Monoid.Closure t \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227 y * z = x", "state_after": "case mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG\u271d : Type ?u.136865\nH : Type ?u.136868\nA : Type ?u.136871\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_1\ninst\u271d : CommGroup G\ns t : Set G\nys : G\nhys : ys \u2208 Monoid.Closure s\nyt : G\nhyt : yt \u2208 Monoid.Closure t\nzs : G\nhzs : zs \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s)\nzt : G\nhzt : zt \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t)\n\u22a2 \u2203 y,\n    (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 y_1 * z = y) \u2227\n      \u2203 z,\n        (\u2203 y, y \u2208 Monoid.Closure t \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227\n          y * z = ys * yt * (zs * zt)"}, {"tactic": "refine' \u27e8_, \u27e8_, hys, _, hzs, rfl\u27e9, _, \u27e8_, hyt, _, hzt, rfl\u27e9, _\u27e9", "state_before": "case mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG\u271d : Type ?u.136865\nH : Type ?u.136868\nA : Type ?u.136871\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_1\ninst\u271d : CommGroup G\ns t : Set G\nys : G\nhys : ys \u2208 Monoid.Closure s\nyt : G\nhyt : yt \u2208 Monoid.Closure t\nzs : G\nhzs : zs \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s)\nzt : G\nhzt : zt \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t)\n\u22a2 \u2203 y,\n    (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 y_1 * z = y) \u2227\n      \u2203 z,\n        (\u2203 y, y \u2208 Monoid.Closure t \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227\n          y * z = ys * yt * (zs * zt)", "state_after": "case mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG\u271d : Type ?u.136865\nH : Type ?u.136868\nA : Type ?u.136871\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_1\ninst\u271d : CommGroup G\ns t : Set G\nys : G\nhys : ys \u2208 Monoid.Closure s\nyt : G\nhyt : yt \u2208 Monoid.Closure t\nzs : G\nhzs : zs \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s)\nzt : G\nhzt : zt \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t)\n\u22a2 ys * zs * (yt * zt) = ys * yt * (zs * zt)"}, {"tactic": "rw [mul_assoc, mul_assoc, mul_left_comm zs]", "state_before": "case mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG\u271d : Type ?u.136865\nH : Type ?u.136868\nA : Type ?u.136871\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_1\ninst\u271d : CommGroup G\ns t : Set G\nys : G\nhys : ys \u2208 Monoid.Closure s\nyt : G\nhyt : yt \u2208 Monoid.Closure t\nzs : G\nhzs : zs \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s)\nzt : G\nhzt : zt \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t)\n\u22a2 ys * zs * (yt * zt) = ys * yt * (zs * zt)", "state_after": "no goals"}, {"tactic": "rintro \u27e8_, \u27e8ys, hys, zs, hzs, rfl\u27e9, _, \u27e8yt, hyt, zt, hzt, rfl\u27e9, rfl\u27e9", "state_before": "case mpr\nG\u271d : Type ?u.136865\nH : Type ?u.136868\nA : Type ?u.136871\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_1\ninst\u271d : CommGroup G\ns t : Set G\nx : G\n\u22a2 (\u2203 y,\n      (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 y_1 * z = y) \u2227\n        \u2203 z, (\u2203 y, y \u2208 Monoid.Closure t \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227 y * z = x) \u2192\n    \u2203 y,\n      (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure t \u2227 y_1 * z = y) \u2227\n        \u2203 z,\n          (\u2203 y, y \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227\n            y * z = x", "state_after": "case mpr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG\u271d : Type ?u.136865\nH : Type ?u.136868\nA : Type ?u.136871\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_1\ninst\u271d : CommGroup G\ns t : Set G\nys : G\nhys : ys \u2208 Monoid.Closure s\nzs : G\nhzs : zs \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s)\nyt : G\nhyt : yt \u2208 Monoid.Closure t\nzt : G\nhzt : zt \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t)\n\u22a2 \u2203 y,\n    (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure t \u2227 y_1 * z = y) \u2227\n      \u2203 z,\n        (\u2203 y, y \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227\n          y * z = ys * zs * (yt * zt)"}, {"tactic": "refine' \u27e8_, \u27e8ys, hys, yt, hyt, rfl\u27e9, _, \u27e8zs, hzs, zt, hzt, rfl\u27e9, _\u27e9", "state_before": "case mpr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG\u271d : Type ?u.136865\nH : Type ?u.136868\nA : Type ?u.136871\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_1\ninst\u271d : CommGroup G\ns t : Set G\nys : G\nhys : ys \u2208 Monoid.Closure s\nzs : G\nhzs : zs \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s)\nyt : G\nhyt : yt \u2208 Monoid.Closure t\nzt : G\nhzt : zt \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t)\n\u22a2 \u2203 y,\n    (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure t \u2227 y_1 * z = y) \u2227\n      \u2203 z,\n        (\u2203 y, y \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227\n          y * z = ys * zs * (yt * zt)", "state_after": "case mpr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG\u271d : Type ?u.136865\nH : Type ?u.136868\nA : Type ?u.136871\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_1\ninst\u271d : CommGroup G\ns t : Set G\nys : G\nhys : ys \u2208 Monoid.Closure s\nzs : G\nhzs : zs \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s)\nyt : G\nhyt : yt \u2208 Monoid.Closure t\nzt : G\nhzt : zt \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t)\n\u22a2 ys * yt * (zs * zt) = ys * zs * (yt * zt)"}, {"tactic": "rw [mul_assoc, mul_assoc, mul_left_comm yt]", "state_before": "case mpr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG\u271d : Type ?u.136865\nH : Type ?u.136868\nA : Type ?u.136871\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_1\ninst\u271d : CommGroup G\ns t : Set G\nys : G\nhys : ys \u2208 Monoid.Closure s\nzs : G\nhzs : zs \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s)\nyt : G\nhyt : yt \u2208 Monoid.Closure t\nzt : G\nhzt : zt \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t)\n\u22a2 ys * yt * (zs * zt) = ys * zs * (yt * zt)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/CharP/Basic.lean", "full_name": "frobenius_neg", "start": [433, 1], "end": [434, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/HahnSeries.lean", "full_name": "HahnSeries.C_ne_zero", "start": [983, 1], "end": [986, 22], "traced_tactics": [{"tactic": "contrapose! h", "state_before": "\u0393 : Type u_2\nR : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nr : R\nh : r \u2260 0\n\u22a2 \u2191C r \u2260 0", "state_after": "\u0393 : Type u_2\nR : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nr : R\nh : \u2191C r = 0\n\u22a2 r = 0"}, {"tactic": "rw [\u2190 C_zero] at h", "state_before": "\u0393 : Type u_2\nR : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nr : R\nh : \u2191C r = 0\n\u22a2 r = 0", "state_after": "\u0393 : Type u_2\nR : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nr : R\nh : \u2191C r = \u2191C 0\n\u22a2 r = 0"}, {"tactic": "exact C_injective h", "state_before": "\u0393 : Type u_2\nR : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393\ninst\u271d : NonAssocSemiring R\nr : R\nh : \u2191C r = \u2191C 0\n\u22a2 r = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Star/Basic.lean", "full_name": "star_ratCast", "start": [353, 1], "end": [354, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Quotient.lean", "full_name": "LinearMap.range_mkQ_comp", "start": [586, 1], "end": [587, 33], "traced_tactics": [{"tactic": "simp", "state_before": "R : Type u_1\nM : Type u_3\nR\u2082 : Type u_2\nM\u2082 : Type u_4\nR\u2083 : Type ?u.414158\nM\u2083 : Type ?u.414161\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : Ring R\u2082\ninst\u271d\u2078 : Ring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nx : M\n\u22a2 \u2191(comp (mkQ (range f)) f) x = \u21910 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Monoidal/Braided.lean", "full_name": "CategoryTheory.tensor_\u03bc_natural", "start": [402, 1], "end": [415, 20], "traced_tactics": [{"tactic": "dsimp [tensor_\u03bc]", "state_before": "C : Type u\u2081\ninst\u271d\u2078 : Category C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n\u22a2 ((f\u2081 \u2297 f\u2082) \u2297 g\u2081 \u2297 g\u2082) \u226b tensor_\u03bc C (Y\u2081, Y\u2082) (V\u2081, V\u2082) = tensor_\u03bc C (X\u2081, X\u2082) (U\u2081, U\u2082) \u226b ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082)", "state_after": "C : Type u\u2081\ninst\u271d\u2078 : Category C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n\u22a2 ((f\u2081 \u2297 f\u2082) \u2297 g\u2081 \u2297 g\u2082) \u226b\n      (\u03b1_ Y\u2081 Y\u2082 (V\u2081 \u2297 V\u2082)).hom \u226b\n        (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Y\u2082 V\u2081 V\u2082).inv) \u226b\n          (\ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 V\u2081).hom \u2297 \ud835\udfd9 V\u2082) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom) \u226b (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv =\n    ((\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (\u03b1_ X\u2081 U\u2081 (X\u2082 \u2297 U\u2082)).inv) \u226b\n      ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082)"}, {"tactic": "slice_lhs 1 2 => rw [associator_naturality]", "state_before": "C : Type u\u2081\ninst\u271d\u2078 : Category C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n\u22a2 ((f\u2081 \u2297 f\u2082) \u2297 g\u2081 \u2297 g\u2082) \u226b\n      (\u03b1_ Y\u2081 Y\u2082 (V\u2081 \u2297 V\u2082)).hom \u226b\n        (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Y\u2082 V\u2081 V\u2082).inv) \u226b\n          (\ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 V\u2081).hom \u2297 \ud835\udfd9 V\u2082) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom) \u226b (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv =\n    ((\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (\u03b1_ X\u2081 U\u2081 (X\u2082 \u2297 U\u2082)).inv) \u226b\n      ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082)", "state_after": "C : Type u\u2081\ninst\u271d\u2078 : Category C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n\u22a2 (((((\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b (f\u2081 \u2297 f\u2082 \u2297 g\u2081 \u2297 g\u2082)) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Y\u2082 V\u2081 V\u2082).inv)) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 V\u2081).hom \u2297 \ud835\udfd9 V\u2082)) \u226b\n        (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom)) \u226b\n      (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv =\n    ((\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (\u03b1_ X\u2081 U\u2081 (X\u2082 \u2297 U\u2082)).inv) \u226b\n      ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082)"}, {"tactic": "slice_lhs 2 3 =>\n  rw [\u2190 tensor_comp, comp_id f\u2081, \u2190 id_comp f\u2081, associator_inv_naturality, tensor_comp]", "state_before": "C : Type u\u2081\ninst\u271d\u2078 : Category C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n\u22a2 (((((\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b (f\u2081 \u2297 f\u2082 \u2297 g\u2081 \u2297 g\u2082)) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Y\u2082 V\u2081 V\u2082).inv)) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 V\u2081).hom \u2297 \ud835\udfd9 V\u2082)) \u226b\n        (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom)) \u226b\n      (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv =\n    ((\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (\u03b1_ X\u2081 U\u2081 (X\u2082 \u2297 U\u2082)).inv) \u226b\n      ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082)", "state_after": "C : Type u\u2081\ninst\u271d\u2078 : Category C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n\u22a2 (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n      ((((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b (f\u2081 \u2297 (f\u2082 \u2297 g\u2081) \u2297 g\u2082)) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 V\u2081).hom \u2297 \ud835\udfd9 V\u2082)) \u226b\n          (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom)) \u226b\n        (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv =\n    ((\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (\u03b1_ X\u2081 U\u2081 (X\u2082 \u2297 U\u2082)).inv) \u226b\n      ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082)"}, {"tactic": "slice_lhs 3 4 =>\n  rw [\u2190 tensor_comp, \u2190 tensor_comp, comp_id f\u2081, \u2190 id_comp f\u2081, comp_id g\u2082, \u2190 id_comp g\u2082,\n    braiding_naturality, tensor_comp, tensor_comp]", "state_before": "C : Type u\u2081\ninst\u271d\u2078 : Category C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n\u22a2 (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n      ((((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b (f\u2081 \u2297 (f\u2082 \u2297 g\u2081) \u2297 g\u2082)) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 V\u2081).hom \u2297 \ud835\udfd9 V\u2082)) \u226b\n          (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom)) \u226b\n        (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv =\n    ((\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (\u03b1_ X\u2081 U\u2081 (X\u2082 \u2297 U\u2082)).inv) \u226b\n      ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082)", "state_after": "C : Type u\u2081\ninst\u271d\u2078 : Category C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n\u22a2 (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n        (((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (f\u2081 \u2297 (g\u2081 \u2297 f\u2082) \u2297 g\u2082)) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom)) \u226b\n          (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv =\n    ((\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (\u03b1_ X\u2081 U\u2081 (X\u2082 \u2297 U\u2082)).inv) \u226b\n      ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082)"}, {"tactic": "slice_lhs 4 5 => rw [\u2190 tensor_comp, comp_id f\u2081, \u2190 id_comp f\u2081, associator_naturality, tensor_comp]", "state_before": "C : Type u\u2081\ninst\u271d\u2078 : Category C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n\u22a2 (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n        (((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (f\u2081 \u2297 (g\u2081 \u2297 f\u2082) \u2297 g\u2082)) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom)) \u226b\n          (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv =\n    ((\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (\u03b1_ X\u2081 U\u2081 (X\u2082 \u2297 U\u2082)).inv) \u226b\n      ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082)", "state_after": "C : Type u\u2081\ninst\u271d\u2078 : Category C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n\u22a2 (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (f\u2081 \u2297 g\u2081 \u2297 f\u2082 \u2297 g\u2082)) \u226b (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv =\n    ((\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (\u03b1_ X\u2081 U\u2081 (X\u2082 \u2297 U\u2082)).inv) \u226b\n      ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082)"}, {"tactic": "slice_lhs 5 6 => rw [associator_inv_naturality]", "state_before": "C : Type u\u2081\ninst\u271d\u2078 : Category C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n\u22a2 (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (f\u2081 \u2297 g\u2081 \u2297 f\u2082 \u2297 g\u2082)) \u226b (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv =\n    ((\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (\u03b1_ X\u2081 U\u2081 (X\u2082 \u2297 U\u2082)).inv) \u226b\n      ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082)", "state_after": "C : Type u\u2081\ninst\u271d\u2078 : Category C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n\u22a2 (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (\u03b1_ X\u2081 U\u2081 (X\u2082 \u2297 U\u2082)).inv \u226b ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082) =\n    ((\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (\u03b1_ X\u2081 U\u2081 (X\u2082 \u2297 U\u2082)).inv) \u226b\n      ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082)"}, {"tactic": "simp only [assoc]", "state_before": "C : Type u\u2081\ninst\u271d\u2078 : Category C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n\u22a2 (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (\u03b1_ X\u2081 U\u2081 (X\u2082 \u2297 U\u2082)).inv \u226b ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082) =\n    ((\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (\u03b1_ X\u2081 U\u2081 (X\u2082 \u2297 U\u2082)).inv) \u226b\n      ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Submonoid/Operations.lean", "full_name": "Submonoid.range_subtype", "start": [1351, 1], "end": [1352, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.map\u2082Right'_nil_left", "start": [4008, 1], "end": [4008, 96], "traced_tactics": [{"tactic": "cases bs <;> rfl", "state_before": "\u03b9 : Type ?u.462841\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\nf : Option \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : \u03b1\nas : List \u03b1\nb : \u03b2\nbs : List \u03b2\n\u22a2 map\u2082Right' f [] bs = (map (f none) bs, [])", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.iInter_subset", "start": [329, 1], "end": [330, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Perm/Support.lean", "full_name": "Equiv.Perm.Disjoint.mul_left", "start": [111, 1], "end": [112, 44], "traced_tactics": [{"tactic": "cases H1 x <;> cases H2 x <;> simp [*]", "state_before": "\u03b1 : Type u_1\nf g h : Perm \u03b1\nH1 : Disjoint f h\nH2 : Disjoint g h\nx : \u03b1\n\u22a2 \u2191(f * g) x = x \u2228 \u2191h x = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "full_name": "Nat.bit0_inj", "start": [117, 11], "end": [130, 14], "traced_tactics": [{"tactic": "contradiction", "state_before": "m : \u2115\nh : bit0 0 = bit0 (m + 1)\n\u22a2 0 = m + 1", "state_after": "no goals"}, {"tactic": "contradiction", "state_before": "n : \u2115\nh : bit0 (n + 1) = bit0 0\n\u22a2 n + 1 = 0", "state_after": "no goals"}, {"tactic": "have h : succ (succ (n + n)) = succ (succ (m + m)) := by\n  unfold bit0 at h\n  simp [add_one, add_succ, succ_add] at h\n  have aux : n + n = m + m := h\n  rw [aux]", "state_before": "n m : \u2115\nh : bit0 (n + 1) = bit0 (m + 1)\n\u22a2 n + 1 = m + 1", "state_after": "n m : \u2115\nh\u271d : bit0 (n + 1) = bit0 (m + 1)\nh : succ (succ (n + n)) = succ (succ (m + m))\n\u22a2 n + 1 = m + 1"}, {"tactic": "have : n + n = m + m := by repeat\n  injection h with h", "state_before": "n m : \u2115\nh\u271d : bit0 (n + 1) = bit0 (m + 1)\nh : succ (succ (n + n)) = succ (succ (m + m))\n\u22a2 n + 1 = m + 1", "state_after": "n m : \u2115\nh\u271d : bit0 (n + 1) = bit0 (m + 1)\nh : succ (succ (n + n)) = succ (succ (m + m))\nthis : n + n = m + m\n\u22a2 n + 1 = m + 1"}, {"tactic": "have : n = m := Nat.bit0_inj this", "state_before": "n m : \u2115\nh\u271d : bit0 (n + 1) = bit0 (m + 1)\nh : succ (succ (n + n)) = succ (succ (m + m))\nthis : n + n = m + m\n\u22a2 n + 1 = m + 1", "state_after": "n m : \u2115\nh\u271d : bit0 (n + 1) = bit0 (m + 1)\nh : succ (succ (n + n)) = succ (succ (m + m))\nthis\u271d : n + n = m + m\nthis : n = m\n\u22a2 n + 1 = m + 1"}, {"tactic": "rw [this]", "state_before": "n m : \u2115\nh\u271d : bit0 (n + 1) = bit0 (m + 1)\nh : succ (succ (n + n)) = succ (succ (m + m))\nthis\u271d : n + n = m + m\nthis : n = m\n\u22a2 n + 1 = m + 1", "state_after": "no goals"}, {"tactic": "unfold bit0 at h", "state_before": "n m : \u2115\nh : bit0 (n + 1) = bit0 (m + 1)\n\u22a2 succ (succ (n + n)) = succ (succ (m + m))", "state_after": "n m : \u2115\nh : n + 1 + (n + 1) = m + 1 + (m + 1)\n\u22a2 succ (succ (n + n)) = succ (succ (m + m))"}, {"tactic": "simp [add_one, add_succ, succ_add] at h", "state_before": "n m : \u2115\nh : n + 1 + (n + 1) = m + 1 + (m + 1)\n\u22a2 succ (succ (n + n)) = succ (succ (m + m))", "state_after": "n m : \u2115\nh : n + n = m + m\n\u22a2 succ (succ (n + n)) = succ (succ (m + m))"}, {"tactic": "have aux : n + n = m + m := h", "state_before": "n m : \u2115\nh : n + n = m + m\n\u22a2 succ (succ (n + n)) = succ (succ (m + m))", "state_after": "n m : \u2115\nh aux : n + n = m + m\n\u22a2 succ (succ (n + n)) = succ (succ (m + m))"}, {"tactic": "rw [aux]", "state_before": "n m : \u2115\nh aux : n + n = m + m\n\u22a2 succ (succ (n + n)) = succ (succ (m + m))", "state_after": "no goals"}, {"tactic": "repeat\ninjection h with h", "state_before": "n m : \u2115\nh\u271d : bit0 (n + 1) = bit0 (m + 1)\nh : succ (succ (n + n)) = succ (succ (m + m))\n\u22a2 n + n = m + m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Hom/NonUnitalAlg.lean", "full_name": "NonUnitalAlgHom.map_add", "start": [236, 11], "end": [237, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean", "full_name": "AffineMap.proj_linear", "start": [709, 1], "end": [710, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Cycle.lean", "full_name": "List.mem_of_nextOr_ne", "start": [80, 1], "end": [88, 28], "traced_tactics": [{"tactic": "induction' xs with y ys IH", "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh : nextOr xs x d \u2260 d\n\u22a2 x \u2208 xs", "state_after": "case nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : nextOr xs x d \u2260 d\nh : nextOr [] x d \u2260 d\n\u22a2 x \u2208 []\n\ncase cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : nextOr xs x d \u2260 d\ny : \u03b1\nys : List \u03b1\nIH : nextOr ys x d \u2260 d \u2192 x \u2208 ys\nh : nextOr (y :: ys) x d \u2260 d\n\u22a2 x \u2208 y :: ys"}, {"tactic": "cases' ys with z zs", "state_before": "case cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : nextOr xs x d \u2260 d\ny : \u03b1\nys : List \u03b1\nIH : nextOr ys x d \u2260 d \u2192 x \u2208 ys\nh : nextOr (y :: ys) x d \u2260 d\n\u22a2 x \u2208 y :: ys", "state_after": "case cons.nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : nextOr xs x d \u2260 d\ny : \u03b1\nIH : nextOr [] x d \u2260 d \u2192 x \u2208 []\nh : nextOr [y] x d \u2260 d\n\u22a2 x \u2208 [y]\n\ncase cons.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : nextOr xs x d \u2260 d\ny z : \u03b1\nzs : List \u03b1\nIH : nextOr (z :: zs) x d \u2260 d \u2192 x \u2208 z :: zs\nh : nextOr (y :: z :: zs) x d \u2260 d\n\u22a2 x \u2208 y :: z :: zs"}, {"tactic": "simp at h", "state_before": "case nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : nextOr xs x d \u2260 d\nh : nextOr [] x d \u2260 d\n\u22a2 x \u2208 []", "state_after": "no goals"}, {"tactic": "simp at h", "state_before": "case cons.nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : nextOr xs x d \u2260 d\ny : \u03b1\nIH : nextOr [] x d \u2260 d \u2192 x \u2208 []\nh : nextOr [y] x d \u2260 d\n\u22a2 x \u2208 [y]", "state_after": "no goals"}, {"tactic": "by_cases hx : x = y", "state_before": "case cons.cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : nextOr xs x d \u2260 d\ny z : \u03b1\nzs : List \u03b1\nIH : nextOr (z :: zs) x d \u2260 d \u2192 x \u2208 z :: zs\nh : nextOr (y :: z :: zs) x d \u2260 d\n\u22a2 x \u2208 y :: z :: zs", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : nextOr xs x d \u2260 d\ny z : \u03b1\nzs : List \u03b1\nIH : nextOr (z :: zs) x d \u2260 d \u2192 x \u2208 z :: zs\nh : nextOr (y :: z :: zs) x d \u2260 d\nhx : x = y\n\u22a2 x \u2208 y :: z :: zs\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : nextOr xs x d \u2260 d\ny z : \u03b1\nzs : List \u03b1\nIH : nextOr (z :: zs) x d \u2260 d \u2192 x \u2208 z :: zs\nh : nextOr (y :: z :: zs) x d \u2260 d\nhx : \u00acx = y\n\u22a2 x \u2208 y :: z :: zs"}, {"tactic": "simp [hx]", "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : nextOr xs x d \u2260 d\ny z : \u03b1\nzs : List \u03b1\nIH : nextOr (z :: zs) x d \u2260 d \u2192 x \u2208 z :: zs\nh : nextOr (y :: z :: zs) x d \u2260 d\nhx : x = y\n\u22a2 x \u2208 y :: z :: zs", "state_after": "no goals"}, {"tactic": "rw [nextOr_cons_of_ne _ _ _ _ hx] at h", "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : nextOr xs x d \u2260 d\ny z : \u03b1\nzs : List \u03b1\nIH : nextOr (z :: zs) x d \u2260 d \u2192 x \u2208 z :: zs\nh : nextOr (y :: z :: zs) x d \u2260 d\nhx : \u00acx = y\n\u22a2 x \u2208 y :: z :: zs", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : nextOr xs x d \u2260 d\ny z : \u03b1\nzs : List \u03b1\nIH : nextOr (z :: zs) x d \u2260 d \u2192 x \u2208 z :: zs\nh : nextOr (z :: zs) x d \u2260 d\nhx : \u00acx = y\n\u22a2 x \u2208 y :: z :: zs"}, {"tactic": "simpa [hx] using IH h", "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs : List \u03b1\nx d : \u03b1\nh\u271d : nextOr xs x d \u2260 d\ny z : \u03b1\nzs : List \u03b1\nIH : nextOr (z :: zs) x d \u2260 d \u2192 x \u2208 z :: zs\nh : nextOr (z :: zs) x d \u2260 d\nhx : \u00acx = y\n\u22a2 x \u2208 y :: z :: zs", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Constructions.lean", "full_name": "isOpenMap_toDual", "start": [166, 1], "end": [166, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "full_name": "CategoryTheory.Limits.coequalizer.isoTargetOfSelf_hom", "start": [1099, 1], "end": [1102, 37], "traced_tactics": [{"tactic": "simp", "state_before": "C : Type u\ninst\u271d : Category C\nX Y : C\nf g : X \u27f6 Y\n\u22a2 f \u226b \ud835\udfd9 Y = f \u226b \ud835\udfd9 Y", "state_after": "no goals"}, {"tactic": "apply coequalizer.hom_ext", "state_before": "C : Type u\ninst\u271d : Category C\nX Y : C\nf g : X \u27f6 Y\n\u22a2 (isoTargetOfSelf f).hom = desc (\ud835\udfd9 Y) (_ : f \u226b \ud835\udfd9 Y = f \u226b \ud835\udfd9 Y)", "state_after": "case h\nC : Type u\ninst\u271d : Category C\nX Y : C\nf g : X \u27f6 Y\n\u22a2 \u03c0 f f \u226b (isoTargetOfSelf f).hom = \u03c0 f f \u226b desc (\ud835\udfd9 Y) (_ : f \u226b \ud835\udfd9 Y = f \u226b \ud835\udfd9 Y)"}, {"tactic": "simp [coequalizer.isoTargetOfSelf]", "state_before": "case h\nC : Type u\ninst\u271d : Category C\nX Y : C\nf g : X \u27f6 Y\n\u22a2 \u03c0 f f \u226b (isoTargetOfSelf f).hom = \u03c0 f f \u226b desc (\ud835\udfd9 Y) (_ : f \u226b \ud835\udfd9 Y = f \u226b \ud835\udfd9 Y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/RelIso/Basic.lean", "full_name": "RelIso.ext_iff", "start": [699, 1], "end": [700, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Basic.lean", "full_name": "Algebra.algebraMap_ofSubsemiring_apply", "start": [522, 1], "end": [523, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Heyting/Basic.lean", "full_name": "hnot_le_iff_codisjoint_left", "start": [1022, 1], "end": [1023, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Control/Traversable/Basic.lean", "full_name": "ApplicativeTransformation.congr_arg", "start": [113, 11], "end": [115, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "full_name": "Set.image_const_add_Iio", "start": [270, 1], "end": [270, 92], "traced_tactics": [{"tactic": "simp [add_comm]", "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedAddCommGroup \u03b1\na b c : \u03b1\n\u22a2 (fun x => a + x) '' Iio b = Iio (a + b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Connected.lean", "full_name": "Sigma.isConnected_iff", "start": [515, 1], "end": [524, 53], "traced_tactics": [{"tactic": "refine' \u27e8fun hs => _, _\u27e9", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : Set ((i : \u03b9) \u00d7 \u03c0 i)\n\u22a2 IsConnected s \u2194 \u2203 i t, IsConnected t \u2227 s = mk i '' t", "state_after": "case refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : Set ((i : \u03b9) \u00d7 \u03c0 i)\nhs : IsConnected s\n\u22a2 \u2203 i t, IsConnected t \u2227 s = mk i '' t\n\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : Set ((i : \u03b9) \u00d7 \u03c0 i)\n\u22a2 (\u2203 i t, IsConnected t \u2227 s = mk i '' t) \u2192 IsConnected s"}, {"tactic": "obtain \u27e8\u27e8i, x\u27e9, hx\u27e9 := hs.nonempty", "state_before": "case refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : Set ((i : \u03b9) \u00d7 \u03c0 i)\nhs : IsConnected s\n\u22a2 \u2203 i t, IsConnected t \u2227 s = mk i '' t", "state_after": "case refine'_1.intro.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : Set ((i : \u03b9) \u00d7 \u03c0 i)\nhs : IsConnected s\ni : \u03b9\nx : \u03c0 i\nhx : { fst := i, snd := x } \u2208 s\n\u22a2 \u2203 i t, IsConnected t \u2227 s = mk i '' t"}, {"tactic": "have : s \u2286 range (Sigma.mk i) :=\n  hs.isPreconnected.subset_clopen isClopen_range_sigmaMk \u27e8\u27e8i, x\u27e9, hx, x, rfl\u27e9", "state_before": "case refine'_1.intro.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : Set ((i : \u03b9) \u00d7 \u03c0 i)\nhs : IsConnected s\ni : \u03b9\nx : \u03c0 i\nhx : { fst := i, snd := x } \u2208 s\n\u22a2 \u2203 i t, IsConnected t \u2227 s = mk i '' t", "state_after": "case refine'_1.intro.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : Set ((i : \u03b9) \u00d7 \u03c0 i)\nhs : IsConnected s\ni : \u03b9\nx : \u03c0 i\nhx : { fst := i, snd := x } \u2208 s\nthis : s \u2286 range (mk i)\n\u22a2 \u2203 i t, IsConnected t \u2227 s = mk i '' t"}, {"tactic": "exact \u27e8i, Sigma.mk i \u207b\u00b9' s, hs.preimage_of_openMap sigma_mk_injective isOpenMap_sigmaMk this,\n  (Set.image_preimage_eq_of_subset this).symm\u27e9", "state_before": "case refine'_1.intro.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : Set ((i : \u03b9) \u00d7 \u03c0 i)\nhs : IsConnected s\ni : \u03b9\nx : \u03c0 i\nhx : { fst := i, snd := x } \u2208 s\nthis : s \u2286 range (mk i)\n\u22a2 \u2203 i t, IsConnected t \u2227 s = mk i '' t", "state_after": "no goals"}, {"tactic": "rintro \u27e8i, t, ht, rfl\u27e9", "state_before": "case refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : Set ((i : \u03b9) \u00d7 \u03c0 i)\n\u22a2 (\u2203 i t, IsConnected t \u2227 s = mk i '' t) \u2192 IsConnected s", "state_after": "case refine'_2.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u v : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ni : \u03b9\nt : Set (\u03c0 i)\nht : IsConnected t\n\u22a2 IsConnected (mk i '' t)"}, {"tactic": "exact ht.image _ continuous_sigmaMk.continuousOn", "state_before": "case refine'_2.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_2\n\u03c0 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u v : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ni : \u03b9\nt : Set (\u03c0 i)\nht : IsConnected t\n\u22a2 IsConnected (mk i '' t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "full_name": "Complex.continuous_sinh", "start": [74, 1], "end": [76, 13], "traced_tactics": [{"tactic": "change Continuous fun z => (exp z - exp (-z)) / 2", "state_before": "\u22a2 Continuous sinh", "state_after": "\u22a2 Continuous fun z => (exp z - exp (-z)) / 2"}, {"tactic": "continuity", "state_before": "\u22a2 Continuous fun z => (exp z - exp (-z)) / 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean", "full_name": "NonUnitalSubsemiring.toSubsemigroup_mono", "start": [157, 1], "end": [158, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/WithBot.lean", "full_name": "Nat.WithBot.add_eq_one_iff", "start": [34, 1], "end": [38, 27], "traced_tactics": [{"tactic": "rcases n, m with \u27e8_ | _, _ | _\u27e9", "state_before": "n m : WithBot \u2115\n\u22a2 n + m = 1 \u2194 n = 0 \u2227 m = 1 \u2228 n = 1 \u2227 m = 0", "state_after": "case none.none\n\n\u22a2 none + none = 1 \u2194 none = 0 \u2227 none = 1 \u2228 none = 1 \u2227 none = 0\n\ncase none.some\nval\u271d : \u2115\n\u22a2 none + some val\u271d = 1 \u2194 none = 0 \u2227 some val\u271d = 1 \u2228 none = 1 \u2227 some val\u271d = 0\n\ncase some.none\nval\u271d : \u2115\n\u22a2 some val\u271d + none = 1 \u2194 some val\u271d = 0 \u2227 none = 1 \u2228 some val\u271d = 1 \u2227 none = 0\n\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 some val\u271d\u00b9 + some val\u271d = 1 \u2194 some val\u271d\u00b9 = 0 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 1 \u2227 some val\u271d = 0"}, {"tactic": "any_goals refine' \u27e8fun h => Option.noConfusion h, fun h => _\u27e9; aesop", "state_before": "case none.none\n\n\u22a2 none + none = 1 \u2194 none = 0 \u2227 none = 1 \u2228 none = 1 \u2227 none = 0\n\ncase none.some\nval\u271d : \u2115\n\u22a2 none + some val\u271d = 1 \u2194 none = 0 \u2227 some val\u271d = 1 \u2228 none = 1 \u2227 some val\u271d = 0\n\ncase some.none\nval\u271d : \u2115\n\u22a2 some val\u271d + none = 1 \u2194 some val\u271d = 0 \u2227 none = 1 \u2228 some val\u271d = 1 \u2227 none = 0\n\ncase some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 some val\u271d\u00b9 + some val\u271d = 1 \u2194 some val\u271d\u00b9 = 0 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 1 \u2227 some val\u271d = 0", "state_after": "case some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 some val\u271d\u00b9 + some val\u271d = 1 \u2194 some val\u271d\u00b9 = 0 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 1 \u2227 some val\u271d = 0"}, {"tactic": "repeat' erw [WithBot.coe_eq_coe]", "state_before": "case some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 some val\u271d\u00b9 + some val\u271d = 1 \u2194 some val\u271d\u00b9 = 0 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 1 \u2227 some val\u271d = 0", "state_after": "case some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = 1 \u2194 val\u271d\u00b9 = 0 \u2227 val\u271d = 1 \u2228 val\u271d\u00b9 = 1 \u2227 val\u271d = 0"}, {"tactic": "exact Nat.add_eq_one_iff", "state_before": "case some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = 1 \u2194 val\u271d\u00b9 = 0 \u2227 val\u271d = 1 \u2228 val\u271d\u00b9 = 1 \u2227 val\u271d = 0", "state_after": "no goals"}, {"tactic": "refine' \u27e8fun h => Option.noConfusion h, fun h => _\u27e9", "state_before": "case some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 some val\u271d\u00b9 + some val\u271d = 1 \u2194 some val\u271d\u00b9 = 0 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 1 \u2227 some val\u271d = 0", "state_after": "case some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 some val\u271d\u00b9 + some val\u271d = 1 \u2194 some val\u271d\u00b9 = 0 \u2227 some val\u271d = 1 \u2228 some val\u271d\u00b9 = 1 \u2227 some val\u271d = 0"}, {"tactic": "aesop", "state_before": "case some.none\nval\u271d : \u2115\nh : some val\u271d = 0 \u2227 none = 1 \u2228 some val\u271d = 1 \u2227 none = 0\n\u22a2 some val\u271d + none = 1", "state_after": "no goals"}, {"tactic": "erw [WithBot.coe_eq_coe]", "state_before": "case some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = 1 \u2194 val\u271d\u00b9 = 0 \u2227 val\u271d = 1 \u2228 val\u271d\u00b9 = 1 \u2227 val\u271d = 0", "state_after": "case some.some\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (fun x x_1 => x + x_1) val\u271d\u00b9 val\u271d = 1 \u2194 val\u271d\u00b9 = 0 \u2227 val\u271d = 1 \u2228 val\u271d\u00b9 = 1 \u2227 val\u271d = 0"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Matrix/Block.lean", "full_name": "Matrix.blockDiagonal'_apply'", "start": [632, 1], "end": [635, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Adjunction/Reflective.lean", "full_name": "CategoryTheory.mem_essImage_of_unit_isIso", "start": [88, 1], "end": [90, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Manifold/LocalInvariantProperties.lean", "full_name": "StructureGroupoid.LocalInvariantProp.left_invariance", "start": [127, 1], "end": [142, 64], "traced_tactics": [{"tactic": "have h2f := hfs.preimage_mem_nhdsWithin (e'.open_source.mem_nhds hxe')", "state_before": "H : Type u_1\nM : Type ?u.5019\nH' : Type u_2\nM' : Type ?u.5025\nX : Type ?u.5028\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ns\u271d t u : Set H\nx\u271d : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 G'\nhfs : ContinuousWithinAt f s x\nhxe' : f x \u2208 e'.source\n\u22a2 P (\u2191e' \u2218 f) s x \u2194 P f s x", "state_after": "H : Type u_1\nM : Type ?u.5019\nH' : Type u_2\nM' : Type ?u.5025\nX : Type ?u.5028\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ns\u271d t u : Set H\nx\u271d : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 G'\nhfs : ContinuousWithinAt f s x\nhxe' : f x \u2208 e'.source\nh2f : f \u207b\u00b9' e'.source \u2208 \ud835\udcdd[s] x\n\u22a2 P (\u2191e' \u2218 f) s x \u2194 P f s x"}, {"tactic": "have h3f :=\n  ((e'.continuousAt hxe').comp_continuousWithinAt hfs).preimage_mem_nhdsWithin <|\n    e'.symm.open_source.mem_nhds <| e'.mapsTo hxe'", "state_before": "H : Type u_1\nM : Type ?u.5019\nH' : Type u_2\nM' : Type ?u.5025\nX : Type ?u.5028\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ns\u271d t u : Set H\nx\u271d : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 G'\nhfs : ContinuousWithinAt f s x\nhxe' : f x \u2208 e'.source\nh2f : f \u207b\u00b9' e'.source \u2208 \ud835\udcdd[s] x\n\u22a2 P (\u2191e' \u2218 f) s x \u2194 P f s x", "state_after": "H : Type u_1\nM : Type ?u.5019\nH' : Type u_2\nM' : Type ?u.5025\nX : Type ?u.5028\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ns\u271d t u : Set H\nx\u271d : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 G'\nhfs : ContinuousWithinAt f s x\nhxe' : f x \u2208 e'.source\nh2f : f \u207b\u00b9' e'.source \u2208 \ud835\udcdd[s] x\nh3f : \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source \u2208 \ud835\udcdd[s] x\n\u22a2 P (\u2191e' \u2218 f) s x \u2194 P f s x"}, {"tactic": "constructor", "state_before": "H : Type u_1\nM : Type ?u.5019\nH' : Type u_2\nM' : Type ?u.5025\nX : Type ?u.5028\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ns\u271d t u : Set H\nx\u271d : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 G'\nhfs : ContinuousWithinAt f s x\nhxe' : f x \u2208 e'.source\nh2f : f \u207b\u00b9' e'.source \u2208 \ud835\udcdd[s] x\nh3f : \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source \u2208 \ud835\udcdd[s] x\n\u22a2 P (\u2191e' \u2218 f) s x \u2194 P f s x", "state_after": "case mp\nH : Type u_1\nM : Type ?u.5019\nH' : Type u_2\nM' : Type ?u.5025\nX : Type ?u.5028\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ns\u271d t u : Set H\nx\u271d : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 G'\nhfs : ContinuousWithinAt f s x\nhxe' : f x \u2208 e'.source\nh2f : f \u207b\u00b9' e'.source \u2208 \ud835\udcdd[s] x\nh3f : \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source \u2208 \ud835\udcdd[s] x\n\u22a2 P (\u2191e' \u2218 f) s x \u2192 P f s x\n\ncase mpr\nH : Type u_1\nM : Type ?u.5019\nH' : Type u_2\nM' : Type ?u.5025\nX : Type ?u.5028\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ns\u271d t u : Set H\nx\u271d : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 G'\nhfs : ContinuousWithinAt f s x\nhxe' : f x \u2208 e'.source\nh2f : f \u207b\u00b9' e'.source \u2208 \ud835\udcdd[s] x\nh3f : \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source \u2208 \ud835\udcdd[s] x\n\u22a2 P f s x \u2192 P (\u2191e' \u2218 f) s x"}, {"tactic": "intro h", "state_before": "case mp\nH : Type u_1\nM : Type ?u.5019\nH' : Type u_2\nM' : Type ?u.5025\nX : Type ?u.5028\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ns\u271d t u : Set H\nx\u271d : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 G'\nhfs : ContinuousWithinAt f s x\nhxe' : f x \u2208 e'.source\nh2f : f \u207b\u00b9' e'.source \u2208 \ud835\udcdd[s] x\nh3f : \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source \u2208 \ud835\udcdd[s] x\n\u22a2 P (\u2191e' \u2218 f) s x \u2192 P f s x", "state_after": "case mp\nH : Type u_1\nM : Type ?u.5019\nH' : Type u_2\nM' : Type ?u.5025\nX : Type ?u.5028\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ns\u271d t u : Set H\nx\u271d : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 G'\nhfs : ContinuousWithinAt f s x\nhxe' : f x \u2208 e'.source\nh2f : f \u207b\u00b9' e'.source \u2208 \ud835\udcdd[s] x\nh3f : \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source \u2208 \ud835\udcdd[s] x\nh : P (\u2191e' \u2218 f) s x\n\u22a2 P f s x"}, {"tactic": "rw [hG.is_local_nhds h3f] at h", "state_before": "case mp\nH : Type u_1\nM : Type ?u.5019\nH' : Type u_2\nM' : Type ?u.5025\nX : Type ?u.5028\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ns\u271d t u : Set H\nx\u271d : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 G'\nhfs : ContinuousWithinAt f s x\nhxe' : f x \u2208 e'.source\nh2f : f \u207b\u00b9' e'.source \u2208 \ud835\udcdd[s] x\nh3f : \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source \u2208 \ud835\udcdd[s] x\nh : P (\u2191e' \u2218 f) s x\n\u22a2 P f s x", "state_after": "case mp\nH : Type u_1\nM : Type ?u.5019\nH' : Type u_2\nM' : Type ?u.5025\nX : Type ?u.5028\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ns\u271d t u : Set H\nx\u271d : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 G'\nhfs : ContinuousWithinAt f s x\nhxe' : f x \u2208 e'.source\nh2f : f \u207b\u00b9' e'.source \u2208 \ud835\udcdd[s] x\nh3f : \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source \u2208 \ud835\udcdd[s] x\nh : P (\u2191e' \u2218 f) (s \u2229 \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source) x\n\u22a2 P f s x"}, {"tactic": "have h2 := hG.left_invariance' (G'.symm he') (inter_subset_right _ _) (e'.mapsTo hxe') h", "state_before": "case mp\nH : Type u_1\nM : Type ?u.5019\nH' : Type u_2\nM' : Type ?u.5025\nX : Type ?u.5028\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ns\u271d t u : Set H\nx\u271d : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 G'\nhfs : ContinuousWithinAt f s x\nhxe' : f x \u2208 e'.source\nh2f : f \u207b\u00b9' e'.source \u2208 \ud835\udcdd[s] x\nh3f : \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source \u2208 \ud835\udcdd[s] x\nh : P (\u2191e' \u2218 f) (s \u2229 \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source) x\n\u22a2 P f s x", "state_after": "case mp\nH : Type u_1\nM : Type ?u.5019\nH' : Type u_2\nM' : Type ?u.5025\nX : Type ?u.5028\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ns\u271d t u : Set H\nx\u271d : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 G'\nhfs : ContinuousWithinAt f s x\nhxe' : f x \u2208 e'.source\nh2f : f \u207b\u00b9' e'.source \u2208 \ud835\udcdd[s] x\nh3f : \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source \u2208 \ud835\udcdd[s] x\nh : P (\u2191e' \u2218 f) (s \u2229 \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source) x\nh2 : P (\u2191(LocalHomeomorph.symm e') \u2218 \u2191e' \u2218 f) (s \u2229 \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source) x\n\u22a2 P f s x"}, {"tactic": "rw [\u2190 hG.is_local_nhds h3f] at h2", "state_before": "case mp\nH : Type u_1\nM : Type ?u.5019\nH' : Type u_2\nM' : Type ?u.5025\nX : Type ?u.5028\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ns\u271d t u : Set H\nx\u271d : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 G'\nhfs : ContinuousWithinAt f s x\nhxe' : f x \u2208 e'.source\nh2f : f \u207b\u00b9' e'.source \u2208 \ud835\udcdd[s] x\nh3f : \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source \u2208 \ud835\udcdd[s] x\nh : P (\u2191e' \u2218 f) (s \u2229 \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source) x\nh2 : P (\u2191(LocalHomeomorph.symm e') \u2218 \u2191e' \u2218 f) (s \u2229 \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source) x\n\u22a2 P f s x", "state_after": "case mp\nH : Type u_1\nM : Type ?u.5019\nH' : Type u_2\nM' : Type ?u.5025\nX : Type ?u.5028\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ns\u271d t u : Set H\nx\u271d : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 G'\nhfs : ContinuousWithinAt f s x\nhxe' : f x \u2208 e'.source\nh2f : f \u207b\u00b9' e'.source \u2208 \ud835\udcdd[s] x\nh3f : \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source \u2208 \ud835\udcdd[s] x\nh : P (\u2191e' \u2218 f) (s \u2229 \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source) x\nh2 : P (\u2191(LocalHomeomorph.symm e') \u2218 \u2191e' \u2218 f) s x\n\u22a2 P f s x"}, {"tactic": "refine' hG.congr_nhdsWithin _ (e'.left_inv hxe') h2", "state_before": "case mp\nH : Type u_1\nM : Type ?u.5019\nH' : Type u_2\nM' : Type ?u.5025\nX : Type ?u.5028\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ns\u271d t u : Set H\nx\u271d : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 G'\nhfs : ContinuousWithinAt f s x\nhxe' : f x \u2208 e'.source\nh2f : f \u207b\u00b9' e'.source \u2208 \ud835\udcdd[s] x\nh3f : \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source \u2208 \ud835\udcdd[s] x\nh : P (\u2191e' \u2218 f) (s \u2229 \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source) x\nh2 : P (\u2191(LocalHomeomorph.symm e') \u2218 \u2191e' \u2218 f) s x\n\u22a2 P f s x", "state_after": "case mp\nH : Type u_1\nM : Type ?u.5019\nH' : Type u_2\nM' : Type ?u.5025\nX : Type ?u.5028\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ns\u271d t u : Set H\nx\u271d : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 G'\nhfs : ContinuousWithinAt f s x\nhxe' : f x \u2208 e'.source\nh2f : f \u207b\u00b9' e'.source \u2208 \ud835\udcdd[s] x\nh3f : \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source \u2208 \ud835\udcdd[s] x\nh : P (\u2191e' \u2218 f) (s \u2229 \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source) x\nh2 : P (\u2191(LocalHomeomorph.symm e') \u2218 \u2191e' \u2218 f) s x\n\u22a2 \u2191(LocalHomeomorph.symm e') \u2218 \u2191e' \u2218 f =\u1da0[\ud835\udcdd[s] x] f"}, {"tactic": "exact eventually_of_mem h2f fun x' \u21a6 e'.left_inv", "state_before": "case mp\nH : Type u_1\nM : Type ?u.5019\nH' : Type u_2\nM' : Type ?u.5025\nX : Type ?u.5028\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ns\u271d t u : Set H\nx\u271d : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 G'\nhfs : ContinuousWithinAt f s x\nhxe' : f x \u2208 e'.source\nh2f : f \u207b\u00b9' e'.source \u2208 \ud835\udcdd[s] x\nh3f : \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source \u2208 \ud835\udcdd[s] x\nh : P (\u2191e' \u2218 f) (s \u2229 \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source) x\nh2 : P (\u2191(LocalHomeomorph.symm e') \u2218 \u2191e' \u2218 f) s x\n\u22a2 \u2191(LocalHomeomorph.symm e') \u2218 \u2191e' \u2218 f =\u1da0[\ud835\udcdd[s] x] f", "state_after": "no goals"}, {"tactic": "simp_rw [hG.is_local_nhds h2f]", "state_before": "case mpr\nH : Type u_1\nM : Type ?u.5019\nH' : Type u_2\nM' : Type ?u.5025\nX : Type ?u.5028\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ns\u271d t u : Set H\nx\u271d : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 G'\nhfs : ContinuousWithinAt f s x\nhxe' : f x \u2208 e'.source\nh2f : f \u207b\u00b9' e'.source \u2208 \ud835\udcdd[s] x\nh3f : \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source \u2208 \ud835\udcdd[s] x\n\u22a2 P f s x \u2192 P (\u2191e' \u2218 f) s x", "state_after": "case mpr\nH : Type u_1\nM : Type ?u.5019\nH' : Type u_2\nM' : Type ?u.5025\nX : Type ?u.5028\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ns\u271d t u : Set H\nx\u271d : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 G'\nhfs : ContinuousWithinAt f s x\nhxe' : f x \u2208 e'.source\nh2f : f \u207b\u00b9' e'.source \u2208 \ud835\udcdd[s] x\nh3f : \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source \u2208 \ud835\udcdd[s] x\n\u22a2 P f (s \u2229 f \u207b\u00b9' e'.source) x \u2192 P (\u2191e' \u2218 f) (s \u2229 f \u207b\u00b9' e'.source) x"}, {"tactic": "exact hG.left_invariance' he' (inter_subset_right _ _) hxe'", "state_before": "case mpr\nH : Type u_1\nM : Type ?u.5019\nH' : Type u_2\nM' : Type ?u.5025\nX : Type ?u.5028\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ns\u271d t u : Set H\nx\u271d : H\nhG : LocalInvariantProp G G' P\ns : Set H\nx : H\nf : H \u2192 H'\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 G'\nhfs : ContinuousWithinAt f s x\nhxe' : f x \u2208 e'.source\nh2f : f \u207b\u00b9' e'.source \u2208 \ud835\udcdd[s] x\nh3f : \u2191e' \u2218 f \u207b\u00b9' (LocalHomeomorph.symm e').toLocalEquiv.source \u2208 \ud835\udcdd[s] x\n\u22a2 P f (s \u2229 f \u207b\u00b9' e'.source) x \u2192 P (\u2191e' \u2218 f) (s \u2229 f \u207b\u00b9' e'.source) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Matrix/PEquiv.lean", "full_name": "PEquiv.toMatrix_bot", "start": [127, 1], "end": [129, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/AlgebraMap.lean", "full_name": "Algebra.adjoin_singleton_eq_range_aeval", "start": [344, 1], "end": [346, 86], "traced_tactics": [{"tactic": "rw [\u2190 Algebra.map_top, \u2190 adjoin_X, AlgHom.map_adjoin, Set.image_singleton, aeval_X]", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1269188\nB' : Type ?u.1269191\na b : R\nn : \u2115\ninst\u271d\u2076 : CommSemiring A'\ninst\u271d\u2075 : Semiring B'\ninst\u271d\u2074 : CommSemiring R\np q : R[X]\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type ?u.1269404\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nx\u271d x : A\n\u22a2 Algebra.adjoin R {x} = AlgHom.range (aeval x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "full_name": "MeasureTheory.LocallyIntegrable.integrableOn_isCompact", "start": [172, 1], "end": [174, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.eq_some_iff", "start": [176, 1], "end": [177, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Lie/Subalgebra.lean", "full_name": "LieSubalgebra.lieSpan_mono", "start": [704, 1], "end": [706, 42], "traced_tactics": [{"tactic": "rw [lieSpan_le]", "state_before": "R : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns t : Set L\nh : s \u2286 t\n\u22a2 lieSpan R L s \u2264 lieSpan R L t", "state_after": "R : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns t : Set L\nh : s \u2286 t\n\u22a2 s \u2286 \u2191(lieSpan R L t)"}, {"tactic": "exact Set.Subset.trans h subset_lieSpan", "state_before": "R : Type u\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nL\u2082 : Type w\ninst\u271d\u00b9 : LieRing L\u2082\ninst\u271d : LieAlgebra R L\u2082\nf : L \u2192\u2097\u2045R\u2046 L\u2082\nK K' : LieSubalgebra R L\nK\u2082 : LieSubalgebra R L\u2082\ns t : Set L\nh : s \u2286 t\n\u22a2 s \u2286 \u2191(lieSpan R L t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Connected.lean", "full_name": "Subtype.connectedSpace", "start": [866, 1], "end": [868, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "full_name": "Polynomial.degree_pow'", "start": [978, 1], "end": [984, 62], "traced_tactics": [{"tactic": "rw [pow_zero, \u2190 C_1] at *", "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u03b9 : Type ?u.684865\nh : leadingCoeff p ^ 0 \u2260 0\n\u22a2 degree (p ^ 0) = 0 \u2022 degree p", "state_after": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u03b9 : Type ?u.684865\nh : 1 \u2260 0\n\u22a2 degree (\u2191C 1) = 0 \u2022 degree p"}, {"tactic": "rw [degree_C h, zero_nsmul]", "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u03b9 : Type ?u.684865\nh : 1 \u2260 0\n\u22a2 degree (\u2191C 1) = 0 \u2022 degree p", "state_after": "no goals"}, {"tactic": "have h\u2081 : leadingCoeff p ^ n \u2260 0 := fun h\u2081 => h <| by rw [pow_succ, h\u2081, mul_zero]", "state_before": "R : Type u\nS : Type v\na b c d : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u03b9 : Type ?u.684865\nn : \u2115\nh : leadingCoeff p ^ (n + 1) \u2260 0\n\u22a2 degree (p ^ (n + 1)) = (n + 1) \u2022 degree p", "state_after": "R : Type u\nS : Type v\na b c d : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u03b9 : Type ?u.684865\nn : \u2115\nh : leadingCoeff p ^ (n + 1) \u2260 0\nh\u2081 : leadingCoeff p ^ n \u2260 0\n\u22a2 degree (p ^ (n + 1)) = (n + 1) \u2022 degree p"}, {"tactic": "have h\u2082 : leadingCoeff p * leadingCoeff (p ^ n) \u2260 0 := by\n  rwa [pow_succ, \u2190 leadingCoeff_pow' h\u2081] at h", "state_before": "R : Type u\nS : Type v\na b c d : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u03b9 : Type ?u.684865\nn : \u2115\nh : leadingCoeff p ^ (n + 1) \u2260 0\nh\u2081 : leadingCoeff p ^ n \u2260 0\n\u22a2 degree (p ^ (n + 1)) = (n + 1) \u2022 degree p", "state_after": "R : Type u\nS : Type v\na b c d : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u03b9 : Type ?u.684865\nn : \u2115\nh : leadingCoeff p ^ (n + 1) \u2260 0\nh\u2081 : leadingCoeff p ^ n \u2260 0\nh\u2082 : leadingCoeff p * leadingCoeff (p ^ n) \u2260 0\n\u22a2 degree (p ^ (n + 1)) = (n + 1) \u2022 degree p"}, {"tactic": "rw [pow_succ, degree_mul' h\u2082, succ_nsmul, degree_pow' h\u2081]", "state_before": "R : Type u\nS : Type v\na b c d : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u03b9 : Type ?u.684865\nn : \u2115\nh : leadingCoeff p ^ (n + 1) \u2260 0\nh\u2081 : leadingCoeff p ^ n \u2260 0\nh\u2082 : leadingCoeff p * leadingCoeff (p ^ n) \u2260 0\n\u22a2 degree (p ^ (n + 1)) = (n + 1) \u2022 degree p", "state_after": "no goals"}, {"tactic": "rw [pow_succ, h\u2081, mul_zero]", "state_before": "R : Type u\nS : Type v\na b c d : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u03b9 : Type ?u.684865\nn : \u2115\nh : leadingCoeff p ^ (n + 1) \u2260 0\nh\u2081 : leadingCoeff p ^ n = 0\n\u22a2 leadingCoeff p ^ (n + 1) = 0", "state_after": "no goals"}, {"tactic": "rwa [pow_succ, \u2190 leadingCoeff_pow' h\u2081] at h", "state_before": "R : Type u\nS : Type v\na b c d : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u03b9 : Type ?u.684865\nn : \u2115\nh : leadingCoeff p ^ (n + 1) \u2260 0\nh\u2081 : leadingCoeff p ^ n \u2260 0\n\u22a2 leadingCoeff p * leadingCoeff (p ^ n) \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.coe_pmap", "start": [1476, 1], "end": [1478, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Manifold/LocalInvariantProperties.lean", "full_name": "StructureGroupoid.LocalInvariantProp.liftPropWithinAt_of_liftPropAt_of_mem_nhds", "start": [381, 1], "end": [383, 53], "traced_tactics": [{"tactic": "rwa [\u2190 univ_inter s, hG.liftPropWithinAt_inter hs]", "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type ?u.37448\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\nh : LiftPropAt P g x\nhs : s \u2208 \ud835\udcdd x\n\u22a2 LiftPropWithinAt P g s x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf_closed_of_inf_closed", "start": [1015, 1], "end": [1017, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Module/Multilinear.lean", "full_name": "ContinuousMultilinearMap.map_piecewise_smul", "start": [463, 1], "end": [465, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/OmegaCompletePartialOrder.lean", "full_name": "CompleteLattice.bot_continuous", "start": [513, 1], "end": [515, 46], "traced_tactics": [{"tactic": "rw [\u2190 sSup_empty]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\n\u22a2 Continuous \u22a5", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\n\u22a2 Continuous (sSup \u2205)"}, {"tactic": "exact sSup_continuous _ fun f hf => hf.elim", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b1\ninst\u271d : CompleteLattice \u03b2\n\u22a2 Continuous (sSup \u2205)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "full_name": "ENNReal.essSup_liminf_le", "start": [330, 1], "end": [334, 64], "traced_tactics": [{"tactic": "simp_rw [essSup]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.5631541\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : MeasureTheory.Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 essSup (fun x => liminf (fun n => f n x) atTop) \u03bc \u2264 liminf (fun n => essSup (fun x => f n x) \u03bc) atTop", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type ?u.5631541\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : MeasureTheory.Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 limsup (fun x => liminf (fun n => f n x) atTop) (Measure.ae \u03bc) \u2264\n    liminf (fun n => limsup (fun x => f n x) (Measure.ae \u03bc)) atTop"}, {"tactic": "exact ENNReal.limsup_liminf_le_liminf_limsup fun a b => f b a", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.5631541\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : MeasureTheory.Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\u22650\u221e\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 limsup (fun x => liminf (fun n => f n x) atTop) (Measure.ae \u03bc) \u2264\n    liminf (fun n => limsup (fun x => f n x) (Measure.ae \u03bc)) atTop", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Coprime/Lemmas.lean", "full_name": "Finset.prod_dvd_of_coprime", "start": [84, 1], "end": [97, 56], "traced_tactics": [{"tactic": "intro a r har ih Hs Hs1", "state_before": "R : Type u\nI : Type v\ninst\u271d : CommSemiring R\nx y z : R\ns : I \u2192 R\nt : Finset I\n\u22a2 \u2200 \u2983a : I\u2984 {s_1 : Finset I},\n    \u00aca \u2208 s_1 \u2192\n      (Set.Pairwise (\u2191s_1) (IsCoprime on s) \u2192 (\u2200 (i : I), i \u2208 s_1 \u2192 s i \u2223 z) \u2192 \u220f x in s_1, s x \u2223 z) \u2192\n        Set.Pairwise (\u2191(insert a s_1)) (IsCoprime on s) \u2192\n          (\u2200 (i : I), i \u2208 insert a s_1 \u2192 s i \u2223 z) \u2192 \u220f x in insert a s_1, s x \u2223 z", "state_after": "R : Type u\nI : Type v\ninst\u271d : CommSemiring R\nx y z : R\ns : I \u2192 R\nt : Finset I\na : I\nr : Finset I\nhar : \u00aca \u2208 r\nih : Set.Pairwise (\u2191r) (IsCoprime on s) \u2192 (\u2200 (i : I), i \u2208 r \u2192 s i \u2223 z) \u2192 \u220f x in r, s x \u2223 z\nHs : Set.Pairwise (\u2191(insert a r)) (IsCoprime on s)\nHs1 : \u2200 (i : I), i \u2208 insert a r \u2192 s i \u2223 z\n\u22a2 \u220f x in insert a r, s x \u2223 z"}, {"tactic": "rw [Finset.prod_insert har]", "state_before": "R : Type u\nI : Type v\ninst\u271d : CommSemiring R\nx y z : R\ns : I \u2192 R\nt : Finset I\na : I\nr : Finset I\nhar : \u00aca \u2208 r\nih : Set.Pairwise (\u2191r) (IsCoprime on s) \u2192 (\u2200 (i : I), i \u2208 r \u2192 s i \u2223 z) \u2192 \u220f x in r, s x \u2223 z\nHs : Set.Pairwise (\u2191(insert a r)) (IsCoprime on s)\nHs1 : \u2200 (i : I), i \u2208 insert a r \u2192 s i \u2223 z\n\u22a2 \u220f x in insert a r, s x \u2223 z", "state_after": "R : Type u\nI : Type v\ninst\u271d : CommSemiring R\nx y z : R\ns : I \u2192 R\nt : Finset I\na : I\nr : Finset I\nhar : \u00aca \u2208 r\nih : Set.Pairwise (\u2191r) (IsCoprime on s) \u2192 (\u2200 (i : I), i \u2208 r \u2192 s i \u2223 z) \u2192 \u220f x in r, s x \u2223 z\nHs : Set.Pairwise (\u2191(insert a r)) (IsCoprime on s)\nHs1 : \u2200 (i : I), i \u2208 insert a r \u2192 s i \u2223 z\n\u22a2 s a * \u220f x in r, s x \u2223 z"}, {"tactic": "have aux1 : a \u2208 (\u2191(insert a r) : Set I) := Finset.mem_insert_self a r", "state_before": "R : Type u\nI : Type v\ninst\u271d : CommSemiring R\nx y z : R\ns : I \u2192 R\nt : Finset I\na : I\nr : Finset I\nhar : \u00aca \u2208 r\nih : Set.Pairwise (\u2191r) (IsCoprime on s) \u2192 (\u2200 (i : I), i \u2208 r \u2192 s i \u2223 z) \u2192 \u220f x in r, s x \u2223 z\nHs : Set.Pairwise (\u2191(insert a r)) (IsCoprime on s)\nHs1 : \u2200 (i : I), i \u2208 insert a r \u2192 s i \u2223 z\n\u22a2 s a * \u220f x in r, s x \u2223 z", "state_after": "R : Type u\nI : Type v\ninst\u271d : CommSemiring R\nx y z : R\ns : I \u2192 R\nt : Finset I\na : I\nr : Finset I\nhar : \u00aca \u2208 r\nih : Set.Pairwise (\u2191r) (IsCoprime on s) \u2192 (\u2200 (i : I), i \u2208 r \u2192 s i \u2223 z) \u2192 \u220f x in r, s x \u2223 z\nHs : Set.Pairwise (\u2191(insert a r)) (IsCoprime on s)\nHs1 : \u2200 (i : I), i \u2208 insert a r \u2192 s i \u2223 z\naux1 : a \u2208 \u2191(insert a r)\n\u22a2 s a * \u220f x in r, s x \u2223 z"}, {"tactic": "refine'\n  (IsCoprime.prod_right fun i hir \u21a6\n        Hs aux1 (Finset.mem_insert_of_mem hir) <| by\n          rintro rfl\n          exact har hir).mul_dvd\n    (Hs1 a aux1) (ih (Hs.mono _) fun i hi \u21a6 Hs1 i <| Finset.mem_insert_of_mem hi)", "state_before": "R : Type u\nI : Type v\ninst\u271d : CommSemiring R\nx y z : R\ns : I \u2192 R\nt : Finset I\na : I\nr : Finset I\nhar : \u00aca \u2208 r\nih : Set.Pairwise (\u2191r) (IsCoprime on s) \u2192 (\u2200 (i : I), i \u2208 r \u2192 s i \u2223 z) \u2192 \u220f x in r, s x \u2223 z\nHs : Set.Pairwise (\u2191(insert a r)) (IsCoprime on s)\nHs1 : \u2200 (i : I), i \u2208 insert a r \u2192 s i \u2223 z\naux1 : a \u2208 \u2191(insert a r)\n\u22a2 s a * \u220f x in r, s x \u2223 z", "state_after": "R : Type u\nI : Type v\ninst\u271d : CommSemiring R\nx y z : R\ns : I \u2192 R\nt : Finset I\na : I\nr : Finset I\nhar : \u00aca \u2208 r\nih : Set.Pairwise (\u2191r) (IsCoprime on s) \u2192 (\u2200 (i : I), i \u2208 r \u2192 s i \u2223 z) \u2192 \u220f x in r, s x \u2223 z\nHs : Set.Pairwise (\u2191(insert a r)) (IsCoprime on s)\nHs1 : \u2200 (i : I), i \u2208 insert a r \u2192 s i \u2223 z\naux1 : a \u2208 \u2191(insert a r)\n\u22a2 \u2191r \u2286 \u2191(insert a r)"}, {"tactic": "simp only [Finset.coe_insert, Set.subset_insert]", "state_before": "R : Type u\nI : Type v\ninst\u271d : CommSemiring R\nx y z : R\ns : I \u2192 R\nt : Finset I\na : I\nr : Finset I\nhar : \u00aca \u2208 r\nih : Set.Pairwise (\u2191r) (IsCoprime on s) \u2192 (\u2200 (i : I), i \u2208 r \u2192 s i \u2223 z) \u2192 \u220f x in r, s x \u2223 z\nHs : Set.Pairwise (\u2191(insert a r)) (IsCoprime on s)\nHs1 : \u2200 (i : I), i \u2208 insert a r \u2192 s i \u2223 z\naux1 : a \u2208 \u2191(insert a r)\n\u22a2 \u2191r \u2286 \u2191(insert a r)", "state_after": "no goals"}, {"tactic": "rintro rfl", "state_before": "R : Type u\nI : Type v\ninst\u271d : CommSemiring R\nx y z : R\ns : I \u2192 R\nt : Finset I\na : I\nr : Finset I\nhar : \u00aca \u2208 r\nih : Set.Pairwise (\u2191r) (IsCoprime on s) \u2192 (\u2200 (i : I), i \u2208 r \u2192 s i \u2223 z) \u2192 \u220f x in r, s x \u2223 z\nHs : Set.Pairwise (\u2191(insert a r)) (IsCoprime on s)\nHs1 : \u2200 (i : I), i \u2208 insert a r \u2192 s i \u2223 z\naux1 : a \u2208 \u2191(insert a r)\ni : I\nhir : i \u2208 r\n\u22a2 a \u2260 i", "state_after": "R : Type u\nI : Type v\ninst\u271d : CommSemiring R\nx y z : R\ns : I \u2192 R\nt : Finset I\na : I\nr : Finset I\nhar : \u00aca \u2208 r\nih : Set.Pairwise (\u2191r) (IsCoprime on s) \u2192 (\u2200 (i : I), i \u2208 r \u2192 s i \u2223 z) \u2192 \u220f x in r, s x \u2223 z\nHs : Set.Pairwise (\u2191(insert a r)) (IsCoprime on s)\nHs1 : \u2200 (i : I), i \u2208 insert a r \u2192 s i \u2223 z\naux1 : a \u2208 \u2191(insert a r)\nhir : a \u2208 r\n\u22a2 False"}, {"tactic": "exact har hir", "state_before": "R : Type u\nI : Type v\ninst\u271d : CommSemiring R\nx y z : R\ns : I \u2192 R\nt : Finset I\na : I\nr : Finset I\nhar : \u00aca \u2208 r\nih : Set.Pairwise (\u2191r) (IsCoprime on s) \u2192 (\u2200 (i : I), i \u2208 r \u2192 s i \u2223 z) \u2192 \u220f x in r, s x \u2223 z\nHs : Set.Pairwise (\u2191(insert a r)) (IsCoprime on s)\nHs1 : \u2200 (i : I), i \u2208 insert a r \u2192 s i \u2223 z\naux1 : a \u2208 \u2191(insert a r)\nhir : a \u2208 r\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/UpperLower.lean", "full_name": "IsUpperSet.div_right", "start": [111, 1], "end": [113, 21], "traced_tactics": [{"tactic": "rw [div_eq_mul_inv]", "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\ns t : Set \u03b1\na : \u03b1\nhs : IsUpperSet s\n\u22a2 IsUpperSet (s / t)", "state_after": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\ns t : Set \u03b1\na : \u03b1\nhs : IsUpperSet s\n\u22a2 IsUpperSet (s * t\u207b\u00b9)"}, {"tactic": "exact hs.mul_right", "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\ns t : Set \u03b1\na : \u03b1\nhs : IsUpperSet s\n\u22a2 IsUpperSet (s * t\u207b\u00b9)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.foldr_fixed'", "start": [2424, 1], "end": [2426, 58], "traced_tactics": [{"tactic": "rw [foldr_cons, foldr_fixed' hf l, hf a]", "state_before": "\u03b9 : Type ?u.231002\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nb : \u03b2\nhf : \u2200 (a : \u03b1), f a b = b\na : \u03b1\nl : List \u03b1\n\u22a2 foldr f b (a :: l) = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/Equiv.lean", "full_name": "UniformEquiv.self_comp_symm", "start": [214, 1], "end": [215, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/MorphismProperty.lean", "full_name": "CategoryTheory.MorphismProperty.epimorphisms.iff", "start": [412, 1], "end": [412, 64], "traced_tactics": [{"tactic": "rfl", "state_before": "C : Type u\ninst\u271d\u00b9 : Category C\nD : Type ?u.59557\ninst\u271d : Category D\nX Y : C\nf : X \u27f6 Y\n\u22a2 epimorphisms C f \u2194 Epi f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "full_name": "Besicovitch.TauPackage.color_lt", "start": [349, 1], "end": [464, 20], "traced_tactics": [{"tactic": "induction' i using Ordinal.induction with i IH", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni : Ordinal\nhi : i < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\n\u22a2 color p i < N", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\n\u22a2 color p i < N"}, {"tactic": "let A : Set \u2115 :=\n  \u22c3 (j : { j // j < i })\n    (_ : (closedBall (p.c (p.index j)) (p.r (p.index j)) \u2229\n      closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),\n    {p.color j}", "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\n\u22a2 color p i < N", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\n\u22a2 color p i < N"}, {"tactic": "have color_i : p.color i = sInf (univ \\ A) := by rw [color]", "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\n\u22a2 color p i < N", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\n\u22a2 color p i < N"}, {"tactic": "rw [color_i]", "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\n\u22a2 color p i < N", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\n\u22a2 sInf (univ \\ A) < N"}, {"tactic": "have N_mem : N \u2208 univ \\ A := by\n  simp only [not_exists, true_and_iff, exists_prop, mem_iUnion, mem_singleton_iff, mem_closedBall,\n    not_and, mem_univ, mem_diff, Subtype.exists, Subtype.coe_mk]\n  intro j ji _\n  exact (IH j ji (ji.trans hi)).ne'", "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\n\u22a2 sInf (univ \\ A) < N", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\n\u22a2 sInf (univ \\ A) < N"}, {"tactic": "intro Inf_eq_N", "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\n\u22a2 sInf (univ \\ A) \u2260 N", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\n\u22a2 False"}, {"tactic": "have :\n  \u2200 k, k < N \u2192 \u2203 j, j < i \u2227\n    (closedBall (p.c (p.index j)) (p.r (p.index j)) \u2229\n      closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty \u2227 k = p.color j := by\n  intro k hk\n  rw [\u2190 Inf_eq_N] at hk\n  have : k \u2208 A := by\n    simpa only [true_and_iff, mem_univ, Classical.not_not, mem_diff] using\n      Nat.not_mem_of_lt_sInf hk\n  simp [and_assoc, -exists_and_left] at this\n  simpa only [exists_prop, mem_iUnion, mem_singleton_iff, mem_closedBall, Subtype.exists,\n    Subtype.coe_mk]", "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\n\u22a2 False", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\nthis :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      \u2203 j,\n        j < i \u2227\n          Set.Nonempty\n              (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) \u2229\n                closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n            k = color p j\n\u22a2 False"}, {"tactic": "choose! g hg using this", "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\nthis :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      \u2203 j,\n        j < i \u2227\n          Set.Nonempty\n              (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) \u2229\n                closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n            k = color p j\n\u22a2 False", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\n\u22a2 False"}, {"tactic": "let G : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n", "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\n\u22a2 False", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\n\u22a2 False"}, {"tactic": "have fGn :\n    \u2200 n, n \u2264 N \u2192\n      p.c (p.index (G n)) \u2209 p.iUnionUpTo (G n) \u2227 p.R (G n) \u2264 p.\u03c4 * p.r (p.index (G n)) := by\n  intro n hn\n  have :\n    p.index (G n) =\n      Classical.epsilon fun t => p.c t \u2209 p.iUnionUpTo (G n) \u2227 p.R (G n) \u2264 p.\u03c4 * p.r t :=\n    by rw [index]; rfl\n  rw [this]\n  have : \u2203 t, p.c t \u2209 p.iUnionUpTo (G n) \u2227 p.R (G n) \u2264 p.\u03c4 * p.r t := by\n    simpa only [not_exists, exists_prop, not_and, not_lt, not_le, mem_setOf_eq, not_forall] using\n      not_mem_of_lt_csInf (G_lt_last n hn) (OrderBot.bddBelow _)\n  exact Classical.epsilon_spec this", "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\n\u22a2 False", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\n\u22a2 False"}, {"tactic": "exact hN.false sc", "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\nsc : SatelliteConfig \u03b1 N p.\u03c4 :=\n  { c := fun k => BallPackage.c p.toBallPackage (index p (G \u2191k)),\n    r := fun k => BallPackage.r p.toBallPackage (index p (G \u2191k)),\n    rpos := (_ : \u2200 (k : Fin (Nat.succ N)), 0 < BallPackage.r p.toBallPackage (index p (G \u2191k))),\n    h :=\n      (_ :\n        \u2200 (a b : Fin (Nat.succ N)),\n          a \u2260 b \u2192\n            (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n                  dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n                    ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b) \u2227\n                (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n                  p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2228\n              (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n                  dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b)\n                    ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a) \u2227\n                (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n                  p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b),\n    hlast :=\n      (_ :\n        \u2200 (a : Fin (N + 1)),\n          a < last N \u2192\n            BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n                dist (BallPackage.c p.toBallPackage (index p (G \u2191a)))\n                  (BallPackage.c p.toBallPackage (index p (G \u2191(last N)))) \u2227\n              BallPackage.r p.toBallPackage (index p (G \u2191(last N))) \u2264\n                p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))),\n    inter :=\n      (_ :\n        \u2200 (a : Fin (N + 1)),\n          a < last N \u2192\n            dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n                ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2264\n              (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a +\n                (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N)) }\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rw [color]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\n\u22a2 color p i = sInf (univ \\ A)", "state_after": "no goals"}, {"tactic": "simp only [not_exists, true_and_iff, exists_prop, mem_iUnion, mem_singleton_iff, mem_closedBall,\n  not_and, mem_univ, mem_diff, Subtype.exists, Subtype.coe_mk]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\n\u22a2 N \u2208 univ \\ A", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\n\u22a2 \u2200 (x : Ordinal),\n    x < i \u2192\n      Set.Nonempty\n          (closedBall (BallPackage.c p.toBallPackage (index p x)) (BallPackage.r p.toBallPackage (index p x)) \u2229\n            closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2192\n        \u00acN = color p x"}, {"tactic": "intro j ji _", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\n\u22a2 \u2200 (x : Ordinal),\n    x < i \u2192\n      Set.Nonempty\n          (closedBall (BallPackage.c p.toBallPackage (index p x)) (BallPackage.r p.toBallPackage (index p x)) \u2229\n            closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2192\n        \u00acN = color p x", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nj : Ordinal\nji : j < i\na\u271d :\n  Set.Nonempty\n    (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) \u2229\n      closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))\n\u22a2 \u00acN = color p j"}, {"tactic": "exact (IH j ji (ji.trans hi)).ne'", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nj : Ordinal\nji : j < i\na\u271d :\n  Set.Nonempty\n    (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) \u2229\n      closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))\n\u22a2 \u00acN = color p j", "state_after": "no goals"}, {"tactic": "rcases(csInf_le (OrderBot.bddBelow (univ \\ A)) N_mem).lt_or_eq with (H | H)", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nthis : sInf (univ \\ A) \u2260 N\n\u22a2 sInf (univ \\ A) < N", "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nthis : sInf (univ \\ A) \u2260 N\nH : sInf (univ \\ A) < N\n\u22a2 sInf (univ \\ A) < N\n\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nthis : sInf (univ \\ A) \u2260 N\nH : sInf (univ \\ A) = N\n\u22a2 sInf (univ \\ A) < N"}, {"tactic": "exact H", "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nthis : sInf (univ \\ A) \u2260 N\nH : sInf (univ \\ A) < N\n\u22a2 sInf (univ \\ A) < N", "state_after": "no goals"}, {"tactic": "exact (this H).elim", "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nthis : sInf (univ \\ A) \u2260 N\nH : sInf (univ \\ A) = N\n\u22a2 sInf (univ \\ A) < N", "state_after": "no goals"}, {"tactic": "intro k hk", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\n\u22a2 \u2200 (k : \u2115),\n    k < N \u2192\n      \u2203 j,\n        j < i \u2227\n          Set.Nonempty\n              (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) \u2229\n                closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n            k = color p j", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\nk : \u2115\nhk : k < N\n\u22a2 \u2203 j,\n    j < i \u2227\n      Set.Nonempty\n          (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) \u2229\n            closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n        k = color p j"}, {"tactic": "rw [\u2190 Inf_eq_N] at hk", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\nk : \u2115\nhk : k < N\n\u22a2 \u2203 j,\n    j < i \u2227\n      Set.Nonempty\n          (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) \u2229\n            closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n        k = color p j", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\nk : \u2115\nhk : k < sInf (univ \\ A)\n\u22a2 \u2203 j,\n    j < i \u2227\n      Set.Nonempty\n          (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) \u2229\n            closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n        k = color p j"}, {"tactic": "have : k \u2208 A := by\n  simpa only [true_and_iff, mem_univ, Classical.not_not, mem_diff] using\n    Nat.not_mem_of_lt_sInf hk", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\nk : \u2115\nhk : k < sInf (univ \\ A)\n\u22a2 \u2203 j,\n    j < i \u2227\n      Set.Nonempty\n          (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) \u2229\n            closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n        k = color p j", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\nk : \u2115\nhk : k < sInf (univ \\ A)\nthis : k \u2208 A\n\u22a2 \u2203 j,\n    j < i \u2227\n      Set.Nonempty\n          (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) \u2229\n            closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n        k = color p j"}, {"tactic": "simp [and_assoc, -exists_and_left] at this", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\nk : \u2115\nhk : k < sInf (univ \\ A)\nthis : k \u2208 A\n\u22a2 \u2203 j,\n    j < i \u2227\n      Set.Nonempty\n          (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) \u2229\n            closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n        k = color p j", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\nk : \u2115\nhk : k < sInf (univ \\ A)\nthis :\n  \u2203 a,\n    a < i \u2227\n      Set.Nonempty\n          (closedBall (BallPackage.c p.toBallPackage (index p a)) (BallPackage.r p.toBallPackage (index p a)) \u2229\n            closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n        k = color p a\n\u22a2 \u2203 j,\n    j < i \u2227\n      Set.Nonempty\n          (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) \u2229\n            closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n        k = color p j"}, {"tactic": "simpa only [exists_prop, mem_iUnion, mem_singleton_iff, mem_closedBall, Subtype.exists,\n  Subtype.coe_mk]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\nk : \u2115\nhk : k < sInf (univ \\ A)\nthis :\n  \u2203 a,\n    a < i \u2227\n      Set.Nonempty\n          (closedBall (BallPackage.c p.toBallPackage (index p a)) (BallPackage.r p.toBallPackage (index p a)) \u2229\n            closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n        k = color p a\n\u22a2 \u2203 j,\n    j < i \u2227\n      Set.Nonempty\n          (closedBall (BallPackage.c p.toBallPackage (index p j)) (BallPackage.r p.toBallPackage (index p j)) \u2229\n            closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n        k = color p j", "state_after": "no goals"}, {"tactic": "simpa only [true_and_iff, mem_univ, Classical.not_not, mem_diff] using\n  Nat.not_mem_of_lt_sInf hk", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\nk : \u2115\nhk : k < sInf (univ \\ A)\n\u22a2 k \u2208 A", "state_after": "no goals"}, {"tactic": "intro n hn", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\n\u22a2 \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\nn : \u2115\nhn : n \u2264 N\n\u22a2 color p (G n) = n"}, {"tactic": "rcases hn.eq_or_lt with (rfl | H)", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\nn : \u2115\nhn : n \u2264 N\n\u22a2 color p (G n) = n", "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\ni : Ordinal\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\ng : \u2115 \u2192 Ordinal\nn : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 n p.\u03c4)\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < n\nN_mem : n \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = n\nhg :\n  \u2200 (k : \u2115),\n    k < n \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n_1 => if n_1 = n then i else g n_1\nhn : n \u2264 n\n\u22a2 color p (G n) = n\n\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\nn : \u2115\nhn : n \u2264 N\nH : n < N\n\u22a2 color p (G n) = n"}, {"tactic": "simp only", "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\ni : Ordinal\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\ng : \u2115 \u2192 Ordinal\nn : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 n p.\u03c4)\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < n\nN_mem : n \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = n\nhg :\n  \u2200 (k : \u2115),\n    k < n \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n_1 => if n_1 = n then i else g n_1\nhn : n \u2264 n\n\u22a2 color p (G n) = n", "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\ni : Ordinal\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\ng : \u2115 \u2192 Ordinal\nn : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 n p.\u03c4)\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < n\nN_mem : n \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = n\nhg :\n  \u2200 (k : \u2115),\n    k < n \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n_1 => if n_1 = n then i else g n_1\nhn : n \u2264 n\n\u22a2 color p (if True then i else g n) = n"}, {"tactic": "simp only [color_i, Inf_eq_N, if_true, eq_self_iff_true]", "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\ni : Ordinal\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\ng : \u2115 \u2192 Ordinal\nn : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 n p.\u03c4)\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < n\nN_mem : n \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = n\nhg :\n  \u2200 (k : \u2115),\n    k < n \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n_1 => if n_1 = n then i else g n_1\nhn : n \u2264 n\n\u22a2 color p (if True then i else g n) = n", "state_after": "no goals"}, {"tactic": "simp only", "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\nn : \u2115\nhn : n \u2264 N\nH : n < N\n\u22a2 color p (G n) = n", "state_after": "case inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\nn : \u2115\nhn : n \u2264 N\nH : n < N\n\u22a2 color p (if n = N then i else g n) = n"}, {"tactic": "simp only [H.ne, (hg n H).right.right.symm, if_false]", "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\nn : \u2115\nhn : n \u2264 N\nH : n < N\n\u22a2 color p (if n = N then i else g n) = n", "state_after": "no goals"}, {"tactic": "intro n hn", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\n\u22a2 \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nn : \u2115\nhn : n \u2264 N\n\u22a2 G n < lastStep p"}, {"tactic": "rcases hn.eq_or_lt with (rfl | H)", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nn : \u2115\nhn : n \u2264 N\n\u22a2 G n < lastStep p", "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\ni : Ordinal\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\ng : \u2115 \u2192 Ordinal\nn : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 n p.\u03c4)\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < n\nN_mem : n \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = n\nhg :\n  \u2200 (k : \u2115),\n    k < n \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n_1 => if n_1 = n then i else g n_1\ncolor_G : \u2200 (n_1 : \u2115), n_1 \u2264 n \u2192 color p (G n_1) = n_1\nhn : n \u2264 n\n\u22a2 G n < lastStep p\n\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nn : \u2115\nhn : n \u2264 N\nH : n < N\n\u22a2 G n < lastStep p"}, {"tactic": "simp only", "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\ni : Ordinal\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\ng : \u2115 \u2192 Ordinal\nn : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 n p.\u03c4)\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < n\nN_mem : n \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = n\nhg :\n  \u2200 (k : \u2115),\n    k < n \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n_1 => if n_1 = n then i else g n_1\ncolor_G : \u2200 (n_1 : \u2115), n_1 \u2264 n \u2192 color p (G n_1) = n_1\nhn : n \u2264 n\n\u22a2 G n < lastStep p", "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\ni : Ordinal\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\ng : \u2115 \u2192 Ordinal\nn : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 n p.\u03c4)\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < n\nN_mem : n \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = n\nhg :\n  \u2200 (k : \u2115),\n    k < n \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n_1 => if n_1 = n then i else g n_1\ncolor_G : \u2200 (n_1 : \u2115), n_1 \u2264 n \u2192 color p (G n_1) = n_1\nhn : n \u2264 n\n\u22a2 (if True then i else g n) < lastStep p"}, {"tactic": "simp only [hi, if_true, eq_self_iff_true]", "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\ni : Ordinal\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\ng : \u2115 \u2192 Ordinal\nn : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 n p.\u03c4)\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < n\nN_mem : n \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = n\nhg :\n  \u2200 (k : \u2115),\n    k < n \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n_1 => if n_1 = n then i else g n_1\ncolor_G : \u2200 (n_1 : \u2115), n_1 \u2264 n \u2192 color p (G n_1) = n_1\nhn : n \u2264 n\n\u22a2 (if True then i else g n) < lastStep p", "state_after": "no goals"}, {"tactic": "simp only", "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nn : \u2115\nhn : n \u2264 N\nH : n < N\n\u22a2 G n < lastStep p", "state_after": "case inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nn : \u2115\nhn : n \u2264 N\nH : n < N\n\u22a2 (if n = N then i else g n) < lastStep p"}, {"tactic": "simp only [H.ne, (hg n H).left.trans hi, if_false]", "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nn : \u2115\nhn : n \u2264 N\nH : n < N\n\u22a2 (if n = N then i else g n) < lastStep p", "state_after": "no goals"}, {"tactic": "intro n hn", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\n\u22a2 \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nn : \u2115\nhn : n \u2264 N\n\u22a2 \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n    R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))"}, {"tactic": "have :\n  p.index (G n) =\n    Classical.epsilon fun t => p.c t \u2209 p.iUnionUpTo (G n) \u2227 p.R (G n) \u2264 p.\u03c4 * p.r t :=\n  by rw [index]; rfl", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nn : \u2115\nhn : n \u2264 N\n\u22a2 \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n    R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nn : \u2115\nhn : n \u2264 N\nthis :\n  index p (G n) =\n    Classical.epsilon fun t =>\n      \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t\n\u22a2 \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n    R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))"}, {"tactic": "rw [this]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nn : \u2115\nhn : n \u2264 N\nthis :\n  index p (G n) =\n    Classical.epsilon fun t =>\n      \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t\n\u22a2 \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n    R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nn : \u2115\nhn : n \u2264 N\nthis :\n  index p (G n) =\n    Classical.epsilon fun t =>\n      \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t\n\u22a2 \u00acBallPackage.c p.toBallPackage\n          (Classical.epsilon fun t =>\n            \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t) \u2208\n        iUnionUpTo p (G n) \u2227\n    R p (G n) \u2264\n      p.\u03c4 *\n        BallPackage.r p.toBallPackage\n          (Classical.epsilon fun t =>\n            \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t)"}, {"tactic": "have : \u2203 t, p.c t \u2209 p.iUnionUpTo (G n) \u2227 p.R (G n) \u2264 p.\u03c4 * p.r t := by\n  simpa only [not_exists, exists_prop, not_and, not_lt, not_le, mem_setOf_eq, not_forall] using\n    not_mem_of_lt_csInf (G_lt_last n hn) (OrderBot.bddBelow _)", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nn : \u2115\nhn : n \u2264 N\nthis :\n  index p (G n) =\n    Classical.epsilon fun t =>\n      \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t\n\u22a2 \u00acBallPackage.c p.toBallPackage\n          (Classical.epsilon fun t =>\n            \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t) \u2208\n        iUnionUpTo p (G n) \u2227\n    R p (G n) \u2264\n      p.\u03c4 *\n        BallPackage.r p.toBallPackage\n          (Classical.epsilon fun t =>\n            \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nn : \u2115\nhn : n \u2264 N\nthis\u271d :\n  index p (G n) =\n    Classical.epsilon fun t =>\n      \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t\nthis : \u2203 t, \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t\n\u22a2 \u00acBallPackage.c p.toBallPackage\n          (Classical.epsilon fun t =>\n            \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t) \u2208\n        iUnionUpTo p (G n) \u2227\n    R p (G n) \u2264\n      p.\u03c4 *\n        BallPackage.r p.toBallPackage\n          (Classical.epsilon fun t =>\n            \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t)"}, {"tactic": "exact Classical.epsilon_spec this", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nn : \u2115\nhn : n \u2264 N\nthis\u271d :\n  index p (G n) =\n    Classical.epsilon fun t =>\n      \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t\nthis : \u2203 t, \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t\n\u22a2 \u00acBallPackage.c p.toBallPackage\n          (Classical.epsilon fun t =>\n            \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t) \u2208\n        iUnionUpTo p (G n) \u2227\n    R p (G n) \u2264\n      p.\u03c4 *\n        BallPackage.r p.toBallPackage\n          (Classical.epsilon fun t =>\n            \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t)", "state_after": "no goals"}, {"tactic": "rw [index]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nn : \u2115\nhn : n \u2264 N\n\u22a2 index p (G n) =\n    Classical.epsilon fun t =>\n      \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nn : \u2115\nhn : n \u2264 N\n\u22a2 (Classical.epsilon fun b =>\n      (\u00acBallPackage.c p.toBallPackage b \u2208\n            \u22c3 (j : { j // j < G n }),\n              ball (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j))) \u2227\n        (\u2a06 (b :\n            { b //\n              \u00acBallPackage.c p.toBallPackage b \u2208\n                  \u22c3 (j : { j // j < G n }),\n                    ball (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) }),\n            BallPackage.r p.toBallPackage \u2191b) \u2264\n          p.\u03c4 * BallPackage.r p.toBallPackage b) =\n    Classical.epsilon fun t =>\n      \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t"}, {"tactic": "rfl", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nn : \u2115\nhn : n \u2264 N\n\u22a2 (Classical.epsilon fun b =>\n      (\u00acBallPackage.c p.toBallPackage b \u2208\n            \u22c3 (j : { j // j < G n }),\n              ball (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j))) \u2227\n        (\u2a06 (b :\n            { b //\n              \u00acBallPackage.c p.toBallPackage b \u2208\n                  \u22c3 (j : { j // j < G n }),\n                    ball (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) }),\n            BallPackage.r p.toBallPackage \u2191b) \u2264\n          p.\u03c4 * BallPackage.r p.toBallPackage b) =\n    Classical.epsilon fun t =>\n      \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t", "state_after": "no goals"}, {"tactic": "simpa only [not_exists, exists_prop, not_and, not_lt, not_le, mem_setOf_eq, not_forall] using\n  not_mem_of_lt_csInf (G_lt_last n hn) (OrderBot.bddBelow _)", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nn : \u2115\nhn : n \u2264 N\nthis :\n  index p (G n) =\n    Classical.epsilon fun t =>\n      \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t\n\u22a2 \u2203 t, \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G n) \u2227 R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage t", "state_after": "no goals"}, {"tactic": "intro a b G_lt", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\n\u22a2 \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n      dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n    BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))"}, {"tactic": "have ha : (a : \u2115) \u2264 N := Nat.lt_succ_iff.1 a.2", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n      dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n    BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n      dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n    BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))"}, {"tactic": "have hb : (b : \u2115) \u2264 N := Nat.lt_succ_iff.1 b.2", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n      dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n    BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n      dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n    BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))"}, {"tactic": "constructor", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n      dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n    BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))", "state_after": "case left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n    dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b)))\n\ncase right\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))"}, {"tactic": "have := (fGn b hb).1", "state_before": "case left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n    dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b)))", "state_after": "case left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nthis : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191b)\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n    dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b)))"}, {"tactic": "simp only [iUnionUpTo, not_exists, exists_prop, mem_iUnion, mem_closedBall, not_and, not_le,\n  Subtype.exists, Subtype.coe_mk] at this", "state_before": "case left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nthis : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191b)\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n    dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b)))", "state_after": "case left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nthis :\n  \u2200 (x : Ordinal),\n    (x < if \u2191b = N then i else g \u2191b) \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (if \u2191b = N then i else g \u2191b)) \u2208\n          ball (BallPackage.c p.toBallPackage (index p x)) (BallPackage.r p.toBallPackage (index p x))\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n    dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b)))"}, {"tactic": "simpa only [dist_comm, mem_ball, not_lt] using this (G a) G_lt", "state_before": "case left\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nthis :\n  \u2200 (x : Ordinal),\n    (x < if \u2191b = N then i else g \u2191b) \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (if \u2191b = N then i else g \u2191b)) \u2208\n          ball (BallPackage.c p.toBallPackage (index p x)) (BallPackage.r p.toBallPackage (index p x))\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n    dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b)))", "state_after": "no goals"}, {"tactic": "apply le_trans _ (fGn a ha).2", "state_before": "case right\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 R p (G \u2191a)"}, {"tactic": "have B : p.c (p.index (G b)) \u2209 p.iUnionUpTo (G a) := by\n  intro H; exact (fGn b hb).1 (p.monotone_iUnionUpTo G_lt.le H)", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 R p (G \u2191a)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nB : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 R p (G \u2191a)"}, {"tactic": "let b' : { t // p.c t \u2209 p.iUnionUpTo (G a) } := \u27e8p.index (G b), B\u27e9", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nB : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 R p (G \u2191a)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nB : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\nb' : { t // \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G \u2191a) } := { val := index p (G \u2191b), property := B }\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 R p (G \u2191a)"}, {"tactic": "apply @le_ciSup _ _ _ (fun t : { t // p.c t \u2209 p.iUnionUpTo (G a) } => p.r t) _ b'", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nB : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\nb' : { t // \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G \u2191a) } := { val := index p (G \u2191b), property := B }\n\u22a2 BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 R p (G \u2191a)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nB : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\nb' : { t // \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G \u2191a) } := { val := index p (G \u2191b), property := B }\n\u22a2 BddAbove (range fun t => BallPackage.r p.toBallPackage \u2191t)"}, {"tactic": "refine' \u27e8p.r_bound, fun t ht => _\u27e9", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nB : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\nb' : { t // \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G \u2191a) } := { val := index p (G \u2191b), property := B }\n\u22a2 BddAbove (range fun t => BallPackage.r p.toBallPackage \u2191t)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nB : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\nb' : { t // \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G \u2191a) } := { val := index p (G \u2191b), property := B }\nt : \u211d\nht : t \u2208 range fun t => BallPackage.r p.toBallPackage \u2191t\n\u22a2 t \u2264 p.r_bound"}, {"tactic": "simp only [exists_prop, mem_range, Subtype.exists, Subtype.coe_mk] at ht", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nB : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\nb' : { t // \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G \u2191a) } := { val := index p (G \u2191b), property := B }\nt : \u211d\nht : t \u2208 range fun t => BallPackage.r p.toBallPackage \u2191t\n\u22a2 t \u2264 p.r_bound", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nB : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\nb' : { t // \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G \u2191a) } := { val := index p (G \u2191b), property := B }\nt : \u211d\nht :\n  \u2203 a_1,\n    \u00acBallPackage.c p.toBallPackage a_1 \u2208 iUnionUpTo p (if \u2191a = N then i else g \u2191a) \u2227\n      BallPackage.r p.toBallPackage a_1 = t\n\u22a2 t \u2264 p.r_bound"}, {"tactic": "rcases ht with \u27e8u, hu\u27e9", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nB : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\nb' : { t // \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G \u2191a) } := { val := index p (G \u2191b), property := B }\nt : \u211d\nht :\n  \u2203 a_1,\n    \u00acBallPackage.c p.toBallPackage a_1 \u2208 iUnionUpTo p (if \u2191a = N then i else g \u2191a) \u2227\n      BallPackage.r p.toBallPackage a_1 = t\n\u22a2 t \u2264 p.r_bound", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nB : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\nb' : { t // \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G \u2191a) } := { val := index p (G \u2191b), property := B }\nt : \u211d\nu : \u03b2\nhu : \u00acBallPackage.c p.toBallPackage u \u2208 iUnionUpTo p (if \u2191a = N then i else g \u2191a) \u2227 BallPackage.r p.toBallPackage u = t\n\u22a2 t \u2264 p.r_bound"}, {"tactic": "rw [\u2190 hu.2]", "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nB : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\nb' : { t // \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G \u2191a) } := { val := index p (G \u2191b), property := B }\nt : \u211d\nu : \u03b2\nhu : \u00acBallPackage.c p.toBallPackage u \u2208 iUnionUpTo p (if \u2191a = N then i else g \u2191a) \u2227 BallPackage.r p.toBallPackage u = t\n\u22a2 t \u2264 p.r_bound", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nB : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\nb' : { t // \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G \u2191a) } := { val := index p (G \u2191b), property := B }\nt : \u211d\nu : \u03b2\nhu : \u00acBallPackage.c p.toBallPackage u \u2208 iUnionUpTo p (if \u2191a = N then i else g \u2191a) \u2227 BallPackage.r p.toBallPackage u = t\n\u22a2 BallPackage.r p.toBallPackage u \u2264 p.r_bound"}, {"tactic": "exact p.r_le _", "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nB : \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\nb' : { t // \u00acBallPackage.c p.toBallPackage t \u2208 iUnionUpTo p (G \u2191a) } := { val := index p (G \u2191b), property := B }\nt : \u211d\nu : \u03b2\nhu : \u00acBallPackage.c p.toBallPackage u \u2208 iUnionUpTo p (if \u2191a = N then i else g \u2191a) \u2227 BallPackage.r p.toBallPackage u = t\n\u22a2 BallPackage.r p.toBallPackage u \u2264 p.r_bound", "state_after": "no goals"}, {"tactic": "intro H", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\n\u22a2 \u00acBallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nH : BallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\n\u22a2 False"}, {"tactic": "exact (fGn b hb).1 (p.monotone_iUnionUpTo G_lt.le H)", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\na b : Fin (Nat.succ N)\nG_lt : G \u2191a < G \u2191b\nha : \u2191a \u2264 N\nhb : \u2191b \u2264 N\nH : BallPackage.c p.toBallPackage (index p (G \u2191b)) \u2208 iUnionUpTo p (G \u2191a)\n\u22a2 False", "state_after": "no goals"}, {"tactic": "intro a b a_ne_b", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\n\u22a2 \u2200 (i j : Fin (Nat.succ N)),\n    i \u2260 j \u2192\n      (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) i \u2264\n            dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) i)\n              ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) j) \u2227\n          (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) j \u2264\n            p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) i \u2228\n        (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) j \u2264\n            dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) j)\n              ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) i) \u2227\n          (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) i \u2264\n            p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) j", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\n\u22a2 (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n        dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n          ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b) \u2227\n      (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n        p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2228\n    (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n        dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b)\n          ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a) \u2227\n      (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n        p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b"}, {"tactic": "wlog G_le : G a \u2264 G b generalizing a b", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\n\u22a2 (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n        dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n          ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b) \u2227\n      (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n        p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2228\n    (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n        dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b)\n          ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a) \u2227\n      (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n        p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b", "state_after": "case inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\nthis :\n  \u2200 (a b : Fin (Nat.succ N)),\n    a \u2260 b \u2192\n      G \u2191a \u2264 G \u2191b \u2192\n        (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n              dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n                ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b) \u2227\n            (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n              p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2228\n          (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n              dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b)\n                ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a) \u2227\n            (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n              p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b\nG_le : \u00acG \u2191a \u2264 G \u2191b\n\u22a2 (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n        dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n          ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b) \u2227\n      (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n        p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2228\n    (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n        dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b)\n          ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a) \u2227\n      (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n        p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b\n\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\nG_le : G \u2191a \u2264 G \u2191b\n\u22a2 (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n        dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n          ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b) \u2227\n      (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n        p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2228\n    (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n        dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b)\n          ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a) \u2227\n      (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n        p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b"}, {"tactic": "exact Or.inl (Gab a b G_lt)", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\nG_le : G \u2191a \u2264 G \u2191b\nG_lt : G \u2191a < G \u2191b\n\u22a2 (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n        dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n          ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b) \u2227\n      (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n        p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2228\n    (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n        dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b)\n          ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a) \u2227\n      (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n        p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b", "state_after": "no goals"}, {"tactic": "exact (this b a a_ne_b.symm (le_of_not_le G_le)).symm", "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\nthis :\n  \u2200 (a b : Fin (Nat.succ N)),\n    a \u2260 b \u2192\n      G \u2191a \u2264 G \u2191b \u2192\n        (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n              dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n                ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b) \u2227\n            (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n              p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2228\n          (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n              dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b)\n                ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a) \u2227\n            (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n              p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b\nG_le : \u00acG \u2191a \u2264 G \u2191b\n\u22a2 (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n        dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n          ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b) \u2227\n      (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n        p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2228\n    (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b \u2264\n        dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) b)\n          ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a) \u2227\n      (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n        p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) b", "state_after": "no goals"}, {"tactic": "rcases G_le.lt_or_eq with (H | H)", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\nG_le : G \u2191a \u2264 G \u2191b\n\u22a2 G \u2191a < G \u2191b", "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\nG_le : G \u2191a \u2264 G \u2191b\nH : G \u2191a < G \u2191b\n\u22a2 G \u2191a < G \u2191b\n\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\nG_le : G \u2191a \u2264 G \u2191b\nH : G \u2191a = G \u2191b\n\u22a2 G \u2191a < G \u2191b"}, {"tactic": "have A : (a : \u2115) \u2260 b := Fin.val_injective.ne a_ne_b", "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\nG_le : G \u2191a \u2264 G \u2191b\nH : G \u2191a = G \u2191b\n\u22a2 G \u2191a < G \u2191b", "state_after": "case inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA\u271d : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A\u271d)\nN_mem : N \u2208 univ \\ A\u271d\nInf_eq_N : sInf (univ \\ A\u271d) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\nG_le : G \u2191a \u2264 G \u2191b\nH : G \u2191a = G \u2191b\nA : \u2191a \u2260 \u2191b\n\u22a2 G \u2191a < G \u2191b"}, {"tactic": "rw [\u2190 color_G a (Nat.lt_succ_iff.1 a.2), \u2190 color_G b (Nat.lt_succ_iff.1 b.2), H] at A", "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA\u271d : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A\u271d)\nN_mem : N \u2208 univ \\ A\u271d\nInf_eq_N : sInf (univ \\ A\u271d) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\nG_le : G \u2191a \u2264 G \u2191b\nH : G \u2191a = G \u2191b\nA : \u2191a \u2260 \u2191b\n\u22a2 G \u2191a < G \u2191b", "state_after": "case inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA\u271d : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A\u271d)\nN_mem : N \u2208 univ \\ A\u271d\nInf_eq_N : sInf (univ \\ A\u271d) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\nG_le : G \u2191a \u2264 G \u2191b\nH : G \u2191a = G \u2191b\nA : color p (G \u2191b) \u2260 color p (G \u2191b)\n\u22a2 G \u2191a < G \u2191b"}, {"tactic": "exact (A rfl).elim", "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA\u271d : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A\u271d)\nN_mem : N \u2208 univ \\ A\u271d\nInf_eq_N : sInf (univ \\ A\u271d) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\nG_le : G \u2191a \u2264 G \u2191b\nH : G \u2191a = G \u2191b\nA : color p (G \u2191b) \u2260 color p (G \u2191b)\n\u22a2 G \u2191a < G \u2191b", "state_after": "no goals"}, {"tactic": "exact H", "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na b : Fin (Nat.succ N)\na_ne_b : a \u2260 b\nG_le : G \u2191a \u2264 G \u2191b\nH : G \u2191a < G \u2191b\n\u22a2 G \u2191a < G \u2191b", "state_after": "no goals"}, {"tactic": "intro a ha", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\n\u22a2 \u2200 (i : Fin (N + 1)),\n    i < last N \u2192\n      (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) i \u2264\n          dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) i)\n            ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2227\n        (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N) \u2264\n          p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) i", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\n\u22a2 (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n      dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n        ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2227\n    (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N) \u2264\n      p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a"}, {"tactic": "have I : (a : \u2115) < N := ha", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\n\u22a2 (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n      dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n        ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2227\n    (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N) \u2264\n      p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\n\u22a2 (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n      dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n        ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2227\n    (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N) \u2264\n      p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a"}, {"tactic": "have : G a < G (Fin.last N) := by dsimp; simp [I.ne, (hg a I).1]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\n\u22a2 (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n      dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n        ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2227\n    (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N) \u2264\n      p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\nthis : G \u2191a < G \u2191(last N)\n\u22a2 (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n      dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n        ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2227\n    (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N) \u2264\n      p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a"}, {"tactic": "exact Gab _ _ this", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\nthis : G \u2191a < G \u2191(last N)\n\u22a2 (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a \u2264\n      dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n        ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2227\n    (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N) \u2264\n      p.\u03c4 * (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a", "state_after": "no goals"}, {"tactic": "dsimp", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\n\u22a2 G \u2191a < G \u2191(last N)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\n\u22a2 (if \u2191a = N then i else g \u2191a) < if N = N then i else g N"}, {"tactic": "simp [I.ne, (hg a I).1]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\n\u22a2 (if \u2191a = N then i else g \u2191a) < if N = N then i else g N", "state_after": "no goals"}, {"tactic": "intro a ha", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\n\u22a2 \u2200 (i : Fin (N + 1)),\n    i < last N \u2192\n      dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) i)\n          ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2264\n        (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) i +\n          (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\n\u22a2 dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n      ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2264\n    (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a +\n      (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N)"}, {"tactic": "have I : (a : \u2115) < N := ha", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\n\u22a2 dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n      ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2264\n    (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a +\n      (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\n\u22a2 dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n      ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2264\n    (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a +\n      (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N)"}, {"tactic": "have J : G (Fin.last N) = i := by dsimp; simp only [if_true, eq_self_iff_true]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\n\u22a2 dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n      ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2264\n    (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a +\n      (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\nJ : G \u2191(last N) = i\n\u22a2 dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n      ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2264\n    (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a +\n      (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N)"}, {"tactic": "have K : G a = g a := by dsimp; simp [I.ne, (hg a I).1]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\nJ : G \u2191(last N) = i\n\u22a2 dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n      ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2264\n    (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a +\n      (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\nJ : G \u2191(last N) = i\nK : G \u2191a = g \u2191a\n\u22a2 dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n      ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2264\n    (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a +\n      (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N)"}, {"tactic": "convert dist_le_add_of_nonempty_closedBall_inter_closedBall (hg _ I).2.1", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\nJ : G \u2191(last N) = i\nK : G \u2191a = g \u2191a\n\u22a2 dist ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) a)\n      ((fun k => BallPackage.c p.toBallPackage (index p (G \u2191k))) (last N)) \u2264\n    (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) a +\n      (fun k => BallPackage.r p.toBallPackage (index p (G \u2191k))) (last N)", "state_after": "no goals"}, {"tactic": "dsimp", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\n\u22a2 G \u2191(last N) = i", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\n\u22a2 (if N = N then i else g N) = i"}, {"tactic": "simp only [if_true, eq_self_iff_true]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\n\u22a2 (if N = N then i else g N) = i", "state_after": "no goals"}, {"tactic": "dsimp", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\nJ : G \u2191(last N) = i\n\u22a2 G \u2191a = g \u2191a", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\nJ : G \u2191(last N) = i\n\u22a2 (if \u2191a = N then i else g \u2191a) = g \u2191a"}, {"tactic": "simp [I.ne, (hg a I).1]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d : Nonempty \u03b2\np : TauPackage \u03b2 \u03b1\ni\u271d : Ordinal\nhi\u271d : i\u271d < lastStep p\nN : \u2115\nhN : IsEmpty (SatelliteConfig \u03b1 N p.\u03c4)\ni : Ordinal\nIH : \u2200 (k : Ordinal), k < i \u2192 k < lastStep p \u2192 color p k < N\nhi : i < lastStep p\nA : Set \u2115 :=\n  \u22c3 (j : { j // j < i }) (_ :\n    Set.Nonempty\n      (closedBall (BallPackage.c p.toBallPackage (index p \u2191j)) (BallPackage.r p.toBallPackage (index p \u2191j)) \u2229\n        closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i)))),\n    {color p \u2191j}\ncolor_i : color p i = sInf (univ \\ A)\nN_mem : N \u2208 univ \\ A\nInf_eq_N : sInf (univ \\ A) = N\ng : \u2115 \u2192 Ordinal\nhg :\n  \u2200 (k : \u2115),\n    k < N \u2192\n      g k < i \u2227\n        Set.Nonempty\n            (closedBall (BallPackage.c p.toBallPackage (index p (g k)))\n                (BallPackage.r p.toBallPackage (index p (g k))) \u2229\n              closedBall (BallPackage.c p.toBallPackage (index p i)) (BallPackage.r p.toBallPackage (index p i))) \u2227\n          k = color p (g k)\nG : \u2115 \u2192 Ordinal := fun n => if n = N then i else g n\ncolor_G : \u2200 (n : \u2115), n \u2264 N \u2192 color p (G n) = n\nG_lt_last : \u2200 (n : \u2115), n \u2264 N \u2192 G n < lastStep p\nfGn :\n  \u2200 (n : \u2115),\n    n \u2264 N \u2192\n      \u00acBallPackage.c p.toBallPackage (index p (G n)) \u2208 iUnionUpTo p (G n) \u2227\n        R p (G n) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G n))\nGab :\n  \u2200 (a b : Fin (Nat.succ N)),\n    G \u2191a < G \u2191b \u2192\n      BallPackage.r p.toBallPackage (index p (G \u2191a)) \u2264\n          dist (BallPackage.c p.toBallPackage (index p (G \u2191a))) (BallPackage.c p.toBallPackage (index p (G \u2191b))) \u2227\n        BallPackage.r p.toBallPackage (index p (G \u2191b)) \u2264 p.\u03c4 * BallPackage.r p.toBallPackage (index p (G \u2191a))\na : Fin (N + 1)\nha : a < last N\nI : \u2191a < N\nJ : G \u2191(last N) = i\n\u22a2 (if \u2191a = N then i else g \u2191a) = g \u2191a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/PiSystem.lean", "full_name": "isPiSystem_image_Iio", "start": [138, 1], "end": [140, 55], "traced_tactics": [{"tactic": "rintro _ \u27e8a, ha, rfl\u27e9 _ \u27e8b, hb, rfl\u27e9 -", "state_before": "\u03b1 : Type u_1\n\u03b9 : Sort ?u.3509\n\u03b9' : Sort ?u.3512\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\n\u22a2 IsPiSystem (Iio '' s)", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Sort ?u.3509\n\u03b9' : Sort ?u.3512\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 s\n\u22a2 Iio a \u2229 Iio b \u2208 Iio '' s"}, {"tactic": "exact \u27e8a \u2293 b, inf_ind a b ha hb, Iio_inter_Iio.symm\u27e9", "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Sort ?u.3509\n\u03b9' : Sort ?u.3512\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 s\n\u22a2 Iio a \u2229 Iio b \u2208 Iio '' s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.atBot_Iio_eq", "start": [1578, 1], "end": [1579, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "full_name": "Asymptotics.isBigO_bot", "start": [615, 1], "end": [616, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Sites/Grothendieck.lean", "full_name": "CategoryTheory.GrothendieckTopology.dense_covering", "start": [366, 1], "end": [367, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Logic.lean", "full_name": "iff_not_self", "start": [90, 1], "end": [90, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Lipschitz.lean", "full_name": "LipschitzWith.of_le_add", "start": [337, 11], "end": [338, 75], "traced_tactics": [{"tactic": "simpa only [NNReal.coe_one, one_mul]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : PseudoMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\nf : \u03b1 \u2192 \u211d\nh : \u2200 (x y : \u03b1), f x \u2264 f y + dist x y\n\u22a2 \u2200 (x y : \u03b1), f x \u2264 f y + \u21911 * dist x y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean", "full_name": "PrimeSpectrum.comap_asIdeal", "start": [599, 1], "end": [600, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/GaloisConnection.lean", "full_name": "GaloisConnection.l_iSup\u2082", "start": [295, 1], "end": [296, 22], "traced_tactics": [{"tactic": "simp_rw [gc.l_iSup]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : CompleteLattice \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\nf : (i : \u03b9) \u2192 \u03ba i \u2192 \u03b1\n\u22a2 l (\u2a06 (i : \u03b9) (j : \u03ba i), f i j) = \u2a06 (i : \u03b9) (j : \u03ba i), l (f i j)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Data/List/Basic.lean", "full_name": "List.length_nil", "start": [38, 9], "end": [39, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "isGLB_of_mem_nhds", "start": [2017, 1], "end": [2019, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "full_name": "MeasureTheory.SignedMeasure.of_diff_eq_zero_of_symmDiff_eq_zero_negative", "start": [322, 1], "end": [328, 16], "traced_tactics": [{"tactic": "rw [\u2190 s.neg_le_neg_iff _ hu, neg_zero] at hsu", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.61556\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict s u \u2264 VectorMeasure.restrict 0 u\nhsv : VectorMeasure.restrict s v \u2264 VectorMeasure.restrict 0 v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.61556\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict (-s) u\nhsv : VectorMeasure.restrict s v \u2264 VectorMeasure.restrict 0 v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0"}, {"tactic": "rw [\u2190 s.neg_le_neg_iff _ hv, neg_zero] at hsv", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.61556\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict (-s) u\nhsv : VectorMeasure.restrict s v \u2264 VectorMeasure.restrict 0 v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.61556\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict (-s) u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict (-s) v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0"}, {"tactic": "have := of_diff_eq_zero_of_symmDiff_eq_zero_positive hu hv hsu hsv", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.61556\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict (-s) u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict (-s) v\nhs : \u2191s (u \u2206 v) = 0\n\u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.61556\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict (-s) u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict (-s) v\nhs : \u2191s (u \u2206 v) = 0\nthis : \u2191(-s) (u \u2206 v) = 0 \u2192 \u2191(-s) (u \\ v) = 0 \u2227 \u2191(-s) (v \\ u) = 0\n\u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0"}, {"tactic": "simp only [Pi.neg_apply, neg_eq_zero, coe_neg] at this", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.61556\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict (-s) u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict (-s) v\nhs : \u2191s (u \u2206 v) = 0\nthis : \u2191(-s) (u \u2206 v) = 0 \u2192 \u2191(-s) (u \\ v) = 0 \u2227 \u2191(-s) (v \\ u) = 0\n\u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.61556\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict (-s) u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict (-s) v\nhs : \u2191s (u \u2206 v) = 0\nthis : \u2191s (u \u2206 v) = 0 \u2192 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0\n\u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0"}, {"tactic": "exact this hs", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.61556\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : IsFiniteMeasure \u03bd\nu v w : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict (-s) u\nhsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict (-s) v\nhs : \u2191s (u \u2206 v) = 0\nthis : \u2191s (u \u2206 v) = 0 \u2192 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0\n\u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean", "full_name": "CircleDeg1Lift.floor_sub_le_translationNumber", "start": [852, 1], "end": [853, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/ModelTheory/ElementaryMaps.lean", "full_name": "FirstOrder.Language.ElementarySubstructure.mem_top", "start": [439, 1], "end": [440, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/TrivSqZeroExt.lean", "full_name": "TrivSqZeroExt.fst_add", "start": [258, 1], "end": [259, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Finprod.lean", "full_name": "finprod_mem_def", "start": [376, 1], "end": [377, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Init/Logic.lean", "full_name": "decidableEq_inr_neg", "start": [335, 1], "end": [338, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "full_name": "Besicovitch.exists_disjoint_closedBall_covering_ae", "start": [857, 1], "end": [914, 53], "traced_tactics": [{"tactic": "let g x := f x \u2229 Ioo 0 (R x)", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)"}, {"tactic": "have hg : \u2200 x \u2208 s, \u2200 \u03b4 > 0, (g x \u2229 Ioo 0 \u03b4).Nonempty := by\n  intro x hx \u03b4 \u03b4pos\n  rcases hf x hx (min \u03b4 (R x)) (lt_min \u03b4pos (hR x hx)) with \u27e8r, hr\u27e9\n  exact\n    \u27e8r,\n      \u27e8\u27e8hr.1, hr.2.1, hr.2.2.trans_le (min_le_right _ _)\u27e9,\n        \u27e8hr.2.1, hr.2.2.trans_le (min_le_left _ _)\u27e9\u27e9\u27e9", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)"}, {"tactic": "rcases exists_disjoint_closedBall_covering_ae_aux \u03bc g s hg with \u27e8v, v_count, vs, vg, \u03bcv, v_disj\u27e9", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)"}, {"tactic": "let t := Prod.fst '' v", "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)"}, {"tactic": "have : \u2200 x \u2208 t, \u2203 r : \u211d, (x, r) \u2208 v := by\n  intro x hx\n  rcases(mem_image _ _ _).1 hx with \u27e8\u27e8p, q\u27e9, hp, rfl\u27e9\n  exact \u27e8q, hp\u27e9", "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nthis : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 r, (x, r) \u2208 v\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)"}, {"tactic": "choose! r hr using this", "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nthis : \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 r, (x, r) \u2208 v\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)"}, {"tactic": "refine' \u27e8t, r, v_count.image _, _, _, _, _\u27e9", "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 \u2203 t r,\n    Set.Countable t \u2227\n      t \u2286 s \u2227\n        (\u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)) \u2227\n          \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0 \u2227 PairwiseDisjoint t fun x => closedBall x (r x)", "state_after": "case intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 t \u2286 s\n\ncase intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)\n\ncase intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0\n\ncase intro.intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 PairwiseDisjoint t fun x => closedBall x (r x)"}, {"tactic": "intro x hx \u03b4 \u03b4pos", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nx : \u03b1\nhx : x \u2208 s\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\n\u22a2 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)"}, {"tactic": "rcases hf x hx (min \u03b4 (R x)) (lt_min \u03b4pos (hR x hx)) with \u27e8r, hr\u27e9", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nx : \u03b1\nhx : x \u2208 s\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\n\u22a2 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nx : \u03b1\nhx : x \u2208 s\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nr : \u211d\nhr : r \u2208 f x \u2229 Ioo 0 (min \u03b4 (R x))\n\u22a2 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)"}, {"tactic": "exact\n  \u27e8r,\n    \u27e8\u27e8hr.1, hr.2.1, hr.2.2.trans_le (min_le_right _ _)\u27e9,\n      \u27e8hr.2.1, hr.2.2.trans_le (min_le_left _ _)\u27e9\u27e9\u27e9", "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nx : \u03b1\nhx : x \u2208 s\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nr : \u211d\nhr : r \u2208 f x \u2229 Ioo 0 (min \u03b4 (R x))\n\u22a2 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)", "state_after": "no goals"}, {"tactic": "intro x hx", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\n\u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 \u2203 r, (x, r) \u2208 v", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nx : \u03b1\nhx : x \u2208 t\n\u22a2 \u2203 r, (x, r) \u2208 v"}, {"tactic": "rcases(mem_image _ _ _).1 hx with \u27e8\u27e8p, q\u27e9, hp, rfl\u27e9", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nx : \u03b1\nhx : x \u2208 t\n\u22a2 \u2203 r, (x, r) \u2208 v", "state_after": "case intro.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\np : \u03b1\nq : \u211d\nhp : (p, q) \u2208 v\nhx : (p, q).fst \u2208 t\n\u22a2 \u2203 r, ((p, q).fst, r) \u2208 v"}, {"tactic": "exact \u27e8q, hp\u27e9", "state_before": "case intro.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\np : \u03b1\nq : \u211d\nhp : (p, q) \u2208 v\nhx : (p, q).fst \u2208 t\n\u22a2 \u2203 r, ((p, q).fst, r) \u2208 v", "state_after": "no goals"}, {"tactic": "have I : \u2200 p : \u03b1 \u00d7 \u211d, p \u2208 v \u2192 0 \u2264 p.2 := fun p hp => (vg p hp).2.1.le", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\n\u22a2 (fun x => (x, r x)) '' t = v", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\n\u22a2 (fun x => (x, r x)) '' t = v"}, {"tactic": "apply Subset.antisymm", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\n\u22a2 (fun x => (x, r x)) '' t = v", "state_after": "case h\u2081\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\n\u22a2 (fun x => (x, r x)) '' t \u2286 v\n\ncase h\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\n\u22a2 v \u2286 (fun x => (x, r x)) '' t"}, {"tactic": "simp only [image_subset_iff]", "state_before": "case h\u2081\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\n\u22a2 (fun x => (x, r x)) '' t \u2286 v", "state_after": "case h\u2081\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\n\u22a2 v \u2286 Prod.fst \u207b\u00b9' ((fun x => (x, r x)) \u207b\u00b9' v)"}, {"tactic": "rintro \u27e8x, p\u27e9 hxp", "state_before": "case h\u2081\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\n\u22a2 v \u2286 Prod.fst \u207b\u00b9' ((fun x => (x, r x)) \u207b\u00b9' v)", "state_after": "case h\u2081.mk\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\n\u22a2 (x, p) \u2208 Prod.fst \u207b\u00b9' ((fun x => (x, r x)) \u207b\u00b9' v)"}, {"tactic": "simp only [mem_preimage]", "state_before": "case h\u2081.mk\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\n\u22a2 (x, p) \u2208 Prod.fst \u207b\u00b9' ((fun x => (x, r x)) \u207b\u00b9' v)", "state_after": "case h\u2081.mk\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\n\u22a2 (x, r x) \u2208 v"}, {"tactic": "exact hr _ (mem_image_of_mem _ hxp)", "state_before": "case h\u2081.mk\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\n\u22a2 (x, r x) \u2208 v", "state_after": "no goals"}, {"tactic": "rintro \u27e8x, p\u27e9 hxp", "state_before": "case h\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\n\u22a2 v \u2286 (fun x => (x, r x)) '' t", "state_after": "case h\u2082.mk\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\n\u22a2 (x, p) \u2208 (fun x => (x, r x)) '' t"}, {"tactic": "have hxrx : (x, r x) \u2208 v := hr _ (mem_image_of_mem _ hxp)", "state_before": "case h\u2082.mk\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\n\u22a2 (x, p) \u2208 (fun x => (x, r x)) '' t", "state_after": "case h\u2082.mk\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\n\u22a2 (x, p) \u2208 (fun x => (x, r x)) '' t"}, {"tactic": "have : p = r x := by\n  by_contra h\n  have A : (x, p) \u2260 (x, r x) := by\n    simpa only [true_and_iff, Prod.mk.inj_iff, eq_self_iff_true, Ne.def] using h\n  have H := v_disj hxp hxrx A\n  contrapose H\n  rw [not_disjoint_iff_nonempty_inter]\n  refine' \u27e8x, by simp (config := { proj := false }) [I _ hxp, I _ hxrx]\u27e9", "state_before": "case h\u2082.mk\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\n\u22a2 (x, p) \u2208 (fun x => (x, r x)) '' t", "state_after": "case h\u2082.mk\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nthis : p = r x\n\u22a2 (x, p) \u2208 (fun x => (x, r x)) '' t"}, {"tactic": "rw [this]", "state_before": "case h\u2082.mk\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nthis : p = r x\n\u22a2 (x, p) \u2208 (fun x => (x, r x)) '' t", "state_after": "case h\u2082.mk\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nthis : p = r x\n\u22a2 (x, r x) \u2208 (fun x => (x, r x)) '' t"}, {"tactic": "apply mem_image_of_mem", "state_before": "case h\u2082.mk\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nthis : p = r x\n\u22a2 (x, r x) \u2208 (fun x => (x, r x)) '' t", "state_after": "case h\u2082.mk.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nthis : p = r x\n\u22a2 x \u2208 t"}, {"tactic": "exact mem_image_of_mem _ hxp", "state_before": "case h\u2082.mk.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nthis : p = r x\n\u22a2 x \u2208 t", "state_after": "no goals"}, {"tactic": "by_contra h", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\n\u22a2 p = r x", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\n\u22a2 False"}, {"tactic": "have A : (x, p) \u2260 (x, r x) := by\n  simpa only [true_and_iff, Prod.mk.inj_iff, eq_self_iff_true, Ne.def] using h", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\n\u22a2 False", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\n\u22a2 False"}, {"tactic": "have H := v_disj hxp hxrx A", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\n\u22a2 False", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\nH : (Disjoint on fun p => closedBall p.fst p.snd) (x, p) (x, r x)\n\u22a2 False"}, {"tactic": "contrapose H", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\nH : (Disjoint on fun p => closedBall p.fst p.snd) (x, p) (x, r x)\n\u22a2 False", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\nH : \u00acFalse\n\u22a2 \u00ac(Disjoint on fun p => closedBall p.fst p.snd) (x, p) (x, r x)"}, {"tactic": "rw [not_disjoint_iff_nonempty_inter]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\nH : \u00acFalse\n\u22a2 \u00ac(Disjoint on fun p => closedBall p.fst p.snd) (x, p) (x, r x)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\nH : \u00acFalse\n\u22a2 Set.Nonempty ((fun p => closedBall p.fst p.snd) (x, p) \u2229 (fun p => closedBall p.fst p.snd) (x, r x))"}, {"tactic": "refine' \u27e8x, by simp (config := { proj := false }) [I _ hxp, I _ hxrx]\u27e9", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\nH : \u00acFalse\n\u22a2 Set.Nonempty ((fun p => closedBall p.fst p.snd) (x, p) \u2229 (fun p => closedBall p.fst p.snd) (x, r x))", "state_after": "no goals"}, {"tactic": "simpa only [true_and_iff, Prod.mk.inj_iff, eq_self_iff_true, Ne.def] using h", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\n\u22a2 (x, p) \u2260 (x, r x)", "state_after": "no goals"}, {"tactic": "simp (config := { proj := false }) [I _ hxp, I _ hxrx]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nI : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 0 \u2264 p.snd\nx : \u03b1\np : \u211d\nhxp : (x, p) \u2208 v\nhxrx : (x, r x) \u2208 v\nh : \u00acp = r x\nA : (x, p) \u2260 (x, r x)\nH : \u00acFalse\n\u22a2 x \u2208 (fun p => closedBall p.fst p.snd) (x, p) \u2229 (fun p => closedBall p.fst p.snd) (x, r x)", "state_after": "no goals"}, {"tactic": "intro x hx", "state_before": "case intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 t \u2286 s", "state_after": "case intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nx : \u03b1\nhx : x \u2208 t\n\u22a2 x \u2208 s"}, {"tactic": "rcases (mem_image _ _ _).1 hx with \u27e8\u27e8p, q\u27e9, hp, rfl\u27e9", "state_before": "case intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nx : \u03b1\nhx : x \u2208 t\n\u22a2 x \u2208 s", "state_after": "case intro.intro.intro.intro.intro.refine'_1.intro.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\np : \u03b1\nq : \u211d\nhp : (p, q) \u2208 v\nhx : (p, q).fst \u2208 t\n\u22a2 (p, q).fst \u2208 s"}, {"tactic": "exact vs _ hp", "state_before": "case intro.intro.intro.intro.intro.refine'_1.intro.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\np : \u03b1\nq : \u211d\nhp : (p, q) \u2208 v\nhx : (p, q).fst \u2208 t\n\u22a2 (p, q).fst \u2208 s", "state_after": "no goals"}, {"tactic": "intro x hx", "state_before": "case intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 r x \u2208 f x \u2229 Ioo 0 (R x)", "state_after": "case intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nx : \u03b1\nhx : x \u2208 t\n\u22a2 r x \u2208 f x \u2229 Ioo 0 (R x)"}, {"tactic": "rcases (mem_image _ _ _).1 hx with \u27e8\u27e8p, q\u27e9, _, rfl\u27e9", "state_before": "case intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nx : \u03b1\nhx : x \u2208 t\n\u22a2 r x \u2208 f x \u2229 Ioo 0 (R x)", "state_after": "case intro.intro.intro.intro.intro.refine'_2.intro.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\np : \u03b1\nq : \u211d\nleft\u271d : (p, q) \u2208 v\nhx : (p, q).fst \u2208 t\n\u22a2 r (p, q).fst \u2208 f (p, q).fst \u2229 Ioo 0 (R (p, q).fst)"}, {"tactic": "exact vg _ (hr _ hx)", "state_before": "case intro.intro.intro.intro.intro.refine'_2.intro.mk.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\np : \u03b1\nq : \u211d\nleft\u271d : (p, q) \u2208 v\nhx : (p, q).fst \u2208 t\n\u22a2 r (p, q).fst \u2208 f (p, q).fst \u2229 Ioo 0 (R (p, q).fst)", "state_after": "no goals"}, {"tactic": "have :\n  (\u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) =\n    \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 (fun x => (x, r x)) '' t), closedBall p.1 p.2 :=\n  by conv_rhs => rw [biUnion_image]", "state_before": "case intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0", "state_after": "case intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nthis :\n  (\u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 (fun x => (x, r x)) '' t), closedBall p.fst p.snd\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0"}, {"tactic": "rw [this, im_t]", "state_before": "case intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nthis :\n  (\u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 (fun x => (x, r x)) '' t), closedBall p.fst p.snd\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = 0", "state_after": "case intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nthis :\n  (\u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 (fun x => (x, r x)) '' t), closedBall p.fst p.snd\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0"}, {"tactic": "exact \u03bcv", "state_before": "case intro.intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nthis :\n  (\u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 (fun x => (x, r x)) '' t), closedBall p.fst p.snd\n\u22a2 \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0", "state_after": "no goals"}, {"tactic": "conv_rhs => rw [biUnion_image]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 (\u22c3 (x : \u03b1) (_ : x \u2208 t), closedBall x (r x)) = \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 (fun x => (x, r x)) '' t), closedBall p.fst p.snd", "state_after": "no goals"}, {"tactic": "have A : InjOn (fun x : \u03b1 => (x, r x)) t := by\n  simp (config := { contextual := true }) only [InjOn, Prod.mk.inj_iff, imp_true_iff,\n    eq_self_iff_true]", "state_before": "case intro.intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 PairwiseDisjoint t fun x => closedBall x (r x)", "state_after": "case intro.intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nA : InjOn (fun x => (x, r x)) t\n\u22a2 PairwiseDisjoint t fun x => closedBall x (r x)"}, {"tactic": "rwa [\u2190 im_t, A.pairwiseDisjoint_image] at v_disj", "state_before": "case intro.intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\nA : InjOn (fun x => (x, r x)) t\n\u22a2 PairwiseDisjoint t fun x => closedBall x (r x)", "state_after": "no goals"}, {"tactic": "simp (config := { contextual := true }) only [InjOn, Prod.mk.inj_iff, imp_true_iff,\n  eq_self_iff_true]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : OpensMeasurableSpace \u03b1\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 Set \u211d\ns : Set \u03b1\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (f x \u2229 Ioo 0 \u03b4)\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\ng : \u03b1 \u2192 Set \u211d := fun x => f x \u2229 Ioo 0 (R x)\nhg : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (\u03b4 : \u211d), \u03b4 > 0 \u2192 Set.Nonempty (g x \u2229 Ioo 0 \u03b4)\nv : Set (\u03b1 \u00d7 \u211d)\nv_count : Set.Countable v\nvs : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.fst \u2208 s\nvg : \u2200 (p : \u03b1 \u00d7 \u211d), p \u2208 v \u2192 p.snd \u2208 g p.fst\n\u03bcv : \u2191\u2191\u03bc (s \\ \u22c3 (p : \u03b1 \u00d7 \u211d) (_ : p \u2208 v), closedBall p.fst p.snd) = 0\nv_disj : PairwiseDisjoint v fun p => closedBall p.fst p.snd\nt : Set \u03b1 := Prod.fst '' v\nr : \u03b1 \u2192 \u211d\nhr : \u2200 (x : \u03b1), x \u2208 t \u2192 (x, r x) \u2208 v\nim_t : (fun x => (x, r x)) '' t = v\n\u22a2 InjOn (fun x => (x, r x)) t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.castPred_mk'", "start": [2331, 1], "end": [2332, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/String/Lemmas.lean", "full_name": "Substring.Valid.get", "start": [981, 1], "end": [984, 50], "traced_tactics": [{"tactic": "let \u27e8l, m, r, h\u27e9 := h.validFor", "state_before": "m\u2081 : List Char\nc : Char\nm\u2082 : List Char\nx\u271d : Substring\nh : Valid x\u271d\ne : (toString x\u271d).data = m\u2081 ++ c :: m\u2082\n\u22a2 Substring.get x\u271d { byteIdx := utf8Len m\u2081 } = c", "state_after": "m\u2081 : List Char\nc : Char\nm\u2082 : List Char\nx\u271d : Substring\nh\u271d : Valid x\u271d\ne : (toString x\u271d).data = m\u2081 ++ c :: m\u2082\nl m r : List Char\nh : ValidFor l m r x\u271d\n\u22a2 Substring.get x\u271d { byteIdx := utf8Len m\u2081 } = c"}, {"tactic": "simp [h.toString] at e", "state_before": "m\u2081 : List Char\nc : Char\nm\u2082 : List Char\nx\u271d : Substring\nh\u271d : Valid x\u271d\ne : (toString x\u271d).data = m\u2081 ++ c :: m\u2082\nl m r : List Char\nh : ValidFor l m r x\u271d\n\u22a2 Substring.get x\u271d { byteIdx := utf8Len m\u2081 } = c", "state_after": "m\u2081 : List Char\nc : Char\nm\u2082 : List Char\nx\u271d : Substring\nh\u271d : Valid x\u271d\nl m r : List Char\nh : ValidFor l m r x\u271d\ne : m = m\u2081 ++ c :: m\u2082\n\u22a2 Substring.get x\u271d { byteIdx := utf8Len m\u2081 } = c"}, {"tactic": "subst e", "state_before": "m\u2081 : List Char\nc : Char\nm\u2082 : List Char\nx\u271d : Substring\nh\u271d : Valid x\u271d\nl m r : List Char\nh : ValidFor l m r x\u271d\ne : m = m\u2081 ++ c :: m\u2082\n\u22a2 Substring.get x\u271d { byteIdx := utf8Len m\u2081 } = c", "state_after": "m\u2081 : List Char\nc : Char\nm\u2082 : List Char\nx\u271d : Substring\nh\u271d : Valid x\u271d\nl r : List Char\nh : ValidFor l (m\u2081 ++ c :: m\u2082) r x\u271d\n\u22a2 Substring.get x\u271d { byteIdx := utf8Len m\u2081 } = c"}, {"tactic": "simp [h.get]", "state_before": "m\u2081 : List Char\nc : Char\nm\u2082 : List Char\nx\u271d : Substring\nh\u271d : Valid x\u271d\nl r : List Char\nh : ValidFor l (m\u2081 ++ c :: m\u2082) r x\u271d\n\u22a2 Substring.get x\u271d { byteIdx := utf8Len m\u2081 } = c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.cons_head!_tail", "start": [923, 1], "end": [924, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Periodic.lean", "full_name": "Function.Periodic.map_vadd_zmultiples", "start": [332, 1], "end": [335, 58], "traced_tactics": [{"tactic": "rcases a with \u27e8_, m, rfl\u27e9", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.133663\nf g : \u03b1 \u2192 \u03b2\nc c\u2081 c\u2082 x\u271d : \u03b1\ninst\u271d : AddCommGroup \u03b1\nhf : Periodic f c\na : { x // x \u2208 AddSubgroup.zmultiples c }\nx : \u03b1\n\u22a2 f (a +\u1d65 x) = f x", "state_after": "case mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.133663\nf g : \u03b1 \u2192 \u03b2\nc c\u2081 c\u2082 x\u271d : \u03b1\ninst\u271d : AddCommGroup \u03b1\nhf : Periodic f c\nx : \u03b1\nm : \u2124\n\u22a2 f ({ val := (fun x => x \u2022 c) m, property := (_ : \u2203 y, (fun x => x \u2022 c) y = (fun x => x \u2022 c) m) } +\u1d65 x) = f x"}, {"tactic": "simp [AddSubgroup.vadd_def, add_comm _ x, hf.zsmul m x]", "state_before": "case mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.133663\nf g : \u03b1 \u2192 \u03b2\nc c\u2081 c\u2082 x\u271d : \u03b1\ninst\u271d : AddCommGroup \u03b1\nhf : Periodic f c\nx : \u03b1\nm : \u2124\n\u22a2 f ({ val := (fun x => x \u2022 c) m, property := (_ : \u2203 y, (fun x => x \u2022 c) y = (fun x => x \u2022 c) m) } +\u1d65 x) = f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Hom/Monoid.lean", "full_name": "OrderMonoidWithZeroHom.mk_coe", "start": [618, 1], "end": [618, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Nodup.lean", "full_name": "List.Nodup.map_on", "start": [244, 1], "end": [246, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/LinearIndependent.lean", "full_name": "LinearIndependent.map'", "start": [227, 1], "end": [229, 29], "traced_tactics": [{"tactic": "simp [hf_inj]", "state_before": "\u03b9 : Type u'\n\u03b9' : Type ?u.105129\nR : Type u_1\nK : Type ?u.105135\nM : Type u_2\nM' : Type u_3\nM'' : Type ?u.105144\nV : Type u\nV' : Type ?u.105149\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nf : M \u2192\u2097[R] M'\nhf_inj : LinearMap.ker f = \u22a5\n\u22a2 Disjoint (span R (range v)) (LinearMap.ker f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/SuccPred/Limit.lean", "full_name": "Order.IsPredLimit.isMax", "start": [404, 11], "end": [405, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Equiv.lean", "full_name": "LinearIsometryEquiv.comp_differentiableWithinAt_iff", "start": [300, 1], "end": [302, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Probability/Kernel/Basic.lean", "full_name": "ProbabilityTheory.kernel.integral_const", "start": [459, 1], "end": [461, 26], "traced_tactics": [{"tactic": "rw [kernel.const_apply]", "state_before": "\u03b1 : Type u_3\n\u03b2 : Type u_2\n\u03b9 : Type ?u.1096105\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u03b2 \u2192 E\n\u03bc : MeasureTheory.Measure \u03b2\na : \u03b1\n\u22a2 (\u222b (x : \u03b2), f x \u2202\u2191(const \u03b1 \u03bc) a) = \u222b (x : \u03b2), f x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "full_name": "Cardinal.aleph'_le_of_limit", "start": [215, 1], "end": [221, 18], "traced_tactics": [{"tactic": "rw [\u2190 aleph'_alephIdx c, aleph'_le, limit_le l]", "state_before": "o : Ordinal\nl : Ordinal.IsLimit o\nc : Cardinal\nh : \u2200 (o' : Ordinal), o' < o \u2192 aleph' o' \u2264 c\n\u22a2 aleph' o \u2264 c", "state_after": "o : Ordinal\nl : Ordinal.IsLimit o\nc : Cardinal\nh : \u2200 (o' : Ordinal), o' < o \u2192 aleph' o' \u2264 c\n\u22a2 \u2200 (x : Ordinal), x < o \u2192 x \u2264 alephIdx c"}, {"tactic": "intro x h'", "state_before": "o : Ordinal\nl : Ordinal.IsLimit o\nc : Cardinal\nh : \u2200 (o' : Ordinal), o' < o \u2192 aleph' o' \u2264 c\n\u22a2 \u2200 (x : Ordinal), x < o \u2192 x \u2264 alephIdx c", "state_after": "o : Ordinal\nl : Ordinal.IsLimit o\nc : Cardinal\nh : \u2200 (o' : Ordinal), o' < o \u2192 aleph' o' \u2264 c\nx : Ordinal\nh' : x < o\n\u22a2 x \u2264 alephIdx c"}, {"tactic": "rw [\u2190 aleph'_le, aleph'_alephIdx]", "state_before": "o : Ordinal\nl : Ordinal.IsLimit o\nc : Cardinal\nh : \u2200 (o' : Ordinal), o' < o \u2192 aleph' o' \u2264 c\nx : Ordinal\nh' : x < o\n\u22a2 x \u2264 alephIdx c", "state_after": "o : Ordinal\nl : Ordinal.IsLimit o\nc : Cardinal\nh : \u2200 (o' : Ordinal), o' < o \u2192 aleph' o' \u2264 c\nx : Ordinal\nh' : x < o\n\u22a2 aleph' x \u2264 c"}, {"tactic": "exact h _ h'", "state_before": "o : Ordinal\nl : Ordinal.IsLimit o\nc : Cardinal\nh : \u2200 (o' : Ordinal), o' < o \u2192 aleph' o' \u2264 c\nx : Ordinal\nh' : x < o\n\u22a2 aleph' x \u2264 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/RingDivision.lean", "full_name": "Polynomial.monic_prod_multiset_X_sub_C", "start": [1065, 1], "end": [1066, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.restrict_union", "start": [1727, 1], "end": [1729, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Submonoid/Operations.lean", "full_name": "Submonoid.comap_strictMono_of_surjective", "start": [500, 1], "end": [501, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Subobject/Lattice.lean", "full_name": "CategoryTheory.Subobject.map_top", "start": [253, 1], "end": [254, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Commute.lean", "full_name": "Commute.pow_pow_self", "start": [209, 1], "end": [210, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fintype/Perm.lean", "full_name": "mem_permsOfList_of_mem", "start": [50, 1], "end": [77, 96], "traced_tactics": [{"tactic": "simp only [not_mem_nil] at h", "state_before": "case nil\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 []\n\u22a2 f \u2208 permsOfList []", "state_after": "case nil\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 False\n\u22a2 f \u2208 permsOfList []"}, {"tactic": "exact List.mem_singleton.2 (Equiv.ext fun x => Decidable.by_contradiction <| h x)", "state_before": "case nil\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 False\n\u22a2 f \u2208 permsOfList []", "state_after": "no goals"}, {"tactic": "by_cases hfa : f a = a", "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\n\u22a2 f \u2208 permsOfList (a :: l)", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u2191f a = a\n\u22a2 f \u2208 permsOfList (a :: l)\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\n\u22a2 f \u2208 permsOfList (a :: l)"}, {"tactic": "have hfa' : f (f a) \u2260 f a := mt (fun h => f.injective h) hfa", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\n\u22a2 f \u2208 permsOfList (a :: l)", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\n\u22a2 f \u2208 permsOfList (a :: l)"}, {"tactic": "have : \u2200 x : \u03b1, (Equiv.swap a (f a) * f) x \u2260 x \u2192 x \u2208 l := by\n  intro x hx\n  have hxa : x \u2260 a := by\n    rintro rfl\n    apply hx\n    simp only [mul_apply, swap_apply_right]\n  refine' List.mem_of_ne_of_mem hxa (h x fun h => _)\n  simp only [mul_apply, swap_apply_def, mul_apply, Ne.def, apply_eq_iff_eq] at hx\n  split_ifs at hx with h_1\n  exacts [hxa (h.symm.trans h_1), hx h]", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\n\u22a2 f \u2208 permsOfList (a :: l)", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\nthis : \u2200 (x : \u03b1), \u2191(Equiv.swap a (\u2191f a) * f) x \u2260 x \u2192 x \u2208 l\n\u22a2 f \u2208 permsOfList (a :: l)"}, {"tactic": "suffices f \u2208 permsOfList l \u2228 \u2203 b \u2208 l, \u2203 g \u2208 permsOfList l, Equiv.swap a b * g = f by\n  simpa only [permsOfList, exists_prop, List.mem_map, mem_append, List.mem_bind]", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\nthis : \u2200 (x : \u03b1), \u2191(Equiv.swap a (\u2191f a) * f) x \u2260 x \u2192 x \u2208 l\n\u22a2 f \u2208 permsOfList (a :: l)", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\nthis : \u2200 (x : \u03b1), \u2191(Equiv.swap a (\u2191f a) * f) x \u2260 x \u2192 x \u2208 l\n\u22a2 f \u2208 permsOfList l \u2228 \u2203 b, b \u2208 l \u2227 \u2203 g, g \u2208 permsOfList l \u2227 Equiv.swap a b * g = f"}, {"tactic": "refine' or_iff_not_imp_left.2 fun _hfl => \u27e8f a, _, Equiv.swap a (f a) * f, IH this, _\u27e9", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\nthis : \u2200 (x : \u03b1), \u2191(Equiv.swap a (\u2191f a) * f) x \u2260 x \u2192 x \u2208 l\n\u22a2 f \u2208 permsOfList l \u2228 \u2203 b, b \u2208 l \u2227 \u2203 g, g \u2208 permsOfList l \u2227 Equiv.swap a b * g = f", "state_after": "case neg.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\nthis : \u2200 (x : \u03b1), \u2191(Equiv.swap a (\u2191f a) * f) x \u2260 x \u2192 x \u2208 l\n_hfl : \u00acf \u2208 permsOfList l\n\u22a2 \u2191f a \u2208 l\n\ncase neg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\nthis : \u2200 (x : \u03b1), \u2191(Equiv.swap a (\u2191f a) * f) x \u2260 x \u2192 x \u2208 l\n_hfl : \u00acf \u2208 permsOfList l\n\u22a2 Equiv.swap a (\u2191f a) * (Equiv.swap a (\u2191f a) * f) = f"}, {"tactic": "refine' mem_append_left _ (IH fun x hx => mem_of_ne_of_mem _ (h x hx))", "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u2191f a = a\n\u22a2 f \u2208 permsOfList (a :: l)", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u2191f a = a\nx : \u03b1\nhx : \u2191f x \u2260 x\n\u22a2 x \u2260 a"}, {"tactic": "rintro rfl", "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u2191f a = a\nx : \u03b1\nhx : \u2191f x \u2260 x\n\u22a2 x \u2260 a", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nx : \u03b1\nhx : \u2191f x \u2260 x\nh : \u2200 (x_1 : \u03b1), \u2191f x_1 \u2260 x_1 \u2192 x_1 \u2208 x :: l\nhfa : \u2191f x = x\n\u22a2 False"}, {"tactic": "exact hx hfa", "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nx : \u03b1\nhx : \u2191f x \u2260 x\nh : \u2200 (x_1 : \u03b1), \u2191f x_1 \u2260 x_1 \u2192 x_1 \u2208 x :: l\nhfa : \u2191f x = x\n\u22a2 False", "state_after": "no goals"}, {"tactic": "intro x hx", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\n\u22a2 \u2200 (x : \u03b1), \u2191(Equiv.swap a (\u2191f a) * f) x \u2260 x \u2192 x \u2208 l", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\nx : \u03b1\nhx : \u2191(Equiv.swap a (\u2191f a) * f) x \u2260 x\n\u22a2 x \u2208 l"}, {"tactic": "have hxa : x \u2260 a := by\n  rintro rfl\n  apply hx\n  simp only [mul_apply, swap_apply_right]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\nx : \u03b1\nhx : \u2191(Equiv.swap a (\u2191f a) * f) x \u2260 x\n\u22a2 x \u2208 l", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\nx : \u03b1\nhx : \u2191(Equiv.swap a (\u2191f a) * f) x \u2260 x\nhxa : x \u2260 a\n\u22a2 x \u2208 l"}, {"tactic": "refine' List.mem_of_ne_of_mem hxa (h x fun h => _)", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\nx : \u03b1\nhx : \u2191(Equiv.swap a (\u2191f a) * f) x \u2260 x\nhxa : x \u2260 a\n\u22a2 x \u2208 l", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh\u271d : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\nx : \u03b1\nhx : \u2191(Equiv.swap a (\u2191f a) * f) x \u2260 x\nhxa : x \u2260 a\nh : \u2191f x = x\n\u22a2 False"}, {"tactic": "simp only [mul_apply, swap_apply_def, mul_apply, Ne.def, apply_eq_iff_eq] at hx", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh\u271d : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\nx : \u03b1\nhx : \u2191(Equiv.swap a (\u2191f a) * f) x \u2260 x\nhxa : x \u2260 a\nh : \u2191f x = x\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh\u271d : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\nx : \u03b1\nhxa : x \u2260 a\nh : \u2191f x = x\nhx : \u00ac(if \u2191f x = a then \u2191f a else if x = a then a else \u2191f x) = x\n\u22a2 False"}, {"tactic": "split_ifs at hx with h_1", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh\u271d : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\nx : \u03b1\nhxa : x \u2260 a\nh : \u2191f x = x\nhx : \u00ac(if \u2191f x = a then \u2191f a else if x = a then a else \u2191f x) = x\n\u22a2 False", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh\u271d : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\nx : \u03b1\nhxa : x \u2260 a\nh : \u2191f x = x\nh_1 : \u2191f x = a\nhx : \u00ac\u2191f a = x\n\u22a2 False\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh\u271d : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\nx : \u03b1\nhxa : x \u2260 a\nh : \u2191f x = x\nh_1 : \u00ac\u2191f x = a\nhx : \u00ac\u2191f x = x\n\u22a2 False"}, {"tactic": "exacts [hxa (h.symm.trans h_1), hx h]", "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh\u271d : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\nx : \u03b1\nhxa : x \u2260 a\nh : \u2191f x = x\nh_1 : \u2191f x = a\nhx : \u00ac\u2191f a = x\n\u22a2 False\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh\u271d : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\nx : \u03b1\nhxa : x \u2260 a\nh : \u2191f x = x\nh_1 : \u00ac\u2191f x = a\nhx : \u00ac\u2191f x = x\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rintro rfl", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\nx : \u03b1\nhx : \u2191(Equiv.swap a (\u2191f a) * f) x \u2260 x\n\u22a2 x \u2260 a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nx : \u03b1\nh : \u2200 (x_1 : \u03b1), \u2191f x_1 \u2260 x_1 \u2192 x_1 \u2208 x :: l\nhfa : \u00ac\u2191f x = x\nhfa' : \u2191f (\u2191f x) \u2260 \u2191f x\nhx : \u2191(Equiv.swap x (\u2191f x) * f) x \u2260 x\n\u22a2 False"}, {"tactic": "apply hx", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nx : \u03b1\nh : \u2200 (x_1 : \u03b1), \u2191f x_1 \u2260 x_1 \u2192 x_1 \u2208 x :: l\nhfa : \u00ac\u2191f x = x\nhfa' : \u2191f (\u2191f x) \u2260 \u2191f x\nhx : \u2191(Equiv.swap x (\u2191f x) * f) x \u2260 x\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nx : \u03b1\nh : \u2200 (x_1 : \u03b1), \u2191f x_1 \u2260 x_1 \u2192 x_1 \u2208 x :: l\nhfa : \u00ac\u2191f x = x\nhfa' : \u2191f (\u2191f x) \u2260 \u2191f x\nhx : \u2191(Equiv.swap x (\u2191f x) * f) x \u2260 x\n\u22a2 \u2191(Equiv.swap x (\u2191f x) * f) x = x"}, {"tactic": "simp only [mul_apply, swap_apply_right]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nx : \u03b1\nh : \u2200 (x_1 : \u03b1), \u2191f x_1 \u2260 x_1 \u2192 x_1 \u2208 x :: l\nhfa : \u00ac\u2191f x = x\nhfa' : \u2191f (\u2191f x) \u2260 \u2191f x\nhx : \u2191(Equiv.swap x (\u2191f x) * f) x \u2260 x\n\u22a2 \u2191(Equiv.swap x (\u2191f x) * f) x = x", "state_after": "no goals"}, {"tactic": "simpa only [permsOfList, exists_prop, List.mem_map, mem_append, List.mem_bind]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\nthis\u271d : \u2200 (x : \u03b1), \u2191(Equiv.swap a (\u2191f a) * f) x \u2260 x \u2192 x \u2208 l\nthis : f \u2208 permsOfList l \u2228 \u2203 b, b \u2208 l \u2227 \u2203 g, g \u2208 permsOfList l \u2227 Equiv.swap a b * g = f\n\u22a2 f \u2208 permsOfList (a :: l)", "state_after": "no goals"}, {"tactic": "exact mem_of_ne_of_mem hfa (h _ hfa')", "state_before": "case neg.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\nthis : \u2200 (x : \u03b1), \u2191(Equiv.swap a (\u2191f a) * f) x \u2260 x \u2192 x \u2208 l\n_hfl : \u00acf \u2208 permsOfList l\n\u22a2 \u2191f a \u2208 l", "state_after": "no goals"}, {"tactic": "rw [\u2190 mul_assoc, mul_def (swap a (f a)) (swap a (f a)), swap_swap, \u2190 Perm.one_def, one_mul]", "state_before": "case neg.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type ?u.4748\n\u03b3 : Type ?u.4751\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\na : \u03b1\nl : List \u03b1\nIH : \u2200 {f : Equiv.Perm \u03b1}, (\u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 l) \u2192 f \u2208 permsOfList l\nf : Equiv.Perm \u03b1\nh : \u2200 (x : \u03b1), \u2191f x \u2260 x \u2192 x \u2208 a :: l\nhfa : \u00ac\u2191f a = a\nhfa' : \u2191f (\u2191f a) \u2260 \u2191f a\nthis : \u2200 (x : \u03b1), \u2191(Equiv.swap a (\u2191f a) * f) x \u2260 x \u2192 x \u2208 l\n_hfl : \u00acf \u2208 permsOfList l\n\u22a2 Equiv.swap a (\u2191f a) * (Equiv.swap a (\u2191f a) * f) = f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Pi.lean", "full_name": "Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle", "start": [577, 1], "end": [607, 53], "traced_tactics": [{"tactic": "refine' \u27e8fun h => _, _\u27e9", "state_before": "\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\n\u22a2 mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v \u2194\n    k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n", "state_after": "case refine'_1\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n\n\ncase refine'_2\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n \u2192\n    mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v"}, {"tactic": "have hk := congr_fun h k", "state_before": "case refine'_1\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n", "state_after": "case refine'_1\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : (mulSingle k u * mulSingle l v) k = (mulSingle m u * mulSingle n v) k\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n"}, {"tactic": "have hl := congr_fun h l", "state_before": "case refine'_1\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : (mulSingle k u * mulSingle l v) k = (mulSingle m u * mulSingle n v) k\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n", "state_after": "case refine'_1\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : (mulSingle k u * mulSingle l v) k = (mulSingle m u * mulSingle n v) k\nhl : (mulSingle k u * mulSingle l v) l = (mulSingle m u * mulSingle n v) l\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n"}, {"tactic": "have hm := (congr_fun h m).symm", "state_before": "case refine'_1\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : (mulSingle k u * mulSingle l v) k = (mulSingle m u * mulSingle n v) k\nhl : (mulSingle k u * mulSingle l v) l = (mulSingle m u * mulSingle n v) l\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n", "state_after": "case refine'_1\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : (mulSingle k u * mulSingle l v) k = (mulSingle m u * mulSingle n v) k\nhl : (mulSingle k u * mulSingle l v) l = (mulSingle m u * mulSingle n v) l\nhm : (mulSingle m u * mulSingle n v) m = (mulSingle k u * mulSingle l v) m\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n"}, {"tactic": "have hn := (congr_fun h n).symm", "state_before": "case refine'_1\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : (mulSingle k u * mulSingle l v) k = (mulSingle m u * mulSingle n v) k\nhl : (mulSingle k u * mulSingle l v) l = (mulSingle m u * mulSingle n v) l\nhm : (mulSingle m u * mulSingle n v) m = (mulSingle k u * mulSingle l v) m\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n", "state_after": "case refine'_1\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : (mulSingle k u * mulSingle l v) k = (mulSingle m u * mulSingle n v) k\nhl : (mulSingle k u * mulSingle l v) l = (mulSingle m u * mulSingle n v) l\nhm : (mulSingle m u * mulSingle n v) m = (mulSingle k u * mulSingle l v) m\nhn : (mulSingle m u * mulSingle n v) n = (mulSingle k u * mulSingle l v) n\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n"}, {"tactic": "simp only [mul_apply, mulSingle_apply, if_pos rfl] at hk hl hm hn", "state_before": "case refine'_1\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : (mulSingle k u * mulSingle l v) k = (mulSingle m u * mulSingle n v) k\nhl : (mulSingle k u * mulSingle l v) l = (mulSingle m u * mulSingle n v) l\nhm : (mulSingle m u * mulSingle n v) m = (mulSingle k u * mulSingle l v) m\nhn : (mulSingle m u * mulSingle n v) n = (mulSingle k u * mulSingle l v) n\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n", "state_after": "case refine'_1\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = n then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = n then v else 1\nhm : ((if True then u else 1) * if m = n then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n"}, {"tactic": "rcases eq_or_ne k m with (rfl | hkm)", "state_before": "case refine'_1\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = n then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = n then v else 1\nhm : ((if True then u else 1) * if m = n then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n", "state_after": "case refine'_1.inl\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle k u * mulSingle n v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = k then u else 1) * if k = n then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = k then u else 1) * if l = n then v else 1\nhm : ((if True then u else 1) * if k = n then v else 1) = (if k = k then u else 1) * if k = l then v else 1\nhn : ((if n = k then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1\n\u22a2 k = k \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = k \u2228 u * v = 1 \u2227 k = l \u2227 k = n\n\ncase refine'_1.inr\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = n then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = n then v else 1\nhm : ((if True then u else 1) * if m = n then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1\nhkm : k \u2260 m\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n"}, {"tactic": "refine' Or.inl \u27e8rfl, not_ne_iff.mp fun hln => (hv _).elim\u27e9", "state_before": "case refine'_1.inl\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle k u * mulSingle n v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = k then u else 1) * if k = n then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = k then u else 1) * if l = n then v else 1\nhm : ((if True then u else 1) * if k = n then v else 1) = (if k = k then u else 1) * if k = l then v else 1\nhn : ((if n = k then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1\n\u22a2 k = k \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = k \u2228 u * v = 1 \u2227 k = l \u2227 k = n", "state_after": "case refine'_1.inl\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle k u * mulSingle n v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = k then u else 1) * if k = n then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = k then u else 1) * if l = n then v else 1\nhm : ((if True then u else 1) * if k = n then v else 1) = (if k = k then u else 1) * if k = l then v else 1\nhn : ((if n = k then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1\nhln : l \u2260 n\n\u22a2 v = 1"}, {"tactic": "rcases eq_or_ne k l with (rfl | hkl)", "state_before": "case refine'_1.inl\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle k u * mulSingle n v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = k then u else 1) * if k = n then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = k then u else 1) * if l = n then v else 1\nhm : ((if True then u else 1) * if k = n then v else 1) = (if k = k then u else 1) * if k = l then v else 1\nhn : ((if n = k then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1\nhln : l \u2260 n\n\u22a2 v = 1", "state_after": "case refine'_1.inl.inl\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle k v = mulSingle k u * mulSingle n v\nhk : ((if True then u else 1) * if k = k then v else 1) = (if k = k then u else 1) * if k = n then v else 1\nhl : ((if k = k then u else 1) * if True then v else 1) = (if k = k then u else 1) * if k = n then v else 1\nhm : ((if True then u else 1) * if k = n then v else 1) = (if k = k then u else 1) * if k = k then v else 1\nhn : ((if n = k then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = k then v else 1\nhln : k \u2260 n\n\u22a2 v = 1\n\ncase refine'_1.inl.inr\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle k u * mulSingle n v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = k then u else 1) * if k = n then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = k then u else 1) * if l = n then v else 1\nhm : ((if True then u else 1) * if k = n then v else 1) = (if k = k then u else 1) * if k = l then v else 1\nhn : ((if n = k then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1\nhln : l \u2260 n\nhkl : k \u2260 l\n\u22a2 v = 1"}, {"tactic": "rwa [if_neg hln.symm, if_neg hln.symm, one_mul, one_mul] at hn", "state_before": "case refine'_1.inl.inl\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle k v = mulSingle k u * mulSingle n v\nhk : ((if True then u else 1) * if k = k then v else 1) = (if k = k then u else 1) * if k = n then v else 1\nhl : ((if k = k then u else 1) * if True then v else 1) = (if k = k then u else 1) * if k = n then v else 1\nhm : ((if True then u else 1) * if k = n then v else 1) = (if k = k then u else 1) * if k = k then v else 1\nhn : ((if n = k then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = k then v else 1\nhln : k \u2260 n\n\u22a2 v = 1", "state_after": "no goals"}, {"tactic": "rwa [if_neg hkl.symm, if_neg hln, one_mul, one_mul] at hl", "state_before": "case refine'_1.inl.inr\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle k u * mulSingle n v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = k then u else 1) * if k = n then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = k then u else 1) * if l = n then v else 1\nhm : ((if True then u else 1) * if k = n then v else 1) = (if k = k then u else 1) * if k = l then v else 1\nhn : ((if n = k then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1\nhln : l \u2260 n\nhkl : k \u2260 l\n\u22a2 v = 1", "state_after": "no goals"}, {"tactic": "rcases eq_or_ne m n with (rfl | hmn)", "state_before": "case refine'_1.inr\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = n then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = n then v else 1\nhm : ((if True then u else 1) * if m = n then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1\nhkm : k \u2260 m\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n", "state_after": "case refine'_1.inr.inl\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle m v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = m then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = m then v else 1\nhm : ((if True then u else 1) * if m = m then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = l then v else 1\n\u22a2 k = m \u2227 l = m \u2228 u = v \u2227 k = m \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = m\n\ncase refine'_1.inr.inr\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = n then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = n then v else 1\nhm : ((if True then u else 1) * if m = n then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1\nhkm : k \u2260 m\nhmn : m \u2260 n\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n"}, {"tactic": "rcases eq_or_ne k l with (rfl | hkl)", "state_before": "case refine'_1.inr.inl\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle m v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = m then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = m then v else 1\nhm : ((if True then u else 1) * if m = m then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = l then v else 1\n\u22a2 k = m \u2227 l = m \u2228 u = v \u2227 k = m \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = m", "state_after": "case refine'_1.inr.inl.inl\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nh : mulSingle k u * mulSingle k v = mulSingle m u * mulSingle m v\nhk : ((if True then u else 1) * if k = k then v else 1) = (if k = m then u else 1) * if k = m then v else 1\nhl : ((if k = k then u else 1) * if True then v else 1) = (if k = m then u else 1) * if k = m then v else 1\nhm : ((if True then u else 1) * if m = m then v else 1) = (if m = k then u else 1) * if m = k then v else 1\nhn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = k then v else 1\n\u22a2 k = m \u2227 k = m \u2228 u = v \u2227 k = m \u2227 k = m \u2228 u * v = 1 \u2227 k = k \u2227 m = m\n\ncase refine'_1.inr.inl.inr\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle m v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = m then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = m then v else 1\nhm : ((if True then u else 1) * if m = m then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhkl : k \u2260 l\n\u22a2 k = m \u2227 l = m \u2228 u = v \u2227 k = m \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = m"}, {"tactic": "rw [if_neg hkm.symm, if_neg hkm.symm, one_mul, if_pos rfl] at hm", "state_before": "case refine'_1.inr.inl.inl\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nh : mulSingle k u * mulSingle k v = mulSingle m u * mulSingle m v\nhk : ((if True then u else 1) * if k = k then v else 1) = (if k = m then u else 1) * if k = m then v else 1\nhl : ((if k = k then u else 1) * if True then v else 1) = (if k = m then u else 1) * if k = m then v else 1\nhm : ((if True then u else 1) * if m = m then v else 1) = (if m = k then u else 1) * if m = k then v else 1\nhn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = k then v else 1\n\u22a2 k = m \u2227 k = m \u2228 u = v \u2227 k = m \u2227 k = m \u2228 u * v = 1 \u2227 k = k \u2227 m = m", "state_after": "case refine'_1.inr.inl.inl\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nh : mulSingle k u * mulSingle k v = mulSingle m u * mulSingle m v\nhk : ((if True then u else 1) * if k = k then v else 1) = (if k = m then u else 1) * if k = m then v else 1\nhl : ((if k = k then u else 1) * if True then v else 1) = (if k = m then u else 1) * if k = m then v else 1\nhm : (if True then u else 1) * v = 1\nhn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = k then v else 1\n\u22a2 k = m \u2227 k = m \u2228 u = v \u2227 k = m \u2227 k = m \u2228 u * v = 1 \u2227 k = k \u2227 m = m"}, {"tactic": "exact Or.inr (Or.inr \u27e8hm, rfl, rfl\u27e9)", "state_before": "case refine'_1.inr.inl.inl\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nh : mulSingle k u * mulSingle k v = mulSingle m u * mulSingle m v\nhk : ((if True then u else 1) * if k = k then v else 1) = (if k = m then u else 1) * if k = m then v else 1\nhl : ((if k = k then u else 1) * if True then v else 1) = (if k = m then u else 1) * if k = m then v else 1\nhm : (if True then u else 1) * v = 1\nhn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = k then v else 1\n\u22a2 k = m \u2227 k = m \u2228 u = v \u2227 k = m \u2227 k = m \u2228 u * v = 1 \u2227 k = k \u2227 m = m", "state_after": "no goals"}, {"tactic": "simp only [if_neg hkm, if_neg hkl, mul_one] at hk", "state_before": "case refine'_1.inr.inl.inr\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle m v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = m then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = m then v else 1\nhm : ((if True then u else 1) * if m = m then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhkl : k \u2260 l\n\u22a2 k = m \u2227 l = m \u2228 u = v \u2227 k = m \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = m", "state_after": "case refine'_1.inr.inl.inr\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle m v\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = m then v else 1\nhm : ((if True then u else 1) * if m = m then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhkl : k \u2260 l\nhk : (if True then u else 1) = 1\n\u22a2 k = m \u2227 l = m \u2228 u = v \u2227 k = m \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = m"}, {"tactic": "dsimp at hk", "state_before": "case refine'_1.inr.inl.inr\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle m v\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = m then v else 1\nhm : ((if True then u else 1) * if m = m then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhkl : k \u2260 l\nhk : (if True then u else 1) = 1\n\u22a2 k = m \u2227 l = m \u2228 u = v \u2227 k = m \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = m", "state_after": "case refine'_1.inr.inl.inr\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle m v\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = m then v else 1\nhm : ((if True then u else 1) * if m = m then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhkl : k \u2260 l\nhk : u = 1\n\u22a2 k = m \u2227 l = m \u2228 u = v \u2227 k = m \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = m"}, {"tactic": "contradiction", "state_before": "case refine'_1.inr.inl.inr\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle m v\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = m then v else 1\nhm : ((if True then u else 1) * if m = m then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhkl : k \u2260 l\nhk : u = 1\n\u22a2 k = m \u2227 l = m \u2228 u = v \u2227 k = m \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = m", "state_after": "no goals"}, {"tactic": "rw [if_neg hkm.symm, if_neg hmn, one_mul, mul_one] at hm", "state_before": "case refine'_1.inr.inr\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = n then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = n then v else 1\nhm : ((if True then u else 1) * if m = n then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1\nhkm : k \u2260 m\nhmn : m \u2260 n\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n", "state_after": "case refine'_1.inr.inr\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = n then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = n then v else 1\nhm : (if True then u else 1) = if m = l then v else 1\nhn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1\nhkm : k \u2260 m\nhmn : m \u2260 n\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n"}, {"tactic": "obtain rfl := (ite_ne_right_iff.mp (ne_of_eq_of_ne hm.symm hu)).1", "state_before": "case refine'_1.inr.inr\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = n then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = n then v else 1\nhm : (if True then u else 1) = if m = l then v else 1\nhn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1\nhkm : k \u2260 m\nhmn : m \u2260 n\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n", "state_after": "case refine'_1.inr.inr\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nhmn : m \u2260 n\nh : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v\nhk : ((if True then u else 1) * if k = m then v else 1) = (if k = m then u else 1) * if k = n then v else 1\nhl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1\nhm : (if True then u else 1) = if m = m then v else 1\nhn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1\n\u22a2 k = m \u2227 m = n \u2228 u = v \u2227 k = n \u2227 m = m \u2228 u * v = 1 \u2227 k = m \u2227 m = n"}, {"tactic": "rw [if_neg hkm, if_neg hkm, one_mul, mul_one] at hk", "state_before": "case refine'_1.inr.inr\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nhmn : m \u2260 n\nh : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v\nhk : ((if True then u else 1) * if k = m then v else 1) = (if k = m then u else 1) * if k = n then v else 1\nhl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1\nhm : (if True then u else 1) = if m = m then v else 1\nhn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1\n\u22a2 k = m \u2227 m = n \u2228 u = v \u2227 k = n \u2227 m = m \u2228 u * v = 1 \u2227 k = m \u2227 m = n", "state_after": "case refine'_1.inr.inr\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nhmn : m \u2260 n\nh : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v\nhk : (if True then u else 1) = if k = n then v else 1\nhl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1\nhm : (if True then u else 1) = if m = m then v else 1\nhn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1\n\u22a2 k = m \u2227 m = n \u2228 u = v \u2227 k = n \u2227 m = m \u2228 u * v = 1 \u2227 k = m \u2227 m = n"}, {"tactic": "obtain rfl := (ite_ne_right_iff.mp (ne_of_eq_of_ne hk.symm hu)).1", "state_before": "case refine'_1.inr.inr\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nhmn : m \u2260 n\nh : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v\nhk : (if True then u else 1) = if k = n then v else 1\nhl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1\nhm : (if True then u else 1) = if m = m then v else 1\nhn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1\n\u22a2 k = m \u2227 m = n \u2228 u = v \u2227 k = n \u2227 m = m \u2228 u * v = 1 \u2227 k = m \u2227 m = n", "state_after": "case refine'_1.inr.inr\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nhm : (if True then u else 1) = if m = m then v else 1\nhmn : m \u2260 k\nh : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle k v\nhk : (if True then u else 1) = if k = k then v else 1\nhl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = k then v else 1\nhn : ((if k = m then u else 1) * if True then v else 1) = (if k = k then u else 1) * if k = m then v else 1\n\u22a2 k = m \u2227 m = k \u2228 u = v \u2227 k = k \u2227 m = m \u2228 u * v = 1 \u2227 k = m \u2227 m = k"}, {"tactic": "exact Or.inr (Or.inl \u27e8hk.trans (if_pos rfl), rfl, rfl\u27e9)", "state_before": "case refine'_1.inr.inr\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nhm : (if True then u else 1) = if m = m then v else 1\nhmn : m \u2260 k\nh : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle k v\nhk : (if True then u else 1) = if k = k then v else 1\nhl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = k then v else 1\nhn : ((if k = m then u else 1) * if True then v else 1) = (if k = k then u else 1) * if k = m then v else 1\n\u22a2 k = m \u2227 m = k \u2228 u = v \u2227 k = k \u2227 m = m \u2228 u * v = 1 \u2227 k = m \u2227 m = k", "state_after": "no goals"}, {"tactic": "rintro (\u27e8rfl, rfl\u27e9 | \u27e8rfl, rfl, rfl\u27e9 | \u27e8h, rfl, rfl\u27e9)", "state_before": "case refine'_2\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n \u2192\n    mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v", "state_after": "case refine'_2.inl.intro\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\n\u22a2 mulSingle k u * mulSingle l v = mulSingle k u * mulSingle l v\n\ncase refine'_2.inr.inl.intro.intro\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l : I\nu : M\nhu hv : u \u2260 1\n\u22a2 mulSingle k u * mulSingle l u = mulSingle l u * mulSingle k u\n\ncase refine'_2.inr.inr.intro.intro\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : u * v = 1\n\u22a2 mulSingle k u * mulSingle k v = mulSingle m u * mulSingle m v"}, {"tactic": "rfl", "state_before": "case refine'_2.inl.intro\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\n\u22a2 mulSingle k u * mulSingle l v = mulSingle k u * mulSingle l v", "state_after": "no goals"}, {"tactic": "apply mul_comm", "state_before": "case refine'_2.inr.inl.intro.intro\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk l : I\nu : M\nhu hv : u \u2260 1\n\u22a2 mulSingle k u * mulSingle l u = mulSingle l u * mulSingle k u", "state_after": "no goals"}, {"tactic": "simp_rw [\u2190 Pi.mulSingle_mul, h, mulSingle_one]", "state_before": "case refine'_2.inr.inr.intro.intro\n\u03b9 : Type ?u.81137\n\u03b1 : Type ?u.81140\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_1\ninst\u271d : CommMonoid M\nk m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : u * v = 1\n\u22a2 mulSingle k u * mulSingle k v = mulSingle m u * mulSingle m v", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Hom/Lattice.lean", "full_name": "InfTopHom.top_apply", "start": [1014, 1], "end": [1015, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "full_name": "Metric.thickening_closure", "start": [1235, 1], "end": [1236, 41], "traced_tactics": [{"tactic": "simp_rw [thickening, infEdist_closure]", "state_before": "\u03b9 : Sort ?u.131156\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4 \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u22a2 thickening \u03b4 (closure s) = thickening \u03b4 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.smul_subset_smul_right", "start": [1370, 1], "end": [1371, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "full_name": "Ideal.Quotient.lift_mk", "start": [276, 1], "end": [278, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.iInter_congr_Prop", "start": [224, 1], "end": [226, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Basic.lean", "full_name": "update_lt_self_iff", "start": [898, 1], "end": [898, 86], "traced_tactics": [{"tactic": "simp [lt_iff_le_not_le]", "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03c0 i)\nx y : (i : \u03b9) \u2192 \u03c0 i\ni : \u03b9\na b : \u03c0 i\n\u22a2 update x i a < x \u2194 a < x i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Cardinal/Cofinality.lean", "full_name": "Cardinal.isRegular_aleph0", "start": [975, 1], "end": [976, 20], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b1 : Type ?u.145122\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u2135\u2080 \u2264 Ordinal.cof (ord \u2135\u2080)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/MinMax.lean", "full_name": "List.not_lt_maximum_of_mem", "start": [323, 1], "end": [324, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Lie/Submodule.lean", "full_name": "LieSubmodule.coe_add", "start": [197, 1], "end": [198, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/QuadraticForm/Basic.lean", "full_name": "QuadraticForm.linMulLin_add", "start": [591, 1], "end": [592, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Vector/Mem.lean", "full_name": "Vector.mem_cons_iff", "start": [51, 1], "end": [52, 41], "traced_tactics": [{"tactic": "rw [Vector.toList_cons, List.mem_cons]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.1633\nn : \u2115\na a' : \u03b1\nv : Vector \u03b1 n\n\u22a2 a' \u2208 toList (a ::\u1d65 v) \u2194 a' = a \u2228 a' \u2208 toList v", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Setoid/Partition.lean", "full_name": "Setoid.nonempty_of_mem_partition", "start": [214, 1], "end": [216, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/RingDivision.lean", "full_name": "Polynomial.ne_zero_of_mem_roots", "start": [584, 1], "end": [585, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Module/Multilinear.lean", "full_name": "ContinuousMultilinearMap.map_smul", "start": [127, 1], "end": [129, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Basic.lean", "full_name": "Set.Iio_union_Ico'", "start": [1383, 1], "end": [1389, 29], "traced_tactics": [{"tactic": "ext1 x", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.96699\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\n\u22a2 Iio b \u222a Ico c d = Iio (max b d)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.96699\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\n\u22a2 x \u2208 Iio b \u222a Ico c d \u2194 x \u2208 Iio (max b d)"}, {"tactic": "simp_rw [mem_union, mem_Iio, mem_Ico, lt_max_iff]", "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.96699\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\n\u22a2 x \u2208 Iio b \u222a Ico c d \u2194 x \u2208 Iio (max b d)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.96699\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\n\u22a2 x < b \u2228 c \u2264 x \u2227 x < d \u2194 x < b \u2228 x < d"}, {"tactic": "by_cases hc : c \u2264 x", "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.96699\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\n\u22a2 x < b \u2228 c \u2264 x \u2227 x < d \u2194 x < b \u2228 x < d", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.96699\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : c \u2264 x\n\u22a2 x < b \u2228 c \u2264 x \u2227 x < d \u2194 x < b \u2228 x < d\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.96699\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : \u00acc \u2264 x\n\u22a2 x < b \u2228 c \u2264 x \u2227 x < d \u2194 x < b \u2228 x < d"}, {"tactic": "simp only [hc, true_and]", "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.96699\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : c \u2264 x\n\u22a2 x < b \u2228 c \u2264 x \u2227 x < d \u2194 x < b \u2228 x < d", "state_after": "no goals"}, {"tactic": "have hxb : x < b := (lt_of_not_ge hc).trans_le h\u2081", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.96699\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : \u00acc \u2264 x\n\u22a2 x < b \u2228 c \u2264 x \u2227 x < d \u2194 x < b \u2228 x < d", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.96699\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : \u00acc \u2264 x\nhxb : x < b\n\u22a2 x < b \u2228 c \u2264 x \u2227 x < d \u2194 x < b \u2228 x < d"}, {"tactic": "simp only [hxb, true_or]", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.96699\ninst\u271d : LinearOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\nh\u2081 : c \u2264 b\nx : \u03b1\nhc : \u00acc \u2264 x\nhxb : x < b\n\u22a2 x < b \u2228 c \u2264 x \u2227 x < d \u2194 x < b \u2228 x < d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Young/SemistandardTableau.lean", "full_name": "Ssyt.row_weak", "start": [114, 1], "end": [116, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Add.lean", "full_name": "DifferentiableOn.const_smul", "start": [96, 1], "end": [97, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "full_name": "CategoryTheory.MonoidalCategory.leftAssocTensor_obj", "start": [422, 1], "end": [423, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Basic.lean", "full_name": "Submodule.range_ofLe", "start": [1650, 1], "end": [1651, 74], "traced_tactics": [{"tactic": "rw [\u2190 map_top, ofLe, LinearMap.map_codRestrict, map_top, range_subtype]", "state_before": "R : Type u_1\nR\u2081 : Type ?u.1546983\nR\u2082 : Type ?u.1546986\nR\u2083 : Type ?u.1546989\nR\u2084 : Type ?u.1546992\nS : Type ?u.1546995\nK : Type ?u.1546998\nK\u2082 : Type ?u.1547001\nM : Type u_2\nM' : Type ?u.1547007\nM\u2081 : Type ?u.1547010\nM\u2082 : Type ?u.1547013\nM\u2083 : Type ?u.1547016\nM\u2084 : Type ?u.1547019\nN : Type ?u.1547022\nN\u2082 : Type ?u.1547025\n\u03b9 : Type ?u.1547028\nV : Type ?u.1547031\nV\u2082 : Type ?u.1547034\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : Semiring R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R\u2082 M\u2082\np\u271d p' : Submodule R M\nq\u271d : Submodule R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\nF : Type ?u.1547208\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\np q : Submodule R M\nh : p \u2264 q\n\u22a2 range (ofLe h) = comap (Submodule.subtype q) p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/ContDiffDef.lean", "full_name": "hasFTaylorSeriesUpToOn_succ_iff_left", "start": [312, 1], "end": [334, 20], "traced_tactics": [{"tactic": "constructor", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 HasFTaylorSeriesUpToOn (\u2191n + 1) f p s \u2194\n    HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n      (\u2200 (x : E),\n          x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n        ContinuousOn (fun x => p x (n + 1)) s", "state_after": "case mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 HasFTaylorSeriesUpToOn (\u2191n + 1) f p s \u2192\n    HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n      (\u2200 (x : E),\n          x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n        ContinuousOn (fun x => p x (n + 1)) s\n\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n      (\u2200 (x : E),\n          x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n        ContinuousOn (fun x => p x (n + 1)) s \u2192\n    HasFTaylorSeriesUpToOn (\u2191n + 1) f p s"}, {"tactic": "exact fun h \u21a6 \u27e8h.of_le (WithTop.coe_le_coe.2 (Nat.le_succ n)),\n  h.fderivWithin _ (WithTop.coe_lt_coe.2 (lt_add_one n)), h.cont (n + 1) le_rfl\u27e9", "state_before": "case mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 HasFTaylorSeriesUpToOn (\u2191n + 1) f p s \u2192\n    HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n      (\u2200 (x : E),\n          x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n        ContinuousOn (fun x => p x (n + 1)) s", "state_after": "no goals"}, {"tactic": "intro h", "state_before": "case mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\n\u22a2 HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n      (\u2200 (x : E),\n          x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n        ContinuousOn (fun x => p x (n + 1)) s \u2192\n    HasFTaylorSeriesUpToOn (\u2191n + 1) f p s", "state_after": "case mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\n\u22a2 HasFTaylorSeriesUpToOn (\u2191n + 1) f p s"}, {"tactic": "constructor", "state_before": "case mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\n\u22a2 HasFTaylorSeriesUpToOn (\u2191n + 1) f p s", "state_after": "case mpr.zero_eq\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\n\ncase mpr.fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\n\u22a2 \u2200 (m : \u2115),\n    \u2191m < \u2191n + 1 \u2192\n      \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x\n\ncase mpr.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 \u2191n + 1 \u2192 ContinuousOn (fun x => p x m) s"}, {"tactic": "exact h.1.zero_eq", "state_before": "case mpr.zero_eq\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x", "state_after": "no goals"}, {"tactic": "intro m hm", "state_before": "case mpr.fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\n\u22a2 \u2200 (m : \u2115),\n    \u2191m < \u2191n + 1 \u2192\n      \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x", "state_after": "case mpr.fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m < \u2191n + 1\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x"}, {"tactic": "by_cases h' : m < n", "state_before": "case mpr.fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m < \u2191n + 1\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x", "state_after": "case pos\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m < \u2191n + 1\nh' : m < n\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x\n\ncase neg\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m < \u2191n + 1\nh' : \u00acm < n\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x"}, {"tactic": "exact h.1.fderivWithin m (WithTop.coe_lt_coe.2 h')", "state_before": "case pos\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m < \u2191n + 1\nh' : m < n\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x", "state_after": "no goals"}, {"tactic": "have : m = n := Nat.eq_of_lt_succ_of_not_lt (WithTop.coe_lt_coe.1 hm) h'", "state_before": "case neg\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m < \u2191n + 1\nh' : \u00acm < n\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x", "state_after": "case neg\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m < \u2191n + 1\nh' : \u00acm < n\nthis : m = n\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x"}, {"tactic": "rw [this]", "state_before": "case neg\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m < \u2191n + 1\nh' : \u00acm < n\nthis : m = n\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x", "state_after": "case neg\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m < \u2191n + 1\nh' : \u00acm < n\nthis : m = n\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun x => p x n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x"}, {"tactic": "exact h.2.1", "state_before": "case neg\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m < \u2191n + 1\nh' : \u00acm < n\nthis : m = n\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt (fun x => p x n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x", "state_after": "no goals"}, {"tactic": "intro m hm", "state_before": "case mpr.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 \u2191n + 1 \u2192 ContinuousOn (fun x => p x m) s", "state_after": "case mpr.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m \u2264 \u2191n + 1\n\u22a2 ContinuousOn (fun x => p x m) s"}, {"tactic": "by_cases h' : m \u2264 n", "state_before": "case mpr.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m \u2264 \u2191n + 1\n\u22a2 ContinuousOn (fun x => p x m) s", "state_after": "case pos\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m \u2264 \u2191n + 1\nh' : m \u2264 n\n\u22a2 ContinuousOn (fun x => p x m) s\n\ncase neg\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m \u2264 \u2191n + 1\nh' : \u00acm \u2264 n\n\u22a2 ContinuousOn (fun x => p x m) s"}, {"tactic": "apply h.1.cont m (WithTop.coe_le_coe.2 h')", "state_before": "case pos\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m \u2264 \u2191n + 1\nh' : m \u2264 n\n\u22a2 ContinuousOn (fun x => p x m) s", "state_after": "no goals"}, {"tactic": "have : m = n + 1 := le_antisymm (WithTop.coe_le_coe.1 hm) (not_le.1 h')", "state_before": "case neg\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m \u2264 \u2191n + 1\nh' : \u00acm \u2264 n\n\u22a2 ContinuousOn (fun x => p x m) s", "state_after": "case neg\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m \u2264 \u2191n + 1\nh' : \u00acm \u2264 n\nthis : m = n + 1\n\u22a2 ContinuousOn (fun x => p x m) s"}, {"tactic": "rw [this]", "state_before": "case neg\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m \u2264 \u2191n + 1\nh' : \u00acm \u2264 n\nthis : m = n + 1\n\u22a2 ContinuousOn (fun x => p x m) s", "state_after": "case neg\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m \u2264 \u2191n + 1\nh' : \u00acm \u2264 n\nthis : m = n + 1\n\u22a2 ContinuousOn (fun x => p x (n + 1)) s"}, {"tactic": "exact h.2.2", "state_before": "case neg\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nh :\n  HasFTaylorSeriesUpToOn (\u2191n) f p s \u2227\n    (\u2200 (x : E),\n        x \u2208 s \u2192 HasFDerivWithinAt (fun y => p y n) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ n))) s x) \u2227\n      ContinuousOn (fun x => p x (n + 1)) s\nm : \u2115\nhm : \u2191m \u2264 \u2191n + 1\nh' : \u00acm \u2264 n\nthis : m = n + 1\n\u22a2 ContinuousOn (fun x => p x (n + 1)) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean", "full_name": "Complex.cpow_two", "start": [129, 1], "end": [131, 29], "traced_tactics": [{"tactic": "rw [\u2190 cpow_nat_cast]", "state_before": "x : \u2102\n\u22a2 x ^ 2 = x ^ 2", "state_after": "x : \u2102\n\u22a2 x ^ 2 = x ^ \u21912"}, {"tactic": "simp only [Nat.cast_ofNat]", "state_before": "x : \u2102\n\u22a2 x ^ 2 = x ^ \u21912", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/ConeCategory.lean", "full_name": "CategoryTheory.Limits.IsColimit.descCoconeMorphism_eq_isInitial_to", "start": [189, 1], "end": [192, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/SModEq.lean", "full_name": "SModEq.trans", "start": [84, 8], "end": [85, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "PosNum.mod'_to_nat", "start": [1634, 1], "end": [1635, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/Basic.lean", "full_name": "Ideal.IsPrime.pow_mem_iff_mem", "start": [583, 1], "end": [585, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Bounds/Basic.lean", "full_name": "lowerBounds_Ioi", "start": [605, 1], "end": [606, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Parity.lean", "full_name": "Odd.ne_two_of_dvd_nat", "start": [371, 1], "end": [373, 46], "traced_tactics": [{"tactic": "rintro rfl", "state_before": "R : Type ?u.29820\ninst\u271d\u00b9 : Monoid R\ninst\u271d : HasDistribNeg R\nn\u271d m n : \u2115\nhn : Odd n\nhm : m \u2223 n\n\u22a2 m \u2260 2", "state_after": "R : Type ?u.29820\ninst\u271d\u00b9 : Monoid R\ninst\u271d : HasDistribNeg R\nn\u271d n : \u2115\nhn : Odd n\nhm : 2 \u2223 n\n\u22a2 False"}, {"tactic": "exact absurd (hn.of_dvd_nat hm) (by decide)", "state_before": "R : Type ?u.29820\ninst\u271d\u00b9 : Monoid R\ninst\u271d : HasDistribNeg R\nn\u271d n : \u2115\nhn : Odd n\nhm : 2 \u2223 n\n\u22a2 False", "state_after": "no goals"}, {"tactic": "decide", "state_before": "R : Type ?u.29820\ninst\u271d\u00b9 : Monoid R\ninst\u271d : HasDistribNeg R\nn\u271d n : \u2115\nhn : Odd n\nhm : 2 \u2223 n\n\u22a2 \u00acOdd 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Lift.lean", "full_name": "Filter.le_lift", "start": [95, 1], "end": [97, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Module/LocalizedModule.lean", "full_name": "IsLocalizedModule.mkOfAlgebra", "start": [1056, 1], "end": [1080, 36], "traced_tactics": [{"tactic": "replace h\u2083 := fun x =>\n  Iff.intro (h\u2083 x) fun \u27e8\u27e8m, hm\u27e9, e\u27e9 =>\n    (h\u2081 m hm).mul_left_cancel <| by\n      rw [\u2190 Algebra.smul_def]\n      simpa [Submonoid.smul_def] using f.congr_arg e", "state_before": "R\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2192 \u2203 m, m \u2022 x = 0\n\u22a2 IsLocalizedModule M (AlgHom.toLinearMap f)", "state_after": "R\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\n\u22a2 IsLocalizedModule M (AlgHom.toLinearMap f)"}, {"tactic": "constructor", "state_before": "R\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\n\u22a2 IsLocalizedModule M (AlgHom.toLinearMap f)", "state_after": "case map_units\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\n\u22a2 \u2200 (x : { x // x \u2208 M }), IsUnit (\u2191(algebraMap R (Module.End R S')) \u2191x)\n\ncase surj'\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\n\u22a2 \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191(AlgHom.toLinearMap f) x.fst\n\ncase eq_iff_exists'\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\n\u22a2 \u2200 {x\u2081 x\u2082 : S}, \u2191(AlgHom.toLinearMap f) x\u2081 = \u2191(AlgHom.toLinearMap f) x\u2082 \u2194 \u2203 c, c \u2022 x\u2082 = c \u2022 x\u2081"}, {"tactic": "rw [\u2190 Algebra.smul_def]", "state_before": "R\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2192 \u2203 m, m \u2022 x = 0\nx : S\nx\u271d : \u2203 m, m \u2022 x = 0\nm : R\nhm : m \u2208 M\ne : { val := m, property := hm } \u2022 x = 0\n\u22a2 \u2191(algebraMap R S') m * \u2191f x = \u2191(algebraMap R S') m * 0", "state_after": "R\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2192 \u2203 m, m \u2022 x = 0\nx : S\nx\u271d : \u2203 m, m \u2022 x = 0\nm : R\nhm : m \u2208 M\ne : { val := m, property := hm } \u2022 x = 0\n\u22a2 m \u2022 \u2191f x = \u2191(algebraMap R S') m * 0"}, {"tactic": "simpa [Submonoid.smul_def] using f.congr_arg e", "state_before": "R\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2192 \u2203 m, m \u2022 x = 0\nx : S\nx\u271d : \u2203 m, m \u2022 x = 0\nm : R\nhm : m \u2208 M\ne : { val := m, property := hm } \u2022 x = 0\n\u22a2 m \u2022 \u2191f x = \u2191(algebraMap R S') m * 0", "state_after": "no goals"}, {"tactic": "intro x", "state_before": "case map_units\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\n\u22a2 \u2200 (x : { x // x \u2208 M }), IsUnit (\u2191(algebraMap R (Module.End R S')) \u2191x)", "state_after": "case map_units\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx : { x // x \u2208 M }\n\u22a2 IsUnit (\u2191(algebraMap R (Module.End R S')) \u2191x)"}, {"tactic": "rw [Module.End_isUnit_iff]", "state_before": "case map_units\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx : { x // x \u2208 M }\n\u22a2 IsUnit (\u2191(algebraMap R (Module.End R S')) \u2191x)", "state_after": "case map_units\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx : { x // x \u2208 M }\n\u22a2 Function.Bijective \u2191(\u2191(algebraMap R (Module.End R S')) \u2191x)"}, {"tactic": "constructor", "state_before": "case map_units\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx : { x // x \u2208 M }\n\u22a2 Function.Bijective \u2191(\u2191(algebraMap R (Module.End R S')) \u2191x)", "state_after": "case map_units.left\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx : { x // x \u2208 M }\n\u22a2 Function.Injective \u2191(\u2191(algebraMap R (Module.End R S')) \u2191x)\n\ncase map_units.right\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx : { x // x \u2208 M }\n\u22a2 Function.Surjective \u2191(\u2191(algebraMap R (Module.End R S')) \u2191x)"}, {"tactic": "rintro a b (e : x \u2022 a = x \u2022 b)", "state_before": "case map_units.left\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx : { x // x \u2208 M }\n\u22a2 Function.Injective \u2191(\u2191(algebraMap R (Module.End R S')) \u2191x)", "state_after": "case map_units.left\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx : { x // x \u2208 M }\na b : S'\ne : x \u2022 a = x \u2022 b\n\u22a2 a = b"}, {"tactic": "simp_rw [Submonoid.smul_def, Algebra.smul_def] at e", "state_before": "case map_units.left\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx : { x // x \u2208 M }\na b : S'\ne : x \u2022 a = x \u2022 b\n\u22a2 a = b", "state_after": "case map_units.left\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx : { x // x \u2208 M }\na b : S'\ne : \u2191(algebraMap R S') \u2191x * a = \u2191(algebraMap R S') \u2191x * b\n\u22a2 a = b"}, {"tactic": "exact (h\u2081 x x.2).mul_left_cancel e", "state_before": "case map_units.left\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx : { x // x \u2208 M }\na b : S'\ne : \u2191(algebraMap R S') \u2191x * a = \u2191(algebraMap R S') \u2191x * b\n\u22a2 a = b", "state_after": "no goals"}, {"tactic": "intro a", "state_before": "case map_units.right\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx : { x // x \u2208 M }\n\u22a2 Function.Surjective \u2191(\u2191(algebraMap R (Module.End R S')) \u2191x)", "state_after": "case map_units.right\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx : { x // x \u2208 M }\na : S'\n\u22a2 \u2203 a_1, \u2191(\u2191(algebraMap R (Module.End R S')) \u2191x) a_1 = a"}, {"tactic": "refine' \u27e8((h\u2081 x x.2).unit\u207b\u00b9 : _) * a, _\u27e9", "state_before": "case map_units.right\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx : { x // x \u2208 M }\na : S'\n\u22a2 \u2203 a_1, \u2191(\u2191(algebraMap R (Module.End R S')) \u2191x) a_1 = a", "state_after": "case map_units.right\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx : { x // x \u2208 M }\na : S'\n\u22a2 \u2191(\u2191(algebraMap R (Module.End R S')) \u2191x) (\u2191(IsUnit.unit (_ : IsUnit (\u2191(algebraMap R S') \u2191x)))\u207b\u00b9 * a) = a"}, {"tactic": "change (x : R) \u2022 (_ * a) = _", "state_before": "case map_units.right\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx : { x // x \u2208 M }\na : S'\n\u22a2 \u2191(\u2191(algebraMap R (Module.End R S')) \u2191x) (\u2191(IsUnit.unit (_ : IsUnit (\u2191(algebraMap R S') \u2191x)))\u207b\u00b9 * a) = a", "state_after": "case map_units.right\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx : { x // x \u2208 M }\na : S'\n\u22a2 \u2191x \u2022 (\u2191(IsUnit.unit (_ : IsUnit (\u2191(algebraMap R S') \u2191x)))\u207b\u00b9 * a) = a"}, {"tactic": "rw [Algebra.smul_def, \u2190 mul_assoc, IsUnit.mul_val_inv, one_mul]", "state_before": "case map_units.right\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx : { x // x \u2208 M }\na : S'\n\u22a2 \u2191x \u2022 (\u2191(IsUnit.unit (_ : IsUnit (\u2191(algebraMap R S') \u2191x)))\u207b\u00b9 * a) = a", "state_after": "no goals"}, {"tactic": "exact h\u2082", "state_before": "case surj'\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\n\u22a2 \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191(AlgHom.toLinearMap f) x.fst", "state_after": "no goals"}, {"tactic": "intros", "state_before": "case eq_iff_exists'\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\n\u22a2 \u2200 {x\u2081 x\u2082 : S}, \u2191(AlgHom.toLinearMap f) x\u2081 = \u2191(AlgHom.toLinearMap f) x\u2082 \u2194 \u2203 c, c \u2022 x\u2082 = c \u2022 x\u2081", "state_after": "case eq_iff_exists'\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx\u2081\u271d x\u2082\u271d : S\n\u22a2 \u2191(AlgHom.toLinearMap f) x\u2081\u271d = \u2191(AlgHom.toLinearMap f) x\u2082\u271d \u2194 \u2203 c, c \u2022 x\u2082\u271d = c \u2022 x\u2081\u271d"}, {"tactic": "dsimp only [AlgHom.toLinearMap_apply]", "state_before": "case eq_iff_exists'\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx\u2081\u271d x\u2082\u271d : S\n\u22a2 \u2191(AlgHom.toLinearMap f) x\u2081\u271d = \u2191(AlgHom.toLinearMap f) x\u2082\u271d \u2194 \u2203 c, c \u2022 x\u2082\u271d = c \u2022 x\u2081\u271d", "state_after": "case eq_iff_exists'\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx\u2081\u271d x\u2082\u271d : S\n\u22a2 \u2191f x\u2081\u271d = \u2191f x\u2082\u271d \u2194 \u2203 c, c \u2022 x\u2082\u271d = c \u2022 x\u2081\u271d"}, {"tactic": "rw [eq_comm, \u2190 sub_eq_zero, \u2190 map_sub, h\u2083]", "state_before": "case eq_iff_exists'\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx\u2081\u271d x\u2082\u271d : S\n\u22a2 \u2191f x\u2081\u271d = \u2191f x\u2082\u271d \u2194 \u2203 c, c \u2022 x\u2082\u271d = c \u2022 x\u2081\u271d", "state_after": "case eq_iff_exists'\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx\u2081\u271d x\u2082\u271d : S\n\u22a2 (\u2203 m, m \u2022 (x\u2082\u271d - x\u2081\u271d) = 0) \u2194 \u2203 c, c \u2022 x\u2082\u271d = c \u2022 x\u2081\u271d"}, {"tactic": "simp_rw [smul_sub, sub_eq_zero]", "state_before": "case eq_iff_exists'\nR\u271d : Type ?u.939670\ninst\u271d\u00b9\u00b2 : CommRing R\u271d\nS\u271d : Submonoid R\u271d\nM\u271d : Type ?u.939862\nM' : Type ?u.939865\nM'' : Type ?u.939868\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : AddCommMonoid M''\ninst\u271d\u2078 : Module R\u271d M\u271d\ninst\u271d\u2077 : Module R\u271d M'\ninst\u271d\u2076 : Module R\u271d M''\nf\u271d : M\u271d \u2192\u2097[R\u271d] M'\ng : M\u271d \u2192\u2097[R\u271d] M''\ninst\u271d\u2075 : IsLocalizedModule S\u271d f\u271d\nR : Type u_1\nS : Type u_2\nS' : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing S'\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R S'\nM : Submonoid R\nf : S \u2192\u2090[R] S'\nh\u2081 : \u2200 (x : R), x \u2208 M \u2192 IsUnit (\u2191(algebraMap R S') x)\nh\u2082 : \u2200 (y : S'), \u2203 x, x.snd \u2022 y = \u2191f x.fst\nh\u2083 : \u2200 (x : S), \u2191f x = 0 \u2194 \u2203 m, m \u2022 x = 0\nx\u2081\u271d x\u2082\u271d : S\n\u22a2 (\u2203 m, m \u2022 (x\u2082\u271d - x\u2081\u271d) = 0) \u2194 \u2203 c, c \u2022 x\u2082\u271d = c \u2022 x\u2081\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/Sub.lean", "full_name": "MeasureTheory.Measure.sub_top", "start": [58, 1], "end": [59, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/MStructure.lean", "full_name": "IsLprojection.compl_mul", "start": [263, 1], "end": [264, 35], "traced_tactics": [{"tactic": "rw [coe_compl, sub_mul, one_mul]", "state_before": "X : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b9 : Ring M\ninst\u271d : Module M X\nP : { P // IsLprojection X P }\nQ : M\n\u22a2 \u2191(P\u1d9c) * Q = Q - \u2191P * Q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "measurableSet_of_continuousAt", "start": [320, 1], "end": [322, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Polynomial/BigOperators.lean", "full_name": "Polynomial.degree_list_sum_le", "start": [65, 1], "end": [76, 13], "traced_tactics": [{"tactic": "by_cases h : l.sum = 0", "state_before": "R : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\n\u22a2 degree (List.sum l) \u2264 List.maximum (List.map natDegree l)", "state_after": "case pos\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : List.sum l = 0\n\u22a2 degree (List.sum l) \u2264 List.maximum (List.map natDegree l)\n\ncase neg\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : \u00acList.sum l = 0\n\u22a2 degree (List.sum l) \u2264 List.maximum (List.map natDegree l)"}, {"tactic": "simp [h]", "state_before": "case pos\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : List.sum l = 0\n\u22a2 degree (List.sum l) \u2264 List.maximum (List.map natDegree l)", "state_after": "no goals"}, {"tactic": "rw [degree_eq_natDegree h]", "state_before": "case neg\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : \u00acList.sum l = 0\n\u22a2 degree (List.sum l) \u2264 List.maximum (List.map natDegree l)", "state_after": "case neg\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : \u00acList.sum l = 0\n\u22a2 \u2191(natDegree (List.sum l)) \u2264 List.maximum (List.map natDegree l)"}, {"tactic": "suffices (l.map natDegree).maximum = ((l.map natDegree).foldr max 0 : \u2115) by\n  rw [this]\n  simpa [this, Nat.cast_withBot] using natDegree_list_sum_le l", "state_before": "case neg\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : \u00acList.sum l = 0\n\u22a2 \u2191(natDegree (List.sum l)) \u2264 List.maximum (List.map natDegree l)", "state_after": "case neg\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : \u00acList.sum l = 0\n\u22a2 List.maximum (List.map natDegree l) = \u2191(List.foldr max 0 (List.map natDegree l))"}, {"tactic": "rw [\u2190 List.foldr_max_of_ne_nil]", "state_before": "case neg\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : \u00acList.sum l = 0\n\u22a2 List.maximum (List.map natDegree l) = \u2191(List.foldr max 0 (List.map natDegree l))", "state_after": "case neg\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : \u00acList.sum l = 0\n\u22a2 \u2191(List.foldr max \u22a5 (List.map natDegree l)) = \u2191(List.foldr max 0 (List.map natDegree l))\n\ncase neg\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : \u00acList.sum l = 0\n\u22a2 List.map natDegree l \u2260 []"}, {"tactic": "contrapose! h", "state_before": "case neg\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : \u00acList.sum l = 0\n\u22a2 List.map natDegree l \u2260 []", "state_after": "case neg\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : List.map natDegree l = []\n\u22a2 List.sum l = 0"}, {"tactic": "rw [List.map_eq_nil] at h", "state_before": "case neg\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : List.map natDegree l = []\n\u22a2 List.sum l = 0", "state_after": "case neg\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : l = []\n\u22a2 List.sum l = 0"}, {"tactic": "simp [h]", "state_before": "case neg\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : l = []\n\u22a2 List.sum l = 0", "state_after": "no goals"}, {"tactic": "rw [this]", "state_before": "R : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : \u00acList.sum l = 0\nthis : List.maximum (List.map natDegree l) = \u2191(List.foldr max 0 (List.map natDegree l))\n\u22a2 \u2191(natDegree (List.sum l)) \u2264 List.maximum (List.map natDegree l)", "state_after": "R : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : \u00acList.sum l = 0\nthis : List.maximum (List.map natDegree l) = \u2191(List.foldr max 0 (List.map natDegree l))\n\u22a2 \u2191(natDegree (List.sum l)) \u2264 \u2191(List.foldr max 0 (List.map natDegree l))"}, {"tactic": "simpa [this, Nat.cast_withBot] using natDegree_list_sum_le l", "state_before": "R : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : \u00acList.sum l = 0\nthis : List.maximum (List.map natDegree l) = \u2191(List.foldr max 0 (List.map natDegree l))\n\u22a2 \u2191(natDegree (List.sum l)) \u2264 \u2191(List.foldr max 0 (List.map natDegree l))", "state_after": "no goals"}, {"tactic": "congr", "state_before": "case neg\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\nh : \u00acList.sum l = 0\n\u22a2 \u2191(List.foldr max \u22a5 (List.map natDegree l)) = \u2191(List.foldr max 0 (List.map natDegree l))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/BilinearForm.lean", "full_name": "BilinForm.toMatrixAux_stdBasis", "start": [153, 1], "end": [157, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "full_name": "Polynomial.degree_add_le", "start": [634, 1], "end": [638, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Int/Order/Basic.lean", "full_name": "Int.ediv_lt_iff_lt_mul", "start": [458, 11], "end": [459, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/RingDivision.lean", "full_name": "Polynomial.rootMultiplicity_X_sub_C_pow", "start": [469, 1], "end": [475, 89], "traced_tactics": [{"tactic": "induction' n with n hn", "state_before": "R : Type u\nS : Type v\nT : Type w\na\u271d b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np q : R[X]\na : R\nn : \u2115\n\u22a2 rootMultiplicity a ((X - \u2191C a) ^ n) = n", "state_after": "case zero\nR : Type u\nS : Type v\nT : Type w\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np q : R[X]\na : R\n\u22a2 rootMultiplicity a ((X - \u2191C a) ^ Nat.zero) = Nat.zero\n\ncase succ\nR : Type u\nS : Type v\nT : Type w\na\u271d b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np q : R[X]\na : R\nn : \u2115\nhn : rootMultiplicity a ((X - \u2191C a) ^ n) = n\n\u22a2 rootMultiplicity a ((X - \u2191C a) ^ Nat.succ n) = Nat.succ n"}, {"tactic": "have hzero := pow_ne_zero n.succ (X_sub_C_ne_zero a)", "state_before": "case succ\nR : Type u\nS : Type v\nT : Type w\na\u271d b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np q : R[X]\na : R\nn : \u2115\nhn : rootMultiplicity a ((X - \u2191C a) ^ n) = n\n\u22a2 rootMultiplicity a ((X - \u2191C a) ^ Nat.succ n) = Nat.succ n", "state_after": "case succ\nR : Type u\nS : Type v\nT : Type w\na\u271d b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np q : R[X]\na : R\nn : \u2115\nhn : rootMultiplicity a ((X - \u2191C a) ^ n) = n\nhzero : (X - \u2191C a) ^ Nat.succ n \u2260 0\n\u22a2 rootMultiplicity a ((X - \u2191C a) ^ Nat.succ n) = Nat.succ n"}, {"tactic": "rw [pow_succ (X - C a) n] at hzero\u22a2", "state_before": "case succ\nR : Type u\nS : Type v\nT : Type w\na\u271d b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np q : R[X]\na : R\nn : \u2115\nhn : rootMultiplicity a ((X - \u2191C a) ^ n) = n\nhzero : (X - \u2191C a) ^ Nat.succ n \u2260 0\n\u22a2 rootMultiplicity a ((X - \u2191C a) ^ Nat.succ n) = Nat.succ n", "state_after": "case succ\nR : Type u\nS : Type v\nT : Type w\na\u271d b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np q : R[X]\na : R\nn : \u2115\nhn : rootMultiplicity a ((X - \u2191C a) ^ n) = n\nhzero : (X - \u2191C a) * (X - \u2191C a) ^ n \u2260 0\n\u22a2 rootMultiplicity a ((X - \u2191C a) * (X - \u2191C a) ^ n) = Nat.succ n"}, {"tactic": "simp only [rootMultiplicity_mul hzero, rootMultiplicity_X_sub_C_self, hn, Nat.one_add]", "state_before": "case succ\nR : Type u\nS : Type v\nT : Type w\na\u271d b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np q : R[X]\na : R\nn : \u2115\nhn : rootMultiplicity a ((X - \u2191C a) ^ n) = n\nhzero : (X - \u2191C a) * (X - \u2191C a) ^ n \u2260 0\n\u22a2 rootMultiplicity a ((X - \u2191C a) * (X - \u2191C a) ^ n) = Nat.succ n", "state_after": "no goals"}, {"tactic": "refine' rootMultiplicity_eq_zero _", "state_before": "case zero\nR : Type u\nS : Type v\nT : Type w\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np q : R[X]\na : R\n\u22a2 rootMultiplicity a ((X - \u2191C a) ^ Nat.zero) = Nat.zero", "state_after": "case zero\nR : Type u\nS : Type v\nT : Type w\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np q : R[X]\na : R\n\u22a2 \u00acIsRoot ((X - \u2191C a) ^ Nat.zero) a"}, {"tactic": "simp only [eval_one, IsRoot.def, not_false_iff, one_ne_zero, pow_zero, Nat.zero_eq]", "state_before": "case zero\nR : Type u\nS : Type v\nT : Type w\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np q : R[X]\na : R\n\u22a2 \u00acIsRoot ((X - \u2191C a) ^ Nat.zero) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Separation.lean", "full_name": "CompactExhaustion.isClosed", "start": [1291, 1], "end": [1292, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Int/DivMod.lean", 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Type ?u.235399\nR : Type u_3\nm_fin : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Field R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\nA : Matrix m n R\nx : n \u2192 R\nh : star x \u2b1d\u1d65 mulVec A\u1d34 (mulVec A x) = star x \u2b1d\u1d65 0\nthis : NoZeroDivisors R\n\u22a2 mulVec A x = 0", "state_after": "no goals"}, {"tactic": "intro h", "state_before": "case h.mpr\nl : Type ?u.235390\nm : Type u_1\nn : Type u_2\no : Type ?u.235399\nR : Type u_3\nm_fin : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Field R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\nA : Matrix m n R\nx : n \u2192 R\n\u22a2 mulVec A x = 0 \u2192 mulVec A\u1d34 (mulVec A x) = 0", "state_after": "case h.mpr\nl : Type ?u.235390\nm : Type u_1\nn : Type u_2\no : Type ?u.235399\nR : Type u_3\nm_fin : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Field R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\nA : Matrix m n R\nx : n \u2192 R\nh : mulVec A x = 0\n\u22a2 mulVec A\u1d34 (mulVec A x) = 0"}, {"tactic": "rw [h, mulVec_zero]", "state_before": "case h.mpr\nl : Type ?u.235390\nm : Type u_1\nn : Type u_2\no : Type ?u.235399\nR : Type u_3\nm_fin : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Field R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\nA : Matrix m n R\nx : n \u2192 R\nh : mulVec A x = 0\n\u22a2 mulVec A\u1d34 (mulVec A x) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Deprecated/Group.lean", "full_name": "IsGroupHom.comp", "start": [323, 1], "end": [325, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Composition.lean", "full_name": "Composition.ones_embedding", "start": [520, 1], "end": [523, 21], "traced_tactics": [{"tactic": "ext", "state_before": "n : \u2115\nc : Composition n\ni : Fin (length (ones n))\nh : 0 < blocksFun (ones n) i\n\u22a2 \u2191(embedding (ones n) i) { val := 0, isLt := h } = { val := \u2191i, isLt := (_ : \u2191i < n) }", "state_after": "case h\nn : \u2115\nc : Composition n\ni : Fin (length (ones n))\nh : 0 < blocksFun (ones n) i\n\u22a2 \u2191(\u2191(embedding (ones n) i) { val := 0, isLt := h }) = \u2191{ val := \u2191i, isLt := (_ : \u2191i < n) }"}, {"tactic": "simpa using i.2.le", "state_before": "case h\nn : \u2115\nc : Composition n\ni : Fin (length (ones n))\nh : 0 < blocksFun (ones n) i\n\u22a2 \u2191(\u2191(embedding (ones n) i) { val := 0, isLt := h }) = \u2191{ val := \u2191i, isLt := (_ : \u2191i < n) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Semiconj.lean", "full_name": "SemiconjBy.units_inv_symm_left", "start": [138, 1], "end": [141, 75], "traced_tactics": [{"tactic": "rw [Units.mul_inv_cancel_right]", "state_before": "M : Type u_1\ninst\u271d : Monoid M\na : M\u02e3\nx y : M\nh : SemiconjBy (\u2191a) x y\n\u22a2 \u2191a\u207b\u00b9 * y = \u2191a\u207b\u00b9 * (y * \u2191a * \u2191a\u207b\u00b9)", "state_after": "no goals"}, {"tactic": "rw [\u2190 h.eq, \u2190 mul_assoc, Units.inv_mul_cancel_left]", "state_before": "M : Type u_1\ninst\u271d : Monoid M\na : M\u02e3\nx y : M\nh : SemiconjBy (\u2191a) x y\n\u22a2 \u2191a\u207b\u00b9 * (y * \u2191a * \u2191a\u207b\u00b9) = x * \u2191a\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/BigOperators/Lemmas.lean", "full_name": "Commute.list_sum_right", "start": [32, 1], "end": [37, 92], "traced_tactics": [{"tactic": "induction' l with x xs ih", "state_before": "\u03b9 : Type ?u.26\n\u03b1 : Type ?u.29\nM : Type ?u.32\nN : Type ?u.35\nP : Type ?u.38\nM\u2080 : Type ?u.41\nG : Type ?u.44\nR : Type u_1\ninst\u271d : NonUnitalNonAssocSemiring R\na : R\nl : List R\nh : \u2200 (b : R), b \u2208 l \u2192 Commute a b\n\u22a2 Commute a (sum l)", "state_after": "case nil\n\u03b9 : Type ?u.26\n\u03b1 : Type ?u.29\nM : Type ?u.32\nN : Type ?u.35\nP : Type ?u.38\nM\u2080 : Type ?u.41\nG : Type ?u.44\nR : Type u_1\ninst\u271d : NonUnitalNonAssocSemiring R\na : R\nl : List R\nh\u271d : \u2200 (b : R), b \u2208 l \u2192 Commute a b\nh : \u2200 (b : R), b \u2208 [] \u2192 Commute a b\n\u22a2 Commute a (sum [])\n\ncase cons\n\u03b9 : Type ?u.26\n\u03b1 : Type ?u.29\nM : Type ?u.32\nN : Type ?u.35\nP : Type ?u.38\nM\u2080 : Type ?u.41\nG : Type ?u.44\nR : Type u_1\ninst\u271d : NonUnitalNonAssocSemiring R\na : R\nl : List R\nh\u271d : \u2200 (b : R), b \u2208 l \u2192 Commute a b\nx : R\nxs : List R\nih : (\u2200 (b : R), b \u2208 xs \u2192 Commute a b) \u2192 Commute a (sum xs)\nh : \u2200 (b : R), b \u2208 x :: xs \u2192 Commute a b\n\u22a2 Commute a (sum (x :: xs))"}, {"tactic": "exact Commute.zero_right _", "state_before": "case nil\n\u03b9 : Type ?u.26\n\u03b1 : Type ?u.29\nM : Type ?u.32\nN : Type ?u.35\nP : Type ?u.38\nM\u2080 : Type ?u.41\nG : Type ?u.44\nR : Type u_1\ninst\u271d : NonUnitalNonAssocSemiring R\na : R\nl : List R\nh\u271d : \u2200 (b : R), b \u2208 l \u2192 Commute a b\nh : \u2200 (b : R), b \u2208 [] \u2192 Commute a b\n\u22a2 Commute a (sum [])", "state_after": "no goals"}, {"tactic": "rw [List.sum_cons]", "state_before": "case cons\n\u03b9 : Type ?u.26\n\u03b1 : Type ?u.29\nM : Type ?u.32\nN : Type ?u.35\nP : Type ?u.38\nM\u2080 : Type ?u.41\nG : Type ?u.44\nR : Type u_1\ninst\u271d : NonUnitalNonAssocSemiring R\na : R\nl : List R\nh\u271d : \u2200 (b : R), b \u2208 l \u2192 Commute a b\nx : R\nxs : List R\nih : (\u2200 (b : R), b \u2208 xs \u2192 Commute a b) \u2192 Commute a (sum xs)\nh : \u2200 (b : R), b \u2208 x :: xs \u2192 Commute a b\n\u22a2 Commute a (sum (x :: xs))", "state_after": "case cons\n\u03b9 : Type ?u.26\n\u03b1 : Type ?u.29\nM : Type ?u.32\nN : Type ?u.35\nP : Type ?u.38\nM\u2080 : Type ?u.41\nG : Type ?u.44\nR : Type u_1\ninst\u271d : NonUnitalNonAssocSemiring R\na : R\nl : List R\nh\u271d : \u2200 (b : R), b \u2208 l \u2192 Commute a b\nx : R\nxs : List R\nih : (\u2200 (b : R), b \u2208 xs \u2192 Commute a b) \u2192 Commute a (sum xs)\nh : \u2200 (b : R), b \u2208 x :: xs \u2192 Commute a b\n\u22a2 Commute a (x + sum xs)"}, {"tactic": "exact (h _ <| mem_cons_self _ _).add_right (ih fun j hj => h _ <| mem_cons_of_mem _ hj)", "state_before": "case cons\n\u03b9 : Type ?u.26\n\u03b1 : Type ?u.29\nM : Type ?u.32\nN : Type ?u.35\nP : Type ?u.38\nM\u2080 : Type ?u.41\nG : Type ?u.44\nR : Type u_1\ninst\u271d : NonUnitalNonAssocSemiring R\na : R\nl : List R\nh\u271d : \u2200 (b : R), b \u2208 l \u2192 Commute a b\nx : R\nxs : List R\nih : (\u2200 (b : R), b \u2208 xs \u2192 Commute a b) \u2192 Commute a (sum xs)\nh : \u2200 (b : R), b \u2208 x :: xs \u2192 Commute a b\n\u22a2 Commute a (x + sum xs)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Bounds/OrderIso.lean", "full_name": "OrderIso.isLUB_image", "start": [36, 1], "end": [38, 67], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2243o \u03b2\ns : Set \u03b1\nx : \u03b2\nh : IsLUB (\u2191f '' s) x\n\u22a2 \u2200 {x y : \u03b1}, \u2191f x \u2264 \u2191f y \u2194 x \u2264 y", "state_after": "no goals"}, {"tactic": "simp", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2243o \u03b2\ns : Set \u03b1\nx : \u03b2\nh : IsLUB s (\u2191(symm f) x)\n\u22a2 \u2200 {x y : \u03b2}, \u2191(symm f) x \u2264 \u2191(symm f) y \u2194 x \u2264 y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Opposite.lean", "full_name": "MulOpposite.unop_intCast", "start": [200, 1], "end": [201, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/WithZero.lean", "full_name": "not_lt_zero'", "start": [98, 1], "end": [99, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "full_name": "Set.add_mul_subset", "start": [1125, 1], "end": [1126, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/FunLike/Equiv.lean", "full_name": "EquivLike.apply_eq_iff_eq", "start": [177, 1], "end": [178, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/SchurZassenhaus.lean", "full_name": "Subgroup.SchurZassenhausInduction.step6", "start": [250, 9], "end": [256, 56], "traced_tactics": [{"tactic": "haveI : Fact (Fintype.card N).minFac.Prime := \u27e8step4 h1 h3\u27e9", "state_before": "G : Type u\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nN : Subgroup G\ninst\u271d : Normal N\nh1 : Nat.coprime (Fintype.card { x // x \u2208 N }) (index N)\nh2 :\n  \u2200 (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n    Fintype.card G' < Fintype.card G \u2192\n      \u2200 {N' : Subgroup G'} [inst_2 : Normal N'],\n        Nat.coprime (Fintype.card { x // x \u2208 N' }) (index N') \u2192 \u2203 H', IsComplement' N' H'\nh3 : \u2200 (H : Subgroup G), \u00acIsComplement' N H\n\u22a2 IsPGroup (Nat.minFac (Fintype.card { x // x \u2208 N })) { x // x \u2208 N }", "state_after": "G : Type u\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nN : Subgroup G\ninst\u271d : Normal N\nh1 : Nat.coprime (Fintype.card { x // x \u2208 N }) (index N)\nh2 :\n  \u2200 (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n    Fintype.card G' < Fintype.card G \u2192\n      \u2200 {N' : Subgroup G'} [inst_2 : Normal N'],\n        Nat.coprime (Fintype.card { x // x \u2208 N' }) (index N') \u2192 \u2203 H', IsComplement' N' H'\nh3 : \u2200 (H : Subgroup G), \u00acIsComplement' N H\nthis : Fact (Nat.Prime (Nat.minFac (Fintype.card { x // x \u2208 N })))\n\u22a2 IsPGroup (Nat.minFac (Fintype.card { x // x \u2208 N })) { x // x \u2208 N }"}, {"tactic": "refine' Sylow.nonempty.elim fun P => P.2.of_surjective P.1.subtype _", "state_before": "G : Type u\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nN : Subgroup G\ninst\u271d : Normal N\nh1 : Nat.coprime (Fintype.card { x // x \u2208 N }) (index N)\nh2 :\n  \u2200 (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n    Fintype.card G' < Fintype.card G \u2192\n      \u2200 {N' : Subgroup G'} [inst_2 : Normal N'],\n        Nat.coprime (Fintype.card { x // x \u2208 N' }) (index N') \u2192 \u2203 H', IsComplement' N' H'\nh3 : \u2200 (H : Subgroup G), \u00acIsComplement' N H\nthis : Fact (Nat.Prime (Nat.minFac (Fintype.card { x // x \u2208 N })))\n\u22a2 IsPGroup (Nat.minFac (Fintype.card { x // x \u2208 N })) { x // x \u2208 N }", "state_after": "G : Type u\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nN : Subgroup G\ninst\u271d : Normal N\nh1 : Nat.coprime (Fintype.card { x // x \u2208 N }) (index N)\nh2 :\n  \u2200 (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n    Fintype.card G' < Fintype.card G \u2192\n      \u2200 {N' : Subgroup G'} [inst_2 : Normal N'],\n        Nat.coprime (Fintype.card { x // x \u2208 N' }) (index N') \u2192 \u2203 H', IsComplement' N' H'\nh3 : \u2200 (H : Subgroup G), \u00acIsComplement' N H\nthis : Fact (Nat.Prime (Nat.minFac (Fintype.card { x // x \u2208 N })))\nP : Sylow (Nat.minFac (Fintype.card { x // x \u2208 N })) { x // x \u2208 N }\n\u22a2 Function.Surjective \u2191(Subgroup.subtype \u2191P)"}, {"tactic": "rw [\u2190 MonoidHom.range_top_iff_surjective, subtype_range]", "state_before": "G : Type u\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nN : Subgroup G\ninst\u271d : Normal N\nh1 : Nat.coprime (Fintype.card { x // x \u2208 N }) (index N)\nh2 :\n  \u2200 (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n    Fintype.card G' < Fintype.card G \u2192\n      \u2200 {N' : Subgroup G'} [inst_2 : Normal N'],\n        Nat.coprime (Fintype.card { x // x \u2208 N' }) (index N') \u2192 \u2203 H', IsComplement' N' H'\nh3 : \u2200 (H : Subgroup G), \u00acIsComplement' N H\nthis : Fact (Nat.Prime (Nat.minFac (Fintype.card { x // x \u2208 N })))\nP : Sylow (Nat.minFac (Fintype.card { x // x \u2208 N })) { x // x \u2208 N }\n\u22a2 Function.Surjective \u2191(Subgroup.subtype \u2191P)", "state_after": "G : Type u\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nN : Subgroup G\ninst\u271d : Normal N\nh1 : Nat.coprime (Fintype.card { x // x \u2208 N }) (index N)\nh2 :\n  \u2200 (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n    Fintype.card G' < Fintype.card G \u2192\n      \u2200 {N' : Subgroup G'} [inst_2 : Normal N'],\n        Nat.coprime (Fintype.card { x // x \u2208 N' }) (index N') \u2192 \u2203 H', IsComplement' N' H'\nh3 : \u2200 (H : Subgroup G), \u00acIsComplement' N H\nthis : Fact (Nat.Prime (Nat.minFac (Fintype.card { x // x \u2208 N })))\nP : Sylow (Nat.minFac (Fintype.card { x // x \u2208 N })) { x // x \u2208 N }\n\u22a2 \u2191P = \u22a4"}, {"tactic": "haveI : (P.1.map N.subtype).Normal :=\n  normalizer_eq_top.mp (step1 h1 h2 h3 (P.1.map N.subtype).normalizer P.normalizer_sup_eq_top)", "state_before": "G : Type u\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nN : Subgroup G\ninst\u271d : Normal N\nh1 : Nat.coprime (Fintype.card { x // x \u2208 N }) (index N)\nh2 :\n  \u2200 (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n    Fintype.card G' < Fintype.card G \u2192\n      \u2200 {N' : Subgroup G'} [inst_2 : Normal N'],\n        Nat.coprime (Fintype.card { x // x \u2208 N' }) (index N') \u2192 \u2203 H', IsComplement' N' H'\nh3 : \u2200 (H : Subgroup G), \u00acIsComplement' N H\nthis : Fact (Nat.Prime (Nat.minFac (Fintype.card { x // x \u2208 N })))\nP : Sylow (Nat.minFac (Fintype.card { x // x \u2208 N })) { x // x \u2208 N }\n\u22a2 \u2191P = \u22a4", "state_after": "G : Type u\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nN : Subgroup G\ninst\u271d : Normal N\nh1 : Nat.coprime (Fintype.card { x // x \u2208 N }) (index N)\nh2 :\n  \u2200 (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n    Fintype.card G' < Fintype.card G \u2192\n      \u2200 {N' : Subgroup G'} [inst_2 : Normal N'],\n        Nat.coprime (Fintype.card { x // x \u2208 N' }) (index N') \u2192 \u2203 H', IsComplement' N' H'\nh3 : \u2200 (H : Subgroup G), \u00acIsComplement' N H\nthis\u271d : Fact (Nat.Prime (Nat.minFac (Fintype.card { x // x \u2208 N })))\nP : Sylow (Nat.minFac (Fintype.card { x // x \u2208 N })) { x // x \u2208 N }\nthis : Normal (map (Subgroup.subtype N) \u2191P)\n\u22a2 \u2191P = \u22a4"}, {"tactic": "exact (step3 h1 h2 h3 P.1).resolve_left (step5 h1 h3)", "state_before": "G : Type u\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\nN : Subgroup G\ninst\u271d : Normal N\nh1 : Nat.coprime (Fintype.card { x // x \u2208 N }) (index N)\nh2 :\n  \u2200 (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],\n    Fintype.card G' < Fintype.card G \u2192\n      \u2200 {N' : Subgroup G'} [inst_2 : Normal N'],\n        Nat.coprime (Fintype.card { x // x \u2208 N' }) (index N') \u2192 \u2203 H', IsComplement' N' H'\nh3 : \u2200 (H : Subgroup G), \u00acIsComplement' N H\nthis\u271d : Fact (Nat.Prime (Nat.minFac (Fintype.card { x // x \u2208 N })))\nP : Sylow (Nat.minFac (Fintype.card { x // x \u2208 N })) { x // x \u2208 N }\nthis : Normal (map (Subgroup.subtype N) \u2191P)\n\u22a2 \u2191P = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "mem_nhdsWithin_Iic_iff_exists_Ioc_subset'", "start": [1819, 1], "end": [1821, 70], "traced_tactics": [{"tactic": "norm_num", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderTopology \u03b1\na l' : \u03b1\ns : Set \u03b1\nhl' : l' < a\n\u22a2 List.get?\n      [s \u2208 \ud835\udcdd[Iic a] a, s \u2208 \ud835\udcdd[Icc l' a] a, s \u2208 \ud835\udcdd[Ioc l' a] a, \u2203 l, l \u2208 Ico l' a \u2227 Ioc l a \u2286 s,\n        \u2203 l, l \u2208 Iio a \u2227 Ioc l a \u2286 s]\n      0 =\n    some (s \u2208 \ud835\udcdd[Iic a] a)", "state_after": "no goals"}, {"tactic": "norm_num", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderTopology \u03b1\na l' : \u03b1\ns : Set \u03b1\nhl' : l' < a\n\u22a2 List.get?\n      [s \u2208 \ud835\udcdd[Iic a] a, s \u2208 \ud835\udcdd[Icc l' a] a, s \u2208 \ud835\udcdd[Ioc l' a] a, \u2203 l, l \u2208 Ico l' a \u2227 Ioc l a \u2286 s,\n        \u2203 l, l \u2208 Iio a \u2227 Ioc l a \u2286 s]\n      4 =\n    some (\u2203 l, l \u2208 Iio a \u2227 Ioc l a \u2286 s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Ring/Equiv.lean", "full_name": "RingEquiv.map_mul", "start": [150, 11], "end": [151, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/EraseLead.lean", "full_name": "Polynomial.mono_map_natDegree_eq", "start": [256, 1], "end": [273, 43], "traced_tactics": [{"tactic": "refine' induction_with_natDegree_le (fun p => (\u03c6 p).natDegree = fu p.natDegree)\n  p.natDegree (by simp [fu0]) _ _ _ rfl.le", "state_before": "R : Type u_3\ninst\u271d\u00b2 : Semiring R\nf : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nk : \u2115\nfu : \u2115 \u2192 \u2115\nfu0 : \u2200 {n : \u2115}, n \u2264 k \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, k \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < k \u2192 \u2191\u03c6 f = 0\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\n\u22a2 natDegree (\u2191\u03c6 p) = fu (natDegree p)", "state_after": "case refine'_1\nR : Type u_3\ninst\u271d\u00b2 : Semiring R\nf : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nk : \u2115\nfu : \u2115 \u2192 \u2115\nfu0 : \u2200 {n : \u2115}, n \u2264 k \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, k \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < k \u2192 \u2191\u03c6 f = 0\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\n\u22a2 \u2200 (n : \u2115) (r : R), r \u2260 0 \u2192 n \u2264 natDegree p \u2192 (fun p => natDegree (\u2191\u03c6 p) = fu (natDegree p)) (\u2191C r * X ^ n)\n\ncase refine'_2\nR : Type u_3\ninst\u271d\u00b2 : Semiring R\nf : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nk : \u2115\nfu : \u2115 \u2192 \u2115\nfu0 : \u2200 {n : \u2115}, n \u2264 k \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, k \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < k \u2192 \u2191\u03c6 f = 0\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\n\u22a2 \u2200 (f g : R[X]),\n    natDegree f < natDegree g \u2192\n      natDegree g \u2264 natDegree p \u2192\n        (fun p => natDegree (\u2191\u03c6 p) = fu (natDegree p)) f \u2192\n          (fun p => natDegree (\u2191\u03c6 p) = fu (natDegree p)) g \u2192 (fun p => natDegree (\u2191\u03c6 p) = fu (natDegree p)) (f + g)"}, {"tactic": "simp [fu0]", "state_before": "R : Type u_3\ninst\u271d\u00b2 : Semiring R\nf : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nk : \u2115\nfu : \u2115 \u2192 \u2115\nfu0 : \u2200 {n : \u2115}, n \u2264 k \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, k \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < k \u2192 \u2191\u03c6 f = 0\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\n\u22a2 (fun p => natDegree (\u2191\u03c6 p) = fu (natDegree p)) 0", "state_after": "no goals"}, {"tactic": "intro n r r0 _", "state_before": "case refine'_1\nR : Type u_3\ninst\u271d\u00b2 : Semiring R\nf : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nk : \u2115\nfu : \u2115 \u2192 \u2115\nfu0 : \u2200 {n : \u2115}, n \u2264 k \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, k \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < k \u2192 \u2191\u03c6 f = 0\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\n\u22a2 \u2200 (n : \u2115) (r : R), r \u2260 0 \u2192 n \u2264 natDegree p \u2192 (fun p => natDegree (\u2191\u03c6 p) = fu (natDegree p)) (\u2191C r * X ^ n)", "state_after": "case refine'_1\nR : Type u_3\ninst\u271d\u00b2 : Semiring R\nf : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nk : \u2115\nfu : \u2115 \u2192 \u2115\nfu0 : \u2200 {n : \u2115}, n \u2264 k \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, k \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < k \u2192 \u2191\u03c6 f = 0\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\nn : \u2115\nr : R\nr0 : r \u2260 0\na\u271d : n \u2264 natDegree p\n\u22a2 natDegree (\u2191\u03c6 (\u2191C r * X ^ n)) = fu (natDegree (\u2191C r * X ^ n))"}, {"tactic": "rw [natDegree_C_mul_X_pow _ _ r0, C_mul_X_pow_eq_monomial, \u03c6_mon_nat _ _ r0]", "state_before": "case refine'_1\nR : Type u_3\ninst\u271d\u00b2 : Semiring R\nf : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nk : \u2115\nfu : \u2115 \u2192 \u2115\nfu0 : \u2200 {n : \u2115}, n \u2264 k \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, k \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < k \u2192 \u2191\u03c6 f = 0\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\nn : \u2115\nr : R\nr0 : r \u2260 0\na\u271d : n \u2264 natDegree p\n\u22a2 natDegree (\u2191\u03c6 (\u2191C r * X ^ n)) = fu (natDegree (\u2191C r * X ^ n))", "state_after": "no goals"}, {"tactic": "intro f g fg _ fk gk", "state_before": "case refine'_2\nR : Type u_3\ninst\u271d\u00b2 : Semiring R\nf : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nk : \u2115\nfu : \u2115 \u2192 \u2115\nfu0 : \u2200 {n : \u2115}, n \u2264 k \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, k \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < k \u2192 \u2191\u03c6 f = 0\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\n\u22a2 \u2200 (f g : R[X]),\n    natDegree f < natDegree g \u2192\n      natDegree g \u2264 natDegree p \u2192\n        (fun p => natDegree (\u2191\u03c6 p) = fu (natDegree p)) f \u2192\n          (fun p => natDegree (\u2191\u03c6 p) = fu (natDegree p)) g \u2192 (fun p => natDegree (\u2191\u03c6 p) = fu (natDegree p)) (f + g)", "state_after": "case refine'_2\nR : Type u_3\ninst\u271d\u00b2 : Semiring R\nf\u271d : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nk : \u2115\nfu : \u2115 \u2192 \u2115\nfu0 : \u2200 {n : \u2115}, n \u2264 k \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, k \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < k \u2192 \u2191\u03c6 f = 0\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\nf g : R[X]\nfg : natDegree f < natDegree g\na\u271d : natDegree g \u2264 natDegree p\nfk : natDegree (\u2191\u03c6 f) = fu (natDegree f)\ngk : natDegree (\u2191\u03c6 g) = fu (natDegree g)\n\u22a2 natDegree (\u2191\u03c6 (f + g)) = fu (natDegree (f + g))"}, {"tactic": "rw [natDegree_add_eq_right_of_natDegree_lt fg, _root_.map_add]", "state_before": "case refine'_2\nR : Type u_3\ninst\u271d\u00b2 : Semiring R\nf\u271d : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nk : \u2115\nfu : \u2115 \u2192 \u2115\nfu0 : \u2200 {n : \u2115}, n \u2264 k \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, k \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < k \u2192 \u2191\u03c6 f = 0\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\nf g : R[X]\nfg : natDegree f < natDegree g\na\u271d : natDegree g \u2264 natDegree p\nfk : natDegree (\u2191\u03c6 f) = fu (natDegree f)\ngk : natDegree (\u2191\u03c6 g) = fu (natDegree g)\n\u22a2 natDegree (\u2191\u03c6 (f + g)) = fu (natDegree (f + g))", "state_after": "case refine'_2\nR : Type u_3\ninst\u271d\u00b2 : Semiring R\nf\u271d : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nk : \u2115\nfu : \u2115 \u2192 \u2115\nfu0 : \u2200 {n : \u2115}, n \u2264 k \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, k \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < k \u2192 \u2191\u03c6 f = 0\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\nf g : R[X]\nfg : natDegree f < natDegree g\na\u271d : natDegree g \u2264 natDegree p\nfk : natDegree (\u2191\u03c6 f) = fu (natDegree f)\ngk : natDegree (\u2191\u03c6 g) = fu (natDegree g)\n\u22a2 natDegree (\u2191\u03c6 f + \u2191\u03c6 g) = fu (natDegree g)"}, {"tactic": "by_cases FG : k \u2264 f.natDegree", "state_before": "case refine'_2\nR : Type u_3\ninst\u271d\u00b2 : Semiring R\nf\u271d : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nk : \u2115\nfu : \u2115 \u2192 \u2115\nfu0 : \u2200 {n : \u2115}, n \u2264 k \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, k \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < k \u2192 \u2191\u03c6 f = 0\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\nf g : R[X]\nfg : natDegree f < natDegree g\na\u271d : natDegree g \u2264 natDegree p\nfk : natDegree (\u2191\u03c6 f) = fu (natDegree f)\ngk : natDegree (\u2191\u03c6 g) = fu (natDegree g)\n\u22a2 natDegree (\u2191\u03c6 f + \u2191\u03c6 g) = fu (natDegree g)", "state_after": "case pos\nR : Type u_3\ninst\u271d\u00b2 : Semiring R\nf\u271d : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nk : \u2115\nfu : \u2115 \u2192 \u2115\nfu0 : \u2200 {n : \u2115}, n \u2264 k \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, k \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < k \u2192 \u2191\u03c6 f = 0\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\nf g : R[X]\nfg : natDegree f < natDegree g\na\u271d : natDegree g \u2264 natDegree p\nfk : natDegree (\u2191\u03c6 f) = fu (natDegree f)\ngk : natDegree (\u2191\u03c6 g) = fu (natDegree g)\nFG : k \u2264 natDegree f\n\u22a2 natDegree (\u2191\u03c6 f + \u2191\u03c6 g) = fu (natDegree g)\n\ncase neg\nR : Type u_3\ninst\u271d\u00b2 : Semiring R\nf\u271d : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nk : \u2115\nfu : \u2115 \u2192 \u2115\nfu0 : \u2200 {n : \u2115}, n \u2264 k \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, k \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < k \u2192 \u2191\u03c6 f = 0\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\nf g : R[X]\nfg : natDegree f < natDegree g\na\u271d : natDegree g \u2264 natDegree p\nfk : natDegree (\u2191\u03c6 f) = fu (natDegree f)\ngk : natDegree (\u2191\u03c6 g) = fu (natDegree g)\nFG : \u00ack \u2264 natDegree f\n\u22a2 natDegree (\u2191\u03c6 f + \u2191\u03c6 g) = fu (natDegree g)"}, {"tactic": "rw [natDegree_add_eq_right_of_natDegree_lt, gk]", "state_before": "case pos\nR : Type u_3\ninst\u271d\u00b2 : Semiring R\nf\u271d : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nk : \u2115\nfu : \u2115 \u2192 \u2115\nfu0 : \u2200 {n : \u2115}, n \u2264 k \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, k \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < k \u2192 \u2191\u03c6 f = 0\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\nf g : R[X]\nfg : natDegree f < natDegree g\na\u271d : natDegree g \u2264 natDegree p\nfk : natDegree (\u2191\u03c6 f) = fu (natDegree f)\ngk : natDegree (\u2191\u03c6 g) = fu (natDegree g)\nFG : k \u2264 natDegree f\n\u22a2 natDegree (\u2191\u03c6 f + \u2191\u03c6 g) = fu (natDegree g)", "state_after": "case pos\nR : Type u_3\ninst\u271d\u00b2 : Semiring R\nf\u271d : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nk : \u2115\nfu : \u2115 \u2192 \u2115\nfu0 : \u2200 {n : \u2115}, n \u2264 k \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, k \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < k \u2192 \u2191\u03c6 f = 0\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\nf g : R[X]\nfg : natDegree f < natDegree g\na\u271d : natDegree g \u2264 natDegree p\nfk : natDegree (\u2191\u03c6 f) = fu (natDegree f)\ngk : natDegree (\u2191\u03c6 g) = fu (natDegree g)\nFG : k \u2264 natDegree f\n\u22a2 natDegree (\u2191\u03c6 f) < natDegree (\u2191\u03c6 g)"}, {"tactic": "rw [fk, gk]", "state_before": "case pos\nR : Type u_3\ninst\u271d\u00b2 : Semiring R\nf\u271d : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nk : \u2115\nfu : \u2115 \u2192 \u2115\nfu0 : \u2200 {n : \u2115}, n \u2264 k \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, k \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < k \u2192 \u2191\u03c6 f = 0\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\nf g : R[X]\nfg : natDegree f < natDegree g\na\u271d : natDegree g \u2264 natDegree p\nfk : natDegree (\u2191\u03c6 f) = fu (natDegree f)\ngk : natDegree (\u2191\u03c6 g) = fu (natDegree g)\nFG : k \u2264 natDegree f\n\u22a2 natDegree (\u2191\u03c6 f) < natDegree (\u2191\u03c6 g)", "state_after": "case pos\nR : Type u_3\ninst\u271d\u00b2 : Semiring R\nf\u271d : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nk : \u2115\nfu : \u2115 \u2192 \u2115\nfu0 : \u2200 {n : \u2115}, n \u2264 k \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, k \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < k \u2192 \u2191\u03c6 f = 0\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\nf g : R[X]\nfg : natDegree f < natDegree g\na\u271d : natDegree g \u2264 natDegree p\nfk : natDegree (\u2191\u03c6 f) = fu (natDegree f)\ngk : natDegree (\u2191\u03c6 g) = fu (natDegree g)\nFG : k \u2264 natDegree f\n\u22a2 fu (natDegree f) < fu (natDegree g)"}, {"tactic": "exact fc FG fg", "state_before": "case pos\nR : Type u_3\ninst\u271d\u00b2 : Semiring R\nf\u271d : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nk : \u2115\nfu : \u2115 \u2192 \u2115\nfu0 : \u2200 {n : \u2115}, n \u2264 k \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, k \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < k \u2192 \u2191\u03c6 f = 0\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\nf g : R[X]\nfg : natDegree f < natDegree g\na\u271d : natDegree g \u2264 natDegree p\nfk : natDegree (\u2191\u03c6 f) = fu (natDegree f)\ngk : natDegree (\u2191\u03c6 g) = fu (natDegree g)\nFG : k \u2264 natDegree f\n\u22a2 fu (natDegree f) < fu (natDegree g)", "state_after": "no goals"}, {"tactic": "cases k", "state_before": "case neg\nR : Type u_3\ninst\u271d\u00b2 : Semiring R\nf\u271d : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nk : \u2115\nfu : \u2115 \u2192 \u2115\nfu0 : \u2200 {n : \u2115}, n \u2264 k \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, k \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < k \u2192 \u2191\u03c6 f = 0\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\nf g : R[X]\nfg : natDegree f < natDegree g\na\u271d : natDegree g \u2264 natDegree p\nfk : natDegree (\u2191\u03c6 f) = fu (natDegree f)\ngk : natDegree (\u2191\u03c6 g) = fu (natDegree g)\nFG : \u00ack \u2264 natDegree f\n\u22a2 natDegree (\u2191\u03c6 f + \u2191\u03c6 g) = fu (natDegree g)", "state_after": "case neg.zero\nR : Type u_3\ninst\u271d\u00b2 : Semiring R\nf\u271d : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nfu : \u2115 \u2192 \u2115\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\nf g : R[X]\nfg : natDegree f < natDegree g\na\u271d : natDegree g \u2264 natDegree p\nfk : natDegree (\u2191\u03c6 f) = fu (natDegree f)\ngk : natDegree (\u2191\u03c6 g) = fu (natDegree g)\nfu0 : \u2200 {n : \u2115}, n \u2264 Nat.zero \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, Nat.zero \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < Nat.zero \u2192 \u2191\u03c6 f = 0\nFG : \u00acNat.zero \u2264 natDegree f\n\u22a2 natDegree (\u2191\u03c6 f + \u2191\u03c6 g) = fu (natDegree g)\n\ncase neg.succ\nR : Type u_3\ninst\u271d\u00b2 : Semiring R\nf\u271d : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nfu : \u2115 \u2192 \u2115\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\nf g : R[X]\nfg : natDegree f < natDegree g\na\u271d : natDegree g \u2264 natDegree p\nfk : natDegree (\u2191\u03c6 f) = fu (natDegree f)\ngk : natDegree (\u2191\u03c6 g) = fu (natDegree g)\nn\u271d : \u2115\nfu0 : \u2200 {n : \u2115}, n \u2264 Nat.succ n\u271d \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, Nat.succ n\u271d \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < Nat.succ n\u271d \u2192 \u2191\u03c6 f = 0\nFG : \u00acNat.succ n\u271d \u2264 natDegree f\n\u22a2 natDegree (\u2191\u03c6 f + \u2191\u03c6 g) = fu (natDegree g)"}, {"tactic": "exact (FG (Nat.zero_le _)).elim", "state_before": "case neg.zero\nR : Type u_3\ninst\u271d\u00b2 : Semiring R\nf\u271d : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nfu : \u2115 \u2192 \u2115\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\nf g : R[X]\nfg : natDegree f < natDegree g\na\u271d : natDegree g \u2264 natDegree p\nfk : natDegree (\u2191\u03c6 f) = fu (natDegree f)\ngk : natDegree (\u2191\u03c6 g) = fu (natDegree g)\nfu0 : \u2200 {n : \u2115}, n \u2264 Nat.zero \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, Nat.zero \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < Nat.zero \u2192 \u2191\u03c6 f = 0\nFG : \u00acNat.zero \u2264 natDegree f\n\u22a2 natDegree (\u2191\u03c6 f + \u2191\u03c6 g) = fu (natDegree g)", "state_after": "no goals"}, {"tactic": "rwa [\u03c6_k (not_le.mp FG), zero_add]", "state_before": "case neg.succ\nR : Type u_3\ninst\u271d\u00b2 : Semiring R\nf\u271d : R[X]\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b9 : Semiring S\ninst\u271d : AddMonoidHomClass F R[X] S[X]\n\u03c6 : F\np : R[X]\nfu : \u2115 \u2192 \u2115\n\u03c6_mon_nat : \u2200 (n : \u2115) (c : R), c \u2260 0 \u2192 natDegree (\u2191\u03c6 (\u2191(monomial n) c)) = fu n\nf g : R[X]\nfg : natDegree f < natDegree g\na\u271d : natDegree g \u2264 natDegree p\nfk : natDegree (\u2191\u03c6 f) = fu (natDegree f)\ngk : natDegree (\u2191\u03c6 g) = fu (natDegree g)\nn\u271d : \u2115\nfu0 : \u2200 {n : \u2115}, n \u2264 Nat.succ n\u271d \u2192 fu n = 0\nfc : \u2200 {n m : \u2115}, Nat.succ n\u271d \u2264 n \u2192 n < m \u2192 fu n < fu m\n\u03c6_k : \u2200 {f : R[X]}, natDegree f < Nat.succ n\u271d \u2192 \u2191\u03c6 f = 0\nFG : \u00acNat.succ n\u271d \u2264 natDegree f\n\u22a2 natDegree (\u2191\u03c6 f + \u2191\u03c6 g) = fu (natDegree g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "full_name": "finrank_eq_one_iff", "start": [1269, 1], "end": [1275, 40], "traced_tactics": [{"tactic": "constructor", "state_before": "K : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\n\u22a2 finrank K V = 1 \u2194 Nonempty (Basis \u03b9 K V)", "state_after": "case mp\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\n\u22a2 finrank K V = 1 \u2192 Nonempty (Basis \u03b9 K V)\n\ncase mpr\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\n\u22a2 Nonempty (Basis \u03b9 K V) \u2192 finrank K V = 1"}, {"tactic": "intro h", "state_before": "case mp\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\n\u22a2 finrank K V = 1 \u2192 Nonempty (Basis \u03b9 K V)", "state_after": "case mp\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\nh : finrank K V = 1\n\u22a2 Nonempty (Basis \u03b9 K V)"}, {"tactic": "haveI := finiteDimensional_of_finrank (_root_.zero_lt_one.trans_le h.symm.le)", "state_before": "case mp\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\nh : finrank K V = 1\n\u22a2 Nonempty (Basis \u03b9 K V)", "state_after": "case mp\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\nh : finrank K V = 1\nthis : FiniteDimensional K V\n\u22a2 Nonempty (Basis \u03b9 K V)"}, {"tactic": "exact \u27e8basisUnique \u03b9 h\u27e9", "state_before": "case mp\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\nh : finrank K V = 1\nthis : FiniteDimensional K V\n\u22a2 Nonempty (Basis \u03b9 K V)", "state_after": "no goals"}, {"tactic": "rintro \u27e8b\u27e9", "state_before": "case mpr\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\n\u22a2 Nonempty (Basis \u03b9 K V) \u2192 finrank K V = 1", "state_after": "case mpr.intro\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\nb : Basis \u03b9 K V\n\u22a2 finrank K V = 1"}, {"tactic": "simpa using finrank_eq_card_basis b", "state_before": "case mpr.intro\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Unique \u03b9\nb : Basis \u03b9 K V\n\u22a2 finrank K V = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "full_name": "CategoryTheory.MonoidalCategory.prodMonoidal_leftUnitor_hom_snd", "start": [632, 1], "end": [635, 6], "traced_tactics": [{"tactic": "cases X", "state_before": "C : Type u\n\ud835\udc9e : Category C\ninst\u271d\u2074 : MonoidalCategory C\nC\u2081 : Type u\u2081\ninst\u271d\u00b3 : Category C\u2081\ninst\u271d\u00b2 : MonoidalCategory C\u2081\nC\u2082 : Type u\u2082\ninst\u271d\u00b9 : Category C\u2082\ninst\u271d : MonoidalCategory C\u2082\nX : C\u2081 \u00d7 C\u2082\n\u22a2 (\u03bb_ X).hom.snd = (\u03bb_ X.snd).hom", "state_after": "case mk\nC : Type u\n\ud835\udc9e : Category C\ninst\u271d\u2074 : MonoidalCategory C\nC\u2081 : Type u\u2081\ninst\u271d\u00b3 : Category C\u2081\ninst\u271d\u00b2 : MonoidalCategory C\u2081\nC\u2082 : Type u\u2082\ninst\u271d\u00b9 : Category C\u2082\ninst\u271d : MonoidalCategory C\u2082\nfst\u271d : C\u2081\nsnd\u271d : C\u2082\n\u22a2 (\u03bb_ (fst\u271d, snd\u271d)).hom.snd = (\u03bb_ (fst\u271d, snd\u271d).snd).hom"}, {"tactic": "rfl", "state_before": "case mk\nC : Type u\n\ud835\udc9e : Category C\ninst\u271d\u2074 : MonoidalCategory C\nC\u2081 : Type u\u2081\ninst\u271d\u00b3 : Category C\u2081\ninst\u271d\u00b2 : MonoidalCategory C\u2081\nC\u2082 : Type u\u2082\ninst\u271d\u00b9 : Category C\u2082\ninst\u271d : MonoidalCategory C\u2082\nfst\u271d : C\u2081\nsnd\u271d : C\u2082\n\u22a2 (\u03bb_ (fst\u271d, snd\u271d)).hom.snd = (\u03bb_ (fst\u271d, snd\u271d).snd).hom", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Ring/BooleanRing.lean", "full_name": "toBoolAlg_symm_eq", "start": [147, 1], "end": [148, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "Complex.tan_neg", "start": [985, 1], "end": [985, 62], "traced_tactics": [{"tactic": "simp [tan, neg_div]", "state_before": "x y : \u2102\n\u22a2 tan (-x) = -tan x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "full_name": "ContinuousLinearMap.projKerOfRightInverse_apply_idem", "start": [1461, 1], "end": [1465, 7], "traced_tactics": [{"tactic": "ext1", "state_before": "R : Type u_2\ninst\u271d\u00b9\u2075 : Ring R\nR\u2082 : Type u_3\ninst\u271d\u00b9\u2074 : Ring R\u2082\nR\u2083 : Type ?u.921374\ninst\u271d\u00b9\u00b3 : Ring R\u2083\nM : Type u_1\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : AddCommGroup M\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2070 : TopologicalSpace M\u2082\ninst\u271d\u2079 : AddCommGroup M\u2082\nM\u2083 : Type ?u.921398\ninst\u271d\u2078 : TopologicalSpace M\u2083\ninst\u271d\u2077 : AddCommGroup M\u2083\nM\u2084 : Type ?u.921407\ninst\u271d\u2076 : TopologicalSpace M\u2084\ninst\u271d\u2075 : AddCommGroup M\u2084\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d : TopologicalAddGroup M\nf\u2081 : M \u2192SL[\u03c3\u2081\u2082] M\u2082\nf\u2082 : M\u2082 \u2192SL[\u03c3\u2082\u2081] M\nh : Function.RightInverse \u2191f\u2082 \u2191f\u2081\nx : { x // x \u2208 ker f\u2081 }\n\u22a2 \u2191(projKerOfRightInverse f\u2081 f\u2082 h) \u2191x = x", "state_after": "case a\nR : Type u_2\ninst\u271d\u00b9\u2075 : Ring R\nR\u2082 : Type u_3\ninst\u271d\u00b9\u2074 : Ring R\u2082\nR\u2083 : Type ?u.921374\ninst\u271d\u00b9\u00b3 : Ring R\u2083\nM : Type u_1\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : AddCommGroup M\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2070 : TopologicalSpace M\u2082\ninst\u271d\u2079 : AddCommGroup M\u2082\nM\u2083 : Type ?u.921398\ninst\u271d\u2078 : TopologicalSpace M\u2083\ninst\u271d\u2077 : AddCommGroup M\u2083\nM\u2084 : Type ?u.921407\ninst\u271d\u2076 : TopologicalSpace M\u2084\ninst\u271d\u2075 : AddCommGroup M\u2084\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d : TopologicalAddGroup M\nf\u2081 : M \u2192SL[\u03c3\u2081\u2082] M\u2082\nf\u2082 : M\u2082 \u2192SL[\u03c3\u2082\u2081] M\nh : Function.RightInverse \u2191f\u2082 \u2191f\u2081\nx : { x // x \u2208 ker f\u2081 }\n\u22a2 \u2191(\u2191(projKerOfRightInverse f\u2081 f\u2082 h) \u2191x) = \u2191x"}, {"tactic": "simp", "state_before": "case a\nR : Type u_2\ninst\u271d\u00b9\u2075 : Ring R\nR\u2082 : Type u_3\ninst\u271d\u00b9\u2074 : Ring R\u2082\nR\u2083 : Type ?u.921374\ninst\u271d\u00b9\u00b3 : Ring R\u2083\nM : Type u_1\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : AddCommGroup M\nM\u2082 : Type u_4\ninst\u271d\u00b9\u2070 : TopologicalSpace M\u2082\ninst\u271d\u2079 : AddCommGroup M\u2082\nM\u2083 : Type ?u.921398\ninst\u271d\u2078 : TopologicalSpace M\u2083\ninst\u271d\u2077 : AddCommGroup M\u2083\nM\u2084 : Type ?u.921407\ninst\u271d\u2076 : TopologicalSpace M\u2084\ninst\u271d\u2075 : AddCommGroup M\u2084\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d : TopologicalAddGroup M\nf\u2081 : M \u2192SL[\u03c3\u2081\u2082] M\u2082\nf\u2082 : M\u2082 \u2192SL[\u03c3\u2082\u2081] M\nh : Function.RightInverse \u2191f\u2082 \u2191f\u2081\nx : { x // x \u2208 ker f\u2081 }\n\u22a2 \u2191(\u2191(projKerOfRightInverse f\u2081 f\u2082 h) \u2191x) = \u2191x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.diff_inter_self_eq_diff", "start": [2034, 1], "end": [2035, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/NaturalOps.lean", "full_name": "Ordinal.nadd_left_cancel_iff", "start": [442, 1], "end": [443, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Basic.lean", "full_name": "LinearMap.map_le_range", "start": [1247, 1], "end": [1248, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Matrix/Kronecker.lean", "full_name": "Matrix.kroneckerMap_assoc", "start": [180, 1], "end": [186, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "full_name": "Real.Angle.tan_eq_inv_of_two_nsmul_add_two_nsmul_eq_pi", "start": [830, 1], "end": [840, 38], "traced_tactics": [{"tactic": "induction \u03b8 using Real.Angle.induction_on", "state_before": "\u03b8 \u03c8 : Angle\nh : 2 \u2022 \u03b8 + 2 \u2022 \u03c8 = \u2191\u03c0\n\u22a2 tan \u03c8 = (tan \u03b8)\u207b\u00b9", "state_after": "case h\n\u03c8 : Angle\nx\u271d : \u211d\nh : 2 \u2022 \u2191x\u271d + 2 \u2022 \u03c8 = \u2191\u03c0\n\u22a2 tan \u03c8 = (tan \u2191x\u271d)\u207b\u00b9"}, {"tactic": "induction \u03c8 using Real.Angle.induction_on", "state_before": "case h\n\u03c8 : Angle\nx\u271d : \u211d\nh : 2 \u2022 \u2191x\u271d + 2 \u2022 \u03c8 = \u2191\u03c0\n\u22a2 tan \u03c8 = (tan \u2191x\u271d)\u207b\u00b9", "state_after": "case h.h\nx\u271d\u00b9 x\u271d : \u211d\nh : 2 \u2022 \u2191x\u271d\u00b9 + 2 \u2022 \u2191x\u271d = \u2191\u03c0\n\u22a2 tan \u2191x\u271d = (tan \u2191x\u271d\u00b9)\u207b\u00b9"}, {"tactic": "rw [\u2190 smul_add, \u2190 coe_add, \u2190 coe_nsmul, two_nsmul, \u2190 two_mul, angle_eq_iff_two_pi_dvd_sub] at h", "state_before": "case h.h\nx\u271d\u00b9 x\u271d : \u211d\nh : 2 \u2022 \u2191x\u271d\u00b9 + 2 \u2022 \u2191x\u271d = \u2191\u03c0\n\u22a2 tan \u2191x\u271d = (tan \u2191x\u271d\u00b9)\u207b\u00b9", "state_after": "case h.h\nx\u271d\u00b9 x\u271d : \u211d\nh : \u2203 k, 2 * (x\u271d\u00b9 + x\u271d) - \u03c0 = 2 * \u03c0 * \u2191k\n\u22a2 tan \u2191x\u271d = (tan \u2191x\u271d\u00b9)\u207b\u00b9"}, {"tactic": "rcases h with \u27e8k, h\u27e9", "state_before": "case h.h\nx\u271d\u00b9 x\u271d : \u211d\nh : \u2203 k, 2 * (x\u271d\u00b9 + x\u271d) - \u03c0 = 2 * \u03c0 * \u2191k\n\u22a2 tan \u2191x\u271d = (tan \u2191x\u271d\u00b9)\u207b\u00b9", "state_after": "case h.h.intro\nx\u271d\u00b9 x\u271d : \u211d\nk : \u2124\nh : 2 * (x\u271d\u00b9 + x\u271d) - \u03c0 = 2 * \u03c0 * \u2191k\n\u22a2 tan \u2191x\u271d = (tan \u2191x\u271d\u00b9)\u207b\u00b9"}, {"tactic": "rw [sub_eq_iff_eq_add, \u2190 mul_inv_cancel_left\u2080 two_ne_zero \u03c0, mul_assoc, \u2190 mul_add,\n  mul_right_inj' (two_ne_zero' \u211d), \u2190 eq_sub_iff_add_eq', mul_inv_cancel_left\u2080 two_ne_zero \u03c0,\n  inv_mul_eq_div, mul_comm] at h", "state_before": "case h.h.intro\nx\u271d\u00b9 x\u271d : \u211d\nk : \u2124\nh : 2 * (x\u271d\u00b9 + x\u271d) - \u03c0 = 2 * \u03c0 * \u2191k\n\u22a2 tan \u2191x\u271d = (tan \u2191x\u271d\u00b9)\u207b\u00b9", "state_after": "case h.h.intro\nx\u271d\u00b9 x\u271d : \u211d\nk : \u2124\nh : x\u271d = \u2191k * \u03c0 + \u03c0 / 2 - x\u271d\u00b9\n\u22a2 tan \u2191x\u271d = (tan \u2191x\u271d\u00b9)\u207b\u00b9"}, {"tactic": "rw [tan_coe, tan_coe, \u2190 tan_pi_div_two_sub, h, add_sub_assoc, add_comm]", "state_before": "case h.h.intro\nx\u271d\u00b9 x\u271d : \u211d\nk : \u2124\nh : x\u271d = \u2191k * \u03c0 + \u03c0 / 2 - x\u271d\u00b9\n\u22a2 tan \u2191x\u271d = (tan \u2191x\u271d\u00b9)\u207b\u00b9", "state_after": "case h.h.intro\nx\u271d\u00b9 x\u271d : \u211d\nk : \u2124\nh : x\u271d = \u2191k * \u03c0 + \u03c0 / 2 - x\u271d\u00b9\n\u22a2 Real.tan (\u03c0 / 2 - x\u271d\u00b9 + \u2191k * \u03c0) = Real.tan (\u03c0 / 2 - x\u271d\u00b9)"}, {"tactic": "exact Real.tan_periodic.int_mul _ _", "state_before": "case h.h.intro\nx\u271d\u00b9 x\u271d : \u211d\nk : \u2124\nh : x\u271d = \u2191k * \u03c0 + \u03c0 / 2 - x\u271d\u00b9\n\u22a2 Real.tan (\u03c0 / 2 - x\u271d\u00b9 + \u2191k * \u03c0) = Real.tan (\u03c0 / 2 - x\u271d\u00b9)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "full_name": "Real.log_prod", "start": [334, 1], "end": [339, 66], "traced_tactics": [{"tactic": "induction' s using Finset.cons_induction_on with a s ha ih", "state_before": "x y : \u211d\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\n\u22a2 log (\u220f i in s, f i) = \u2211 i in s, log (f i)", "state_after": "case h\u2081\nx y : \u211d\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nhf : \u2200 (x : \u03b1), x \u2208 \u2205 \u2192 f x \u2260 0\n\u22a2 log (\u220f i in \u2205, f i) = \u2211 i in \u2205, log (f i)\n\ncase h\u2082\nx y : \u211d\n\u03b1 : Type u_1\ns\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf\u271d : \u2200 (x : \u03b1), x \u2208 s\u271d \u2192 f x \u2260 0\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : (\u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0) \u2192 log (\u220f i in s, f i) = \u2211 i in s, log (f i)\nhf : \u2200 (x : \u03b1), x \u2208 Finset.cons a s ha \u2192 f x \u2260 0\n\u22a2 log (\u220f i in Finset.cons a s ha, f i) = \u2211 i in Finset.cons a s ha, log (f i)"}, {"tactic": "simp", "state_before": "case h\u2081\nx y : \u211d\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nhf : \u2200 (x : \u03b1), x \u2208 \u2205 \u2192 f x \u2260 0\n\u22a2 log (\u220f i in \u2205, f i) = \u2211 i in \u2205, log (f i)", "state_after": "no goals"}, {"tactic": "rw [Finset.forall_mem_cons] at hf", "state_before": "case h\u2082\nx y : \u211d\n\u03b1 : Type u_1\ns\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf\u271d : \u2200 (x : \u03b1), x \u2208 s\u271d \u2192 f x \u2260 0\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : (\u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0) \u2192 log (\u220f i in s, f i) = \u2211 i in s, log (f i)\nhf : \u2200 (x : \u03b1), x \u2208 Finset.cons a s ha \u2192 f x \u2260 0\n\u22a2 log (\u220f i in Finset.cons a s ha, f i) = \u2211 i in Finset.cons a s ha, log (f i)", "state_after": "case h\u2082\nx y : \u211d\n\u03b1 : Type u_1\ns\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf\u271d : \u2200 (x : \u03b1), x \u2208 s\u271d \u2192 f x \u2260 0\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : (\u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0) \u2192 log (\u220f i in s, f i) = \u2211 i in s, log (f i)\nhf : f a \u2260 0 \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\n\u22a2 log (\u220f i in Finset.cons a s ha, f i) = \u2211 i in Finset.cons a s ha, log (f i)"}, {"tactic": "simp [ih hf.2, log_mul hf.1 (Finset.prod_ne_zero_iff.2 hf.2)]", "state_before": "case h\u2082\nx y : \u211d\n\u03b1 : Type u_1\ns\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf\u271d : \u2200 (x : \u03b1), x \u2208 s\u271d \u2192 f x \u2260 0\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : (\u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0) \u2192 log (\u220f i in s, f i) = \u2211 i in s, log (f i)\nhf : f a \u2260 0 \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\n\u22a2 log (\u220f i in Finset.cons a s ha, f i) = \u2211 i in Finset.cons a s ha, log (f i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "isClosed_frontier", "start": [769, 1], "end": [770, 97], "traced_tactics": [{"tactic": "rw [frontier_eq_closure_inter_closure]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\na : \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\np p\u2081 p\u2082 : \u03b1 \u2192 Prop\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\n\u22a2 IsClosed (frontier s)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\na : \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\np p\u2081 p\u2082 : \u03b1 \u2192 Prop\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\n\u22a2 IsClosed (closure s \u2229 closure (s\u1d9c))"}, {"tactic": "exact IsClosed.inter isClosed_closure isClosed_closure", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\na : \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\np p\u2081 p\u2082 : \u03b1 \u2192 Prop\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b1\n\u22a2 IsClosed (closure s \u2229 closure (s\u1d9c))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Derivative.lean", "full_name": "Polynomial.mem_support_derivative", "start": [366, 1], "end": [371, 45], "traced_tactics": [{"tactic": "suffices \u00acp.coeff (n + 1) * (n + 1 : \u2115) = 0 \u2194 coeff p (n + 1) \u2260 0 by\n  simpa only [mem_support_iff, coeff_derivative, Ne.def, Nat.cast_succ]", "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nn : \u2115\n\u22a2 n \u2208 support (\u2191derivative p) \u2194 n + 1 \u2208 support p", "state_after": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nn : \u2115\n\u22a2 \u00accoeff p (n + 1) * \u2191(n + 1) = 0 \u2194 coeff p (n + 1) \u2260 0"}, {"tactic": "rw [\u2190 nsmul_eq_mul', smul_eq_zero]", "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nn : \u2115\n\u22a2 \u00accoeff p (n + 1) * \u2191(n + 1) = 0 \u2194 coeff p (n + 1) \u2260 0", "state_after": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nn : \u2115\n\u22a2 \u00ac(n + 1 = 0 \u2228 coeff p (n + 1) = 0) \u2194 coeff p (n + 1) \u2260 0"}, {"tactic": "simp only [Nat.succ_ne_zero, false_or_iff]", "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nn : \u2115\n\u22a2 \u00ac(n + 1 = 0 \u2228 coeff p (n + 1) = 0) \u2194 coeff p (n + 1) \u2260 0", "state_after": "no goals"}, {"tactic": "simpa only [mem_support_iff, coeff_derivative, Ne.def, Nat.cast_succ]", "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroSMulDivisors \u2115 R\np : R[X]\nn : \u2115\nthis : \u00accoeff p (n + 1) * \u2191(n + 1) = 0 \u2194 coeff p (n + 1) \u2260 0\n\u22a2 n \u2208 support (\u2191derivative p) \u2194 n + 1 \u2208 support p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Subsingleton.eq_univ_of_nonempty", "start": [2827, 1], "end": [2828, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/IntegralClosure.lean", "full_name": "isIntegral_iff_isIntegral_closure_finite", "start": [190, 1], "end": [197, 35], "traced_tactics": [{"tactic": "constructor <;> intro hr", "state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.284040\nS : Type ?u.284043\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nr : A\n\u22a2 IsIntegral R r \u2194 \u2203 s, Set.Finite s \u2227 IsIntegral { x // x \u2208 Subring.closure s } r", "state_after": "case mp\nR : Type u_1\nA : Type u_2\nB : Type ?u.284040\nS : Type ?u.284043\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nr : A\nhr : IsIntegral R r\n\u22a2 \u2203 s, Set.Finite s \u2227 IsIntegral { x // x \u2208 Subring.closure s } r\n\ncase mpr\nR : Type u_1\nA : Type u_2\nB : Type ?u.284040\nS : Type ?u.284043\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nr : A\nhr : \u2203 s, Set.Finite s \u2227 IsIntegral { x // x \u2208 Subring.closure s } r\n\u22a2 IsIntegral R r"}, {"tactic": "rcases hr with \u27e8s, _, hsr\u27e9", "state_before": "case mpr\nR : Type u_1\nA : Type u_2\nB : Type ?u.284040\nS : Type ?u.284043\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nr : A\nhr : \u2203 s, Set.Finite s \u2227 IsIntegral { x // x \u2208 Subring.closure s } r\n\u22a2 IsIntegral R r", "state_after": "case mpr.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.284040\nS : Type ?u.284043\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nr : A\ns : Set R\nleft\u271d : Set.Finite s\nhsr : IsIntegral { x // x \u2208 Subring.closure s } r\n\u22a2 IsIntegral R r"}, {"tactic": "exact isIntegral_ofSubring _ hsr", "state_before": "case mpr.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.284040\nS : Type ?u.284043\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nr : A\ns : Set R\nleft\u271d : Set.Finite s\nhsr : IsIntegral { x // x \u2208 Subring.closure s } r\n\u22a2 IsIntegral R r", "state_after": "no goals"}, {"tactic": "rcases hr with \u27e8p, hmp, hpr\u27e9", "state_before": "case mp\nR : Type u_1\nA : Type u_2\nB : Type ?u.284040\nS : Type ?u.284043\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nr : A\nhr : IsIntegral R r\n\u22a2 \u2203 s, Set.Finite s \u2227 IsIntegral { x // x \u2208 Subring.closure s } r", "state_after": "case mp.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.284040\nS : Type ?u.284043\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nr : A\np : R[X]\nhmp : Monic p\nhpr : eval\u2082 (algebraMap R A) r p = 0\n\u22a2 \u2203 s, Set.Finite s \u2227 IsIntegral { x // x \u2208 Subring.closure s } r"}, {"tactic": "refine' \u27e8_, Finset.finite_toSet _, p.restriction, monic_restriction.2 hmp, _\u27e9", "state_before": "case mp.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.284040\nS : Type ?u.284043\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nr : A\np : R[X]\nhmp : Monic p\nhpr : eval\u2082 (algebraMap R A) r p = 0\n\u22a2 \u2203 s, Set.Finite s \u2227 IsIntegral { x // x \u2208 Subring.closure s } r", "state_after": "case mp.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.284040\nS : Type ?u.284043\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nr : A\np : R[X]\nhmp : Monic p\nhpr : eval\u2082 (algebraMap R A) r p = 0\n\u22a2 eval\u2082 (algebraMap { x // x \u2208 Subring.closure \u2191(frange p) } A) r (restriction p) = 0"}, {"tactic": "rw [\u2190 aeval_def, \u2190 aeval_map_algebraMap R r p.restriction, map_restriction, aeval_def, hpr]", "state_before": "case mp.intro.intro\nR : Type u_1\nA : Type u_2\nB : Type ?u.284040\nS : Type ?u.284043\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : R \u2192+* S\nr : A\np : R[X]\nhmp : Monic p\nhpr : eval\u2082 (algebraMap R A) r p = 0\n\u22a2 eval\u2082 (algebraMap { x // x \u2208 Subring.closure \u2191(frange p) } A) r (restriction p) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "full_name": "LocalHomeomorph.singleton_hasGroupoid", "start": [1011, 1], "end": [1020, 72], "traced_tactics": [{"tactic": "intro e' e'' he' he''", "state_before": "H : Type u\nH' : Type ?u.75941\nM : Type ?u.75944\nM' : Type ?u.75947\nM'' : Type ?u.75950\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ne : LocalHomeomorph \u03b1 H\nh : e.source = univ\nG : StructureGroupoid H\ninst\u271d : ClosedUnderRestriction G\nsrc\u271d : ChartedSpace H \u03b1 := singletonChartedSpace e h\n\u22a2 \u2200 {e_1 e' : LocalHomeomorph \u03b1 H}, e_1 \u2208 atlas H \u03b1 \u2192 e' \u2208 atlas H \u03b1 \u2192 LocalHomeomorph.symm e_1 \u226b\u2095 e' \u2208 G", "state_after": "H : Type u\nH' : Type ?u.75941\nM : Type ?u.75944\nM' : Type ?u.75947\nM'' : Type ?u.75950\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ne : LocalHomeomorph \u03b1 H\nh : e.source = univ\nG : StructureGroupoid H\ninst\u271d : ClosedUnderRestriction G\nsrc\u271d : ChartedSpace H \u03b1 := singletonChartedSpace e h\ne' e'' : LocalHomeomorph \u03b1 H\nhe' : e' \u2208 atlas H \u03b1\nhe'' : e'' \u2208 atlas H \u03b1\n\u22a2 LocalHomeomorph.symm e' \u226b\u2095 e'' \u2208 G"}, {"tactic": "rw [e.singletonChartedSpace_mem_atlas_eq h e' he',\n  e.singletonChartedSpace_mem_atlas_eq h e'' he'']", "state_before": "H : Type u\nH' : Type ?u.75941\nM : Type ?u.75944\nM' : Type ?u.75947\nM'' : Type ?u.75950\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ne : LocalHomeomorph \u03b1 H\nh : e.source = univ\nG : StructureGroupoid H\ninst\u271d : ClosedUnderRestriction G\nsrc\u271d : ChartedSpace H \u03b1 := singletonChartedSpace e h\ne' e'' : LocalHomeomorph \u03b1 H\nhe' : e' \u2208 atlas H \u03b1\nhe'' : e'' \u2208 atlas H \u03b1\n\u22a2 LocalHomeomorph.symm e' \u226b\u2095 e'' \u2208 G", "state_after": "H : Type u\nH' : Type ?u.75941\nM : Type ?u.75944\nM' : Type ?u.75947\nM'' : Type ?u.75950\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ne : LocalHomeomorph \u03b1 H\nh : e.source = univ\nG : StructureGroupoid H\ninst\u271d : ClosedUnderRestriction G\nsrc\u271d : ChartedSpace H \u03b1 := singletonChartedSpace e h\ne' e'' : LocalHomeomorph \u03b1 H\nhe' : e' \u2208 atlas H \u03b1\nhe'' : e'' \u2208 atlas H \u03b1\n\u22a2 LocalHomeomorph.symm e \u226b\u2095 e \u2208 G"}, {"tactic": "refine' G.eq_on_source _ e.trans_symm_self", "state_before": "H : Type u\nH' : Type ?u.75941\nM : Type ?u.75944\nM' : Type ?u.75947\nM'' : Type ?u.75950\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ne : LocalHomeomorph \u03b1 H\nh : e.source = univ\nG : StructureGroupoid H\ninst\u271d : ClosedUnderRestriction G\nsrc\u271d : ChartedSpace H \u03b1 := singletonChartedSpace e h\ne' e'' : LocalHomeomorph \u03b1 H\nhe' : e' \u2208 atlas H \u03b1\nhe'' : e'' \u2208 atlas H \u03b1\n\u22a2 LocalHomeomorph.symm e \u226b\u2095 e \u2208 G", "state_after": "H : Type u\nH' : Type ?u.75941\nM : Type ?u.75944\nM' : Type ?u.75947\nM'' : Type ?u.75950\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ne : LocalHomeomorph \u03b1 H\nh : e.source = univ\nG : StructureGroupoid H\ninst\u271d : ClosedUnderRestriction G\nsrc\u271d : ChartedSpace H \u03b1 := singletonChartedSpace e h\ne' e'' : LocalHomeomorph \u03b1 H\nhe' : e' \u2208 atlas H \u03b1\nhe'' : e'' \u2208 atlas H \u03b1\n\u22a2 ofSet e.target (_ : IsOpen e.target) \u2208 G"}, {"tactic": "have hle : idRestrGroupoid \u2264 G := (closedUnderRestriction_iff_id_le G).mp (by assumption)", "state_before": "H : Type u\nH' : Type ?u.75941\nM : Type ?u.75944\nM' : Type ?u.75947\nM'' : Type ?u.75950\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ne : LocalHomeomorph \u03b1 H\nh : e.source = univ\nG : StructureGroupoid H\ninst\u271d : ClosedUnderRestriction G\nsrc\u271d : ChartedSpace H \u03b1 := singletonChartedSpace e h\ne' e'' : LocalHomeomorph \u03b1 H\nhe' : e' \u2208 atlas H \u03b1\nhe'' : e'' \u2208 atlas H \u03b1\n\u22a2 ofSet e.target (_ : IsOpen e.target) \u2208 G", "state_after": "H : Type u\nH' : Type ?u.75941\nM : Type ?u.75944\nM' : Type ?u.75947\nM'' : Type ?u.75950\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ne : LocalHomeomorph \u03b1 H\nh : e.source = univ\nG : StructureGroupoid H\ninst\u271d : ClosedUnderRestriction G\nsrc\u271d : ChartedSpace H \u03b1 := singletonChartedSpace e h\ne' e'' : LocalHomeomorph \u03b1 H\nhe' : e' \u2208 atlas H \u03b1\nhe'' : e'' \u2208 atlas H \u03b1\nhle : idRestrGroupoid \u2264 G\n\u22a2 ofSet e.target (_ : IsOpen e.target) \u2208 G"}, {"tactic": "exact StructureGroupoid.le_iff.mp hle _ (idRestrGroupoid_mem _)", "state_before": "H : Type u\nH' : Type ?u.75941\nM : Type ?u.75944\nM' : Type ?u.75947\nM'' : Type ?u.75950\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ne : LocalHomeomorph \u03b1 H\nh : e.source = univ\nG : StructureGroupoid H\ninst\u271d : ClosedUnderRestriction G\nsrc\u271d : ChartedSpace H \u03b1 := singletonChartedSpace e h\ne' e'' : LocalHomeomorph \u03b1 H\nhe' : e' \u2208 atlas H \u03b1\nhe'' : e'' \u2208 atlas H \u03b1\nhle : idRestrGroupoid \u2264 G\n\u22a2 ofSet e.target (_ : IsOpen e.target) \u2208 G", "state_after": "no goals"}, {"tactic": "assumption", "state_before": "H : Type u\nH' : Type ?u.75941\nM : Type ?u.75944\nM' : Type ?u.75947\nM'' : Type ?u.75950\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ne : LocalHomeomorph \u03b1 H\nh : e.source = univ\nG : StructureGroupoid H\ninst\u271d : ClosedUnderRestriction G\nsrc\u271d : ChartedSpace H \u03b1 := singletonChartedSpace e h\ne' e'' : LocalHomeomorph \u03b1 H\nhe' : e' \u2208 atlas H \u03b1\nhe'' : e'' \u2208 atlas H \u03b1\n\u22a2 ClosedUnderRestriction G", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.OuterMeasure.trim_iSup", "start": [1768, 1], "end": [1778, 27], "traced_tactics": [{"tactic": "simp_rw [\u2190 @iSup_plift_down _ \u03b9]", "state_before": "\u03b1 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\n\u22a2 trim (\u2a06 (i : \u03b9), \u03bc i) = \u2a06 (i : \u03b9), trim (\u03bc i)", "state_after": "\u03b1 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\n\u22a2 trim (\u2a06 (i : PLift \u03b9), \u03bc i.down) = \u2a06 (i : PLift \u03b9), trim (\u03bc i.down)"}, {"tactic": "ext1 s", "state_before": "\u03b1 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\n\u22a2 trim (\u2a06 (i : PLift \u03b9), \u03bc i.down) = \u2a06 (i : PLift \u03b9), trim (\u03bc i.down)", "state_after": "case h\n\u03b1 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2191(trim (\u2a06 (i : PLift \u03b9), \u03bc i.down)) s = \u2191(\u2a06 (i : PLift \u03b9), trim (\u03bc i.down)) s"}, {"tactic": "haveI : Countable (Option <| PLift \u03b9) := by exact instCountableOption", "state_before": "case h\n\u03b1 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 \u2191(trim (\u2a06 (i : PLift \u03b9), \u03bc i.down)) s = \u2191(\u2a06 (i : PLift \u03b9), trim (\u03bc i.down)) s", "state_after": "case h\n\u03b1 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\nthis : Countable (Option (PLift \u03b9))\n\u22a2 \u2191(trim (\u2a06 (i : PLift \u03b9), \u03bc i.down)) s = \u2191(\u2a06 (i : PLift \u03b9), trim (\u03bc i.down)) s"}, {"tactic": "obtain \u27e8t, _, _, h\u03bct\u27e9 :=\n  exists_measurable_superset_forall_eq_trim\n    (Option.elim' (\u2a06 i, \u03bc (PLift.down i)) (\u03bc \u2218 PLift.down)) s", "state_before": "case h\n\u03b1 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\nthis : Countable (Option (PLift \u03b9))\n\u22a2 \u2191(trim (\u2a06 (i : PLift \u03b9), \u03bc i.down)) s = \u2191(\u2a06 (i : PLift \u03b9), trim (\u03bc i.down)) s", "state_after": "case h.intro.intro.intro\n\u03b1 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\nthis : Countable (Option (PLift \u03b9))\nt : Set \u03b1\nleft\u271d\u00b9 : s \u2286 t\nleft\u271d : MeasurableSet t\nh\u03bct :\n  \u2200 (i : Option (PLift \u03b9)),\n    \u2191(Option.elim' (\u2a06 (i : PLift \u03b9), \u03bc i.down) (\u03bc \u2218 PLift.down) i) t =\n      \u2191(trim (Option.elim' (\u2a06 (i : PLift \u03b9), \u03bc i.down) (\u03bc \u2218 PLift.down) i)) s\n\u22a2 \u2191(trim (\u2a06 (i : PLift \u03b9), \u03bc i.down)) s = \u2191(\u2a06 (i : PLift \u03b9), trim (\u03bc i.down)) s"}, {"tactic": "simp only [Option.forall, Option.elim'] at h\u03bct", "state_before": "case h.intro.intro.intro\n\u03b1 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\nthis : Countable (Option (PLift \u03b9))\nt : Set \u03b1\nleft\u271d\u00b9 : s \u2286 t\nleft\u271d : MeasurableSet t\nh\u03bct :\n  \u2200 (i : Option (PLift \u03b9)),\n    \u2191(Option.elim' (\u2a06 (i : PLift \u03b9), \u03bc i.down) (\u03bc \u2218 PLift.down) i) t =\n      \u2191(trim (Option.elim' (\u2a06 (i : PLift \u03b9), \u03bc i.down) (\u03bc \u2218 PLift.down) i)) s\n\u22a2 \u2191(trim (\u2a06 (i : PLift \u03b9), \u03bc i.down)) s = \u2191(\u2a06 (i : PLift \u03b9), trim (\u03bc i.down)) s", "state_after": "case h.intro.intro.intro\n\u03b1 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\nthis : Countable (Option (PLift \u03b9))\nt : Set \u03b1\nleft\u271d\u00b9 : s \u2286 t\nleft\u271d : MeasurableSet t\nh\u03bct :\n  \u2191(\u2a06 (i : PLift \u03b9), \u03bc i.down) t = \u2191(trim (\u2a06 (i : PLift \u03b9), \u03bc i.down)) s \u2227\n    \u2200 (x : PLift \u03b9), \u2191((\u03bc \u2218 PLift.down) x) t = \u2191(trim ((\u03bc \u2218 PLift.down) x)) s\n\u22a2 \u2191(trim (\u2a06 (i : PLift \u03b9), \u03bc i.down)) s = \u2191(\u2a06 (i : PLift \u03b9), trim (\u03bc i.down)) s"}, {"tactic": "simp only [iSup_apply, \u2190 h\u03bct.1]", "state_before": "case h.intro.intro.intro\n\u03b1 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\nthis : Countable (Option (PLift \u03b9))\nt : Set \u03b1\nleft\u271d\u00b9 : s \u2286 t\nleft\u271d : MeasurableSet t\nh\u03bct :\n  \u2191(\u2a06 (i : PLift \u03b9), \u03bc i.down) t = \u2191(trim (\u2a06 (i : PLift \u03b9), \u03bc i.down)) s \u2227\n    \u2200 (x : PLift \u03b9), \u2191((\u03bc \u2218 PLift.down) x) t = \u2191(trim ((\u03bc \u2218 PLift.down) x)) s\n\u22a2 \u2191(trim (\u2a06 (i : PLift \u03b9), \u03bc i.down)) s = \u2191(\u2a06 (i : PLift \u03b9), trim (\u03bc i.down)) s", "state_after": "case h.intro.intro.intro\n\u03b1 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\nthis : Countable (Option (PLift \u03b9))\nt : Set \u03b1\nleft\u271d\u00b9 : s \u2286 t\nleft\u271d : MeasurableSet t\nh\u03bct :\n  \u2191(\u2a06 (i : PLift \u03b9), \u03bc i.down) t = \u2191(trim (\u2a06 (i : PLift \u03b9), \u03bc i.down)) s \u2227\n    \u2200 (x : PLift \u03b9), \u2191((\u03bc \u2218 PLift.down) x) t = \u2191(trim ((\u03bc \u2218 PLift.down) x)) s\n\u22a2 (\u2a06 (i : PLift \u03b9), \u2191(\u03bc i.down) t) = \u2a06 (i : PLift \u03b9), \u2191(trim (\u03bc i.down)) s"}, {"tactic": "exact iSup_congr h\u03bct.2", "state_before": "case h.intro.intro.intro\n\u03b1 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\nthis : Countable (Option (PLift \u03b9))\nt : Set \u03b1\nleft\u271d\u00b9 : s \u2286 t\nleft\u271d : MeasurableSet t\nh\u03bct :\n  \u2191(\u2a06 (i : PLift \u03b9), \u03bc i.down) t = \u2191(trim (\u2a06 (i : PLift \u03b9), \u03bc i.down)) s \u2227\n    \u2200 (x : PLift \u03b9), \u2191((\u03bc \u2218 PLift.down) x) t = \u2191(trim ((\u03bc \u2218 PLift.down) x)) s\n\u22a2 (\u2a06 (i : PLift \u03b9), \u2191(\u03bc i.down) t) = \u2a06 (i : PLift \u03b9), \u2191(trim (\u03bc i.down)) s", "state_after": "no goals"}, {"tactic": "exact instCountableOption", "state_before": "\u03b1 : Type u_2\ninst\u271d\u00b9 : MeasurableSpace \u03b1\nm : OuterMeasure \u03b1\n\u03b9 : Sort u_1\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 OuterMeasure \u03b1\ns : Set \u03b1\n\u22a2 Countable (Option (PLift \u03b9))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fintype/Fin.lean", "full_name": "Fin.card_filter_univ_succ", "start": [81, 1], "end": [84, 72], "traced_tactics": [{"tactic": "split_ifs <;> simp [add_comm 1]", "state_before": "\u03b1 : Type ?u.16258\n\u03b2 : Type ?u.16261\nn : \u2115\np : Fin (n + 1) \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 (if p 0 then 1 else 0) + Finset.card (filter (p \u2218 succ) univ) =\n    if p 0 then Finset.card (filter (p \u2218 succ) univ) + 1 else Finset.card (filter (p \u2218 succ) univ)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/Minpoly/Field.lean", "full_name": "minpoly.coeff_zero_eq_zero", "start": [265, 1], "end": [272, 9], "traced_tactics": [{"tactic": "constructor", "state_before": "A : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : Field A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : IsDomain B\ninst\u271d : Algebra A B\nx : B\nhx : IsIntegral A x\n\u22a2 coeff (minpoly A x) 0 = 0 \u2194 x = 0", "state_after": "case mp\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : Field A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : IsDomain B\ninst\u271d : Algebra A B\nx : B\nhx : IsIntegral A x\n\u22a2 coeff (minpoly A x) 0 = 0 \u2192 x = 0\n\ncase mpr\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : Field A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : IsDomain B\ninst\u271d : Algebra A B\nx : B\nhx : IsIntegral A x\n\u22a2 x = 0 \u2192 coeff (minpoly A x) 0 = 0"}, {"tactic": "intro h", "state_before": "case mp\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : Field A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : IsDomain B\ninst\u271d : Algebra A B\nx : B\nhx : IsIntegral A x\n\u22a2 coeff (minpoly A x) 0 = 0 \u2192 x = 0", "state_after": "case mp\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : Field A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : IsDomain B\ninst\u271d : Algebra A B\nx : B\nhx : IsIntegral A x\nh : coeff (minpoly A x) 0 = 0\n\u22a2 x = 0"}, {"tactic": "have zero_root := zero_isRoot_of_coeff_zero_eq_zero h", "state_before": "case mp\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : Field A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : IsDomain B\ninst\u271d : Algebra A B\nx : B\nhx : IsIntegral A x\nh : coeff (minpoly A x) 0 = 0\n\u22a2 x = 0", "state_after": "case mp\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : Field A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : IsDomain B\ninst\u271d : Algebra A B\nx : B\nhx : IsIntegral A x\nh : coeff (minpoly A x) 0 = 0\nzero_root : IsRoot (minpoly A x) 0\n\u22a2 x = 0"}, {"tactic": "rw [\u2190 root hx zero_root]", "state_before": "case mp\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : Field A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : IsDomain B\ninst\u271d : Algebra A B\nx : B\nhx : IsIntegral A x\nh : coeff (minpoly A x) 0 = 0\nzero_root : IsRoot (minpoly A x) 0\n\u22a2 x = 0", "state_after": "case mp\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : Field A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : IsDomain B\ninst\u271d : Algebra A B\nx : B\nhx : IsIntegral A x\nh : coeff (minpoly A x) 0 = 0\nzero_root : IsRoot (minpoly A x) 0\n\u22a2 \u2191(algebraMap A B) 0 = 0"}, {"tactic": "exact RingHom.map_zero _", "state_before": "case mp\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : Field A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : IsDomain B\ninst\u271d : Algebra A B\nx : B\nhx : IsIntegral A x\nh : coeff (minpoly A x) 0 = 0\nzero_root : IsRoot (minpoly A x) 0\n\u22a2 \u2191(algebraMap A B) 0 = 0", "state_after": "no goals"}, {"tactic": "rintro rfl", "state_before": "case mpr\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : Field A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : IsDomain B\ninst\u271d : Algebra A B\nx : B\nhx : IsIntegral A x\n\u22a2 x = 0 \u2192 coeff (minpoly A x) 0 = 0", "state_after": "case mpr\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : Field A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : IsDomain B\ninst\u271d : Algebra A B\nhx : IsIntegral A 0\n\u22a2 coeff (minpoly A 0) 0 = 0"}, {"tactic": "simp", "state_before": "case mpr\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : Field A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : IsDomain B\ninst\u271d : Algebra A B\nhx : IsIntegral A 0\n\u22a2 coeff (minpoly A 0) 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Order/Lattice.lean", "full_name": "Continuous.sup", "start": [104, 1], "end": [106, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Support.lean", "full_name": "Function.support_smul", "start": [348, 1], "end": [350, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Closure.lean", "full_name": "LowerAdjoint.le_iff_subset", "start": [488, 1], "end": [489, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/Orthogonal.lean", "full_name": "Submodule.mem_orthogonal_singleton_iff_inner_right", "start": [79, 1], "end": [84, 29], "traced_tactics": [{"tactic": "refine' \u27e8inner_right_of_mem_orthogonal (mem_span_singleton_self u), _\u27e9", "state_before": "\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.25024\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nu v : E\n\u22a2 v \u2208 (span \ud835\udd5c {u})\u15ee \u2194 inner u v = 0", "state_after": "\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.25024\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nu v : E\n\u22a2 inner u v = 0 \u2192 v \u2208 (span \ud835\udd5c {u})\u15ee"}, {"tactic": "intro hv w hw", "state_before": "\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.25024\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nu v : E\n\u22a2 inner u v = 0 \u2192 v \u2208 (span \ud835\udd5c {u})\u15ee", "state_after": "\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.25024\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : inner u v = 0\nw : E\nhw : w \u2208 span \ud835\udd5c {u}\n\u22a2 inner w v = 0"}, {"tactic": "rw [mem_span_singleton] at hw", "state_before": "\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.25024\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : inner u v = 0\nw : E\nhw : w \u2208 span \ud835\udd5c {u}\n\u22a2 inner w v = 0", "state_after": "\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.25024\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : inner u v = 0\nw : E\nhw : \u2203 a, a \u2022 u = w\n\u22a2 inner w v = 0"}, {"tactic": "obtain \u27e8c, rfl\u27e9 := hw", "state_before": "\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.25024\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : inner u v = 0\nw : E\nhw : \u2203 a, a \u2022 u = w\n\u22a2 inner w v = 0", "state_after": "case intro\n\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.25024\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : inner u v = 0\nc : \ud835\udd5c\n\u22a2 inner (c \u2022 u) v = 0"}, {"tactic": "simp [inner_smul_left, hv]", "state_before": "case intro\n\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.25024\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : inner u v = 0\nc : \ud835\udd5c\n\u22a2 inner (c \u2022 u) v = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "full_name": "SimpleGraph.Walk.mem_tail_support_append_iff", "start": [595, 1], "end": [597, 44], "traced_tactics": [{"tactic": "rw [tail_support_append, List.mem_append]", "state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nt u v w : V\np : Walk G u v\np' : Walk G v w\n\u22a2 t \u2208 List.tail (support (append p p')) \u2194 t \u2208 List.tail (support p) \u2228 t \u2208 List.tail (support p')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Dfinsupp/Basic.lean", "full_name": "Dfinsupp.liftAddHom_comp_single", "start": [2040, 1], "end": [2041, 79], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ninst\u271d : AddCommMonoid \u03b3\nf : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\ni : \u03b9\n\u22a2 AddMonoidHom.comp (\u2191liftAddHom f) (singleAddHom \u03b2 i) = f i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "full_name": "UniqueFactorizationMonoid.zero_not_mem_normalizedFactors", "start": [745, 1], "end": [746, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Sums/Associator.lean", "full_name": "CategoryTheory.sum.inverseAssociator_obj_inr_inr", "start": [112, 1], "end": [113, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.disjoint_empty", "start": [655, 1], "end": [656, 7], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GroupWithZero/Divisibility.lean", "full_name": "dvd_antisymm", "start": [116, 1], "end": [119, 40], "traced_tactics": [{"tactic": "rintro \u27e8c, rfl\u27e9 \u27e8d, hcd\u27e9", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Subsingleton \u03b1\u02e3\na b : \u03b1\n\u22a2 a \u2223 b \u2192 b \u2223 a \u2192 a = b", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Subsingleton \u03b1\u02e3\na c d : \u03b1\nhcd : a = a * c * d\n\u22a2 a = a * c"}, {"tactic": "rw [mul_assoc, eq_comm, mul_right_eq_self\u2080, mul_eq_one] at hcd", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Subsingleton \u03b1\u02e3\na c d : \u03b1\nhcd : a = a * c * d\n\u22a2 a = a * c", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Subsingleton \u03b1\u02e3\na c d : \u03b1\nhcd : c = 1 \u2227 d = 1 \u2228 a = 0\n\u22a2 a = a * c"}, {"tactic": "obtain \u27e8rfl, -\u27e9 | rfl := hcd <;> simp", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : Subsingleton \u03b1\u02e3\na c d : \u03b1\nhcd : c = 1 \u2227 d = 1 \u2228 a = 0\n\u22a2 a = a * c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.smul_set_inter\u2080", "start": [1046, 1], "end": [1047, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Hom/Ring.lean", "full_name": "OrderRingHom.coe_copy", "start": [256, 1], "end": [257, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Indexes.lean", "full_name": "List.mapIdx_eq_enum_map", "start": [167, 1], "end": [173, 57], "traced_tactics": [{"tactic": "rw [List.new_def_eq_old_def]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nl : List \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\n\u22a2 mapIdx f l = map (uncurry f) (enum l)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nl : List \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\n\u22a2 List.oldMapIdx f l = map (uncurry f) (enum l)"}, {"tactic": "induction' l with hd tl hl generalizing f", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nl : List \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\n\u22a2 List.oldMapIdx f l = map (uncurry f) (enum l)", "state_after": "case nil\n\u03b1 : Type u\n\u03b2 : Type v\nf\u271d f : \u2115 \u2192 \u03b1 \u2192 \u03b2\n\u22a2 List.oldMapIdx f [] = map (uncurry f) (enum [])\n\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (f : \u2115 \u2192 \u03b1 \u2192 \u03b2), List.oldMapIdx f tl = map (uncurry f) (enum tl)\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\n\u22a2 List.oldMapIdx f (hd :: tl) = map (uncurry f) (enum (hd :: tl))"}, {"tactic": "rfl", "state_before": "case nil\n\u03b1 : Type u\n\u03b2 : Type v\nf\u271d f : \u2115 \u2192 \u03b1 \u2192 \u03b2\n\u22a2 List.oldMapIdx f [] = map (uncurry f) (enum [])", "state_after": "no goals"}, {"tactic": "rw [List.oldMapIdx, List.oldMapIdxCore, List.oldMapIdxCore_eq, hl]", "state_before": "case cons\n\u03b1 : Type u\n\u03b2 : Type v\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (f : \u2115 \u2192 \u03b1 \u2192 \u03b2), List.oldMapIdx f tl = map (uncurry f) (enum tl)\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\n\u22a2 List.oldMapIdx f (hd :: tl) = map (uncurry f) (enum (hd :: tl))", "state_after": "case cons\n\u03b1 : Type u\n\u03b2 : Type v\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (f : \u2115 \u2192 \u03b1 \u2192 \u03b2), List.oldMapIdx f tl = map (uncurry f) (enum tl)\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\n\u22a2 f 0 hd :: map (uncurry fun i a => f (i + (0 + 1)) a) (enum tl) = map (uncurry f) (enum (hd :: tl))"}, {"tactic": "simp [enum_eq_zip_range, map_uncurry_zip_eq_zipWith]", "state_before": "case cons\n\u03b1 : Type u\n\u03b2 : Type v\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (f : \u2115 \u2192 \u03b1 \u2192 \u03b2), List.oldMapIdx f tl = map (uncurry f) (enum tl)\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\n\u22a2 f 0 hd :: map (uncurry fun i a => f (i + (0 + 1)) a) (enum tl) = map (uncurry f) (enum (hd :: tl))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/NaturalOps.lean", "full_name": "NatOrdinal.induction", "start": [143, 1], "end": [144, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.eq_nil_of_sublist_nil", "start": [1077, 1], "end": [1078, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/ContinuousOn.lean", "full_name": "continuousWithinAt_inter'", "start": [706, 1], "end": [708, 55], "traced_tactics": [{"tactic": "simp [ContinuousWithinAt, nhdsWithin_restrict'' s h]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.321337\n\u03b4 : Type ?u.321340\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nx : \u03b1\nh : t \u2208 \ud835\udcdd[s] x\n\u22a2 ContinuousWithinAt f (s \u2229 t) x \u2194 ContinuousWithinAt f s x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Basic.lean", "full_name": "div_eq_self", "start": [833, 1], "end": [833, 97], "traced_tactics": [{"tactic": "rw [div_eq_mul_inv, mul_right_eq_self, inv_eq_one]", "state_before": "\u03b1 : Type ?u.61853\n\u03b2 : Type ?u.61856\nG : Type u_1\ninst\u271d : Group G\na b c d : G\n\u22a2 a / b = a \u2194 b = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Sort.lean", "full_name": "List.perm_orderedInsert", "start": [229, 1], "end": [234, 95], "traced_tactics": [{"tactic": "by_cases h : a \u227c b", "state_before": "\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl : List \u03b1\n\u22a2 orderedInsert r a (b :: l) ~ a :: b :: l", "state_after": "case pos\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl : List \u03b1\nh : r a b\n\u22a2 orderedInsert r a (b :: l) ~ a :: b :: l\n\ncase neg\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl : List \u03b1\nh : \u00acr a b\n\u22a2 orderedInsert r a (b :: l) ~ a :: b :: l"}, {"tactic": "simp [orderedInsert, h]", "state_before": "case pos\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl : List \u03b1\nh : r a b\n\u22a2 orderedInsert r a (b :: l) ~ a :: b :: l", "state_after": "no goals"}, {"tactic": "simpa [orderedInsert, h] using ((perm_orderedInsert a l).cons _).trans (Perm.swap _ _ _)", "state_before": "case neg\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\na b : \u03b1\nl : List \u03b1\nh : \u00acr a b\n\u22a2 orderedInsert r a (b :: l) ~ a :: b :: l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/Ackermann.lean", "full_name": "ack_succ_right_le_ack_succ_left", "start": [255, 1], "end": [260, 13], "traced_tactics": [{"tactic": "cases' n with n n", "state_before": "m n : \u2115\n\u22a2 ack m (n + 1) \u2264 ack (m + 1) n", "state_after": "case zero\nm : \u2115\n\u22a2 ack m (zero + 1) \u2264 ack (m + 1) zero\n\ncase succ\nm n : \u2115\n\u22a2 ack m (succ n + 1) \u2264 ack (m + 1) (succ n)"}, {"tactic": "simp", "state_before": "case zero\nm : \u2115\n\u22a2 ack m (zero + 1) \u2264 ack (m + 1) zero", "state_after": "no goals"}, {"tactic": "rw [ack_succ_succ, succ_eq_add_one]", "state_before": "case succ\nm n : \u2115\n\u22a2 ack m (succ n + 1) \u2264 ack (m + 1) (succ n)", "state_after": "case succ\nm n : \u2115\n\u22a2 ack m (n + 1 + 1) \u2264 ack m (ack (m + 1) n)"}, {"tactic": "apply ack_mono_right m (le_trans _ <| add_add_one_le_ack _ n)", "state_before": "case succ\nm n : \u2115\n\u22a2 ack m (n + 1 + 1) \u2264 ack m (ack (m + 1) n)", "state_after": "m n : \u2115\n\u22a2 n + 1 + 1 \u2264 m + 1 + n + 1"}, {"tactic": "linarith", "state_before": "m n : \u2115\n\u22a2 n + 1 + 1 \u2264 m + 1 + n + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Compare.lean", "full_name": "cmp_compares", "start": [170, 1], "end": [171, 78], "traced_tactics": [{"tactic": "obtain h | h | h := lt_trichotomy a b <;> simp [cmp, cmpUsing, h, h.not_lt]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.11521\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 Compares (cmp a b) a b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.inter_add_distrib", "start": [1859, 1], "end": [1872, 69], "traced_tactics": [{"tactic": "by_contra h", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.188231\n\u03b3 : Type ?u.188234\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u\u271d : Multiset \u03b1\na b : \u03b1\ns t u : Multiset \u03b1\n\u22a2 s \u2229 t + u = (s + u) \u2229 (t + u)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.188231\n\u03b3 : Type ?u.188234\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u\u271d : Multiset \u03b1\na b : \u03b1\ns t u : Multiset \u03b1\nh : \u00acs \u2229 t + u = (s + u) \u2229 (t + u)\n\u22a2 False"}, {"tactic": "cases'\n  lt_iff_cons_le.1\n    (lt_of_le_of_ne\n      (le_inter (add_le_add_right (inter_le_left s t) u)\n        (add_le_add_right (inter_le_right s t) u))\n      h) with\n  a hl", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.188231\n\u03b3 : Type ?u.188234\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u\u271d : Multiset \u03b1\na b : \u03b1\ns t u : Multiset \u03b1\nh : \u00acs \u2229 t + u = (s + u) \u2229 (t + u)\n\u22a2 False", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.188231\n\u03b3 : Type ?u.188234\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u\u271d : Multiset \u03b1\na\u271d b : \u03b1\ns t u : Multiset \u03b1\nh : \u00acs \u2229 t + u = (s + u) \u2229 (t + u)\na : \u03b1\nhl : a ::\u2098 (s \u2229 t + u) \u2264 (s + u) \u2229 (t + u)\n\u22a2 False"}, {"tactic": "rw [\u2190 cons_add] at hl", "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.188231\n\u03b3 : Type ?u.188234\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u\u271d : Multiset \u03b1\na\u271d b : \u03b1\ns t u : Multiset \u03b1\nh : \u00acs \u2229 t + u = (s + u) \u2229 (t + u)\na : \u03b1\nhl : a ::\u2098 (s \u2229 t + u) \u2264 (s + u) \u2229 (t + u)\n\u22a2 False", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.188231\n\u03b3 : Type ?u.188234\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u\u271d : Multiset \u03b1\na\u271d b : \u03b1\ns t u : Multiset \u03b1\nh : \u00acs \u2229 t + u = (s + u) \u2229 (t + u)\na : \u03b1\nhl : a ::\u2098 s \u2229 t + u \u2264 (s + u) \u2229 (t + u)\n\u22a2 False"}, {"tactic": "exact\n  not_le_of_lt (lt_cons_self (s \u2229 t) a)\n    (le_inter (le_of_add_le_add_right (le_trans hl (inter_le_left _ _)))\n      (le_of_add_le_add_right (le_trans hl (inter_le_right _ _))))", "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.188231\n\u03b3 : Type ?u.188234\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u\u271d : Multiset \u03b1\na\u271d b : \u03b1\ns t u : Multiset \u03b1\nh : \u00acs \u2229 t + u = (s + u) \u2229 (t + u)\na : \u03b1\nhl : a ::\u2098 s \u2229 t + u \u2264 (s + u) \u2229 (t + u)\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/ModelTheory/Satisfiability.lean", "full_name": "FirstOrder.Language.BoundedFormula.IsQF.induction_on_inf_not", "start": [592, 1], "end": [602, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/CharZero/Lemmas.lean", "full_name": "bit0_injective", "start": [123, 1], "end": [127, 19], "traced_tactics": [{"tactic": "dsimp [bit0] at h", "state_before": "R : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\na b : R\nh : bit0 a = bit0 b\n\u22a2 a = b", "state_after": "R : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\na b : R\nh : a + a = b + b\n\u22a2 a = b"}, {"tactic": "simp only [(two_mul a).symm, (two_mul b).symm] at h", "state_before": "R : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\na b : R\nh : a + a = b + b\n\u22a2 a = b", "state_after": "R : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\na b : R\nh : 2 * a = 2 * b\n\u22a2 a = b"}, {"tactic": "refine' nat_mul_inj' _ two_ne_zero", "state_before": "R : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\na b : R\nh : 2 * a = 2 * b\n\u22a2 a = b", "state_after": "R : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\na b : R\nh : 2 * a = 2 * b\n\u22a2 \u21912 * a = \u21912 * b"}, {"tactic": "exact_mod_cast h", "state_before": "R : Type u_1\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : CharZero R\na b : R\nh : 2 * a = 2 * b\n\u22a2 \u21912 * a = \u21912 * b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/MonoidAlgebra/ToDirectSum.lean", "full_name": "DirectSum.toAddMonoidAlgebra_toDirectSum", "start": [107, 1], "end": [109, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Size.lean", "full_name": "Nat.shiftl'_tt_eq_mul_pow", "start": [30, 1], "end": [36, 62], "traced_tactics": [{"tactic": "simp [shiftl, shiftl', pow_zero, Nat.one_mul]", "state_before": "m : \u2115\n\u22a2 shiftl' true m 0 + 1 = (m + 1) * 2 ^ 0", "state_after": "no goals"}, {"tactic": "change bit1 (shiftl' true m k) + 1 = (m + 1) * (2 ^ k * 2)", "state_before": "m k : \u2115\n\u22a2 shiftl' true m (k + 1) + 1 = (m + 1) * 2 ^ (k + 1)", "state_after": "m k : \u2115\n\u22a2 bit1 (shiftl' true m k) + 1 = (m + 1) * (2 ^ k * 2)"}, {"tactic": "rw [bit1_val]", "state_before": "m k : \u2115\n\u22a2 bit1 (shiftl' true m k) + 1 = (m + 1) * (2 ^ k * 2)", "state_after": "m k : \u2115\n\u22a2 2 * shiftl' true m k + 1 + 1 = (m + 1) * (2 ^ k * 2)"}, {"tactic": "change 2 * (shiftl' true m k + 1) = _", "state_before": "m k : \u2115\n\u22a2 2 * shiftl' true m k + 1 + 1 = (m + 1) * (2 ^ k * 2)", "state_after": "m k : \u2115\n\u22a2 2 * (shiftl' true m k + 1) = (m + 1) * (2 ^ k * 2)"}, {"tactic": "rw [shiftl'_tt_eq_mul_pow m k, mul_left_comm, mul_comm 2]", "state_before": "m k : \u2115\n\u22a2 2 * (shiftl' true m k + 1) = (m + 1) * (2 ^ k * 2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Nat/Gcd.lean", "full_name": "Nat.coprime.gcd_eq_one", "start": [242, 1], "end": [242, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Sets/Opens.lean", "full_name": "TopologicalSpace.Opens.isOpen", "start": [120, 11], "end": [121, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/ModEq.lean", "full_name": "Nat.odd_mod_four_iff", "start": [522, 1], "end": [526, 72], "traced_tactics": [{"tactic": "decide", "state_before": "m n\u271d a b c d n : \u2115\n\u22a2 \u2200 (m : \u2115), m < 4 \u2192 m % 2 = 1 \u2192 m = 1 \u2228 m = 3", "state_after": "no goals"}, {"tactic": "norm_num", "state_before": "m n\u271d a b c d n : \u2115\nhelp : \u2200 (m : \u2115), m < 4 \u2192 m % 2 = 1 \u2192 m = 1 \u2228 m = 3\nhn : n % 2 = 1\n\u22a2 4 > 0", "state_after": "no goals"}, {"tactic": "norm_num", "state_before": "m n\u271d a b c d n : \u2115\nhelp : \u2200 (m : \u2115), m < 4 \u2192 m % 2 = 1 \u2192 m = 1 \u2228 m = 3\nhn : n % 2 = 1\n\u22a2 2 \u2223 4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Manifold/LocalInvariantProperties.lean", "full_name": "StructureGroupoid.isLocalStructomorphWithinAt_localInvariantProp", "start": [565, 1], "end": [603, 64], "traced_tactics": [{"tactic": "intro s x u f hu hux", "state_before": "H : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\n\u22a2 \u2200 {s : Set H} {x : H} {u : Set H} {f : H \u2192 H},\n    IsOpen u \u2192 x \u2208 u \u2192 (IsLocalStructomorphWithinAt G f s x \u2194 IsLocalStructomorphWithinAt G f (s \u2229 u) x)", "state_after": "H : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\n\u22a2 IsLocalStructomorphWithinAt G f s x \u2194 IsLocalStructomorphWithinAt G f (s \u2229 u) x"}, {"tactic": "constructor", "state_before": "H : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\n\u22a2 IsLocalStructomorphWithinAt G f s x \u2194 IsLocalStructomorphWithinAt G f (s \u2229 u) x", "state_after": "case mp\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\n\u22a2 IsLocalStructomorphWithinAt G f s x \u2192 IsLocalStructomorphWithinAt G f (s \u2229 u) x\n\ncase mpr\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\n\u22a2 IsLocalStructomorphWithinAt G f (s \u2229 u) x \u2192 IsLocalStructomorphWithinAt G f s x"}, {"tactic": "rintro h hx", "state_before": "case mp\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\n\u22a2 IsLocalStructomorphWithinAt G f s x \u2192 IsLocalStructomorphWithinAt G f (s \u2229 u) x", "state_after": "case mp\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s \u2229 u\n\u22a2 \u2203 e, e \u2208 G \u2227 EqOn f (\u2191e.toLocalEquiv) (s \u2229 u \u2229 e.source) \u2227 x \u2208 e.source"}, {"tactic": "rcases h hx.1 with \u27e8e, heG, hef, hex\u27e9", "state_before": "case mp\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s \u2229 u\n\u22a2 \u2203 e, e \u2208 G \u2227 EqOn f (\u2191e.toLocalEquiv) (s \u2229 u \u2229 e.source) \u2227 x \u2208 e.source", "state_after": "case mp.intro.intro.intro\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s \u2229 u\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 \u2203 e, e \u2208 G \u2227 EqOn f (\u2191e.toLocalEquiv) (s \u2229 u \u2229 e.source) \u2227 x \u2208 e.source"}, {"tactic": "have : s \u2229 u \u2229 e.source \u2286 s \u2229 e.source := by mfld_set_tac", "state_before": "case mp.intro.intro.intro\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s \u2229 u\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 \u2203 e, e \u2208 G \u2227 EqOn f (\u2191e.toLocalEquiv) (s \u2229 u \u2229 e.source) \u2227 x \u2208 e.source", "state_after": "case mp.intro.intro.intro\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s \u2229 u\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\nthis : s \u2229 u \u2229 e.source \u2286 s \u2229 e.source\n\u22a2 \u2203 e, e \u2208 G \u2227 EqOn f (\u2191e.toLocalEquiv) (s \u2229 u \u2229 e.source) \u2227 x \u2208 e.source"}, {"tactic": "exact \u27e8e, heG, hef.mono this, hex\u27e9", "state_before": "case mp.intro.intro.intro\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s \u2229 u\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\nthis : s \u2229 u \u2229 e.source \u2286 s \u2229 e.source\n\u22a2 \u2203 e, e \u2208 G \u2227 EqOn f (\u2191e.toLocalEquiv) (s \u2229 u \u2229 e.source) \u2227 x \u2208 e.source", "state_after": "no goals"}, {"tactic": "mfld_set_tac", "state_before": "H : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s \u2229 u\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 s \u2229 u \u2229 e.source \u2286 s \u2229 e.source", "state_after": "no goals"}, {"tactic": "rintro h hx", "state_before": "case mpr\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\n\u22a2 IsLocalStructomorphWithinAt G f (s \u2229 u) x \u2192 IsLocalStructomorphWithinAt G f s x", "state_after": "case mpr\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\nh : IsLocalStructomorphWithinAt G f (s \u2229 u) x\nhx : x \u2208 s\n\u22a2 \u2203 e, e \u2208 G \u2227 EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source) \u2227 x \u2208 e.source"}, {"tactic": "rcases h \u27e8hx, hux\u27e9 with \u27e8e, heG, hef, hex\u27e9", "state_before": "case mpr\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\nh : IsLocalStructomorphWithinAt G f (s \u2229 u) x\nhx : x \u2208 s\n\u22a2 \u2203 e, e \u2208 G \u2227 EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source) \u2227 x \u2208 e.source", "state_after": "case mpr.intro.intro.intro\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\nh : IsLocalStructomorphWithinAt G f (s \u2229 u) x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 u \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 \u2203 e, e \u2208 G \u2227 EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source) \u2227 x \u2208 e.source"}, {"tactic": "refine' \u27e8e.restr (interior u), _, _, _\u27e9", "state_before": "case mpr.intro.intro.intro\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\nh : IsLocalStructomorphWithinAt G f (s \u2229 u) x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 u \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 \u2203 e, e \u2208 G \u2227 EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source) \u2227 x \u2208 e.source", "state_after": "case mpr.intro.intro.intro.refine'_1\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\nh : IsLocalStructomorphWithinAt G f (s \u2229 u) x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 u \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 LocalHomeomorph.restr e (interior u) \u2208 G\n\ncase mpr.intro.intro.intro.refine'_2\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\nh : IsLocalStructomorphWithinAt G f (s \u2229 u) x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 u \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 EqOn f (\u2191(LocalHomeomorph.restr e (interior u)).toLocalEquiv)\n    (s \u2229 (LocalHomeomorph.restr e (interior u)).toLocalEquiv.source)\n\ncase mpr.intro.intro.intro.refine'_3\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\nh : IsLocalStructomorphWithinAt G f (s \u2229 u) x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 u \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 x \u2208 (LocalHomeomorph.restr e (interior u)).toLocalEquiv.source"}, {"tactic": "exact closedUnderRestriction' heG isOpen_interior", "state_before": "case mpr.intro.intro.intro.refine'_1\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\nh : IsLocalStructomorphWithinAt G f (s \u2229 u) x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 u \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 LocalHomeomorph.restr e (interior u) \u2208 G", "state_after": "no goals"}, {"tactic": "have : s \u2229 u \u2229 e.source = s \u2229 (e.source \u2229 u) := by mfld_set_tac", "state_before": "case mpr.intro.intro.intro.refine'_2\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\nh : IsLocalStructomorphWithinAt G f (s \u2229 u) x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 u \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 EqOn f (\u2191(LocalHomeomorph.restr e (interior u)).toLocalEquiv)\n    (s \u2229 (LocalHomeomorph.restr e (interior u)).toLocalEquiv.source)", "state_after": "case mpr.intro.intro.intro.refine'_2\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\nh : IsLocalStructomorphWithinAt G f (s \u2229 u) x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 u \u2229 e.source)\nhex : x \u2208 e.source\nthis : s \u2229 u \u2229 e.source = s \u2229 (e.source \u2229 u)\n\u22a2 EqOn f (\u2191(LocalHomeomorph.restr e (interior u)).toLocalEquiv)\n    (s \u2229 (LocalHomeomorph.restr e (interior u)).toLocalEquiv.source)"}, {"tactic": "simpa only [this, interior_interior, hu.interior_eq, mfld_simps] using hef", "state_before": "case mpr.intro.intro.intro.refine'_2\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\nh : IsLocalStructomorphWithinAt G f (s \u2229 u) x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 u \u2229 e.source)\nhex : x \u2208 e.source\nthis : s \u2229 u \u2229 e.source = s \u2229 (e.source \u2229 u)\n\u22a2 EqOn f (\u2191(LocalHomeomorph.restr e (interior u)).toLocalEquiv)\n    (s \u2229 (LocalHomeomorph.restr e (interior u)).toLocalEquiv.source)", "state_after": "no goals"}, {"tactic": "mfld_set_tac", "state_before": "H : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\nh : IsLocalStructomorphWithinAt G f (s \u2229 u) x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 u \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 s \u2229 u \u2229 e.source = s \u2229 (e.source \u2229 u)", "state_after": "no goals"}, {"tactic": "simp only [*, interior_interior, hu.interior_eq, mfld_simps]", "state_before": "case mpr.intro.intro.intro.refine'_3\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nu : Set H\nf : H \u2192 H\nhu : IsOpen u\nhux : x \u2208 u\nh : IsLocalStructomorphWithinAt G f (s \u2229 u) x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 u \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 x \u2208 (LocalHomeomorph.restr e (interior u)).toLocalEquiv.source", "state_after": "no goals"}, {"tactic": "intro s x f e' he'G he'x h hx", "state_before": "H : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\n\u22a2 \u2200 {s : Set H} {x : H} {f : H \u2192 H} {e : LocalHomeomorph H H},\n    e \u2208 G \u2192\n      x \u2208 e.source \u2192\n        IsLocalStructomorphWithinAt G f s x \u2192\n          IsLocalStructomorphWithinAt G (f \u2218 \u2191(LocalHomeomorph.symm e)) (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) (\u2191e x)", "state_after": "H : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\nhe'x : x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : \u2191e' x \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s\n\u22a2 \u2203 e,\n    e \u2208 G \u2227\n      EqOn (f \u2218 \u2191(LocalHomeomorph.symm e')) (\u2191e.toLocalEquiv) (\u2191(LocalHomeomorph.symm e') \u207b\u00b9' s \u2229 e.source) \u2227\n        \u2191e' x \u2208 e.source"}, {"tactic": "have hxs : x \u2208 s := by simpa only [e'.left_inv he'x, mfld_simps] using hx", "state_before": "H : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\nhe'x : x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : \u2191e' x \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s\n\u22a2 \u2203 e,\n    e \u2208 G \u2227\n      EqOn (f \u2218 \u2191(LocalHomeomorph.symm e')) (\u2191e.toLocalEquiv) (\u2191(LocalHomeomorph.symm e') \u207b\u00b9' s \u2229 e.source) \u2227\n        \u2191e' x \u2208 e.source", "state_after": "H : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\nhe'x : x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : \u2191e' x \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s\nhxs : x \u2208 s\n\u22a2 \u2203 e,\n    e \u2208 G \u2227\n      EqOn (f \u2218 \u2191(LocalHomeomorph.symm e')) (\u2191e.toLocalEquiv) (\u2191(LocalHomeomorph.symm e') \u207b\u00b9' s \u2229 e.source) \u2227\n        \u2191e' x \u2208 e.source"}, {"tactic": "rcases h hxs with \u27e8e, heG, hef, hex\u27e9", "state_before": "H : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\nhe'x : x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : \u2191e' x \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s\nhxs : x \u2208 s\n\u22a2 \u2203 e,\n    e \u2208 G \u2227\n      EqOn (f \u2218 \u2191(LocalHomeomorph.symm e')) (\u2191e.toLocalEquiv) (\u2191(LocalHomeomorph.symm e') \u207b\u00b9' s \u2229 e.source) \u2227\n        \u2191e' x \u2208 e.source", "state_after": "case intro.intro.intro\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\nhe'x : x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : \u2191e' x \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s\nhxs : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 \u2203 e,\n    e \u2208 G \u2227\n      EqOn (f \u2218 \u2191(LocalHomeomorph.symm e')) (\u2191e.toLocalEquiv) (\u2191(LocalHomeomorph.symm e') \u207b\u00b9' s \u2229 e.source) \u2227\n        \u2191e' x \u2208 e.source"}, {"tactic": "refine' \u27e8e'.symm.trans e, G.trans (G.symm he'G) heG, _, _\u27e9", "state_before": "case intro.intro.intro\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\nhe'x : x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : \u2191e' x \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s\nhxs : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 \u2203 e,\n    e \u2208 G \u2227\n      EqOn (f \u2218 \u2191(LocalHomeomorph.symm e')) (\u2191e.toLocalEquiv) (\u2191(LocalHomeomorph.symm e') \u207b\u00b9' s \u2229 e.source) \u2227\n        \u2191e' x \u2208 e.source", "state_after": "case intro.intro.intro.refine'_1\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\nhe'x : x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : \u2191e' x \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s\nhxs : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 EqOn (f \u2218 \u2191(LocalHomeomorph.symm e')) (\u2191(LocalHomeomorph.symm e' \u226b\u2095 e).toLocalEquiv)\n    (\u2191(LocalHomeomorph.symm e') \u207b\u00b9' s \u2229 (LocalHomeomorph.symm e' \u226b\u2095 e).toLocalEquiv.source)\n\ncase intro.intro.intro.refine'_2\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\nhe'x : x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : \u2191e' x \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s\nhxs : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 \u2191e' x \u2208 (LocalHomeomorph.symm e' \u226b\u2095 e).toLocalEquiv.source"}, {"tactic": "simpa only [e'.left_inv he'x, mfld_simps] using hx", "state_before": "H : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\nhe'x : x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : \u2191e' x \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s\n\u22a2 x \u2208 s", "state_after": "no goals"}, {"tactic": "intro y hy", "state_before": "case intro.intro.intro.refine'_1\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\nhe'x : x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : \u2191e' x \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s\nhxs : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 EqOn (f \u2218 \u2191(LocalHomeomorph.symm e')) (\u2191(LocalHomeomorph.symm e' \u226b\u2095 e).toLocalEquiv)\n    (\u2191(LocalHomeomorph.symm e') \u207b\u00b9' s \u2229 (LocalHomeomorph.symm e' \u226b\u2095 e).toLocalEquiv.source)", "state_after": "case intro.intro.intro.refine'_1\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\nhe'x : x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : \u2191e' x \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s\nhxs : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\ny : H\nhy : y \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s \u2229 (LocalHomeomorph.symm e' \u226b\u2095 e).toLocalEquiv.source\n\u22a2 (f \u2218 \u2191(LocalHomeomorph.symm e')) y = \u2191(LocalHomeomorph.symm e' \u226b\u2095 e).toLocalEquiv y"}, {"tactic": "simp only [mfld_simps] at hy", "state_before": "case intro.intro.intro.refine'_1\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\nhe'x : x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : \u2191e' x \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s\nhxs : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\ny : H\nhy : y \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s \u2229 (LocalHomeomorph.symm e' \u226b\u2095 e).toLocalEquiv.source\n\u22a2 (f \u2218 \u2191(LocalHomeomorph.symm e')) y = \u2191(LocalHomeomorph.symm e' \u226b\u2095 e).toLocalEquiv y", "state_after": "case intro.intro.intro.refine'_1\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\nhe'x : x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : \u2191e' x \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s\nhxs : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\ny : H\nhy : \u2191(LocalHomeomorph.symm e') y \u2208 s \u2227 y \u2208 e'.target \u2227 \u2191(LocalHomeomorph.symm e') y \u2208 e.source\n\u22a2 (f \u2218 \u2191(LocalHomeomorph.symm e')) y = \u2191(LocalHomeomorph.symm e' \u226b\u2095 e).toLocalEquiv y"}, {"tactic": "simp only [hef \u27e8hy.1, hy.2.2\u27e9, mfld_simps]", "state_before": "case intro.intro.intro.refine'_1\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\nhe'x : x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : \u2191e' x \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s\nhxs : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\ny : H\nhy : \u2191(LocalHomeomorph.symm e') y \u2208 s \u2227 y \u2208 e'.target \u2227 \u2191(LocalHomeomorph.symm e') y \u2208 e.source\n\u22a2 (f \u2218 \u2191(LocalHomeomorph.symm e')) y = \u2191(LocalHomeomorph.symm e' \u226b\u2095 e).toLocalEquiv y", "state_after": "no goals"}, {"tactic": "simp only [hex, he'x, mfld_simps]", "state_before": "case intro.intro.intro.refine'_2\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\nhe'x : x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : \u2191e' x \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s\nhxs : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 \u2191e' x \u2208 (LocalHomeomorph.symm e' \u226b\u2095 e).toLocalEquiv.source", "state_after": "no goals"}, {"tactic": "intro s x f g hfgs _ h hx", "state_before": "H : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\n\u22a2 \u2200 {s : Set H} {x : H} {f g : H \u2192 H},\n    (\u2200 (y : H), y \u2208 s \u2192 f y = g y) \u2192\n      f x = g x \u2192 IsLocalStructomorphWithinAt G f s x \u2192 IsLocalStructomorphWithinAt G g s x", "state_after": "H : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf g : H \u2192 H\nhfgs : \u2200 (y : H), y \u2208 s \u2192 f y = g y\na\u271d : f x = g x\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s\n\u22a2 \u2203 e, e \u2208 G \u2227 EqOn g (\u2191e.toLocalEquiv) (s \u2229 e.source) \u2227 x \u2208 e.source"}, {"tactic": "rcases h hx with \u27e8e, heG, hef, hex\u27e9", "state_before": "H : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf g : H \u2192 H\nhfgs : \u2200 (y : H), y \u2208 s \u2192 f y = g y\na\u271d : f x = g x\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s\n\u22a2 \u2203 e, e \u2208 G \u2227 EqOn g (\u2191e.toLocalEquiv) (s \u2229 e.source) \u2227 x \u2208 e.source", "state_after": "case intro.intro.intro\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf g : H \u2192 H\nhfgs : \u2200 (y : H), y \u2208 s \u2192 f y = g y\na\u271d : f x = g x\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 \u2203 e, e \u2208 G \u2227 EqOn g (\u2191e.toLocalEquiv) (s \u2229 e.source) \u2227 x \u2208 e.source"}, {"tactic": "refine' \u27e8e, heG, _, hex\u27e9", "state_before": "case intro.intro.intro\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf g : H \u2192 H\nhfgs : \u2200 (y : H), y \u2208 s \u2192 f y = g y\na\u271d : f x = g x\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 \u2203 e, e \u2208 G \u2227 EqOn g (\u2191e.toLocalEquiv) (s \u2229 e.source) \u2227 x \u2208 e.source", "state_after": "case intro.intro.intro\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf g : H \u2192 H\nhfgs : \u2200 (y : H), y \u2208 s \u2192 f y = g y\na\u271d : f x = g x\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 EqOn g (\u2191e.toLocalEquiv) (s \u2229 e.source)"}, {"tactic": "intro y hy", "state_before": "case intro.intro.intro\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf g : H \u2192 H\nhfgs : \u2200 (y : H), y \u2208 s \u2192 f y = g y\na\u271d : f x = g x\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 EqOn g (\u2191e.toLocalEquiv) (s \u2229 e.source)", "state_after": "case intro.intro.intro\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf g : H \u2192 H\nhfgs : \u2200 (y : H), y \u2208 s \u2192 f y = g y\na\u271d : f x = g x\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\ny : H\nhy : y \u2208 s \u2229 e.source\n\u22a2 g y = \u2191e.toLocalEquiv y"}, {"tactic": "rw [\u2190 hef hy, hfgs y hy.1]", "state_before": "case intro.intro.intro\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e' : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf g : H \u2192 H\nhfgs : \u2200 (y : H), y \u2208 s \u2192 f y = g y\na\u271d : f x = g x\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\ny : H\nhy : y \u2208 s \u2229 e.source\n\u22a2 g y = \u2191e.toLocalEquiv y", "state_after": "no goals"}, {"tactic": "intro s x f e' he'G _ hfx h hx", "state_before": "H : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\n\u22a2 \u2200 {s : Set H} {x : H} {f : H \u2192 H} {e' : LocalHomeomorph H H},\n    e' \u2208 G \u2192\n      s \u2286 f \u207b\u00b9' e'.source \u2192\n        f x \u2208 e'.source \u2192 IsLocalStructomorphWithinAt G f s x \u2192 IsLocalStructomorphWithinAt G (\u2191e' \u2218 f) s x", "state_after": "H : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\na\u271d : s \u2286 f \u207b\u00b9' e'.source\nhfx : f x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s\n\u22a2 \u2203 e, e \u2208 G \u2227 EqOn (\u2191e' \u2218 f) (\u2191e.toLocalEquiv) (s \u2229 e.source) \u2227 x \u2208 e.source"}, {"tactic": "rcases h hx with \u27e8e, heG, hef, hex\u27e9", "state_before": "H : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\na\u271d : s \u2286 f \u207b\u00b9' e'.source\nhfx : f x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s\n\u22a2 \u2203 e, e \u2208 G \u2227 EqOn (\u2191e' \u2218 f) (\u2191e.toLocalEquiv) (s \u2229 e.source) \u2227 x \u2208 e.source", "state_after": "case intro.intro.intro\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\na\u271d : s \u2286 f \u207b\u00b9' e'.source\nhfx : f x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 \u2203 e, e \u2208 G \u2227 EqOn (\u2191e' \u2218 f) (\u2191e.toLocalEquiv) (s \u2229 e.source) \u2227 x \u2208 e.source"}, {"tactic": "refine' \u27e8e.trans e', G.trans heG he'G, _, _\u27e9", "state_before": "case intro.intro.intro\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\na\u271d : s \u2286 f \u207b\u00b9' e'.source\nhfx : f x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 \u2203 e, e \u2208 G \u2227 EqOn (\u2191e' \u2218 f) (\u2191e.toLocalEquiv) (s \u2229 e.source) \u2227 x \u2208 e.source", "state_after": "case intro.intro.intro.refine'_1\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\na\u271d : s \u2286 f \u207b\u00b9' e'.source\nhfx : f x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 EqOn (\u2191e' \u2218 f) (\u2191(e \u226b\u2095 e').toLocalEquiv) (s \u2229 (e \u226b\u2095 e').toLocalEquiv.source)\n\ncase intro.intro.intro.refine'_2\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\na\u271d : s \u2286 f \u207b\u00b9' e'.source\nhfx : f x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 x \u2208 (e \u226b\u2095 e').toLocalEquiv.source"}, {"tactic": "intro y hy", "state_before": "case intro.intro.intro.refine'_1\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\na\u271d : s \u2286 f \u207b\u00b9' e'.source\nhfx : f x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 EqOn (\u2191e' \u2218 f) (\u2191(e \u226b\u2095 e').toLocalEquiv) (s \u2229 (e \u226b\u2095 e').toLocalEquiv.source)", "state_after": "case intro.intro.intro.refine'_1\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\na\u271d : s \u2286 f \u207b\u00b9' e'.source\nhfx : f x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\ny : H\nhy : y \u2208 s \u2229 (e \u226b\u2095 e').toLocalEquiv.source\n\u22a2 (\u2191e' \u2218 f) y = \u2191(e \u226b\u2095 e').toLocalEquiv y"}, {"tactic": "simp only [mfld_simps] at hy", "state_before": "case intro.intro.intro.refine'_1\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\na\u271d : s \u2286 f \u207b\u00b9' e'.source\nhfx : f x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\ny : H\nhy : y \u2208 s \u2229 (e \u226b\u2095 e').toLocalEquiv.source\n\u22a2 (\u2191e' \u2218 f) y = \u2191(e \u226b\u2095 e').toLocalEquiv y", "state_after": "case intro.intro.intro.refine'_1\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\na\u271d : s \u2286 f \u207b\u00b9' e'.source\nhfx : f x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\ny : H\nhy : y \u2208 s \u2227 y \u2208 e.source \u2227 \u2191e y \u2208 e'.source\n\u22a2 (\u2191e' \u2218 f) y = \u2191(e \u226b\u2095 e').toLocalEquiv y"}, {"tactic": "simp only [hef \u27e8hy.1, hy.2.1\u27e9, mfld_simps]", "state_before": "case intro.intro.intro.refine'_1\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\na\u271d : s \u2286 f \u207b\u00b9' e'.source\nhfx : f x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\ny : H\nhy : y \u2208 s \u2227 y \u2208 e.source \u2227 \u2191e y \u2208 e'.source\n\u22a2 (\u2191e' \u2218 f) y = \u2191(e \u226b\u2095 e').toLocalEquiv y", "state_after": "no goals"}, {"tactic": "simpa only [hex, hef \u27e8hx, hex\u27e9, mfld_simps] using hfx", "state_before": "case intro.intro.intro.refine'_2\nH : Type u_1\nM : Type ?u.65722\nH' : Type ?u.65725\nM' : Type ?u.65728\nX : Type ?u.65731\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : ChartedSpace H M\ninst\u271d\u2074 : TopologicalSpace H'\ninst\u271d\u00b3 : TopologicalSpace M'\ninst\u271d\u00b2 : ChartedSpace H' M'\ninst\u271d\u00b9 : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne\u271d e'\u271d : LocalHomeomorph M H\nf\u271d f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns\u271d t : Set M\nx\u271d : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\ninst\u271d : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H \u2192 H\ne' : LocalHomeomorph H H\nhe'G : e' \u2208 G\na\u271d : s \u2286 f \u207b\u00b9' e'.source\nhfx : f x \u2208 e'.source\nh : IsLocalStructomorphWithinAt G f s x\nhx : x \u2208 s\ne : LocalHomeomorph H H\nheG : e \u2208 G\nhef : EqOn f (\u2191e.toLocalEquiv) (s \u2229 e.source)\nhex : x \u2208 e.source\n\u22a2 x \u2208 (e \u226b\u2095 e').toLocalEquiv.source", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean", "full_name": "PrimeSpectrum.primeSpectrumProd_symm_inr_asIdeal", "start": [122, 1], "end": [125, 6], "traced_tactics": [{"tactic": "cases x", "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nx : PrimeSpectrum S\n\u22a2 (\u2191(primeSpectrumProd R S).symm (Sum.inr x)).asIdeal = Ideal.prod \u22a4 x.asIdeal", "state_after": "case mk\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nasIdeal\u271d : Ideal S\nIsPrime\u271d : Ideal.IsPrime asIdeal\u271d\n\u22a2 (\u2191(primeSpectrumProd R S).symm (Sum.inr { asIdeal := asIdeal\u271d, IsPrime := IsPrime\u271d })).asIdeal =\n    Ideal.prod \u22a4 { asIdeal := asIdeal\u271d, IsPrime := IsPrime\u271d }.asIdeal"}, {"tactic": "rfl", "state_before": "case mk\nR : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nasIdeal\u271d : Ideal S\nIsPrime\u271d : Ideal.IsPrime asIdeal\u271d\n\u22a2 (\u2191(primeSpectrumProd R S).symm (Sum.inr { asIdeal := asIdeal\u271d, IsPrime := IsPrime\u271d })).asIdeal =\n    Ideal.prod \u22a4 { asIdeal := asIdeal\u271d, IsPrime := IsPrime\u271d }.asIdeal", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/LocallyConstant/Basic.lean", "full_name": "LocallyConstant.ext_iff", "start": [295, 1], "end": [295, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Prod.lean", "full_name": "Finset.filter_product_right", "start": [157, 1], "end": [159, 51], "traced_tactics": [{"tactic": "simpa using filter_product (fun _ : \u03b1 => true) q", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.45778\ns s' : Finset \u03b1\nt t' : Finset \u03b2\na : \u03b1\nb : \u03b2\nq : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred q\n\u22a2 filter (fun x => q x.snd) (s \u00d7\u02e2 t) = s \u00d7\u02e2 filter q t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean", "full_name": "CategoryTheory.Limits.pullbackDiagonalMapIdIso_inv_fst", "start": [257, 1], "end": [260, 7], "traced_tactics": [{"tactic": "rw [Iso.inv_comp_eq, \u2190 Category.comp_id pullback.fst, \u2190 diagonal_fst i, pullback.condition_assoc]", "state_before": "C : Type u_2\ninst\u271d\u2075 : Category C\nX Y Z : C\ninst\u271d\u2074 : HasPullbacks C\nS T : C\nf : X \u27f6 T\ng : Y \u27f6 T\ni : T \u27f6 S\ninst\u271d\u00b3 : HasPullback i i\ninst\u271d\u00b2 : HasPullback f g\ninst\u271d\u00b9 : HasPullback (f \u226b i) (g \u226b i)\ninst\u271d :\n  HasPullback (diagonal i) (map (f \u226b i) (g \u226b i) i i f g (\ud835\udfd9 S) (_ : (f \u226b i) \u226b \ud835\udfd9 S = f \u226b i) (_ : (g \u226b i) \u226b \ud835\udfd9 S = g \u226b i))\n\u22a2 (pullbackDiagonalMapIdIso f g i).inv \u226b fst = fst \u226b f", "state_after": "C : Type u_2\ninst\u271d\u2075 : Category C\nX Y Z : C\ninst\u271d\u2074 : HasPullbacks C\nS T : C\nf : X \u27f6 T\ng : Y \u27f6 T\ni : T \u27f6 S\ninst\u271d\u00b3 : HasPullback i i\ninst\u271d\u00b2 : HasPullback f g\ninst\u271d\u00b9 : HasPullback (f \u226b i) (g \u226b i)\ninst\u271d :\n  HasPullback (diagonal i) (map (f \u226b i) (g \u226b i) i i f g (\ud835\udfd9 S) (_ : (f \u226b i) \u226b \ud835\udfd9 S = f \u226b i) (_ : (g \u226b i) \u226b \ud835\udfd9 S = g \u226b i))\n\u22a2 snd \u226b map (f \u226b i) (g \u226b i) i i f g (\ud835\udfd9 S) (_ : (f \u226b i) \u226b \ud835\udfd9 S = f \u226b i) (_ : (g \u226b i) \u226b \ud835\udfd9 S = g \u226b i) \u226b fst =\n    (pullbackDiagonalMapIdIso f g i).hom \u226b fst \u226b f"}, {"tactic": "simp", "state_before": "C : Type u_2\ninst\u271d\u2075 : Category C\nX Y Z : C\ninst\u271d\u2074 : HasPullbacks C\nS T : C\nf : X \u27f6 T\ng : Y \u27f6 T\ni : T \u27f6 S\ninst\u271d\u00b3 : HasPullback i i\ninst\u271d\u00b2 : HasPullback f g\ninst\u271d\u00b9 : HasPullback (f \u226b i) (g \u226b i)\ninst\u271d :\n  HasPullback (diagonal i) (map (f \u226b i) (g \u226b i) i i f g (\ud835\udfd9 S) (_ : (f \u226b i) \u226b \ud835\udfd9 S = f \u226b i) (_ : (g \u226b i) \u226b \ud835\udfd9 S = g \u226b i))\n\u22a2 snd \u226b map (f \u226b i) (g \u226b i) i i f g (\ud835\udfd9 S) (_ : (f \u226b i) \u226b \ud835\udfd9 S = f \u226b i) (_ : (g \u226b i) \u226b \ud835\udfd9 S = g \u226b i) \u226b fst =\n    (pullbackDiagonalMapIdIso f g i).hom \u226b fst \u226b f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Deprecated/Subfield.lean", "full_name": "Image.isSubfield", "start": [71, 1], "end": [74, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/UnionFind.lean", "full_name": "UFModel.Agrees.push", "start": [92, 1], "end": [102, 13], "traced_tactics": [{"tactic": "cases H", "state_before": "\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nn : \u2115\nm : Fin n \u2192 \u03b2\nH : Agrees arr f m\nk : \u2115\nhk : k = n + 1\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < n), m' i = m { val := \u2191i, isLt := h }\nhm\u2082 : \u2200 (h : n < k), f x = m' { val := n, isLt := h }\n\u22a2 Agrees (Array.push arr x) f m'", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = Array.size arr + 1\nhm\u2082 : \u2200 (h : Array.size arr < k), f x = m' { val := Array.size arr, isLt := h }\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < Array.size arr), m' i = (fun i => f (Array.get arr i)) { val := \u2191i, isLt := h }\n\u22a2 Agrees (Array.push arr x) f m'"}, {"tactic": "have : k = (arr.push x).size := by simp [hk]", "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = Array.size arr + 1\nhm\u2082 : \u2200 (h : Array.size arr < k), f x = m' { val := Array.size arr, isLt := h }\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < Array.size arr), m' i = (fun i => f (Array.get arr i)) { val := \u2191i, isLt := h }\n\u22a2 Agrees (Array.push arr x) f m'", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = Array.size arr + 1\nhm\u2082 : \u2200 (h : Array.size arr < k), f x = m' { val := Array.size arr, isLt := h }\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < Array.size arr), m' i = (fun i => f (Array.get arr i)) { val := \u2191i, isLt := h }\nthis : k = Array.size (Array.push arr x)\n\u22a2 Agrees (Array.push arr x) f m'"}, {"tactic": "refine mk' this fun i h\u2081 h\u2082 \u21a6 ?_", "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = Array.size arr + 1\nhm\u2082 : \u2200 (h : Array.size arr < k), f x = m' { val := Array.size arr, isLt := h }\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < Array.size arr), m' i = (fun i => f (Array.get arr i)) { val := \u2191i, isLt := h }\nthis : k = Array.size (Array.push arr x)\n\u22a2 Agrees (Array.push arr x) f m'", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = Array.size arr + 1\nhm\u2082 : \u2200 (h : Array.size arr < k), f x = m' { val := Array.size arr, isLt := h }\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < Array.size arr), m' i = (fun i => f (Array.get arr i)) { val := \u2191i, isLt := h }\nthis : k = Array.size (Array.push arr x)\ni : \u2115\nh\u2081 : i < Array.size (Array.push arr x)\nh\u2082 : i < k\n\u22a2 f (Array.get (Array.push arr x) { val := i, isLt := h\u2081 }) = m' { val := i, isLt := h\u2082 }"}, {"tactic": "simp [Array.get_push]", "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = Array.size arr + 1\nhm\u2082 : \u2200 (h : Array.size arr < k), f x = m' { val := Array.size arr, isLt := h }\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < Array.size arr), m' i = (fun i => f (Array.get arr i)) { val := \u2191i, isLt := h }\nthis : k = Array.size (Array.push arr x)\ni : \u2115\nh\u2081 : i < Array.size (Array.push arr x)\nh\u2082 : i < k\n\u22a2 f (Array.get (Array.push arr x) { val := i, isLt := h\u2081 }) = m' { val := i, isLt := h\u2082 }", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = Array.size arr + 1\nhm\u2082 : \u2200 (h : Array.size arr < k), f x = m' { val := Array.size arr, isLt := h }\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < Array.size arr), m' i = (fun i => f (Array.get arr i)) { val := \u2191i, isLt := h }\nthis : k = Array.size (Array.push arr x)\ni : \u2115\nh\u2081 : i < Array.size (Array.push arr x)\nh\u2082 : i < k\n\u22a2 f (if h : i < Array.size arr then arr[i] else x) = m' { val := i, isLt := h\u2082 }"}, {"tactic": "split <;> (rename_i h; simp at hm\u2081 \u22a2)", "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = Array.size arr + 1\nhm\u2082 : \u2200 (h : Array.size arr < k), f x = m' { val := Array.size arr, isLt := h }\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < Array.size arr), m' i = (fun i => f (Array.get arr i)) { val := \u2191i, isLt := h }\nthis : k = Array.size (Array.push arr x)\ni : \u2115\nh\u2081 : i < Array.size (Array.push arr x)\nh\u2082 : i < k\n\u22a2 f (if h : i < Array.size arr then arr[i] else x) = m' { val := i, isLt := h\u2082 }", "state_after": "case mk.inl\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = Array.size arr + 1\nhm\u2082 : \u2200 (h : Array.size arr < k), f x = m' { val := Array.size arr, isLt := h }\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < Array.size arr), m' i = f arr[\u2191i]\nthis : k = Array.size (Array.push arr x)\ni : \u2115\nh\u2081 : i < Array.size (Array.push arr x)\nh\u2082 : i < k\nh : i < Array.size arr\n\u22a2 f arr[i] = m' { val := i, isLt := h\u2082 }\n\ncase mk.inr\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = Array.size arr + 1\nhm\u2082 : \u2200 (h : Array.size arr < k), f x = m' { val := Array.size arr, isLt := h }\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < Array.size arr), m' i = f arr[\u2191i]\nthis : k = Array.size (Array.push arr x)\ni : \u2115\nh\u2081 : i < Array.size (Array.push arr x)\nh\u2082 : i < k\nh : \u00aci < Array.size arr\n\u22a2 f x = m' { val := i, isLt := h\u2082 }"}, {"tactic": "simp [hk]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = Array.size arr + 1\nhm\u2082 : \u2200 (h : Array.size arr < k), f x = m' { val := Array.size arr, isLt := h }\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < Array.size arr), m' i = (fun i => f (Array.get arr i)) { val := \u2191i, isLt := h }\n\u22a2 k = Array.size (Array.push arr x)", "state_after": "no goals"}, {"tactic": "rename_i h", "state_before": "case mk.inr\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = Array.size arr + 1\nhm\u2082 : \u2200 (h : Array.size arr < k), f x = m' { val := Array.size arr, isLt := h }\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < Array.size arr), m' i = (fun i => f (Array.get arr i)) { val := \u2191i, isLt := h }\nthis : k = Array.size (Array.push arr x)\ni : \u2115\nh\u2081 : i < Array.size (Array.push arr x)\nh\u2082 : i < k\nh\u271d : \u00aci < Array.size arr\n\u22a2 f x = m' { val := i, isLt := h\u2082 }", "state_after": "case mk.inr\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = Array.size arr + 1\nhm\u2082 : \u2200 (h : Array.size arr < k), f x = m' { val := Array.size arr, isLt := h }\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < Array.size arr), m' i = (fun i => f (Array.get arr i)) { val := \u2191i, isLt := h }\nthis : k = Array.size (Array.push arr x)\ni : \u2115\nh\u2081 : i < Array.size (Array.push arr x)\nh\u2082 : i < k\nh : \u00aci < Array.size arr\n\u22a2 f x = m' { val := i, isLt := h\u2082 }"}, {"tactic": "simp at hm\u2081 \u22a2", "state_before": "case mk.inr\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = Array.size arr + 1\nhm\u2082 : \u2200 (h : Array.size arr < k), f x = m' { val := Array.size arr, isLt := h }\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < Array.size arr), m' i = (fun i => f (Array.get arr i)) { val := \u2191i, isLt := h }\nthis : k = Array.size (Array.push arr x)\ni : \u2115\nh\u2081 : i < Array.size (Array.push arr x)\nh\u2082 : i < k\nh : \u00aci < Array.size arr\n\u22a2 f x = m' { val := i, isLt := h\u2082 }", "state_after": "case mk.inr\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = Array.size arr + 1\nhm\u2082 : \u2200 (h : Array.size arr < k), f x = m' { val := Array.size arr, isLt := h }\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < Array.size arr), m' i = f arr[\u2191i]\nthis : k = Array.size (Array.push arr x)\ni : \u2115\nh\u2081 : i < Array.size (Array.push arr x)\nh\u2082 : i < k\nh : \u00aci < Array.size arr\n\u22a2 f x = m' { val := i, isLt := h\u2082 }"}, {"tactic": "rw [\u2190 hm\u2081 \u27e8i, h\u2082\u27e9]", "state_before": "case mk.inl\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = Array.size arr + 1\nhm\u2082 : \u2200 (h : Array.size arr < k), f x = m' { val := Array.size arr, isLt := h }\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < Array.size arr), m' i = f arr[\u2191i]\nthis : k = Array.size (Array.push arr x)\ni : \u2115\nh\u2081 : i < Array.size (Array.push arr x)\nh\u2082 : i < k\nh : i < Array.size arr\n\u22a2 f arr[i] = m' { val := i, isLt := h\u2082 }", "state_after": "case mk.inl\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = Array.size arr + 1\nhm\u2082 : \u2200 (h : Array.size arr < k), f x = m' { val := Array.size arr, isLt := h }\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < Array.size arr), m' i = f arr[\u2191i]\nthis : k = Array.size (Array.push arr x)\ni : \u2115\nh\u2081 : i < Array.size (Array.push arr x)\nh\u2082 : i < k\nh : i < Array.size arr\n\u22a2 \u2191{ val := i, isLt := h\u2082 } < Array.size arr"}, {"tactic": "assumption", "state_before": "case mk.inl\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = Array.size arr + 1\nhm\u2082 : \u2200 (h : Array.size arr < k), f x = m' { val := Array.size arr, isLt := h }\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < Array.size arr), m' i = f arr[\u2191i]\nthis : k = Array.size (Array.push arr x)\ni : \u2115\nh\u2081 : i < Array.size (Array.push arr x)\nh\u2082 : i < k\nh : i < Array.size arr\n\u22a2 \u2191{ val := i, isLt := h\u2082 } < Array.size arr", "state_after": "no goals"}, {"tactic": "cases show i = arr.size by apply le_antisymm <;> simp_all [Nat.lt_succ]", "state_before": "case mk.inr\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = Array.size arr + 1\nhm\u2082 : \u2200 (h : Array.size arr < k), f x = m' { val := Array.size arr, isLt := h }\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < Array.size arr), m' i = f arr[\u2191i]\nthis : k = Array.size (Array.push arr x)\ni : \u2115\nh\u2081 : i < Array.size (Array.push arr x)\nh\u2082 : i < k\nh : \u00aci < Array.size arr\n\u22a2 f x = m' { val := i, isLt := h\u2082 }", "state_after": "case mk.inr.refl\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = Array.size arr + 1\nhm\u2082 : \u2200 (h : Array.size arr < k), f x = m' { val := Array.size arr, isLt := h }\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < Array.size arr), m' i = f arr[\u2191i]\nthis : k = Array.size (Array.push arr x)\nh\u2081 : Array.size arr < Array.size (Array.push arr x)\nh\u2082 : Array.size arr < k\nh : \u00acArray.size arr < Array.size arr\n\u22a2 f x = m' { val := Array.size arr, isLt := h\u2082 }"}, {"tactic": "rw [hm\u2082]", "state_before": "case mk.inr.refl\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = Array.size arr + 1\nhm\u2082 : \u2200 (h : Array.size arr < k), f x = m' { val := Array.size arr, isLt := h }\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < Array.size arr), m' i = f arr[\u2191i]\nthis : k = Array.size (Array.push arr x)\nh\u2081 : Array.size arr < Array.size (Array.push arr x)\nh\u2082 : Array.size arr < k\nh : \u00acArray.size arr < Array.size arr\n\u22a2 f x = m' { val := Array.size arr, isLt := h\u2082 }", "state_after": "no goals"}, {"tactic": "apply le_antisymm <;> simp_all [Nat.lt_succ]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nk : \u2115\nx : \u03b1\nm' : Fin k \u2192 \u03b2\nhk : k = Array.size arr + 1\nhm\u2082 : \u2200 (h : Array.size arr < k), f x = m' { val := Array.size arr, isLt := h }\nhm\u2081 : \u2200 (i : Fin k) (h : \u2191i < Array.size arr), m' i = f arr[\u2191i]\nthis : k = Array.size (Array.push arr x)\ni : \u2115\nh\u2081 : i < Array.size (Array.push arr x)\nh\u2082 : i < k\nh : \u00aci < Array.size arr\n\u22a2 i = Array.size arr", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBNode.IsCut.gt_trans", "start": [126, 1], "end": [128, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "full_name": "AddLECancellable.tsub_tsub_cancel_of_le", "start": [190, 11], "end": [192, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean", "full_name": "AffineEquiv.constVAdd_zsmul", "start": [517, 1], "end": [518, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Opposite.lean", "full_name": "Units.coe_opEquiv_symm", "start": [457, 1], "end": [459, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "tendsto_comp_coe_Ioo_atBot", "start": [2538, 1], "end": [2540, 52], "traced_tactics": [{"tactic": "rw [\u2190 map_coe_Ioo_atBot h, tendsto_map'_iff]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : DenselyOrdered \u03b1\na b : \u03b1\ns : Set \u03b1\nl : Filter \u03b2\nf : \u03b1 \u2192 \u03b2\nh : a < b\n\u22a2 Tendsto (fun x => f \u2191x) atBot l \u2194 Tendsto f (\ud835\udcdd[Ioi a] a) l", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : DenselyOrdered \u03b1\na b : \u03b1\ns : Set \u03b1\nl : Filter \u03b2\nf : \u03b1 \u2192 \u03b2\nh : a < b\n\u22a2 Tendsto (fun x => f \u2191x) atBot l \u2194 Tendsto (f \u2218 Subtype.val) atBot l"}, {"tactic": "rfl", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : DenselyOrdered \u03b1\na b : \u03b1\ns : Set \u03b1\nl : Filter \u03b2\nf : \u03b1 \u2192 \u03b2\nh : a < b\n\u22a2 Tendsto (fun x => f \u2191x) atBot l \u2194 Tendsto (f \u2218 Subtype.val) atBot l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean", "full_name": "exists_extension_norm_eq", "start": [72, 1], "end": [105, 96], "traced_tactics": [{"tactic": "letI : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\n\u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\n\u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016"}, {"tactic": "letI : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower _ _ _", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\n\u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\n\u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016"}, {"tactic": "letI : NormedSpace \u211d F := NormedSpace.restrictScalars _ \ud835\udd5c _", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\n\u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\n\u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016"}, {"tactic": "let fr := reClm.comp (f.restrictScalars \u211d)", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\n\u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\n\u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016"}, {"tactic": "rcases Real.exists_extension_norm_eq (p.restrictScalars \u211d) fr with \u27e8g, \u27e8hextends, hnormeq\u27e9\u27e9", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\n\u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\n\u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016"}, {"tactic": "refine' \u27e8g.extendTo\ud835\udd5c, _\u27e9", "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\n\u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\n\u22a2 (\u2200 (x : { x // x \u2208 p }), \u2191(ContinuousLinearMap.extendTo\ud835\udd5c g) \u2191x = \u2191f x) \u2227 \u2016ContinuousLinearMap.extendTo\ud835\udd5c g\u2016 = \u2016f\u2016"}, {"tactic": "refine' \u27e8h, le_antisymm _ _\u27e9", "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nh : \u2200 (x : { x // x \u2208 p }), \u2191(ContinuousLinearMap.extendTo\ud835\udd5c g) \u2191x = \u2191f x\n\u22a2 (\u2200 (x : { x // x \u2208 p }), \u2191(ContinuousLinearMap.extendTo\ud835\udd5c g) \u2191x = \u2191f x) \u2227 \u2016ContinuousLinearMap.extendTo\ud835\udd5c g\u2016 = \u2016f\u2016", "state_after": "case intro.intro.refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nh : \u2200 (x : { x // x \u2208 p }), \u2191(ContinuousLinearMap.extendTo\ud835\udd5c g) \u2191x = \u2191f x\n\u22a2 \u2016ContinuousLinearMap.extendTo\ud835\udd5c g\u2016 \u2264 \u2016f\u2016\n\ncase intro.intro.refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nh : \u2200 (x : { x // x \u2208 p }), \u2191(ContinuousLinearMap.extendTo\ud835\udd5c g) \u2191x = \u2191f x\n\u22a2 \u2016f\u2016 \u2264 \u2016ContinuousLinearMap.extendTo\ud835\udd5c g\u2016"}, {"tactic": "intro x", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\n\u22a2 \u2200 (x : { x // x \u2208 p }), \u2191(ContinuousLinearMap.extendTo\ud835\udd5c g) \u2191x = \u2191f x", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : { x // x \u2208 p }\n\u22a2 \u2191(ContinuousLinearMap.extendTo\ud835\udd5c g) \u2191x = \u2191f x"}, {"tactic": "rw [ContinuousLinearMap.extendTo\ud835\udd5c_apply, \u2190 Submodule.coe_smul, hextends, hextends]", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : { x // x \u2208 p }\n\u22a2 \u2191(ContinuousLinearMap.extendTo\ud835\udd5c g) \u2191x = \u2191f x", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : { x // x \u2208 p }\n\u22a2 \u2191(\u2191fr x) - I * \u2191(\u2191fr (I \u2022 x)) = \u2191f x"}, {"tactic": "have : (fr x : \ud835\udd5c) - I * \u2191(fr (I \u2022 x)) = (re (f x) : \ud835\udd5c) - (I : \ud835\udd5c) * re (f ((I : \ud835\udd5c) \u2022 x)) := by\n  rfl", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : { x // x \u2208 p }\n\u22a2 \u2191(\u2191fr x) - I * \u2191(\u2191fr (I \u2022 x)) = \u2191f x", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b2 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d\u00b9 : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis\u271d : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : { x // x \u2208 p }\nthis : \u2191(\u2191fr x) - I * \u2191(\u2191fr (I \u2022 x)) = \u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x)))\n\u22a2 \u2191(\u2191fr x) - I * \u2191(\u2191fr (I \u2022 x)) = \u2191f x"}, {"tactic": "rw [this]", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b2 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d\u00b9 : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis\u271d : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : { x // x \u2208 p }\nthis : \u2191(\u2191fr x) - I * \u2191(\u2191fr (I \u2022 x)) = \u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x)))\n\u22a2 \u2191(\u2191fr x) - I * \u2191(\u2191fr (I \u2022 x)) = \u2191f x", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b2 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d\u00b9 : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis\u271d : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : { x // x \u2208 p }\nthis : \u2191(\u2191fr x) - I * \u2191(\u2191fr (I \u2022 x)) = \u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x)))\n\u22a2 \u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x))) = \u2191f x"}, {"tactic": "apply ext", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b2 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d\u00b9 : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis\u271d : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : { x // x \u2208 p }\nthis : \u2191(\u2191fr x) - I * \u2191(\u2191fr (I \u2022 x)) = \u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x)))\n\u22a2 \u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x))) = \u2191f x", "state_after": "case hre\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b2 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d\u00b9 : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis\u271d : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : { x // x \u2208 p }\nthis : \u2191(\u2191fr x) - I * \u2191(\u2191fr (I \u2022 x)) = \u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x)))\n\u22a2 \u2191re (\u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x)))) = \u2191re (\u2191f x)\n\ncase him\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b2 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d\u00b9 : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis\u271d : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : { x // x \u2208 p }\nthis : \u2191(\u2191fr x) - I * \u2191(\u2191fr (I \u2022 x)) = \u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x)))\n\u22a2 \u2191im (\u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x)))) = \u2191im (\u2191f x)"}, {"tactic": "rfl", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : { x // x \u2208 p }\n\u22a2 \u2191(\u2191fr x) - I * \u2191(\u2191fr (I \u2022 x)) = \u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x)))", "state_after": "no goals"}, {"tactic": "simp only [add_zero, Algebra.id.smul_eq_mul, I_re, ofReal_im, AddMonoidHom.map_add, zero_sub,\n  I_im', MulZeroClass.zero_mul, ofReal_re, eq_self_iff_true, sub_zero, mul_neg, ofReal_neg,\n  mul_re, MulZeroClass.mul_zero, sub_neg_eq_add, ContinuousLinearMap.map_smul]", "state_before": "case hre\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b2 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d\u00b9 : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis\u271d : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : { x // x \u2208 p }\nthis : \u2191(\u2191fr x) - I * \u2191(\u2191fr (I \u2022 x)) = \u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x)))\n\u22a2 \u2191re (\u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x)))) = \u2191re (\u2191f x)", "state_after": "no goals"}, {"tactic": "simp only [Algebra.id.smul_eq_mul, I_re, ofReal_im, AddMonoidHom.map_add, zero_sub, I_im',\n  MulZeroClass.zero_mul, ofReal_re, mul_neg, mul_im, zero_add, ofReal_neg, mul_re,\n  sub_neg_eq_add, ContinuousLinearMap.map_smul]", "state_before": "case him\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b2 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d\u00b9 : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis\u271d : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : { x // x \u2208 p }\nthis : \u2191(\u2191fr x) - I * \u2191(\u2191fr (I \u2022 x)) = \u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x)))\n\u22a2 \u2191im (\u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x)))) = \u2191im (\u2191f x)", "state_after": "no goals"}, {"tactic": "calc\n  \u2016g.extendTo\ud835\udd5c\u2016 = \u2016g\u2016 := g.norm_extendTo\ud835\udd5c\n  _ = \u2016fr\u2016 := hnormeq\n  _ \u2264 \u2016reClm\u2016 * \u2016f\u2016 := (ContinuousLinearMap.op_norm_comp_le _ _)\n  _ = \u2016f\u2016 := by rw [reClm_norm, one_mul]", "state_before": "case intro.intro.refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nh : \u2200 (x : { x // x \u2208 p }), \u2191(ContinuousLinearMap.extendTo\ud835\udd5c g) \u2191x = \u2191f x\n\u22a2 \u2016ContinuousLinearMap.extendTo\ud835\udd5c g\u2016 \u2264 \u2016f\u2016", "state_after": "no goals"}, {"tactic": "rw [reClm_norm, one_mul]", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nh : \u2200 (x : { x // x \u2208 p }), \u2191(ContinuousLinearMap.extendTo\ud835\udd5c g) \u2191x = \u2191f x\n\u22a2 \u2016reClm\u2016 * \u2016f\u2016 = \u2016f\u2016", "state_after": "no goals"}, {"tactic": "exact f.op_norm_le_bound g.extendTo\ud835\udd5c.op_norm_nonneg fun x => h x \u25b8 g.extendTo\ud835\udd5c.le_op_norm x", "state_before": "case intro.intro.refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c F\nf : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F\nthis\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F\nthis : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F\nfr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f)\ng : F \u2192L[\u211d] \u211d\nhextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nh : \u2200 (x : { x // x \u2208 p }), \u2191(ContinuousLinearMap.extendTo\ud835\udd5c g) \u2191x = \u2191f x\n\u22a2 \u2016f\u2016 \u2264 \u2016ContinuousLinearMap.extendTo\ud835\udd5c g\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/Separable.lean", "full_name": "Polynomial.separable_prod_X_sub_C_iff'", "start": [294, 1], "end": [303, 38], "traced_tactics": [{"tactic": "rw [\u2190 prod_attach]", "state_before": "F : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\n\u03b9 : Type u_1\nf : \u03b9 \u2192 F\ns : Finset \u03b9\nH : \u2200 (x : \u03b9), x \u2208 s \u2192 \u2200 (y : \u03b9), y \u2208 s \u2192 f x = f y \u2192 x = y\n\u22a2 Separable (\u220f i in s, (X - \u2191C (f i)))", "state_after": "F : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\n\u03b9 : Type u_1\nf : \u03b9 \u2192 F\ns : Finset \u03b9\nH : \u2200 (x : \u03b9), x \u2208 s \u2192 \u2200 (y : \u03b9), y \u2208 s \u2192 f x = f y \u2192 x = y\n\u22a2 Separable (\u220f x in attach s, (X - \u2191C (f \u2191x)))"}, {"tactic": "exact\n  separable_prod'\n    (fun x _hx y _hy hxy =>\n      @pairwise_coprime_X_sub_C _ _ { x // x \u2208 s } (fun x => f x)\n        (fun x y hxy => Subtype.eq <| H x.1 x.2 y.1 y.2 hxy) _ _ hxy)\n    fun _ _ => separable_X_sub_C", "state_before": "F : Type u\ninst\u271d\u00b9 : Field F\nK : Type v\ninst\u271d : Field K\n\u03b9 : Type u_1\nf : \u03b9 \u2192 F\ns : Finset \u03b9\nH : \u2200 (x : \u03b9), x \u2208 s \u2192 \u2200 (y : \u03b9), y \u2208 s \u2192 f x = f y \u2192 x = y\n\u22a2 Separable (\u220f x in attach s, (X - \u2191C (f \u2191x)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Order.lean", "full_name": "Finset.card_le_card_biUnion", "start": [369, 1], "end": [372, 49], "traced_tactics": [{"tactic": "rw [card_biUnion hs, card_eq_sum_ones]", "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type ?u.112113\nM : Type ?u.112116\nN : Type ?u.112119\nG : Type ?u.112122\nk : Type ?u.112125\nR : Type ?u.112128\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nB : Finset (Finset \u03b1)\nn : \u2115\ns : Finset \u03b9\nf : \u03b9 \u2192 Finset \u03b1\nhs : Set.PairwiseDisjoint (\u2191s) f\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Finset.Nonempty (f i)\n\u22a2 card s \u2264 card (Finset.biUnion s f)", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type ?u.112113\nM : Type ?u.112116\nN : Type ?u.112119\nG : Type ?u.112122\nk : Type ?u.112125\nR : Type ?u.112128\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nB : Finset (Finset \u03b1)\nn : \u2115\ns : Finset \u03b9\nf : \u03b9 \u2192 Finset \u03b1\nhs : Set.PairwiseDisjoint (\u2191s) f\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Finset.Nonempty (f i)\n\u22a2 \u2211 x in s, 1 \u2264 \u2211 u in s, card (f u)"}, {"tactic": "exact sum_le_sum fun i hi \u21a6 (hf i hi).card_pos", "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type ?u.112113\nM : Type ?u.112116\nN : Type ?u.112119\nG : Type ?u.112122\nk : Type ?u.112125\nR : Type ?u.112128\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nB : Finset (Finset \u03b1)\nn : \u2115\ns : Finset \u03b9\nf : \u03b9 \u2192 Finset \u03b1\nhs : Set.PairwiseDisjoint (\u2191s) f\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Finset.Nonempty (f i)\n\u22a2 \u2211 x in s, 1 \u2264 \u2211 u in s, card (f u)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "full_name": "ciSup_set_le_iff", "start": [522, 1], "end": [524, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/IsROrC/Basic.lean", "full_name": "IsROrC.ofReal_re", "start": [117, 1], "end": [118, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Synonym.lean", "full_name": "OrderDual.toDual_inj", "start": [82, 1], "end": [83, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/MulAction.lean", "full_name": "Continuous.smul", "start": [123, 1], "end": [124, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Powerset.lean", "full_name": "Multiset.powerset_zero", "start": [100, 1], "end": [101, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/Irrational.lean", "full_name": "irrational_int_div_iff", "start": [643, 1], "end": [644, 24], "traced_tactics": [{"tactic": "simp [div_eq_mul_inv]", "state_before": "q : \u211a\nm : \u2124\nn : \u2115\nx : \u211d\n\u22a2 Irrational (\u2191m / x) \u2194 m \u2260 0 \u2227 Irrational x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/Deriv/Inv.lean", "full_name": "differentiableAt_inv", "start": [82, 1], "end": [84, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.AEFinStronglyMeasurable.add", "start": [1880, 11], "end": [1883, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "full_name": "real_inner_sub_sub_self", "start": [683, 1], "end": [687, 7], "traced_tactics": [{"tactic": "have : \u27eay, x\u27eb_\u211d = \u27eax, y\u27eb_\u211d := by rw [\u2190 inner_conj_symm]; rfl", "state_before": "\ud835\udd5c : Type ?u.1810605\nE : Type ?u.1810608\nF : Type u_1\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 inner (x - y) (x - y) = inner x x - 2 * inner x y + inner y y", "state_after": "\ud835\udd5c : Type ?u.1810605\nE : Type ?u.1810608\nF : Type u_1\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nthis : inner y x = inner x y\n\u22a2 inner (x - y) (x - y) = inner x x - 2 * inner x y + inner y y"}, {"tactic": "simp only [inner_sub_sub_self, this, add_left_inj]", "state_before": "\ud835\udd5c : Type ?u.1810605\nE : Type ?u.1810608\nF : Type u_1\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nthis : inner y x = inner x y\n\u22a2 inner (x - y) (x - y) = inner x x - 2 * inner x y + inner y y", "state_after": "\ud835\udd5c : Type ?u.1810605\nE : Type ?u.1810608\nF : Type u_1\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nthis : inner y x = inner x y\n\u22a2 inner x x - inner x y - inner x y = inner x x - 2 * inner x y"}, {"tactic": "ring", "state_before": "\ud835\udd5c : Type ?u.1810605\nE : Type ?u.1810608\nF : Type u_1\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nthis : inner y x = inner x y\n\u22a2 inner x x - inner x y - inner x y = inner x x - 2 * inner x y", "state_after": "no goals"}, {"tactic": "rw [\u2190 inner_conj_symm]", "state_before": "\ud835\udd5c : Type ?u.1810605\nE : Type ?u.1810608\nF : Type u_1\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 inner y x = inner x y", "state_after": "\ud835\udd5c : Type ?u.1810605\nE : Type ?u.1810608\nF : Type u_1\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 \u2191(starRingEnd \u211d) (inner x y) = inner x y"}, {"tactic": "rfl", "state_before": "\ud835\udd5c : Type ?u.1810605\nE : Type ?u.1810608\nF : Type u_1\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 \u2191(starRingEnd \u211d) (inner x y) = inner x y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Basic.lean", "full_name": "Equiv.Perm.IsCycle.sameCycle", "start": [298, 11], "end": [303, 69], "traced_tactics": [{"tactic": "rw [\u2190 ha, \u2190 mul_apply, \u2190 zpow_add, sub_add_cancel, hb]", "state_before": "\u03b9 : Type ?u.309988\n\u03b1 : Type u_1\n\u03b2 : Type ?u.309994\nf g\u271d : Perm \u03b1\nx y : \u03b1\nhf : IsCycle f\nhx : \u2191f x \u2260 x\nhy : \u2191f y \u2260 y\ng : \u03b1\nhg : \u2191f g \u2260 g \u2227 \u2200 \u2983y : \u03b1\u2984, \u2191f y \u2260 y \u2192 SameCycle f g y\na : \u2124\nha : \u2191(f ^ a) g = x\nb : \u2124\nhb : \u2191(f ^ b) g = y\n\u22a2 \u2191(f ^ (b - a)) x = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Euclidean/Sphere/Power.lean", "full_name": "EuclideanGeometry.mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_pi", "start": [116, 1], "end": [121, 92], "traced_tactics": [{"tactic": "obtain \u27e8-, k\u2081, _, hab\u27e9 := angle_eq_pi_iff.mp hapb", "state_before": "V : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_1\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : \u2220 a p b = \u03c0\nhcpd : \u2220 c p d = \u03c0\n\u22a2 dist a p * dist b p = dist c p * dist d p", "state_after": "case intro.intro.intro\nV : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_1\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : \u2220 a p b = \u03c0\nhcpd : \u2220 c p d = \u03c0\nk\u2081 : \u211d\nleft\u271d : k\u2081 < 0\nhab : b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\n\u22a2 dist a p * dist b p = dist c p * dist d p"}, {"tactic": "obtain \u27e8-, k\u2082, _, hcd\u27e9 := angle_eq_pi_iff.mp hcpd", "state_before": "case intro.intro.intro\nV : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_1\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : \u2220 a p b = \u03c0\nhcpd : \u2220 c p d = \u03c0\nk\u2081 : \u211d\nleft\u271d : k\u2081 < 0\nhab : b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\n\u22a2 dist a p * dist b p = dist c p * dist d p", "state_after": "case intro.intro.intro.intro.intro.intro\nV : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_1\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : \u2220 a p b = \u03c0\nhcpd : \u2220 c p d = \u03c0\nk\u2081 : \u211d\nleft\u271d\u00b9 : k\u2081 < 0\nhab : b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\nk\u2082 : \u211d\nleft\u271d : k\u2082 < 0\nhcd : d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p)\n\u22a2 dist a p * dist b p = dist c p * dist d p"}, {"tactic": "exact mul_dist_eq_mul_dist_of_cospherical h \u27e8k\u2081, by linarith, hab\u27e9 \u27e8k\u2082, by linarith, hcd\u27e9", "state_before": "case intro.intro.intro.intro.intro.intro\nV : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_1\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : \u2220 a p b = \u03c0\nhcpd : \u2220 c p d = \u03c0\nk\u2081 : \u211d\nleft\u271d\u00b9 : k\u2081 < 0\nhab : b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\nk\u2082 : \u211d\nleft\u271d : k\u2082 < 0\nhcd : d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p)\n\u22a2 dist a p * dist b p = dist c p * dist d p", "state_after": "no goals"}, {"tactic": "linarith", "state_before": "V : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_1\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : \u2220 a p b = \u03c0\nhcpd : \u2220 c p d = \u03c0\nk\u2081 : \u211d\nleft\u271d\u00b9 : k\u2081 < 0\nhab : b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\nk\u2082 : \u211d\nleft\u271d : k\u2082 < 0\nhcd : d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p)\n\u22a2 k\u2081 \u2260 1", "state_after": "no goals"}, {"tactic": "linarith", "state_before": "V : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\nP : Type u_1\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhapb : \u2220 a p b = \u03c0\nhcpd : \u2220 c p d = \u03c0\nk\u2081 : \u211d\nleft\u271d\u00b9 : k\u2081 < 0\nhab : b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p)\nk\u2082 : \u211d\nleft\u271d : k\u2082 < 0\nhcd : d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p)\n\u22a2 k\u2082 \u2260 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Normed/Order/UpperLower.lean", "full_name": "IsLowerSet.exists_subset_ball", "start": [139, 1], "end": [154, 11], "traced_tactics": [{"tactic": "refine' \u27e8x - const _ (3 / 4 * \u03b4), closedBall_subset_closedBall' _, _\u27e9", "state_before": "\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\n\u22a2 \u2203 y, closedBall y (\u03b4 / 4) \u2286 closedBall x \u03b4 \u2227 closedBall y (\u03b4 / 4) \u2286 interior s", "state_after": "case refine'_1\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\n\u22a2 \u03b4 / 4 + dist (x - const \u03b9 (3 / 4 * \u03b4)) x \u2264 \u03b4\n\ncase refine'_2\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\n\u22a2 closedBall (x - const \u03b9 (3 / 4 * \u03b4)) (\u03b4 / 4) \u2286 interior s"}, {"tactic": "obtain \u27e8y, hy, hxy\u27e9 := Metric.mem_closure_iff.1 hx _ (div_pos h\u03b4 zero_lt_four)", "state_before": "case refine'_2\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\n\u22a2 closedBall (x - const \u03b9 (3 / 4 * \u03b4)) (\u03b4 / 4) \u2286 interior s", "state_after": "case refine'_2.intro.intro\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y\u271d : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\ny : \u03b9 \u2192 \u211d\nhy : y \u2208 s\nhxy : dist x y < \u03b4 / 4\n\u22a2 closedBall (x - const \u03b9 (3 / 4 * \u03b4)) (\u03b4 / 4) \u2286 interior s"}, {"tactic": "refine' fun z hz => hs.mem_interior_of_forall_lt (subset_closure hy) fun i => _", "state_before": "case refine'_2.intro.intro\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y\u271d : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\ny : \u03b9 \u2192 \u211d\nhy : y \u2208 s\nhxy : dist x y < \u03b4 / 4\n\u22a2 closedBall (x - const \u03b9 (3 / 4 * \u03b4)) (\u03b4 / 4) \u2286 interior s", "state_after": "case refine'_2.intro.intro\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y\u271d : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\ny : \u03b9 \u2192 \u211d\nhy : y \u2208 s\nhxy : dist x y < \u03b4 / 4\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 closedBall (x - const \u03b9 (3 / 4 * \u03b4)) (\u03b4 / 4)\ni : \u03b9\n\u22a2 z i < y i"}, {"tactic": "rw [mem_closedBall, dist_eq_norm'] at hz", "state_before": "case refine'_2.intro.intro\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y\u271d : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\ny : \u03b9 \u2192 \u211d\nhy : y \u2208 s\nhxy : dist x y < \u03b4 / 4\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 closedBall (x - const \u03b9 (3 / 4 * \u03b4)) (\u03b4 / 4)\ni : \u03b9\n\u22a2 z i < y i", "state_after": "case refine'_2.intro.intro\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y\u271d : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\ny : \u03b9 \u2192 \u211d\nhy : y \u2208 s\nhxy : dist x y < \u03b4 / 4\nz : \u03b9 \u2192 \u211d\nhz : \u2016x - const \u03b9 (3 / 4 * \u03b4) - z\u2016 \u2264 \u03b4 / 4\ni : \u03b9\n\u22a2 z i < y i"}, {"tactic": "rw [dist_eq_norm] at hxy", "state_before": "case refine'_2.intro.intro\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y\u271d : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\ny : \u03b9 \u2192 \u211d\nhy : y \u2208 s\nhxy : dist x y < \u03b4 / 4\nz : \u03b9 \u2192 \u211d\nhz : \u2016x - const \u03b9 (3 / 4 * \u03b4) - z\u2016 \u2264 \u03b4 / 4\ni : \u03b9\n\u22a2 z i < y i", "state_after": "case refine'_2.intro.intro\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y\u271d : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\ny : \u03b9 \u2192 \u211d\nhy : y \u2208 s\nhxy : \u2016x - y\u2016 < \u03b4 / 4\nz : \u03b9 \u2192 \u211d\nhz : \u2016x - const \u03b9 (3 / 4 * \u03b4) - z\u2016 \u2264 \u03b4 / 4\ni : \u03b9\n\u22a2 z i < y i"}, {"tactic": "replace hxy := (norm_le_pi_norm _ i).trans hxy.le", "state_before": "case refine'_2.intro.intro\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y\u271d : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\ny : \u03b9 \u2192 \u211d\nhy : y \u2208 s\nhxy : \u2016x - y\u2016 < \u03b4 / 4\nz : \u03b9 \u2192 \u211d\nhz : \u2016x - const \u03b9 (3 / 4 * \u03b4) - z\u2016 \u2264 \u03b4 / 4\ni : \u03b9\n\u22a2 z i < y i", "state_after": "case refine'_2.intro.intro\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y\u271d : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\ny : \u03b9 \u2192 \u211d\nhy : y \u2208 s\nz : \u03b9 \u2192 \u211d\nhz : \u2016x - const \u03b9 (3 / 4 * \u03b4) - z\u2016 \u2264 \u03b4 / 4\ni : \u03b9\nhxy : \u2016(x - y) i\u2016 \u2264 \u03b4 / 4\n\u22a2 z i < y i"}, {"tactic": "replace hz := (norm_le_pi_norm _ i).trans hz", "state_before": "case refine'_2.intro.intro\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y\u271d : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\ny : \u03b9 \u2192 \u211d\nhy : y \u2208 s\nz : \u03b9 \u2192 \u211d\nhz : \u2016x - const \u03b9 (3 / 4 * \u03b4) - z\u2016 \u2264 \u03b4 / 4\ni : \u03b9\nhxy : \u2016(x - y) i\u2016 \u2264 \u03b4 / 4\n\u22a2 z i < y i", "state_after": "case refine'_2.intro.intro\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y\u271d : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\ny : \u03b9 \u2192 \u211d\nhy : y \u2208 s\nz : \u03b9 \u2192 \u211d\ni : \u03b9\nhxy : \u2016(x - y) i\u2016 \u2264 \u03b4 / 4\nhz : \u2016(x - const \u03b9 (3 / 4 * \u03b4) - z) i\u2016 \u2264 \u03b4 / 4\n\u22a2 z i < y i"}, {"tactic": "dsimp at hxy hz", "state_before": "case refine'_2.intro.intro\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y\u271d : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\ny : \u03b9 \u2192 \u211d\nhy : y \u2208 s\nz : \u03b9 \u2192 \u211d\ni : \u03b9\nhxy : \u2016(x - y) i\u2016 \u2264 \u03b4 / 4\nhz : \u2016(x - const \u03b9 (3 / 4 * \u03b4) - z) i\u2016 \u2264 \u03b4 / 4\n\u22a2 z i < y i", "state_after": "case refine'_2.intro.intro\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y\u271d : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\ny : \u03b9 \u2192 \u211d\nhy : y \u2208 s\nz : \u03b9 \u2192 \u211d\ni : \u03b9\nhxy : abs (x i - y i) \u2264 \u03b4 / 4\nhz : abs (x i - 3 / 4 * \u03b4 - z i) \u2264 \u03b4 / 4\n\u22a2 z i < y i"}, {"tactic": "rw [abs_sub_le_iff] at hxy hz", "state_before": "case refine'_2.intro.intro\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y\u271d : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\ny : \u03b9 \u2192 \u211d\nhy : y \u2208 s\nz : \u03b9 \u2192 \u211d\ni : \u03b9\nhxy : abs (x i - y i) \u2264 \u03b4 / 4\nhz : abs (x i - 3 / 4 * \u03b4 - z i) \u2264 \u03b4 / 4\n\u22a2 z i < y i", "state_after": "case refine'_2.intro.intro\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y\u271d : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\ny : \u03b9 \u2192 \u211d\nhy : y \u2208 s\nz : \u03b9 \u2192 \u211d\ni : \u03b9\nhxy : x i - y i \u2264 \u03b4 / 4 \u2227 y i - x i \u2264 \u03b4 / 4\nhz : x i - 3 / 4 * \u03b4 - z i \u2264 \u03b4 / 4 \u2227 z i - (x i - 3 / 4 * \u03b4) \u2264 \u03b4 / 4\n\u22a2 z i < y i"}, {"tactic": "linarith", "state_before": "case refine'_2.intro.intro\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y\u271d : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\ny : \u03b9 \u2192 \u211d\nhy : y \u2208 s\nz : \u03b9 \u2192 \u211d\ni : \u03b9\nhxy : x i - y i \u2264 \u03b4 / 4 \u2227 y i - x i \u2264 \u03b4 / 4\nhz : x i - 3 / 4 * \u03b4 - z i \u2264 \u03b4 / 4 \u2227 z i - (x i - 3 / 4 * \u03b4) \u2264 \u03b4 / 4\n\u22a2 z i < y i", "state_after": "no goals"}, {"tactic": "rw [dist_self_sub_left]", "state_before": "case refine'_1\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\n\u22a2 \u03b4 / 4 + dist (x - const \u03b9 (3 / 4 * \u03b4)) x \u2264 \u03b4", "state_after": "case refine'_1\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\n\u22a2 \u03b4 / 4 + \u2016const \u03b9 (3 / 4 * \u03b4)\u2016 \u2264 \u03b4"}, {"tactic": "refine' (add_le_add_left (pi_norm_const_le <| 3 / 4 * \u03b4) _).trans_eq _", "state_before": "case refine'_1\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\n\u22a2 \u03b4 / 4 + \u2016const \u03b9 (3 / 4 * \u03b4)\u2016 \u2264 \u03b4", "state_after": "case refine'_1\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\n\u22a2 \u03b4 / 4 + \u20163 / 4 * \u03b4\u2016 = \u03b4"}, {"tactic": "simp [abs_of_pos, abs_of_pos h\u03b4]", "state_before": "case refine'_1\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\n\u03b4 : \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh\u03b4 : 0 < \u03b4\n\u22a2 \u03b4 / 4 + \u20163 / 4 * \u03b4\u2016 = \u03b4", "state_after": "case refine'_1\n\u03b1 : Type ?u.18237\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set 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\u03b2\ninst\u271d\u00b3 : MulAction H \u03b1\ninst\u271d\u00b2 : SMul H \u03b2\ninst\u271d\u00b9 : SMulCommClass G H \u03b2\ninst\u271d : SMulCommClass G H \u03b1\ns t : Set \u03b1\nb : \u03b2\ng : H\nh : Supports G s b\ng' : G\nhg' : \u2200 \u2983a : \u03b1\u2984, a \u2208 g \u2022 s \u2192 g' \u2022 a = a\n\u22a2 g' \u2022 g \u2022 b = g \u2022 b"}, {"tactic": "rw [smul_comm, h]", "state_before": "G : Type u_1\nH : Type u_4\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2076 : Group H\ninst\u271d\u2075 : SMul G \u03b1\ninst\u271d\u2074 : SMul G \u03b2\ninst\u271d\u00b3 : MulAction H \u03b1\ninst\u271d\u00b2 : SMul H \u03b2\ninst\u271d\u00b9 : SMulCommClass G H \u03b2\ninst\u271d : SMulCommClass G H \u03b1\ns t : Set \u03b1\nb : \u03b2\ng : H\nh : Supports G s b\ng' : G\nhg' : \u2200 \u2983a : \u03b1\u2984, a \u2208 g \u2022 s \u2192 g' \u2022 a = a\n\u22a2 g' \u2022 g \u2022 b = g \u2022 b", "state_after": "case a\nG : Type u_1\nH : Type u_4\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2076 : Group H\ninst\u271d\u2075 : SMul G \u03b1\ninst\u271d\u2074 : SMul G \u03b2\ninst\u271d\u00b3 : MulAction H \u03b1\ninst\u271d\u00b2 : SMul H \u03b2\ninst\u271d\u00b9 : SMulCommClass G H \u03b2\ninst\u271d : SMulCommClass G H \u03b1\ns t : Set \u03b1\nb : \u03b2\ng : H\nh : Supports G s b\ng' : G\nhg' : \u2200 \u2983a : \u03b1\u2984, a \u2208 g \u2022 s \u2192 g' \u2022 a = a\n\u22a2 \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2192 g' \u2022 a = a"}, {"tactic": "rintro a ha", "state_before": "case a\nG : Type u_1\nH : Type u_4\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2076 : Group H\ninst\u271d\u2075 : SMul G \u03b1\ninst\u271d\u2074 : SMul G \u03b2\ninst\u271d\u00b3 : MulAction H \u03b1\ninst\u271d\u00b2 : SMul H \u03b2\ninst\u271d\u00b9 : SMulCommClass G H \u03b2\ninst\u271d : SMulCommClass G H \u03b1\ns t : Set \u03b1\nb : \u03b2\ng : H\nh : Supports G s b\ng' : G\nhg' : \u2200 \u2983a : \u03b1\u2984, a \u2208 g \u2022 s \u2192 g' \u2022 a = a\n\u22a2 \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2192 g' \u2022 a = a", "state_after": "case a\nG : Type u_1\nH : Type u_4\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2076 : Group H\ninst\u271d\u2075 : SMul G \u03b1\ninst\u271d\u2074 : SMul G \u03b2\ninst\u271d\u00b3 : MulAction H \u03b1\ninst\u271d\u00b2 : SMul H \u03b2\ninst\u271d\u00b9 : SMulCommClass G H \u03b2\ninst\u271d : SMulCommClass G H \u03b1\ns t : Set \u03b1\nb : \u03b2\ng : H\nh : Supports G s b\ng' : G\nhg' : \u2200 \u2983a : \u03b1\u2984, a \u2208 g \u2022 s \u2192 g' \u2022 a = a\na : \u03b1\nha : a \u2208 s\n\u22a2 g' \u2022 a = a"}, {"tactic": "have := Set.ball_image_iff.1 hg' a ha", "state_before": "case a\nG : Type u_1\nH : Type u_4\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2076 : Group H\ninst\u271d\u2075 : SMul G \u03b1\ninst\u271d\u2074 : SMul G \u03b2\ninst\u271d\u00b3 : MulAction H \u03b1\ninst\u271d\u00b2 : SMul H \u03b2\ninst\u271d\u00b9 : SMulCommClass G H \u03b2\ninst\u271d : SMulCommClass G H \u03b1\ns t : Set \u03b1\nb : \u03b2\ng : H\nh : Supports G s b\ng' : G\nhg' : \u2200 \u2983a : \u03b1\u2984, a \u2208 g \u2022 s \u2192 g' \u2022 a = a\na : \u03b1\nha : a \u2208 s\n\u22a2 g' \u2022 a = a", "state_after": "case a\nG : Type u_1\nH : Type u_4\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2076 : Group H\ninst\u271d\u2075 : SMul G \u03b1\ninst\u271d\u2074 : SMul G \u03b2\ninst\u271d\u00b3 : MulAction H \u03b1\ninst\u271d\u00b2 : SMul H \u03b2\ninst\u271d\u00b9 : SMulCommClass G H \u03b2\ninst\u271d : SMulCommClass G H \u03b1\ns t : Set \u03b1\nb : \u03b2\ng : H\nh : Supports G s b\ng' : G\nhg' : \u2200 \u2983a : \u03b1\u2984, a \u2208 g \u2022 s \u2192 g' \u2022 a = a\na : \u03b1\nha : a \u2208 s\nthis : g' \u2022 g \u2022 a = g \u2022 a\n\u22a2 g' \u2022 a = a"}, {"tactic": "rwa [smul_comm, smul_left_cancel_iff] at this", "state_before": "case a\nG : Type u_1\nH : Type u_4\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2076 : Group H\ninst\u271d\u2075 : SMul G \u03b1\ninst\u271d\u2074 : SMul G \u03b2\ninst\u271d\u00b3 : MulAction H \u03b1\ninst\u271d\u00b2 : SMul H \u03b2\ninst\u271d\u00b9 : SMulCommClass G H \u03b2\ninst\u271d : SMulCommClass G H \u03b1\ns t : Set \u03b1\nb : \u03b2\ng : H\nh : Supports G s b\ng' : G\nhg' : \u2200 \u2983a : \u03b1\u2984, a \u2208 g \u2022 s \u2192 g' \u2022 a = a\na : \u03b1\nha : a \u2208 s\nthis : g' \u2022 g \u2022 a = g \u2022 a\n\u22a2 g' \u2022 a = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/MeasurableSpace.lean", "full_name": "Measurable.of_uncurry_left", "start": [694, 1], "end": [696, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Deprecated/Subgroup.lean", "full_name": "IsGroupHom.injective_iff_trivial_ker", "start": [472, 1], "end": [474, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Order/Floor.lean", "full_name": "ContinuousOn.comp_fract''", "start": [224, 1], "end": [227, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/ToIntervalMod.lean", "full_name": "toIcoDiv_add_left'", "start": [324, 1], "end": [325, 37], "traced_tactics": [{"tactic": "rw [add_comm, toIcoDiv_add_right']", "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nh\u03b1 : Archimedean \u03b1\np : \u03b1\nhp : 0 < p\na\u271d b\u271d c : \u03b1\nn : \u2124\na b : \u03b1\n\u22a2 toIcoDiv hp (p + a) b = toIcoDiv hp a b - 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Dfinsupp/Basic.lean", "full_name": "Dfinsupp.piecewise_single_erase", "start": [759, 1], "end": [764, 21], "traced_tactics": [{"tactic": "ext j", "state_before": "\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\nx : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\ninst\u271d : (i' : \u03b9) \u2192 Decidable (i' \u2208 {i})\n\u22a2 piecewise (single i (\u2191x i)) (erase i x) {i} = x", "state_after": "case h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\nx : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\ninst\u271d : (i' : \u03b9) \u2192 Decidable (i' \u2208 {i})\nj : \u03b9\n\u22a2 \u2191(piecewise (single i (\u2191x i)) (erase i x) {i}) j = \u2191x j"}, {"tactic": "rw [piecewise_apply]", "state_before": "case h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\nx : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\ninst\u271d : (i' : \u03b9) \u2192 Decidable (i' \u2208 {i})\nj : \u03b9\n\u22a2 \u2191(piecewise (single i (\u2191x i)) (erase i x) {i}) j = \u2191x j", "state_after": "case h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\nx : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\ninst\u271d : (i' : \u03b9) \u2192 Decidable (i' \u2208 {i})\nj : \u03b9\n\u22a2 (if j \u2208 {i} then \u2191(single i (\u2191x i)) j else \u2191(erase i x) j) = \u2191x j"}, {"tactic": "split_ifs with h", "state_before": "case h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\nx : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\ninst\u271d : (i' : \u03b9) \u2192 Decidable (i' \u2208 {i})\nj : \u03b9\n\u22a2 (if j \u2208 {i} then \u2191(single i (\u2191x i)) j else \u2191(erase i x) j) = \u2191x j", "state_after": "case h.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\nx : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\ninst\u271d : (i' : \u03b9) \u2192 Decidable (i' \u2208 {i})\nj : \u03b9\nh : j \u2208 {i}\n\u22a2 \u2191(single i (\u2191x i)) j = \u2191x j\n\ncase h.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\nx : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\ninst\u271d : (i' : \u03b9) \u2192 Decidable (i' \u2208 {i})\nj : \u03b9\nh : \u00acj \u2208 {i}\n\u22a2 \u2191(erase i x) j = \u2191x j"}, {"tactic": "rw [(id h : j = i), single_eq_same]", "state_before": "case h.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\nx : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\ninst\u271d : (i' : \u03b9) \u2192 Decidable (i' \u2208 {i})\nj : \u03b9\nh : j \u2208 {i}\n\u22a2 \u2191(single i (\u2191x i)) j = \u2191x j", "state_after": "no goals"}, {"tactic": "exact erase_ne h", "state_before": "case h.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\nx : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni : \u03b9\ninst\u271d : (i' : \u03b9) \u2192 Decidable (i' \u2208 {i})\nj : \u03b9\nh : \u00acj \u2208 {i}\n\u22a2 \u2191(erase i x) j = \u2191x j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "isOpenMap_div_right", "start": [1184, 1], "end": [1185, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Tactic/NormNum/Core.lean", "full_name": "Mathlib.Meta.NormNum.IsNat.to_isInt", "start": [80, 1], "end": [81, 29], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nn\u271d : \u2115\n\u22a2 \u2191n\u271d = \u2191(Int.ofNat n\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/BilinearForm.lean", "full_name": "BilinForm.add_right", "start": [105, 1], "end": [106, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.Measure.zero_le_toSignedMeasure", "start": [1422, 1], "end": [1426, 51], "traced_tactics": [{"tactic": "rw [\u2190 le_restrict_univ_iff_le]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.719217\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u22a2 0 \u2264 toSignedMeasure \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.719217\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u22a2 VectorMeasure.restrict 0 Set.univ \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc) Set.univ"}, {"tactic": "refine' restrict_le_restrict_of_subset_le _ _ fun j hj\u2081 _ => _", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.719217\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\n\u22a2 VectorMeasure.restrict 0 Set.univ \u2264 VectorMeasure.restrict (toSignedMeasure \u03bc) Set.univ", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.719217\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nx\u271d : j \u2286 Set.univ\n\u22a2 \u21910 j \u2264 \u2191(toSignedMeasure \u03bc) j"}, {"tactic": "simp only [Measure.toSignedMeasure_apply_measurable hj\u2081, coe_zero, Pi.zero_apply,\n  ENNReal.toReal_nonneg, VectorMeasure.coe_zero]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.719217\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nx\u271d : j \u2286 Set.univ\n\u22a2 \u21910 j \u2264 \u2191(toSignedMeasure \u03bc) j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Homeomorph.lean", "full_name": "Homeomorph.openEmbedding", "start": [331, 11], "end": [332, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/Basic.lean", "full_name": "uniformity_prod_eq_comap_prod", "start": [1574, 1], "end": [1578, 78], "traced_tactics": [{"tactic": "dsimp [SProd.sprod]", "state_before": "\u03b1 : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort ?u.153564\ninst\u271d\u00b9 : UniformSpace 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"full_name": "measurable_from_nat", "start": [423, 1], "end": [424, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/UnionLift.lean", "full_name": "Set.preimage_liftCover", "start": [179, 1], "end": [182, 12], "traced_tactics": [{"tactic": "change (iUnionLift S f hf univ hS.symm.subset \u2218 fun a => \u27e8a, mem_univ a\u27e9) \u207b\u00b9' t = _", "state_before": "\u03b1 : Type u_2\n\u03b9 : Sort u_3\n\u03b2 : Type u_1\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n  \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n    f i { val := x, property := hxi } = f j { val := x, property := hxj }\nhS : iUnion S = univ\nt : Set \u03b2\n\u22a2 liftCover S f hf hS \u207b\u00b9' t = \u22c3 (i : \u03b9), Subtype.val '' (f i \u207b\u00b9' t)", "state_after": "\u03b1 : Type u_2\n\u03b9 : Sort u_3\n\u03b2 : Type u_1\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n  \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n    f i { val := x, property := hxi } = f j { val := x, property := hxj }\nhS : iUnion S = univ\nt : Set \u03b2\n\u22a2 (iUnionLift S f hf univ (_ : univ \u2286 iUnion S) \u2218 fun a => { val := a, property := (_ : a \u2208 univ) }) \u207b\u00b9' t =\n    \u22c3 (i : \u03b9), Subtype.val '' (f i \u207b\u00b9' t)"}, {"tactic": "rw [preimage_comp, preimage_iUnionLift]", "state_before": "\u03b1 : Type u_2\n\u03b9 : Sort u_3\n\u03b2 : Type u_1\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n  \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n    f i { val := x, property := hxi } = f j { val := x, property := hxj }\nhS : iUnion S = univ\nt : Set \u03b2\n\u22a2 (iUnionLift S f hf univ (_ : univ \u2286 iUnion S) \u2218 fun a => { val := a, property := (_ : a \u2208 univ) }) \u207b\u00b9' t =\n    \u22c3 (i : \u03b9), Subtype.val '' (f i \u207b\u00b9' t)", "state_after": "\u03b1 : Type u_2\n\u03b9 : Sort u_3\n\u03b2 : Type u_1\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n  \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n    f i { val := x, property := hxi } = f j { val := x, property := hxj }\nhS : iUnion S = univ\nt : Set \u03b2\n\u22a2 (fun a => { val := a, property := (_ : a \u2208 univ) }) \u207b\u00b9'\n      (inclusion (_ : univ \u2286 iUnion S) \u207b\u00b9' \u22c3 (i : \u03b9), inclusion (_ : S i \u2286 \u22c3 (i : \u03b9), S i) '' (f i \u207b\u00b9' t)) =\n    \u22c3 (i : \u03b9), Subtype.val '' (f i \u207b\u00b9' t)"}, {"tactic": "ext", "state_before": "\u03b1 : Type u_2\n\u03b9 : Sort u_3\n\u03b2 : Type u_1\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n  \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n    f i { val := x, property := hxi } = f j { val := x, property := hxj }\nhS : iUnion S = univ\nt : Set \u03b2\n\u22a2 (fun a => { val := a, property := (_ : a \u2208 univ) }) \u207b\u00b9'\n      (inclusion (_ : univ \u2286 iUnion S) \u207b\u00b9' \u22c3 (i : \u03b9), inclusion (_ : S i \u2286 \u22c3 (i : \u03b9), S i) '' (f i \u207b\u00b9' t)) =\n    \u22c3 (i : \u03b9), Subtype.val '' (f i \u207b\u00b9' t)", "state_after": "case h\n\u03b1 : Type u_2\n\u03b9 : Sort u_3\n\u03b2 : Type u_1\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n  \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n    f i { val := x, property := hxi } = f j { val := x, property := hxj }\nhS : iUnion S = univ\nt : Set \u03b2\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208\n      (fun a => { val := a, property := (_ : a \u2208 univ) }) \u207b\u00b9'\n        (inclusion (_ : univ \u2286 iUnion S) \u207b\u00b9' \u22c3 (i : \u03b9), inclusion (_ : S i \u2286 \u22c3 (i : \u03b9), S i) '' (f i \u207b\u00b9' t)) \u2194\n    x\u271d \u2208 \u22c3 (i : \u03b9), Subtype.val '' (f i \u207b\u00b9' t)"}, {"tactic": "simp", "state_before": "case h\n\u03b1 : Type u_2\n\u03b9 : Sort u_3\n\u03b2 : Type u_1\nS : \u03b9 \u2192 Set \u03b1\nf : (i : \u03b9) \u2192 \u2191(S i) \u2192 \u03b2\nhf :\n  \u2200 (i j : \u03b9) (x : \u03b1) (hxi : x \u2208 S i) (hxj : x \u2208 S j),\n    f i { val := x, property := hxi } = f j { val := x, property := hxj }\nhS : iUnion S = univ\nt : Set \u03b2\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208\n      (fun a => { val := a, property := (_ : a \u2208 univ) }) \u207b\u00b9'\n        (inclusion (_ : univ \u2286 iUnion S) \u207b\u00b9' \u22c3 (i : \u03b9), inclusion (_ : S i \u2286 \u22c3 (i : \u03b9), S i) '' (f i \u207b\u00b9' t)) \u2194\n    x\u271d \u2208 \u22c3 (i : \u03b9), Subtype.val '' (f i \u207b\u00b9' t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Basic.lean", "full_name": "Equiv.Perm.IsCycle.exists_pow_eq_one", "start": [387, 1], "end": [397, 34], "traced_tactics": [{"tactic": "have : IsOfFinOrder f := by exact _root_.exists_pow_eq_one f", "state_before": "\u03b9 : Type ?u.418876\n\u03b1 : Type ?u.418879\n\u03b2 : Type u_1\nf\u271d g : Perm \u03b1\nx y : \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Finite \u03b2\nf : Perm \u03b2\nhf : IsCycle f\n\u22a2 \u2203 k x, f ^ k = 1", "state_after": "\u03b9 : Type ?u.418876\n\u03b1 : Type ?u.418879\n\u03b2 : Type u_1\nf\u271d g : Perm \u03b1\nx y : \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Finite \u03b2\nf : Perm \u03b2\nhf : IsCycle f\nthis : IsOfFinOrder f\n\u22a2 \u2203 k x, f ^ k = 1"}, {"tactic": "rw [isOfFinOrder_iff_pow_eq_one] at this", "state_before": "\u03b9 : Type ?u.418876\n\u03b1 : Type ?u.418879\n\u03b2 : Type u_1\nf\u271d g : Perm \u03b1\nx y : \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Finite \u03b2\nf : Perm \u03b2\nhf : IsCycle f\nthis : IsOfFinOrder f\n\u22a2 \u2203 k x, f ^ k = 1", "state_after": "\u03b9 : Type ?u.418876\n\u03b1 : Type ?u.418879\n\u03b2 : Type u_1\nf\u271d g : Perm \u03b1\nx y : \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Finite \u03b2\nf : Perm \u03b2\nhf : IsCycle f\nthis : \u2203 n, 0 < n \u2227 f ^ n = 1\n\u22a2 \u2203 k x, f ^ k = 1"}, {"tactic": "obtain \u27e8x, hx, _\u27e9 := hf", "state_before": "\u03b9 : Type ?u.418876\n\u03b1 : Type ?u.418879\n\u03b2 : Type u_1\nf\u271d g : Perm \u03b1\nx y : \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Finite \u03b2\nf : Perm \u03b2\nhf : IsCycle f\nthis : \u2203 n, 0 < n \u2227 f ^ n = 1\n\u22a2 \u2203 k x, f ^ k = 1", "state_after": "case intro.intro\n\u03b9 : Type ?u.418876\n\u03b1 : Type ?u.418879\n\u03b2 : Type u_1\nf\u271d g : Perm \u03b1\nx\u271d y : \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Finite \u03b2\nf : Perm \u03b2\nthis : \u2203 n, 0 < n \u2227 f ^ n = 1\nx : \u03b2\nhx : \u2191f x \u2260 x\nright\u271d : \u2200 \u2983y : \u03b2\u2984, \u2191f y \u2260 y \u2192 SameCycle f x y\n\u22a2 \u2203 k x, f ^ k = 1"}, {"tactic": "obtain \u27e8_ | _ | k, hk, hk'\u27e9 := this", "state_before": "case intro.intro\n\u03b9 : Type ?u.418876\n\u03b1 : Type ?u.418879\n\u03b2 : Type u_1\nf\u271d g : Perm \u03b1\nx\u271d y : \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Finite \u03b2\nf : Perm \u03b2\nthis : \u2203 n, 0 < n \u2227 f ^ n = 1\nx : \u03b2\nhx : \u2191f x \u2260 x\nright\u271d : \u2200 \u2983y : \u03b2\u2984, \u2191f y \u2260 y \u2192 SameCycle f x y\n\u22a2 \u2203 k x, f ^ k = 1", "state_after": "case intro.intro.intro.zero.intro\n\u03b9 : Type ?u.418876\n\u03b1 : Type ?u.418879\n\u03b2 : Type u_1\nf\u271d g : Perm \u03b1\nx\u271d y : \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Finite \u03b2\nf : Perm \u03b2\nx : \u03b2\nhx : \u2191f x \u2260 x\nright\u271d : \u2200 \u2983y : \u03b2\u2984, \u2191f y \u2260 y \u2192 SameCycle f x y\nhk : 0 < Nat.zero\nhk' : f ^ Nat.zero = 1\n\u22a2 \u2203 k x, f ^ k = 1\n\ncase intro.intro.intro.succ.zero.intro\n\u03b9 : Type ?u.418876\n\u03b1 : Type ?u.418879\n\u03b2 : Type u_1\nf\u271d g : Perm \u03b1\nx\u271d y : \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Finite \u03b2\nf : Perm \u03b2\nx : \u03b2\nhx : \u2191f x \u2260 x\nright\u271d : \u2200 \u2983y : \u03b2\u2984, \u2191f y \u2260 y \u2192 SameCycle f x y\nhk : 0 < Nat.succ Nat.zero\nhk' : f ^ Nat.succ Nat.zero = 1\n\u22a2 \u2203 k x, f ^ k = 1\n\ncase intro.intro.intro.succ.succ.intro\n\u03b9 : Type ?u.418876\n\u03b1 : Type ?u.418879\n\u03b2 : Type u_1\nf\u271d g : Perm \u03b1\nx\u271d y : \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Finite \u03b2\nf : Perm \u03b2\nx : \u03b2\nhx : \u2191f x \u2260 x\nright\u271d : \u2200 \u2983y : \u03b2\u2984, \u2191f y \u2260 y \u2192 SameCycle f x y\nk : \u2115\nhk : 0 < Nat.succ (Nat.succ k)\nhk' : f ^ Nat.succ (Nat.succ k) = 1\n\u22a2 \u2203 k x, f ^ k = 1"}, {"tactic": "exact _root_.exists_pow_eq_one f", "state_before": "\u03b9 : Type ?u.418876\n\u03b1 : Type ?u.418879\n\u03b2 : Type u_1\nf\u271d g : Perm \u03b1\nx y : \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Finite \u03b2\nf : Perm \u03b2\nhf : IsCycle f\n\u22a2 IsOfFinOrder f", "state_after": "no goals"}, {"tactic": "exact absurd hk (lt_asymm hk)", "state_before": "case intro.intro.intro.zero.intro\n\u03b9 : Type ?u.418876\n\u03b1 : Type ?u.418879\n\u03b2 : Type u_1\nf\u271d g : Perm \u03b1\nx\u271d y : \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Finite \u03b2\nf : Perm \u03b2\nx : \u03b2\nhx : \u2191f x \u2260 x\nright\u271d : \u2200 \u2983y : \u03b2\u2984, \u2191f y \u2260 y \u2192 SameCycle f x y\nhk : 0 < Nat.zero\nhk' : f ^ Nat.zero = 1\n\u22a2 \u2203 k x, f ^ k = 1", "state_after": "no goals"}, {"tactic": "rw [pow_one] at hk'", "state_before": "case intro.intro.intro.succ.zero.intro\n\u03b9 : Type ?u.418876\n\u03b1 : Type ?u.418879\n\u03b2 : Type u_1\nf\u271d g : Perm \u03b1\nx\u271d y : \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Finite \u03b2\nf : Perm \u03b2\nx : \u03b2\nhx : \u2191f x \u2260 x\nright\u271d : \u2200 \u2983y : \u03b2\u2984, \u2191f y \u2260 y \u2192 SameCycle f x y\nhk : 0 < Nat.succ Nat.zero\nhk' : f ^ Nat.succ Nat.zero = 1\n\u22a2 \u2203 k x, f ^ k = 1", "state_after": "case intro.intro.intro.succ.zero.intro\n\u03b9 : Type ?u.418876\n\u03b1 : Type ?u.418879\n\u03b2 : Type u_1\nf\u271d g : Perm \u03b1\nx\u271d y : \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Finite \u03b2\nf : Perm \u03b2\nx : \u03b2\nhx : \u2191f x \u2260 x\nright\u271d : \u2200 \u2983y : \u03b2\u2984, \u2191f y \u2260 y \u2192 SameCycle f x y\nhk : 0 < Nat.succ Nat.zero\nhk' : f = 1\n\u22a2 \u2203 k x, f ^ k = 1"}, {"tactic": "simp [hk'] at hx", "state_before": "case intro.intro.intro.succ.zero.intro\n\u03b9 : Type ?u.418876\n\u03b1 : Type ?u.418879\n\u03b2 : Type u_1\nf\u271d g : Perm \u03b1\nx\u271d y : \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Finite \u03b2\nf : Perm \u03b2\nx : \u03b2\nhx : \u2191f x \u2260 x\nright\u271d : \u2200 \u2983y : \u03b2\u2984, \u2191f y \u2260 y \u2192 SameCycle f x y\nhk : 0 < Nat.succ Nat.zero\nhk' : f = 1\n\u22a2 \u2203 k x, f ^ k = 1", "state_after": "no goals"}, {"tactic": "exact \u27e8k + 2, by simp, hk'\u27e9", "state_before": "case intro.intro.intro.succ.succ.intro\n\u03b9 : Type ?u.418876\n\u03b1 : Type ?u.418879\n\u03b2 : Type u_1\nf\u271d g : Perm \u03b1\nx\u271d y : \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Finite \u03b2\nf : Perm \u03b2\nx : \u03b2\nhx : \u2191f x \u2260 x\nright\u271d : \u2200 \u2983y : \u03b2\u2984, \u2191f y \u2260 y \u2192 SameCycle f x y\nk : \u2115\nhk : 0 < Nat.succ (Nat.succ k)\nhk' : f ^ Nat.succ (Nat.succ k) = 1\n\u22a2 \u2203 k x, f ^ k = 1", "state_after": "no goals"}, {"tactic": "simp", "state_before": "\u03b9 : Type ?u.418876\n\u03b1 : Type ?u.418879\n\u03b2 : Type u_1\nf\u271d g : Perm \u03b1\nx\u271d y : \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Finite \u03b2\nf : Perm \u03b2\nx : \u03b2\nhx : \u2191f x \u2260 x\nright\u271d : \u2200 \u2983y : \u03b2\u2984, \u2191f y \u2260 y \u2192 SameCycle f x y\nk : \u2115\nhk : 0 < Nat.succ (Nat.succ k)\nhk' : f ^ Nat.succ (Nat.succ k) = 1\n\u22a2 1 < k + 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/StrictConvexSpace.lean", "full_name": "sameRay_iff_norm_sub", "start": [226, 1], "end": [228, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Normed/Group/Hom.lean", "full_name": "NormedAddGroupHom.coe_inj", "start": [104, 1], "end": [105, 26], "traced_tactics": [{"tactic": "cases f", "state_before": "V : Type ?u.108968\nV\u2081 : Type u_1\nV\u2082 : Type u_2\nV\u2083 : Type ?u.108977\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : SeminormedAddCommGroup V\u2081\ninst\u271d\u00b9 : SeminormedAddCommGroup V\u2082\ninst\u271d : SeminormedAddCommGroup V\u2083\nf g : NormedAddGroupHom V\u2081 V\u2082\nH : \u2191f = \u2191g\n\u22a2 f = g", "state_after": "case mk\nV : Type ?u.108968\nV\u2081 : Type u_1\nV\u2082 : Type u_2\nV\u2083 : Type ?u.108977\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : SeminormedAddCommGroup V\u2081\ninst\u271d\u00b9 : SeminormedAddCommGroup V\u2082\ninst\u271d : SeminormedAddCommGroup V\u2083\ng : NormedAddGroupHom V\u2081 V\u2082\ntoFun\u271d : V\u2081 \u2192 V\u2082\nmap_add'\u271d : \u2200 (v\u2081 v\u2082 : V\u2081), toFun\u271d (v\u2081 + v\u2082) = toFun\u271d v\u2081 + toFun\u271d v\u2082\nbound'\u271d : \u2203 C, \u2200 (v : V\u2081), \u2016toFun\u271d v\u2016 \u2264 C * \u2016v\u2016\nH : \u2191{ toFun := toFun\u271d, map_add' := map_add'\u271d, bound' := bound'\u271d } = \u2191g\n\u22a2 { toFun := toFun\u271d, map_add' := map_add'\u271d, bound' := bound'\u271d } = g"}, {"tactic": "cases g", "state_before": "case mk\nV : Type ?u.108968\nV\u2081 : Type u_1\nV\u2082 : Type u_2\nV\u2083 : Type ?u.108977\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : SeminormedAddCommGroup V\u2081\ninst\u271d\u00b9 : SeminormedAddCommGroup V\u2082\ninst\u271d : SeminormedAddCommGroup V\u2083\ng : NormedAddGroupHom V\u2081 V\u2082\ntoFun\u271d : V\u2081 \u2192 V\u2082\nmap_add'\u271d : \u2200 (v\u2081 v\u2082 : V\u2081), toFun\u271d (v\u2081 + v\u2082) = toFun\u271d v\u2081 + toFun\u271d v\u2082\nbound'\u271d : \u2203 C, \u2200 (v : V\u2081), \u2016toFun\u271d v\u2016 \u2264 C * \u2016v\u2016\nH : \u2191{ toFun := toFun\u271d, map_add' := map_add'\u271d, bound' := bound'\u271d } = \u2191g\n\u22a2 { toFun := toFun\u271d, map_add' := map_add'\u271d, bound' := bound'\u271d } = g", "state_after": "case mk.mk\nV : Type ?u.108968\nV\u2081 : Type u_1\nV\u2082 : Type u_2\nV\u2083 : Type ?u.108977\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : SeminormedAddCommGroup V\u2081\ninst\u271d\u00b9 : SeminormedAddCommGroup V\u2082\ninst\u271d : SeminormedAddCommGroup V\u2083\ntoFun\u271d\u00b9 : V\u2081 \u2192 V\u2082\nmap_add'\u271d\u00b9 : \u2200 (v\u2081 v\u2082 : V\u2081), toFun\u271d\u00b9 (v\u2081 + v\u2082) = toFun\u271d\u00b9 v\u2081 + toFun\u271d\u00b9 v\u2082\nbound'\u271d\u00b9 : \u2203 C, \u2200 (v : V\u2081), \u2016toFun\u271d\u00b9 v\u2016 \u2264 C * \u2016v\u2016\ntoFun\u271d : V\u2081 \u2192 V\u2082\nmap_add'\u271d : \u2200 (v\u2081 v\u2082 : V\u2081), toFun\u271d (v\u2081 + v\u2082) = toFun\u271d v\u2081 + toFun\u271d v\u2082\nbound'\u271d : \u2203 C, \u2200 (v : V\u2081), \u2016toFun\u271d v\u2016 \u2264 C * \u2016v\u2016\nH :\n  \u2191{ toFun := toFun\u271d\u00b9, map_add' := map_add'\u271d\u00b9, bound' := bound'\u271d\u00b9 } =\n    \u2191{ toFun := toFun\u271d, map_add' := map_add'\u271d, bound' := bound'\u271d }\n\u22a2 { toFun := toFun\u271d\u00b9, map_add' := map_add'\u271d\u00b9, bound' := bound'\u271d\u00b9 } =\n    { toFun := toFun\u271d, map_add' := map_add'\u271d, bound' := bound'\u271d }"}, {"tactic": "congr", "state_before": "case mk.mk\nV : Type ?u.108968\nV\u2081 : Type u_1\nV\u2082 : Type u_2\nV\u2083 : Type ?u.108977\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : SeminormedAddCommGroup V\u2081\ninst\u271d\u00b9 : SeminormedAddCommGroup V\u2082\ninst\u271d : SeminormedAddCommGroup V\u2083\ntoFun\u271d\u00b9 : V\u2081 \u2192 V\u2082\nmap_add'\u271d\u00b9 : \u2200 (v\u2081 v\u2082 : V\u2081), toFun\u271d\u00b9 (v\u2081 + v\u2082) = toFun\u271d\u00b9 v\u2081 + toFun\u271d\u00b9 v\u2082\nbound'\u271d\u00b9 : \u2203 C, \u2200 (v : V\u2081), \u2016toFun\u271d\u00b9 v\u2016 \u2264 C * \u2016v\u2016\ntoFun\u271d : V\u2081 \u2192 V\u2082\nmap_add'\u271d : \u2200 (v\u2081 v\u2082 : V\u2081), toFun\u271d (v\u2081 + v\u2082) = toFun\u271d v\u2081 + toFun\u271d v\u2082\nbound'\u271d : \u2203 C, \u2200 (v : V\u2081), \u2016toFun\u271d v\u2016 \u2264 C * \u2016v\u2016\nH :\n  \u2191{ toFun := toFun\u271d\u00b9, map_add' := map_add'\u271d\u00b9, bound' := bound'\u271d\u00b9 } =\n    \u2191{ toFun := toFun\u271d, map_add' := map_add'\u271d, bound' := bound'\u271d }\n\u22a2 { toFun := toFun\u271d\u00b9, map_add' := map_add'\u271d\u00b9, bound' := bound'\u271d\u00b9 } =\n    { toFun := toFun\u271d, map_add' := map_add'\u271d, bound' := bound'\u271d }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "full_name": "WithBot.unbot_one'", "start": [502, 1], "end": [503, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "full_name": "le_of_mul_le_of_one_le_nonneg_right", "start": [949, 1], "end": [951, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Prod.lean", "full_name": "Filter.prod_mono_right", "start": [240, 1], "end": [241, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Array/Lemmas.lean", "full_name": "Array.back_push", "start": [76, 1], "end": [76, 79], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b1 : Type u_1\nx : \u03b1\ninst\u271d : Inhabited \u03b1\na : Array \u03b1\n\u22a2 back (push a x) = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/Padics/RingHoms.lean", "full_name": "PadicInt.norm_sub_modPart", "start": [126, 1], "end": [136, 33], "traced_tactics": [{"tactic": "let n := modPart p r", "state_before": "p : \u2115\nhp_prime : Fact (Nat.Prime p)\nr : \u211a\nh : \u2016\u2191r\u2016 \u2264 1\n\u22a2 \u2016{ val := \u2191r, property := h } - \u2191(modPart p r)\u2016 < 1", "state_after": "p : \u2115\nhp_prime : Fact (Nat.Prime p)\nr : \u211a\nh : \u2016\u2191r\u2016 \u2264 1\nn : \u2124 := modPart p r\n\u22a2 \u2016{ val := \u2191r, property := h } - \u2191(modPart p r)\u2016 < 1"}, {"tactic": "rw [norm_lt_one_iff_dvd, \u2190 (isUnit_den r h).dvd_mul_right]", "state_before": "p : \u2115\nhp_prime : Fact (Nat.Prime p)\nr : \u211a\nh : \u2016\u2191r\u2016 \u2264 1\nn : \u2124 := modPart p r\n\u22a2 \u2016{ val := \u2191r, property := h } - \u2191(modPart p r)\u2016 < 1", "state_after": "p : \u2115\nhp_prime : Fact (Nat.Prime p)\nr : \u211a\nh : \u2016\u2191r\u2016 \u2264 1\nn : \u2124 := modPart p r\n\u22a2 \u2191p \u2223 ({ val := \u2191r, property := h } - \u2191(modPart p r)) * \u2191r.den"}, {"tactic": "suffices \u2191p \u2223 r.num - n * r.den by\n  convert(Int.castRingHom \u2124_[p]).map_dvd this\n  simp only [sub_mul, Int.cast_ofNat, eq_intCast, Int.cast_mul, sub_left_inj, Int.cast_sub]\n  apply Subtype.coe_injective\n  simp only [coe_mul, Subtype.coe_mk, coe_nat_cast]\n  rw_mod_cast [@Rat.mul_den_eq_num r]\n  rfl", "state_before": "p : \u2115\nhp_prime : Fact (Nat.Prime p)\nr : \u211a\nh : \u2016\u2191r\u2016 \u2264 1\nn : \u2124 := modPart p r\n\u22a2 \u2191p \u2223 ({ val := \u2191r, property := h } - \u2191(modPart p r)) * \u2191r.den", "state_after": "p : \u2115\nhp_prime : Fact (Nat.Prime p)\nr : \u211a\nh : \u2016\u2191r\u2016 \u2264 1\nn : \u2124 := modPart p r\n\u22a2 \u2191p \u2223 r.num - n * \u2191r.den"}, {"tactic": "exact norm_sub_modPart_aux r h", "state_before": "p : \u2115\nhp_prime : Fact (Nat.Prime p)\nr : \u211a\nh : \u2016\u2191r\u2016 \u2264 1\nn : \u2124 := modPart p r\n\u22a2 \u2191p \u2223 r.num - n * \u2191r.den", "state_after": "no goals"}, {"tactic": "convert(Int.castRingHom \u2124_[p]).map_dvd this", "state_before": "p : \u2115\nhp_prime : Fact (Nat.Prime p)\nr : \u211a\nh : \u2016\u2191r\u2016 \u2264 1\nn : \u2124 := modPart p r\nthis : \u2191p \u2223 r.num - n * \u2191r.den\n\u22a2 \u2191p \u2223 ({ val := \u2191r, property := h } - \u2191(modPart p r)) * \u2191r.den", "state_after": "case h.e'_4\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nr : \u211a\nh : \u2016\u2191r\u2016 \u2264 1\nn : \u2124 := modPart p r\nthis : \u2191p \u2223 r.num - n * \u2191r.den\n\u22a2 ({ val := \u2191r, property := h } - \u2191(modPart p r)) * \u2191r.den = \u2191(Int.castRingHom \u2124_[p]) (r.num - n * \u2191r.den)"}, {"tactic": "simp only [sub_mul, Int.cast_ofNat, eq_intCast, Int.cast_mul, sub_left_inj, Int.cast_sub]", "state_before": "case h.e'_4\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nr : \u211a\nh : \u2016\u2191r\u2016 \u2264 1\nn : \u2124 := modPart p r\nthis : \u2191p \u2223 r.num - n * \u2191r.den\n\u22a2 ({ val := \u2191r, property := h } - \u2191(modPart p r)) * \u2191r.den = \u2191(Int.castRingHom \u2124_[p]) (r.num - n * \u2191r.den)", "state_after": "case h.e'_4\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nr : \u211a\nh : \u2016\u2191r\u2016 \u2264 1\nn : \u2124 := modPart p r\nthis : \u2191p \u2223 r.num - n * \u2191r.den\n\u22a2 { val := \u2191r, property := h } * \u2191r.den = \u2191r.num"}, {"tactic": "apply Subtype.coe_injective", "state_before": "case h.e'_4\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nr : \u211a\nh : \u2016\u2191r\u2016 \u2264 1\nn : \u2124 := modPart p r\nthis : \u2191p \u2223 r.num - n * \u2191r.den\n\u22a2 { val := \u2191r, property := h } * \u2191r.den = \u2191r.num", "state_after": "case h.e'_4.a\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nr : \u211a\nh : \u2016\u2191r\u2016 \u2264 1\nn : \u2124 := modPart p r\nthis : \u2191p \u2223 r.num - n * \u2191r.den\n\u22a2 (fun a => \u2191a) ({ val := \u2191r, property := h } * \u2191r.den) = (fun a => \u2191a) \u2191r.num"}, {"tactic": "simp only [coe_mul, Subtype.coe_mk, coe_nat_cast]", "state_before": "case h.e'_4.a\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nr : \u211a\nh : \u2016\u2191r\u2016 \u2264 1\nn : \u2124 := modPart p r\nthis : \u2191p \u2223 r.num - n * \u2191r.den\n\u22a2 (fun a => \u2191a) ({ val := \u2191r, property := h } * \u2191r.den) = (fun a => \u2191a) \u2191r.num", "state_after": "case h.e'_4.a\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nr : \u211a\nh : \u2016\u2191r\u2016 \u2264 1\nn : \u2124 := modPart p r\nthis : \u2191p \u2223 r.num - n * \u2191r.den\n\u22a2 \u2191r * \u2191r.den = \u2191\u2191r.num"}, {"tactic": "rw_mod_cast [@Rat.mul_den_eq_num r]", "state_before": "case h.e'_4.a\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nr : \u211a\nh : \u2016\u2191r\u2016 \u2264 1\nn : \u2124 := modPart p r\nthis : \u2191p \u2223 r.num - n * \u2191r.den\n\u22a2 \u2191r * \u2191r.den = \u2191\u2191r.num", "state_after": "case h.e'_4.a\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nr : \u211a\nn : \u2124 := modPart p r\nthis : \u2191p \u2223 r.num - n * \u2191r.den\nh : \u2016\u2191r\u2016 \u2264 1\n\u22a2 \u2191\u2191r.num = \u2191r.num"}, {"tactic": "rfl", "state_before": "case h.e'_4.a\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nr : \u211a\nn : \u2124 := modPart p r\nthis : \u2191p \u2223 r.num - n * \u2191r.den\nh : \u2016\u2191r\u2016 \u2264 1\n\u22a2 \u2191\u2191r.num = \u2191r.num", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean", "full_name": "MeasureTheory.ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite", "start": [228, 1], "end": [235, 49], "traced_tactics": [{"tactic": "have A : f \u2264\u1d50[\u03bc] g :=\n  ae_le_of_forall_set_lintegral_le_of_sigmaFinite hf hg fun s hs h's => le_of_eq (h s hs h's)", "state_before": "\u03b1 : Type u_1\nE : Type ?u.67365\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 (\u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc) = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\n\u22a2 f =\u1d50[\u03bc] g", "state_after": "\u03b1 : Type u_1\nE : Type ?u.67365\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 (\u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc) = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : f \u2264\u1d50[\u03bc] g\n\u22a2 f =\u1d50[\u03bc] g"}, {"tactic": "have B : g \u2264\u1d50[\u03bc] f :=\n  ae_le_of_forall_set_lintegral_le_of_sigmaFinite hg hf fun s hs h's => ge_of_eq (h s hs h's)", "state_before": "\u03b1 : Type u_1\nE : Type ?u.67365\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 (\u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc) = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : f \u2264\u1d50[\u03bc] g\n\u22a2 f =\u1d50[\u03bc] g", "state_after": "\u03b1 : Type u_1\nE : Type ?u.67365\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 (\u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc) = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : f \u2264\u1d50[\u03bc] g\nB : g \u2264\u1d50[\u03bc] f\n\u22a2 f =\u1d50[\u03bc] g"}, {"tactic": "filter_upwards [A, B] with x using le_antisymm", "state_before": "\u03b1 : Type u_1\nE : Type ?u.67365\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\np : \u211d\u22650\u221e\ninst\u271d : SigmaFinite \u03bc\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 (\u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc) = \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc\nA : f \u2264\u1d50[\u03bc] g\nB : g \u2264\u1d50[\u03bc] f\n\u22a2 f =\u1d50[\u03bc] g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/ModelTheory/Satisfiability.lean", "full_name": "FirstOrder.Language.Theory.models_formula_iff", "start": [316, 1], "end": [318, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "binary_relation_sInf_iff", "start": [1799, 1], "end": [1802, 22], "traced_tactics": [{"tactic": "rw [sInf_apply]", "state_before": "\u03b1\u271d : Type ?u.205042\n\u03b2\u271d : Type ?u.205045\n\u03b2\u2082 : Type ?u.205048\n\u03b3 : Type ?u.205051\n\u03b9 : Sort ?u.205054\n\u03b9' : Sort ?u.205057\n\u03ba : \u03b9 \u2192 Sort ?u.205062\n\u03ba' : \u03b9' \u2192 Sort ?u.205067\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set (\u03b1 \u2192 \u03b2 \u2192 Prop)\na : \u03b1\nb : \u03b2\n\u22a2 sInf s a b \u2194 \u2200 (r : \u03b1 \u2192 \u03b2 \u2192 Prop), r \u2208 s \u2192 r a b", "state_after": "\u03b1\u271d : Type ?u.205042\n\u03b2\u271d : Type ?u.205045\n\u03b2\u2082 : Type ?u.205048\n\u03b3 : Type ?u.205051\n\u03b9 : Sort ?u.205054\n\u03b9' : Sort ?u.205057\n\u03ba : \u03b9 \u2192 Sort ?u.205062\n\u03ba' : \u03b9' \u2192 Sort ?u.205067\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set (\u03b1 \u2192 \u03b2 \u2192 Prop)\na : \u03b1\nb : \u03b2\n\u22a2 iInf (fun f => \u2191f a) b \u2194 \u2200 (r : \u03b1 \u2192 \u03b2 \u2192 Prop), r \u2208 s \u2192 r a b"}, {"tactic": "simp [\u2190 eq_iff_iff]", "state_before": "\u03b1\u271d : Type ?u.205042\n\u03b2\u271d : Type ?u.205045\n\u03b2\u2082 : Type ?u.205048\n\u03b3 : Type ?u.205051\n\u03b9 : Sort ?u.205054\n\u03b9' : Sort ?u.205057\n\u03ba : \u03b9 \u2192 Sort ?u.205062\n\u03ba' : \u03b9' \u2192 Sort ?u.205067\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Set (\u03b1 \u2192 \u03b2 \u2192 Prop)\na : \u03b1\nb : \u03b2\n\u22a2 iInf (fun f => \u2191f a) b \u2194 \u2200 (r : \u03b1 \u2192 \u03b2 \u2192 Prop), r \u2208 s \u2192 r a b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Sign.lean", "full_name": "SignType.lt_one_iff", "start": [183, 1], "end": [183, 75], "traced_tactics": [{"tactic": "cases a <;> decide", "state_before": "a : SignType\n\u22a2 a < 1 \u2194 a \u2264 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Concrete.lean", "full_name": "Equiv.Perm.pow_apply_mem_toList_iff_mem_support", "start": [340, 1], "end": [343, 26], "traced_tactics": [{"tactic": "rw [mem_toList_iff, and_iff_right_iff_imp]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx : \u03b1\nn : \u2115\n\u22a2 \u2191(p ^ n) x \u2208 toList p x \u2194 x \u2208 support p", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx : \u03b1\nn : \u2115\n\u22a2 x \u2208 support p \u2192 SameCycle p x (\u2191(p ^ n) x)"}, {"tactic": "refine' fun _ => SameCycle.symm _", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx : \u03b1\nn : \u2115\n\u22a2 x \u2208 support p \u2192 SameCycle p x (\u2191(p ^ n) x)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx : \u03b1\nn : \u2115\nx\u271d : x \u2208 support p\n\u22a2 SameCycle p (\u2191(p ^ n) x) x"}, {"tactic": "rw [sameCycle_pow_left]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx : \u03b1\nn : \u2115\nx\u271d : x \u2208 support p\n\u22a2 SameCycle p (\u2191(p ^ n) x) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "full_name": "Zsqrtd.mul_star", "start": [340, 1], "end": [341, 39], "traced_tactics": [{"tactic": "simp [ext, sub_eq_add_neg, mul_comm]", "state_before": "d x y : \u2124\n\u22a2 { re := x, im := y } * star { re := x, im := y } = \u2191x * \u2191x - \u2191d * \u2191y * \u2191y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/RingDivision.lean", "full_name": "Polynomial.rootMultiplicity_le_iff", "start": [380, 1], "end": [382, 70], "traced_tactics": [{"tactic": "rw [\u2190 (le_rootMultiplicity_iff p0).not, not_le, Nat.lt_add_one_iff]", "state_before": "R : Type u\nS : Type v\nT : Type w\na\u271d b : R\nn\u271d : \u2115\ninst\u271d : CommRing R\np : R[X]\np0 : p \u2260 0\na : R\nn : \u2115\n\u22a2 rootMultiplicity a p \u2264 n \u2194 \u00ac(X - \u2191C a) ^ (n + 1) \u2223 p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.lift_sum", "start": [898, 1], "end": [904, 81], "traced_tactics": [{"tactic": "rw [\u2190 lift_mk_eq.{_,_,v}, mk_out, mk_out, lift_lift]", "state_before": "\u03b1 \u03b2 \u03b9 : Type u\nf : \u03b9 \u2192 Cardinal\na : \u03b9\n\u22a2 Nonempty (Quotient.out (f a) \u2243 Quotient.out ((fun i => lift (f i)) a))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "full_name": "DifferentiableWithinAt.fderivWithin_congr_mono", "start": [936, 1], "end": [939, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/RootsOfUnity/Basic.lean", "full_name": "mem_primitiveRoots", "start": [318, 1], "end": [320, 35], "traced_tactics": [{"tactic": "rw [primitiveRoots, mem_filter, Multiset.mem_toFinset, mem_nthRoots h0, and_iff_right_iff_imp]", "state_before": "M : Type ?u.1450185\nN : Type ?u.1450188\nG : Type ?u.1450191\nR : Type u_1\nS : Type ?u.1450197\nF : Type ?u.1450200\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\nh0 : 0 < k\n\u22a2 \u03b6 \u2208 primitiveRoots k R \u2194 IsPrimitiveRoot \u03b6 k", "state_after": "M : Type ?u.1450185\nN : Type ?u.1450188\nG : Type ?u.1450191\nR : Type u_1\nS : Type ?u.1450197\nF : Type ?u.1450200\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\nh0 : 0 < k\n\u22a2 IsPrimitiveRoot \u03b6 k \u2192 \u03b6 ^ k = 1"}, {"tactic": "exact IsPrimitiveRoot.pow_eq_one", "state_before": "M : Type ?u.1450185\nN : Type ?u.1450188\nG : Type ?u.1450191\nR : Type u_1\nS : Type ?u.1450197\nF : Type ?u.1450200\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\nh0 : 0 < k\n\u22a2 IsPrimitiveRoot \u03b6 k \u2192 \u03b6 ^ k = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Basic.lean", "full_name": "Nat.find_le", "start": [857, 1], "end": [858, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Dfinsupp.lean", "full_name": "CompleteLattice.independent_iff_forall_dfinsupp", "start": [431, 1], "end": [438, 34], "traced_tactics": [{"tactic": "simp_rw [CompleteLattice.independent_def, Submodule.disjoint_def,\n  Submodule.mem_biSup_iff_exists_dfinsupp, exists_imp, filter_ne_eq_erase]", "state_before": "\u03b9 : Type u_3\nR : Type u_1\nS : Type ?u.465837\nM : \u03b9 \u2192 Type ?u.465842\nN : Type u_2\ndec_\u03b9 : DecidableEq \u03b9\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : Module R N\np : \u03b9 \u2192 Submodule R N\n\u22a2 Independent p \u2194\n    \u2200 (i : \u03b9) (x : { x // x \u2208 p i }) (v : \u03a0\u2080 (i : \u03b9), { x // x \u2208 p i }),\n      \u2191(\u2191(lsum \u2115) fun i => Submodule.subtype (p i)) (erase i v) = \u2191x \u2192 x = 0", "state_after": "\u03b9 : Type u_3\nR : Type u_1\nS : Type ?u.465837\nM : \u03b9 \u2192 Type ?u.465842\nN : Type u_2\ndec_\u03b9 : DecidableEq \u03b9\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : Module R N\np : \u03b9 \u2192 Submodule R N\n\u22a2 (\u2200 (i : \u03b9) (x : N),\n      x \u2208 p i \u2192\n        \u2200 (x_1 : \u03a0\u2080 (i : \u03b9), { x // x \u2208 p i }),\n          \u2191(\u2191(lsum \u2115) fun i => Submodule.subtype (p i)) (erase i x_1) = x \u2192 x = 0) \u2194\n    \u2200 (i : \u03b9) (x : { x // x \u2208 p i }) (v : \u03a0\u2080 (i : \u03b9), { x // x \u2208 p i }),\n      \u2191(\u2191(lsum \u2115) fun i => Submodule.subtype (p i)) (erase i v) = \u2191x \u2192 x = 0"}, {"tactic": "refine' forall_congr' fun i => Subtype.forall'.trans _", "state_before": "\u03b9 : Type u_3\nR : Type u_1\nS : Type ?u.465837\nM : \u03b9 \u2192 Type ?u.465842\nN : Type u_2\ndec_\u03b9 : DecidableEq \u03b9\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : Module R N\np : \u03b9 \u2192 Submodule R N\n\u22a2 (\u2200 (i : \u03b9) (x : N),\n      x \u2208 p i \u2192\n        \u2200 (x_1 : \u03a0\u2080 (i : \u03b9), { x // x \u2208 p i }),\n          \u2191(\u2191(lsum \u2115) fun i => Submodule.subtype (p i)) (erase i x_1) = x \u2192 x = 0) \u2194\n    \u2200 (i : \u03b9) (x : { x // x \u2208 p i }) (v : \u03a0\u2080 (i : \u03b9), { x // x \u2208 p i }),\n      \u2191(\u2191(lsum \u2115) fun i => Submodule.subtype (p i)) (erase i v) = \u2191x \u2192 x = 0", "state_after": "\u03b9 : Type u_3\nR : Type u_1\nS : Type ?u.465837\nM : \u03b9 \u2192 Type ?u.465842\nN : Type u_2\ndec_\u03b9 : DecidableEq \u03b9\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : Module R N\np : \u03b9 \u2192 Submodule R N\ni : \u03b9\n\u22a2 (\u2200 (x : { a // a \u2208 p i }) (x_1 : \u03a0\u2080 (i : \u03b9), { x // x \u2208 p i }),\n      \u2191(\u2191(lsum \u2115) fun i => Submodule.subtype (p i)) (erase i x_1) = \u2191x \u2192 \u2191x = 0) \u2194\n    \u2200 (x : { x // x \u2208 p i }) (v : \u03a0\u2080 (i : \u03b9), { x // x \u2208 p i }),\n      \u2191(\u2191(lsum \u2115) fun i => Submodule.subtype (p i)) (erase i v) = \u2191x \u2192 x = 0"}, {"tactic": "simp_rw [Submodule.coe_eq_zero]", "state_before": "\u03b9 : Type u_3\nR : Type u_1\nS : Type ?u.465837\nM : \u03b9 \u2192 Type ?u.465842\nN : Type u_2\ndec_\u03b9 : DecidableEq \u03b9\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : Module R N\np : \u03b9 \u2192 Submodule R N\ni : \u03b9\n\u22a2 (\u2200 (x : { a // a \u2208 p i }) (x_1 : \u03a0\u2080 (i : \u03b9), { x // x \u2208 p i }),\n      \u2191(\u2191(lsum \u2115) fun i => Submodule.subtype (p i)) (erase i x_1) = \u2191x \u2192 \u2191x = 0) \u2194\n    \u2200 (x : { x // x \u2208 p i }) (v : \u03a0\u2080 (i : \u03b9), { x // x \u2208 p i }),\n      \u2191(\u2191(lsum \u2115) fun i => Submodule.subtype (p i)) (erase i v) = \u2191x \u2192 x = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Function/Basic.lean", "full_name": "Function.bijective_iff_existsUnique", "start": [241, 1], "end": [245, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Separation.lean", "full_name": "tendsto_nhds_unique_of_frequently_eq", "start": [977, 1], "end": [980, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/AddTorsor.lean", "full_name": "Equiv.coe_vaddConst_symm", "start": [378, 1], "end": [379, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.subsingleton_of_image", "start": [1228, 1], "end": [1230, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/NoncommProd.lean", "full_name": "Multiset.noncommProd_commute", "start": [216, 1], "end": [220, 38], "traced_tactics": [{"tactic": "induction s using Quotient.inductionOn", "state_before": "F : Type ?u.136558\n\u03b9 : Type ?u.136561\n\u03b1 : Type u_1\n\u03b2 : Type ?u.136567\n\u03b3 : Type ?u.136570\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ns : Multiset \u03b1\ncomm : Set.Pairwise {x | x \u2208 s} Commute\ny : \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 Commute y x\n\u22a2 Commute y (noncommProd s comm)", "state_after": "case h\nF : Type ?u.136558\n\u03b9 : Type ?u.136561\n\u03b1 : Type u_1\n\u03b2 : Type ?u.136567\n\u03b3 : Type ?u.136570\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ny : \u03b1\na\u271d : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} Commute\nh : \u2200 (x : \u03b1), x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d \u2192 Commute y x\n\u22a2 Commute y (noncommProd (Quotient.mk (List.isSetoid \u03b1) a\u271d) comm)"}, {"tactic": "simp only [quot_mk_to_coe, noncommProd_coe]", "state_before": "case h\nF : Type ?u.136558\n\u03b9 : Type ?u.136561\n\u03b1 : Type u_1\n\u03b2 : Type ?u.136567\n\u03b3 : Type ?u.136570\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ny : \u03b1\na\u271d : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} Commute\nh : \u2200 (x : \u03b1), x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d \u2192 Commute y x\n\u22a2 Commute y (noncommProd (Quotient.mk (List.isSetoid \u03b1) a\u271d) comm)", "state_after": "case h\nF : Type ?u.136558\n\u03b9 : Type ?u.136561\n\u03b1 : Type u_1\n\u03b2 : Type ?u.136567\n\u03b3 : Type ?u.136570\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ny : \u03b1\na\u271d : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} Commute\nh : \u2200 (x : \u03b1), x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d \u2192 Commute y x\n\u22a2 Commute y (List.prod a\u271d)"}, {"tactic": "exact Commute.list_prod_right _ _ h", "state_before": "case h\nF : Type ?u.136558\n\u03b9 : Type ?u.136561\n\u03b1 : Type u_1\n\u03b2 : Type ?u.136567\n\u03b3 : Type ?u.136570\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\ny : \u03b1\na\u271d : List \u03b1\ncomm : Set.Pairwise {x | x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d} Commute\nh : \u2200 (x : \u03b1), x \u2208 Quotient.mk (List.isSetoid \u03b1) a\u271d \u2192 Commute y x\n\u22a2 Commute y (List.prod a\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Gluing.lean", "full_name": "Metric.Sigma.dist_same", "start": [329, 1], "end": [330, 31], "traced_tactics": [{"tactic": "simp [Dist.dist, Sigma.dist]", "state_before": "\u03b9 : Type u_2\nE : \u03b9 \u2192 Type u_1\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (E i)\ni : \u03b9\nx y : E i\n\u22a2 dist { fst := i, snd := x } { fst := i, snd := y } = dist x y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/BumpFunctionInner.lean", "full_name": "Real.smoothTransition.zero", "start": [196, 11], "end": [197, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/Multilinear.lean", "full_name": "MultilinearMap.norm_image_sub_le_of_bound'", "start": [149, 1], "end": [191, 7], "traced_tactics": [{"tactic": "convert A univ", "state_before": "\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\nA :\n  \u2200 (s : Finset \u03b9),\n    \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n\u22a2 \u2016\u2191f m\u2081 - \u2191f m\u2082\u2016 \u2264 C * \u2211 i : \u03b9, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016", "state_after": "case h.e'_3.h.e'_3.h.e'_6.h.e'_6\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\nA :\n  \u2200 (s : Finset \u03b9),\n    \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n\u22a2 m\u2082 = piecewise univ m\u2082 m\u2081"}, {"tactic": "simp", "state_before": "case h.e'_3.h.e'_3.h.e'_6.h.e'_6\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\nA :\n  \u2200 (s : Finset \u03b9),\n    \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n\u22a2 m\u2082 = piecewise univ m\u2082 m\u2081", "state_after": "no goals"}, {"tactic": "intro s", "state_before": "\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\n\u22a2 \u2200 (s : Finset \u03b9),\n    \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016", "state_after": "\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ns : Finset \u03b9\n\u22a2 \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016"}, {"tactic": "induction' s using Finset.induction with i s his Hrec", "state_before": "\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ns : Finset \u03b9\n\u22a2 \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016", "state_after": "case empty\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191f m\u2081 - \u2191f (piecewise \u2205 m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in \u2205, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n\ncase insert\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n\u22a2 \u2016\u2191f m\u2081 - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016 \u2264\n    C * \u2211 i in insert i s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016"}, {"tactic": "calc\n  \u2016f m\u2081 - f ((insert i s).piecewise m\u2082 m\u2081)\u2016 \u2264\n      \u2016f m\u2081 - f (s.piecewise m\u2082 m\u2081)\u2016 +\n        \u2016f (s.piecewise m\u2082 m\u2081) - f ((insert i s).piecewise m\u2082 m\u2081)\u2016 := by\n    rw [\u2190 dist_eq_norm, \u2190 dist_eq_norm, \u2190 dist_eq_norm]\n    exact dist_triangle _ _ _\n  _ \u2264\n      (C * \u2211 i in s, \u220f j, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) +\n        C * \u220f j, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016 :=\n    (add_le_add Hrec I)\n  _ = C * \u2211 i in insert i s, \u220f j, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016 := by\n    simp [his, add_comm, left_distrib]", "state_before": "case insert\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nI :\n  \u2016\u2191f (piecewise s m\u2082 m\u2081) - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016 \u2264\n    C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n\u22a2 \u2016\u2191f m\u2081 - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016 \u2264\n    C * \u2211 i in insert i s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016", "state_after": "no goals"}, {"tactic": "simp", "state_before": "case empty\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\n\u22a2 \u2016\u2191f m\u2081 - \u2191f (piecewise \u2205 m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in \u2205, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016", "state_after": "no goals"}, {"tactic": "have A : (insert i s).piecewise m\u2082 m\u2081 = Function.update (s.piecewise m\u2082 m\u2081) i (m\u2082 i) :=\n  s.piecewise_insert _ _ _", "state_before": "\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n\u22a2 \u2016\u2191f (piecewise s m\u2082 m\u2081) - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016 \u2264\n    C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016", "state_after": "\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\n\u22a2 \u2016\u2191f (piecewise s m\u2082 m\u2081) - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016 \u2264\n    C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016"}, {"tactic": "rw [B, A, \u2190 f.map_sub]", "state_before": "\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\n\u22a2 \u2016\u2191f (piecewise s m\u2082 m\u2081) - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016 \u2264\n    C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016", "state_after": "\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\n\u22a2 \u2016\u2191f (Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i))\u2016 \u2264\n    C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016"}, {"tactic": "apply le_trans (H _)", "state_before": "\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\n\u22a2 \u2016\u2191f (Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i))\u2016 \u2264\n    C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016", "state_after": "\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\n\u22a2 C * \u220f i_1 : \u03b9, \u2016Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i) i_1\u2016 \u2264\n    C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016"}, {"tactic": "gcongr with j", "state_before": "\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\n\u22a2 C * \u220f i_1 : \u03b9, \u2016Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i) i_1\u2016 \u2264\n    C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016", "state_after": "case h.h0\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\n\u22a2 \u2200 (i_1 : \u03b9), i_1 \u2208 univ \u2192 0 \u2264 \u2016Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i) i_1\u2016\n\ncase h.h1\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\nj : \u03b9\na\u271d : j \u2208 univ\n\u22a2 \u2016Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i) j\u2016 \u2264 if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016"}, {"tactic": "by_cases h : j = i", "state_before": "case h.h1\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\nj : \u03b9\na\u271d : j \u2208 univ\n\u22a2 \u2016Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i) j\u2016 \u2264 if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016", "state_after": "case pos\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\nj : \u03b9\na\u271d : j \u2208 univ\nh : j = i\n\u22a2 \u2016Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i) j\u2016 \u2264 if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n\ncase neg\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\nj : \u03b9\na\u271d : j \u2208 univ\nh : \u00acj = i\n\u22a2 \u2016Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i) j\u2016 \u2264 if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016"}, {"tactic": "ext j", "state_before": "\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\n\u22a2 piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)", "state_after": "case h\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nj : \u03b9\n\u22a2 piecewise s m\u2082 m\u2081 j = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i) j"}, {"tactic": "by_cases h : j = i", "state_before": "case h\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nj : \u03b9\n\u22a2 piecewise s m\u2082 m\u2081 j = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i) j", "state_after": "case pos\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nj : \u03b9\nh : j = i\n\u22a2 piecewise s m\u2082 m\u2081 j = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i) j\n\ncase neg\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nj : \u03b9\nh : \u00acj = i\n\u22a2 piecewise s m\u2082 m\u2081 j = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i) j"}, {"tactic": "rw [h]", "state_before": "case pos\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nj : \u03b9\nh : j = i\n\u22a2 piecewise s m\u2082 m\u2081 j = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i) j", "state_after": "case pos\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nj : \u03b9\nh : j = i\n\u22a2 piecewise s m\u2082 m\u2081 i = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i) i"}, {"tactic": "simp [his]", "state_before": "case pos\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nj : \u03b9\nh : j = i\n\u22a2 piecewise s m\u2082 m\u2081 i = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i) i", "state_after": "no goals"}, {"tactic": "simp [h]", "state_before": "case neg\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nj : \u03b9\nh : \u00acj = i\n\u22a2 piecewise s m\u2082 m\u2081 j = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i) j", "state_after": "no goals"}, {"tactic": "exact fun j _ => norm_nonneg _", "state_before": "case h.h0\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\n\u22a2 \u2200 (i_1 : \u03b9), i_1 \u2208 univ \u2192 0 \u2264 \u2016Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i) i_1\u2016", "state_after": "no goals"}, {"tactic": "rw [h]", "state_before": "case pos\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\nj : \u03b9\na\u271d : j \u2208 univ\nh : j = i\n\u22a2 \u2016Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i) j\u2016 \u2264 if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016", "state_after": "case pos\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\nj : \u03b9\na\u271d : j \u2208 univ\nh : j = i\n\u22a2 \u2016Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i) i\u2016 \u2264 if i = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 i\u2016 \u2016m\u2082 i\u2016"}, {"tactic": "simp", "state_before": "case pos\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\nj : \u03b9\na\u271d : j \u2208 univ\nh : j = i\n\u22a2 \u2016Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i) i\u2016 \u2264 if i = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 i\u2016 \u2016m\u2082 i\u2016", "state_after": "no goals"}, {"tactic": "by_cases h' : j \u2208 s <;> simp [h', h, le_refl]", "state_before": "case neg\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nA : piecewise (insert i s) m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2082 i)\nB : piecewise s m\u2082 m\u2081 = Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i)\nj : \u03b9\na\u271d : j \u2208 univ\nh : \u00acj = i\n\u22a2 \u2016Function.update (piecewise s m\u2082 m\u2081) i (m\u2081 i - m\u2082 i) j\u2016 \u2264 if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016", "state_after": "no goals"}, {"tactic": "rw [\u2190 dist_eq_norm, \u2190 dist_eq_norm, \u2190 dist_eq_norm]", "state_before": "\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nI :\n  \u2016\u2191f (piecewise s m\u2082 m\u2081) - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016 \u2264\n    C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n\u22a2 \u2016\u2191f m\u2081 - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016 \u2264\n    \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 + \u2016\u2191f (piecewise s m\u2082 m\u2081) - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016", "state_after": "\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nI :\n  \u2016\u2191f (piecewise s m\u2082 m\u2081) - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016 \u2264\n    C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n\u22a2 dist (\u2191f m\u2081) (\u2191f (piecewise (insert i s) m\u2082 m\u2081)) \u2264\n    dist (\u2191f m\u2081) (\u2191f (piecewise s m\u2082 m\u2081)) + dist (\u2191f (piecewise s m\u2082 m\u2081)) (\u2191f (piecewise (insert i s) m\u2082 m\u2081))"}, {"tactic": "exact dist_triangle _ _ _", "state_before": "\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nI :\n  \u2016\u2191f (piecewise s m\u2082 m\u2081) - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016 \u2264\n    C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n\u22a2 dist (\u2191f m\u2081) (\u2191f (piecewise (insert i s) m\u2082 m\u2081)) \u2264\n    dist (\u2191f m\u2081) (\u2191f (piecewise s m\u2082 m\u2081)) + dist (\u2191f (piecewise s m\u2082 m\u2081)) (\u2191f (piecewise (insert i s) m\u2082 m\u2081))", "state_after": "no goals"}, {"tactic": "simp [his, add_comm, left_distrib]", "state_before": "\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03b9'\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2076 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\nf : MultilinearMap \ud835\udd5c E G\ninst\u271d : DecidableEq \u03b9\nC : \u211d\nhC : 0 \u2264 C\nH : \u2200 (m : (i : \u03b9) \u2192 E i), \u2016\u2191f m\u2016 \u2264 C * \u220f i : \u03b9, \u2016m i\u2016\nm\u2081 m\u2082 : (i : \u03b9) \u2192 E i\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nHrec : \u2016\u2191f m\u2081 - \u2191f (piecewise s m\u2082 m\u2081)\u2016 \u2264 C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\nI :\n  \u2016\u2191f (piecewise s m\u2082 m\u2081) - \u2191f (piecewise (insert i s) m\u2082 m\u2081)\u2016 \u2264\n    C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016\n\u22a2 ((C * \u2211 i in s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) +\n      C * \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016) =\n    C * \u2211 i in insert i s, \u220f j : \u03b9, if j = i then \u2016m\u2081 i - m\u2082 i\u2016 else max \u2016m\u2081 j\u2016 \u2016m\u2082 j\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.disjoint_of_subset_left", "start": [939, 1], "end": [940, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/ModEq.lean", "full_name": "Nat.mod_mul_right_mod", "start": [401, 1], "end": [402, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Game/PGame.lean", "full_name": "PGame.le_iff_forall_lf", "start": [425, 1], "end": [430, 6], "traced_tactics": [{"tactic": "unfold LE.le le", "state_before": "x y : PGame\n\u22a2 x \u2264 y \u2194 (\u2200 (i : LeftMoves x), moveLeft x i \u29cf y) \u2227 \u2200 (j : RightMoves y), x \u29cf moveRight y j", "state_after": "x y : PGame\n\u22a2 {\n          le :=\n            Sym2.GameAdd.fix wf_isOption fun x y le =>\n              (\u2200 (i : LeftMoves x),\n                  \u00acle y (moveLeft x i)\n                      (_ :\n                        Sym2.GameAdd IsOption (Quotient.mk (Sym2.Rel.setoid PGame) (y, moveLeft x i))\n                          (Quotient.mk (Sym2.Rel.setoid PGame) (x, y)))) \u2227\n                \u2200 (j : RightMoves y),\n                  \u00acle (moveRight y j) x\n                      (_ :\n                        Sym2.GameAdd IsOption (Quotient.mk (Sym2.Rel.setoid PGame) (moveRight y j, x))\n                          (Quotient.mk (Sym2.Rel.setoid PGame) (x, y))) }.1\n      x y \u2194\n    (\u2200 (i : LeftMoves x), moveLeft x i \u29cf y) \u2227 \u2200 (j : RightMoves y), x \u29cf moveRight y j"}, {"tactic": "simp only", "state_before": "x y : PGame\n\u22a2 {\n          le :=\n            Sym2.GameAdd.fix wf_isOption fun x y le =>\n              (\u2200 (i : LeftMoves x),\n                  \u00acle y (moveLeft x i)\n                      (_ :\n                        Sym2.GameAdd IsOption (Quotient.mk (Sym2.Rel.setoid PGame) (y, moveLeft x i))\n                          (Quotient.mk (Sym2.Rel.setoid PGame) (x, y)))) \u2227\n                \u2200 (j : RightMoves y),\n                  \u00acle (moveRight y j) x\n                      (_ :\n                        Sym2.GameAdd IsOption (Quotient.mk (Sym2.Rel.setoid PGame) (moveRight y j, x))\n                          (Quotient.mk (Sym2.Rel.setoid PGame) (x, y))) }.1\n      x y \u2194\n    (\u2200 (i : LeftMoves x), moveLeft x i \u29cf y) \u2227 \u2200 (j : RightMoves y), x \u29cf moveRight y j", "state_after": "x y : PGame\n\u22a2 Sym2.GameAdd.fix wf_isOption\n      (fun x y le =>\n        (\u2200 (i : LeftMoves x),\n            \u00acle y (moveLeft x i)\n                (_ :\n                  Sym2.GameAdd IsOption (Quotient.mk (Sym2.Rel.setoid PGame) (y, moveLeft x i))\n                    (Quotient.mk (Sym2.Rel.setoid PGame) (x, y)))) \u2227\n          \u2200 (j : RightMoves y),\n            \u00acle (moveRight y j) x\n                (_ :\n                  Sym2.GameAdd IsOption (Quotient.mk (Sym2.Rel.setoid PGame) (moveRight y j, x))\n                    (Quotient.mk (Sym2.Rel.setoid PGame) (x, y))))\n      x y \u2194\n    (\u2200 (i : LeftMoves x), moveLeft x i \u29cf y) \u2227 \u2200 (j : RightMoves y), x \u29cf moveRight y j"}, {"tactic": "rw [Sym2.GameAdd.fix_eq]", "state_before": "x y : PGame\n\u22a2 Sym2.GameAdd.fix wf_isOption\n      (fun x y le =>\n        (\u2200 (i : LeftMoves x),\n            \u00acle y (moveLeft x i)\n                (_ :\n                  Sym2.GameAdd IsOption (Quotient.mk (Sym2.Rel.setoid PGame) (y, moveLeft x i))\n                    (Quotient.mk (Sym2.Rel.setoid PGame) (x, y)))) \u2227\n          \u2200 (j : RightMoves y),\n            \u00acle (moveRight y j) x\n                (_ :\n                  Sym2.GameAdd IsOption (Quotient.mk (Sym2.Rel.setoid PGame) (moveRight y j, x))\n                    (Quotient.mk (Sym2.Rel.setoid PGame) (x, y))))\n      x y \u2194\n    (\u2200 (i : LeftMoves x), moveLeft x i \u29cf y) \u2227 \u2200 (j : RightMoves y), x \u29cf moveRight y j", "state_after": "x y : PGame\n\u22a2 ((\u2200 (i : LeftMoves x),\n        \u00ac(fun a' b' x =>\n              Sym2.GameAdd.fix wf_isOption\n                (fun x y le =>\n                  (\u2200 (i : LeftMoves x),\n                      \u00acle y (moveLeft x i)\n                          (_ :\n                            Sym2.GameAdd IsOption (Quotient.mk (Sym2.Rel.setoid PGame) (y, moveLeft x i))\n                              (Quotient.mk (Sym2.Rel.setoid PGame) (x, y)))) \u2227\n                    \u2200 (j : RightMoves y),\n                      \u00acle (moveRight y j) x\n                          (_ :\n                            Sym2.GameAdd IsOption (Quotient.mk (Sym2.Rel.setoid PGame) (moveRight y j, x))\n                              (Quotient.mk (Sym2.Rel.setoid PGame) (x, y))))\n                a' b')\n            y (moveLeft x i)\n            (_ :\n              Sym2.GameAdd IsOption (Quotient.mk (Sym2.Rel.setoid PGame) (y, moveLeft x i))\n                (Quotient.mk (Sym2.Rel.setoid PGame) (x, y)))) \u2227\n      \u2200 (j : RightMoves y),\n        \u00ac(fun a' b' x =>\n              Sym2.GameAdd.fix wf_isOption\n                (fun x y le =>\n                  (\u2200 (i : LeftMoves x),\n                      \u00acle y (moveLeft x i)\n                          (_ :\n                            Sym2.GameAdd IsOption (Quotient.mk (Sym2.Rel.setoid PGame) (y, moveLeft x i))\n                              (Quotient.mk (Sym2.Rel.setoid PGame) (x, y)))) \u2227\n                    \u2200 (j : RightMoves y),\n                      \u00acle (moveRight y j) x\n                          (_ :\n                            Sym2.GameAdd IsOption (Quotient.mk (Sym2.Rel.setoid PGame) (moveRight y j, x))\n                              (Quotient.mk (Sym2.Rel.setoid PGame) (x, y))))\n                a' b')\n            (moveRight y j) x\n            (_ :\n              Sym2.GameAdd IsOption (Quotient.mk (Sym2.Rel.setoid PGame) (moveRight y j, x))\n                (Quotient.mk (Sym2.Rel.setoid PGame) (x, y)))) \u2194\n    (\u2200 (i : LeftMoves x), moveLeft x i \u29cf y) \u2227 \u2200 (j : RightMoves y), x \u29cf moveRight y j"}, {"tactic": "rfl", "state_before": "x y : PGame\n\u22a2 ((\u2200 (i : LeftMoves x),\n        \u00ac(fun a' b' x =>\n              Sym2.GameAdd.fix wf_isOption\n                (fun x y le =>\n                  (\u2200 (i : LeftMoves x),\n                      \u00acle y (moveLeft x i)\n                          (_ :\n                            Sym2.GameAdd IsOption (Quotient.mk (Sym2.Rel.setoid PGame) (y, moveLeft x i))\n                              (Quotient.mk (Sym2.Rel.setoid PGame) (x, y)))) \u2227\n                    \u2200 (j : RightMoves y),\n                      \u00acle (moveRight y j) x\n                          (_ :\n                            Sym2.GameAdd IsOption (Quotient.mk (Sym2.Rel.setoid PGame) (moveRight y j, x))\n                              (Quotient.mk (Sym2.Rel.setoid PGame) (x, y))))\n                a' b')\n            y (moveLeft x i)\n            (_ :\n              Sym2.GameAdd IsOption (Quotient.mk (Sym2.Rel.setoid PGame) (y, moveLeft x i))\n                (Quotient.mk (Sym2.Rel.setoid PGame) (x, y)))) \u2227\n      \u2200 (j : RightMoves y),\n        \u00ac(fun a' b' x =>\n              Sym2.GameAdd.fix wf_isOption\n                (fun x y le =>\n                  (\u2200 (i : LeftMoves x),\n                      \u00acle y (moveLeft x i)\n                          (_ :\n                            Sym2.GameAdd IsOption (Quotient.mk (Sym2.Rel.setoid PGame) (y, moveLeft x i))\n                              (Quotient.mk (Sym2.Rel.setoid PGame) (x, y)))) \u2227\n                    \u2200 (j : RightMoves y),\n                      \u00acle (moveRight y j) x\n                          (_ :\n                            Sym2.GameAdd IsOption (Quotient.mk (Sym2.Rel.setoid PGame) (moveRight y j, x))\n                              (Quotient.mk (Sym2.Rel.setoid PGame) (x, y))))\n                a' b')\n            (moveRight y j) x\n            (_ :\n              Sym2.GameAdd IsOption (Quotient.mk (Sym2.Rel.setoid PGame) (moveRight y j, x))\n                (Quotient.mk (Sym2.Rel.setoid PGame) (x, y)))) \u2194\n    (\u2200 (i : LeftMoves x), moveLeft x i \u29cf y) \u2227 \u2200 (j : RightMoves y), x \u29cf moveRight y j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Multiplicity.lean", "full_name": "Nat.Prime.multiplicity_choose", "start": [191, 1], "end": [206, 88], "traced_tactics": [{"tactic": "have h\u2081 :\n  multiplicity p (choose n k) + multiplicity p (k ! * (n - k)!) =\n    ((Finset.Ico 1 b).filter fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i).card +\n      multiplicity p (k ! * (n - k)!) := by\n  rw [\u2190 hp.multiplicity_mul, \u2190 mul_assoc, choose_mul_factorial_mul_factorial hkn,\n    hp.multiplicity_factorial hnb, hp.multiplicity_mul,\n    hp.multiplicity_factorial ((log_mono_right hkn).trans_lt hnb),\n    hp.multiplicity_factorial (lt_of_le_of_lt (log_mono_right tsub_le_self) hnb),\n    multiplicity_choose_aux hp hkn]\n  simp [add_comm]", "state_before": "p n k b : \u2115\nhp : Prime p\nhkn : k \u2264 n\nhnb : log p n < b\n\u22a2 multiplicity p (choose n k) = \u2191(card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b)))", "state_after": "p n k b : \u2115\nhp : Prime p\nhkn : k \u2264 n\nhnb : log p n < b\nh\u2081 :\n  multiplicity p (choose n k) + multiplicity p (k ! * (n - k)!) =\n    \u2191(card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b))) + multiplicity p (k ! * (n - k)!)\n\u22a2 multiplicity p (choose n k) = \u2191(card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b)))"}, {"tactic": "refine (PartENat.add_right_cancel_iff ?_).1 h\u2081", "state_before": "p n k b : \u2115\nhp : Prime p\nhkn : k \u2264 n\nhnb : log p n < b\nh\u2081 :\n  multiplicity p (choose n k) + multiplicity p (k ! * (n - k)!) =\n    \u2191(card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b))) + multiplicity p (k ! * (n - k)!)\n\u22a2 multiplicity p (choose n k) = \u2191(card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b)))", "state_after": "p n k b : \u2115\nhp : Prime p\nhkn : k \u2264 n\nhnb : log p n < b\nh\u2081 :\n  multiplicity p (choose n k) + multiplicity p (k ! * (n - k)!) =\n    \u2191(card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b))) + multiplicity p (k ! * (n - k)!)\n\u22a2 multiplicity p (k ! * (n - k)!) \u2260 \u22a4"}, {"tactic": "apply PartENat.ne_top_iff_dom.2", "state_before": "p n k b : \u2115\nhp : Prime p\nhkn : k \u2264 n\nhnb : log p n < b\nh\u2081 :\n  multiplicity p (choose n k) + multiplicity p (k ! * (n - k)!) =\n    \u2191(card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b))) + multiplicity p (k ! * (n - k)!)\n\u22a2 multiplicity p (k ! * (n - k)!) \u2260 \u22a4", "state_after": "p n k b : \u2115\nhp : Prime p\nhkn : k \u2264 n\nhnb : log p n < b\nh\u2081 :\n  multiplicity p (choose n k) + multiplicity p (k ! * (n - k)!) =\n    \u2191(card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b))) + multiplicity p (k ! * (n - k)!)\n\u22a2 (multiplicity p (k ! * (n - k)!)).Dom"}, {"tactic": "exact finite_nat_iff.2 \u27e8hp.ne_one, mul_pos (factorial_pos k) (factorial_pos (n - k))\u27e9", "state_before": "p n k b : \u2115\nhp : Prime p\nhkn : k \u2264 n\nhnb : log p n < b\nh\u2081 :\n  multiplicity p (choose n k) + multiplicity p (k ! * (n - k)!) =\n    \u2191(card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b))) + multiplicity p (k ! * (n - k)!)\n\u22a2 (multiplicity p (k ! * (n - k)!)).Dom", "state_after": "no goals"}, {"tactic": "rw [\u2190 hp.multiplicity_mul, \u2190 mul_assoc, choose_mul_factorial_mul_factorial hkn,\n  hp.multiplicity_factorial hnb, hp.multiplicity_mul,\n  hp.multiplicity_factorial ((log_mono_right hkn).trans_lt hnb),\n  hp.multiplicity_factorial (lt_of_le_of_lt (log_mono_right tsub_le_self) hnb),\n  multiplicity_choose_aux hp hkn]", "state_before": "p n k b : \u2115\nhp : Prime p\nhkn : k \u2264 n\nhnb : log p n < b\n\u22a2 multiplicity p (choose n k) + multiplicity p (k ! * (n - k)!) =\n    \u2191(card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b))) + multiplicity p (k ! * (n - k)!)", "state_after": "p n k b : \u2115\nhp : Prime p\nhkn : k \u2264 n\nhnb : log p n < b\n\u22a2 \u2191(\u2211 i in Ico 1 b, k / p ^ i + \u2211 i in Ico 1 b, (n - k) / p ^ i +\n        card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b))) =\n    \u2191(card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b))) +\n      (\u2191(\u2211 i in Ico 1 b, k / p ^ i) + \u2191(\u2211 i in Ico 1 b, (n - k) / p ^ i))"}, {"tactic": "simp [add_comm]", "state_before": "p n k b : \u2115\nhp : Prime p\nhkn : k \u2264 n\nhnb : log p n < b\n\u22a2 \u2191(\u2211 i in Ico 1 b, k / p ^ i + \u2211 i in Ico 1 b, (n - k) / p ^ i +\n        card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b))) =\n    \u2191(card (filter (fun i => p ^ i \u2264 k % p ^ i + (n - k) % p ^ i) (Ico 1 b))) +\n      (\u2191(\u2211 i in Ico 1 b, k / p ^ i) + \u2191(\u2211 i in Ico 1 b, (n - k) / p ^ i))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.splitOnP_spec", "start": [2945, 1], "end": [2962, 57], "traced_tactics": [{"tactic": "rfl", "state_before": "case nil\n\u03b9 : Type ?u.296730\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nxs ys : List \u03b1\nls : List (List \u03b1)\nf : List \u03b1 \u2192 List \u03b1\n\u22a2 join (zipWith (fun x x_1 => x ++ x_1) (splitOnP p []) (map (fun x => [x]) (filter p []) ++ [[]])) = []", "state_after": "no goals"}, {"tactic": "rw [splitOnP_cons, filter]", "state_before": "case cons\n\u03b9 : Type ?u.296730\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nxs ys : List \u03b1\nls : List (List \u03b1)\nf : List \u03b1 \u2192 List \u03b1\na : \u03b1\nas' : List \u03b1\nih : join (zipWith (fun x x_1 => x ++ x_1) (splitOnP p as') (map (fun x => [x]) (filter p as') ++ [[]])) = as'\n\u22a2 join (zipWith (fun x x_1 => x ++ x_1) (splitOnP p (a :: as')) (map (fun x => [x]) (filter p (a :: as')) ++ [[]])) =\n    a :: as'", "state_after": "case cons\n\u03b9 : Type ?u.296730\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nxs ys : List \u03b1\nls : List (List \u03b1)\nf : List \u03b1 \u2192 List \u03b1\na : \u03b1\nas' : List \u03b1\nih : join (zipWith (fun x x_1 => x ++ x_1) (splitOnP p as') (map (fun x => [x]) (filter p as') ++ [[]])) = as'\n\u22a2 join\n      (zipWith (fun x x_1 => x ++ x_1)\n        (if p a = true then [] :: splitOnP p as' else modifyHead (cons a) (splitOnP p as'))\n        (map (fun x => [x])\n            (match p a with\n            | true => a :: filter p as'\n            | false => filter p as') ++\n          [[]])) =\n    a :: as'"}, {"tactic": "by_cases h : p a", "state_before": "case cons\n\u03b9 : Type ?u.296730\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nxs ys : List \u03b1\nls : List (List \u03b1)\nf : List \u03b1 \u2192 List \u03b1\na : \u03b1\nas' : List \u03b1\nih : join (zipWith (fun x x_1 => x ++ x_1) (splitOnP p as') (map (fun x => [x]) (filter p as') ++ [[]])) = as'\n\u22a2 join\n      (zipWith (fun x x_1 => x ++ x_1)\n        (if p a = true then [] :: splitOnP p as' else modifyHead (cons a) (splitOnP p as'))\n        (map (fun x => [x])\n            (match p a with\n            | true => a :: filter p as'\n            | false => filter p as') ++\n          [[]])) =\n    a :: as'", "state_after": "case pos\n\u03b9 : Type ?u.296730\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nxs ys : List \u03b1\nls : List (List \u03b1)\nf : List \u03b1 \u2192 List \u03b1\na : \u03b1\nas' : List \u03b1\nih : join (zipWith (fun x x_1 => x ++ x_1) (splitOnP p as') (map (fun x => [x]) (filter p as') ++ [[]])) = as'\nh : p a = true\n\u22a2 join\n      (zipWith (fun x x_1 => x ++ x_1)\n        (if p a = true then [] :: splitOnP p as' else modifyHead (cons a) (splitOnP p as'))\n        (map (fun x => [x])\n            (match p a with\n            | true => a :: filter p as'\n            | false => filter p as') ++\n          [[]])) =\n    a :: as'\n\ncase neg\n\u03b9 : Type ?u.296730\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nxs ys : List \u03b1\nls : List (List \u03b1)\nf : List \u03b1 \u2192 List \u03b1\na : \u03b1\nas' : List \u03b1\nih : join (zipWith (fun x x_1 => x ++ x_1) (splitOnP p as') (map (fun x => [x]) (filter p as') ++ [[]])) = as'\nh : \u00acp a = true\n\u22a2 join\n      (zipWith (fun x x_1 => x ++ x_1)\n        (if p a = true then [] :: splitOnP p as' else modifyHead (cons a) (splitOnP p as'))\n        (map (fun x => [x])\n            (match p a with\n            | true => a :: filter p as'\n            | false => filter p as') ++\n          [[]])) =\n    a :: as'"}, {"tactic": "rw [if_pos h, h, map, cons_append, zipWith, nil_append, join, cons_append, cons_inj]", "state_before": "case pos\n\u03b9 : Type ?u.296730\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nxs ys : List \u03b1\nls : List (List \u03b1)\nf : List \u03b1 \u2192 List \u03b1\na : \u03b1\nas' : List \u03b1\nih : join (zipWith (fun x x_1 => x ++ x_1) (splitOnP p as') (map (fun x => [x]) (filter p as') ++ [[]])) = as'\nh : p a = true\n\u22a2 join\n      (zipWith (fun x x_1 => x ++ x_1)\n        (if p a = true then [] :: splitOnP p as' else modifyHead (cons a) (splitOnP p as'))\n        (map (fun x => [x])\n            (match p a with\n            | true => a :: filter p as'\n            | false => filter p as') ++\n          [[]])) =\n    a :: as'", "state_after": "case pos\n\u03b9 : Type ?u.296730\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nxs ys : List \u03b1\nls : List (List \u03b1)\nf : List \u03b1 \u2192 List \u03b1\na : \u03b1\nas' : List \u03b1\nih : join (zipWith (fun x x_1 => x ++ x_1) (splitOnP p as') (map (fun x => [x]) (filter p as') ++ [[]])) = as'\nh : p a = true\n\u22a2 [] ++ join (zipWith (fun x x_1 => x ++ x_1) (splitOnP p as') (map (fun x => [x]) (filter p as') ++ [[]])) = as'"}, {"tactic": "exact ih", "state_before": "case pos\n\u03b9 : Type ?u.296730\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nxs ys : List \u03b1\nls : List (List \u03b1)\nf : List \u03b1 \u2192 List \u03b1\na : \u03b1\nas' : List \u03b1\nih : join (zipWith (fun x x_1 => x ++ x_1) (splitOnP p as') (map (fun x => [x]) (filter p as') ++ [[]])) = as'\nh : p a = true\n\u22a2 [] ++ join (zipWith (fun x x_1 => x ++ x_1) (splitOnP p as') (map (fun x => [x]) (filter p as') ++ [[]])) = as'", "state_after": "no goals"}, {"tactic": "rw [if_neg h, eq_false_of_ne_true h, join_zipWith (splitOnP_ne_nil _ _)\n  (append_ne_nil_of_ne_nil_right _ [[]] (cons_ne_nil [] [])), cons_inj]", "state_before": "case neg\n\u03b9 : Type ?u.296730\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nxs ys : List \u03b1\nls : List (List \u03b1)\nf : List \u03b1 \u2192 List \u03b1\na : \u03b1\nas' : List \u03b1\nih : join (zipWith (fun x x_1 => x ++ x_1) (splitOnP p as') (map (fun x => [x]) (filter p as') ++ [[]])) = as'\nh : \u00acp a = true\n\u22a2 join\n      (zipWith (fun x x_1 => x ++ x_1)\n        (if p a = true then [] :: splitOnP p as' else modifyHead (cons a) (splitOnP p as'))\n        (map (fun x => [x])\n            (match p a with\n            | true => a :: filter p as'\n            | false => filter p as') ++\n          [[]])) =\n    a :: as'", "state_after": "case neg\n\u03b9 : Type ?u.296730\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nxs ys : List \u03b1\nls : List (List \u03b1)\nf : List \u03b1 \u2192 List \u03b1\na : \u03b1\nas' : List \u03b1\nih : join (zipWith (fun x x_1 => x ++ x_1) (splitOnP p as') (map (fun x => [x]) (filter p as') ++ [[]])) = as'\nh : \u00acp a = true\n\u22a2 join\n      (zipWith (fun x x_1 => x ++ x_1) (splitOnP p as')\n        (map (fun x => [x])\n            (match false with\n            | true => a :: filter p as'\n            | false => filter p as') ++\n          [[]])) =\n    as'"}, {"tactic": "exact ih", "state_before": "case neg\n\u03b9 : Type ?u.296730\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nxs ys : List \u03b1\nls : List (List \u03b1)\nf : List \u03b1 \u2192 List \u03b1\na : \u03b1\nas' : List \u03b1\nih : join (zipWith (fun x x_1 => x ++ x_1) (splitOnP p as') (map (fun x => [x]) (filter p as') ++ [[]])) = as'\nh : \u00acp a = true\n\u22a2 join\n      (zipWith (fun x x_1 => x ++ x_1) (splitOnP p as')\n        (map (fun x => [x])\n            (match false with\n            | true => a :: filter p as'\n            | false => filter p as') ++\n          [[]])) =\n    as'", "state_after": "no goals"}, {"tactic": "cases xs with | nil => contradiction | cons =>\n  cases ys with | nil => contradiction | cons => rfl", "state_before": "\u03b9 : Type ?u.296730\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nxs\u271d ys\u271d : List \u03b1\nls : List (List \u03b1)\nf : List \u03b1 \u2192 List \u03b1\nas : List \u03b1\nxs ys : List (List \u03b1)\na : \u03b1\nhxs : xs \u2260 []\nhys : ys \u2260 []\n\u22a2 join (zipWith (fun x x_1 => x ++ x_1) (modifyHead (cons a) xs) ys) = a :: join (zipWith (fun x x_1 => x ++ x_1) xs ys)", "state_after": "no goals"}, {"tactic": "contradiction", "state_before": "case nil\n\u03b9 : Type ?u.296730\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nxs ys\u271d : List \u03b1\nls : List (List \u03b1)\nf : List \u03b1 \u2192 List \u03b1\nas : List \u03b1\nys : List (List \u03b1)\na : \u03b1\nhys : ys \u2260 []\nhxs : [] \u2260 []\n\u22a2 join (zipWith (fun x x_1 => x ++ x_1) (modifyHead (cons a) []) ys) = a :: join (zipWith (fun x x_1 => x ++ x_1) [] ys)", "state_after": "no goals"}, {"tactic": "cases ys with | nil => contradiction | cons => rfl", "state_before": "case cons\n\u03b9 : Type ?u.296730\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nxs ys\u271d : List \u03b1\nls : List (List \u03b1)\nf : List \u03b1 \u2192 List \u03b1\nas : List \u03b1\nys : List (List \u03b1)\na : \u03b1\nhys : ys \u2260 []\nhead\u271d : List \u03b1\ntail\u271d : List (List \u03b1)\nhxs : head\u271d :: tail\u271d \u2260 []\n\u22a2 join (zipWith (fun x x_1 => x ++ x_1) (modifyHead (cons a) (head\u271d :: tail\u271d)) ys) =\n    a :: join (zipWith (fun x x_1 => x ++ x_1) (head\u271d :: tail\u271d) ys)", "state_after": "no goals"}, {"tactic": "contradiction", "state_before": "case cons.nil\n\u03b9 : Type ?u.296730\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nxs ys : List \u03b1\nls : List (List \u03b1)\nf : List \u03b1 \u2192 List \u03b1\nas : List \u03b1\na : \u03b1\nhead\u271d : List \u03b1\ntail\u271d : List (List \u03b1)\nhxs : head\u271d :: tail\u271d \u2260 []\nhys : [] \u2260 []\n\u22a2 join (zipWith (fun x x_1 => x ++ x_1) (modifyHead (cons a) (head\u271d :: tail\u271d)) []) =\n    a :: join (zipWith (fun x x_1 => x ++ x_1) (head\u271d :: tail\u271d) [])", "state_after": "no goals"}, {"tactic": "rfl", "state_before": "case cons.cons\n\u03b9 : Type ?u.296730\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nxs ys : List \u03b1\nls : List (List \u03b1)\nf : List \u03b1 \u2192 List \u03b1\nas : List \u03b1\na : \u03b1\nhead\u271d\u00b9 : List \u03b1\ntail\u271d\u00b9 : List (List \u03b1)\nhxs : head\u271d\u00b9 :: tail\u271d\u00b9 \u2260 []\nhead\u271d : List \u03b1\ntail\u271d : List (List \u03b1)\nhys : head\u271d :: tail\u271d \u2260 []\n\u22a2 join (zipWith (fun x x_1 => x ++ x_1) (modifyHead (cons a) (head\u271d\u00b9 :: tail\u271d\u00b9)) (head\u271d :: tail\u271d)) =\n    a :: join (zipWith (fun x x_1 => x ++ x_1) (head\u271d\u00b9 :: tail\u271d\u00b9) (head\u271d :: tail\u271d))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finsupp/Basic.lean", "full_name": "Finsupp.apply_eq_of_mem_graph", "start": [90, 1], "end": [91, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/Padics/PadicNumbers.lean", "full_name": "PadicSeq.lift_index_left", "start": [187, 1], "end": [194, 17], "traced_tactics": [{"tactic": "apply stationaryPoint_spec hf", "state_before": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv1 v3 : \u2115\n\u22a2 padicNorm p (\u2191f (stationaryPoint hf)) = padicNorm p (\u2191f (max v1 (max (stationaryPoint hf) v3)))", "state_after": "case a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv1 v3 : \u2115\n\u22a2 stationaryPoint hf \u2264 max v1 (max (stationaryPoint hf) v3)\n\ncase a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv1 v3 : \u2115\n\u22a2 stationaryPoint hf \u2264 stationaryPoint hf"}, {"tactic": "apply le_trans", "state_before": "case a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv1 v3 : \u2115\n\u22a2 stationaryPoint hf \u2264 max v1 (max (stationaryPoint hf) v3)", "state_after": "case a.a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv1 v3 : \u2115\n\u22a2 stationaryPoint hf \u2264 ?a.b\u271d\n\ncase a.a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv1 v3 : \u2115\n\u22a2 ?a.b\u271d \u2264 max v1 (max (stationaryPoint hf) v3)\n\ncase a.b\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv1 v3 : \u2115\n\u22a2 \u2115"}, {"tactic": "apply le_max_left _ v3", "state_before": "case a.a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv1 v3 : \u2115\n\u22a2 stationaryPoint hf \u2264 ?a.b\u271d", "state_after": "no goals"}, {"tactic": "apply le_max_right", "state_before": "case a.a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv1 v3 : \u2115\n\u22a2 max (stationaryPoint hf) v3 \u2264 max v1 (max (stationaryPoint hf) v3)", "state_after": "no goals"}, {"tactic": "exact le_rfl", "state_before": "case a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv1 v3 : \u2115\n\u22a2 stationaryPoint hf \u2264 stationaryPoint hf", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicTopology/SplitSimplicialObject.lean", "full_name": "SimplicialObject.Split.Hom.ext", "start": [389, 1], "end": [400, 14], "traced_tactics": [{"tactic": "rcases \u03a6\u2081 with \u27e8F\u2081, f\u2081, c\u2081\u27e9", "state_before": "C : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6\u2081 \u03a6\u2082 : Hom S\u2081 S\u2082\nh : \u2200 (n : \u2115), f \u03a6\u2081 n = f \u03a6\u2082 n\n\u22a2 \u03a6\u2081 = \u03a6\u2082", "state_after": "case mk\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6\u2082 : Hom S\u2081 S\u2082\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2081.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f \u03a6\u2082 n\n\u22a2 mk F\u2081 f\u2081 = \u03a6\u2082"}, {"tactic": "rcases \u03a6\u2082 with \u27e8F\u2082, f\u2082, c\u2082\u27e9", "state_before": "case mk\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\n\u03a6\u2082 : Hom S\u2081 S\u2082\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2081.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f \u03a6\u2082 n\n\u22a2 mk F\u2081 f\u2081 = \u03a6\u2082", "state_after": "case mk.mk\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2081.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nf\u2082 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2082.app [n].op = f\u2082 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2082) n\n\u22a2 mk F\u2081 f\u2081 = mk F\u2082 f\u2082"}, {"tactic": "have h' : f\u2081 = f\u2082 := by\n  ext\n  apply h", "state_before": "case mk.mk\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2081.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nf\u2082 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2082.app [n].op = f\u2082 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2082) n\n\u22a2 mk F\u2081 f\u2081 = mk F\u2082 f\u2082", "state_after": "case mk.mk\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2081.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nf\u2082 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2082.app [n].op = f\u2082 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2082) n\nh' : f\u2081 = f\u2082\n\u22a2 mk F\u2081 f\u2081 = mk F\u2082 f\u2082"}, {"tactic": "subst h'", "state_before": "case mk.mk\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2081.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nf\u2082 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2082.app [n].op = f\u2082 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2082) n\nh' : f\u2081 = f\u2082\n\u22a2 mk F\u2081 f\u2081 = mk F\u2082 f\u2082", "state_after": "case mk.mk\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2081.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2082.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2081) n\n\u22a2 mk F\u2081 f\u2081 = mk F\u2082 f\u2081"}, {"tactic": "simp only [mk.injEq, and_true]", "state_before": "case mk.mk\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2081.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2082.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2081) n\n\u22a2 mk F\u2081 f\u2081 = mk F\u2082 f\u2081", "state_after": "case mk.mk\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2081.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2082.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2081) n\n\u22a2 F\u2081 = F\u2082"}, {"tactic": "apply S\u2081.s.hom_ext", "state_before": "case mk.mk\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2081.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2082.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2081) n\n\u22a2 F\u2081 = F\u2082", "state_after": "case mk.mk.h\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2081.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2082.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2081) n\n\u22a2 \u2200 (n : \u2115), Splitting.\u03c6 S\u2081.s F\u2081 n = Splitting.\u03c6 S\u2081.s F\u2082 n"}, {"tactic": "intro n", "state_before": "case mk.mk.h\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2081.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2082.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2081) n\n\u22a2 \u2200 (n : \u2115), Splitting.\u03c6 S\u2081.s F\u2081 n = Splitting.\u03c6 S\u2081.s F\u2082 n", "state_after": "case mk.mk.h\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2081.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2082.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2081) n\nn : \u2115\n\u22a2 Splitting.\u03c6 S\u2081.s F\u2081 n = Splitting.\u03c6 S\u2081.s F\u2082 n"}, {"tactic": "dsimp", "state_before": "case mk.mk.h\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2081.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2082.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2081) n\nn : \u2115\n\u22a2 Splitting.\u03c6 S\u2081.s F\u2081 n = Splitting.\u03c6 S\u2081.s F\u2082 n", "state_after": "case mk.mk.h\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2081.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2082.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2081) n\nn : \u2115\n\u22a2 Splitting.\u03b9 S\u2081.s n \u226b F\u2081.app [n].op = Splitting.\u03b9 S\u2081.s n \u226b F\u2082.app [n].op"}, {"tactic": "rw [c\u2081, c\u2082]", "state_before": "case mk.mk.h\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2081.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2082.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2081) n\nn : \u2115\n\u22a2 Splitting.\u03b9 S\u2081.s n \u226b F\u2081.app [n].op = Splitting.\u03b9 S\u2081.s n \u226b F\u2082.app [n].op", "state_after": "no goals"}, {"tactic": "ext", "state_before": "C : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2081.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nf\u2082 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2082.app [n].op = f\u2082 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2082) n\n\u22a2 f\u2081 = f\u2082", "state_after": "case h\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2081.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nf\u2082 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2082.app [n].op = f\u2082 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2082) n\nx\u271d : \u2115\n\u22a2 f\u2081 x\u271d = f\u2082 x\u271d"}, {"tactic": "apply h", "state_before": "case h\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : HasFiniteCoproducts C\nS\u2081 S\u2082 : Split C\nF\u2081 : S\u2081.X \u27f6 S\u2082.X\nf\u2081 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2081 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2081.app [n].op = f\u2081 n \u226b Splitting.\u03b9 S\u2082.s n\nF\u2082 : S\u2081.X \u27f6 S\u2082.X\nf\u2082 : (n : \u2115) \u2192 Splitting.N S\u2081.s n \u27f6 Splitting.N S\u2082.s n\nc\u2082 : \u2200 (n : \u2115), Splitting.\u03b9 S\u2081.s n \u226b F\u2082.app [n].op = f\u2082 n \u226b Splitting.\u03b9 S\u2082.s n\nh : \u2200 (n : \u2115), f (mk F\u2081 f\u2081) n = f (mk F\u2082 f\u2082) n\nx\u271d : \u2115\n\u22a2 f\u2081 x\u271d = f\u2082 x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.indep_iSup_of_disjoint", "start": [444, 1], "end": [453, 68], "traced_tactics": [{"tactic": "refine'\n  IndepSets.indep (iSup\u2082_le fun i _ => h_le i) (iSup\u2082_le fun i _ => h_le i) _ _\n    (generateFrom_piiUnionInter_measurableSet m S).symm\n    (generateFrom_piiUnionInter_measurableSet m T).symm _", "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_indep : iIndep m\nS T : Set \u03b9\nhST : Disjoint S T\n\u22a2 Indep (\u2a06 (i : \u03b9) (_ : i \u2208 S), m i) (\u2a06 (i : \u03b9) (_ : i \u2208 T), m i)", "state_after": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_indep : iIndep m\nS T : Set \u03b9\nhST : Disjoint S T\n\u22a2 IsPiSystem (piiUnionInter (fun n => {s | MeasurableSet s}) S)\n\ncase refine'_2\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_indep : iIndep m\nS T : Set \u03b9\nhST : Disjoint S T\n\u22a2 IsPiSystem (piiUnionInter (fun n => {s | MeasurableSet s}) T)\n\ncase refine'_3\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_indep : iIndep m\nS T : Set \u03b9\nhST : Disjoint S T\n\u22a2 IndepSets (piiUnionInter (fun n => {s | MeasurableSet s}) S) (piiUnionInter (fun n => {s | MeasurableSet s}) T)"}, {"tactic": "exact isPiSystem_piiUnionInter _ (fun n => @isPiSystem_measurableSet \u03a9 (m n)) _", "state_before": "case refine'_1\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_indep : iIndep m\nS T : Set \u03b9\nhST : Disjoint S T\n\u22a2 IsPiSystem (piiUnionInter (fun n => {s | MeasurableSet s}) S)", "state_after": "no goals"}, {"tactic": "exact isPiSystem_piiUnionInter _ (fun n => @isPiSystem_measurableSet \u03a9 (m n)) _", "state_before": "case refine'_2\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_indep : iIndep m\nS T : Set \u03b9\nhST : Disjoint S T\n\u22a2 IsPiSystem (piiUnionInter (fun n => {s | MeasurableSet s}) T)", "state_after": "no goals"}, {"tactic": "classical exact indepSets_piiUnionInter_of_disjoint h_indep hST", "state_before": "case refine'_3\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_indep : iIndep m\nS T : Set \u03b9\nhST : Disjoint S T\n\u22a2 IndepSets (piiUnionInter (fun n => {s | MeasurableSet s}) S) (piiUnionInter (fun n => {s | MeasurableSet s}) T)", "state_after": "no goals"}, {"tactic": "exact indepSets_piiUnionInter_of_disjoint h_indep hST", "state_before": "case refine'_3\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nm : \u03b9 \u2192 MeasurableSpace \u03a9\nh_le : \u2200 (i : \u03b9), m i \u2264 m0\nh_indep : iIndep m\nS T : Set \u03b9\nhST : Disjoint S T\n\u22a2 IndepSets (piiUnionInter (fun n => {s | MeasurableSet s}) S) (piiUnionInter (fun n => {s | MeasurableSet s}) T)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Generator.lean", "full_name": "CategoryTheory.isCodetector_op_iff", "start": [426, 1], "end": [427, 74], "traced_tactics": [{"tactic": "rw [IsDetector, IsCodetector, \u2190 isCodetecting_op_iff, Set.singleton_op]", "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category C\nD : Type u\u2082\ninst\u271d : Category D\nG : C\n\u22a2 IsCodetector G.op \u2194 IsDetector G", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/OrderOfElement.lean", "full_name": "orderOf_pos'", "start": [151, 1], "end": [152, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Ordmap/Ordset.lean", "full_name": "Ordnode.erase.valid", "start": [1635, 1], "end": [1636, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "Complex.tanh_ofReal_im", "start": [719, 1], "end": [719, 95], "traced_tactics": [{"tactic": "rw [\u2190 ofReal_tanh_ofReal_re, ofReal_im]", "state_before": "x\u271d y : \u2102\nx : \u211d\n\u22a2 (tanh \u2191x).im = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Stream/Init.lean", "full_name": "Stream'.zip_inits_tails", "start": [731, 1], "end": [734, 17], "traced_tactics": [{"tactic": "apply Stream'.ext", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 zip appendStream' (inits s) (tails s) = const s", "state_after": "case a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 \u2200 (n : \u2115), nth (zip appendStream' (inits s) (tails s)) n = nth (const s) n"}, {"tactic": "intro n", "state_before": "case a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\n\u22a2 \u2200 (n : \u2115), nth (zip appendStream' (inits s) (tails s)) n = nth (const s) n", "state_after": "case a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\nn : \u2115\n\u22a2 nth (zip appendStream' (inits s) (tails s)) n = nth (const s) n"}, {"tactic": "rw [nth_zip, nth_inits, nth_tails, nth_const, take_succ, cons_append_stream, append_take_drop,\n  Stream'.eta]", "state_before": "case a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns : Stream' \u03b1\nn : \u2115\n\u22a2 nth (zip appendStream' (inits s) (tails s)) n = nth (const s) n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Powerset.lean", "full_name": "Finset.card_powersetLen", "start": [217, 1], "end": [218, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "full_name": "AlgHom.coe_range", "start": [619, 1], "end": [622, 6], "traced_tactics": [{"tactic": "ext", "state_before": "R' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\n\u03c6\u271d \u03c6 : A \u2192\u2090[R] B\n\u22a2 \u2191(AlgHom.range \u03c6) = Set.range \u2191\u03c6", "state_after": "case h\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\n\u03c6\u271d \u03c6 : A \u2192\u2090[R] B\nx\u271d : B\n\u22a2 x\u271d \u2208 \u2191(AlgHom.range \u03c6) \u2194 x\u271d \u2208 Set.range \u2191\u03c6"}, {"tactic": "rw [SetLike.mem_coe, mem_range]", "state_before": "case h\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\n\u03c6\u271d \u03c6 : A \u2192\u2090[R] B\nx\u271d : B\n\u22a2 x\u271d \u2208 \u2191(AlgHom.range \u03c6) \u2194 x\u271d \u2208 Set.range \u2191\u03c6", "state_after": "case h\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\n\u03c6\u271d \u03c6 : A \u2192\u2090[R] B\nx\u271d : B\n\u22a2 (\u2203 x, \u2191\u03c6 x = x\u271d) \u2194 x\u271d \u2208 Set.range \u2191\u03c6"}, {"tactic": "rfl", "state_before": "case h\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Semiring C\ninst\u271d : Algebra R C\n\u03c6\u271d \u03c6 : A \u2192\u2090[R] B\nx\u271d : B\n\u22a2 (\u2203 x, \u2191\u03c6 x = x\u271d) \u2194 x\u271d \u2208 Set.range \u2191\u03c6", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "full_name": "PowerSeries.rescale_X", "start": [1956, 1], "end": [1959, 32], "traced_tactics": [{"tactic": "ext", "state_before": "R : Type ?u.3514033\nA : Type u_1\ninst\u271d : CommRing A\na : A\n\u22a2 \u2191(rescale a) X = \u2191(C A) a * X", "state_after": "case h\nR : Type ?u.3514033\nA : Type u_1\ninst\u271d : CommRing A\na : A\nn\u271d : \u2115\n\u22a2 \u2191(coeff A n\u271d) (\u2191(rescale a) X) = \u2191(coeff A n\u271d) (\u2191(C A) a * X)"}, {"tactic": "simp only [coeff_rescale, coeff_C_mul, coeff_X]", "state_before": "case h\nR : Type ?u.3514033\nA : Type u_1\ninst\u271d : CommRing A\na : A\nn\u271d : \u2115\n\u22a2 \u2191(coeff A n\u271d) (\u2191(rescale a) X) = \u2191(coeff A n\u271d) (\u2191(C A) a * X)", "state_after": "case h\nR : Type ?u.3514033\nA : Type u_1\ninst\u271d : CommRing A\na : A\nn\u271d : \u2115\n\u22a2 (a ^ n\u271d * if n\u271d = 1 then 1 else 0) = a * if n\u271d = 1 then 1 else 0"}, {"tactic": "split_ifs with h <;> simp [h]", "state_before": "case h\nR : Type ?u.3514033\nA : Type u_1\ninst\u271d : CommRing A\na : A\nn\u271d : \u2115\n\u22a2 (a ^ n\u271d * if n\u271d = 1 then 1 else 0) = a * if n\u271d = 1 then 1 else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.range_id", "start": [856, 1], "end": [857, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.injOn_preimage", "start": [702, 1], "end": [703, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/NonIntegrable.lean", "full_name": "not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_within_diff_singleton", "start": [106, 1], "end": [123, 19], "traced_tactics": [{"tactic": "have : l \u2264 \ud835\udcdd[[[a, b]] \\ {c}] c := le_inf hle (le_principal_iff.2 hmem)", "state_before": "case intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SecondCountableTopology E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedAddCommGroup F\nf : \u211d \u2192 E\ng : \u211d \u2192 F\na b c : \u211d\nhne : a \u2260 b\nhc : c \u2208 [[a, b]]\nh_deriv : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[[[a, b]] \\ {c}] c, DifferentiableAt \u211d f x\nh_infty : Tendsto (fun x => \u2016f x\u2016) (\ud835\udcdd[[[a, b]] \\ {c}] c) atTop\nhg : deriv f =O[\ud835\udcdd[[[a, b]] \\ {c}] c] g\nl : Filter \u211d\nhl : TendstoIxxClass Icc l l\nhl' : NeBot l\nhle : l \u2264 \ud835\udcdd c\nhmem : [[a, b]] \\ {c} \u2208 l\n\u22a2 \u00acIntervalIntegrable g volume a b", "state_after": "case intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SecondCountableTopology E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedAddCommGroup F\nf : \u211d \u2192 E\ng : \u211d \u2192 F\na b c : \u211d\nhne : a \u2260 b\nhc : c \u2208 [[a, b]]\nh_deriv : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[[[a, b]] \\ {c}] c, DifferentiableAt \u211d f x\nh_infty : Tendsto (fun x => \u2016f x\u2016) (\ud835\udcdd[[[a, b]] \\ {c}] c) atTop\nhg : deriv f =O[\ud835\udcdd[[[a, b]] \\ {c}] c] g\nl : Filter \u211d\nhl : TendstoIxxClass Icc l l\nhl' : NeBot l\nhle : l \u2264 \ud835\udcdd c\nhmem : [[a, b]] \\ {c} \u2208 l\nthis : l \u2264 \ud835\udcdd[[[a, b]] \\ {c}] c\n\u22a2 \u00acIntervalIntegrable g volume a b"}, {"tactic": "exact not_intervalIntegrable_of_tendsto_norm_atTop_of_deriv_isBigO_filter l\n  (mem_of_superset hmem (diff_subset _ _)) (h_deriv.filter_mono this) (h_infty.mono_left this)\n  (hg.mono this)", "state_before": "case intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SecondCountableTopology E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedAddCommGroup F\nf : \u211d \u2192 E\ng : \u211d \u2192 F\na b c : \u211d\nhne : a \u2260 b\nhc : c \u2208 [[a, b]]\nh_deriv : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[[[a, b]] \\ {c}] c, DifferentiableAt \u211d f x\nh_infty : Tendsto (fun x => \u2016f x\u2016) (\ud835\udcdd[[[a, b]] \\ {c}] c) atTop\nhg : deriv f =O[\ud835\udcdd[[[a, b]] \\ {c}] c] g\nl : Filter \u211d\nhl : TendstoIxxClass Icc l l\nhl' : NeBot l\nhle : l \u2264 \ud835\udcdd c\nhmem : [[a, b]] \\ {c} \u2208 l\nthis : l \u2264 \ud835\udcdd[[[a, b]] \\ {c}] c\n\u22a2 \u00acIntervalIntegrable g volume a b", "state_after": "no goals"}, {"tactic": "cases' (min_lt_max.2 hne).lt_or_lt c with hlt hlt", "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SecondCountableTopology E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedAddCommGroup F\nf : \u211d \u2192 E\ng : \u211d \u2192 F\na b c : \u211d\nhne : a \u2260 b\nhc : c \u2208 [[a, b]]\nh_deriv : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[[[a, b]] \\ {c}] c, DifferentiableAt \u211d f x\nh_infty : Tendsto (fun x => \u2016f x\u2016) (\ud835\udcdd[[[a, b]] \\ {c}] c) atTop\nhg : deriv f =O[\ud835\udcdd[[[a, b]] \\ {c}] c] g\n\u22a2 \u2203 l, TendstoIxxClass Icc l l \u2227 NeBot l \u2227 l \u2264 \ud835\udcdd c \u2227 [[a, b]] \\ {c} \u2208 l", "state_after": "case inl\nE : Type u_1\nF : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SecondCountableTopology E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedAddCommGroup F\nf : \u211d \u2192 E\ng : \u211d \u2192 F\na b c : \u211d\nhne : a \u2260 b\nhc : c \u2208 [[a, b]]\nh_deriv : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[[[a, b]] \\ {c}] c, DifferentiableAt \u211d f x\nh_infty : Tendsto (fun x => \u2016f x\u2016) (\ud835\udcdd[[[a, b]] \\ {c}] c) atTop\nhg : deriv f =O[\ud835\udcdd[[[a, b]] \\ {c}] c] g\nhlt : min a b < c\n\u22a2 \u2203 l, TendstoIxxClass Icc l l \u2227 NeBot l \u2227 l \u2264 \ud835\udcdd c \u2227 [[a, b]] \\ {c} \u2208 l\n\ncase inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SecondCountableTopology E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedAddCommGroup F\nf : \u211d \u2192 E\ng : \u211d \u2192 F\na b c : \u211d\nhne : a \u2260 b\nhc : c \u2208 [[a, b]]\nh_deriv : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[[[a, b]] \\ {c}] c, DifferentiableAt \u211d f x\nh_infty : Tendsto (fun x => \u2016f x\u2016) (\ud835\udcdd[[[a, b]] \\ {c}] c) atTop\nhg : deriv f =O[\ud835\udcdd[[[a, b]] \\ {c}] c] g\nhlt : c < max a b\n\u22a2 \u2203 l, TendstoIxxClass Icc l l \u2227 NeBot l \u2227 l \u2264 \ud835\udcdd c \u2227 [[a, b]] \\ {c} \u2208 l"}, {"tactic": "refine' \u27e8\ud835\udcdd[<] c, inferInstance, inferInstance, inf_le_left, _\u27e9", "state_before": "case inl\nE : Type u_1\nF : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SecondCountableTopology E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedAddCommGroup F\nf : \u211d \u2192 E\ng : \u211d \u2192 F\na b c : \u211d\nhne : a \u2260 b\nhc : c \u2208 [[a, b]]\nh_deriv : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[[[a, b]] \\ {c}] c, DifferentiableAt \u211d f x\nh_infty : Tendsto (fun x => \u2016f x\u2016) (\ud835\udcdd[[[a, b]] \\ {c}] c) atTop\nhg : deriv f =O[\ud835\udcdd[[[a, b]] \\ {c}] c] g\nhlt : min a b < c\n\u22a2 \u2203 l, TendstoIxxClass Icc l l \u2227 NeBot l \u2227 l \u2264 \ud835\udcdd c \u2227 [[a, b]] \\ {c} \u2208 l", "state_after": "case inl\nE : Type u_1\nF : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SecondCountableTopology E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedAddCommGroup F\nf : \u211d \u2192 E\ng : \u211d \u2192 F\na b c : \u211d\nhne : a \u2260 b\nhc : c \u2208 [[a, b]]\nh_deriv : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[[[a, b]] \\ {c}] c, DifferentiableAt \u211d f x\nh_infty : Tendsto (fun x => \u2016f x\u2016) (\ud835\udcdd[[[a, b]] \\ {c}] c) atTop\nhg : deriv f =O[\ud835\udcdd[[[a, b]] \\ {c}] c] g\nhlt : min a b < c\n\u22a2 [[a, b]] \\ {c} \u2208 \ud835\udcdd[Iio c] c"}, {"tactic": "rw [\u2190 Iic_diff_right]", "state_before": "case inl\nE : Type u_1\nF : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SecondCountableTopology E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedAddCommGroup F\nf : \u211d \u2192 E\ng : \u211d \u2192 F\na b c : \u211d\nhne : a \u2260 b\nhc : c \u2208 [[a, b]]\nh_deriv : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[[[a, b]] \\ {c}] c, DifferentiableAt \u211d f x\nh_infty : Tendsto (fun x => \u2016f x\u2016) (\ud835\udcdd[[[a, b]] \\ {c}] c) atTop\nhg : deriv f =O[\ud835\udcdd[[[a, b]] \\ {c}] c] g\nhlt : min a b < c\n\u22a2 [[a, b]] \\ {c} \u2208 \ud835\udcdd[Iio c] c", "state_after": "case inl\nE : Type u_1\nF : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SecondCountableTopology E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedAddCommGroup F\nf : \u211d \u2192 E\ng : \u211d \u2192 F\na b c : \u211d\nhne : a \u2260 b\nhc : c \u2208 [[a, b]]\nh_deriv : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[[[a, b]] \\ {c}] c, DifferentiableAt \u211d f x\nh_infty : Tendsto (fun x => \u2016f x\u2016) (\ud835\udcdd[[[a, b]] \\ {c}] c) atTop\nhg : deriv f =O[\ud835\udcdd[[[a, b]] \\ {c}] c] g\nhlt : min a b < c\n\u22a2 [[a, b]] \\ {c} \u2208 \ud835\udcdd[Iic c \\ {c}] c"}, {"tactic": "exact diff_mem_nhdsWithin_diff (Icc_mem_nhdsWithin_Iic \u27e8hlt, hc.2\u27e9) _", "state_before": "case inl\nE : Type u_1\nF : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SecondCountableTopology E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedAddCommGroup F\nf : \u211d \u2192 E\ng : \u211d \u2192 F\na b c : \u211d\nhne : a \u2260 b\nhc : c \u2208 [[a, b]]\nh_deriv : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[[[a, b]] \\ {c}] c, DifferentiableAt \u211d f x\nh_infty : Tendsto (fun x => \u2016f x\u2016) (\ud835\udcdd[[[a, b]] \\ {c}] c) atTop\nhg : deriv f =O[\ud835\udcdd[[[a, b]] \\ {c}] c] g\nhlt : min a b < c\n\u22a2 [[a, b]] \\ {c} \u2208 \ud835\udcdd[Iic c \\ {c}] c", "state_after": "no goals"}, {"tactic": "refine' \u27e8\ud835\udcdd[>] c, inferInstance, inferInstance, inf_le_left, _\u27e9", "state_before": "case inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SecondCountableTopology E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedAddCommGroup F\nf : \u211d \u2192 E\ng : \u211d \u2192 F\na b c : \u211d\nhne : a \u2260 b\nhc : c \u2208 [[a, b]]\nh_deriv : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[[[a, b]] \\ {c}] c, DifferentiableAt \u211d f x\nh_infty : Tendsto (fun x => \u2016f x\u2016) (\ud835\udcdd[[[a, b]] \\ {c}] c) atTop\nhg : deriv f =O[\ud835\udcdd[[[a, b]] \\ {c}] c] g\nhlt : c < max a b\n\u22a2 \u2203 l, TendstoIxxClass Icc l l \u2227 NeBot l \u2227 l \u2264 \ud835\udcdd c \u2227 [[a, b]] \\ {c} \u2208 l", "state_after": "case inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SecondCountableTopology E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedAddCommGroup F\nf : \u211d \u2192 E\ng : \u211d \u2192 F\na b c : \u211d\nhne : a \u2260 b\nhc : c \u2208 [[a, b]]\nh_deriv : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[[[a, b]] \\ {c}] c, DifferentiableAt \u211d f x\nh_infty : Tendsto (fun x => \u2016f x\u2016) (\ud835\udcdd[[[a, b]] \\ {c}] c) atTop\nhg : deriv f =O[\ud835\udcdd[[[a, b]] \\ {c}] c] g\nhlt : c < max a b\n\u22a2 [[a, b]] \\ {c} \u2208 \ud835\udcdd[Ioi c] c"}, {"tactic": "rw [\u2190 Ici_diff_left]", "state_before": "case inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SecondCountableTopology E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedAddCommGroup F\nf : \u211d \u2192 E\ng : \u211d \u2192 F\na b c : \u211d\nhne : a \u2260 b\nhc : c \u2208 [[a, b]]\nh_deriv : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[[[a, b]] \\ {c}] c, DifferentiableAt \u211d f x\nh_infty : Tendsto (fun x => \u2016f x\u2016) (\ud835\udcdd[[[a, b]] \\ {c}] c) atTop\nhg : deriv f =O[\ud835\udcdd[[[a, b]] \\ {c}] c] g\nhlt : c < max a b\n\u22a2 [[a, b]] \\ {c} \u2208 \ud835\udcdd[Ioi c] c", "state_after": "case inr\nE : Type u_1\nF : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace 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\u2286 \u22c3 (x : \u2191t), f \u2191x\n\u22a2 Dense (\u22c3 (x : \u2191t), interior (f \u2191x))"}, {"tactic": "exact hs.dense_iUnion_interior_of_closed hd hc hU", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.25899\n\u03b3 : Type ?u.25902\n\u03b9 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\nt : Set \u03b9\ns : Set \u03b1\nhs : IsG\u03b4 s\nhd : Dense s\nht : Set.Countable t\nf : \u03b9 \u2192 Set \u03b1\nthis : Encodable \u2191t\nhc : \u2200 (x : \u2191t), IsClosed (f \u2191x)\nhU : s \u2286 \u22c3 (x : \u2191t), f \u2191x\n\u22a2 Dense (\u22c3 (x : \u2191t), interior (f \u2191x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Coprime/Basic.lean", "full_name": "IsCoprime.isUnit_of_dvd'", "start": [171, 1], "end": [173, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/PiSystem.lean", "full_name": "IsPiSystem.comap", "start": [107, 1], "end": [115, 14], "traced_tactics": [{"tactic": "rintro _ \u27e8s, hs_mem, rfl\u27e9 _ \u27e8t, ht_mem, rfl\u27e9 hst", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nS : Set (Set \u03b2)\nh_pi : IsPiSystem S\nf : \u03b1 \u2192 \u03b2\n\u22a2 IsPiSystem {s | \u2203 t, t \u2208 S \u2227 f \u207b\u00b9' t = s}", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nS : Set (Set \u03b2)\nh_pi : IsPiSystem S\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhs_mem : s \u2208 S\nt : Set \u03b2\nht_mem : t \u2208 S\nhst : Set.Nonempty (f \u207b\u00b9' s \u2229 f \u207b\u00b9' t)\n\u22a2 f \u207b\u00b9' s \u2229 f \u207b\u00b9' t \u2208 {s | \u2203 t, t \u2208 S \u2227 f \u207b\u00b9' t = s}"}, {"tactic": "rw [\u2190 Set.preimage_inter] at hst\u22a2", "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nS : Set (Set \u03b2)\nh_pi : IsPiSystem S\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhs_mem : s \u2208 S\nt : Set \u03b2\nht_mem : t \u2208 S\nhst : Set.Nonempty (f \u207b\u00b9' s \u2229 f \u207b\u00b9' t)\n\u22a2 f \u207b\u00b9' s \u2229 f \u207b\u00b9' t \u2208 {s | \u2203 t, t \u2208 S \u2227 f \u207b\u00b9' t = s}", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nS : Set (Set \u03b2)\nh_pi : IsPiSystem S\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhs_mem : s \u2208 S\nt : Set \u03b2\nht_mem : t \u2208 S\nhst : Set.Nonempty (f \u207b\u00b9' (s \u2229 t))\n\u22a2 f \u207b\u00b9' (s \u2229 t) \u2208 {s | \u2203 t, t \u2208 S \u2227 f \u207b\u00b9' t = s}"}, {"tactic": "refine' \u27e8s \u2229 t, h_pi s hs_mem t ht_mem _, rfl\u27e9", "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nS : Set (Set \u03b2)\nh_pi : IsPiSystem S\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhs_mem : s \u2208 S\nt : Set \u03b2\nht_mem : t \u2208 S\nhst : Set.Nonempty (f \u207b\u00b9' (s \u2229 t))\n\u22a2 f \u207b\u00b9' (s \u2229 t) \u2208 {s | \u2203 t, t \u2208 S \u2227 f \u207b\u00b9' t = s}", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nS : Set (Set \u03b2)\nh_pi : IsPiSystem S\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhs_mem : s \u2208 S\nt : Set \u03b2\nht_mem : t \u2208 S\nhst : Set.Nonempty (f \u207b\u00b9' (s \u2229 t))\n\u22a2 Set.Nonempty (s \u2229 t)"}, {"tactic": "by_contra h", "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nS : Set (Set \u03b2)\nh_pi : IsPiSystem S\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhs_mem : s \u2208 S\nt : Set \u03b2\nht_mem : t \u2208 S\nhst : Set.Nonempty (f \u207b\u00b9' (s \u2229 t))\n\u22a2 Set.Nonempty (s \u2229 t)", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nS : Set (Set \u03b2)\nh_pi : IsPiSystem S\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhs_mem : s \u2208 S\nt : Set \u03b2\nht_mem : t \u2208 S\nhst : Set.Nonempty (f \u207b\u00b9' (s \u2229 t))\nh : \u00acSet.Nonempty (s \u2229 t)\n\u22a2 False"}, {"tactic": "rw [Set.not_nonempty_iff_eq_empty] at h", "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nS : Set (Set \u03b2)\nh_pi : IsPiSystem S\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhs_mem : s \u2208 S\nt : Set \u03b2\nht_mem : t \u2208 S\nhst : Set.Nonempty (f \u207b\u00b9' (s \u2229 t))\nh : \u00acSet.Nonempty (s \u2229 t)\n\u22a2 False", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nS : Set (Set \u03b2)\nh_pi : IsPiSystem S\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhs_mem : s \u2208 S\nt : Set \u03b2\nht_mem : t \u2208 S\nhst : Set.Nonempty (f \u207b\u00b9' (s \u2229 t))\nh : s \u2229 t = \u2205\n\u22a2 False"}, {"tactic": "rw [h] at hst", "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nS : Set (Set \u03b2)\nh_pi : IsPiSystem S\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhs_mem : s \u2208 S\nt : Set \u03b2\nht_mem : t \u2208 S\nhst : Set.Nonempty (f \u207b\u00b9' (s \u2229 t))\nh : s \u2229 t = \u2205\n\u22a2 False", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nS : Set (Set \u03b2)\nh_pi : IsPiSystem S\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhs_mem : s \u2208 S\nt : Set \u03b2\nht_mem : t \u2208 S\nhst : Set.Nonempty (f \u207b\u00b9' \u2205)\nh : s \u2229 t = \u2205\n\u22a2 False"}, {"tactic": "simp at hst", "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nS : Set (Set \u03b2)\nh_pi : IsPiSystem S\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\nhs_mem : s \u2208 S\nt : Set \u03b2\nht_mem : t \u2208 S\nhst : Set.Nonempty (f \u207b\u00b9' \u2205)\nh : s \u2229 t = \u2205\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Isometry.lean", "full_name": "Isometry.embedding", "start": [207, 11], "end": [208, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean", "full_name": "AddMonoidHom.coe_ofMapMidpoint", "start": [261, 1], "end": [264, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/Basic.lean", "full_name": "Equiv.swap_eq_refl_iff", "start": [1587, 1], "end": [1589, 43], "traced_tactics": [{"tactic": "refine' \u27e8fun h => (Equiv.refl _).injective _, fun h => h \u25b8 swap_self _\u27e9", "state_before": "\u03b1 : Sort u_1\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\n\u22a2 swap x y = Equiv.refl \u03b1 \u2194 x = y", "state_after": "\u03b1 : Sort u_1\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nh : swap x y = Equiv.refl \u03b1\n\u22a2 \u2191(Equiv.refl \u03b1) x = \u2191(Equiv.refl \u03b1) y"}, {"tactic": "rw [\u2190 h, swap_apply_left, h, refl_apply]", "state_before": "\u03b1 : Sort u_1\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\nh : swap x y = Equiv.refl \u03b1\n\u22a2 \u2191(Equiv.refl \u03b1) x = \u2191(Equiv.refl \u03b1) y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Init/Algebra/Order.lean", "full_name": "eq_or_lt_of_not_lt", "start": [390, 1], "end": [391, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/AffineIsometry.lean", "full_name": "AffineSubspace.isometryEquivMap.apply_symm_apply", "start": [905, 1], "end": [907, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Basic.lean", "full_name": "Metric.isOpen_singleton_iff", "start": [1113, 1], "end": [1115, 52], "traced_tactics": [{"tactic": "simp [isOpen_iff, subset_singleton_iff, mem_ball]", "state_before": "\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type ?u.146352\n\u03b9 : Type ?u.146355\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\u271d\nx\u271d y z : \u03b1\u271d\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\n\u22a2 IsOpen {x} \u2194 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : \u03b1), dist y x < \u03b5 \u2192 y = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.integral_comp_add_left", "start": [747, 8], "end": [749, 60], "traced_tactics": [{"tactic": "simpa only [add_comm d] using integral_comp_add_right f d", "state_before": "\u03b9 : Type ?u.14871374\n\ud835\udd5c : Type ?u.14871377\nE : Type u_1\nF : Type ?u.14871383\nA : Type ?u.14871386\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na b c d\u271d : \u211d\nf : \u211d \u2192 E\nd : \u211d\n\u22a2 (\u222b (x : \u211d) in a..b, f (d + x)) = \u222b (x : \u211d) in d + a..d + b, f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Quaternion.lean", "full_name": "Quaternion.normSq_neg", "start": [1225, 1], "end": [1225, 96], "traced_tactics": [{"tactic": "simp only [normSq_def, star_neg, neg_mul_neg]", "state_before": "S : Type ?u.680334\nT : Type ?u.680337\nR : Type u_1\ninst\u271d : CommRing R\nr x y z : R\na b c : \u210d[R]\n\u22a2 \u2191normSq (-a) = \u2191normSq a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/PEquiv.lean", "full_name": "PEquiv.mem_single_iff", "start": [350, 1], "end": [351, 50], "traced_tactics": [{"tactic": "dsimp [single]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : \u03b2\n\u22a2 b\u2081 \u2208 \u2191(single a\u2082 b\u2082) a\u2081 \u2194 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : \u03b2\n\u22a2 (b\u2081 \u2208 if a\u2081 = a\u2082 then some b\u2082 else none) \u2194 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082"}, {"tactic": "split_ifs <;> simp [*, eq_comm]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\ninst\u271d : DecidableEq \u03b3\na\u2081 a\u2082 : \u03b1\nb\u2081 b\u2082 : \u03b2\n\u22a2 (b\u2081 \u2208 if a\u2081 = a\u2082 then some b\u2082 else none) \u2194 a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/QuadraticForm/Basic.lean", "full_name": "QuadraticForm.map_smul", "start": [209, 1], "end": [210, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.MeasurePreserving.lintegral_comp_emb", "start": [1337, 1], "end": [1339, 80], "traced_tactics": [{"tactic": "rw [\u2190 hg.map_eq, hge.lintegral_map]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.1686960\n\u03b4 : Type ?u.1686963\nm : MeasurableSpace \u03b1\n\u03bc \u03bd\u271d : Measure \u03b1\nmb : MeasurableSpace \u03b2\n\u03bd : Measure \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g\nhge : MeasurableEmbedding g\nf : \u03b2 \u2192 \u211d\u22650\u221e\n\u22a2 (\u222b\u207b (a : \u03b1), f (g a) \u2202\u03bc) = \u222b\u207b (b : \u03b2), f b \u2202\u03bd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.eval_preimage", "start": [836, 1], "end": [839, 80], "traced_tactics": [{"tactic": "ext x", "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type ?u.158390\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\ns : Set (\u03b1 i)\n\u22a2 eval i \u207b\u00b9' s = pi univ (update (fun i => univ) i s)", "state_after": "case h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type ?u.158390\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\ns : Set (\u03b1 i)\nx : (x : \u03b9) \u2192 \u03b1 x\n\u22a2 x \u2208 eval i \u207b\u00b9' s \u2194 x \u2208 pi univ (update (fun i => univ) i s)"}, {"tactic": "simp [@forall_update_iff _ (fun i => Set (\u03b1 i)) _ _ _ _ fun i' y => x i' \u2208 y]", "state_before": "case h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type ?u.158390\ns\u271d s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\ns : Set (\u03b1 i)\nx : (x : \u03b9) \u2192 \u03b1 x\n\u22a2 x \u2208 eval i \u207b\u00b9' s \u2194 x \u2208 pi univ (update (fun i => univ) i s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "full_name": "MeasureTheory.le_extend", "start": [1338, 1], "end": [1341, 6], "traced_tactics": [{"tactic": "simp only [extend, le_iInf_iff]", "state_before": "\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\ns : \u03b1\nh : P s\n\u22a2 m s h \u2264 extend m s", "state_after": "\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\ns : \u03b1\nh : P s\n\u22a2 \u2200 (i : P s), m s h \u2264 m s i"}, {"tactic": "intro", "state_before": "\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\ns : \u03b1\nh : P s\n\u22a2 \u2200 (i : P s), m s h \u2264 m s i", "state_after": "\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\ns : \u03b1\nh i\u271d : P s\n\u22a2 m s h \u2264 m s i\u271d"}, {"tactic": "rfl", "state_before": "\u03b1 : Type u_1\nP : \u03b1 \u2192 Prop\nm : (s : \u03b1) \u2192 P s \u2192 \u211d\u22650\u221e\ns : \u03b1\nh i\u271d : P s\n\u22a2 m s h \u2264 m s i\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/LocallyConstant/Basic.lean", "full_name": "IsLocallyConstant.tfae", "start": [48, 11], "end": [66, 14], "traced_tactics": [{"tactic": "tfae_have 1 \u2192 4", "state_before": "X : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\n\u22a2 TFAE\n    [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]", "state_after": "case tfae_1_to_4\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\n\u22a2 IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\n\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\n\u22a2 TFAE\n    [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]"}, {"tactic": "exact fun h y => h {y}", "state_before": "case tfae_1_to_4\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\n\u22a2 IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\n\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\n\u22a2 TFAE\n    [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]", "state_after": "X : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\n\u22a2 TFAE\n    [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]"}, {"tactic": "tfae_have 4 \u2192 3", "state_before": "X : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\n\u22a2 TFAE\n    [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]", "state_after": "case tfae_4_to_3\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\n\u22a2 (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\n\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\n\u22a2 TFAE\n    [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]"}, {"tactic": "exact fun h x => h (f x)", "state_before": "case tfae_4_to_3\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\n\u22a2 (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\n\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\n\u22a2 TFAE\n    [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]", "state_after": "X : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\n\u22a2 TFAE\n    [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]"}, {"tactic": "tfae_have 3 \u2192 2", "state_before": "X : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\n\u22a2 TFAE\n    [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]", "state_after": "case tfae_3_to_2\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\n\u22a2 (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\n\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\n\u22a2 TFAE\n    [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]"}, {"tactic": "exact fun h x => IsOpen.mem_nhds (h x) rfl", "state_before": "case tfae_3_to_2\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\n\u22a2 (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\n\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\n\u22a2 TFAE\n    [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]", "state_after": "X : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\n\u22a2 TFAE\n    [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]"}, {"tactic": "tfae_have 2 \u2192 5", "state_before": "X : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\n\u22a2 TFAE\n    [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]", "state_after": "case tfae_2_to_5\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\n\u22a2 (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\n\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 :\n  (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\n\u22a2 TFAE\n    [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]"}, {"tactic": "tfae_have 5 \u2192 1", "state_before": "X : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 :\n  (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\n\u22a2 TFAE\n    [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]", "state_after": "case tfae_5_to_1\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 :\n  (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\n\u22a2 (\u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x) \u2192 IsLocallyConstant f\n\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 :\n  (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\ntfae_5_to_1 : (\u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x) \u2192 IsLocallyConstant f\n\u22a2 TFAE\n    [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]"}, {"tactic": "tfae_finish", "state_before": "X : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 :\n  (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\ntfae_5_to_1 : (\u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x) \u2192 IsLocallyConstant f\n\u22a2 TFAE\n    [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x]", "state_after": "no goals"}, {"tactic": "intro h x", "state_before": "case tfae_2_to_5\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\n\u22a2 (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x", "state_after": "case tfae_2_to_5\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\nh : \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\nx : X\n\u22a2 \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x"}, {"tactic": "rcases mem_nhds_iff.1 (h x) with \u27e8U, eq, hU, hx\u27e9", "state_before": "case tfae_2_to_5\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\nh : \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\nx : X\n\u22a2 \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x", "state_after": "case tfae_2_to_5.intro.intro.intro\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\nh : \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\nx : X\nU : Set X\neq : U \u2286 {x_1 | (fun x' => f x' = f x) x_1}\nhU : IsOpen U\nhx : x \u2208 U\n\u22a2 \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x"}, {"tactic": "exact \u27e8U, hU, hx, eq\u27e9", "state_before": "case tfae_2_to_5.intro.intro.intro\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\nh : \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\nx : X\nU : Set X\neq : U \u2286 {x_1 | (fun x' => f x' = f x) x_1}\nhU : IsOpen U\nhx : x \u2208 U\n\u22a2 \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x", "state_after": "no goals"}, {"tactic": "intro h s", "state_before": "case tfae_5_to_1\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 :\n  (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\n\u22a2 (\u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x) \u2192 IsLocallyConstant f", "state_after": "case tfae_5_to_1\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 :\n  (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\nh : \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\ns : Set Y\n\u22a2 IsOpen (f \u207b\u00b9' s)"}, {"tactic": "refine' isOpen_iff_forall_mem_open.2 fun x hx => _", "state_before": "case tfae_5_to_1\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 :\n  (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\nh : \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\ns : Set Y\n\u22a2 IsOpen (f \u207b\u00b9' s)", "state_after": "case tfae_5_to_1\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 :\n  (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\nh : \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\ns : Set Y\nx : X\nhx : x \u2208 f \u207b\u00b9' s\n\u22a2 \u2203 t, t \u2286 f \u207b\u00b9' s \u2227 IsOpen t \u2227 x \u2208 t"}, {"tactic": "rcases h x with \u27e8U, hU, hxU, eq\u27e9", "state_before": "case tfae_5_to_1\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 :\n  (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\nh : \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\ns : Set Y\nx : X\nhx : x \u2208 f \u207b\u00b9' s\n\u22a2 \u2203 t, t \u2286 f \u207b\u00b9' s \u2227 IsOpen t \u2227 x \u2208 t", "state_after": "case tfae_5_to_1.intro.intro.intro\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 :\n  (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\nh : \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\ns : Set Y\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nU : Set X\nhU : IsOpen U\nhxU : x \u2208 U\neq : \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\n\u22a2 \u2203 t, t \u2286 f \u207b\u00b9' s \u2227 IsOpen t \u2227 x \u2208 t"}, {"tactic": "exact \u27e8U, fun x' hx' => mem_preimage.2 <| (eq x' hx').symm \u25b8 hx, hU, hxU\u27e9", "state_before": "case tfae_5_to_1.intro.intro.intro\nX : Type u_1\nY : Type u_2\nZ : Type ?u.104\n\u03b1 : Type ?u.107\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 :\n  (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\nh : \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\ns : Set Y\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nU : Set X\nhU : IsOpen U\nhxU : x \u2208 U\neq : \u2200 (x' : X), x' \u2208 U \u2192 f x' = f x\n\u22a2 \u2203 t, t \u2286 f \u207b\u00b9' s \u2227 IsOpen t \u2227 x \u2208 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fin/Tuple/Basic.lean", "full_name": "Fin.repeat_one", "start": [388, 1], "end": [393, 42], "traced_tactics": [{"tactic": "generalize_proofs h", "state_before": "m n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d 0\nq : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (succ i)\ni : Fin n\ny : \u03b1\u271d (succ i)\nz : \u03b1\u271d 0\n\u03b1 : Type u_1\na : Fin n \u2192 \u03b1\n\u22a2 repeat 1 a = a \u2218 \u2191(cast (_ : 1 * n = n))", "state_after": "m n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d 0\nq : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (succ i)\ni : Fin n\ny : \u03b1\u271d (succ i)\nz : \u03b1\u271d 0\n\u03b1 : Type u_1\na : Fin n \u2192 \u03b1\nh : 1 * n = n\n\u22a2 repeat 1 a = a \u2218 \u2191(cast h)"}, {"tactic": "apply funext", "state_before": "m n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d 0\nq : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (succ i)\ni : Fin n\ny : \u03b1\u271d (succ i)\nz : \u03b1\u271d 0\n\u03b1 : Type u_1\na : Fin n \u2192 \u03b1\nh : 1 * n = n\n\u22a2 repeat 1 a = a \u2218 \u2191(cast h)", "state_after": "case h\nm n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d 0\nq : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (succ i)\ni : Fin n\ny : \u03b1\u271d (succ i)\nz : \u03b1\u271d 0\n\u03b1 : Type u_1\na : Fin n \u2192 \u03b1\nh : 1 * n = n\n\u22a2 \u2200 (x : Fin (1 * n)), repeat 1 a x = (a \u2218 \u2191(cast h)) x"}, {"tactic": "rw [(Fin.cast h.symm).surjective.forall]", "state_before": "case h\nm n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d 0\nq : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (succ i)\ni : Fin n\ny : \u03b1\u271d (succ i)\nz : \u03b1\u271d 0\n\u03b1 : Type u_1\na : Fin n \u2192 \u03b1\nh : 1 * n = n\n\u22a2 \u2200 (x : Fin (1 * n)), repeat 1 a x = (a \u2218 \u2191(cast h)) x", "state_after": "case h\nm n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d 0\nq : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (succ i)\ni : Fin n\ny : \u03b1\u271d (succ i)\nz : \u03b1\u271d 0\n\u03b1 : Type u_1\na : Fin n \u2192 \u03b1\nh : 1 * n = n\n\u22a2 \u2200 (x : Fin n), repeat 1 a (\u2191(cast (_ : n = 1 * n)) x) = (a \u2218 \u2191(cast h)) (\u2191(cast (_ : n = 1 * n)) x)"}, {"tactic": "intro i", "state_before": "case h\nm n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d 0\nq : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (succ i)\ni : Fin n\ny : \u03b1\u271d (succ i)\nz : \u03b1\u271d 0\n\u03b1 : Type u_1\na : Fin n \u2192 \u03b1\nh : 1 * n = n\n\u22a2 \u2200 (x : Fin n), repeat 1 a (\u2191(cast (_ : n = 1 * n)) x) = (a \u2218 \u2191(cast h)) (\u2191(cast (_ : n = 1 * n)) x)", "state_after": "case h\nm n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d 0\nq : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (succ i)\ni\u271d : Fin n\ny : \u03b1\u271d (succ i\u271d)\nz : \u03b1\u271d 0\n\u03b1 : Type u_1\na : Fin n \u2192 \u03b1\nh : 1 * n = n\ni : Fin n\n\u22a2 repeat 1 a (\u2191(cast (_ : n = 1 * n)) i) = (a \u2218 \u2191(cast h)) (\u2191(cast (_ : n = 1 * n)) i)"}, {"tactic": "simp [modNat, Nat.mod_eq_of_lt i.is_lt]", "state_before": "case h\nm n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d 0\nq : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (succ i)\ni\u271d : Fin n\ny : \u03b1\u271d (succ i\u271d)\nz : \u03b1\u271d 0\n\u03b1 : Type u_1\na : Fin n \u2192 \u03b1\nh : 1 * n = n\ni : Fin n\n\u22a2 repeat 1 a (\u2191(cast (_ : n = 1 * n)) i) = (a \u2218 \u2191(cast h)) (\u2191(cast (_ : n = 1 * n)) i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/DoubleCoset.lean", "full_name": "Doset.left_bot_eq_left_quot", "start": [200, 1], "end": [206, 6], "traced_tactics": [{"tactic": "unfold Quotient", "state_before": "G : Type u_1\ninst\u271d\u00b9 : Group G\n\u03b1 : Type ?u.69926\ninst\u271d : Mul \u03b1\nJ : Subgroup G\ng : G\nH : Subgroup G\n\u22a2 Quotient \u2191\u22a5.toSubmonoid \u2191H = (G \u29f8 H)", "state_after": "G : Type u_1\ninst\u271d\u00b9 : Group G\n\u03b1 : Type ?u.69926\ninst\u271d : Mul \u03b1\nJ : Subgroup G\ng : G\nH : Subgroup G\n\u22a2 _root_.Quotient (setoid \u2191\u22a5.toSubmonoid \u2191H) = (G \u29f8 H)"}, {"tactic": "congr", "state_before": "G : Type u_1\ninst\u271d\u00b9 : Group G\n\u03b1 : Type ?u.69926\ninst\u271d : Mul \u03b1\nJ : Subgroup G\ng : G\nH : Subgroup G\n\u22a2 _root_.Quotient (setoid \u2191\u22a5.toSubmonoid \u2191H) = (G \u29f8 H)", "state_after": "case e_s\nG : Type u_1\ninst\u271d\u00b9 : Group G\n\u03b1 : Type ?u.69926\ninst\u271d : Mul \u03b1\nJ : Subgroup G\ng : G\nH : Subgroup G\n\u22a2 setoid \u2191\u22a5.toSubmonoid \u2191H = QuotientGroup.leftRel H"}, {"tactic": "ext", "state_before": "case e_s\nG : Type u_1\ninst\u271d\u00b9 : Group G\n\u03b1 : Type ?u.69926\ninst\u271d : Mul \u03b1\nJ : Subgroup G\ng : G\nH : Subgroup G\n\u22a2 setoid \u2191\u22a5.toSubmonoid \u2191H = QuotientGroup.leftRel H", "state_after": "case e_s.H\nG : Type u_1\ninst\u271d\u00b9 : Group G\n\u03b1 : Type ?u.69926\ninst\u271d : Mul \u03b1\nJ : Subgroup G\ng : G\nH : Subgroup G\na\u271d b\u271d : G\n\u22a2 Setoid.Rel (setoid \u2191\u22a5.toSubmonoid \u2191H) a\u271d b\u271d \u2194 Setoid.Rel (QuotientGroup.leftRel H) a\u271d b\u271d"}, {"tactic": "simp_rw [\u2190 bot_rel_eq_leftRel H]", "state_before": "case e_s.H\nG : Type u_1\ninst\u271d\u00b9 : Group G\n\u03b1 : Type ?u.69926\ninst\u271d : Mul \u03b1\nJ : Subgroup G\ng : G\nH : Subgroup G\na\u271d b\u271d : G\n\u22a2 Setoid.Rel (setoid \u2191\u22a5.toSubmonoid \u2191H) a\u271d b\u271d \u2194 Setoid.Rel (QuotientGroup.leftRel H) a\u271d b\u271d", "state_after": "case e_s.H\nG : Type u_1\ninst\u271d\u00b9 : Group G\n\u03b1 : Type ?u.69926\ninst\u271d : Mul \u03b1\nJ : Subgroup G\ng : G\nH : Subgroup G\na\u271d b\u271d : G\n\u22a2 Setoid.Rel (setoid \u2191\u22a5.toSubmonoid \u2191H) a\u271d b\u271d \u2194 Setoid.Rel (setoid \u2191\u22a5 \u2191H) a\u271d b\u271d"}, {"tactic": "rfl", "state_before": "case e_s.H\nG : Type u_1\ninst\u271d\u00b9 : Group G\n\u03b1 : Type ?u.69926\ninst\u271d : Mul \u03b1\nJ : Subgroup G\ng : G\nH : Subgroup G\na\u271d b\u271d : G\n\u22a2 Setoid.Rel (setoid \u2191\u22a5.toSubmonoid \u2191H) a\u271d b\u271d \u2194 Setoid.Rel (setoid \u2191\u22a5 \u2191H) a\u271d b\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/NAry.lean", "full_name": "Filter.map_prod_eq_map\u2082'", "start": [73, 1], "end": [75, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Int/GCD.lean", "full_name": "pow_gcd_eq_one", "start": [504, 1], "end": [510, 79], "traced_tactics": [{"tactic": "rcases m with (rfl | m)", "state_before": "M : Type u_1\ninst\u271d : Monoid M\nx : M\nm n : \u2115\nhm : x ^ m = 1\nhn : x ^ n = 1\n\u22a2 x ^ Nat.gcd m n = 1", "state_after": "case zero\nM : Type u_1\ninst\u271d : Monoid M\nx : M\nn : \u2115\nhn : x ^ n = 1\nhm : x ^ Nat.zero = 1\n\u22a2 x ^ Nat.gcd Nat.zero n = 1\n\ncase succ\nM : Type u_1\ninst\u271d : Monoid M\nx : M\nn : \u2115\nhn : x ^ n = 1\nm : \u2115\nhm : x ^ Nat.succ m = 1\n\u22a2 x ^ Nat.gcd (Nat.succ m) n = 1"}, {"tactic": "obtain \u27e8y, rfl\u27e9 := isUnit_ofPowEqOne hm m.succ_ne_zero", "state_before": "case succ\nM : Type u_1\ninst\u271d : Monoid M\nx : M\nn : \u2115\nhn : x ^ n = 1\nm : \u2115\nhm : x ^ Nat.succ m = 1\n\u22a2 x ^ Nat.gcd (Nat.succ m) n = 1", "state_after": "case succ.intro\nM : Type u_1\ninst\u271d : Monoid M\nn m : \u2115\ny : M\u02e3\nhn : \u2191y ^ n = 1\nhm : \u2191y ^ Nat.succ m = 1\n\u22a2 \u2191y ^ Nat.gcd (Nat.succ m) n = 1"}, {"tactic": "simp only [\u2190 Units.val_pow_eq_pow_val] at *", "state_before": "case succ.intro\nM : Type u_1\ninst\u271d : Monoid M\nn m : \u2115\ny : M\u02e3\nhn : \u2191y ^ n = 1\nhm : \u2191y ^ Nat.succ m = 1\n\u22a2 \u2191y ^ Nat.gcd (Nat.succ m) n = 1", "state_after": "case succ.intro\nM : Type u_1\ninst\u271d : Monoid M\nn m : \u2115\ny : M\u02e3\nhn : \u2191(y ^ n) = 1\nhm : \u2191(y ^ Nat.succ m) = 1\n\u22a2 \u2191(y ^ Nat.gcd (Nat.succ m) n) = 1"}, {"tactic": "rw [\u2190 Units.val_one, \u2190 zpow_coe_nat, \u2190 Units.ext_iff] at *", "state_before": "case succ.intro\nM : Type u_1\ninst\u271d : Monoid M\nn m : \u2115\ny : M\u02e3\nhn : \u2191(y ^ n) = 1\nhm : \u2191(y ^ Nat.succ m) = 1\n\u22a2 \u2191(y ^ Nat.gcd (Nat.succ m) n) = 1", "state_after": "case succ.intro\nM : Type u_1\ninst\u271d : Monoid M\nn m : \u2115\ny : M\u02e3\nhn : y ^ \u2191n = 1\nhm : y ^ \u2191(Nat.succ m) = 1\n\u22a2 y ^ \u2191(Nat.gcd (Nat.succ m) n) = 1"}, {"tactic": "simp only [Nat.gcd_eq_gcd_ab, zpow_add, zpow_mul, hm, hn, one_zpow, one_mul]", "state_before": "case succ.intro\nM : Type u_1\ninst\u271d : Monoid M\nn m : \u2115\ny : M\u02e3\nhn : y ^ \u2191n = 1\nhm : y ^ \u2191(Nat.succ m) = 1\n\u22a2 y ^ \u2191(Nat.gcd (Nat.succ m) n) = 1", "state_after": "no goals"}, {"tactic": "simp [hn]", "state_before": "case zero\nM : Type u_1\ninst\u271d : Monoid M\nx : M\nn : \u2115\nhn : x ^ n = 1\nhm : x ^ Nat.zero = 1\n\u22a2 x ^ Nat.gcd Nat.zero n = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Category/GroupCat/EpiMono.lean", "full_name": "CommGroupCat.epi_iff_range_eq_top", "start": [446, 1], "end": [448, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.coe_of_injective_castSucc_symm", "start": [1330, 1], "end": [1333, 57], "traced_tactics": [{"tactic": "rw [\u2190 coe_castSucc]", "state_before": "n\u271d m n : \u2115\ni : Fin (Nat.succ n)\nhi : i \u2208 Set.range \u2191castSucc\n\u22a2 \u2191(\u2191(Equiv.ofInjective \u2191castSucc (_ : Injective \u2191castSucc)).symm { val := i, property := hi }) = \u2191i", "state_after": "n\u271d m n : \u2115\ni : Fin (Nat.succ n)\nhi : i \u2208 Set.range \u2191castSucc\n\u22a2 \u2191(\u2191castSucc (\u2191(Equiv.ofInjective \u2191castSucc (_ : Injective \u2191castSucc)).symm { val := i, property := hi })) = \u2191i"}, {"tactic": "exact congr_arg val (Equiv.apply_ofInjective_symm _ _)", "state_before": "n\u271d m n : \u2115\ni : Fin (Nat.succ n)\nhi : i \u2208 Set.range \u2191castSucc\n\u22a2 \u2191(\u2191castSucc (\u2191(Equiv.ofInjective \u2191castSucc (_ : Injective \u2191castSucc)).symm { val := i, property := hi })) = \u2191i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Bind.lean", "full_name": "Multiset.coe_sigma", "start": [315, 1], "end": [318, 7], "traced_tactics": [{"tactic": "rw [Multiset.sigma, List.sigma, \u2190 coe_bind]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.67678\n\u03b3 : Type ?u.67681\n\u03b4 : Type ?u.67684\n\u03c3 : \u03b1 \u2192 Type u_2\na : \u03b1\ns : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03c3 a)\nl\u2081 : List \u03b1\nl\u2082 : (a : \u03b1) \u2192 List (\u03c3 a)\n\u22a2 (Multiset.sigma \u2191l\u2081 fun a => \u2191(l\u2082 a)) = \u2191(List.sigma l\u2081 l\u2082)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.67678\n\u03b3 : Type ?u.67681\n\u03b4 : Type ?u.67684\n\u03c3 : \u03b1 \u2192 Type u_2\na : \u03b1\ns : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03c3 a)\nl\u2081 : List \u03b1\nl\u2082 : (a : \u03b1) \u2192 List (\u03c3 a)\n\u22a2 (bind \u2191l\u2081 fun a => map (Sigma.mk a) \u2191(l\u2082 a)) = bind \u2191l\u2081 fun a => \u2191(List.map (Sigma.mk a) (l\u2082 a))"}, {"tactic": "simp", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.67678\n\u03b3 : Type ?u.67681\n\u03b4 : Type ?u.67684\n\u03c3 : \u03b1 \u2192 Type u_2\na : \u03b1\ns : Multiset \u03b1\nt : (a : \u03b1) \u2192 Multiset (\u03c3 a)\nl\u2081 : List \u03b1\nl\u2082 : (a : \u03b1) \u2192 List (\u03c3 a)\n\u22a2 (bind \u2191l\u2081 fun a => map (Sigma.mk a) \u2191(l\u2082 a)) = bind \u2191l\u2081 fun a => \u2191(List.map (Sigma.mk a) (l\u2082 a))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.L1.integral_zero", "start": [698, 1], "end": [700, 29], "traced_tactics": [{"tactic": "simp only [integral]", "state_before": "\u03b1 : Type u_2\nE : Type u_1\nF : Type ?u.429938\n\ud835\udd5c : Type ?u.429941\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace E\n\u22a2 integral 0 = 0", "state_after": "\u03b1 : Type u_2\nE : Type u_1\nF : Type ?u.429938\n\ud835\udd5c : Type ?u.429941\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace E\n\u22a2 \u2191integralCLM 0 = 0"}, {"tactic": "exact map_zero integralCLM", "state_before": "\u03b1 : Type u_2\nE : Type u_1\nF : Type ?u.429938\n\ud835\udd5c : Type ?u.429941\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace E\n\u22a2 \u2191integralCLM 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/IsTensorProduct.lean", "full_name": "TensorProduct.isBaseChange", "start": [225, 1], "end": [232, 18], "traced_tactics": [{"tactic": "delta IsBaseChange", "state_before": "R : Type u_1\nM : Type v\u2081\nN : Type v\u2082\nS : Type v\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R N\ninst\u271d\u2075 : Module S N\ninst\u271d\u2074 : IsScalarTower R S N\nf : M \u2192\u2097[R] N\nh : IsBaseChange S f\nP : Type ?u.260871\nQ : Type ?u.260874\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : AddCommMonoid Q\ninst\u271d : Module S Q\n\u22a2 IsBaseChange S (\u2191(mk R S M) 1)", "state_after": "R : Type u_1\nM : Type v\u2081\nN : Type v\u2082\nS : Type v\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R N\ninst\u271d\u2075 : Module S N\ninst\u271d\u2074 : IsScalarTower R S N\nf : M \u2192\u2097[R] N\nh : IsBaseChange S f\nP : Type ?u.260871\nQ : Type ?u.260874\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : AddCommMonoid Q\ninst\u271d : Module S Q\n\u22a2 IsTensorProduct\n    (\u2191R (\u2191(LinearMap.flip (AlgHom.toLinearMap (Algebra.ofId S (Module.End S (M \u2192\u2097[R] S \u2297[R] M))))) (\u2191(mk R S M) 1)))"}, {"tactic": "convert TensorProduct.isTensorProduct R S M using 1", "state_before": "R : Type u_1\nM : Type v\u2081\nN : Type v\u2082\nS : Type v\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R N\ninst\u271d\u2075 : Module S N\ninst\u271d\u2074 : IsScalarTower R S N\nf : M \u2192\u2097[R] N\nh : IsBaseChange S f\nP : Type ?u.260871\nQ : Type ?u.260874\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : AddCommMonoid Q\ninst\u271d : Module S Q\n\u22a2 IsTensorProduct\n    (\u2191R (\u2191(LinearMap.flip (AlgHom.toLinearMap (Algebra.ofId S (Module.End S (M \u2192\u2097[R] S \u2297[R] M))))) (\u2191(mk R S M) 1)))", "state_after": "case h.e'_12\nR : Type u_1\nM : Type v\u2081\nN : Type v\u2082\nS : Type v\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R N\ninst\u271d\u2075 : Module S N\ninst\u271d\u2074 : IsScalarTower R S N\nf : M \u2192\u2097[R] N\nh : IsBaseChange S f\nP : Type ?u.260871\nQ : Type ?u.260874\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : AddCommMonoid Q\ninst\u271d : Module S Q\n\u22a2 \u2191R (\u2191(LinearMap.flip (AlgHom.toLinearMap (Algebra.ofId S (Module.End S (M \u2192\u2097[R] S \u2297[R] M))))) (\u2191(mk R S M) 1)) =\n    mk R S M"}, {"tactic": "ext (s x)", "state_before": "case h.e'_12\nR : Type u_1\nM : Type v\u2081\nN : Type v\u2082\nS : Type v\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R N\ninst\u271d\u2075 : Module S N\ninst\u271d\u2074 : IsScalarTower R S N\nf : M \u2192\u2097[R] N\nh : IsBaseChange S f\nP : Type ?u.260871\nQ : Type ?u.260874\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : AddCommMonoid Q\ninst\u271d : Module S Q\n\u22a2 \u2191R (\u2191(LinearMap.flip (AlgHom.toLinearMap (Algebra.ofId S (Module.End S (M \u2192\u2097[R] S \u2297[R] M))))) (\u2191(mk R S M) 1)) =\n    mk R S M", "state_after": "case h.e'_12.h.h\nR : Type u_1\nM : Type v\u2081\nN : Type v\u2082\nS : Type v\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R N\ninst\u271d\u2075 : Module S N\ninst\u271d\u2074 : IsScalarTower R S N\nf : M \u2192\u2097[R] N\nh : IsBaseChange S f\nP : Type ?u.260871\nQ : Type ?u.260874\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : AddCommMonoid Q\ninst\u271d : Module S Q\ns : S\nx : M\n\u22a2 \u2191(\u2191(\u2191R (\u2191(LinearMap.flip (AlgHom.toLinearMap (Algebra.ofId S (Module.End S (M \u2192\u2097[R] S \u2297[R] M))))) (\u2191(mk R S M) 1))) s)\n      x =\n    \u2191(\u2191(mk R S M) s) x"}, {"tactic": "change s \u2022 (1 : S) \u2297\u209c[R] x = s \u2297\u209c[R] x", "state_before": "case h.e'_12.h.h\nR : Type u_1\nM : Type v\u2081\nN : Type v\u2082\nS : Type v\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R N\ninst\u271d\u2075 : Module S N\ninst\u271d\u2074 : IsScalarTower R S N\nf : M \u2192\u2097[R] N\nh : IsBaseChange S f\nP : Type ?u.260871\nQ : Type ?u.260874\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : AddCommMonoid Q\ninst\u271d : Module S Q\ns : S\nx : M\n\u22a2 \u2191(\u2191(\u2191R (\u2191(LinearMap.flip (AlgHom.toLinearMap (Algebra.ofId S (Module.End S (M \u2192\u2097[R] S \u2297[R] M))))) (\u2191(mk R S M) 1))) s)\n      x =\n    \u2191(\u2191(mk R S M) s) x", "state_after": "case h.e'_12.h.h\nR : Type u_1\nM : Type v\u2081\nN : Type v\u2082\nS : Type v\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R N\ninst\u271d\u2075 : Module S N\ninst\u271d\u2074 : IsScalarTower R S N\nf : M \u2192\u2097[R] N\nh : IsBaseChange S f\nP : Type ?u.260871\nQ : Type ?u.260874\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : AddCommMonoid Q\ninst\u271d : Module S Q\ns : S\nx : M\n\u22a2 s \u2022 1 \u2297\u209c[R] x = s \u2297\u209c[R] x"}, {"tactic": "rw [TensorProduct.smul_tmul']", "state_before": "case h.e'_12.h.h\nR : Type u_1\nM : Type v\u2081\nN : Type v\u2082\nS : Type v\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R N\ninst\u271d\u2075 : Module S N\ninst\u271d\u2074 : IsScalarTower R S N\nf : M \u2192\u2097[R] N\nh : IsBaseChange S f\nP : Type ?u.260871\nQ : Type ?u.260874\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : AddCommMonoid Q\ninst\u271d : Module S Q\ns : S\nx : M\n\u22a2 s \u2022 1 \u2297\u209c[R] x = s \u2297\u209c[R] x", "state_after": "case h.e'_12.h.h\nR : Type u_1\nM : Type v\u2081\nN : Type v\u2082\nS : Type v\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R N\ninst\u271d\u2075 : Module S N\ninst\u271d\u2074 : IsScalarTower R S N\nf : M \u2192\u2097[R] N\nh : IsBaseChange S f\nP : Type ?u.260871\nQ : Type ?u.260874\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : AddCommMonoid Q\ninst\u271d : Module S Q\ns : S\nx : M\n\u22a2 (s \u2022 1) \u2297\u209c[R] x = s \u2297\u209c[R] x"}, {"tactic": "congr 1", "state_before": "case h.e'_12.h.h\nR : Type u_1\nM : Type v\u2081\nN : Type v\u2082\nS : Type v\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R N\ninst\u271d\u2075 : Module S N\ninst\u271d\u2074 : IsScalarTower R S N\nf : M \u2192\u2097[R] N\nh : IsBaseChange S f\nP : Type ?u.260871\nQ : Type ?u.260874\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : AddCommMonoid Q\ninst\u271d : Module S Q\ns : S\nx : M\n\u22a2 (s \u2022 1) \u2297\u209c[R] x = s \u2297\u209c[R] x", "state_after": "case h.e'_12.h.h.e_m\nR : Type u_1\nM : Type v\u2081\nN : Type v\u2082\nS : Type v\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R N\ninst\u271d\u2075 : Module S N\ninst\u271d\u2074 : IsScalarTower R S N\nf : M \u2192\u2097[R] N\nh : IsBaseChange S f\nP : Type ?u.260871\nQ : Type ?u.260874\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : AddCommMonoid Q\ninst\u271d : Module S Q\ns : S\nx : M\n\u22a2 s \u2022 1 = s"}, {"tactic": "exact mul_one _", "state_before": "case h.e'_12.h.h.e_m\nR : Type u_1\nM : Type v\u2081\nN : Type v\u2082\nS : Type v\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R N\ninst\u271d\u2075 : Module S N\ninst\u271d\u2074 : IsScalarTower R S N\nf : M \u2192\u2097[R] N\nh : IsBaseChange S f\nP : Type ?u.260871\nQ : Type ?u.260874\ninst\u271d\u00b3 : AddCommMonoid P\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : AddCommMonoid Q\ninst\u271d : Module S Q\ns : S\nx : M\n\u22a2 s \u2022 1 = s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/FiberBundle/Trivialization.lean", "full_name": "Trivialization.apply_symm_apply", "start": [407, 1], "end": [408, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "full_name": "Cardinal.mul_mk_eq_max", "start": [560, 1], "end": [561, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Complex/PhragmenLindelof.lean", "full_name": "PhragmenLindelof.quadrant_IV", "start": [612, 1], "end": [631, 37], "traced_tactics": [{"tactic": "obtain \u27e8z, rfl\u27e9 : \u2203 z', -z' = z := \u27e8-z, neg_neg z\u27e9", "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nhz_re : 0 \u2264 z.re\nhz_im : z.im \u2264 0\n\u22a2 \u2016f z\u2016 \u2264 C", "state_after": "case intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 (-z).re\nhz_im : (-z).im \u2264 0\n\u22a2 \u2016f (-z)\u2016 \u2264 C"}, {"tactic": "simp only [neg_re, neg_im, neg_nonpos, neg_nonneg] at hz_re hz_im", "state_before": "case intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 (-z).re\nhz_im : (-z).im \u2264 0\n\u22a2 \u2016f (-z)\u2016 \u2264 C", "state_after": "case intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\n\u22a2 \u2016f (-z)\u2016 \u2264 C"}, {"tactic": "change \u2016(f \u2218 Neg.neg) z\u2016 \u2264 C", "state_before": "case intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\n\u22a2 \u2016f (-z)\u2016 \u2264 C", "state_after": "case intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\n\u22a2 \u2016(f \u2218 Neg.neg) z\u2016 \u2264 C"}, {"tactic": "have H : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0) := fun w hw \u21a6 by\n  simpa only [mem_reProdIm, neg_re, neg_im, neg_lt_zero, neg_pos, mem_Ioi, mem_Iio] using hw", "state_before": "case intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\n\u22a2 \u2016(f \u2218 Neg.neg) z\u2016 \u2264 C", "state_after": "case intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\n\u22a2 \u2016(f \u2218 Neg.neg) z\u2016 \u2264 C"}, {"tactic": "refine' quadrant_II (hd.comp differentiable_neg.diffContOnCl H) _ (fun x hx => _) (fun x hx => _)\n  hz_re hz_im", "state_before": "case intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\n\u22a2 \u2016(f \u2218 Neg.neg) z\u2016 \u2264 C", "state_after": "case intro.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\n\u22a2 \u2203 c,\n    c < 2 \u2227\n      \u2203 B, (f \u2218 Neg.neg) =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n\ncase intro.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\nx : \u211d\nhx : x \u2264 0\n\u22a2 \u2016(f \u2218 Neg.neg) \u2191x\u2016 \u2264 C\n\ncase intro.refine'_3\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\nx : \u211d\nhx : 0 \u2264 x\n\u22a2 \u2016(f \u2218 Neg.neg) (\u2191x * I)\u2016 \u2264 C"}, {"tactic": "simpa only [mem_reProdIm, neg_re, neg_im, neg_lt_zero, neg_pos, mem_Ioi, mem_Iio] using hw", "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nw : \u2102\nhw : w \u2208 Iio 0 \u00d7\u2102 Ioi 0\n\u22a2 -w \u2208 Ioi 0 \u00d7\u2102 Iio 0", "state_after": "no goals"}, {"tactic": "rcases hB with \u27e8c, hc, B, hO\u27e9", "state_before": "case intro.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\n\u22a2 \u2203 c,\n    c < 2 \u2227\n      \u2203 B, (f \u2218 Neg.neg) =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)", "state_after": "case intro.refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 \u2203 c,\n    c < 2 \u2227\n      \u2203 B, (f \u2218 Neg.neg) =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)"}, {"tactic": "refine \u27e8c, hc, B, ?_\u27e9", "state_before": "case intro.refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 \u2203 c,\n    c < 2 \u2227\n      \u2203 B, (f \u2218 Neg.neg) =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)", "state_after": "case intro.refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 (f \u2218 Neg.neg) =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)"}, {"tactic": "rw [comp_apply, \u2190 ofReal_neg]", "state_before": "case intro.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\nx : \u211d\nhx : x \u2264 0\n\u22a2 \u2016(f \u2218 Neg.neg) \u2191x\u2016 \u2264 C", "state_after": "case intro.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\nx : \u211d\nhx : x \u2264 0\n\u22a2 \u2016f \u2191(-x)\u2016 \u2264 C"}, {"tactic": "exact hre (-x) (neg_nonneg.2 hx)", "state_before": "case intro.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\nx : \u211d\nhx : x \u2264 0\n\u22a2 \u2016f \u2191(-x)\u2016 \u2264 C", "state_after": "no goals"}, {"tactic": "rw [comp_apply, \u2190 neg_mul, \u2190 ofReal_neg]", "state_before": "case intro.refine'_3\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\nx : \u211d\nhx : 0 \u2264 x\n\u22a2 \u2016(f \u2218 Neg.neg) (\u2191x * I)\u2016 \u2264 C", "state_after": "case intro.refine'_3\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\nx : \u211d\nhx : 0 \u2264 x\n\u22a2 \u2016f (\u2191(-x) * I)\u2016 \u2264 C"}, {"tactic": "exact him (-x) (neg_nonpos.2 hx)", "state_before": "case intro.refine'_3\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\nx : \u211d\nhx : 0 \u2264 x\n\u22a2 \u2016f (\u2191(-x) * I)\u2016 \u2264 C", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Vector/Basic.lean", "full_name": "Vector.empty_toList_eq_ff", "start": [214, 1], "end": [216, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Eval.lean", "full_name": "Polynomial.coe_compRingHom_apply", "start": [1095, 1], "end": [1096, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "mem_map_indicator_ae_iff_of_zero_nmem", "start": [4644, 1], "end": [4648, 96], "traced_tactics": [{"tactic": "rw [mem_map, mem_ae_iff, Set.indicator_preimage, Set.ite, Set.compl_union, Set.compl_inter]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.4552540\n\u03b4 : Type ?u.4552543\n\u03b9 : Type ?u.4552546\nR : Type ?u.4552549\nR' : Type ?u.4552552\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nt : Set \u03b2\nht : \u00ac0 \u2208 t\n\u22a2 t \u2208 Filter.map (indicator s f) (ae \u03bc) \u2194 \u2191\u2191\u03bc ((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) = 0", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.4552540\n\u03b4 : Type ?u.4552543\n\u03b9 : Type ?u.4552546\nR : Type ?u.4552549\nR' : Type ?u.4552552\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nt : Set \u03b2\nht : \u00ac0 \u2208 t\n\u22a2 \u2191\u2191\u03bc (((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) \u2229 (0 \u207b\u00b9' t \\ s)\u1d9c) = 0 \u2194 \u2191\u2191\u03bc ((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) = 0"}, {"tactic": "change \u03bc (((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) \u2229 ((fun _ => (0 : \u03b2)) \u207b\u00b9' t \\ s)\u1d9c) = 0 \u2194 \u03bc ((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) = 0", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.4552540\n\u03b4 : Type ?u.4552543\n\u03b9 : Type ?u.4552546\nR : Type ?u.4552549\nR' : Type ?u.4552552\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nt : Set \u03b2\nht : \u00ac0 \u2208 t\n\u22a2 \u2191\u2191\u03bc (((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) \u2229 (0 \u207b\u00b9' t \\ s)\u1d9c) = 0 \u2194 \u2191\u2191\u03bc ((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) = 0", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.4552540\n\u03b4 : Type ?u.4552543\n\u03b9 : Type ?u.4552546\nR : Type ?u.4552549\nR' : Type ?u.4552552\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nt : Set \u03b2\nht : \u00ac0 \u2208 t\n\u22a2 \u2191\u2191\u03bc (((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) \u2229 ((fun x => 0) \u207b\u00b9' t \\ s)\u1d9c) = 0 \u2194 \u2191\u2191\u03bc ((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) = 0"}, {"tactic": "simp only [ht, if_false, Set.compl_empty, Set.empty_diff, Set.inter_univ, Set.preimage_const]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.4552540\n\u03b4 : Type ?u.4552543\n\u03b9 : Type ?u.4552546\nR : Type ?u.4552549\nR' : Type ?u.4552552\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ns t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ninst\u271d : Zero \u03b2\nt : Set \u03b2\nht : \u00ac0 \u2208 t\n\u22a2 \u2191\u2191\u03bc (((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) \u2229 ((fun x => 0) \u207b\u00b9' t \\ s)\u1d9c) = 0 \u2194 \u2191\u2191\u03bc ((f \u207b\u00b9' t)\u1d9c \u222a s\u1d9c) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/ReesAlgebra.lean", "full_name": "reesAlgebra.fg", "start": [116, 1], "end": [126, 49], "traced_tactics": [{"tactic": "classical\n  obtain \u27e8s, hs\u27e9 := hI\n  rw [\u2190 adjoin_monomial_eq_reesAlgebra, \u2190 hs]\n  use s.image (monomial 1)\n  rw [Finset.coe_image]\n  change\n    _ =\n      Algebra.adjoin R\n        (Submodule.map (monomial 1 : R \u2192\u2097[R] R[X]) (Submodule.span R \u2191s) : Set R[X])\n  rw [Submodule.map_span, Algebra.adjoin_span]", "state_before": "R M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nhI : Ideal.FG I\n\u22a2 Subalgebra.FG (reesAlgebra I)", "state_after": "no goals"}, {"tactic": "obtain \u27e8s, hs\u27e9 := hI", "state_before": "R M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nhI : Ideal.FG I\n\u22a2 Subalgebra.FG (reesAlgebra I)", "state_after": "case intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\ns : Finset R\nhs : Ideal.span \u2191s = I\n\u22a2 Subalgebra.FG (reesAlgebra I)"}, {"tactic": "rw [\u2190 adjoin_monomial_eq_reesAlgebra, \u2190 hs]", "state_before": "case intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\ns : Finset R\nhs : Ideal.span \u2191s = I\n\u22a2 Subalgebra.FG (reesAlgebra I)", "state_after": "case intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\ns : Finset R\nhs : Ideal.span \u2191s = I\n\u22a2 Subalgebra.FG (Algebra.adjoin R \u2191(Submodule.map (monomial 1) (Ideal.span \u2191s)))"}, {"tactic": "use s.image (monomial 1)", "state_before": "case intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\ns : Finset R\nhs : Ideal.span \u2191s = I\n\u22a2 Subalgebra.FG (Algebra.adjoin R \u2191(Submodule.map (monomial 1) (Ideal.span \u2191s)))", "state_after": "case intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\ns : Finset R\nhs : Ideal.span \u2191s = I\n\u22a2 Algebra.adjoin R \u2191(Finset.image (\u2191(monomial 1)) s) = Algebra.adjoin R \u2191(Submodule.map (monomial 1) (Ideal.span \u2191s))"}, {"tactic": "rw [Finset.coe_image]", "state_before": "case intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\ns : Finset R\nhs : Ideal.span \u2191s = I\n\u22a2 Algebra.adjoin R \u2191(Finset.image (\u2191(monomial 1)) s) = Algebra.adjoin R \u2191(Submodule.map (monomial 1) (Ideal.span \u2191s))", "state_after": "case intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\ns : Finset R\nhs : Ideal.span \u2191s = I\n\u22a2 Algebra.adjoin R (\u2191(monomial 1) '' \u2191s) = Algebra.adjoin R \u2191(Submodule.map (monomial 1) (Ideal.span \u2191s))"}, {"tactic": "change\n  _ =\n    Algebra.adjoin R\n      (Submodule.map (monomial 1 : R \u2192\u2097[R] R[X]) (Submodule.span R \u2191s) : Set R[X])", "state_before": "case intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\ns : Finset R\nhs : Ideal.span \u2191s = I\n\u22a2 Algebra.adjoin R (\u2191(monomial 1) '' \u2191s) = Algebra.adjoin R \u2191(Submodule.map (monomial 1) (Ideal.span \u2191s))", "state_after": "case intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\ns : Finset R\nhs : Ideal.span \u2191s = I\n\u22a2 Algebra.adjoin R (\u2191(monomial 1) '' \u2191s) = Algebra.adjoin R \u2191(Submodule.map (monomial 1) (Submodule.span R \u2191s))"}, {"tactic": "rw [Submodule.map_span, Algebra.adjoin_span]", "state_before": "case intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\ns : Finset R\nhs : Ideal.span \u2191s = I\n\u22a2 Algebra.adjoin R (\u2191(monomial 1) '' \u2191s) = Algebra.adjoin R \u2191(Submodule.map (monomial 1) (Submodule.span R \u2191s))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Spectrum.lean", "full_name": "spectrum.singleton_sub_eq", "start": [314, 1], "end": [315, 64], "traced_tactics": [{"tactic": "rw [sub_eq_add_neg, neg_eq, singleton_add_eq, sub_eq_add_neg]", "state_before": "R : Type u\nA : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\nr : R\n\u22a2 {r} - \u03c3 a = \u03c3 (\u2191\u2191\u2090 r - a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/ContinuousOn.lean", "full_name": "ContinuousOn.preimage_interior_subset_interior_preimage", "start": [1025, 1], "end": [1031, 73], "traced_tactics": [{"tactic": "rw [interior_inter, hs.interior_eq]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.346669\n\u03b4 : Type ?u.346672\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhf : ContinuousOn f s\nhs : IsOpen s\n\u22a2 interior (s \u2229 f \u207b\u00b9' t) = s \u2229 interior (f \u207b\u00b9' t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.differentiableOn_integral_of_continuous", "start": [983, 1], "end": [987, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Fintype.lean", "full_name": "Multiset.prod_eq_prod_toEnumFinset", "start": [271, 1], "end": [274, 7], "traced_tactics": [{"tactic": "congr", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\nm\u271d : Multiset \u03b1\ninst\u271d : CommMonoid \u03b1\nm : Multiset \u03b1\n\u22a2 prod m = \u220f x in toEnumFinset m, x.fst", "state_after": "case e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\nm\u271d : Multiset \u03b1\ninst\u271d : CommMonoid \u03b1\nm : Multiset \u03b1\n\u22a2 m = map (fun x => x.fst) (toEnumFinset m).val"}, {"tactic": "simp", "state_before": "case e_a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\nm\u271d : Multiset \u03b1\ninst\u271d : CommMonoid \u03b1\nm : Multiset \u03b1\n\u22a2 m = map (fun x => x.fst) (toEnumFinset m).val", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/PUnitInstances.lean", "full_name": "PUnit.smul_eq", "start": [137, 1], "end": [138, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.L1.SimpleFunc.norm_integral_le_norm", "start": [555, 1], "end": [557, 82], "traced_tactics": [{"tactic": "rw [integral, norm_eq_integral]", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.291771\n\ud835\udd5c : Type ?u.291774\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type ?u.294149\ninst\u271d\u00b9 : NormedAddCommGroup F'\ninst\u271d : NormedSpace \u211d F'\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2016integral f\u2016 \u2264 \u2016f\u2016", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.291771\n\ud835\udd5c : Type ?u.291774\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type ?u.294149\ninst\u271d\u00b9 : NormedAddCommGroup F'\ninst\u271d : NormedSpace \u211d F'\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2016MeasureTheory.SimpleFunc.integral \u03bc (toSimpleFunc f)\u2016 \u2264\n    MeasureTheory.SimpleFunc.integral \u03bc (SimpleFunc.map norm (toSimpleFunc f))"}, {"tactic": "exact (toSimpleFunc f).norm_integral_le_integral_norm (SimpleFunc.integrable f)", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.291771\n\ud835\udd5c : Type ?u.291774\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c E\nF' : Type ?u.294149\ninst\u271d\u00b9 : NormedAddCommGroup F'\ninst\u271d : NormedSpace \u211d F'\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2016MeasureTheory.SimpleFunc.integral \u03bc (toSimpleFunc f)\u2016 \u2264\n    MeasureTheory.SimpleFunc.integral \u03bc (SimpleFunc.map norm (toSimpleFunc f))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean", "full_name": "Complex.cpow_eq_zero_iff", "start": [52, 1], "end": [54, 38], "traced_tactics": [{"tactic": "simp only [cpow_def]", "state_before": "x y : \u2102\n\u22a2 x ^ y = 0 \u2194 x = 0 \u2227 y \u2260 0", "state_after": "x y : \u2102\n\u22a2 (if x = 0 then if y = 0 then 1 else 0 else exp (log x * y)) = 0 \u2194 x = 0 \u2227 y \u2260 0"}, {"tactic": "split_ifs <;> simp [*, exp_ne_zero]", "state_before": "x y : \u2102\n\u22a2 (if x = 0 then if y = 0 then 1 else 0 else exp (log x * y)) = 0 \u2194 x = 0 \u2227 y \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "BoundedContinuousFunction.range_toLpHom", "start": [1634, 1], "end": [1640, 60], "traced_tactics": [{"tactic": "symm", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.10310262\nG : Type ?u.10310265\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : MeasureTheory.Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : Fact (1 \u2264 p)\n\u22a2 NormedAddGroupHom.range (toLpHom p \u03bc) = Lp.boundedContinuousFunction E p \u03bc", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.10310262\nG : Type ?u.10310265\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : MeasureTheory.Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : Fact (1 \u2264 p)\n\u22a2 Lp.boundedContinuousFunction E p \u03bc = NormedAddGroupHom.range (toLpHom p \u03bc)"}, {"tactic": "convert AddMonoidHom.addSubgroupOf_range_eq_of_le\n    ((ContinuousMap.toAEEqFunAddHom \u03bc).comp (toContinuousMapAddHom \u03b1 E))\n    (by rintro - \u27e8f, rfl\u27e9; exact mem_Lp f : _ \u2264 Lp E p \u03bc)", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.10310262\nG : Type ?u.10310265\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : MeasureTheory.Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : Fact (1 \u2264 p)\n\u22a2 Lp.boundedContinuousFunction E p \u03bc = NormedAddGroupHom.range (toLpHom p \u03bc)", "state_after": "no goals"}, {"tactic": "rintro - \u27e8f, rfl\u27e9", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.10310262\nG : Type ?u.10310265\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : MeasureTheory.Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : Fact (1 \u2264 p)\n\u22a2 AddMonoidHom.range (AddMonoidHom.comp (ContinuousMap.toAEEqFunAddHom \u03bc) (toContinuousMapAddHom \u03b1 E)) \u2264 Lp E p", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.10310262\nG : Type ?u.10310265\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : MeasureTheory.Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : Fact (1 \u2264 p)\nf : \u03b1 \u2192\u1d47 E\n\u22a2 \u2191(AddMonoidHom.comp (ContinuousMap.toAEEqFunAddHom \u03bc) (toContinuousMapAddHom \u03b1 E)) f \u2208 Lp E p"}, {"tactic": "exact mem_Lp f", "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.10310262\nG : Type ?u.10310265\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : MeasureTheory.Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\ninst\u271d\u00b2 : SecondCountableTopologyEither \u03b1 E\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\ninst\u271d : Fact (1 \u2264 p)\nf : \u03b1 \u2192\u1d47 E\n\u22a2 \u2191(AddMonoidHom.comp (ContinuousMap.toAEEqFunAddHom \u03bc) (toContinuousMapAddHom \u03b1 E)) f \u2208 Lp E p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.Measure.univ_pi_Ico_ae_eq_Icc", "start": [539, 1], "end": [541, 48], "traced_tactics": [{"tactic": "rw [\u2190 pi_univ_Icc]", "state_before": "\u03b9 : Type u_2\n\u03b9' : Type ?u.4891185\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoAtoms (\u03bc i)\nf g : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (Set.pi univ fun i => Ico (f i) (g i)) =\u1da0[ae (Measure.pi \u03bc)] Icc f g", "state_after": "\u03b9 : Type u_2\n\u03b9' : Type ?u.4891185\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoAtoms (\u03bc i)\nf g : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (Set.pi univ fun i => Ico (f i) (g i)) =\u1da0[ae (Measure.pi \u03bc)] Set.pi univ fun i => Icc (f i) (g i)"}, {"tactic": "exact pi_Ico_ae_eq_pi_Icc", "state_before": "\u03b9 : Type u_2\n\u03b9' : Type ?u.4891185\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoAtoms (\u03bc i)\nf g : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (Set.pi univ fun i => Ico (f i) (g i)) =\u1da0[ae (Measure.pi \u03bc)] Set.pi univ fun i => Icc (f i) (g i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "mem_nhdsWithin_Ioi_iff_exists_Ioo_subset", "start": [1638, 1], "end": [1641, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/MvPolynomial/Homogeneous.lean", "full_name": "MvPolynomial.IsHomogeneous.add", "start": [181, 1], "end": [182, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Basic.lean", "full_name": "Metric.isBounded_iff_exists_ge", "start": [704, 1], "end": [707, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/PathConnected.lean", "full_name": "IsPathConnected.subset_pathComponent", "start": [993, 1], "end": [994, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.insert_diff_eq_singleton", "start": [1994, 1], "end": [1999, 10], "traced_tactics": [{"tactic": "ext", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na\u271d b : \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u : Set \u03b1\na : \u03b1\ns : Set \u03b1\nh : \u00aca \u2208 s\n\u22a2 insert a s \\ s = {a}", "state_after": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na\u271d b : \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u : Set \u03b1\na : \u03b1\ns : Set \u03b1\nh : \u00aca \u2208 s\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 insert a s \\ s \u2194 x\u271d \u2208 {a}"}, {"tactic": "rw [Set.mem_diff, Set.mem_insert_iff, Set.mem_singleton_iff, or_and_right, and_not_self_iff,\n  or_false_iff, and_iff_left_iff_imp]", "state_before": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na\u271d b : \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u : Set \u03b1\na : \u03b1\ns : Set \u03b1\nh : \u00aca \u2208 s\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 insert a s \\ s \u2194 x\u271d \u2208 {a}", "state_after": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na\u271d b : \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u : Set \u03b1\na : \u03b1\ns : Set \u03b1\nh : \u00aca \u2208 s\nx\u271d : \u03b1\n\u22a2 x\u271d = a \u2192 \u00acx\u271d \u2208 s"}, {"tactic": "rintro rfl", "state_before": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na\u271d b : \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u : Set \u03b1\na : \u03b1\ns : Set \u03b1\nh : \u00aca \u2208 s\nx\u271d : \u03b1\n\u22a2 x\u271d = a \u2192 \u00acx\u271d \u2208 s", "state_after": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u s : Set \u03b1\nx\u271d : \u03b1\nh : \u00acx\u271d \u2208 s\n\u22a2 \u00acx\u271d \u2208 s"}, {"tactic": "exact h", "state_before": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u s : Set \u03b1\nx\u271d : \u03b1\nh : \u00acx\u271d \u2208 s\n\u22a2 \u00acx\u271d \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Rat/NNRat.lean", "full_name": "NNRat.ext_num_den_iff", "start": [480, 1], "end": [481, 65], "traced_tactics": [{"tactic": "rintro rfl", "state_before": "p q : \u211a\u22650\n\u22a2 p = q \u2192 num p = num q \u2227 den p = den q", "state_after": "p : \u211a\u22650\n\u22a2 num p = num p \u2227 den p = den p"}, {"tactic": "exact \u27e8rfl, rfl\u27e9", "state_before": "p : \u211a\u22650\n\u22a2 num p = num p \u2227 den p = den p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Pairing.lean", "full_name": "Nat.max_sq_add_min_le_pair", "start": [164, 1], "end": [169, 20], "traced_tactics": [{"tactic": "rw [pair]", "state_before": "m n : \u2115\n\u22a2 max m n ^ 2 + min m n \u2264 pair m n", "state_after": "m n : \u2115\n\u22a2 max m n ^ 2 + min m n \u2264 if m < n then n * n + m else m * m + m + n"}, {"tactic": "cases' lt_or_le m n with h h", "state_before": "m n : \u2115\n\u22a2 max m n ^ 2 + min m n \u2264 if m < n then n * n + m else m * m + m + n", "state_after": "case inl\nm n : \u2115\nh : m < n\n\u22a2 max m n ^ 2 + min m n \u2264 if m < n then n * n + m else m * m + m + n\n\ncase inr\nm n : \u2115\nh : n \u2264 m\n\u22a2 max m n ^ 2 + min m n \u2264 if m < n then n * n + m else m * m + m + n"}, {"tactic": "rw [if_pos h, max_eq_right h.le, min_eq_left h.le, sq]", "state_before": "case inl\nm n : \u2115\nh : m < n\n\u22a2 max m n ^ 2 + min m n \u2264 if m < n then n * n + m else m * m + m + n\n\ncase inr\nm n : \u2115\nh : n \u2264 m\n\u22a2 max m n ^ 2 + min m n \u2264 if m < n then n * n + m else m * m + m + n", "state_after": "case inr\nm n : \u2115\nh : n \u2264 m\n\u22a2 max m n ^ 2 + min m n \u2264 if m < n then n * n + m else m * m + m + n"}, {"tactic": "rw [if_neg h.not_lt, max_eq_left h, min_eq_right h, sq, add_assoc, add_le_add_iff_left]", "state_before": "case inr\nm n : \u2115\nh : n \u2264 m\n\u22a2 max m n ^ 2 + min m n \u2264 if m < n then n * n + m else m * m + m + n", "state_after": "case inr\nm n : \u2115\nh : n \u2264 m\n\u22a2 n \u2264 m + n"}, {"tactic": "exact le_add_self", "state_before": "case inr\nm n : \u2115\nh : n \u2264 m\n\u22a2 n \u2264 m + n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.max'_mem", "start": [1350, 1], "end": [1351, 70], "traced_tactics": [{"tactic": "simp only [max', Finset.max, id_eq, coe_sup']", "state_before": "F : Type ?u.336690\n\u03b1 : Type u_1\n\u03b2 : Type ?u.336696\n\u03b3 : Type ?u.336699\n\u03b9 : Type ?u.336702\n\u03ba : Type ?u.336705\ninst\u271d : LinearOrder \u03b1\ns : Finset \u03b1\nH : Finset.Nonempty s\nx : \u03b1\n\u22a2 Finset.max s = \u2191(max' s H)", "state_after": "F : Type ?u.336690\n\u03b1 : Type u_1\n\u03b2 : Type ?u.336696\n\u03b3 : Type ?u.336699\n\u03b9 : Type ?u.336702\n\u03ba : Type ?u.336705\ninst\u271d : LinearOrder \u03b1\ns : Finset \u03b1\nH : Finset.Nonempty s\nx : \u03b1\n\u22a2 sup s WithBot.some = sup s (WithBot.some \u2218 fun x => x)"}, {"tactic": "rfl", "state_before": "F : Type ?u.336690\n\u03b1 : Type u_1\n\u03b2 : Type ?u.336696\n\u03b3 : Type ?u.336699\n\u03b9 : Type ?u.336702\n\u03ba : Type ?u.336705\ninst\u271d : LinearOrder \u03b1\ns : Finset \u03b1\nH : Finset.Nonempty s\nx : \u03b1\n\u22a2 sup s WithBot.some = sup s (WithBot.some \u2218 fun x => x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snorm_const'", "start": [298, 1], "end": [300, 77], "traced_tactics": [{"tactic": "simp [snorm_eq_snorm' h0 h_top, snorm'_const, ENNReal.toReal_pos h0 h_top]", "state_before": "\u03b1 : Type u_1\nE : Type ?u.1186891\nF : Type u_2\nG : Type ?u.1186897\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc : F\nh0 : p \u2260 0\nh_top : p \u2260 \u22a4\n\u22a2 snorm (fun x => c) p \u03bc = \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc Set.univ ^ (1 / ENNReal.toReal p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "IsLUB.isLUB_of_tendsto", "start": [2039, 1], "end": [2044, 101], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "full_name": "norm_inv'", "start": [429, 1], "end": [429, 75], "traced_tactics": [{"tactic": "simpa using norm_div_rev 1 a", "state_before": "\ud835\udcd5 : Type ?u.54204\n\ud835\udd5c : Type ?u.54207\n\u03b1 : Type ?u.54210\n\u03b9 : Type ?u.54213\n\u03ba : Type ?u.54216\nE : Type u_1\nF : Type ?u.54222\nG : Type ?u.54225\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na : E\n\u22a2 \u2016a\u207b\u00b9\u2016 = \u2016a\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/ShrinkingLemma.lean", "full_name": "exists_iUnion_ball_eq_radius_lt", "start": [55, 1], "end": [59, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Complex/ReImTopology.lean", "full_name": "Complex.frontier_preimage_re", "start": [88, 1], "end": [89, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Integrals.lean", "full_name": "integral_sin_pow_mul_cos_pow_odd", "start": [751, 1], "end": [762, 80], "traced_tactics": [{"tactic": "simp only [_root_.pow_zero, _root_.pow_succ', mul_assoc, pow_mul, one_mul]", "state_before": "a b : \u211d\nn\u271d m n : \u2115\nhc : Continuous fun u => u ^ m * (1 - u ^ 2) ^ n\n\u22a2 (\u222b (x : \u211d) in a..b, sin x ^ m * cos x ^ (2 * n + 1)) = \u222b (x : \u211d) in a..b, sin x ^ m * (1 - sin x ^ 2) ^ n * cos x", "state_after": "a b : \u211d\nn\u271d m n : \u2115\nhc : Continuous fun u => u ^ m * (1 - u ^ 2) ^ n\n\u22a2 (\u222b (x : \u211d) in a..b, sin x ^ m * ((cos x * cos x) ^ n * cos x)) =\n    \u222b (x : \u211d) in a..b, sin x ^ m * ((1 - sin x * sin x) ^ n * cos x)"}, {"tactic": "congr! 5", "state_before": "a b : \u211d\nn\u271d m n : \u2115\nhc : Continuous fun u => u ^ m * (1 - u ^ 2) ^ n\n\u22a2 (\u222b (x : \u211d) in a..b, sin x ^ m * ((cos x * cos x) ^ n * cos x)) =\n    \u222b (x : \u211d) in a..b, sin x ^ m * ((1 - sin x * sin x) ^ n * cos x)", "state_after": "case h.e'_5.h.h.e'_6.h.e'_5.h.e'_5\na b : \u211d\nn\u271d m n : \u2115\nhc : Continuous fun u => u ^ m * (1 - u ^ 2) ^ n\nx\u271d : \u211d\n\u22a2 cos x\u271d * cos x\u271d = 1 - sin x\u271d * sin x\u271d"}, {"tactic": "rw [\u2190 sq, \u2190 sq, cos_sq']", "state_before": "case h.e'_5.h.h.e'_6.h.e'_5.h.e'_5\na b : \u211d\nn\u271d m n : \u2115\nhc : Continuous fun u => u ^ m * (1 - u ^ 2) ^ n\nx\u271d : \u211d\n\u22a2 cos x\u271d * cos x\u271d = 1 - sin x\u271d * sin x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Between.lean", "full_name": "sbtw_one_zero_iff", "start": [465, 1], "end": [466, 36], "traced_tactics": [{"tactic": "rw [sbtw_comm, sbtw_zero_one_iff]", "state_before": "R : Type u_1\nV : Type ?u.269565\nV' : Type ?u.269568\nP : Type ?u.269571\nP' : Type ?u.269574\ninst\u271d\u2076 : OrderedRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx : R\n\u22a2 Sbtw R 1 x 0 \u2194 x \u2208 Set.Ioo 0 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Card.lean", "full_name": "List.toFinset_card_le", "start": [200, 1], "end": [201, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean", "full_name": "AlgebraicTopology.DoldKan.h\u03c3'_eq'", "start": [127, 1], "end": [130, 47], "traced_tactics": [{"tactic": "rw [h\u03c3'_eq ha rfl, eqToHom_refl, comp_id]", "state_before": "C : Type u_2\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nX : SimplicialObject C\nq n a : \u2115\nha : n = a + q\n\u22a2 h\u03c3' q n (n + 1) (_ : n + 1 = n + 1) = (-1) ^ a \u2022 \u03c3 X { val := a, isLt := (_ : a < Nat.succ n) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Units.lean", "full_name": "Units.mul_inv_cancel_left", "start": [284, 1], "end": [285, 37], "traced_tactics": [{"tactic": "rw [\u2190 mul_assoc, mul_inv, one_mul]", "state_before": "\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b\u271d c u a : \u03b1\u02e3\nb : \u03b1\n\u22a2 \u2191a * (\u2191a\u207b\u00b9 * b) = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.edist_toLp_zero", "start": [304, 1], "end": [306, 7], "traced_tactics": [{"tactic": "convert edist_toLp_toLp f 0 hf zero_mem\u2112p", "state_before": "\u03b1 : Type u_2\nE : Type u_1\nF : Type ?u.390075\nG : Type ?u.390078\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u22a2 edist (Mem\u2112p.toLp f hf) 0 = snorm f p \u03bc", "state_after": "case h.e'_3.h.e'_5\n\u03b1 : Type u_2\nE : Type u_1\nF : Type ?u.390075\nG : Type ?u.390078\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u22a2 f = f - 0"}, {"tactic": "simp", "state_before": "case h.e'_3.h.e'_5\n\u03b1 : Type u_2\nE : Type u_1\nF : Type ?u.390075\nG : Type ?u.390078\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u22a2 f = f - 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/NaturalOps.lean", "full_name": "Ordinal.nadd_comm", "start": [248, 1], "end": [253, 40], "traced_tactics": [{"tactic": "rw [nadd_def, nadd_def, max_comm]", "state_before": "a\u271d b\u271d c : Ordinal\na b : Ordinal\n\u22a2 a \u266f b = b \u266f a", "state_after": "a\u271d b\u271d c : Ordinal\na b : Ordinal\n\u22a2 max (blsub b fun b' x => a \u266f b') (blsub a fun a' x => a' \u266f b) =\n    max (blsub b fun a' x => a' \u266f a) (blsub a fun b' x => b \u266f b')"}, {"tactic": "congr <;> ext <;> apply nadd_comm", "state_before": "a\u271d b\u271d c : Ordinal\na b : Ordinal\n\u22a2 max (blsub b fun b' x => a \u266f b') (blsub a fun a' x => a' \u266f b) =\n    max (blsub b fun a' x => a' \u266f a) (blsub a fun b' x => b \u266f b')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/IndicatorFunction.lean", "full_name": "Set.mulIndicator_univ", "start": [201, 1], "end": [202, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean", "full_name": "continuousAt_clog", "start": [237, 1], "end": [246, 29], "traced_tactics": [{"tactic": "refine' ContinuousAt.add _ _", "state_before": "\u03b1 : Type ?u.271373\nx : \u2102\nh : 0 < x.re \u2228 x.im \u2260 0\n\u22a2 ContinuousAt log x", "state_after": "case refine'_1\n\u03b1 : Type ?u.271373\nx : \u2102\nh : 0 < x.re \u2228 x.im \u2260 0\n\u22a2 ContinuousAt (fun x => \u2191(Real.log (\u2191Complex.abs x))) x\n\ncase refine'_2\n\u03b1 : Type ?u.271373\nx : \u2102\nh : 0 < x.re \u2228 x.im \u2260 0\n\u22a2 ContinuousAt (fun x => \u2191(arg x) * I) x"}, {"tactic": "refine' continuous_ofReal.continuousAt.comp _", "state_before": "case refine'_1\n\u03b1 : Type ?u.271373\nx : \u2102\nh : 0 < x.re \u2228 x.im \u2260 0\n\u22a2 ContinuousAt (fun x => \u2191(Real.log (\u2191Complex.abs x))) x", "state_after": "case refine'_1\n\u03b1 : Type ?u.271373\nx : \u2102\nh : 0 < x.re \u2228 x.im \u2260 0\n\u22a2 ContinuousAt (fun x => Real.log (\u2191Complex.abs x)) x"}, {"tactic": "refine' (Real.continuousAt_log _).comp Complex.continuous_abs.continuousAt", "state_before": "case refine'_1\n\u03b1 : Type ?u.271373\nx : \u2102\nh : 0 < x.re \u2228 x.im \u2260 0\n\u22a2 ContinuousAt (fun x => Real.log (\u2191Complex.abs x)) x", "state_after": "case refine'_1\n\u03b1 : Type ?u.271373\nx : \u2102\nh : 0 < x.re \u2228 x.im \u2260 0\n\u22a2 \u2191Complex.abs x \u2260 0"}, {"tactic": "rw [Complex.abs.ne_zero_iff]", "state_before": "case refine'_1\n\u03b1 : Type ?u.271373\nx : \u2102\nh : 0 < x.re \u2228 x.im \u2260 0\n\u22a2 \u2191Complex.abs x \u2260 0", "state_after": "case refine'_1\n\u03b1 : Type ?u.271373\nx : \u2102\nh : 0 < x.re \u2228 x.im \u2260 0\n\u22a2 x \u2260 0"}, {"tactic": "rintro rfl", "state_before": "case refine'_1\n\u03b1 : Type ?u.271373\nx : \u2102\nh : 0 < x.re \u2228 x.im \u2260 0\n\u22a2 x \u2260 0", "state_after": "case refine'_1\n\u03b1 : Type ?u.271373\nh : 0 < 0.re \u2228 0.im \u2260 0\n\u22a2 False"}, {"tactic": "simp at h", "state_before": "case refine'_1\n\u03b1 : Type ?u.271373\nh : 0 < 0.re \u2228 0.im \u2260 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "have h_cont_mul : Continuous fun x : \u2102 => x * I := continuous_id'.mul continuous_const", "state_before": "case refine'_2\n\u03b1 : Type ?u.271373\nx : \u2102\nh : 0 < x.re \u2228 x.im \u2260 0\n\u22a2 ContinuousAt (fun x => \u2191(arg x) * I) x", "state_after": "case refine'_2\n\u03b1 : Type ?u.271373\nx : \u2102\nh : 0 < x.re \u2228 x.im \u2260 0\nh_cont_mul : Continuous fun x => x * I\n\u22a2 ContinuousAt (fun x => \u2191(arg x) * I) x"}, {"tactic": "refine' h_cont_mul.continuousAt.comp (continuous_ofReal.continuousAt.comp _)", "state_before": "case refine'_2\n\u03b1 : Type ?u.271373\nx : \u2102\nh : 0 < x.re \u2228 x.im \u2260 0\nh_cont_mul : Continuous fun x => x * I\n\u22a2 ContinuousAt (fun x => \u2191(arg x) * I) x", "state_after": "case refine'_2\n\u03b1 : Type ?u.271373\nx : \u2102\nh : 0 < x.re \u2228 x.im \u2260 0\nh_cont_mul : Continuous fun x => x * I\n\u22a2 ContinuousAt (fun x => arg x) x"}, {"tactic": "exact continuousAt_arg h", "state_before": "case refine'_2\n\u03b1 : Type ?u.271373\nx : \u2102\nh : 0 < x.re \u2228 x.im \u2260 0\nh_cont_mul : Continuous fun x => x * I\n\u22a2 ContinuousAt (fun x => arg x) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Dynamics/PeriodicPts.lean", "full_name": "Function.IsPeriodicPt.eq_of_apply_eq_same", "start": [180, 1], "end": [182, 84], "traced_tactics": [{"tactic": "rw [\u2190 hx.eq, \u2190 hy.eq, \u2190 iterate_pred_comp_of_pos f hn, comp_apply, comp_apply, h]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.4108\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhx : IsPeriodicPt f n x\nhy : IsPeriodicPt f n y\nhn : 0 < n\nh : f x = f y\n\u22a2 x = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "Real.cosh_abs", "start": [1374, 1], "end": [1375, 71], "traced_tactics": [{"tactic": "cases le_total x 0 <;> simp [*, _root_.abs_of_nonneg, abs_of_nonpos]", "state_before": "x y : \u211d\n\u22a2 cosh (abs' x) = cosh x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean", "full_name": "isLittleO_pow_exp_pos_mul_atTop", "start": [262, 1], "end": [264, 52], "traced_tactics": [{"tactic": "simpa using isLittleO_zpow_exp_pos_mul_atTop k hb", "state_before": "k : \u2115\nb : \u211d\nhb : 0 < b\n\u22a2 (fun x => x ^ \u2191k) =o[atTop] fun x => exp (b * x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "measurable_liminf'", "start": [1310, 1], "end": [1315, 46], "traced_tactics": [{"tactic": "simp_rw [hu.toHasBasis.liminf_eq_iSup_iInf]", "state_before": "\u03b1 : Type u_4\n\u03b2 : Type ?u.1636243\n\u03b3 : Type ?u.1636246\n\u03b3\u2082 : Type ?u.1636249\n\u03b4 : Type u_3\n\u03b9\u271d : Sort y\ns\u271d t u\u271d : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : CompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nu : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhu : HasCountableBasis u p s\nhs : \u2200 (i : \u03b9'), Set.Countable (s i)\n\u22a2 Measurable fun x => liminf (fun i => f i x) u", "state_after": "\u03b1 : Type u_4\n\u03b2 : Type ?u.1636243\n\u03b3 : Type ?u.1636246\n\u03b3\u2082 : Type ?u.1636249\n\u03b4 : Type u_3\n\u03b9\u271d : Sort y\ns\u271d t u\u271d : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : CompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nu : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhu : HasCountableBasis u p s\nhs : \u2200 (i : \u03b9'), Set.Countable (s i)\n\u22a2 Measurable fun x => \u2a06 (i : \u03b9') (_ : p i), \u2a05 (a : \u03b9) (_ : a \u2208 s i), f a x"}, {"tactic": "refine' measurable_biSup _ hu.countable _", "state_before": "\u03b1 : Type u_4\n\u03b2 : Type ?u.1636243\n\u03b3 : Type ?u.1636246\n\u03b3\u2082 : Type ?u.1636249\n\u03b4 : Type u_3\n\u03b9\u271d : Sort y\ns\u271d t u\u271d : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : CompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nu : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhu : HasCountableBasis u p s\nhs : \u2200 (i : \u03b9'), Set.Countable (s i)\n\u22a2 Measurable fun x => \u2a06 (i : \u03b9') (_ : p i), \u2a05 (a : \u03b9) (_ : a \u2208 s i), f a x", "state_after": "\u03b1 : Type u_4\n\u03b2 : Type ?u.1636243\n\u03b3 : Type ?u.1636246\n\u03b3\u2082 : Type ?u.1636249\n\u03b4 : Type u_3\n\u03b9\u271d : Sort y\ns\u271d t u\u271d : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : CompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nu : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhu : HasCountableBasis u p s\nhs : \u2200 (i : \u03b9'), Set.Countable (s i)\n\u22a2 \u2200 (i : \u03b9'), Measurable fun x => \u2a05 (a : \u03b9) (_ : a \u2208 s i), f a x"}, {"tactic": "exact fun i => measurable_biInf _ (hs i) hf", "state_before": "\u03b1 : Type u_4\n\u03b2 : Type ?u.1636243\n\u03b3 : Type ?u.1636246\n\u03b3\u2082 : Type ?u.1636249\n\u03b4 : Type u_3\n\u03b9\u271d : Sort y\ns\u271d t u\u271d : Set \u03b1\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1\ninst\u271d\u00b9\u2070 : BorelSpace \u03b1\ninst\u271d\u2079 : TopologicalSpace \u03b2\ninst\u271d\u2078 : MeasurableSpace \u03b2\ninst\u271d\u2077 : BorelSpace \u03b2\ninst\u271d\u2076 : TopologicalSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\ninst\u271d\u2074 : BorelSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : CompleteLinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nf : \u03b9 \u2192 \u03b4 \u2192 \u03b1\nu : Filter \u03b9\nhf : \u2200 (i : \u03b9), Measurable (f i)\np : \u03b9' \u2192 Prop\ns : \u03b9' \u2192 Set \u03b9\nhu : HasCountableBasis u p s\nhs : \u2200 (i : \u03b9'), Set.Countable (s i)\n\u22a2 \u2200 (i : \u03b9'), Measurable fun x => \u2a05 (a : \u03b9) (_ : a \u2208 s i), f a x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/IsROrC.lean", "full_name": "IsROrC.norm_coe_norm", "start": [39, 1], "end": [39, 68], "traced_tactics": [{"tactic": "simp", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d : NormedAddCommGroup E\nz : E\n\u22a2 \u2016\u2191\u2016z\u2016\u2016 = \u2016z\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.tendsto_iSup", "start": [3031, 1], "end": [3032, 101], "traced_tactics": [{"tactic": "simp only [Tendsto, map_iSup, iSup_le_iff]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.316422\n\u03b9 : Sort x\nf : \u03b1 \u2192 \u03b2\nx : \u03b9 \u2192 Filter \u03b1\ny : Filter \u03b2\n\u22a2 Tendsto f (\u2a06 (i : \u03b9), x i) y \u2194 \u2200 (i : \u03b9), Tendsto f (x i) y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean", "full_name": "LocalRing.specializes_closedPoint", "start": [999, 1], "end": [1000, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.univ_pi_nonempty_iff", "start": [687, 1], "end": [688, 40], "traced_tactics": [{"tactic": "simp [Classical.skolem, Set.Nonempty]", "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type ?u.119820\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\n\u22a2 Set.Nonempty (pi univ t) \u2194 \u2200 (i : \u03b9), Set.Nonempty (t i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Lie/Classical.lean", "full_name": "LieAlgebra.Orthogonal.soIndefiniteEquiv_apply", "start": [228, 1], "end": [232, 49], "traced_tactics": [{"tactic": "rw [soIndefiniteEquiv, LieEquiv.trans_apply, LieEquiv.ofEq_apply,\n  skewAdjointMatricesLieSubalgebraEquiv_apply]", "state_before": "n : Type ?u.71302\np : Type u_1\nq : Type u_2\nl : Type ?u.71311\nR : Type u\u2082\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : DecidableEq q\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Fintype p\ninst\u271d : Fintype q\ni : R\nhi : i * i = -1\nA : { x // x \u2208 so' p q R }\n\u22a2 \u2191(\u2191(soIndefiniteEquiv p q R hi) A) = (Pso p q R i)\u207b\u00b9 \u2b1d \u2191A \u2b1d Pso p q R i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "full_name": "UniqueDiffWithinAt.eq_deriv", "start": [251, 1], "end": [253, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Regularity/Equitabilise.lean", "full_name": "Finpartition.card_parts_equitabilise_subset_le", "start": [198, 1], "end": [200, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Polynomial/ScaleRoots.lean", "full_name": "Polynomial.scaleRoots_ne_zero", "start": [52, 1], "end": [58, 16], "traced_tactics": [{"tactic": "intro h", "state_before": "A : Type ?u.25744\nK : Type ?u.25747\nR : Type u_1\nS : Type ?u.25753\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : IsDomain A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nM : Submonoid A\np : R[X]\nhp : p \u2260 0\ns : R\n\u22a2 scaleRoots p s \u2260 0", "state_after": "A : Type ?u.25744\nK : Type ?u.25747\nR : Type u_1\nS : Type ?u.25753\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : IsDomain A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nM : Submonoid A\np : R[X]\nhp : p \u2260 0\ns : R\nh : scaleRoots p s = 0\n\u22a2 False"}, {"tactic": "have : p.coeff p.natDegree \u2260 0 := mt leadingCoeff_eq_zero.mp hp", "state_before": "A : Type ?u.25744\nK : Type ?u.25747\nR : Type u_1\nS : Type ?u.25753\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : IsDomain A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nM : Submonoid A\np : R[X]\nhp : p \u2260 0\ns : R\nh : scaleRoots p s = 0\n\u22a2 False", "state_after": "A : Type ?u.25744\nK : Type ?u.25747\nR : Type u_1\nS : Type ?u.25753\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : IsDomain A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nM : Submonoid A\np : R[X]\nhp : p \u2260 0\ns : R\nh : scaleRoots p s = 0\nthis : coeff p (natDegree p) \u2260 0\n\u22a2 False"}, {"tactic": "have : (scaleRoots p s).coeff p.natDegree = 0 :=\n  congr_fun (congr_arg (coeff : R[X] \u2192 \u2115 \u2192 R) h) p.natDegree", "state_before": "A : Type ?u.25744\nK : Type ?u.25747\nR : Type u_1\nS : Type ?u.25753\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : IsDomain A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nM : Submonoid A\np : R[X]\nhp : p \u2260 0\ns : R\nh : scaleRoots p s = 0\nthis : coeff p (natDegree p) \u2260 0\n\u22a2 False", "state_after": "A : Type ?u.25744\nK : Type ?u.25747\nR : Type u_1\nS : Type ?u.25753\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : IsDomain A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nM : Submonoid A\np : R[X]\nhp : p \u2260 0\ns : R\nh : scaleRoots p s = 0\nthis\u271d : coeff p (natDegree p) \u2260 0\nthis : coeff (scaleRoots p s) (natDegree p) = 0\n\u22a2 False"}, {"tactic": "rw [coeff_scaleRoots_natDegree] at this", "state_before": "A : Type ?u.25744\nK : Type ?u.25747\nR : Type u_1\nS : Type ?u.25753\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : IsDomain A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nM : Submonoid A\np : R[X]\nhp : p \u2260 0\ns : R\nh : scaleRoots p s = 0\nthis\u271d : coeff p (natDegree p) \u2260 0\nthis : coeff (scaleRoots p s) (natDegree p) = 0\n\u22a2 False", "state_after": "A : Type ?u.25744\nK : Type ?u.25747\nR : Type u_1\nS : Type ?u.25753\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : IsDomain A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nM : Submonoid A\np : R[X]\nhp : p \u2260 0\ns : R\nh : scaleRoots p s = 0\nthis\u271d : coeff p (natDegree p) \u2260 0\nthis : leadingCoeff p = 0\n\u22a2 False"}, {"tactic": "contradiction", "state_before": "A : Type ?u.25744\nK : Type ?u.25747\nR : Type u_1\nS : Type ?u.25753\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : IsDomain A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nM : Submonoid A\np : R[X]\nhp : p \u2260 0\ns : R\nh : scaleRoots p s = 0\nthis\u271d : coeff p (natDegree p) \u2260 0\nthis : leadingCoeff p = 0\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "topologicalGroup_induced", "start": [685, 1], "end": [688, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.support_eq_empty", "start": [819, 1], "end": [820, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/OrderIsoNat.lean", "full_name": "Nat.orderEmbeddingOfSet_apply", "start": [138, 1], "end": [140, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "full_name": "Asymptotics.isBigO_pure", "start": [1265, 1], "end": [1268, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/ContDiffDef.lean", "full_name": "contDiffOn_top", "start": [689, 1], "end": [690, 83], "traced_tactics": [{"tactic": "simp only [le_top, forall_prop_of_true]", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 (\u2200 (m : \u2115), \u2191m \u2264 \u22a4 \u2192 ContDiffOn \ud835\udd5c (\u2191m) f s) \u2194 \u2200 (n : \u2115), ContDiffOn \ud835\udd5c (\u2191n) f s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/ContinuousMonoidHom.lean", "full_name": "ContinuousMonoidHom.continuous_comp_left", "start": [368, 1], "end": [371, 90], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", 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Balanced l\nhr : Balanced r\nsl : Sized l\nsr : Sized r\nH : (\u2203 l', Raised (size l) l' \u2227 BalancedSz l' (size r)) \u2228 \u2203 r', Raised r' (size r) \u2227 BalancedSz (size l) r'\n\u22a2 balanceR l x r = balance' l x r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Monoidal/Braided.lean", "full_name": "CategoryTheory.rightUnitor_inv_braiding", "start": [216, 1], "end": [218, 57], "traced_tactics": [{"tactic": "apply (cancel_mono (\u03bb_ X).hom).1", "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\u03c1_ X).inv \u226b (\u03b2_ X (\ud835\udfd9_ C)).hom = (\u03bb_ X).inv", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 ((\u03c1_ X).inv \u226b (\u03b2_ X (\ud835\udfd9_ C)).hom) \u226b (\u03bb_ X).hom = (\u03bb_ X).inv \u226b (\u03bb_ X).hom"}, {"tactic": "simp only [assoc, braiding_leftUnitor, Iso.inv_hom_id]", "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 ((\u03c1_ X).inv \u226b (\u03b2_ X (\ud835\udfd9_ C)).hom) \u226b (\u03bb_ X).hom = (\u03bb_ X).inv \u226b (\u03bb_ X).hom", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Core.lean", "full_name": "ofBoolUsing_eq_true", "start": [709, 1], "end": [710, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/LocallyConstant/Basic.lean", "full_name": "LocallyConstant.comap_id", "start": [484, 1], "end": [486, 71], "traced_tactics": [{"tactic": "ext", "state_before": "X : Type u_1\nY : Type ?u.32577\nZ : Type u_2\n\u03b1 : Type ?u.32583\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\n\u22a2 comap id = id", "state_after": "case h.h\nX : Type u_1\nY : Type ?u.32577\nZ : Type u_2\n\u03b1 : Type ?u.32583\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u271d\u00b9 : LocallyConstant X Z\nx\u271d : X\n\u22a2 \u2191(comap id x\u271d\u00b9) x\u271d = \u2191(id x\u271d\u00b9) x\u271d"}, {"tactic": "simp only [continuous_id, id.def, Function.comp.right_id, coe_comap]", "state_before": "case h.h\nX : Type u_1\nY : Type ?u.32577\nZ : Type u_2\n\u03b1 : Type ?u.32583\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u271d\u00b9 : LocallyConstant X Z\nx\u271d : X\n\u22a2 \u2191(comap id x\u271d\u00b9) x\u271d = \u2191(id x\u271d\u00b9) x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "full_name": "continuousSMul_induced", "start": [107, 1], "end": [111, 73], "traced_tactics": [{"tactic": "let _ : TopologicalSpace M\u2081 := t.induced f", "state_before": "\u03b9 : Type ?u.48066\nR : Type u_1\nM\u2081 : Type u_2\nM\u2082 : Type u_3\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\nu : TopologicalSpace R\nt : TopologicalSpace M\u2082\ninst\u271d : ContinuousSMul R M\u2082\nf : M\u2081 \u2192\u2097[R] M\u2082\n\u22a2 ContinuousSMul R M\u2081", "state_after": "\u03b9 : Type ?u.48066\nR : Type u_1\nM\u2081 : Type u_2\nM\u2082 : Type u_3\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\nu : TopologicalSpace R\nt : TopologicalSpace M\u2082\ninst\u271d : ContinuousSMul R M\u2082\nf : M\u2081 \u2192\u2097[R] M\u2082\nx\u271d : TopologicalSpace M\u2081 := TopologicalSpace.induced (\u2191f) t\n\u22a2 ContinuousSMul R M\u2081"}, {"tactic": "refine' \u27e8continuous_induced_rng.2 _\u27e9", "state_before": "\u03b9 : Type ?u.48066\nR : Type u_1\nM\u2081 : Type u_2\nM\u2082 : Type u_3\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\nu : TopologicalSpace R\nt : TopologicalSpace M\u2082\ninst\u271d : ContinuousSMul R M\u2082\nf : M\u2081 \u2192\u2097[R] M\u2082\nx\u271d : TopologicalSpace M\u2081 := TopologicalSpace.induced (\u2191f) t\n\u22a2 ContinuousSMul R M\u2081", "state_after": "\u03b9 : Type ?u.48066\nR : Type u_1\nM\u2081 : Type u_2\nM\u2082 : Type u_3\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\nu : TopologicalSpace R\nt : TopologicalSpace M\u2082\ninst\u271d : ContinuousSMul R M\u2082\nf : M\u2081 \u2192\u2097[R] M\u2082\nx\u271d : TopologicalSpace M\u2081 := TopologicalSpace.induced (\u2191f) t\n\u22a2 Continuous (\u2191f \u2218 fun p => p.fst \u2022 p.snd)"}, {"tactic": "simp_rw [Function.comp, f.map_smul]", "state_before": "\u03b9 : Type ?u.48066\nR : Type u_1\nM\u2081 : Type u_2\nM\u2082 : Type u_3\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\nu : TopologicalSpace R\nt : TopologicalSpace M\u2082\ninst\u271d : ContinuousSMul R M\u2082\nf : M\u2081 \u2192\u2097[R] M\u2082\nx\u271d : TopologicalSpace M\u2081 := TopologicalSpace.induced (\u2191f) t\n\u22a2 Continuous (\u2191f \u2218 fun p => p.fst \u2022 p.snd)", "state_after": "\u03b9 : Type ?u.48066\nR : Type u_1\nM\u2081 : Type u_2\nM\u2082 : Type u_3\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\nu : TopologicalSpace R\nt : TopologicalSpace M\u2082\ninst\u271d : ContinuousSMul R M\u2082\nf : M\u2081 \u2192\u2097[R] M\u2082\nx\u271d : TopologicalSpace M\u2081 := TopologicalSpace.induced (\u2191f) t\n\u22a2 Continuous fun x => x.fst \u2022 \u2191f x.snd"}, {"tactic": "exact continuous_fst.smul (continuous_induced_dom.comp continuous_snd)", "state_before": "\u03b9 : Type ?u.48066\nR : Type u_1\nM\u2081 : Type u_2\nM\u2082 : Type u_3\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : Module R M\u2082\nu : TopologicalSpace R\nt : TopologicalSpace M\u2082\ninst\u271d : ContinuousSMul R M\u2082\nf : M\u2081 \u2192\u2097[R] M\u2082\nx\u271d : TopologicalSpace M\u2081 := TopologicalSpace.induced (\u2191f) t\n\u22a2 Continuous fun x => x.fst \u2022 \u2191f x.snd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/IteratedDeriv.lean", "full_name": "iteratedDerivWithin_one", "start": [124, 1], "end": [126, 73], "traced_tactics": [{"tactic": "simp only [iteratedDerivWithin, iteratedFDerivWithin_one_apply h]", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type ?u.32314\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nn : \u2115\nf : \ud835\udd5c \u2192 F\ns : Set \ud835\udd5c\nx\u271d x : \ud835\udd5c\nh : UniqueDiffWithinAt \ud835\udd5c s x\n\u22a2 iteratedDerivWithin 1 f s x = derivWithin f s x", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type ?u.32314\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nn : \u2115\nf : \ud835\udd5c \u2192 F\ns : Set \ud835\udd5c\nx\u271d x : \ud835\udd5c\nh : UniqueDiffWithinAt \ud835\udd5c s x\n\u22a2 \u2191(fderivWithin \ud835\udd5c f s x) 1 = derivWithin f s x"}, {"tactic": "rfl", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type ?u.32314\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nn : \u2115\nf : \ud835\udd5c \u2192 F\ns : Set \ud835\udd5c\nx\u271d x : \ud835\udd5c\nh : UniqueDiffWithinAt \ud835\udd5c s x\n\u22a2 \u2191(fderivWithin \ud835\udd5c f s x) 1 = derivWithin f s x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Basic.lean", "full_name": "Finset.sum_multiset_singleton", "start": [200, 1], "end": [201, 62], "traced_tactics": [{"tactic": "simp only [sum_eq_multiset_sum, Multiset.sum_map_singleton]", "state_before": "\u03b9 : Type ?u.92407\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf g : \u03b1 \u2192 \u03b2\ns : Finset \u03b1\n\u22a2 \u2211 x in s, {x} = s.val", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/RingDivision.lean", "full_name": "Polynomial.exists_prod_multiset_X_sub_C_mul", "start": [1100, 1], "end": [1115, 88], "traced_tactics": [{"tactic": "obtain \u27e8q, he\u27e9 := p.prod_multiset_X_sub_C_dvd", "state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q p : R[X]\n\u22a2 \u2203 q,\n    Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q = p \u2227\n      \u2191Multiset.card (roots p) + natDegree q = natDegree p \u2227 roots q = 0", "state_after": "case intro\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q\u271d p q : R[X]\nhe : p = Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q\n\u22a2 \u2203 q,\n    Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q = p \u2227\n      \u2191Multiset.card (roots p) + natDegree q = natDegree p \u2227 roots q = 0"}, {"tactic": "use q, he.symm", "state_before": "case intro\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q\u271d p q : R[X]\nhe : p = Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q\n\u22a2 \u2203 q,\n    Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q = p \u2227\n      \u2191Multiset.card (roots p) + natDegree q = natDegree p \u2227 roots q = 0", "state_after": "case intro\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q\u271d p q : R[X]\nhe : p = Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q\n\u22a2 \u2191Multiset.card (roots p) + natDegree q = natDegree p \u2227 roots q = 0"}, {"tactic": "obtain rfl | hq := eq_or_ne q 0", "state_before": "case intro\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q\u271d p q : R[X]\nhe : p = Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q\n\u22a2 \u2191Multiset.card (roots p) + natDegree q = natDegree p \u2227 roots q = 0", "state_after": "case intro.inl\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q p : R[X]\nhe : p = Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * 0\n\u22a2 \u2191Multiset.card (roots p) + natDegree 0 = natDegree p \u2227 roots 0 = 0\n\ncase intro.inr\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q\u271d p q : R[X]\nhe : p = Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q\nhq : q \u2260 0\n\u22a2 \u2191Multiset.card (roots p) + natDegree q = natDegree p \u2227 roots q = 0"}, {"tactic": "constructor", "state_before": "case intro.inr\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q\u271d p q : R[X]\nhe : p = Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q\nhq : q \u2260 0\n\u22a2 \u2191Multiset.card (roots p) + natDegree q = natDegree p \u2227 roots q = 0", "state_after": "case intro.inr.left\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q\u271d p q : R[X]\nhe : p = Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q\nhq : q \u2260 0\n\u22a2 \u2191Multiset.card (roots p) + natDegree q = natDegree p\n\ncase intro.inr.right\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q\u271d p q : R[X]\nhe : p = Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q\nhq : q \u2260 0\n\u22a2 roots q = 0"}, {"tactic": "rw [MulZeroClass.mul_zero] at he", "state_before": "case intro.inl\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q p : R[X]\nhe : p = Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * 0\n\u22a2 \u2191Multiset.card (roots p) + natDegree 0 = natDegree p \u2227 roots 0 = 0", "state_after": "case intro.inl\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q p : R[X]\nhe : p = 0\n\u22a2 \u2191Multiset.card (roots p) + natDegree 0 = natDegree p \u2227 roots 0 = 0"}, {"tactic": "subst he", "state_before": "case intro.inl\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q p : R[X]\nhe : p = 0\n\u22a2 \u2191Multiset.card (roots p) + natDegree 0 = natDegree p \u2227 roots 0 = 0", "state_after": "case intro.inl\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np q : R[X]\n\u22a2 \u2191Multiset.card (roots 0) + natDegree 0 = natDegree 0 \u2227 roots 0 = 0"}, {"tactic": "simp", "state_before": "case intro.inl\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np q : R[X]\n\u22a2 \u2191Multiset.card (roots 0) + natDegree 0 = natDegree 0 \u2227 roots 0 = 0", "state_after": "no goals"}, {"tactic": "conv_rhs => rw [he]", "state_before": "case intro.inr.left\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q\u271d p q : R[X]\nhe : p = Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q\nhq : q \u2260 0\n\u22a2 \u2191Multiset.card (roots p) + natDegree q = natDegree p", "state_after": "case intro.inr.left\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q\u271d p q : R[X]\nhe : p = Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q\nhq : q \u2260 0\n\u22a2 \u2191Multiset.card (roots p) + natDegree q = natDegree (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q)"}, {"tactic": "rw [monic_prod_multiset_X_sub_C.natDegree_mul' hq, natDegree_multiset_prod_X_sub_C_eq_card]", "state_before": "case intro.inr.left\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q\u271d p q : R[X]\nhe : p = Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q\nhq : q \u2260 0\n\u22a2 \u2191Multiset.card (roots p) + natDegree q = natDegree (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q)", "state_after": "no goals"}, {"tactic": "replace he := congr_arg roots he.symm", "state_before": "case intro.inr.right\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q\u271d p q : R[X]\nhe : p = Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q\nhq : q \u2260 0\n\u22a2 roots q = 0", "state_after": "case intro.inr.right\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q\u271d p q : R[X]\nhq : q \u2260 0\nhe : roots (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q) = roots p\n\u22a2 roots q = 0"}, {"tactic": "rw [roots_mul, roots_multiset_prod_X_sub_C] at he", "state_before": "case intro.inr.right\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q\u271d p q : R[X]\nhq : q \u2260 0\nhe : roots (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q) = roots p\n\u22a2 roots q = 0", "state_after": "case intro.inr.right\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q\u271d p q : R[X]\nhq : q \u2260 0\nhe : roots p + roots q = roots p\n\u22a2 roots q = 0\n\ncase intro.inr.right\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q\u271d p q : R[X]\nhq : q \u2260 0\nhe : roots (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q) = roots p\n\u22a2 Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q \u2260 0"}, {"tactic": "exacts [add_right_eq_self.1 he, mul_ne_zero monic_prod_multiset_X_sub_C.ne_zero hq]", "state_before": "case intro.inr.right\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q\u271d p q : R[X]\nhq : q \u2260 0\nhe : roots p + roots q = roots p\n\u22a2 roots q = 0\n\ncase intro.inr.right\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q\u271d p q : R[X]\nhq : q \u2260 0\nhe : roots (Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q) = roots p\n\u22a2 Multiset.prod (Multiset.map (fun a => X - \u2191C a) (roots p)) * q \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/SubsetProperties.lean", "full_name": "nhdsContainBoxes_of_compact", "start": [666, 1], "end": [688, 66], "traced_tactics": [{"tactic": "simpa", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.58082\n\u03c0 : \u03b9 \u2192 Type ?u.58087\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d s : Set \u03b1\nhs : IsCompact s\nt : Set \u03b2\nH : \u2200 (x : \u03b1), x \u2208 s \u2192 NhdsContainBoxes {x} t\nn : Set (\u03b1 \u00d7 \u03b2)\nhn : IsOpen n\nhp : s \u00d7\u02e2 t \u2286 n\nx\u271d : \u2191s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 {x} \u2286 s", "state_after": "no goals"}, {"tactic": "simpa using (h \u27e8x, hx\u27e9).2.2.1", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.58082\n\u03c0 : \u03b9 \u2192 Type ?u.58087\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d s : Set \u03b1\nhs : IsCompact s\nt : Set \u03b2\nH : \u2200 (x : \u03b1), x \u2208 s \u2192 NhdsContainBoxes {x} t\nn : Set (\u03b1 \u00d7 \u03b2)\nhn : IsOpen n\nhp : s \u00d7\u02e2 t \u2286 n\nthis : \u2200 (x : \u2191s), \u2203 uv, IsOpen uv.fst \u2227 IsOpen uv.snd \u2227 {\u2191x} \u2286 uv.fst \u2227 t \u2286 uv.snd \u2227 uv.fst \u00d7\u02e2 uv.snd \u2286 n\nuvs : \u2191s \u2192 Set \u03b1 \u00d7 Set \u03b2\nh :\n  \u2200 (x : \u2191s),\n    IsOpen (uvs x).fst \u2227 IsOpen (uvs x).snd \u2227 {\u2191x} \u2286 (uvs x).fst \u2227 t \u2286 (uvs x).snd \u2227 (uvs x).fst \u00d7\u02e2 (uvs x).snd \u2286 n\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 (uvs { val := x, property := hx }).fst", "state_after": "no goals"}, {"tactic": "simpa using hx'", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.58082\n\u03c0 : \u03b9 \u2192 Type ?u.58087\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d s : Set \u03b1\nhs : IsCompact s\nt : Set \u03b2\nH : \u2200 (x : \u03b1), x \u2208 s \u2192 NhdsContainBoxes {x} t\nn : Set (\u03b1 \u00d7 \u03b2)\nhn : IsOpen n\nhp : s \u00d7\u02e2 t \u2286 n\nthis\u271d\u00b2 : \u2200 (x : \u2191s), \u2203 uv, IsOpen uv.fst \u2227 IsOpen uv.snd \u2227 {\u2191x} \u2286 uv.fst \u2227 t \u2286 uv.snd \u2227 uv.fst \u00d7\u02e2 uv.snd \u2286 n\nuvs : \u2191s \u2192 Set \u03b1 \u00d7 Set \u03b2\nh :\n  \u2200 (x : \u2191s),\n    IsOpen (uvs x).fst \u2227 IsOpen (uvs x).snd \u2227 {\u2191x} \u2286 (uvs x).fst \u2227 t \u2286 (uvs x).snd \u2227 (uvs x).fst \u00d7\u02e2 (uvs x).snd \u2286 n\nus_cover : s \u2286 \u22c3 (i : \u2191s), (uvs i).fst\ns0 : Finset \u2191s\ns0_cover : s \u2286 \u22c3 (i : \u2191s) (_ : i \u2208 s0), (uvs i).fst\nu : Set \u03b1 := \u22c3 (i : \u2191s) (_ : i \u2208 s0), (uvs i).fst\nv : Set \u03b2 := \u22c2 (i : \u2191s) (_ : i \u2208 s0), (uvs i).snd\nthis\u271d\u00b9 : IsOpen u\nthis\u271d : IsOpen v\nthis : t \u2286 v\nx\u271d\u00b9 : \u03b1 \u00d7 \u03b2\nx' : \u03b1\ny' : \u03b2\nx\u271d : (x', y') \u2208 u \u00d7\u02e2 v\nhx' : (x', y').fst \u2208 u\nhy' : (x', y').snd \u2208 v\n\u22a2 \u2203 i, i \u2208 s0 \u2227 x' \u2208 (uvs i).fst", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Quiver/Path.lean", "full_name": "Quiver.Path.eq_of_length_zero", "start": [85, 1], "end": [88, 35], "traced_tactics": [{"tactic": "cases p", "state_before": "V : Type u\ninst\u271d : Quiver V\na b c d : V\np : Path a b\nhzero : length p = 0\n\u22a2 a = b", "state_after": "case nil\nV : Type u\ninst\u271d : Quiver V\na c d : V\nhzero : length nil = 0\n\u22a2 a = a\n\ncase cons\nV : Type u\ninst\u271d : Quiver V\na b c d b\u271d : V\na\u271d\u00b9 : Path a b\u271d\na\u271d : b\u271d \u27f6 b\nhzero : length (cons a\u271d\u00b9 a\u271d) = 0\n\u22a2 a = b"}, {"tactic": "rfl", "state_before": "case nil\nV : Type u\ninst\u271d : Quiver V\na c d : V\nhzero : length nil = 0\n\u22a2 a = a", "state_after": "no goals"}, {"tactic": "cases Nat.succ_ne_zero _ hzero", "state_before": "case cons\nV : Type u\ninst\u271d : Quiver V\na b c d b\u271d : V\na\u271d\u00b9 : Path a b\u271d\na\u271d : b\u271d \u27f6 b\nhzero : length (cons a\u271d\u00b9 a\u271d) = 0\n\u22a2 a = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/ContDiff.lean", "full_name": "HasFTaylorSeriesUpToOn.prod", "start": [506, 1], "end": [516, 82], "traced_tactics": [{"tactic": "set L := fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun i : Fin m => E) F G", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.591998\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\n\u22a2 HasFTaylorSeriesUpToOn n (fun y => (f y, g y)) (fun y k => ContinuousMultilinearMap.prod (p y k) (q y k)) s", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.591998\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun i => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun i => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun i => E) F G\n\u22a2 HasFTaylorSeriesUpToOn n (fun y => (f y, g y)) (fun y k => ContinuousMultilinearMap.prod (p y k) (q y k)) s"}, {"tactic": "constructor", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.591998\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun i => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun i => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun i => E) F G\n\u22a2 HasFTaylorSeriesUpToOn n (fun y => (f y, g y)) (fun y k => ContinuousMultilinearMap.prod (p y k) (q y k)) s", "state_after": "case zero_eq\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.591998\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun i => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun i => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun i => E) F G\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (ContinuousMultilinearMap.prod (p x 0) (q x 0)) = (f x, g x)\n\ncase fderivWithin\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.591998\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun i => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun i => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun i => E) F G\n\u22a2 \u2200 (m : \u2115),\n    \u2191m < n \u2192\n      \u2200 (x : E),\n        x \u2208 s \u2192\n          HasFDerivWithinAt (fun x => ContinuousMultilinearMap.prod (p x m) (q x m))\n            (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.prod (p x (Nat.succ m)) (q x (Nat.succ m)))) s\n            x\n\ncase cont\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.591998\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun i => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun i => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun i => E) F G\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => ContinuousMultilinearMap.prod (p x m) (q x m)) s"}, {"tactic": "intro x hx", "state_before": "case zero_eq\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.591998\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun i => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun i => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun i => E) F G\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (ContinuousMultilinearMap.prod (p x 0) (q x 0)) = (f x, g x)", "state_after": "case zero_eq\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.591998\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun i => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun i => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun i => E) F G\nx : E\nhx : x \u2208 s\n\u22a2 ContinuousMultilinearMap.uncurry0 (ContinuousMultilinearMap.prod (p x 0) (q x 0)) = (f x, g x)"}, {"tactic": "rw [\u2190 hf.zero_eq x hx, \u2190 hg.zero_eq x hx]", "state_before": "case zero_eq\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.591998\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun i => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun i => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun i => E) F G\nx : E\nhx : x \u2208 s\n\u22a2 ContinuousMultilinearMap.uncurry0 (ContinuousMultilinearMap.prod (p x 0) (q x 0)) = (f x, g x)", "state_after": "case zero_eq\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.591998\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun i => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun i => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun i => E) F G\nx : E\nhx : x \u2208 s\n\u22a2 ContinuousMultilinearMap.uncurry0 (ContinuousMultilinearMap.prod (p x 0) (q x 0)) =\n    (ContinuousMultilinearMap.uncurry0 (p x 0), ContinuousMultilinearMap.uncurry0 (q x 0))"}, {"tactic": "rfl", "state_before": "case zero_eq\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.591998\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun i => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun i => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun i => E) F G\nx : E\nhx : x \u2208 s\n\u22a2 ContinuousMultilinearMap.uncurry0 (ContinuousMultilinearMap.prod (p x 0) (q x 0)) =\n    (ContinuousMultilinearMap.uncurry0 (p x 0), ContinuousMultilinearMap.uncurry0 (q x 0))", "state_after": "no goals"}, {"tactic": "intro m hm x hx", "state_before": "case fderivWithin\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.591998\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun i => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun i => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun i => E) F G\n\u22a2 \u2200 (m : \u2115),\n    \u2191m < n \u2192\n      \u2200 (x : E),\n        x \u2208 s \u2192\n          HasFDerivWithinAt (fun x => ContinuousMultilinearMap.prod (p x m) (q x m))\n            (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.prod (p x (Nat.succ m)) (q x (Nat.succ m)))) s\n            x", "state_after": "case fderivWithin\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.591998\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun i => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun i => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun i => E) F G\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\n\u22a2 HasFDerivWithinAt (fun x => ContinuousMultilinearMap.prod (p x m) (q x m))\n    (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.prod (p x (Nat.succ m)) (q x (Nat.succ m)))) s x"}, {"tactic": "convert (L m).hasFDerivAt.comp_hasFDerivWithinAt x\n    ((hf.fderivWithin m hm x hx).prod (hg.fderivWithin m hm x hx))", "state_before": "case fderivWithin\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.591998\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun i => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun i => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun i => E) F G\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\n\u22a2 HasFDerivWithinAt (fun x => ContinuousMultilinearMap.prod (p x m) (q x m))\n    (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.prod (p x (Nat.succ m)) (q x (Nat.succ m)))) s x", "state_after": "no goals"}, {"tactic": "intro m hm", "state_before": "case cont\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.591998\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun i => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun i => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun i => E) F G\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => ContinuousMultilinearMap.prod (p x m) (q x m)) s", "state_after": "case cont\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.591998\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun i => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun i => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun i => E) F G\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 ContinuousOn (fun x => ContinuousMultilinearMap.prod (p x m) (q x m)) s"}, {"tactic": "exact (L m).continuous.comp_continuousOn ((hf.cont m hm).prod (hg.cont m hm))", "state_before": "case cont\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.591998\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun i => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun i => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun i => E) F G\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 ContinuousOn (fun x => ContinuousMultilinearMap.prod (p x m) (q x m)) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Equiv.lean", "full_name": "ContinuousLinearEquiv.hasFDerivWithinAt", "start": [66, 11], "end": [67, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.castLE_zero", "start": [1028, 1], "end": [1028, 93], "traced_tactics": [{"tactic": "simp [eq_iff_veq]", "state_before": "n\u271d m\u271d n m : \u2115\nh : Nat.succ n \u2264 Nat.succ m\n\u22a2 \u2191(castLE h) 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Control/LawfulFix.lean", "full_name": "Part.Fix.exists_fix_le_approx", "start": [102, 1], "end": [115, 20], "traced_tactics": [{"tactic": "by_cases hh : \u2203 i b, b \u2208 approx f i x", "state_before": "\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\n\u22a2 \u2203 i, Part.fix (\u2191f) x \u2264 approx (\u2191f) i x", "state_after": "case pos\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nhh : \u2203 i b, b \u2208 approx (\u2191f) i x\n\u22a2 \u2203 i, Part.fix (\u2191f) x \u2264 approx (\u2191f) i x\n\ncase neg\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nhh : \u00ac\u2203 i b, b \u2208 approx (\u2191f) i x\n\u22a2 \u2203 i, Part.fix (\u2191f) x \u2264 approx (\u2191f) i x"}, {"tactic": "rcases hh with \u27e8i, b, hb\u27e9", "state_before": "case pos\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nhh : \u2203 i b, b \u2208 approx (\u2191f) i x\n\u22a2 \u2203 i, Part.fix (\u2191f) x \u2264 approx (\u2191f) i x", "state_after": "case pos.intro.intro\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\ni : \u2115\nb : \u03b2 x\nhb : b \u2208 approx (\u2191f) i x\n\u22a2 \u2203 i, Part.fix (\u2191f) x \u2264 approx (\u2191f) i x"}, {"tactic": "exists i", "state_before": "case pos.intro.intro\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\ni : \u2115\nb : \u03b2 x\nhb : b \u2208 approx (\u2191f) i x\n\u22a2 \u2203 i, Part.fix (\u2191f) x \u2264 approx (\u2191f) i x", "state_after": "case pos.intro.intro\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\ni : \u2115\nb : \u03b2 x\nhb : b \u2208 approx (\u2191f) i x\n\u22a2 Part.fix (\u2191f) x \u2264 approx (\u2191f) i x"}, {"tactic": "intro b' h'", "state_before": "case pos.intro.intro\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\ni : \u2115\nb : \u03b2 x\nhb : b \u2208 approx (\u2191f) i x\n\u22a2 Part.fix (\u2191f) x \u2264 approx (\u2191f) i x", "state_after": "case pos.intro.intro\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\ni : \u2115\nb : \u03b2 x\nhb : b \u2208 approx (\u2191f) i x\nb' : \u03b2 x\nh' : b' \u2208 Part.fix (\u2191f) x\n\u22a2 b' \u2208 approx (\u2191f) i x"}, {"tactic": "have hb' := approx_le_fix f i _ _ hb", "state_before": "case pos.intro.intro\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\ni : \u2115\nb : \u03b2 x\nhb : b \u2208 approx (\u2191f) i x\nb' : \u03b2 x\nh' : b' \u2208 Part.fix (\u2191f) x\n\u22a2 b' \u2208 approx (\u2191f) i x", "state_after": "case pos.intro.intro\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\ni : \u2115\nb : \u03b2 x\nhb : b \u2208 approx (\u2191f) i x\nb' : \u03b2 x\nh' : b' \u2208 Part.fix (\u2191f) x\nhb' : b \u2208 Part.fix (\u2191f) x\n\u22a2 b' \u2208 approx (\u2191f) i x"}, {"tactic": "obtain rfl := Part.mem_unique h' hb'", "state_before": "case pos.intro.intro\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\ni : \u2115\nb : \u03b2 x\nhb : b \u2208 approx (\u2191f) i x\nb' : \u03b2 x\nh' : b' \u2208 Part.fix (\u2191f) x\nhb' : b \u2208 Part.fix (\u2191f) x\n\u22a2 b' \u2208 approx (\u2191f) i x", "state_after": "case pos.intro.intro\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\ni : \u2115\nb' : \u03b2 x\nh' : b' \u2208 Part.fix (\u2191f) x\nhb : b' \u2208 approx (\u2191f) i x\nhb' : b' \u2208 Part.fix (\u2191f) x\n\u22a2 b' \u2208 approx (\u2191f) i x"}, {"tactic": "exact hb", "state_before": "case pos.intro.intro\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\ni : \u2115\nb' : \u03b2 x\nh' : b' \u2208 Part.fix (\u2191f) x\nhb : b' \u2208 approx (\u2191f) i x\nhb' : b' \u2208 Part.fix (\u2191f) x\n\u22a2 b' \u2208 approx (\u2191f) i x", "state_after": "no goals"}, {"tactic": "simp only [not_exists] at hh", "state_before": "case neg\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nhh : \u00ac\u2203 i b, b \u2208 approx (\u2191f) i x\n\u22a2 \u2203 i, Part.fix (\u2191f) x \u2264 approx (\u2191f) i x", "state_after": "case neg\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nhh : \u2200 (x_1 : \u2115) (x_2 : \u03b2 x), \u00acx_2 \u2208 approx (\u2191f) x_1 x\n\u22a2 \u2203 i, Part.fix (\u2191f) x \u2264 approx (\u2191f) i x"}, {"tactic": "exists 0", "state_before": "case neg\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nhh : \u2200 (x_1 : \u2115) (x_2 : \u03b2 x), \u00acx_2 \u2208 approx (\u2191f) x_1 x\n\u22a2 \u2203 i, Part.fix (\u2191f) x \u2264 approx (\u2191f) i x", "state_after": "case neg\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nhh : \u2200 (x_1 : \u2115) (x_2 : \u03b2 x), \u00acx_2 \u2208 approx (\u2191f) x_1 x\n\u22a2 Part.fix (\u2191f) x \u2264 approx (\u2191f) 0 x"}, {"tactic": "intro b' h'", "state_before": "case neg\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nhh : \u2200 (x_1 : \u2115) (x_2 : \u03b2 x), \u00acx_2 \u2208 approx (\u2191f) x_1 x\n\u22a2 Part.fix (\u2191f) x \u2264 approx (\u2191f) 0 x", "state_after": "case neg\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nhh : \u2200 (x_1 : \u2115) (x_2 : \u03b2 x), \u00acx_2 \u2208 approx (\u2191f) x_1 x\nb' : \u03b2 x\nh' : b' \u2208 Part.fix (\u2191f) x\n\u22a2 b' \u2208 approx (\u2191f) 0 x"}, {"tactic": "simp only [mem_iff f] at h'", "state_before": "case neg\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nhh : \u2200 (x_1 : \u2115) (x_2 : \u03b2 x), \u00acx_2 \u2208 approx (\u2191f) x_1 x\nb' : \u03b2 x\nh' : b' \u2208 Part.fix (\u2191f) x\n\u22a2 b' \u2208 approx (\u2191f) 0 x", "state_after": "case neg\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nhh : \u2200 (x_1 : \u2115) (x_2 : \u03b2 x), \u00acx_2 \u2208 approx (\u2191f) x_1 x\nb' : \u03b2 x\nh' : \u2203 i, b' \u2208 approx (\u2191f) i x\n\u22a2 b' \u2208 approx (\u2191f) 0 x"}, {"tactic": "cases' h' with i h'", "state_before": "case neg\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nhh : \u2200 (x_1 : \u2115) (x_2 : \u03b2 x), \u00acx_2 \u2208 approx (\u2191f) x_1 x\nb' : \u03b2 x\nh' : \u2203 i, b' \u2208 approx (\u2191f) i x\n\u22a2 b' \u2208 approx (\u2191f) 0 x", "state_after": "case neg.intro\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nhh : \u2200 (x_1 : \u2115) (x_2 : \u03b2 x), \u00acx_2 \u2208 approx (\u2191f) x_1 x\nb' : \u03b2 x\ni : \u2115\nh' : b' \u2208 approx (\u2191f) i x\n\u22a2 b' \u2208 approx (\u2191f) 0 x"}, {"tactic": "cases hh _ _ h'", "state_before": "case neg.intro\n\u03b1 : Type u_2\n\u03b2 : \u03b1 \u2192 Type u_1\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192o (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nhh : \u2200 (x_1 : \u2115) (x_2 : \u03b2 x), \u00acx_2 \u2208 approx (\u2191f) x_1 x\nb' : \u03b2 x\ni : \u2115\nh' : b' \u2208 approx (\u2191f) i x\n\u22a2 b' \u2208 approx (\u2191f) 0 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/RelClasses.lean", "full_name": "IsTotal.swap", "start": [91, 1], "end": [92, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Relation.lean", "full_name": "Relation.ReflTransGen.trans_induction_on", "start": [311, 1], "end": [317, 70], "traced_tactics": [{"tactic": "induction h", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.17364\n\u03b3 : Type ?u.17367\n\u03b4 : Type ?u.17370\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d b\u271d c d : \u03b1\nP : {a b : \u03b1} \u2192 ReflTransGen r a b \u2192 Prop\na b : \u03b1\nh : ReflTransGen r a b\nih\u2081 : \u2200 (a : \u03b1), P (_ : ReflTransGen r a a)\nih\u2082 : \u2200 {a b : \u03b1} (h : r a b), P (_ : ReflTransGen r a b)\nih\u2083 : \u2200 {a b c : \u03b1} (h\u2081 : ReflTransGen r a b) (h\u2082 : ReflTransGen r b c), P h\u2081 \u2192 P h\u2082 \u2192 P (_ : ReflTransGen r a c)\n\u22a2 P h", "state_after": "case refl\n\u03b1 : Type u_1\n\u03b2 : Type ?u.17364\n\u03b3 : Type ?u.17367\n\u03b4 : Type ?u.17370\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d b\u271d c d : \u03b1\nP : {a b : \u03b1} \u2192 ReflTransGen r a b \u2192 Prop\na b : \u03b1\nih\u2081 : \u2200 (a : \u03b1), P (_ : ReflTransGen r a a)\nih\u2082 : \u2200 {a b : \u03b1} (h : r a b), P (_ : ReflTransGen r a b)\nih\u2083 : \u2200 {a b c : \u03b1} (h\u2081 : ReflTransGen r a b) (h\u2082 : ReflTransGen r b c), P h\u2081 \u2192 P h\u2082 \u2192 P (_ : ReflTransGen r a c)\n\u22a2 P (_ : ReflTransGen r a a)\n\ncase tail\n\u03b1 : Type u_1\n\u03b2 : Type ?u.17364\n\u03b3 : Type ?u.17367\n\u03b4 : Type ?u.17370\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d\u00b2 b\u271d\u00b9 c d : \u03b1\nP : {a b : \u03b1} \u2192 ReflTransGen r a b \u2192 Prop\na b : \u03b1\nih\u2081 : \u2200 (a : \u03b1), P (_ : ReflTransGen r a a)\nih\u2082 : \u2200 {a b : \u03b1} (h : r a b), P (_ : ReflTransGen r a b)\nih\u2083 : \u2200 {a b c : \u03b1} (h\u2081 : ReflTransGen r a b) (h\u2082 : ReflTransGen r b c), P h\u2081 \u2192 P h\u2082 \u2192 P (_ : ReflTransGen 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H j).app k\nj : J\n\u22a2 (f \u226b (limitObjIsoLimitCompEvaluation H k).hom) \u226b limit.\u03c0 (H \u22d9 (evaluation K C).obj k) j =\n    (g \u226b (limitObjIsoLimitCompEvaluation H k).hom) \u226b limit.\u03c0 (H \u22d9 (evaluation K C).obj k) j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/List/Lemmas.lean", "full_name": "List.filterMap_cons_some", "start": [1159, 1], "end": [1160, 77], "traced_tactics": [{"tactic": "simp only [filterMap, h]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 Option \u03b2\na : \u03b1\nl : List \u03b1\nb : \u03b2\nh : f a = some b\n\u22a2 filterMap f (a :: l) = b :: filterMap f l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Equiv.lean", "full_name": "LinearIsometryEquiv.differentiableOn", "start": [296, 11], "end": [297, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/UniformConvergence.lean", "full_name": "tendstoLocallyUniformly_iff_tendstoUniformly_of_compactSpace", "start": [690, 1], "end": [701, 33], "traced_tactics": [{"tactic": "refine' \u27e8fun h V hV => _, TendstoUniformly.tendstoLocallyUniformly\u27e9", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : CompactSpace \u03b1\n\u22a2 TendstoLocallyUniformly F f p \u2194 TendstoUniformly F f p", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : CompactSpace \u03b1\nh : TendstoLocallyUniformly F f p\nV : Set (\u03b2 \u00d7 \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\n\u22a2 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b1), (f x, F n x) \u2208 V"}, {"tactic": "choose U hU using h V hV", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : CompactSpace \u03b1\nh : TendstoLocallyUniformly F f p\nV : Set (\u03b2 \u00d7 \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\n\u22a2 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b1), (f x, F n x) \u2208 V", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : CompactSpace \u03b1\nh : TendstoLocallyUniformly F f p\nV : Set (\u03b2 \u00d7 \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\nU : \u03b1 \u2192 Set \u03b1\nhU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x \u2227 \u2200\u1da0 (n : \u03b9) in p, \u2200 (y : \u03b1), y \u2208 U x \u2192 (f y, F n y) \u2208 V\n\u22a2 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b1), (f x, F n x) \u2208 V"}, {"tactic": "obtain \u27e8t, ht\u27e9 := isCompact_univ.elim_nhds_subcover' (fun k _ => U k) fun k _ => (hU k).1", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : CompactSpace \u03b1\nh : TendstoLocallyUniformly F f p\nV : Set (\u03b2 \u00d7 \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\nU : \u03b1 \u2192 Set \u03b1\nhU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x \u2227 \u2200\u1da0 (n : \u03b9) in p, \u2200 (y : \u03b1), y \u2208 U x \u2192 (f y, F n y) \u2208 V\n\u22a2 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b1), (f x, F n x) \u2208 V", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : CompactSpace \u03b1\nh : TendstoLocallyUniformly F f p\nV : Set (\u03b2 \u00d7 \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\nU : \u03b1 \u2192 Set \u03b1\nhU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x \u2227 \u2200\u1da0 (n : \u03b9) in p, \u2200 (y : \u03b1), y \u2208 U x \u2192 (f y, F n y) \u2208 V\nt : Finset \u2191univ\nht : univ \u2286 \u22c3 (x : \u2191univ) (_ : x \u2208 t), U \u2191x\n\u22a2 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b1), (f x, F n x) \u2208 V"}, {"tactic": "replace hU := fun x : t => (hU x).2", "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : CompactSpace \u03b1\nh : TendstoLocallyUniformly F f p\nV : Set (\u03b2 \u00d7 \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\nU : \u03b1 \u2192 Set \u03b1\nhU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x \u2227 \u2200\u1da0 (n : \u03b9) in p, \u2200 (y : \u03b1), y \u2208 U x \u2192 (f y, F n y) \u2208 V\nt : Finset \u2191univ\nht : univ \u2286 \u22c3 (x : \u2191univ) (_ : x \u2208 t), U \u2191x\n\u22a2 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b1), (f x, F n x) \u2208 V", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : CompactSpace \u03b1\nh : TendstoLocallyUniformly F f p\nV : Set (\u03b2 \u00d7 \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\nU : \u03b1 \u2192 Set \u03b1\nt : Finset \u2191univ\nht : univ \u2286 \u22c3 (x : \u2191univ) (_ : x \u2208 t), U \u2191x\nhU : \u2200 (x : { x // x \u2208 t }), \u2200\u1da0 (n : \u03b9) in p, \u2200 (y : \u03b1), y \u2208 U \u2191\u2191x \u2192 (f y, F n y) \u2208 V\n\u22a2 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b1), (f x, F n x) \u2208 V"}, {"tactic": "rw [\u2190 eventually_all] at hU", "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : CompactSpace \u03b1\nh : TendstoLocallyUniformly F f p\nV : Set (\u03b2 \u00d7 \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\nU : \u03b1 \u2192 Set \u03b1\nt : Finset \u2191univ\nht : univ \u2286 \u22c3 (x : \u2191univ) (_ : x \u2208 t), U \u2191x\nhU : \u2200 (x : { x // x \u2208 t }), \u2200\u1da0 (n : \u03b9) in p, \u2200 (y : \u03b1), y \u2208 U \u2191\u2191x \u2192 (f y, F n y) \u2208 V\n\u22a2 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b1), (f x, F n x) \u2208 V", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : CompactSpace \u03b1\nh : TendstoLocallyUniformly F f p\nV : Set (\u03b2 \u00d7 \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\nU : \u03b1 \u2192 Set \u03b1\nt : Finset \u2191univ\nht : univ \u2286 \u22c3 (x : \u2191univ) (_ : x \u2208 t), U \u2191x\nhU : \u2200\u1da0 (x : \u03b9) in p, \u2200 (i : { x // x \u2208 t }) (y : \u03b1), y \u2208 U \u2191\u2191i \u2192 (f y, F x y) \u2208 V\n\u22a2 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b1), (f x, F n x) \u2208 V"}, {"tactic": "refine' hU.mono fun i hi x => _", "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : CompactSpace \u03b1\nh : TendstoLocallyUniformly F f p\nV : Set (\u03b2 \u00d7 \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\nU : \u03b1 \u2192 Set \u03b1\nt : Finset \u2191univ\nht : univ \u2286 \u22c3 (x : \u2191univ) (_ : x \u2208 t), U \u2191x\nhU : \u2200\u1da0 (x : \u03b9) in p, \u2200 (i : { x // x \u2208 t }) (y : \u03b1), y \u2208 U \u2191\u2191i \u2192 (f y, F x y) \u2208 V\n\u22a2 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b1), (f x, F n x) \u2208 V", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx\u271d : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : CompactSpace \u03b1\nh : TendstoLocallyUniformly F f p\nV : Set (\u03b2 \u00d7 \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\nU : \u03b1 \u2192 Set \u03b1\nt : Finset \u2191univ\nht : univ \u2286 \u22c3 (x : \u2191univ) (_ : x \u2208 t), U \u2191x\nhU : \u2200\u1da0 (x : \u03b9) in p, \u2200 (i : { x // x \u2208 t }) (y : \u03b1), y \u2208 U \u2191\u2191i \u2192 (f y, F x y) \u2208 V\ni : \u03b9\nhi : \u2200 (i_1 : { x // x \u2208 t }) (y : \u03b1), y \u2208 U \u2191\u2191i_1 \u2192 (f y, F i y) \u2208 V\nx : \u03b1\n\u22a2 (f x, F i x) \u2208 V"}, {"tactic": "specialize ht (mem_univ x)", "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx\u271d : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : CompactSpace \u03b1\nh : TendstoLocallyUniformly F f p\nV : Set (\u03b2 \u00d7 \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\nU : \u03b1 \u2192 Set \u03b1\nt : Finset \u2191univ\nht : univ \u2286 \u22c3 (x : \u2191univ) (_ : x \u2208 t), U \u2191x\nhU : \u2200\u1da0 (x : \u03b9) in p, \u2200 (i : { x // x \u2208 t }) (y : \u03b1), y \u2208 U \u2191\u2191i \u2192 (f y, F x y) \u2208 V\ni : \u03b9\nhi : \u2200 (i_1 : { x // x \u2208 t }) (y : \u03b1), y \u2208 U \u2191\u2191i_1 \u2192 (f y, F i y) \u2208 V\nx : \u03b1\n\u22a2 (f x, F i x) \u2208 V", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx\u271d : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : CompactSpace \u03b1\nh : TendstoLocallyUniformly F f p\nV : Set (\u03b2 \u00d7 \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\nU : \u03b1 \u2192 Set \u03b1\nt : Finset \u2191univ\nhU : \u2200\u1da0 (x : \u03b9) in p, \u2200 (i : { x // x \u2208 t }) (y : \u03b1), y \u2208 U \u2191\u2191i \u2192 (f y, F x y) \u2208 V\ni : \u03b9\nhi : \u2200 (i_1 : { x // x \u2208 t }) (y : \u03b1), y \u2208 U \u2191\u2191i_1 \u2192 (f y, F i y) \u2208 V\nx : \u03b1\nht : x \u2208 \u22c3 (x : \u2191univ) (_ : x \u2208 t), U \u2191x\n\u22a2 (f x, F i x) \u2208 V"}, {"tactic": "simp only [exists_prop, mem_iUnion, SetCoe.exists, exists_and_right, Subtype.coe_mk] at ht", "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx\u271d : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : CompactSpace \u03b1\nh : TendstoLocallyUniformly F f p\nV : Set (\u03b2 \u00d7 \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\nU : \u03b1 \u2192 Set \u03b1\nt : Finset \u2191univ\nhU : \u2200\u1da0 (x : \u03b9) in p, \u2200 (i : { x // x \u2208 t }) (y : \u03b1), y \u2208 U \u2191\u2191i \u2192 (f y, F x y) \u2208 V\ni : \u03b9\nhi : \u2200 (i_1 : { x // x \u2208 t }) (y : \u03b1), y \u2208 U \u2191\u2191i_1 \u2192 (f y, F i y) \u2208 V\nx : \u03b1\nht : x \u2208 \u22c3 (x : \u2191univ) (_ : x \u2208 t), U \u2191x\n\u22a2 (f x, F i x) \u2208 V", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx\u271d : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : CompactSpace \u03b1\nh : TendstoLocallyUniformly F f p\nV : Set (\u03b2 \u00d7 \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\nU : \u03b1 \u2192 Set \u03b1\nt : Finset \u2191univ\nhU : \u2200\u1da0 (x : \u03b9) in p, \u2200 (i : { x // x \u2208 t }) (y : \u03b1), y \u2208 U \u2191\u2191i \u2192 (f y, F x y) \u2208 V\ni : \u03b9\nhi : \u2200 (i_1 : { x // x \u2208 t }) (y : \u03b1), y \u2208 U \u2191\u2191i_1 \u2192 (f y, F i y) \u2208 V\nx : \u03b1\nht : \u2203 x_1, (\u2203 x, { val := x_1, property := (_ : x_1 \u2208 univ) } \u2208 t) \u2227 x \u2208 U x_1\n\u22a2 (f x, F i x) \u2208 V"}, {"tactic": "obtain \u27e8y, \u27e8hy\u2081, hy\u2082\u27e9, hy\u2083\u27e9 := ht", "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx\u271d : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : CompactSpace \u03b1\nh : TendstoLocallyUniformly F f p\nV : Set (\u03b2 \u00d7 \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\nU : \u03b1 \u2192 Set \u03b1\nt : Finset \u2191univ\nhU : \u2200\u1da0 (x : \u03b9) in p, \u2200 (i : { x // x \u2208 t }) (y : \u03b1), y \u2208 U \u2191\u2191i \u2192 (f y, F x y) \u2208 V\ni : \u03b9\nhi : \u2200 (i_1 : { x // x \u2208 t }) (y : \u03b1), y \u2208 U \u2191\u2191i_1 \u2192 (f y, F i y) \u2208 V\nx : \u03b1\nht : \u2203 x_1, (\u2203 x, { val := x_1, property := (_ : x_1 \u2208 univ) } \u2208 t) \u2227 x \u2208 U x_1\n\u22a2 (f x, F i x) \u2208 V", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx\u271d : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : CompactSpace \u03b1\nh : TendstoLocallyUniformly F f p\nV : Set (\u03b2 \u00d7 \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\nU : \u03b1 \u2192 Set \u03b1\nt : Finset \u2191univ\nhU : \u2200\u1da0 (x : \u03b9) in p, \u2200 (i : { x // x \u2208 t }) (y : \u03b1), y \u2208 U \u2191\u2191i \u2192 (f y, F x y) \u2208 V\ni : \u03b9\nhi : \u2200 (i_1 : { x // x \u2208 t }) (y : \u03b1), y \u2208 U \u2191\u2191i_1 \u2192 (f y, F i y) \u2208 V\nx y : \u03b1\nhy\u2083 : x \u2208 U y\nhy\u2081 : y \u2208 univ\nhy\u2082 : { val := y, property := (_ : y \u2208 univ) } \u2208 t\n\u22a2 (f x, F i x) \u2208 V"}, {"tactic": "exact hi \u27e8\u27e8y, hy\u2081\u27e9, hy\u2082\u27e9 x hy\u2083", "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx\u271d : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : CompactSpace \u03b1\nh : TendstoLocallyUniformly F f p\nV : Set (\u03b2 \u00d7 \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\nU : \u03b1 \u2192 Set \u03b1\nt : Finset \u2191univ\nhU : \u2200\u1da0 (x : \u03b9) in p, \u2200 (i : { x // x \u2208 t }) (y : \u03b1), y \u2208 U \u2191\u2191i \u2192 (f y, F x y) \u2208 V\ni : \u03b9\nhi : \u2200 (i_1 : { x // x \u2208 t }) (y : \u03b1), y \u2208 U \u2191\u2191i_1 \u2192 (f y, F i y) \u2208 V\nx y : \u03b1\nhy\u2083 : x \u2208 U y\nhy\u2081 : y \u2208 univ\nhy\u2082 : { val := y, property := (_ : y \u2208 univ) } \u2208 t\n\u22a2 (f x, F i x) \u2208 V", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/WithZeroTopology.lean", "full_name": "WithZeroTopology.nhds_zero_of_units", "start": [77, 1], "end": [78, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Lie/OfAssociative.lean", "full_name": "LieAlgebra.ad_apply", "start": [238, 1], "end": [239, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/IntermediateField.lean", "full_name": "IntermediateField.isAlgebraic_iff", "start": [729, 1], "end": [730, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.eventually_imp_distrib_right", "start": [1368, 1], "end": [1370, 74], "traced_tactics": [{"tactic": "simp only [imp_iff_not_or, eventually_or_distrib_right, not_frequently]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.169127\n\u03b9 : Sort x\nf : Filter \u03b1\np : \u03b1 \u2192 Prop\nq : Prop\n\u22a2 (\u2200\u1da0 (x : \u03b1) in f, p x \u2192 q) \u2194 (\u2203\u1da0 (x : \u03b1) in f, p x) \u2192 q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.mem_map_seq_iff", "start": [2621, 1], "end": [2625, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicGeometry/SheafedSpace.lean", "full_name": "AlgebraicGeometry.SheafedSpace.comp_c_app'", "start": [151, 1], "end": [153, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/WeakDual.lean", "full_name": "WeakDual.toNormedDual_eq_iff", "start": [178, 1], "end": [179, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.cauchySeq_Lp_iff_cauchySeq_\u2112p", "start": [1275, 1], "end": [1281, 23], "traced_tactics": [{"tactic": "simp_rw [cauchySeq_iff_tendsto_dist_atTop_0, dist_def]", "state_before": "\u03b1 : Type u_2\nE : Type u_3\nF : Type ?u.8132571\nG : Type ?u.8132574\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : SemilatticeSup \u03b9\nhp : Fact (1 \u2264 p)\nf : \u03b9 \u2192 { x // x \u2208 Lp E p }\n\u22a2 CauchySeq f \u2194 Tendsto (fun n => snorm (\u2191\u2191(f n.fst) - \u2191\u2191(f n.snd)) p \u03bc) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_2\nE : Type u_3\nF : Type ?u.8132571\nG : Type ?u.8132574\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : SemilatticeSup \u03b9\nhp : Fact (1 \u2264 p)\nf : \u03b9 \u2192 { x // x \u2208 Lp E p }\n\u22a2 Tendsto (fun n => ENNReal.toReal (snorm (\u2191\u2191(f n.fst) - \u2191\u2191(f n.snd)) p \u03bc)) atTop (\ud835\udcdd 0) \u2194\n    Tendsto (fun n => snorm (\u2191\u2191(f n.fst) - \u2191\u2191(f n.snd)) p \u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "rw [\u2190 ENNReal.zero_toReal, ENNReal.tendsto_toReal_iff (fun n => ?_) ENNReal.zero_ne_top]", "state_before": "\u03b1 : Type u_2\nE : Type u_3\nF : Type ?u.8132571\nG : Type ?u.8132574\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : SemilatticeSup \u03b9\nhp : Fact (1 \u2264 p)\nf : \u03b9 \u2192 { x // x \u2208 Lp E p }\n\u22a2 Tendsto (fun n => ENNReal.toReal (snorm (\u2191\u2191(f n.fst) - \u2191\u2191(f n.snd)) p \u03bc)) atTop (\ud835\udcdd 0) \u2194\n    Tendsto (fun n => snorm (\u2191\u2191(f n.fst) - \u2191\u2191(f n.snd)) p \u03bc) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_2\nE : Type u_3\nF : Type ?u.8132571\nG : Type ?u.8132574\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : SemilatticeSup \u03b9\nhp : Fact (1 \u2264 p)\nf : \u03b9 \u2192 { x // x \u2208 Lp E p }\nn : \u03b9 \u00d7 \u03b9\n\u22a2 snorm (\u2191\u2191(f n.fst) - \u2191\u2191(f n.snd)) p \u03bc \u2260 \u22a4"}, {"tactic": "rw [snorm_congr_ae (Lp.coeFn_sub _ _).symm]", "state_before": "\u03b1 : Type u_2\nE : Type u_3\nF : Type ?u.8132571\nG : Type ?u.8132574\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : SemilatticeSup \u03b9\nhp : Fact (1 \u2264 p)\nf : \u03b9 \u2192 { x // x \u2208 Lp E p }\nn : \u03b9 \u00d7 \u03b9\n\u22a2 snorm (\u2191\u2191(f n.fst) - \u2191\u2191(f n.snd)) p \u03bc \u2260 \u22a4", "state_after": "\u03b1 : Type u_2\nE : Type u_3\nF : Type ?u.8132571\nG : Type ?u.8132574\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : SemilatticeSup \u03b9\nhp : Fact (1 \u2264 p)\nf : \u03b9 \u2192 { x // x \u2208 Lp E p }\nn : \u03b9 \u00d7 \u03b9\n\u22a2 snorm (\u2191\u2191(f n.fst - f n.snd)) p \u03bc \u2260 \u22a4"}, {"tactic": "exact snorm_ne_top _", "state_before": "\u03b1 : Type u_2\nE : Type u_3\nF : Type ?u.8132571\nG : Type ?u.8132574\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : SemilatticeSup \u03b9\nhp : Fact (1 \u2264 p)\nf : \u03b9 \u2192 { x // x \u2208 Lp E p }\nn : \u03b9 \u00d7 \u03b9\n\u22a2 snorm (\u2191\u2191(f n.fst - f n.snd)) p \u03bc \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Ring/BooleanRing.lean", "full_name": "ofBoolAlg_inf", "start": [298, 1], "end": [299, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/Slope.lean", "full_name": "AffineMap.slope_comp", "start": [85, 1], "end": [87, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "full_name": "Metric.thickening_singleton", "start": [984, 1], "end": [986, 28], "traced_tactics": [{"tactic": "ext", "state_before": "\u03b9 : Sort ?u.95281\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\n\u03b4\u271d : \u211d\ns : Set \u03b1\nx\u271d : \u03b1\nX : Type u\ninst\u271d : PseudoMetricSpace X\n\u03b4 : \u211d\nx : X\n\u22a2 thickening \u03b4 {x} = ball x \u03b4", "state_after": "case h\n\u03b9 : Sort ?u.95281\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\n\u03b4\u271d : \u211d\ns : Set \u03b1\nx\u271d\u00b9 : \u03b1\nX : Type u\ninst\u271d : PseudoMetricSpace X\n\u03b4 : \u211d\nx x\u271d : X\n\u22a2 x\u271d \u2208 thickening \u03b4 {x} \u2194 x\u271d \u2208 ball x \u03b4"}, {"tactic": "simp [mem_thickening_iff]", "state_before": "case h\n\u03b9 : Sort ?u.95281\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\n\u03b4\u271d : \u211d\ns : Set \u03b1\nx\u271d\u00b9 : \u03b1\nX : Type u\ninst\u271d : PseudoMetricSpace X\n\u03b4 : \u211d\nx x\u271d : X\n\u22a2 x\u271d \u2208 thickening \u03b4 {x} \u2194 x\u271d \u2208 ball x \u03b4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "full_name": "Subgroup.normalCore_mono", "start": [2553, 1], "end": [2554, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Tactic/Abel.lean", "full_name": "Mathlib.Tactic.Abel.unfold_zsmul", "start": [260, 1], "end": [261, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finsupp/ToDfinsupp.lean", "full_name": "Finsupp.toDfinsupp_smul", "start": [176, 1], "end": [178, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/FieldDivision.lean", "full_name": "Polynomial.quotient_mul_add_remainder_eq_aux", "start": [176, 9], "end": [182, 39], "traced_tactics": [{"tactic": "simp only [h, MulZeroClass.zero_mul, mod, modByMonic_zero, zero_add]", "state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np\u271d q\u271d p q : R[X]\nh : q = 0\n\u22a2 q * div p q + mod p q = p", "state_after": "no goals"}, {"tactic": "conv =>\n  rhs\n  rw [\u2190 modByMonic_add_div p (monic_mul_leadingCoeff_inv h)]", "state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np\u271d q\u271d p q : R[X]\nh : \u00acq = 0\n\u22a2 q * div p q + mod p q = p", "state_after": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np\u271d q\u271d p q : R[X]\nh : \u00acq = 0\n\u22a2 q * div p q + mod p q =\n    p %\u2098 (q * \u2191C (leadingCoeff q)\u207b\u00b9) + q * \u2191C (leadingCoeff q)\u207b\u00b9 * (p /\u2098 (q * \u2191C (leadingCoeff q)\u207b\u00b9))"}, {"tactic": "rw [div, mod, add_comm, mul_assoc]", "state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np\u271d q\u271d p q : R[X]\nh : \u00acq = 0\n\u22a2 q * div p q + mod p q =\n    p %\u2098 (q * \u2191C (leadingCoeff q)\u207b\u00b9) + q * \u2191C (leadingCoeff q)\u207b\u00b9 * (p /\u2098 (q * \u2191C (leadingCoeff q)\u207b\u00b9))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.strongDownwardInduction_eq", "start": [849, 1], "end": [853, 31], "traced_tactics": [{"tactic": "rw [strongDownwardInduction]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.78622\n\u03b3 : Type ?u.78625\np : Multiset \u03b1 \u2192 Sort u_2\nn : \u2115\nH : (t\u2081 : Multiset \u03b1) \u2192 ({t\u2082 : Multiset \u03b1} \u2192 \u2191card t\u2082 \u2264 n \u2192 t\u2081 < t\u2082 \u2192 p t\u2082) \u2192 \u2191card t\u2081 \u2264 n \u2192 p t\u2081\ns : Multiset \u03b1\n\u22a2 strongDownwardInduction H s = H s fun {t\u2082} ht _hst => strongDownwardInduction H t\u2082 ht", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Disjoint.lean", "full_name": "disjoint_left_comm", "start": [144, 1], "end": [145, 47], "traced_tactics": [{"tactic": "simp_rw [disjoint_iff_inf_le, inf_left_comm]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\na b c d : \u03b1\n\u22a2 Disjoint a (b \u2293 c) \u2194 Disjoint b (a \u2293 c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Factorization/Basic.lean", "full_name": "Nat.factorization_def", "start": [69, 1], "end": [70, 40], "traced_tactics": [{"tactic": "simpa [factorization] using absurd pp", "state_before": "n p : \u2115\npp : Prime p\n\u22a2 \u2191(factorization n) p = padicValNat p n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Perm/Support.lean", "full_name": "Equiv.Perm.support_extend_domain", "start": [546, 1], "end": [568, 30], "traced_tactics": [{"tactic": "ext b", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\n\u22a2 support (extendDomain g f) = map (asEmbedding f) (support g)", "state_after": "case a\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\n\u22a2 b \u2208 support (extendDomain g f) \u2194 b \u2208 map (asEmbedding f) (support g)"}, {"tactic": "simp only [exists_prop, Function.Embedding.coeFn_mk, toEmbedding_apply, mem_map, Ne.def,\n  Function.Embedding.trans_apply, mem_support]", "state_before": "case a\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\n\u22a2 b \u2208 support (extendDomain g f) \u2194 b \u2208 map (asEmbedding f) (support g)", "state_after": "case a\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\n\u22a2 \u00ac\u2191(extendDomain g f) b = b \u2194 \u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b"}, {"tactic": "by_cases pb : p b", "state_before": "case a\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\n\u22a2 \u00ac\u2191(extendDomain g f) b = b \u2194 \u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\n\u22a2 \u00ac\u2191(extendDomain g f) b = b \u2194 \u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : \u00acp b\n\u22a2 \u00ac\u2191(extendDomain g f) b = b \u2194 \u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b"}, {"tactic": "rw [extendDomain_apply_subtype _ _ pb]", "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\n\u22a2 \u00ac\u2191(extendDomain g f) b = b \u2194 \u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\n\u22a2 \u00ac\u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b \u2194 \u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b"}, {"tactic": "constructor", "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\n\u22a2 \u00ac\u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b \u2194 \u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b", "state_after": "case pos.mp\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\n\u22a2 \u00ac\u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b \u2192 \u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b\n\ncase pos.mpr\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\n\u22a2 (\u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b) \u2192 \u00ac\u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b"}, {"tactic": "rintro h", "state_before": "case pos.mp\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\n\u22a2 \u00ac\u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b \u2192 \u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b", "state_after": "case pos.mp\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\nh : \u00ac\u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\n\u22a2 \u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b"}, {"tactic": "refine' \u27e8f.symm \u27e8b, pb\u27e9, _, by simp\u27e9", "state_before": "case pos.mp\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\nh : \u00ac\u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\n\u22a2 \u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b", "state_after": "case pos.mp\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\nh : \u00ac\u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\n\u22a2 \u00ac\u2191g (\u2191f.symm { val := b, property := pb }) = \u2191f.symm { val := b, property := pb }"}, {"tactic": "contrapose! h", "state_before": "case pos.mp\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\nh : \u00ac\u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\n\u22a2 \u00ac\u2191g (\u2191f.symm { val := b, property := pb }) = \u2191f.symm { val := b, property := pb }", "state_after": "case pos.mp\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\nh : \u2191g (\u2191f.symm { val := b, property := pb }) = \u2191f.symm { val := b, property := pb }\n\u22a2 \u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b"}, {"tactic": "simp [h]", "state_before": "case pos.mp\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\nh : \u2191g (\u2191f.symm { val := b, property := pb }) = \u2191f.symm { val := b, property := pb }\n\u22a2 \u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b", "state_after": "no goals"}, {"tactic": "simp", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\nh : \u00ac\u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\n\u22a2 \u2191(asEmbedding f) (\u2191f.symm { val := b, property := pb }) = b", "state_after": "no goals"}, {"tactic": "rintro \u27e8a, ha, hb\u27e9", "state_before": "case pos.mpr\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\n\u22a2 (\u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b) \u2192 \u00ac\u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b", "state_after": "case pos.mpr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\na : \u03b1\nha : \u00ac\u2191g a = a\nhb : \u2191(asEmbedding f) a = b\n\u22a2 \u00ac\u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b"}, {"tactic": "contrapose! ha", "state_before": "case pos.mpr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\na : \u03b1\nha : \u00ac\u2191g a = a\nhb : \u2191(asEmbedding f) a = b\n\u22a2 \u00ac\u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b", "state_after": "case pos.mpr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\na : \u03b1\nhb : \u2191(asEmbedding f) a = b\nha : \u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\n\u22a2 \u2191g a = a"}, {"tactic": "obtain rfl : a = f.symm \u27e8b, pb\u27e9 := by\n  rw [eq_symm_apply]\n  exact Subtype.coe_injective hb", "state_before": "case pos.mpr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\na : \u03b1\nhb : \u2191(asEmbedding f) a = b\nha : \u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\n\u22a2 \u2191g a = a", "state_after": "case pos.mpr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\nha : \u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\nhb : \u2191(asEmbedding f) (\u2191f.symm { val := b, property := pb }) = b\n\u22a2 \u2191g (\u2191f.symm { val := b, property := pb }) = \u2191f.symm { val := b, property := pb }"}, {"tactic": "rw [eq_symm_apply]", "state_before": "case pos.mpr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\nha : \u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\nhb : \u2191(asEmbedding f) (\u2191f.symm { val := b, property := pb }) = b\n\u22a2 \u2191g (\u2191f.symm { val := b, property := pb }) = \u2191f.symm { val := b, property := pb }", "state_after": "case pos.mpr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\nha : \u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\nhb : \u2191(asEmbedding f) (\u2191f.symm { val := b, property := pb }) = b\n\u22a2 \u2191f (\u2191g (\u2191f.symm { val := b, property := pb })) = { val := b, property := pb }"}, {"tactic": "exact Subtype.coe_injective ha", "state_before": "case pos.mpr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\nha : \u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\nhb : \u2191(asEmbedding f) (\u2191f.symm { val := b, property := pb }) = b\n\u22a2 \u2191f (\u2191g (\u2191f.symm { val := b, property := pb })) = { val := b, property := pb }", "state_after": "no goals"}, {"tactic": "rw [eq_symm_apply]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\na : \u03b1\nhb : \u2191(asEmbedding f) a = b\nha : \u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\n\u22a2 a = \u2191f.symm { val := b, property := pb }", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\na : \u03b1\nhb : \u2191(asEmbedding f) a = b\nha : \u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\n\u22a2 \u2191f a = { val := b, property := pb }"}, {"tactic": "exact Subtype.coe_injective hb", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : p b\na : \u03b1\nhb : \u2191(asEmbedding f) a = b\nha : \u2191(\u2191f (\u2191g (\u2191f.symm { val := b, property := pb }))) = b\n\u22a2 \u2191f a = { val := b, property := pb }", "state_after": "no goals"}, {"tactic": "rw [extendDomain_apply_not_subtype _ _ pb]", "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : \u00acp b\n\u22a2 \u00ac\u2191(extendDomain g f) b = b \u2194 \u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : \u00acp b\n\u22a2 \u00acb = b \u2194 \u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b"}, {"tactic": "simp only [not_exists, false_iff_iff, not_and, eq_self_iff_true, not_true]", "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : \u00acp b\n\u22a2 \u00acb = b \u2194 \u2203 a, \u00ac\u2191g a = a \u2227 \u2191(asEmbedding f) a = b", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : \u00acp b\n\u22a2 \u2200 (x : \u03b1), \u00ac\u2191g x = x \u2192 \u00ac\u2191(asEmbedding f) x = b"}, {"tactic": "rintro a _ rfl", "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\nb : \u03b2\npb : \u00acp b\n\u22a2 \u2200 (x : \u03b1), \u00ac\u2191g x = x \u2192 \u00ac\u2191(asEmbedding f) x = b", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\na : \u03b1\na\u271d : \u00ac\u2191g a = a\npb : \u00acp (\u2191(asEmbedding f) a)\n\u22a2 False"}, {"tactic": "exact pb (Subtype.prop _)", "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u2074 : DecidableEq \u03b1\ninst\u271d\u00b3 : Fintype \u03b1\nf\u271d g\u271d : Perm \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\ninst\u271d\u00b9 : Fintype \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng : Perm \u03b1\na : \u03b1\na\u271d : \u00ac\u2191g a = a\npb : \u00acp (\u2191(asEmbedding f) a)\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Group/Defs.lean", "full_name": "lt_inv_mul_iff_mul_lt", "start": [174, 1], "end": [176, 7], "traced_tactics": [{"tactic": "rw [\u2190 mul_lt_mul_iff_left a]", "state_before": "\u03b1 : Type u\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c : \u03b1\n\u22a2 b < a\u207b\u00b9 * c \u2194 a * b < c", "state_after": "\u03b1 : Type u\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c : \u03b1\n\u22a2 a * b < a * (a\u207b\u00b9 * c) \u2194 a * b < c"}, {"tactic": "simp", "state_before": "\u03b1 : Type u\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c : \u03b1\n\u22a2 a * b < a * (a\u207b\u00b9 * c) \u2194 a * b < c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/SymmDiff.lean", "full_name": "toDual_symmDiff", "start": [108, 1], "end": [109, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/Irrational.lean", "full_name": "irrational_sub_int_iff", "start": [582, 1], "end": [583, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/Basis.lean", "full_name": "AffineBasis.coord_apply_combination_of_mem", "start": [190, 1], "end": [194, 38], "traced_tactics": [{"tactic": "classical simp only [coord_apply, hi, Finset.affineCombination_eq_linear_combination, if_true,\n    mul_boole, hw, Function.comp_apply, smul_eq_mul, s.sum_ite_eq,\n    s.map_affineCombination b w hw]", "state_before": "\u03b9 : Type u_1\n\u03b9' : Type ?u.100622\nk : Type u_2\nV : Type u_3\nP : Type u_4\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AffineSpace V P\ninst\u271d\u00b9 : Ring k\ninst\u271d : Module k V\nb : AffineBasis \u03b9 k P\ns : Finset \u03b9\ni j : \u03b9\ne : \u03b9 \u2243 \u03b9'\nhi : i \u2208 s\nw : \u03b9 \u2192 k\nhw : Finset.sum s w = 1\n\u22a2 \u2191(coord b i) (\u2191(Finset.affineCombination k s \u2191b) w) = w i", "state_after": "no goals"}, {"tactic": "simp only [coord_apply, hi, Finset.affineCombination_eq_linear_combination, if_true,\nmul_boole, hw, Function.comp_apply, smul_eq_mul, s.sum_ite_eq,\ns.map_affineCombination b w hw]", "state_before": "\u03b9 : Type u_1\n\u03b9' : Type ?u.100622\nk : Type u_2\nV : Type u_3\nP : Type u_4\ninst\u271d\u00b3 : AddCommGroup 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f)\n\u22a2 Set.Infinite (mulSupport fun i => mulIndicator s (fun i => f i) i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Associated.lean", "full_name": "not_irreducible_of_not_unit_dvdNotUnit", "start": [1163, 1], "end": [1165, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/LocallyFinite.lean", "full_name": "Finset.subtype_Ioo_eq", "start": [1279, 1], "end": [1280, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Module/LocalizedModule.lean", "full_name": "IsLocalizedModule.mk'_surjective", "start": [1048, 1], "end": [1051, 40], "traced_tactics": [{"tactic": "intro x", "state_before": "R : Type u_1\ninst\u271d\u2077 : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type ?u.936326\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\nf : M \u2192\u2097[R] M'\ng : M \u2192\u2097[R] M''\ninst\u271d : IsLocalizedModule S f\n\u22a2 Function.Surjective (Function.uncurry (mk' f))", "state_after": "R : Type u_1\ninst\u271d\u2077 : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type ?u.936326\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\nf : M \u2192\u2097[R] M'\ng : M \u2192\u2097[R] M''\ninst\u271d : IsLocalizedModule S f\nx : M'\n\u22a2 \u2203 a, Function.uncurry (mk' f) a = x"}, {"tactic": "obtain \u27e8\u27e8m, s\u27e9, e : s \u2022 x = f m\u27e9 := IsLocalizedModule.surj S f x", "state_before": "R : Type u_1\ninst\u271d\u2077 : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type ?u.936326\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\nf : M \u2192\u2097[R] M'\ng : M \u2192\u2097[R] M''\ninst\u271d : IsLocalizedModule S f\nx : M'\n\u22a2 \u2203 a, Function.uncurry (mk' f) a = x", "state_after": "case intro.mk\nR : Type u_1\ninst\u271d\u2077 : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type ?u.936326\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\nf : M \u2192\u2097[R] M'\ng : M \u2192\u2097[R] M''\ninst\u271d : IsLocalizedModule S f\nx : M'\nm : M\ns : { x // x \u2208 S }\ne : s \u2022 x = \u2191f m\n\u22a2 \u2203 a, Function.uncurry (mk' f) a = x"}, {"tactic": "exact \u27e8\u27e8m, s\u27e9, mk'_eq_iff.mpr e.symm\u27e9", "state_before": "case intro.mk\nR : Type u_1\ninst\u271d\u2077 : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type u_3\nM'' : Type ?u.936326\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\nf : M \u2192\u2097[R] M'\ng : M \u2192\u2097[R] M''\ninst\u271d : IsLocalizedModule S f\nx : M'\nm : M\ns : { x // x \u2208 S }\ne : s \u2022 x = \u2191f m\n\u22a2 \u2203 a, Function.uncurry (mk' f) a = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/SuccPred/Limit.lean", "full_name": "Order.isSuccLimitRecOn_succ'", "start": [180, 1], "end": [188, 36], "traced_tactics": [{"tactic": "have hb' := not_isSuccLimit_succ_of_not_isMax hb", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nb : \u03b1\nhb : \u00acIsMax b\n\u22a2 isSuccLimitRecOn (succ b) hs hl = hs b hb", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nb : \u03b1\nhb : \u00acIsMax b\nhb' : \u00acIsSuccLimit (succ b)\n\u22a2 isSuccLimitRecOn (succ b) hs hl = hs b hb"}, {"tactic": "have H := Classical.choose_spec (not_isSuccLimit_iff.1 hb')", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nb : \u03b1\nhb : \u00acIsMax b\nhb' : \u00acIsSuccLimit (succ b)\n\u22a2 isSuccLimitRecOn (succ b) hs hl = hs b hb", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nb : \u03b1\nhb : \u00acIsMax b\nhb' : \u00acIsSuccLimit (succ b)\nH :\n  \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) \u2227\n    succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) = succ b\n\u22a2 isSuccLimitRecOn (succ b) hs hl = hs b hb"}, {"tactic": "rw [isSuccLimitRecOn]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nb : \u03b1\nhb : \u00acIsMax b\nhb' : \u00acIsSuccLimit (succ b)\nH :\n  \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) \u2227\n    succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) = succ b\n\u22a2 isSuccLimitRecOn (succ b) hs hl = hs b hb", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nb : \u03b1\nhb : \u00acIsMax b\nhb' : \u00acIsSuccLimit (succ b)\nH :\n  \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) \u2227\n    succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) = succ b\n\u22a2 (if hb : IsSuccLimit (succ b) then hl (succ b) hb\n    else\n      let_fun H :=\n        (_ :\n          \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) \u2227\n            succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) = succ b);\n      Eq.mpr (_ : C (succ b) = C (succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b))))\n        (hs (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b))\n          (_ : \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b))))) =\n    hs b hb"}, {"tactic": "simp only [cast_eq_iff_heq, hb', not_false_iff, eq_mpr_eq_cast, dif_neg]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nb : \u03b1\nhb : \u00acIsMax b\nhb' : \u00acIsSuccLimit (succ b)\nH :\n  \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) \u2227\n    succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) = succ b\n\u22a2 (if hb : IsSuccLimit (succ b) then hl (succ b) hb\n    else\n      let_fun H :=\n        (_ :\n          \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) \u2227\n            succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) = succ b);\n      Eq.mpr (_ : C (succ b) = C (succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b))))\n        (hs (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b))\n          (_ : \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b))))) =\n    hs b hb", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nb : \u03b1\nhb : \u00acIsMax b\nhb' : \u00acIsSuccLimit (succ b)\nH :\n  \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) \u2227\n    succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) = succ b\n\u22a2 HEq\n    (hs (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b))\n      (_ : \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b))))\n    (hs b hb)"}, {"tactic": "congr 1 <;> first |\n  exact (succ_eq_succ_iff_of_not_isMax H.left hb).mp H.right |\n  exact proof_irrel_heq H.left hb", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nb : \u03b1\nhb : \u00acIsMax b\nhb' : \u00acIsSuccLimit (succ b)\nH :\n  \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) \u2227\n    succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) = succ b\n\u22a2 HEq\n    (hs (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b))\n      (_ : \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b))))\n    (hs b hb)", "state_after": "no goals"}, {"tactic": "exact (succ_eq_succ_iff_of_not_isMax H.left hb).mp H.right", "state_before": "case e_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nb : \u03b1\nhb : \u00acIsMax b\nhb' : \u00acIsSuccLimit (succ b)\nH :\n  \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) \u2227\n    succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) = succ b\n\u22a2 HEq (_ : \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b))) hb", "state_after": "case e_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nb : \u03b1\nhb : \u00acIsMax b\nhb' : \u00acIsSuccLimit (succ b)\nH :\n  \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) \u2227\n    succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) = succ b\n\u22a2 HEq (_ : \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b))) hb"}, {"tactic": "exact proof_irrel_heq H.left hb", "state_before": "case e_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b\u271d : \u03b1\nC : \u03b1 \u2192 Sort u_2\nhs : (a : \u03b1) \u2192 \u00acIsMax a \u2192 C (succ a)\nhl : (a : \u03b1) \u2192 IsSuccLimit a \u2192 C a\nb : \u03b1\nhb : \u00acIsMax b\nhb' : \u00acIsSuccLimit (succ b)\nH :\n  \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) \u2227\n    succ (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b)) = succ b\n\u22a2 HEq (_ : \u00acIsMax (Classical.choose (_ : \u2203 b_1, \u00acIsMax b_1 \u2227 succ b_1 = succ b))) hb", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.lift_mk_shrink", "start": [1017, 1], "end": [1020, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Int/GCD.lean", "full_name": "Int.lcm_comm", "start": [443, 1], "end": [445, 25], "traced_tactics": [{"tactic": "rw [Int.lcm, Int.lcm]", "state_before": "i j : \u2124\n\u22a2 lcm i j = lcm j i", "state_after": "i j : \u2124\n\u22a2 Nat.lcm (natAbs i) (natAbs j) = Nat.lcm (natAbs j) (natAbs i)"}, {"tactic": "exact Nat.lcm_comm _ _", "state_before": "i j : \u2124\n\u22a2 Nat.lcm (natAbs i) (natAbs j) = Nat.lcm (natAbs j) (natAbs i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "MeasureTheory.Measure.ext_of_Ici", "start": [762, 1], "end": [765, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Simple.lean", "full_name": "CategoryTheory.isIso_of_epi_of_nonzero", "start": [167, 1], "end": [171, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Prod.lean", "full_name": "Filter.eventually_prod_principal_iff", "start": [106, 1], "end": [109, 27], "traced_tactics": [{"tactic": "rw [eventually_iff, eventually_iff, mem_prod_principal]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.4102\n\u03b4 : Type ?u.4105\n\u03b9 : Sort ?u.4108\ns\u271d : Set \u03b1\nt : Set \u03b2\nf : Filter \u03b1\ng : Filter \u03b2\np : \u03b1 \u00d7 \u03b2 \u2192 Prop\ns : Set \u03b2\n\u22a2 (\u2200\u1da0 (x : \u03b1 \u00d7 \u03b2) in f \u00d7\u02e2 \ud835\udcdf s, p x) \u2194 \u2200\u1da0 (x : \u03b1) in f, \u2200 (y : \u03b2), y \u2208 s \u2192 p (x, y)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.4102\n\u03b4 : Type ?u.4105\n\u03b9 : Sort ?u.4108\ns\u271d : Set \u03b1\nt : Set \u03b2\nf : Filter \u03b1\ng : Filter \u03b2\np : \u03b1 \u00d7 \u03b2 \u2192 Prop\ns : Set \u03b2\n\u22a2 {a | \u2200 (b : \u03b2), b \u2208 s \u2192 (a, b) \u2208 {x | p x}} \u2208 f \u2194 {x | \u2200 (y : \u03b2), y \u2208 s \u2192 p (x, y)} \u2208 f"}, {"tactic": "simp only [mem_setOf_eq]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.4102\n\u03b4 : Type ?u.4105\n\u03b9 : Sort ?u.4108\ns\u271d : Set \u03b1\nt : Set \u03b2\nf : Filter \u03b1\ng : Filter \u03b2\np : \u03b1 \u00d7 \u03b2 \u2192 Prop\ns : Set \u03b2\n\u22a2 {a | \u2200 (b : \u03b2), b \u2208 s \u2192 (a, b) \u2208 {x | p x}} \u2208 f \u2194 {x | \u2200 (y : \u03b2), y \u2208 s \u2192 p (x, y)} \u2208 f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SetFamily/Compression/UV.lean", "full_name": "UV.card_compress", "start": [307, 1], "end": [312, 8], "traced_tactics": [{"tactic": "unfold compress", "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nu v a\u271d : Finset \u03b1\nhuv : card u = card v\na : Finset \u03b1\n\u22a2 card (compress u v a) = card a", "state_after": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nu v a\u271d : Finset \u03b1\nhuv : card u = card v\na : Finset \u03b1\n\u22a2 card (if Disjoint u a \u2227 v \u2264 a then (a \u2294 u) \\ v else a) = card a"}, {"tactic": "split_ifs with h", "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nu v a\u271d : Finset \u03b1\nhuv : card u = card v\na : Finset \u03b1\n\u22a2 card (if Disjoint u a \u2227 v \u2264 a then (a \u2294 u) \\ v else a) = card a", "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nu v a\u271d : Finset \u03b1\nhuv : card u = card v\na : Finset \u03b1\nh : Disjoint u a \u2227 v \u2264 a\n\u22a2 card ((a \u2294 u) \\ v) = card a\n\ncase inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nu v a\u271d : Finset \u03b1\nhuv : card u = card v\na : Finset \u03b1\nh : \u00ac(Disjoint u a \u2227 v \u2264 a)\n\u22a2 card a = card a"}, {"tactic": "rw [card_sdiff (h.2.trans le_sup_left), sup_eq_union, card_disjoint_union h.1.symm, huv,\n  add_tsub_cancel_right]", "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nu v a\u271d : Finset \u03b1\nhuv : card u = card v\na : Finset \u03b1\nh : Disjoint u a \u2227 v \u2264 a\n\u22a2 card ((a \u2294 u) \\ v) = card a", "state_after": "no goals"}, {"tactic": "rfl", "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\nu v a\u271d : Finset \u03b1\nhuv : card u = card v\na : Finset \u03b1\nh : \u00ac(Disjoint u a \u2227 v \u2264 a)\n\u22a2 card a = card a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Basic.lean", "full_name": "Equiv.Perm.cycleOf_one", "start": [1079, 1], "end": [1080, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/CauSeq.lean", "full_name": "CauSeq.inf_equiv_inf", "start": [875, 1], "end": [883, 76], "traced_tactics": [{"tactic": "intro \u03b5 \u03b50", "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u2081 b\u2081 a\u2082 b\u2082 : CauSeq \u03b1 abs\nha : a\u2081 \u2248 a\u2082\nhb : b\u2081 \u2248 b\u2082\n\u22a2 a\u2081 \u2293 b\u2081 \u2248 a\u2082 \u2293 b\u2082", "state_after": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u2081 b\u2081 a\u2082 b\u2082 : CauSeq \u03b1 abs\nha : a\u2081 \u2248 a\u2082\nhb : b\u2081 \u2248 b\u2082\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 abs (\u2191(a\u2081 \u2293 b\u2081 - a\u2082 \u2293 b\u2082) j) < \u03b5"}, {"tactic": "obtain \u27e8ai, hai\u27e9 := ha \u03b5 \u03b50", "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u2081 b\u2081 a\u2082 b\u2082 : CauSeq \u03b1 abs\nha : a\u2081 \u2248 a\u2082\nhb : b\u2081 \u2248 b\u2082\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 abs (\u2191(a\u2081 \u2293 b\u2081 - a\u2082 \u2293 b\u2082) j) < \u03b5", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u2081 b\u2081 a\u2082 b\u2082 : CauSeq \u03b1 abs\nha : a\u2081 \u2248 a\u2082\nhb : b\u2081 \u2248 b\u2082\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nai : \u2115\nhai : \u2200 (j : \u2115), j \u2265 ai \u2192 abs (\u2191(a\u2081 - a\u2082) j) < \u03b5\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 abs (\u2191(a\u2081 \u2293 b\u2081 - a\u2082 \u2293 b\u2082) j) < \u03b5"}, {"tactic": "obtain \u27e8bi, hbi\u27e9 := hb \u03b5 \u03b50", "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u2081 b\u2081 a\u2082 b\u2082 : CauSeq \u03b1 abs\nha : a\u2081 \u2248 a\u2082\nhb : b\u2081 \u2248 b\u2082\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nai : \u2115\nhai : \u2200 (j : \u2115), j \u2265 ai \u2192 abs (\u2191(a\u2081 - a\u2082) j) < \u03b5\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 abs (\u2191(a\u2081 \u2293 b\u2081 - a\u2082 \u2293 b\u2082) j) < \u03b5", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u2081 b\u2081 a\u2082 b\u2082 : CauSeq \u03b1 abs\nha : a\u2081 \u2248 a\u2082\nhb : b\u2081 \u2248 b\u2082\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nai : \u2115\nhai : \u2200 (j : \u2115), j \u2265 ai \u2192 abs (\u2191(a\u2081 - a\u2082) j) < \u03b5\nbi : \u2115\nhbi : \u2200 (j : \u2115), j \u2265 bi \u2192 abs (\u2191(b\u2081 - b\u2082) j) < \u03b5\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 abs (\u2191(a\u2081 \u2293 b\u2081 - a\u2082 \u2293 b\u2082) j) < \u03b5"}, {"tactic": "exact\n  \u27e8ai \u2294 bi, fun i hi =>\n    (abs_min_sub_min_le_max (a\u2081 i) (b\u2081 i) (a\u2082 i) (b\u2082 i)).trans_lt\n      (max_lt (hai i (sup_le_iff.mp hi).1) (hbi i (sup_le_iff.mp hi).2))\u27e9", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u2081 b\u2081 a\u2082 b\u2082 : CauSeq \u03b1 abs\nha : a\u2081 \u2248 a\u2082\nhb : b\u2081 \u2248 b\u2082\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nai : \u2115\nhai : \u2200 (j : \u2115), j \u2265 ai \u2192 abs (\u2191(a\u2081 - a\u2082) j) < \u03b5\nbi : \u2115\nhbi : \u2200 (j : \u2115), j \u2265 bi \u2192 abs (\u2191(b\u2081 - b\u2082) j) < \u03b5\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 abs (\u2191(a\u2081 \u2293 b\u2081 - a\u2082 \u2293 b\u2082) j) < \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/UniformGroup.lean", "full_name": "tendsto_div_comap_self", "start": [756, 1], "end": [766, 50], "traced_tactics": [{"tactic": "have comm : ((fun x : \u03b1 \u00d7 \u03b1 => x.2 / x.1) \u2218 fun t : \u03b2 \u00d7 \u03b2 => (e t.1, e t.2)) =\n    e \u2218 fun t : \u03b2 \u00d7 \u03b2 => t.2 / t.1 := by\n  ext t\n  change e t.2 / e t.1 = e (t.2 / t.1)\n  rw [\u2190 map_div e t.2 t.1]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\nhom : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : Group \u03b1\ninst\u271d\u00b3 : TopologicalGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Group \u03b2\ninst\u271d : MonoidHomClass hom \u03b2 \u03b1\ne : hom\nde : DenseInducing \u2191e\nx\u2080 : \u03b1\n\u22a2 Tendsto (fun t => t.snd / t.fst) (comap (fun p => (\u2191e p.fst, \u2191e p.snd)) (\ud835\udcdd (x\u2080, x\u2080))) (\ud835\udcdd 1)", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_1\nhom : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : Group \u03b1\ninst\u271d\u00b3 : TopologicalGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Group \u03b2\ninst\u271d : MonoidHomClass hom \u03b2 \u03b1\ne : hom\nde : DenseInducing \u2191e\nx\u2080 : \u03b1\ncomm : ((fun x => x.snd / x.fst) \u2218 fun t => (\u2191e t.fst, \u2191e t.snd)) = \u2191e \u2218 fun t => t.snd / t.fst\n\u22a2 Tendsto (fun t => t.snd / t.fst) (comap (fun p => (\u2191e p.fst, \u2191e p.snd)) (\ud835\udcdd (x\u2080, x\u2080))) (\ud835\udcdd 1)"}, {"tactic": "have lim : Tendsto (fun x : \u03b1 \u00d7 \u03b1 => x.2 / x.1) (\ud835\udcdd (x\u2080, x\u2080)) (\ud835\udcdd (e 1)) := by\n  simpa using (continuous_div'.comp (@continuous_swap \u03b1 \u03b1 _ _)).tendsto (x\u2080, x\u2080)", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\nhom : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : Group \u03b1\ninst\u271d\u00b3 : TopologicalGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Group \u03b2\ninst\u271d : MonoidHomClass hom \u03b2 \u03b1\ne : hom\nde : DenseInducing \u2191e\nx\u2080 : \u03b1\ncomm : ((fun x => x.snd / x.fst) \u2218 fun t => (\u2191e t.fst, \u2191e t.snd)) = \u2191e \u2218 fun t => t.snd / t.fst\n\u22a2 Tendsto (fun t => t.snd / t.fst) (comap (fun p => (\u2191e p.fst, \u2191e p.snd)) (\ud835\udcdd (x\u2080, x\u2080))) (\ud835\udcdd 1)", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_1\nhom : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : Group \u03b1\ninst\u271d\u00b3 : TopologicalGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Group \u03b2\ninst\u271d : MonoidHomClass hom \u03b2 \u03b1\ne : hom\nde : DenseInducing \u2191e\nx\u2080 : \u03b1\ncomm : ((fun x => x.snd / x.fst) \u2218 fun t => (\u2191e t.fst, \u2191e t.snd)) = \u2191e \u2218 fun t => t.snd / t.fst\nlim : Tendsto (fun x => x.snd / x.fst) (\ud835\udcdd (x\u2080, x\u2080)) (\ud835\udcdd (\u2191e 1))\n\u22a2 Tendsto (fun t => t.snd / t.fst) (comap (fun p => (\u2191e p.fst, \u2191e p.snd)) (\ud835\udcdd (x\u2080, x\u2080))) (\ud835\udcdd 1)"}, {"tactic": "simpa using de.tendsto_comap_nhds_nhds lim comm", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\nhom : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : Group \u03b1\ninst\u271d\u00b3 : TopologicalGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Group \u03b2\ninst\u271d : MonoidHomClass hom \u03b2 \u03b1\ne : hom\nde : DenseInducing \u2191e\nx\u2080 : \u03b1\ncomm : ((fun x => x.snd / x.fst) \u2218 fun t => (\u2191e t.fst, \u2191e t.snd)) = \u2191e \u2218 fun t => t.snd / t.fst\nlim : Tendsto (fun x => x.snd / x.fst) (\ud835\udcdd (x\u2080, x\u2080)) (\ud835\udcdd (\u2191e 1))\n\u22a2 Tendsto (fun t => t.snd / t.fst) (comap (fun p => (\u2191e p.fst, \u2191e p.snd)) (\ud835\udcdd (x\u2080, x\u2080))) (\ud835\udcdd 1)", "state_after": "no goals"}, {"tactic": "ext t", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\nhom : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : Group \u03b1\ninst\u271d\u00b3 : TopologicalGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Group \u03b2\ninst\u271d : MonoidHomClass hom \u03b2 \u03b1\ne : hom\nde : DenseInducing \u2191e\nx\u2080 : \u03b1\n\u22a2 ((fun x => x.snd / x.fst) \u2218 fun t => (\u2191e t.fst, \u2191e t.snd)) = \u2191e \u2218 fun t => t.snd / t.fst", "state_after": "case h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\nhom : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : Group \u03b1\ninst\u271d\u00b3 : TopologicalGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Group \u03b2\ninst\u271d : MonoidHomClass hom \u03b2 \u03b1\ne : hom\nde : DenseInducing \u2191e\nx\u2080 : \u03b1\nt : \u03b2 \u00d7 \u03b2\n\u22a2 ((fun x => x.snd / x.fst) \u2218 fun t => (\u2191e t.fst, \u2191e t.snd)) t = (\u2191e \u2218 fun t => t.snd / t.fst) t"}, {"tactic": "change e t.2 / e t.1 = e (t.2 / t.1)", "state_before": "case h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\nhom : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : Group \u03b1\ninst\u271d\u00b3 : TopologicalGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Group \u03b2\ninst\u271d : MonoidHomClass hom \u03b2 \u03b1\ne : hom\nde : DenseInducing \u2191e\nx\u2080 : \u03b1\nt : \u03b2 \u00d7 \u03b2\n\u22a2 ((fun x => x.snd / x.fst) \u2218 fun t => (\u2191e t.fst, \u2191e t.snd)) t = (\u2191e \u2218 fun t => t.snd / t.fst) t", "state_after": "case h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\nhom : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : Group \u03b1\ninst\u271d\u00b3 : TopologicalGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Group \u03b2\ninst\u271d : MonoidHomClass hom \u03b2 \u03b1\ne : hom\nde : DenseInducing \u2191e\nx\u2080 : \u03b1\nt : \u03b2 \u00d7 \u03b2\n\u22a2 \u2191e t.snd / \u2191e t.fst = \u2191e (t.snd / t.fst)"}, {"tactic": "rw [\u2190 map_div e t.2 t.1]", "state_before": "case h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\nhom : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : Group \u03b1\ninst\u271d\u00b3 : TopologicalGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Group \u03b2\ninst\u271d : MonoidHomClass hom \u03b2 \u03b1\ne : hom\nde : DenseInducing \u2191e\nx\u2080 : \u03b1\nt : \u03b2 \u00d7 \u03b2\n\u22a2 \u2191e t.snd / \u2191e t.fst = \u2191e (t.snd / t.fst)", "state_after": "no goals"}, {"tactic": "simpa using (continuous_div'.comp (@continuous_swap \u03b1 \u03b1 _ _)).tendsto (x\u2080, x\u2080)", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\nhom : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : Group \u03b1\ninst\u271d\u00b3 : TopologicalGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Group \u03b2\ninst\u271d : MonoidHomClass hom \u03b2 \u03b1\ne : hom\nde : DenseInducing \u2191e\nx\u2080 : \u03b1\ncomm : ((fun x => x.snd / x.fst) \u2218 fun t => (\u2191e t.fst, \u2191e t.snd)) = \u2191e \u2218 fun t => t.snd / t.fst\n\u22a2 Tendsto (fun x => x.snd / x.fst) (\ud835\udcdd (x\u2080, x\u2080)) (\ud835\udcdd (\u2191e 1))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Hom/Basic.lean", "full_name": "WithTop.toDualBotEquiv_top", "start": [1295, 1], "end": [1296, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/NaturalOps.lean", "full_name": "Ordinal.nadd_lt_nadd_iff_left", "start": [418, 1], "end": [419, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.comp_eval\u2082Hom", "start": [1051, 1], "end": [1057, 54], "traced_tactics": [{"tactic": "apply MvPolynomial.ringHom_ext", "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf\u271d : R \u2192+* S\u2081\ng\u271d : \u03c3 \u2192 S\u2081\ninst\u271d : CommSemiring S\u2082\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u03c6 : S\u2081 \u2192+* S\u2082\n\u22a2 RingHom.comp \u03c6 (eval\u2082Hom f g) = eval\u2082Hom (RingHom.comp \u03c6 f) fun i => \u2191\u03c6 (g i)", "state_after": "case hC\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf\u271d : R \u2192+* S\u2081\ng\u271d : \u03c3 \u2192 S\u2081\ninst\u271d : CommSemiring S\u2082\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u03c6 : S\u2081 \u2192+* S\u2082\n\u22a2 \u2200 (r : R), \u2191(RingHom.comp \u03c6 (eval\u2082Hom f g)) (\u2191C r) = \u2191(eval\u2082Hom (RingHom.comp \u03c6 f) fun i => \u2191\u03c6 (g i)) (\u2191C r)\n\ncase hX\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf\u271d : R \u2192+* S\u2081\ng\u271d : \u03c3 \u2192 S\u2081\ninst\u271d : CommSemiring S\u2082\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u03c6 : S\u2081 \u2192+* S\u2082\n\u22a2 \u2200 (i : \u03c3), \u2191(RingHom.comp \u03c6 (eval\u2082Hom f g)) (X i) = \u2191(eval\u2082Hom (RingHom.comp \u03c6 f) fun i => \u2191\u03c6 (g i)) (X i)"}, {"tactic": "intro r", "state_before": "case hC\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf\u271d : R \u2192+* S\u2081\ng\u271d : \u03c3 \u2192 S\u2081\ninst\u271d : CommSemiring S\u2082\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u03c6 : S\u2081 \u2192+* S\u2082\n\u22a2 \u2200 (r : R), \u2191(RingHom.comp \u03c6 (eval\u2082Hom f g)) (\u2191C r) = \u2191(eval\u2082Hom (RingHom.comp \u03c6 f) fun i => \u2191\u03c6 (g i)) (\u2191C r)", "state_after": "case hC\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf\u271d : R \u2192+* S\u2081\ng\u271d : \u03c3 \u2192 S\u2081\ninst\u271d : CommSemiring S\u2082\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u03c6 : S\u2081 \u2192+* S\u2082\nr : R\n\u22a2 \u2191(RingHom.comp \u03c6 (eval\u2082Hom f g)) (\u2191C r) = \u2191(eval\u2082Hom (RingHom.comp \u03c6 f) fun i => \u2191\u03c6 (g i)) (\u2191C r)"}, {"tactic": "rw [RingHom.comp_apply, eval\u2082Hom_C, eval\u2082Hom_C, RingHom.comp_apply]", "state_before": "case hC\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf\u271d : R \u2192+* S\u2081\ng\u271d : \u03c3 \u2192 S\u2081\ninst\u271d : CommSemiring S\u2082\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u03c6 : S\u2081 \u2192+* S\u2082\nr : R\n\u22a2 \u2191(RingHom.comp \u03c6 (eval\u2082Hom f g)) (\u2191C r) = \u2191(eval\u2082Hom (RingHom.comp \u03c6 f) fun i => \u2191\u03c6 (g i)) (\u2191C r)", "state_after": "no goals"}, {"tactic": "intro i", "state_before": "case hX\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf\u271d : R \u2192+* S\u2081\ng\u271d : \u03c3 \u2192 S\u2081\ninst\u271d : CommSemiring S\u2082\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u03c6 : S\u2081 \u2192+* S\u2082\n\u22a2 \u2200 (i : \u03c3), \u2191(RingHom.comp \u03c6 (eval\u2082Hom f g)) (X i) = \u2191(eval\u2082Hom (RingHom.comp \u03c6 f) fun i => \u2191\u03c6 (g i)) (X i)", "state_after": "case hX\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf\u271d : R \u2192+* S\u2081\ng\u271d : \u03c3 \u2192 S\u2081\ninst\u271d : CommSemiring S\u2082\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u03c6 : S\u2081 \u2192+* S\u2082\ni : \u03c3\n\u22a2 \u2191(RingHom.comp \u03c6 (eval\u2082Hom f g)) (X i) = \u2191(eval\u2082Hom (RingHom.comp \u03c6 f) fun i => \u2191\u03c6 (g i)) (X i)"}, {"tactic": "rw [RingHom.comp_apply, eval\u2082Hom_X', eval\u2082Hom_X']", "state_before": "case hX\nR : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nf\u271d : R \u2192+* S\u2081\ng\u271d : \u03c3 \u2192 S\u2081\ninst\u271d : CommSemiring S\u2082\nf : R \u2192+* S\u2081\ng : \u03c3 \u2192 S\u2081\n\u03c6 : S\u2081 \u2192+* S\u2082\ni : \u03c3\n\u22a2 \u2191(RingHom.comp \u03c6 (eval\u2082Hom f g)) (X i) = \u2191(eval\u2082Hom (RingHom.comp \u03c6 f) fun i => \u2191\u03c6 (g i)) (X i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Homology/Homotopy.lean", "full_name": "Homotopy.nullHomotopicMap_f", "start": [368, 1], "end": [372, 42], "traced_tactics": [{"tactic": "dsimp only [nullHomotopicMap]", "state_before": "\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category V\ninst\u271d : Preadditive V\nc : ComplexShape \u03b9\nC D E : HomologicalComplex V c\nf g : C \u27f6 D\nh k : D \u27f6 E\ni k\u2082 k\u2081 k\u2080 : \u03b9\nr\u2082\u2081 : ComplexShape.Rel c k\u2082 k\u2081\nr\u2081\u2080 : ComplexShape.Rel c k\u2081 k\u2080\nhom : (i j : \u03b9) \u2192 X C i \u27f6 X D j\n\u22a2 Hom.f (nullHomotopicMap hom) k\u2081 = d C k\u2081 k\u2080 \u226b hom k\u2080 k\u2081 + hom k\u2081 k\u2082 \u226b d D k\u2082 k\u2081", "state_after": "\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category V\ninst\u271d : Preadditive V\nc : ComplexShape \u03b9\nC D E : HomologicalComplex V c\nf g : C \u27f6 D\nh k : D \u27f6 E\ni k\u2082 k\u2081 k\u2080 : \u03b9\nr\u2082\u2081 : ComplexShape.Rel c k\u2082 k\u2081\nr\u2081\u2080 : ComplexShape.Rel c k\u2081 k\u2080\nhom : (i j : \u03b9) \u2192 X C i \u27f6 X D j\n\u22a2 \u2191(dNext k\u2081) hom + \u2191(prevD k\u2081) hom = d C k\u2081 k\u2080 \u226b hom k\u2080 k\u2081 + hom k\u2081 k\u2082 \u226b d D k\u2082 k\u2081"}, {"tactic": "rw [dNext_eq hom r\u2081\u2080, prevD_eq hom r\u2082\u2081]", "state_before": "\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category V\ninst\u271d : Preadditive V\nc : ComplexShape \u03b9\nC D E : HomologicalComplex V c\nf g : C \u27f6 D\nh k : D \u27f6 E\ni k\u2082 k\u2081 k\u2080 : \u03b9\nr\u2082\u2081 : ComplexShape.Rel c k\u2082 k\u2081\nr\u2081\u2080 : ComplexShape.Rel c k\u2081 k\u2080\nhom : (i j : \u03b9) \u2192 X C i \u27f6 X D j\n\u22a2 \u2191(dNext k\u2081) hom + \u2191(prevD k\u2081) hom = d C k\u2081 k\u2080 \u226b hom k\u2080 k\u2081 + hom k\u2081 k\u2082 \u226b d D k\u2082 k\u2081", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Complex/Circle.lean", "full_name": "circle_def", "start": [56, 1], "end": [57, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.FinMeasSupp.pair", "start": [1223, 11], "end": [1228, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Conj.lean", "full_name": "isConj_iff\u2080", "start": [134, 1], "end": [142, 17], "traced_tactics": [{"tactic": "rw [\u2190 Units.val_inv_eq_inv_val, Units.mul_inv_eq_iff_eq_mul]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : GroupWithZero \u03b1\na b : \u03b1\nx\u271d : IsConj a b\nc : \u03b1\u02e3\nhc : SemiconjBy (\u2191c) a b\n\u22a2 \u2191c \u2260 0 \u2227 \u2191c * a * (\u2191c)\u207b\u00b9 = b", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : GroupWithZero \u03b1\na b : \u03b1\nx\u271d : IsConj a b\nc : \u03b1\u02e3\nhc : SemiconjBy (\u2191c) a b\n\u22a2 \u2191c \u2260 0 \u2227 \u2191c * a = b * \u2191c"}, {"tactic": "exact \u27e8c.ne_zero, hc\u27e9", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : GroupWithZero \u03b1\na b : \u03b1\nx\u271d : IsConj a b\nc : \u03b1\u02e3\nhc : SemiconjBy (\u2191c) a b\n\u22a2 \u2191c \u2260 0 \u2227 \u2191c * a = b * \u2191c", "state_after": "no goals"}, {"tactic": "rw [SemiconjBy, \u2190 Units.mul_inv_eq_iff_eq_mul, Units.val_inv_eq_inv_val, Units.val_mk0]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : GroupWithZero \u03b1\na b : \u03b1\nx\u271d : \u2203 c, c \u2260 0 \u2227 c * a * c\u207b\u00b9 = b\nc : \u03b1\nc0 : c \u2260 0\nhc : c * a * c\u207b\u00b9 = b\n\u22a2 SemiconjBy (\u2191(Units.mk0 c c0)) a b", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : GroupWithZero \u03b1\na b : \u03b1\nx\u271d : \u2203 c, c \u2260 0 \u2227 c * a * c\u207b\u00b9 = b\nc : \u03b1\nc0 : c \u2260 0\nhc : c * a * c\u207b\u00b9 = b\n\u22a2 c * a * c\u207b\u00b9 = b"}, {"tactic": "exact hc", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : GroupWithZero \u03b1\na b : \u03b1\nx\u271d : \u2203 c, c \u2260 0 \u2227 c * a * c\u207b\u00b9 = b\nc : \u03b1\nc0 : c \u2260 0\nhc : c * a * c\u207b\u00b9 = b\n\u22a2 c * a * c\u207b\u00b9 = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/SuccPred/Relation.lean", "full_name": "reflTransGen_of_pred_of_le", "start": [108, 1], "end": [110, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/RelIso/Basic.lean", "full_name": "RelHom.coe_fn_toFun", "start": [121, 1], "end": [122, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/Principal.lean", "full_name": "Ordinal.principal_add_of_principal_mul_opow", "start": [383, 1], "end": [386, 71], "traced_tactics": [{"tactic": "have := ho ((opow_lt_opow_iff_right hb).2 hx) ((opow_lt_opow_iff_right hb).2 hy)", "state_before": "o b : Ordinal\nhb : 1 < b\nho : Principal (fun x x_1 => x * x_1) (b ^ o)\nx y : Ordinal\nhx : x < o\nhy : y < o\n\u22a2 (fun x x_1 => x + x_1) x y < o", "state_after": "o b : Ordinal\nhb : 1 < b\nho : Principal (fun x x_1 => x * x_1) (b ^ o)\nx y : Ordinal\nhx : x < o\nhy : y < o\nthis : (fun x x_1 => x * x_1) (b ^ x) (b ^ y) < b ^ o\n\u22a2 (fun x x_1 => x + x_1) x y < o"}, {"tactic": "dsimp only at *", "state_before": "o b : Ordinal\nhb : 1 < b\nho : Principal (fun x x_1 => x * x_1) (b ^ o)\nx y : Ordinal\nhx : x < o\nhy : y < o\nthis : (fun x x_1 => x * x_1) (b ^ x) (b ^ y) < b ^ o\n\u22a2 (fun x x_1 => x + x_1) x y < o", "state_after": "o b : Ordinal\nhb : 1 < b\nho : Principal (fun x x_1 => x * x_1) (b ^ o)\nx y : Ordinal\nhx : x < o\nhy : y < o\nthis : b ^ x * b ^ y < b ^ o\n\u22a2 x + y < o"}, {"tactic": "rwa [\u2190 opow_add, opow_lt_opow_iff_right hb] at this", "state_before": "o b : Ordinal\nhb : 1 < b\nho : Principal (fun x x_1 => x * x_1) (b ^ o)\nx y : Ordinal\nhx : x < o\nhy : y < o\nthis : b ^ x * b ^ y < b ^ o\n\u22a2 x + y < o", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Disjoint.lean", "full_name": "Set.iUnion_Iio", "start": [97, 1], "end": [98, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/LiminfLimsup.lean", "full_name": "OrderIso.liminf_apply", "start": [1235, 1], "end": [1242, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/lpSpace.lean", "full_name": "lp.norm_apply_le_of_tendsto", "start": [1130, 1], "end": [1136, 68], "traced_tactics": [{"tactic": "have : Tendsto (fun k => \u2016F k a\u2016) l (\ud835\udcdd \u2016f a\u2016) :=\n  (Tendsto.comp (continuous_apply a).continuousAt hf).norm", "state_before": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E \u22a4 }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\na : \u03b1\n\u22a2 \u2016f a\u2016 \u2264 C", "state_after": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E \u22a4 }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\na : \u03b1\nthis : Tendsto (fun k => \u2016\u2191(F k) a\u2016) l (\ud835\udcdd \u2016f a\u2016)\n\u22a2 \u2016f a\u2016 \u2264 C"}, {"tactic": "refine' le_of_tendsto this (hCF.mono _)", "state_before": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E \u22a4 }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\na : \u03b1\nthis : Tendsto (fun k => \u2016\u2191(F k) a\u2016) l (\ud835\udcdd \u2016f a\u2016)\n\u22a2 \u2016f a\u2016 \u2264 C", "state_after": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E \u22a4 }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\na : \u03b1\nthis : Tendsto (fun k => \u2016\u2191(F k) a\u2016) l (\ud835\udcdd \u2016f a\u2016)\n\u22a2 \u2200 (x : \u03b9), \u2016F x\u2016 \u2264 C \u2192 \u2016\u2191(F x) a\u2016 \u2264 C"}, {"tactic": "intro k hCFk", "state_before": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E \u22a4 }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\na : \u03b1\nthis : Tendsto (fun k => \u2016\u2191(F k) a\u2016) l (\ud835\udcdd \u2016f a\u2016)\n\u22a2 \u2200 (x : \u03b9), \u2016F x\u2016 \u2264 C \u2192 \u2016\u2191(F x) a\u2016 \u2264 C", "state_after": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E \u22a4 }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\na : \u03b1\nthis : Tendsto (fun k => \u2016\u2191(F k) a\u2016) l (\ud835\udcdd \u2016f a\u2016)\nk : \u03b9\nhCFk : \u2016F k\u2016 \u2264 C\n\u22a2 \u2016\u2191(F k) a\u2016 \u2264 C"}, {"tactic": "exact (norm_apply_le_norm ENNReal.top_ne_zero (F k) a).trans hCFk", "state_before": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d\u00b9 : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n\u03b9 : Type u_3\nl : Filter \u03b9\ninst\u271d : NeBot l\nC : \u211d\nF : \u03b9 \u2192 { x // x \u2208 lp E \u22a4 }\nhCF : \u2200\u1da0 (k : \u03b9) in l, \u2016F k\u2016 \u2264 C\nf : (a : \u03b1) \u2192 E a\nhf : Tendsto (id fun i => \u2191(F i)) l (\ud835\udcdd f)\na : \u03b1\nthis : Tendsto (fun k => \u2016\u2191(F k) a\u2016) l (\ud835\udcdd \u2016f a\u2016)\nk : \u03b9\nhCFk : \u2016F k\u2016 \u2264 C\n\u22a2 \u2016\u2191(F k) a\u2016 \u2264 C", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Logic.lean", "full_name": "And.imp_right", "start": [154, 1], "end": [154, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm.lean", "full_name": "ContinuousLinearMap.restrictScalarsIsometry_toLinearMap", "start": [1265, 1], "end": [1267, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Field.lean", "full_name": "IsLocalMin.inv", "start": [114, 1], "end": [116, 82], "traced_tactics": [{"tactic": "filter_upwards [h1, h2]with z h3 h4 using(inv_le_inv h4 h2.self_of_nhds).mpr h3", "state_before": "K : Type ?u.14947\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : TopologicalSpace K\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : LinearOrderedSemifield \u03b2\na\u271d : \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nh1 : IsLocalMin f a\nh2 : \u2200\u1da0 (z : \u03b1) in \ud835\udcdd a, 0 < f z\n\u22a2 IsLocalMax f\u207b\u00b9 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Data/Nat/Basic.lean", "full_name": "Nat.add_sub_self_left", "start": [532, 1], "end": [537, 13], "traced_tactics": [{"tactic": "induction a with\n| zero => simp\n| succ a ih =>\n  rw [Nat.succ_add, Nat.succ_sub_succ]\n  apply ih", "state_before": "a b : Nat\n\u22a2 a + b - a = b", "state_after": "no goals"}, {"tactic": "simp", "state_before": "case zero\nb : Nat\n\u22a2 zero + b - zero = b", "state_after": "no goals"}, {"tactic": "rw [Nat.succ_add, Nat.succ_sub_succ]", "state_before": "case succ\nb a : Nat\nih : a + b - a = b\n\u22a2 succ a + b - succ a = b", "state_after": "case succ\nb a : Nat\nih : a + b - a = b\n\u22a2 a + b - a = b"}, {"tactic": "apply ih", "state_before": "case succ\nb a : Nat\nih : a + b - a = b\n\u22a2 a + b - a = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "Complex.two_cos", "start": [808, 1], "end": [809, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/Irrational.lean", "full_name": "irrational_rat_sub_iff", "start": [562, 1], "end": [563, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Subring/Basic.lean", "full_name": "Subring.closure_iUnion", "start": [1054, 1], "end": [1055, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fintype/Order.lean", "full_name": "Fintype.bddBelow_range", "start": [219, 1], "end": [224, 13], "traced_tactics": [{"tactic": "obtain \u27e8M, hM\u27e9 := Fintype.exists_ge f", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : Nonempty \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : IsDirected \u03b1 fun x x_1 => x \u2265 x_1\n\u03b2 : Type u_2\ninst\u271d : Fintype \u03b2\nf : \u03b2 \u2192 \u03b1\n\u22a2 BddBelow (Set.range f)", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Nonempty \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : IsDirected \u03b1 fun x x_1 => x \u2265 x_1\n\u03b2 : Type u_2\ninst\u271d : Fintype \u03b2\nf : \u03b2 \u2192 \u03b1\nM : \u03b1\nhM : \u2200 (i : \u03b2), M \u2264 f i\n\u22a2 BddBelow (Set.range f)"}, {"tactic": "refine' \u27e8M, fun a ha => _\u27e9", "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Nonempty \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : IsDirected \u03b1 fun x x_1 => x \u2265 x_1\n\u03b2 : Type u_2\ninst\u271d : Fintype \u03b2\nf : \u03b2 \u2192 \u03b1\nM : \u03b1\nhM : \u2200 (i : \u03b2), M \u2264 f i\n\u22a2 BddBelow (Set.range f)", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Nonempty \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : IsDirected \u03b1 fun x x_1 => x \u2265 x_1\n\u03b2 : Type u_2\ninst\u271d : Fintype \u03b2\nf : \u03b2 \u2192 \u03b1\nM : \u03b1\nhM : \u2200 (i : \u03b2), M \u2264 f i\na : \u03b1\nha : a \u2208 Set.range f\n\u22a2 M \u2264 a"}, {"tactic": "obtain \u27e8b, rfl\u27e9 := ha", "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Nonempty \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : IsDirected \u03b1 fun x x_1 => x \u2265 x_1\n\u03b2 : Type u_2\ninst\u271d : Fintype \u03b2\nf : \u03b2 \u2192 \u03b1\nM : \u03b1\nhM : \u2200 (i : \u03b2), M \u2264 f i\na : \u03b1\nha : a \u2208 Set.range f\n\u22a2 M \u2264 a", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Nonempty \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : IsDirected \u03b1 fun x x_1 => x \u2265 x_1\n\u03b2 : Type u_2\ninst\u271d : Fintype \u03b2\nf : \u03b2 \u2192 \u03b1\nM : \u03b1\nhM : \u2200 (i : \u03b2), M \u2264 f i\nb : \u03b2\n\u22a2 M \u2264 f b"}, {"tactic": "exact hM b", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : Nonempty \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : IsDirected \u03b1 fun x x_1 => x \u2265 x_1\n\u03b2 : Type u_2\ninst\u271d : Fintype \u03b2\nf : \u03b2 \u2192 \u03b1\nM : \u03b1\nhM : \u2200 (i : \u03b2), M \u2264 f i\nb : \u03b2\n\u22a2 M \u2264 f b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Prelude.lean", "full_name": "Nat.eq_of_beq_eq_true", "start": [1504, 1], "end": [1511, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm.lean", "full_name": "ContinuousLinearMap.nnnorm_smulRight_apply", "start": [1852, 1], "end": [1853, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/BigOperators/Basic.lean", "full_name": "List.eq_of_prod_take_eq", "start": [470, 1], "end": [475, 31], "traced_tactics": [{"tactic": "refine ext_get h fun i h\u2081 h\u2082 => ?_", "state_before": "\u03b9 : Type ?u.124973\n\u03b1 : Type ?u.124976\nM : Type u_1\nN : Type ?u.124982\nP : Type ?u.124985\nM\u2080 : Type ?u.124988\nG : Type ?u.124991\nR : Type ?u.124994\ninst\u271d : LeftCancelMonoid M\nL L' : List M\nh : length L = length L'\nh' : \u2200 (i : \u2115), i \u2264 length L \u2192 prod (take i L) = prod (take i L')\n\u22a2 L = L'", "state_after": "\u03b9 : Type ?u.124973\n\u03b1 : Type ?u.124976\nM : Type u_1\nN : Type ?u.124982\nP : Type ?u.124985\nM\u2080 : Type ?u.124988\nG : Type ?u.124991\nR : Type ?u.124994\ninst\u271d : LeftCancelMonoid M\nL L' : List M\nh : length L = length L'\nh' : \u2200 (i : \u2115), i \u2264 length L \u2192 prod (take i L) = prod (take i L')\ni : \u2115\nh\u2081 : i < length L\nh\u2082 : i < length L'\n\u22a2 get L { val := i, isLt := h\u2081 } = get L' { val := i, isLt := h\u2082 }"}, {"tactic": "have : (L.take (i + 1)).prod = (L'.take (i + 1)).prod := h' _ (Nat.succ_le_of_lt h\u2081)", "state_before": "\u03b9 : Type ?u.124973\n\u03b1 : Type ?u.124976\nM : Type u_1\nN : Type ?u.124982\nP : Type ?u.124985\nM\u2080 : Type ?u.124988\nG : Type ?u.124991\nR : Type ?u.124994\ninst\u271d : LeftCancelMonoid M\nL L' : List M\nh : length L = length L'\nh' : \u2200 (i : \u2115), i \u2264 length L \u2192 prod (take i L) = prod (take i L')\ni : \u2115\nh\u2081 : i < length L\nh\u2082 : i < length L'\n\u22a2 get L { val := i, isLt := h\u2081 } = get L' { val := i, isLt := h\u2082 }", "state_after": "\u03b9 : Type ?u.124973\n\u03b1 : Type ?u.124976\nM : Type u_1\nN : Type ?u.124982\nP : Type ?u.124985\nM\u2080 : Type ?u.124988\nG : Type ?u.124991\nR : Type ?u.124994\ninst\u271d : LeftCancelMonoid M\nL L' : List M\nh : length L = length L'\nh' : \u2200 (i : \u2115), i \u2264 length L \u2192 prod (take i L) = prod (take i L')\ni : \u2115\nh\u2081 : i < length L\nh\u2082 : i < length L'\nthis : prod (take (i + 1) L) = prod (take (i + 1) L')\n\u22a2 get L { val := i, isLt := h\u2081 } = get L' { val := i, isLt := h\u2082 }"}, {"tactic": "rw [prod_take_succ L i h\u2081, prod_take_succ L' i h\u2082, h' i (le_of_lt h\u2081)] at this", "state_before": "\u03b9 : Type ?u.124973\n\u03b1 : Type ?u.124976\nM : Type u_1\nN : Type ?u.124982\nP : Type ?u.124985\nM\u2080 : Type ?u.124988\nG : Type ?u.124991\nR : Type ?u.124994\ninst\u271d : LeftCancelMonoid M\nL L' : List M\nh : length L = length L'\nh' : \u2200 (i : \u2115), i \u2264 length L \u2192 prod (take i L) = prod (take i L')\ni : \u2115\nh\u2081 : i < length L\nh\u2082 : i < length L'\nthis : prod (take (i + 1) L) = prod (take (i + 1) L')\n\u22a2 get L { val := i, isLt := h\u2081 } = get L' { val := i, isLt := h\u2082 }", "state_after": "\u03b9 : Type ?u.124973\n\u03b1 : Type ?u.124976\nM : Type u_1\nN : Type ?u.124982\nP : Type ?u.124985\nM\u2080 : Type ?u.124988\nG : Type ?u.124991\nR : Type ?u.124994\ninst\u271d : LeftCancelMonoid M\nL L' : List M\nh : length L = length L'\nh' : \u2200 (i : \u2115), i \u2264 length L \u2192 prod (take i L) = prod (take i L')\ni : \u2115\nh\u2081 : i < length L\nh\u2082 : i < length L'\nthis : prod (take i L') * nthLe L i h\u2081 = prod (take i L') * nthLe L' i h\u2082\n\u22a2 get L { val := i, isLt := h\u2081 } = get L' { val := i, isLt := h\u2082 }"}, {"tactic": "convert mul_left_cancel this", "state_before": "\u03b9 : Type ?u.124973\n\u03b1 : Type ?u.124976\nM : Type u_1\nN : Type ?u.124982\nP : Type ?u.124985\nM\u2080 : Type ?u.124988\nG : Type ?u.124991\nR : Type ?u.124994\ninst\u271d : LeftCancelMonoid M\nL L' : List M\nh : length L = length L'\nh' : \u2200 (i : \u2115), i \u2264 length L \u2192 prod (take i L) = prod (take i L')\ni : \u2115\nh\u2081 : i < length L\nh\u2082 : i < length L'\nthis : prod (take i L') * nthLe L i h\u2081 = prod (take i L') * nthLe L' i h\u2082\n\u22a2 get L { val := i, isLt := h\u2081 } = get L' { val := i, isLt := h\u2082 }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Sort.lean", "full_name": "Finset.orderEmbOfFin_unique'", "start": [228, 1], "end": [230, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Field/Basic.lean", "full_name": "add_self_div_two", "start": [502, 1], "end": [503, 47], "traced_tactics": [{"tactic": "rw [\u2190 mul_two, mul_div_cancel a two_ne_zero]", "state_before": "\u03b9 : Type ?u.90856\n\u03b1 : Type u_1\n\u03b2 : Type ?u.90862\ninst\u271d : LinearOrderedSemifield \u03b1\na\u271d b c d e : \u03b1\nm n : \u2124\na : \u03b1\n\u22a2 (a + a) / 2 = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Normed/Group/AddTorsor.lean", "full_name": "nndist_vadd_right", "start": [132, 1], "end": [133, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.ne_neg_self", "start": [817, 1], "end": [820, 61], "traced_tactics": [{"tactic": "have : a.val \u2264 n / 2 \u2194 (n : \u2115) / 2 < (-a).val := le_div_two_iff_lt_neg n ha", "state_before": "n : \u2115\nhn : Fact (n % 2 = 1)\na : ZMod n\nha : a \u2260 0\nh : a = -a\n\u22a2 False", "state_after": "n : \u2115\nhn : Fact (n % 2 = 1)\na : ZMod n\nha : a \u2260 0\nh : a = -a\nthis : val a \u2264 n / 2 \u2194 n / 2 < val (-a)\n\u22a2 False"}, {"tactic": "rwa [\u2190 h, \u2190 not_lt, \u2190 not_iff, iff_self, not_true] at this", "state_before": "n : \u2115\nhn : Fact (n % 2 = 1)\na : ZMod n\nha : a \u2260 0\nh : a = -a\nthis : val a \u2264 n / 2 \u2194 n / 2 < val (-a)\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "isCauSeq_series_of_abv_isCauSeq", "start": [120, 1], "end": [122, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "full_name": "smul_cancel_of_non_zero_divisor", "start": [466, 1], "end": [471, 30], "traced_tactics": [{"tactic": "rw [\u2190 sub_eq_zero]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nM : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : NonUnitalNonAssocRing R\ninst\u271d : DistribMulAction M R\nk : M\nh : \u2200 (x : R), k \u2022 x = 0 \u2192 x = 0\na b : R\nh' : k \u2022 a = k \u2022 b\n\u22a2 a = b", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nM : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : NonUnitalNonAssocRing R\ninst\u271d : DistribMulAction M R\nk : M\nh : \u2200 (x : R), k \u2022 x = 0 \u2192 x = 0\na b : R\nh' : k \u2022 a = k \u2022 b\n\u22a2 a - b = 0"}, {"tactic": "refine' h _ _", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nM : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : NonUnitalNonAssocRing R\ninst\u271d : DistribMulAction M R\nk : M\nh : \u2200 (x : R), k \u2022 x = 0 \u2192 x = 0\na b : R\nh' : k \u2022 a = k \u2022 b\n\u22a2 a - b = 0", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nM : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : NonUnitalNonAssocRing R\ninst\u271d : DistribMulAction M R\nk : M\nh : \u2200 (x : R), k \u2022 x = 0 \u2192 x = 0\na b : R\nh' : k \u2022 a = k \u2022 b\n\u22a2 k \u2022 (a - b) = 0"}, {"tactic": "rw [smul_sub, h', sub_self]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nM : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : NonUnitalNonAssocRing R\ninst\u271d : DistribMulAction M R\nk : M\nh : \u2200 (x : R), k \u2022 x = 0 \u2192 x = 0\na b : R\nh' : k \u2022 a = k \u2022 b\n\u22a2 k \u2022 (a - b) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/PeakFunction.lean", "full_name": "integrableOn_peak_smul_of_integrableOn_of_continuousWithinAt", "start": [57, 1], "end": [80, 31], "traced_tactics": [{"tactic": "obtain \u27e8u, u_open, x\u2080u, hu\u27e9 : \u2203 u, IsOpen u \u2227 x\u2080 \u2208 u \u2227 \u2200 x \u2208 u \u2229 s, g x \u2208 ball (g x\u2080) 1", "state_before": "\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, IntegrableOn (fun x => \u03c6 i x \u2022 g x) s", "state_after": "\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u22a2 \u2203 u, IsOpen u \u2227 x\u2080 \u2208 u \u2227 \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\n\ncase intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, IntegrableOn (fun x => \u03c6 i x \u2022 g x) s"}, {"tactic": "exact mem_nhdsWithin.1 (hcg (ball_mem_nhds _ zero_lt_one))", "state_before": "\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\n\u22a2 \u2203 u, IsOpen u \u2227 x\u2080 \u2208 u \u2227 \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\n\ncase intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, IntegrableOn (fun x => \u03c6 i x \u2022 g x) s", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, IntegrableOn (fun x => \u03c6 i x \u2022 g x) s"}, {"tactic": "filter_upwards [tendstoUniformlyOn_iff.1 (hl\u03c6 u u_open x\u2080u) 1 zero_lt_one, hi\u03c6] with i hi h'i", "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\n\u22a2 \u2200\u1da0 (i : \u03b9) in l, IntegrableOn (fun x => \u03c6 i x \u2022 g x) s", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\n\u22a2 IntegrableOn (fun x => \u03c6 i x \u2022 g x) s"}, {"tactic": "have A : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u) \u03bc := by\n  refine' Integrable.smul_of_top_right (hmg.mono (diff_subset _ _) le_rfl) _\n  apply\n    mem\u2112p_top_of_bound\n      ((integrable_of_integral_eq_one h'i).aestronglyMeasurable.mono_set (diff_subset _ _)) 1\n  filter_upwards [self_mem_ae_restrict (hs.diff u_open.measurableSet)] with x hx\n  simpa only [Pi.zero_apply, dist_zero_left] using (hi x hx).le", "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\n\u22a2 IntegrableOn (fun x => \u03c6 i x \u2022 g x) s", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\n\u22a2 IntegrableOn (fun x => \u03c6 i x \u2022 g x) s"}, {"tactic": "convert A.union B", "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\nB : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \u2229 u)\n\u22a2 IntegrableOn (fun x => \u03c6 i x \u2022 g x) s", "state_after": "case h.e'_6\n\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\nB : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \u2229 u)\n\u22a2 s = s \\ u \u222a s \u2229 u"}, {"tactic": "simp only [diff_union_inter]", "state_before": "case h.e'_6\n\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\nB : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \u2229 u)\n\u22a2 s = s \\ u \u222a s \u2229 u", "state_after": "no goals"}, {"tactic": "refine' Integrable.smul_of_top_right (hmg.mono (diff_subset _ _) le_rfl) _", "state_before": "\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\n\u22a2 IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)", "state_after": "\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\n\u22a2 Mem\u2112p (fun x => \u03c6 i x) \u22a4"}, {"tactic": "apply\n  mem\u2112p_top_of_bound\n    ((integrable_of_integral_eq_one h'i).aestronglyMeasurable.mono_set (diff_subset _ _)) 1", "state_before": "\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\n\u22a2 Mem\u2112p (fun x => \u03c6 i x) \u22a4", "state_after": "\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (s \\ u), \u2016\u03c6 i x\u2016 \u2264 1"}, {"tactic": "filter_upwards [self_mem_ae_restrict (hs.diff u_open.measurableSet)] with x hx", "state_before": "\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (s \\ u), \u2016\u03c6 i x\u2016 \u2264 1", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u2016\u03c6 i x\u2016 \u2264 1"}, {"tactic": "simpa only [Pi.zero_apply, dist_zero_left] using (hi x hx).le", "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nx : \u03b1\nhx : x \u2208 s \\ u\n\u22a2 \u2016\u03c6 i x\u2016 \u2264 1", "state_after": "no goals"}, {"tactic": "apply Integrable.smul_of_top_left", "state_before": "\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\n\u22a2 IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \u2229 u)", "state_after": "case h\u03c6\n\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\n\u22a2 Integrable fun x => \u03c6 i x\n\ncase hf\n\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\n\u22a2 Mem\u2112p (fun x => g x) \u22a4"}, {"tactic": "exact IntegrableOn.mono_set (integrable_of_integral_eq_one h'i) (inter_subset_left _ _)", "state_before": "case h\u03c6\n\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\n\u22a2 Integrable fun x => \u03c6 i x", "state_after": "no goals"}, {"tactic": "apply\n  mem\u2112p_top_of_bound (hmg.mono_set (inter_subset_left _ _)).aestronglyMeasurable (\u2016g x\u2080\u2016 + 1)", "state_before": "case hf\n\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\n\u22a2 Mem\u2112p (fun x => g x) \u22a4", "state_after": "case hf\n\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (s \u2229 u), \u2016g x\u2016 \u2264 \u2016g x\u2080\u2016 + 1"}, {"tactic": "filter_upwards [self_mem_ae_restrict (hs.inter u_open.measurableSet)] with x hx", "state_before": "case hf\n\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (s \u2229 u), \u2016g x\u2016 \u2264 \u2016g x\u2080\u2016 + 1", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\nx : \u03b1\nhx : x \u2208 s \u2229 u\n\u22a2 \u2016g x\u2016 \u2264 \u2016g x\u2080\u2016 + 1"}, {"tactic": "rw [inter_comm] at hx", "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\nx : \u03b1\nhx : x \u2208 s \u2229 u\n\u22a2 \u2016g x\u2016 \u2264 \u2016g x\u2080\u2016 + 1", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\nx : \u03b1\nhx : x \u2208 u \u2229 s\n\u22a2 \u2016g x\u2016 \u2264 \u2016g x\u2080\u2016 + 1"}, {"tactic": "exact (norm_lt_of_mem_ball (hu x hx)).le", "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_3\n\u03b9 : Type u_2\nhm : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\ng : \u03b1 \u2192 E\nl : Filter \u03b9\nx\u2080 : \u03b1\ns : Set \u03b1\n\u03c6 : \u03b9 \u2192 \u03b1 \u2192 \u211d\nhs : MeasurableSet s\nhl\u03c6 : \u2200 (u : Set \u03b1), IsOpen u \u2192 x\u2080 \u2208 u \u2192 TendstoUniformlyOn \u03c6 0 l (s \\ u)\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nhmg : IntegrableOn g s\nhcg : ContinuousWithinAt g s x\u2080\nu : Set \u03b1\nu_open : IsOpen u\nx\u2080u : x\u2080 \u2208 u\nhu : \u2200 (x : \u03b1), x \u2208 u \u2229 s \u2192 g x \u2208 ball (g x\u2080) 1\ni : \u03b9\nhi : \u2200 (x : \u03b1), x \u2208 s \\ u \u2192 dist (OfNat.ofNat 0 x) (\u03c6 i x) < 1\nh'i : (\u222b (x : \u03b1) in s, \u03c6 i x \u2202\u03bc) = 1\nA : IntegrableOn (fun x => \u03c6 i x \u2022 g x) (s \\ u)\nx : \u03b1\nhx : x \u2208 u \u2229 s\n\u22a2 \u2016g x\u2016 \u2264 \u2016g x\u2080\u2016 + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "full_name": "Ordinal.enum_zero_eq_bot", "start": [1233, 1], "end": [1237, 6], "traced_tactics": [{"tactic": "rwa [type_lt]", "state_before": "\u03b1 : Type ?u.207159\n\u03b2 : Type ?u.207162\n\u03b3 : Type ?u.207165\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal\nho : 0 < o\n\u22a2 0 < type fun x x_1 => x < x_1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.Nonempty.preimage'", "start": [713, 1], "end": [717, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Type.lean", "full_name": "Equiv.Perm.card_cycleType_pos", "start": [93, 1], "end": [94, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "full_name": "cthickening_ball", "start": [335, 1], "end": [338, 53], "traced_tactics": [{"tactic": "rw [\u2190 thickening_singleton, cthickening_thickening h\u03b5 h\u03b4,\n    cthickening_singleton _ (add_nonneg h\u03b5 h\u03b4.le)]", "state_before": "\ud835\udd5c : Type ?u.447208\nE : Type u_1\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : NormedSpace \u211d E\nx\u271d y z : E\n\u03b4 \u03b5 : \u211d\nh\u03b5 : 0 \u2264 \u03b5\nh\u03b4 : 0 < \u03b4\nx : E\n\u22a2 cthickening \u03b5 (Metric.ball x \u03b4) = Metric.closedBall x (\u03b5 + \u03b4)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Support.lean", "full_name": "Function.disjoint_mulSupport_iff", "start": [104, 1], "end": [105, 49], "traced_tactics": [{"tactic": "rw [disjoint_comm, mulSupport_disjoint_iff]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.4262\nA : Type ?u.4265\nB : Type ?u.4268\nM : Type u_2\nN : Type ?u.4274\nP : Type ?u.4277\nR : Type ?u.4280\nS : Type ?u.4283\nG : Type ?u.4286\nM\u2080 : Type ?u.4289\nG\u2080 : Type ?u.4292\n\u03b9 : Sort ?u.4295\ninst\u271d\u00b2 : One M\ninst\u271d\u00b9 : One N\ninst\u271d : One P\nf : \u03b1 \u2192 M\ns : Set \u03b1\n\u22a2 Disjoint s (mulSupport f) \u2194 EqOn f 1 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Cover.lean", "full_name": "Covby.lt", "start": [225, 1], "end": [226, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Semiconj.lean", "full_name": "SemiconjBy.units_val", "start": [152, 1], "end": [153, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Segment.lean", "full_name": "openSegment_translate_image", "start": [287, 1], "end": [289, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Basic.lean", "full_name": "Metric.mem_sphere", "start": [491, 9], "end": [491, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Paracompact.lean", "full_name": "refinement_of_locallyCompact_sigmaCompact_of_nhds_basis", "start": [273, 1], "end": [280, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Basic.lean", "full_name": "Polynomial.toFinsupp_zero", "start": [202, 1], "end": [203, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.count_map_eq_count'", "start": [2553, 1], "end": [2560, 18], "traced_tactics": [{"tactic": "by_cases H : x \u2208 s", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.418813\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Injective f\nx : \u03b1\n\u22a2 count (f x) (map f s) = count x s", "state_after": "case pos\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.418813\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Injective f\nx : \u03b1\nH : x \u2208 s\n\u22a2 count (f x) (map f s) = count x s\n\ncase neg\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.418813\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Injective f\nx : \u03b1\nH : \u00acx \u2208 s\n\u22a2 count (f x) (map f s) = count x s"}, {"tactic": "exact count_map_eq_count f _ (Set.injOn_of_injective hf _) _ H", "state_before": "case pos\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.418813\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Injective f\nx : \u03b1\nH : x \u2208 s\n\u22a2 count (f x) (map f s) = count x s", "state_after": "no goals"}, {"tactic": "rw [count_eq_zero_of_not_mem H, count_eq_zero, mem_map]", "state_before": "case neg\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.418813\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Injective f\nx : \u03b1\nH : \u00acx \u2208 s\n\u22a2 count (f x) (map f s) = count x s", "state_after": "case neg\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.418813\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Injective f\nx : \u03b1\nH : \u00acx \u2208 s\n\u22a2 \u00ac\u2203 a, a \u2208 s \u2227 f a = f x"}, {"tactic": "rintro \u27e8k, hks, hkx\u27e9", "state_before": "case neg\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.418813\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Injective f\nx : \u03b1\nH : \u00acx \u2208 s\n\u22a2 \u00ac\u2203 a, a \u2208 s \u2227 f a = f x", "state_after": "case neg.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.418813\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Injective f\nx : \u03b1\nH : \u00acx \u2208 s\nk : \u03b1\nhks : k \u2208 s\nhkx : f k = f x\n\u22a2 False"}, {"tactic": "rw [hf hkx] at hks", "state_before": "case neg.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.418813\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Injective f\nx : \u03b1\nH : \u00acx \u2208 s\nk : \u03b1\nhks : k \u2208 s\nhkx : f k = f x\n\u22a2 False", "state_after": "case neg.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.418813\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Injective f\nx : \u03b1\nH : \u00acx \u2208 s\nk : \u03b1\nhks : x \u2208 s\nhkx : f k = f x\n\u22a2 False"}, {"tactic": "contradiction", "state_before": "case neg.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.418813\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\nhf : Injective f\nx : \u03b1\nH : \u00acx \u2208 s\nk : \u03b1\nhks : x \u2208 s\nhkx : f k = f x\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/ContinuousFunction/Bounded.lean", "full_name": "BoundedContinuousFunction.const_apply'", "start": [304, 1], "end": [304, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "full_name": "NormedSpace.sphere_nonempty", "start": [418, 1], "end": [425, 56], "traced_tactics": [{"tactic": "obtain \u27e8y, hy\u27e9 := exists_ne x", "state_before": "\ud835\udd5c : Type ?u.1764162\nE : Type u_1\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\n\u22a2 Set.Nonempty (sphere x r) \u2194 0 \u2264 r", "state_after": "case intro\n\ud835\udd5c : Type ?u.1764162\nE : Type u_1\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\ny : E\nhy : y \u2260 x\n\u22a2 Set.Nonempty (sphere x r) \u2194 0 \u2264 r"}, {"tactic": "refine' \u27e8fun h => nonempty_closedBall.1 (h.mono sphere_subset_closedBall), fun hr =>\n  \u27e8r \u2022 \u2016y - x\u2016\u207b\u00b9 \u2022 (y - x) + x, _\u27e9\u27e9", "state_before": "case intro\n\ud835\udd5c : Type ?u.1764162\nE : Type u_1\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\ny : E\nhy : y \u2260 x\n\u22a2 Set.Nonempty (sphere x r) \u2194 0 \u2264 r", "state_after": "case intro\n\ud835\udd5c : Type ?u.1764162\nE : Type u_1\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\ny : E\nhy : y \u2260 x\nhr : 0 \u2264 r\n\u22a2 r \u2022 \u2016y - x\u2016\u207b\u00b9 \u2022 (y - x) + x \u2208 sphere x r"}, {"tactic": "have : \u2016y - x\u2016 \u2260 0 := by simpa [sub_eq_zero]", "state_before": "case intro\n\ud835\udd5c : Type ?u.1764162\nE : Type u_1\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\ny : E\nhy : y \u2260 x\nhr : 0 \u2264 r\n\u22a2 r \u2022 \u2016y - x\u2016\u207b\u00b9 \u2022 (y - x) + x \u2208 sphere x r", "state_after": "case intro\n\ud835\udd5c : Type ?u.1764162\nE : Type u_1\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\ny : E\nhy : y \u2260 x\nhr : 0 \u2264 r\nthis : \u2016y - x\u2016 \u2260 0\n\u22a2 r \u2022 \u2016y - x\u2016\u207b\u00b9 \u2022 (y - x) + x \u2208 sphere x r"}, {"tactic": "simp [norm_smul, this, Real.norm_of_nonneg hr]", "state_before": "case intro\n\ud835\udd5c : Type ?u.1764162\nE : Type u_1\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\ny : E\nhy : y \u2260 x\nhr : 0 \u2264 r\nthis : \u2016y - x\u2016 \u2260 0\n\u22a2 r \u2022 \u2016y - x\u2016\u207b\u00b9 \u2022 (y - x) + x \u2208 sphere x r", "state_after": "case intro\n\ud835\udd5c : Type ?u.1764162\nE : Type u_1\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\ny : E\nhy : y \u2260 x\nhr : 0 \u2264 r\nthis : \u2016y - x\u2016 \u2260 0\n\u22a2 abs r * (\u2016y - x\u2016\u207b\u00b9 * \u2016y - x\u2016) = r"}, {"tactic": "rw [inv_mul_cancel this, mul_one, abs_eq_self.mpr hr]", "state_before": "case intro\n\ud835\udd5c : Type ?u.1764162\nE : Type u_1\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\ny : E\nhy : y \u2260 x\nhr : 0 \u2264 r\nthis : \u2016y - x\u2016 \u2260 0\n\u22a2 abs r * (\u2016y - x\u2016\u207b\u00b9 * \u2016y - x\u2016) = r", "state_after": "no goals"}, {"tactic": "simpa [sub_eq_zero]", "state_before": "\ud835\udd5c : Type ?u.1764162\nE : Type u_1\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\ny : E\nhy : y \u2260 x\nhr : 0 \u2264 r\n\u22a2 \u2016y - x\u2016 \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "full_name": "MvPolynomial.degree_finSuccEquiv", "start": [475, 1], "end": [485, 31], "traced_tactics": [{"tactic": "have h\u2080 : \u2200 {\u03b1 \u03b2 : Type _} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f := fun f => rfl", "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type ?u.1335767\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\n\u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f)", "state_after": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type ?u.1335767\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\nh\u2080 : \u2200 {\u03b1 : Type ?u.1350795} {\u03b2 : Type ?u.1350798} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f\n\u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f)"}, {"tactic": "have h\u2081 : \u2200 {\u03b1 \u03b2 : Type _} (f : \u03b1 \u2192 \u03b2), f \u2218 (fun x => x) = f := fun f => rfl", "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type ?u.1335767\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\nh\u2080 : \u2200 {\u03b1 : Type ?u.1350795} {\u03b2 : Type ?u.1350798} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f\n\u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f)", "state_after": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type ?u.1335767\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\nh\u2080 : \u2200 {\u03b1 : Type ?u.1350795} {\u03b2 : Type ?u.1350798} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f\nh\u2081 : \u2200 {\u03b1 : Type ?u.1350860} {\u03b2 : Type ?u.1350863} (f : \u03b1 \u2192 \u03b2), (f \u2218 fun x => x) = f\n\u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f)"}, {"tactic": "have h\u2082 : WithBot.some = Nat.cast := rfl", "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type ?u.1335767\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\nh\u2080 : \u2200 {\u03b1 : Type ?u.1350795} {\u03b2 : Type ?u.1350798} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f\nh\u2081 : \u2200 {\u03b1 : Type ?u.1350860} {\u03b2 : Type ?u.1350863} (f : \u03b1 \u2192 \u03b2), (f \u2218 fun x => x) = f\n\u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f)", "state_after": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type ?u.1335767\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\nh\u2080 : \u2200 {\u03b1 : Type ?u.1350795} {\u03b2 : Type ?u.1350798} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f\nh\u2081 : \u2200 {\u03b1 : Type ?u.1350860} {\u03b2 : Type ?u.1350863} (f : \u03b1 \u2192 \u03b2), (f \u2218 fun x => x) = f\nh\u2082 : WithBot.some = Nat.cast\n\u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f)"}, {"tactic": "have h' : ((finSuccEquiv R n f).support.sup fun x => x) = degreeOf 0 f := by\n  rw [degreeOf_eq_sup, finSuccEquiv_support f, Finset.sup_image, h\u2080]", "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type ?u.1335767\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\nh\u2080 : \u2200 {\u03b1 : Type ?u.1350795} {\u03b2 : Type ?u.1350798} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f\nh\u2081 : \u2200 {\u03b1 : Type ?u.1350860} {\u03b2 : Type ?u.1350863} (f : \u03b1 \u2192 \u03b2), (f \u2218 fun x => x) = f\nh\u2082 : WithBot.some = Nat.cast\n\u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f)", "state_after": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type ?u.1335767\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f\nh\u2081 : \u2200 {\u03b1 : Type ?u.1350860} {\u03b2 : Type ?u.1350863} (f : \u03b1 \u2192 \u03b2), (f \u2218 fun x => x) = f\nh\u2082 : WithBot.some = Nat.cast\nh' : (Finset.sup (Polynomial.support (\u2191(finSuccEquiv R n) f)) fun x => x) = degreeOf 0 f\n\u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f)"}, {"tactic": "rw [Polynomial.degree, \u2190 h', \u2190 h\u2082, Finset.coe_sup_of_nonempty (support_finSuccEquiv_nonempty h),\n  Finset.max_eq_sup_coe, h\u2081]", "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type ?u.1335767\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\nh\u2080 : \u2200 {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f\nh\u2081 : \u2200 {\u03b1 : Type ?u.1350860} {\u03b2 : Type ?u.1350863} (f : \u03b1 \u2192 \u03b2), (f \u2218 fun x => x) = f\nh\u2082 : WithBot.some = Nat.cast\nh' : (Finset.sup (Polynomial.support (\u2191(finSuccEquiv R n) f)) fun x => x) = degreeOf 0 f\n\u22a2 degree (\u2191(finSuccEquiv R n) f) = \u2191(degreeOf 0 f)", "state_after": "no goals"}, {"tactic": "rw [degreeOf_eq_sup, finSuccEquiv_support f, Finset.sup_image, h\u2080]", "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type ?u.1335767\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\nf : MvPolynomial (Fin (n + 1)) R\nh : f \u2260 0\nh\u2080 : \u2200 {\u03b1 : Type ?u.1350795} {\u03b2 : Type ?u.1350798} (f : \u03b1 \u2192 \u03b2), (fun x => x) \u2218 f = f\nh\u2081 : \u2200 {\u03b1 : Type ?u.1350860} {\u03b2 : Type ?u.1350863} (f : \u03b1 \u2192 \u03b2), (f \u2218 fun x => x) = f\nh\u2082 : WithBot.some = Nat.cast\n\u22a2 (Finset.sup (Polynomial.support (\u2191(finSuccEquiv R n) f)) fun x => x) = degreeOf 0 f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Bounds/Basic.lean", "full_name": "le_glb_Ioi", "start": [554, 1], "end": [555, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Equiv.lean", "full_name": "AlgEquiv.toLinearMap_injective", "start": [587, 1], "end": [588, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Module/Submodule/Lattice.lean", "full_name": "Submodule.disjoint_def'", "start": [331, 1], "end": [334, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Periodic.lean", "full_name": "Function.Periodic.const_smul\u2080", "start": [116, 11], "end": [120, 61], "traced_tactics": [{"tactic": "by_cases ha : a = 0", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_3\n\u03b3 : Type u_2\nf g : \u03b1 \u2192 \u03b2\nc c\u2081 c\u2082 x\u271d : \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : DivisionSemiring \u03b3\ninst\u271d : Module \u03b3 \u03b1\nh : Periodic f c\na : \u03b3\nx : \u03b1\n\u22a2 (fun x => f (a \u2022 x)) (x + a\u207b\u00b9 \u2022 c) = (fun x => f (a \u2022 x)) x", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_3\n\u03b3 : Type u_2\nf g : \u03b1 \u2192 \u03b2\nc c\u2081 c\u2082 x\u271d : \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : DivisionSemiring \u03b3\ninst\u271d : Module \u03b3 \u03b1\nh : Periodic f c\na : \u03b3\nx : \u03b1\nha : a = 0\n\u22a2 (fun x => f (a \u2022 x)) (x + a\u207b\u00b9 \u2022 c) = (fun x => f (a \u2022 x)) x\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_3\n\u03b3 : Type u_2\nf g : \u03b1 \u2192 \u03b2\nc c\u2081 c\u2082 x\u271d : \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : DivisionSemiring \u03b3\ninst\u271d : Module \u03b3 \u03b1\nh : Periodic f c\na : \u03b3\nx : \u03b1\nha : \u00aca = 0\n\u22a2 (fun x => f (a \u2022 x)) (x + a\u207b\u00b9 \u2022 c) = (fun x => f (a \u2022 x)) x"}, {"tactic": "simp only [ha, zero_smul]", "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_3\n\u03b3 : Type u_2\nf g : \u03b1 \u2192 \u03b2\nc c\u2081 c\u2082 x\u271d : \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : DivisionSemiring \u03b3\ninst\u271d : Module \u03b3 \u03b1\nh : Periodic f c\na : \u03b3\nx : \u03b1\nha : a = 0\n\u22a2 (fun x => f (a \u2022 x)) (x + a\u207b\u00b9 \u2022 c) = (fun x => f (a \u2022 x)) x", "state_after": "no goals"}, {"tactic": "simpa only [smul_add, smul_inv_smul\u2080 ha] using h (a \u2022 x)", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_3\n\u03b3 : Type u_2\nf g : \u03b1 \u2192 \u03b2\nc c\u2081 c\u2082 x\u271d : \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : DivisionSemiring \u03b3\ninst\u271d : Module \u03b3 \u03b1\nh : Periodic f c\na : \u03b3\nx : \u03b1\nha : \u00aca = 0\n\u22a2 (fun x => f (a \u2022 x)) (x + a\u207b\u00b9 \u2022 c) = (fun x => f (a \u2022 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\ud835\udd5c f i]", "state_before": "\ud835\udd5c : Type u_3\nE : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nj i : \u03b9\nhij : i \u2264 j\n\u22a2 gramSchmidt \ud835\udd5c f i \u2208 span \ud835\udd5c (f '' Set.Iic j)", "state_after": "\ud835\udd5c : Type u_3\nE : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nj i : \u03b9\nhij : i \u2264 j\n\u22a2 f i - \u2211 i_1 in Finset.Iio i, \u2191(\u2191(orthogonalProjection (span \ud835\udd5c {gramSchmidt \ud835\udd5c f i_1})) (f i)) \u2208 span \ud835\udd5c (f '' Set.Iic j)"}, {"tactic": "simp_rw [orthogonalProjection_singleton]", "state_before": "\ud835\udd5c : Type u_3\nE : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nj i : \u03b9\nhij : i \u2264 j\n\u22a2 f i - \u2211 i_1 in Finset.Iio i, \u2191(\u2191(orthogonalProjection (span \ud835\udd5c {gramSchmidt \ud835\udd5c f i_1})) (f i)) \u2208 span \ud835\udd5c (f '' Set.Iic j)", "state_after": "\ud835\udd5c : Type u_3\nE : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nj i : \u03b9\nhij : i \u2264 j\n\u22a2 f i - \u2211 x in Finset.Iio i, (inner (gramSchmidt \ud835\udd5c f x) (f i) / \u2191(\u2016gramSchmidt \ud835\udd5c f x\u2016 ^ 2)) \u2022 gramSchmidt \ud835\udd5c f x \u2208\n    span \ud835\udd5c (f '' Set.Iic j)"}, {"tactic": "refine' Submodule.sub_mem _ (subset_span (mem_image_of_mem _ hij))\n  (Submodule.sum_mem _ fun k hk => _)", "state_before": "\ud835\udd5c : Type u_3\nE : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nj i : \u03b9\nhij : i \u2264 j\n\u22a2 f i - \u2211 x in Finset.Iio i, (inner (gramSchmidt \ud835\udd5c f x) (f i) / \u2191(\u2016gramSchmidt \ud835\udd5c f x\u2016 ^ 2)) \u2022 gramSchmidt \ud835\udd5c f x \u2208\n    span \ud835\udd5c (f '' Set.Iic j)", "state_after": "\ud835\udd5c : Type u_3\nE : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nj i : \u03b9\nhij : i \u2264 j\nk : \u03b9\nhk : k \u2208 Finset.Iio i\n\u22a2 (inner (gramSchmidt \ud835\udd5c f k) (f i) / \u2191(\u2016gramSchmidt \ud835\udd5c f k\u2016 ^ 2)) \u2022 gramSchmidt \ud835\udd5c f k \u2208 span \ud835\udd5c (f '' Set.Iic j)"}, {"tactic": "let hkj : k < j := (Finset.mem_Iio.1 hk).trans_le hij", "state_before": "\ud835\udd5c : Type u_3\nE : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nj i : \u03b9\nhij : i \u2264 j\nk : \u03b9\nhk : k \u2208 Finset.Iio i\n\u22a2 (inner (gramSchmidt \ud835\udd5c f k) (f i) / \u2191(\u2016gramSchmidt \ud835\udd5c f k\u2016 ^ 2)) \u2022 gramSchmidt \ud835\udd5c f k \u2208 span \ud835\udd5c (f '' Set.Iic j)", "state_after": "\ud835\udd5c : Type u_3\nE : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nj i : \u03b9\nhij : i \u2264 j\nk : \u03b9\nhk : k \u2208 Finset.Iio i\nhkj : k < j := LT.lt.trans_le (Iff.mp Finset.mem_Iio hk) hij\n\u22a2 (inner (gramSchmidt \ud835\udd5c f k) (f i) / \u2191(\u2016gramSchmidt \ud835\udd5c f k\u2016 ^ 2)) \u2022 gramSchmidt \ud835\udd5c f k \u2208 span \ud835\udd5c (f '' Set.Iic j)"}, {"tactic": "exact smul_mem _ _\n  (span_mono (image_subset f <| Iic_subset_Iic.2 hkj.le) <| gramSchmidt_mem_span _ le_rfl)", "state_before": "\ud835\udd5c : Type u_3\nE : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\nj i : \u03b9\nhij : i \u2264 j\nk : \u03b9\nhk : k \u2208 Finset.Iio i\nhkj : k < j := LT.lt.trans_le (Iff.mp Finset.mem_Iio hk) hij\n\u22a2 (inner (gramSchmidt \ud835\udd5c f k) (f i) / \u2191(\u2016gramSchmidt \ud835\udd5c f k\u2016 ^ 2)) \u2022 gramSchmidt \ud835\udd5c f k \u2208 span \ud835\udd5c (f '' Set.Iic j)", "state_after": "no goals"}, {"tactic": "exact hkj", "state_before": "\ud835\udd5c : Type u_3\nE : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\n_x\u271d : (j : \u03b9) \u00d7' (i : \u03b9) \u00d7' i \u2264 j\na\u271d\u00b2 :\n  \u2200 (y : (j : \u03b9) \u00d7' (i : \u03b9) \u00d7' i \u2264 j),\n    (invImage (fun a => PSigma.casesOn a fun j snd => PSigma.casesOn snd fun i snd => j) IsWellOrder.toHasWellFounded).1\n        y _x\u271d \u2192\n      gramSchmidt \ud835\udd5c f y.2.1 \u2208 span \ud835\udd5c (f '' Set.Iic y.1)\nj : \u03b9\ni\u271d : (i : \u03b9) \u00d7' i \u2264 j\na\u271d\u00b9 :\n  \u2200 (y : (j : \u03b9) \u00d7' (i : \u03b9) \u00d7' i \u2264 j),\n    (invImage (fun a => PSigma.casesOn a fun j snd => PSigma.casesOn snd fun i snd => j) IsWellOrder.toHasWellFounded).1\n        y { fst := j, snd := i\u271d } \u2192\n      gramSchmidt \ud835\udd5c f y.2.1 \u2208 span \ud835\udd5c (f '' Set.Iic y.1)\ni : \u03b9\nhij : i \u2264 j\na\u271d :\n  \u2200 (y : (j : \u03b9) \u00d7' (i : \u03b9) \u00d7' i \u2264 j),\n    (invImage (fun a => PSigma.casesOn a fun j snd => PSigma.casesOn snd fun i snd => j) IsWellOrder.toHasWellFounded).1\n        y { fst := j, snd := { fst := i, snd := hij } } \u2192\n      gramSchmidt \ud835\udd5c f y.2.1 \u2208 span \ud835\udd5c (f '' Set.Iic y.1)\nk : \u03b9\nhk : k \u2208 Finset.Iio i\nhkj : k < j := LT.lt.trans_le (Iff.mp Finset.mem_Iio hk) hij\n\u22a2 (invImage (fun a => PSigma.casesOn a fun j snd => PSigma.casesOn snd fun i snd => j) IsWellOrder.toHasWellFounded).1\n    { fst := k, snd := { fst := k, snd := (_ : k \u2264 k) } } { fst := j, snd := { fst := i, snd := hij } }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/QuotientOperations.lean", "full_name": "Ideal.quotientMap_mk", "start": [343, 1], "end": [345, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Sylow.lean", "full_name": "Sylow.pointwise_smul_def", "start": [231, 1], "end": [233, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "full_name": "hasSum_pi_single", "start": [214, 1], "end": [216, 25], "traced_tactics": [{"tactic": "convert hasSum_ite_eq b a", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.43306\n\u03b4 : Type ?u.43309\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d b\u271d : \u03b1\ns : Finset \u03b2\ninst\u271d : DecidableEq \u03b2\nb : \u03b2\na : \u03b1\n\u22a2 HasSum (Pi.single b a) a", "state_after": "case h.e'_5.h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.43306\n\u03b4 : Type ?u.43309\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d b\u271d : \u03b1\ns : Finset \u03b2\ninst\u271d : DecidableEq \u03b2\nb : \u03b2\na : \u03b1\nx\u271d : \u03b2\n\u22a2 Pi.single b a x\u271d = if x\u271d = b then a else 0"}, {"tactic": "simp [Pi.single_apply]", "state_before": "case h.e'_5.h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.43306\n\u03b4 : Type ?u.43309\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf g : \u03b2 \u2192 \u03b1\na\u271d b\u271d : \u03b1\ns : Finset \u03b2\ninst\u271d : DecidableEq \u03b2\nb : \u03b2\na : \u03b1\nx\u271d : \u03b2\n\u22a2 Pi.single b a x\u271d = if x\u271d = b then a else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Core.lean", "full_name": "false_of_ne", "start": [578, 1], "end": [578, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/ContinuousOn.lean", "full_name": "inter_mem_nhdsWithin", "start": [160, 1], "end": [161, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Sylow.lean", "full_name": "Sylow.smul_eq_iff_mem_normalizer", "start": [262, 1], "end": [270, 90], "traced_tactics": [{"tactic": "rw [eq_comm, SetLike.ext_iff, \u2190 inv_mem_iff (G := G) (H := normalizer P.toSubgroup),\n    mem_normalizer_iff, inv_inv]", "state_before": "p : 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: Sylow p G\nh : G\nx\u271d : h \u2208 g \u2022 P\na : G\nb : a \u2208 \u2191\u2191P\nc : \u2191(\u2191(MulDistribMulAction.toMonoidEnd (MulAut G) G) (\u2191MulAut.conj g)) a = h\n\u22a2 g\u207b\u00b9 * \u2191(\u2191(MulDistribMulAction.toMonoidEnd (MulAut G) G) (\u2191MulAut.conj g)) a * g \u2208 \u2191P", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Subobject/FactorThru.lean", "full_name": "CategoryTheory.Subobject.factors_zero", "start": [107, 1], "end": [108, 37], "traced_tactics": [{"tactic": "simp", "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category C\nX\u271d Y\u271d Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category D\ninst\u271d : HasZeroMorphisms C\nX Y : C\nP : Subobject Y\n\u22a2 0 \u226b MonoOver.arrow (representative.obj P) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.preimage_surjective", "start": [1541, 1], "end": [1544, 87], "traced_tactics": [{"tactic": "refine' \u27e8fun h x x' hx => _, Injective.preimage_surjective\u27e9", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\n\u22a2 Surjective (preimage f) \u2194 Injective f", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (preimage f)\nx x' : \u03b1\nhx : f x = f x'\n\u22a2 x = x'"}, {"tactic": "cases' h {x} with s hs", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (preimage f)\nx x' : \u03b1\nhx : f x = f x'\n\u22a2 x = x'", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (preimage f)\nx x' : \u03b1\nhx : f x = f x'\ns : Set \u03b2\nhs : f \u207b\u00b9' s = {x}\n\u22a2 x = x'"}, {"tactic": "have := mem_singleton x", "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (preimage f)\nx x' : \u03b1\nhx : f x = f x'\ns : Set \u03b2\nhs : f \u207b\u00b9' s = {x}\n\u22a2 x = x'", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (preimage f)\nx x' : \u03b1\nhx : f x = f x'\ns : Set \u03b2\nhs : f \u207b\u00b9' s = {x}\nthis : x \u2208 {x}\n\u22a2 x = x'"}, {"tactic": "rwa [\u2190 hs, mem_preimage, hx, \u2190 mem_preimage, hs, mem_singleton_iff, eq_comm] at this", "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (preimage f)\nx x' : \u03b1\nhx : f x = f x'\ns : Set \u03b2\nhs : f \u207b\u00b9' s = {x}\nthis : x \u2208 {x}\n\u22a2 x = x'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.coe_cons", "start": [875, 1], "end": [877, 7], "traced_tactics": [{"tactic": "ext", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.38954\n\u03b3 : Type ?u.38957\ns\u271d t : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\nh : \u00aca \u2208 s\n\u22a2 \u2191(cons a s h) = insert a \u2191s", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.38954\n\u03b3 : Type ?u.38957\ns\u271d t : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\nh : \u00aca \u2208 s\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u2191(cons a s h) \u2194 x\u271d \u2208 insert a \u2191s"}, {"tactic": "simp", "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.38954\n\u03b3 : Type ?u.38957\ns\u271d t : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\nh : \u00aca \u2208 s\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u2191(cons a s h) \u2194 x\u271d \u2208 insert a \u2191s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Add.lean", "full_name": "HasStrictFDerivAt.add", "start": [122, 8], "end": [126, 9], "traced_tactics": [{"tactic": "simp only [LinearMap.sub_apply, LinearMap.add_apply, map_sub, map_add, add_apply]", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type ?u.108128\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type ?u.108223\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhf : HasStrictFDerivAt f f' x\nhg : HasStrictFDerivAt g g' x\ny : E \u00d7 E\n\u22a2 f y.fst - f y.snd - \u2191f' (y.fst - y.snd) + (g y.fst - g y.snd - \u2191g' (y.fst - y.snd)) =\n    (fun y => f y + g y) y.fst - (fun y => f y + g y) y.snd - \u2191(f' + g') (y.fst - y.snd)", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type ?u.108128\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type ?u.108223\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhf : HasStrictFDerivAt f f' x\nhg : HasStrictFDerivAt g g' x\ny : E \u00d7 E\n\u22a2 f y.fst - f y.snd - (\u2191f' y.fst - \u2191f' y.snd) + (g y.fst - g y.snd - (\u2191g' y.fst - \u2191g' y.snd)) =\n    f y.fst + g y.fst - (f y.snd + g y.snd) - (\u2191f' y.fst + \u2191g' y.fst - (\u2191f' y.snd + \u2191g' y.snd))"}, {"tactic": "abel", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type ?u.108128\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type ?u.108223\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nhf : HasStrictFDerivAt f f' x\nhg : HasStrictFDerivAt g g' x\ny : E \u00d7 E\n\u22a2 f y.fst - f y.snd - (\u2191f' y.fst - \u2191f' y.snd) + (g y.fst - g y.snd - (\u2191g' y.fst - \u2191g' y.snd)) =\n    f y.fst + g y.fst - (f y.snd + g y.snd) - (\u2191f' y.fst + \u2191g' y.fst - (\u2191f' y.snd + \u2191g' y.snd))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Basic.lean", "full_name": "Metric.uniformContinuous_iff", "start": [819, 1], "end": [821, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Valuation/Basic.lean", "full_name": "AddValuation.comap_id", "start": [751, 1], "end": [752, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Sober.lean", "full_name": "isGenericPoint_closure", "start": [50, 1], "end": [51, 9], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/Fermat4.lean", "full_name": "Fermat42.exists_minimal", "start": [74, 1], "end": [89, 8], "traced_tactics": [{"tactic": "let S : Set \u2115 := { n | \u2203 s : \u2124 \u00d7 \u2124 \u00d7 \u2124, Fermat42 s.1 s.2.1 s.2.2 \u2227 n = Int.natAbs s.2.2 }", "state_before": "a b c : \u2124\nh : Fermat42 a b c\n\u22a2 \u2203 a0 b0 c0, Minimal a0 b0 c0", "state_after": "a b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\n\u22a2 \u2203 a0 b0 c0, Minimal a0 b0 c0"}, {"tactic": "have S_nonempty : S.Nonempty := by\n  use Int.natAbs c\n  rw [Set.mem_setOf_eq]\n  use \u27e8a, \u27e8b, c\u27e9\u27e9\n  tauto", "state_before": "a b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\n\u22a2 \u2203 a0 b0 c0, Minimal a0 b0 c0", "state_after": "a b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\nS_nonempty : Set.Nonempty S\n\u22a2 \u2203 a0 b0 c0, Minimal a0 b0 c0"}, {"tactic": "let m : \u2115 := Nat.find S_nonempty", "state_before": "a b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\nS_nonempty : Set.Nonempty S\n\u22a2 \u2203 a0 b0 c0, Minimal a0 b0 c0", "state_after": "a b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\nS_nonempty : Set.Nonempty S\nm : \u2115 := Nat.find S_nonempty\n\u22a2 \u2203 a0 b0 c0, Minimal a0 b0 c0"}, {"tactic": "have m_mem : m \u2208 S := Nat.find_spec S_nonempty", "state_before": "a b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\nS_nonempty : Set.Nonempty S\nm : \u2115 := Nat.find S_nonempty\n\u22a2 \u2203 a0 b0 c0, Minimal a0 b0 c0", "state_after": "a b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\nS_nonempty : Set.Nonempty S\nm : \u2115 := Nat.find S_nonempty\nm_mem : m \u2208 S\n\u22a2 \u2203 a0 b0 c0, Minimal a0 b0 c0"}, {"tactic": "rcases m_mem with \u27e8s0, hs0, hs1\u27e9", "state_before": "a b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\nS_nonempty : Set.Nonempty S\nm : \u2115 := Nat.find S_nonempty\nm_mem : m \u2208 S\n\u22a2 \u2203 a0 b0 c0, Minimal a0 b0 c0", "state_after": "case intro.intro\na b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\nS_nonempty : Set.Nonempty S\nm : \u2115 := Nat.find S_nonempty\ns0 : \u2124 \u00d7 \u2124 \u00d7 \u2124\nhs0 : Fermat42 s0.fst s0.snd.fst s0.snd.snd\nhs1 : m = Int.natAbs s0.snd.snd\n\u22a2 \u2203 a0 b0 c0, Minimal a0 b0 c0"}, {"tactic": "use s0.1, s0.2.1, s0.2.2, hs0", "state_before": "case intro.intro\na b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\nS_nonempty : Set.Nonempty S\nm : \u2115 := Nat.find S_nonempty\ns0 : \u2124 \u00d7 \u2124 \u00d7 \u2124\nhs0 : Fermat42 s0.fst s0.snd.fst s0.snd.snd\nhs1 : m = Int.natAbs s0.snd.snd\n\u22a2 \u2203 a0 b0 c0, Minimal a0 b0 c0", "state_after": "case intro.intro\na b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\nS_nonempty : Set.Nonempty S\nm : \u2115 := Nat.find S_nonempty\ns0 : \u2124 \u00d7 \u2124 \u00d7 \u2124\nhs0 : Fermat42 s0.fst s0.snd.fst s0.snd.snd\nhs1 : m = Int.natAbs s0.snd.snd\n\u22a2 \u2200 (a1 b1 c1 : \u2124), Fermat42 a1 b1 c1 \u2192 Int.natAbs s0.snd.snd \u2264 Int.natAbs c1"}, {"tactic": "intro a1 b1 c1 h1", "state_before": "case intro.intro\na b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\nS_nonempty : Set.Nonempty S\nm : \u2115 := Nat.find S_nonempty\ns0 : \u2124 \u00d7 \u2124 \u00d7 \u2124\nhs0 : Fermat42 s0.fst s0.snd.fst s0.snd.snd\nhs1 : m = Int.natAbs s0.snd.snd\n\u22a2 \u2200 (a1 b1 c1 : \u2124), Fermat42 a1 b1 c1 \u2192 Int.natAbs s0.snd.snd \u2264 Int.natAbs c1", "state_after": "case intro.intro\na b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\nS_nonempty : Set.Nonempty S\nm : \u2115 := Nat.find S_nonempty\ns0 : \u2124 \u00d7 \u2124 \u00d7 \u2124\nhs0 : Fermat42 s0.fst s0.snd.fst s0.snd.snd\nhs1 : m = Int.natAbs s0.snd.snd\na1 b1 c1 : \u2124\nh1 : Fermat42 a1 b1 c1\n\u22a2 Int.natAbs s0.snd.snd \u2264 Int.natAbs c1"}, {"tactic": "rw [\u2190 hs1]", "state_before": "case intro.intro\na b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\nS_nonempty : Set.Nonempty S\nm : \u2115 := Nat.find S_nonempty\ns0 : \u2124 \u00d7 \u2124 \u00d7 \u2124\nhs0 : Fermat42 s0.fst s0.snd.fst s0.snd.snd\nhs1 : m = Int.natAbs s0.snd.snd\na1 b1 c1 : \u2124\nh1 : Fermat42 a1 b1 c1\n\u22a2 Int.natAbs s0.snd.snd \u2264 Int.natAbs c1", "state_after": "case intro.intro\na b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\nS_nonempty : Set.Nonempty S\nm : \u2115 := Nat.find S_nonempty\ns0 : \u2124 \u00d7 \u2124 \u00d7 \u2124\nhs0 : Fermat42 s0.fst s0.snd.fst s0.snd.snd\nhs1 : m = Int.natAbs s0.snd.snd\na1 b1 c1 : \u2124\nh1 : Fermat42 a1 b1 c1\n\u22a2 m \u2264 Int.natAbs c1"}, {"tactic": "apply Nat.find_min'", "state_before": "case intro.intro\na b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\nS_nonempty : Set.Nonempty S\nm : \u2115 := Nat.find S_nonempty\ns0 : \u2124 \u00d7 \u2124 \u00d7 \u2124\nhs0 : Fermat42 s0.fst s0.snd.fst s0.snd.snd\nhs1 : m = Int.natAbs s0.snd.snd\na1 b1 c1 : \u2124\nh1 : Fermat42 a1 b1 c1\n\u22a2 m \u2264 Int.natAbs c1", "state_after": "case intro.intro.h\na b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\nS_nonempty : Set.Nonempty S\nm : \u2115 := Nat.find S_nonempty\ns0 : \u2124 \u00d7 \u2124 \u00d7 \u2124\nhs0 : Fermat42 s0.fst s0.snd.fst s0.snd.snd\nhs1 : m = Int.natAbs s0.snd.snd\na1 b1 c1 : \u2124\nh1 : Fermat42 a1 b1 c1\n\u22a2 Int.natAbs c1 \u2208 S"}, {"tactic": "use \u27e8a1, \u27e8b1, c1\u27e9\u27e9", "state_before": "case intro.intro.h\na b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\nS_nonempty : Set.Nonempty S\nm : \u2115 := Nat.find S_nonempty\ns0 : \u2124 \u00d7 \u2124 \u00d7 \u2124\nhs0 : Fermat42 s0.fst s0.snd.fst s0.snd.snd\nhs1 : m = Int.natAbs s0.snd.snd\na1 b1 c1 : \u2124\nh1 : Fermat42 a1 b1 c1\n\u22a2 Int.natAbs c1 \u2208 S", "state_after": "case intro.intro.h\na b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\nS_nonempty : Set.Nonempty S\nm : \u2115 := Nat.find S_nonempty\ns0 : \u2124 \u00d7 \u2124 \u00d7 \u2124\nhs0 : Fermat42 s0.fst s0.snd.fst s0.snd.snd\nhs1 : m = Int.natAbs s0.snd.snd\na1 b1 c1 : \u2124\nh1 : Fermat42 a1 b1 c1\n\u22a2 Fermat42 (a1, b1, c1).fst (a1, b1, c1).snd.fst (a1, b1, c1).snd.snd \u2227 Int.natAbs c1 = Int.natAbs (a1, b1, c1).snd.snd"}, {"tactic": "tauto", "state_before": "case intro.intro.h\na b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\nS_nonempty : Set.Nonempty S\nm : \u2115 := Nat.find S_nonempty\ns0 : \u2124 \u00d7 \u2124 \u00d7 \u2124\nhs0 : Fermat42 s0.fst s0.snd.fst s0.snd.snd\nhs1 : m = Int.natAbs s0.snd.snd\na1 b1 c1 : \u2124\nh1 : Fermat42 a1 b1 c1\n\u22a2 Fermat42 (a1, b1, c1).fst (a1, b1, c1).snd.fst (a1, b1, c1).snd.snd \u2227 Int.natAbs c1 = Int.natAbs (a1, b1, c1).snd.snd", "state_after": "no goals"}, {"tactic": "use Int.natAbs c", "state_before": "a b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\n\u22a2 Set.Nonempty S", "state_after": "a b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\n\u22a2 Int.natAbs c \u2208 S"}, {"tactic": "rw [Set.mem_setOf_eq]", "state_before": "a b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\n\u22a2 Int.natAbs c \u2208 S", "state_after": "a b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\n\u22a2 \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 Int.natAbs c = Int.natAbs s.snd.snd"}, {"tactic": "use \u27e8a, \u27e8b, c\u27e9\u27e9", "state_before": "a b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\n\u22a2 \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 Int.natAbs c = Int.natAbs s.snd.snd", "state_after": "a b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\n\u22a2 Fermat42 (a, b, c).fst (a, b, c).snd.fst (a, b, c).snd.snd \u2227 Int.natAbs c = Int.natAbs (a, b, c).snd.snd"}, {"tactic": "tauto", "state_before": "a b c : \u2124\nh : Fermat42 a b c\nS : Set \u2115 := {n | \u2203 s, Fermat42 s.fst s.snd.fst s.snd.snd \u2227 n = Int.natAbs s.snd.snd}\n\u22a2 Fermat42 (a, b, c).fst (a, b, c).snd.fst (a, b, c).snd.snd \u2227 Int.natAbs c = Int.natAbs (a, b, c).snd.snd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Bases.lean", "full_name": "Filter.hasBasis_iff", "start": [274, 1], "end": [275, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.const_apply", "start": [151, 1], "end": [152, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Submonoid/Center.lean", "full_name": "Submonoid.center_toSubsemigroup", "start": [50, 1], "end": [51, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Hom/Basic.lean", "full_name": "OrderHom.orderHom_eq_id", "start": [557, 1], "end": [558, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Normed/Group/Hom.lean", "full_name": "NormedAddGroupHom.norm_id", "start": [441, 1], "end": [444, 42], "traced_tactics": [{"tactic": "refine' norm_id_of_nontrivial_seminorm V _", "state_before": "V\u271d : Type ?u.320637\nV\u2081 : Type ?u.320640\nV\u2082 : Type ?u.320643\nV\u2083 : Type ?u.320646\ninst\u271d\u2075 : SeminormedAddCommGroup V\u271d\ninst\u271d\u2074 : SeminormedAddCommGroup V\u2081\ninst\u271d\u00b3 : SeminormedAddCommGroup V\u2082\ninst\u271d\u00b2 : SeminormedAddCommGroup V\u2083\nf g : NormedAddGroupHom V\u2081 V\u2082\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : Nontrivial V\n\u22a2 \u2016id V\u2016 = 1", "state_after": "V\u271d : Type ?u.320637\nV\u2081 : Type ?u.320640\nV\u2082 : Type ?u.320643\nV\u2083 : Type ?u.320646\ninst\u271d\u2075 : SeminormedAddCommGroup V\u271d\ninst\u271d\u2074 : SeminormedAddCommGroup V\u2081\ninst\u271d\u00b3 : SeminormedAddCommGroup V\u2082\ninst\u271d\u00b2 : SeminormedAddCommGroup V\u2083\nf g : NormedAddGroupHom V\u2081 V\u2082\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : Nontrivial V\n\u22a2 \u2203 x, \u2016x\u2016 \u2260 0"}, {"tactic": "obtain \u27e8x, hx\u27e9 := exists_ne (0 : V)", "state_before": "V\u271d : Type ?u.320637\nV\u2081 : Type ?u.320640\nV\u2082 : Type ?u.320643\nV\u2083 : Type ?u.320646\ninst\u271d\u2075 : SeminormedAddCommGroup V\u271d\ninst\u271d\u2074 : SeminormedAddCommGroup V\u2081\ninst\u271d\u00b3 : SeminormedAddCommGroup V\u2082\ninst\u271d\u00b2 : SeminormedAddCommGroup V\u2083\nf g : NormedAddGroupHom V\u2081 V\u2082\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : Nontrivial V\n\u22a2 \u2203 x, \u2016x\u2016 \u2260 0", "state_after": "case intro\nV\u271d : Type ?u.320637\nV\u2081 : Type ?u.320640\nV\u2082 : Type ?u.320643\nV\u2083 : Type ?u.320646\ninst\u271d\u2075 : SeminormedAddCommGroup V\u271d\ninst\u271d\u2074 : SeminormedAddCommGroup V\u2081\ninst\u271d\u00b3 : SeminormedAddCommGroup V\u2082\ninst\u271d\u00b2 : SeminormedAddCommGroup V\u2083\nf g : NormedAddGroupHom V\u2081 V\u2082\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : Nontrivial V\nx : V\nhx : x \u2260 0\n\u22a2 \u2203 x, \u2016x\u2016 \u2260 0"}, {"tactic": "exact \u27e8x, ne_of_gt (norm_pos_iff.2 hx)\u27e9", "state_before": "case intro\nV\u271d : Type ?u.320637\nV\u2081 : Type ?u.320640\nV\u2082 : Type ?u.320643\nV\u2083 : Type ?u.320646\ninst\u271d\u2075 : SeminormedAddCommGroup V\u271d\ninst\u271d\u2074 : SeminormedAddCommGroup V\u2081\ninst\u271d\u00b3 : SeminormedAddCommGroup V\u2082\ninst\u271d\u00b2 : SeminormedAddCommGroup V\u2083\nf g : NormedAddGroupHom V\u2081 V\u2082\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : Nontrivial V\nx : V\nhx : x \u2260 0\n\u22a2 \u2203 x, \u2016x\u2016 \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "full_name": "Ordinal.lift_one", "start": [787, 1], "end": [788, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/NAry.lean", "full_name": "Set.image2_right", "start": [306, 1], "end": [307, 36], "traced_tactics": [{"tactic": "simp [nonempty_def.mp h, ext_iff]", "state_before": "\u03b1 : Type u_1\n\u03b1' : Type ?u.37462\n\u03b2 : Type u_2\n\u03b2' : Type ?u.37468\n\u03b3 : Type ?u.37471\n\u03b3' : Type ?u.37474\n\u03b4 : Type ?u.37477\n\u03b4' : Type ?u.37480\n\u03b5 : Type ?u.37483\n\u03b5' : Type ?u.37486\n\u03b6 : Type ?u.37489\n\u03b6' : Type ?u.37492\n\u03bd : Type ?u.37495\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nh : Set.Nonempty s\n\u22a2 image2 (fun x y => y) s t = t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Polynomial/Vieta.lean", "full_name": "MvPolynomial.prod_X_add_C_coeff", "start": [184, 1], "end": [194, 50], "traced_tactics": [{"tactic": "let s := Finset.univ.val.map fun i => (MvPolynomial.X i : MvPolynomial \u03c3 R)", "state_before": "R : Type u_2\n\u03c3 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Fintype \u03c3\nk : \u2115\nh : k \u2264 Fintype.card \u03c3\n\u22a2 Polynomial.coeff (\u220f i : \u03c3, (Polynomial.X + \u2191Polynomial.C (X i))) k = esymm \u03c3 R (Fintype.card \u03c3 - k)", "state_after": "R : Type u_2\n\u03c3 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Fintype \u03c3\nk : \u2115\nh : k \u2264 Fintype.card \u03c3\ns : Multiset (MvPolynomial \u03c3 R) := Multiset.map (fun i => X i) univ.val\n\u22a2 Polynomial.coeff (\u220f i : \u03c3, (Polynomial.X + \u2191Polynomial.C (X i))) k = esymm \u03c3 R (Fintype.card \u03c3 - k)"}, {"tactic": "have : Fintype.card \u03c3 = Multiset.card s := by\n  rw [Multiset.card_map, \u2190Finset.card_univ, Finset.card_def]", "state_before": "R : Type u_2\n\u03c3 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Fintype \u03c3\nk : \u2115\nh : k \u2264 Fintype.card \u03c3\ns : Multiset (MvPolynomial \u03c3 R) := Multiset.map (fun i => X i) univ.val\n\u22a2 Polynomial.coeff (\u220f i : \u03c3, (Polynomial.X + \u2191Polynomial.C (X i))) k = esymm \u03c3 R (Fintype.card \u03c3 - k)", "state_after": "R : Type u_2\n\u03c3 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Fintype \u03c3\nk : \u2115\nh : k \u2264 Fintype.card \u03c3\ns : Multiset (MvPolynomial \u03c3 R) := Multiset.map (fun i => X i) univ.val\nthis : Fintype.card \u03c3 = \u2191Multiset.card s\n\u22a2 Polynomial.coeff (\u220f i : \u03c3, (Polynomial.X + \u2191Polynomial.C (X i))) k = esymm \u03c3 R (Fintype.card \u03c3 - k)"}, {"tactic": "rw [this] at h \u22a2", "state_before": "R : Type u_2\n\u03c3 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Fintype \u03c3\nk : \u2115\nh : k \u2264 Fintype.card \u03c3\ns : Multiset (MvPolynomial \u03c3 R) := Multiset.map (fun i => X i) univ.val\nthis : Fintype.card \u03c3 = \u2191Multiset.card s\n\u22a2 Polynomial.coeff (\u220f i : \u03c3, (Polynomial.X + \u2191Polynomial.C (X i))) k = esymm \u03c3 R (Fintype.card \u03c3 - k)", "state_after": "R : Type u_2\n\u03c3 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Fintype \u03c3\nk : \u2115\ns : Multiset (MvPolynomial \u03c3 R) := Multiset.map (fun i => X i) univ.val\nh : k \u2264 \u2191Multiset.card s\nthis : Fintype.card \u03c3 = \u2191Multiset.card s\n\u22a2 Polynomial.coeff (\u220f i : \u03c3, (Polynomial.X + \u2191Polynomial.C (X i))) k = esymm \u03c3 R (\u2191Multiset.card s - k)"}, {"tactic": "rw [MvPolynomial.esymm_eq_multiset_esymm \u03c3 R, Finset.prod_eq_multiset_prod]", "state_before": "R : Type u_2\n\u03c3 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Fintype \u03c3\nk : \u2115\ns : Multiset (MvPolynomial \u03c3 R) := Multiset.map (fun i => X i) univ.val\nh : k \u2264 \u2191Multiset.card s\nthis : Fintype.card \u03c3 = \u2191Multiset.card s\n\u22a2 Polynomial.coeff (\u220f i : \u03c3, (Polynomial.X + \u2191Polynomial.C (X i))) k = esymm \u03c3 R (\u2191Multiset.card s - k)", "state_after": "R : Type u_2\n\u03c3 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Fintype \u03c3\nk : \u2115\ns : Multiset (MvPolynomial \u03c3 R) := Multiset.map (fun i => X i) univ.val\nh : k \u2264 \u2191Multiset.card s\nthis : Fintype.card \u03c3 = \u2191Multiset.card s\n\u22a2 Polynomial.coeff (Multiset.prod (Multiset.map (fun i => Polynomial.X + \u2191Polynomial.C (X i)) univ.val)) k =\n    Multiset.esymm (Multiset.map X univ.val) (\u2191Multiset.card s - k)"}, {"tactic": "convert Multiset.prod_X_add_C_coeff s h", "state_before": "R : Type u_2\n\u03c3 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Fintype \u03c3\nk : \u2115\ns : Multiset (MvPolynomial \u03c3 R) := Multiset.map (fun i => X i) univ.val\nh : k \u2264 \u2191Multiset.card s\nthis : Fintype.card \u03c3 = \u2191Multiset.card s\n\u22a2 Polynomial.coeff (Multiset.prod (Multiset.map (fun i => Polynomial.X + \u2191Polynomial.C (X i)) univ.val)) k =\n    Multiset.esymm (Multiset.map X univ.val) (\u2191Multiset.card s - k)", "state_after": "case h.e'_2.h.e'_3.h.e'_3\nR : Type u_2\n\u03c3 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Fintype \u03c3\nk : \u2115\ns : Multiset (MvPolynomial \u03c3 R) := Multiset.map (fun i => X i) univ.val\nh : k \u2264 \u2191Multiset.card s\nthis : Fintype.card \u03c3 = \u2191Multiset.card s\n\u22a2 Multiset.map (fun i => Polynomial.X + \u2191Polynomial.C (X i)) univ.val =\n    Multiset.map (fun r => Polynomial.X + \u2191Polynomial.C r) s"}, {"tactic": "dsimp", "state_before": "case h.e'_2.h.e'_3.h.e'_3\nR : Type u_2\n\u03c3 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Fintype \u03c3\nk : \u2115\ns : Multiset (MvPolynomial \u03c3 R) := Multiset.map (fun i => X i) univ.val\nh : k \u2264 \u2191Multiset.card s\nthis : Fintype.card \u03c3 = \u2191Multiset.card s\n\u22a2 Multiset.map (fun i => Polynomial.X + \u2191Polynomial.C (X i)) univ.val =\n    Multiset.map (fun r => Polynomial.X + \u2191Polynomial.C r) s", "state_after": "case h.e'_2.h.e'_3.h.e'_3\nR : Type u_2\n\u03c3 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Fintype \u03c3\nk : \u2115\ns : Multiset (MvPolynomial \u03c3 R) := Multiset.map (fun i => X i) univ.val\nh : k \u2264 \u2191Multiset.card s\nthis : Fintype.card \u03c3 = \u2191Multiset.card s\n\u22a2 Multiset.map (fun i => Polynomial.X + \u2191Polynomial.C (X i)) univ.val =\n    Multiset.map (fun r => Polynomial.X + \u2191Polynomial.C r) (Multiset.map (fun i => X i) univ.val)"}, {"tactic": "simp_rw [Multiset.map_map, Function.comp_apply]", "state_before": "case h.e'_2.h.e'_3.h.e'_3\nR : Type u_2\n\u03c3 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Fintype \u03c3\nk : \u2115\ns : Multiset (MvPolynomial \u03c3 R) := Multiset.map (fun i => X i) univ.val\nh : k \u2264 \u2191Multiset.card s\nthis : Fintype.card \u03c3 = \u2191Multiset.card s\n\u22a2 Multiset.map (fun i => Polynomial.X + \u2191Polynomial.C (X i)) univ.val =\n    Multiset.map (fun r => Polynomial.X + \u2191Polynomial.C r) (Multiset.map (fun i => X i) univ.val)", "state_after": "no goals"}, {"tactic": "rw [Multiset.card_map, \u2190Finset.card_univ, Finset.card_def]", "state_before": "R : Type u_2\n\u03c3 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Fintype \u03c3\nk : \u2115\nh : k \u2264 Fintype.card \u03c3\ns : Multiset (MvPolynomial \u03c3 R) := Multiset.map (fun i => X i) univ.val\n\u22a2 Fintype.card \u03c3 = \u2191Multiset.card s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean", "full_name": "Real.differentiableAt_arctan", "start": [97, 1], "end": [98, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "full_name": "ae_le_essSup", "start": [117, 1], "end": [120, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/RingDivision.lean", "full_name": "Polynomial.rootMultiplicity_X_sub_C_self", "start": [453, 1], "end": [455, 40], "traced_tactics": [{"tactic": "rw [rootMultiplicity_eq_multiplicity, dif_neg (X_sub_C_ne_zero x),\n  multiplicity.get_multiplicity_self]", "state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np q : R[X]\nx : R\n\u22a2 rootMultiplicity x (X - \u2191C x) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.coe_smul", "start": [831, 1], "end": [832, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Simple.lean", "full_name": "CategoryTheory.Biprod.isIso_inl_iff_isZero", "start": [197, 1], "end": [205, 22], "traced_tactics": [{"tactic": "rw [biprod.isIso_inl_iff_id_eq_fst_comp_inl, \u2190 biprod.total, add_right_eq_self]", "state_before": "C : Type u\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\n\u22a2 IsIso biprod.inl \u2194 IsZero Y", "state_after": "C : Type u\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\n\u22a2 biprod.snd \u226b biprod.inr = 0 \u2194 IsZero Y"}, {"tactic": "constructor", "state_before": "C : Type u\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\n\u22a2 biprod.snd \u226b biprod.inr = 0 \u2194 IsZero Y", "state_after": "case mp\nC : Type u\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\n\u22a2 biprod.snd \u226b biprod.inr = 0 \u2192 IsZero Y\n\ncase mpr\nC : Type u\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\n\u22a2 IsZero Y \u2192 biprod.snd \u226b biprod.inr = 0"}, {"tactic": "intro h", "state_before": "case mp\nC : Type u\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\n\u22a2 biprod.snd \u226b biprod.inr = 0 \u2192 IsZero Y", "state_after": "case mp\nC : Type u\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nh : biprod.snd \u226b biprod.inr = 0\n\u22a2 IsZero Y"}, {"tactic": "replace h := h =\u226b biprod.snd", "state_before": "case mp\nC : Type u\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nh : biprod.snd \u226b biprod.inr = 0\n\u22a2 IsZero Y", "state_after": "case mp\nC : Type u\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nh : (biprod.snd \u226b biprod.inr) \u226b biprod.snd = 0 \u226b biprod.snd\n\u22a2 IsZero Y"}, {"tactic": "simpa [\u2190 IsZero.iff_isSplitEpi_eq_zero (biprod.snd : X \u229e Y \u27f6 Y)] using h", "state_before": "case mp\nC : Type u\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nh : (biprod.snd \u226b biprod.inr) \u226b biprod.snd = 0 \u226b biprod.snd\n\u22a2 IsZero Y", "state_after": "no goals"}, {"tactic": "intro h", "state_before": "case mpr\nC : Type u\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\n\u22a2 IsZero Y \u2192 biprod.snd \u226b biprod.inr = 0", "state_after": "case mpr\nC : Type u\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nh : IsZero Y\n\u22a2 biprod.snd \u226b biprod.inr = 0"}, {"tactic": "rw [IsZero.iff_isSplitEpi_eq_zero (biprod.snd : X \u229e Y \u27f6 Y)] at h", "state_before": "case mpr\nC : Type u\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nh : IsZero Y\n\u22a2 biprod.snd \u226b biprod.inr = 0", "state_after": "case mpr\nC : Type u\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nh : biprod.snd = 0\n\u22a2 biprod.snd \u226b biprod.inr = 0"}, {"tactic": "rw [h, zero_comp]", "state_before": "case mpr\nC : Type u\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasBinaryBiproducts C\nX Y : C\nh : biprod.snd = 0\n\u22a2 biprod.snd \u226b biprod.inr = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "full_name": "differentiableWithinAt_inter", "start": [584, 1], "end": [586, 65], "traced_tactics": [{"tactic": "simp only [DifferentiableWithinAt, hasFDerivWithinAt_inter ht]", "state_before": "\ud835\udd5c : Type u_2\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type ?u.304437\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type ?u.304532\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nht : t \u2208 \ud835\udcdd x\n\u22a2 DifferentiableWithinAt \ud835\udd5c f (s \u2229 t) x \u2194 DifferentiableWithinAt \ud835\udd5c f s x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/StructuredArrow.lean", "full_name": "CategoryTheory.StructuredArrow.epi_of_epi_right", "start": [159, 1], "end": [160, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Ring/Prod.lean", "full_name": "RingHom.coe_snd", "start": [210, 1], "end": [211, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Array/Lemmas.lean", "full_name": "Array.swapAt!_def", "start": [130, 1], "end": [131, 67], "traced_tactics": [{"tactic": "simp [swapAt!, h]", "state_before": "\u03b1 : Type u_1\na : Array \u03b1\ni : Nat\nv : \u03b1\nh : i < size a\n\u22a2 swapAt! a i v = (a[i], set a { val := i, isLt := h } v)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Congruence.lean", "full_name": "Con.induction_on\u2082", "start": [349, 11], "end": [351, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "full_name": "zero_eq_inv", "start": [403, 1], "end": [404, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.monomial_add_single", "start": [326, 1], "end": [327, 48], "traced_tactics": [{"tactic": "rw [X_pow_eq_monomial, monomial_mul, mul_one]", "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\n\u22a2 \u2191(monomial (s + Finsupp.single n e)) a = \u2191(monomial s) a * X n ^ e", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/OmegaCompletePartialOrder.lean", "full_name": "OmegaCompletePartialOrder.continuous_comp", "start": [287, 1], "end": [289, 32], "traced_tactics": [{"tactic": "dsimp [Continuous] at *", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b2\ninst\u271d : OmegaCompletePartialOrder \u03b3\nf : \u03b1 \u2192o \u03b2\ng : \u03b2 \u2192o \u03b3\nhfc : Continuous f\nhgc : Continuous g\n\u22a2 Continuous (OrderHom.comp g f)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b2\ninst\u271d : OmegaCompletePartialOrder \u03b3\nf : \u03b1 \u2192o \u03b2\ng : \u03b2 \u2192o \u03b3\nhfc : \u2200 (c : Chain \u03b1), \u2191f (\u03c9Sup c) = \u03c9Sup (map c f)\nhgc : \u2200 (c : Chain \u03b2), \u2191g (\u03c9Sup c) = \u03c9Sup (map c g)\n\u22a2 \u2200 (c : Chain \u03b1), \u2191g (\u2191f (\u03c9Sup c)) = \u03c9Sup (map c (OrderHom.comp g f))"}, {"tactic": "intro", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b2\ninst\u271d : OmegaCompletePartialOrder \u03b3\nf : \u03b1 \u2192o \u03b2\ng : \u03b2 \u2192o \u03b3\nhfc : \u2200 (c : Chain \u03b1), \u2191f (\u03c9Sup c) = \u03c9Sup (map c f)\nhgc : \u2200 (c : Chain \u03b2), \u2191g (\u03c9Sup c) = \u03c9Sup (map c g)\n\u22a2 \u2200 (c : Chain \u03b1), \u2191g (\u2191f (\u03c9Sup c)) = \u03c9Sup (map c (OrderHom.comp g f))", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b2\ninst\u271d : OmegaCompletePartialOrder \u03b3\nf : \u03b1 \u2192o \u03b2\ng : \u03b2 \u2192o \u03b3\nhfc : \u2200 (c : Chain \u03b1), \u2191f (\u03c9Sup c) = \u03c9Sup (map c f)\nhgc : \u2200 (c : Chain \u03b2), \u2191g (\u03c9Sup c) = \u03c9Sup (map c g)\nc\u271d : Chain \u03b1\n\u22a2 \u2191g (\u2191f (\u03c9Sup c\u271d)) = \u03c9Sup (map c\u271d (OrderHom.comp g f))"}, {"tactic": "rw [hfc, hgc, Chain.map_comp]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\ninst\u271d\u00b2 : OmegaCompletePartialOrder \u03b1\ninst\u271d\u00b9 : OmegaCompletePartialOrder \u03b2\ninst\u271d : OmegaCompletePartialOrder \u03b3\nf : \u03b1 \u2192o \u03b2\ng : \u03b2 \u2192o \u03b3\nhfc : \u2200 (c : Chain \u03b1), \u2191f (\u03c9Sup c) = \u03c9Sup (map c f)\nhgc : \u2200 (c : Chain \u03b2), \u2191g (\u03c9Sup c) = \u03c9Sup (map c g)\nc\u271d : Chain \u03b1\n\u22a2 \u2191g (\u2191f (\u03c9Sup c\u271d)) = \u03c9Sup (map c\u271d (OrderHom.comp g f))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/AffineIsometry.lean", "full_name": "AffineIsometry.map_vsub", "start": [140, 1], "end": [141, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "Finset.nullMeasurableSet_biInter", "start": [395, 1], "end": [397, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subsemigroup/Operations.lean", "full_name": "AddSubsemigroup.toSubsemigroup_closure", "start": [151, 1], "end": [157, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Control/Fold.lean", "full_name": "Traversable.foldl_map", "start": [376, 1], "end": [378, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Complex/CauchyIntegral.lean", "full_name": "DifferentiableOn.analyticAt", "start": [597, 11], "end": [601, 61], "traced_tactics": [{"tactic": "rcases nhds_basis_closedBall.mem_iff.1 hz with \u27e8R, hR0, hRs\u27e9", "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\ns : Set \u2102\nf : \u2102 \u2192 E\nz : \u2102\nhd : DifferentiableOn \u2102 f s\nhz : s \u2208 \ud835\udcdd z\n\u22a2 AnalyticAt \u2102 f z", "state_after": "case intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\ns : Set \u2102\nf : \u2102 \u2192 E\nz : \u2102\nhd : DifferentiableOn \u2102 f s\nhz : s \u2208 \ud835\udcdd z\nR : \u211d\nhR0 : 0 < R\nhRs : closedBall z R \u2286 s\n\u22a2 AnalyticAt \u2102 f z"}, {"tactic": "lift R to \u211d\u22650 using hR0.le", "state_before": "case intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\ns : Set \u2102\nf : \u2102 \u2192 E\nz : \u2102\nhd : DifferentiableOn \u2102 f s\nhz : s \u2208 \ud835\udcdd z\nR : \u211d\nhR0 : 0 < R\nhRs : closedBall z R \u2286 s\n\u22a2 AnalyticAt \u2102 f z", "state_after": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\ns : Set \u2102\nf : \u2102 \u2192 E\nz : \u2102\nhd : DifferentiableOn \u2102 f s\nhz : s \u2208 \ud835\udcdd z\nR : \u211d\u22650\nhR0 : 0 < \u2191R\nhRs : closedBall z \u2191R \u2286 s\n\u22a2 AnalyticAt \u2102 f z"}, {"tactic": "exact ((hd.mono hRs).hasFPowerSeriesOnBall hR0).analyticAt", "state_before": "case intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\ns : Set \u2102\nf : \u2102 \u2192 E\nz : \u2102\nhd : DifferentiableOn \u2102 f s\nhz : s \u2208 \ud835\udcdd z\nR : \u211d\u22650\nhR0 : 0 < \u2191R\nhRs : closedBall z \u2191R \u2286 s\n\u22a2 AnalyticAt \u2102 f z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean", "full_name": "CategoryTheory.Limits.coprod.inr_map", "start": [848, 1], "end": [850, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/MvPolynomial/Tower.lean", "full_name": "Subalgebra.mvPolynomial_aeval_coe", "start": [88, 1], "end": [89, 88], "traced_tactics": [{"tactic": "convert aeval_algebraMap_apply A x p", "state_before": "R : Type u_1\nA : Type u_2\nB : Type ?u.261678\n\u03c3 : Type u_3\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\nS : Subalgebra R A\nx : \u03c3 \u2192 { x // x \u2208 S }\np : MvPolynomial \u03c3 R\n\u22a2 \u2191(aeval fun i => \u2191(x i)) p = \u2191(\u2191(aeval x) p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Group/Prod.lean", "full_name": "MeasureTheory.absolutelyContinuous_map_mul_right", "start": [230, 1], "end": [232, 79], "traced_tactics": [{"tactic": "refine' AbsolutelyContinuous.mk fun s hs => _", "state_before": "G : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ng : G\n\u22a2 \u03bc \u226a map (fun x => x * g) \u03bc", "state_after": "G : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ng : G\ns : Set G\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(map (fun x => x * g) \u03bc) s = 0 \u2192 \u2191\u2191\u03bc s = 0"}, {"tactic": "rw [map_apply (measurable_mul_const g) hs, measure_mul_right_null]", "state_before": "G : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ng : G\ns : Set G\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(map (fun x => x * g) \u03bc) s = 0 \u2192 \u2191\u2191\u03bc s = 0", "state_after": "G : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ng : G\ns : Set G\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s = 0 \u2192 \u2191\u2191\u03bc s = 0"}, {"tactic": "exact id", "state_before": "G : Type u_1\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : SigmaFinite \u03bd\ninst\u271d\u00b2 : SigmaFinite \u03bc\ns\u271d : Set G\ninst\u271d\u00b9 : MeasurableInv G\ninst\u271d : IsMulLeftInvariant \u03bc\ng : G\ns : Set G\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u03bc s = 0 \u2192 \u2191\u2191\u03bc s = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "full_name": "csInf_pair", "start": [667, 1], "end": [668, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Analytic/Basic.lean", "full_name": "FormalMultilinearSeries.hasFPowerSeriesOnBall", "start": [892, 11], "end": [899, 26], "traced_tactics": [{"tactic": "rw [zero_add]", "state_before": "\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_1\nG : Type ?u.1057089\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np\u271d pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx : E\nr r' : \u211d\u22650\u221e\ninst\u271d : CompleteSpace F\np : FormalMultilinearSeries \ud835\udd5c E F\nh : 0 < radius p\ny\u271d : E\nhy : y\u271d \u2208 EMetric.ball 0 (radius p)\n\u22a2 HasSum (fun n => \u2191(p n) fun x => y\u271d) (FormalMultilinearSeries.sum p (0 + y\u271d))", "state_after": "\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_1\nG : Type ?u.1057089\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np\u271d pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx : E\nr r' : \u211d\u22650\u221e\ninst\u271d : CompleteSpace F\np : FormalMultilinearSeries \ud835\udd5c E F\nh : 0 < radius p\ny\u271d : E\nhy : y\u271d \u2208 EMetric.ball 0 (radius p)\n\u22a2 HasSum (fun n => \u2191(p n) fun x => y\u271d) (FormalMultilinearSeries.sum p y\u271d)"}, {"tactic": "exact p.hasSum hy", "state_before": "\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_1\nG : Type ?u.1057089\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np\u271d pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx : E\nr r' : \u211d\u22650\u221e\ninst\u271d : CompleteSpace F\np : FormalMultilinearSeries \ud835\udd5c E F\nh : 0 < radius p\ny\u271d : E\nhy : y\u271d \u2208 EMetric.ball 0 (radius p)\n\u22a2 HasSum (fun n => \u2191(p n) fun x => y\u271d) (FormalMultilinearSeries.sum p y\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/EraseLead.lean", "full_name": "Polynomial.card_support_eq_one", "start": [354, 1], "end": [360, 50], "traced_tactics": [{"tactic": "refine' \u27e8fun h => _, _\u27e9", "state_before": "R : Type u_1\ninst\u271d : Semiring R\nf : R[X]\n\u22a2 card (support f) = 1 \u2194 \u2203 k x hx, f = \u2191C x * X ^ k", "state_after": "case refine'_1\nR : Type u_1\ninst\u271d : Semiring R\nf : R[X]\nh : card (support f) = 1\n\u22a2 \u2203 k x hx, f = \u2191C x * X ^ k\n\ncase refine'_2\nR : Type u_1\ninst\u271d : Semiring R\nf : R[X]\n\u22a2 (\u2203 k x hx, f = \u2191C x * X ^ k) \u2192 card (support f) = 1"}, {"tactic": "obtain \u27e8k, x, _, hx, rfl\u27e9 := card_support_eq.mp h", "state_before": "case refine'_1\nR : Type u_1\ninst\u271d : Semiring R\nf : R[X]\nh : card (support f) = 1\n\u22a2 \u2203 k x hx, f = \u2191C x * X ^ k", "state_after": "case refine'_1.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : Semiring R\nk : Fin 1 \u2192 \u2115\nx : Fin 1 \u2192 R\nw\u271d : StrictMono k\nhx : \u2200 (i : Fin 1), x i \u2260 0\nh : card (support (\u2211 i : Fin 1, \u2191C (x i) * X ^ k i)) = 1\n\u22a2 \u2203 k_1 x_1 hx, \u2211 i : Fin 1, \u2191C (x i) * X ^ k i = \u2191C x_1 * X ^ k_1"}, {"tactic": "exact \u27e8k 0, x 0, hx 0, Fin.sum_univ_one _\u27e9", "state_before": "case refine'_1.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : Semiring R\nk : Fin 1 \u2192 \u2115\nx : Fin 1 \u2192 R\nw\u271d : StrictMono k\nhx : \u2200 (i : Fin 1), x i \u2260 0\nh : card (support (\u2211 i : Fin 1, \u2191C (x i) * X ^ k i)) = 1\n\u22a2 \u2203 k_1 x_1 hx, \u2211 i : Fin 1, \u2191C (x i) * X ^ k i = \u2191C x_1 * X ^ k_1", "state_after": "no goals"}, {"tactic": "rintro \u27e8k, x, hx, rfl\u27e9", "state_before": "case refine'_2\nR : Type u_1\ninst\u271d : Semiring R\nf : R[X]\n\u22a2 (\u2203 k x hx, f = \u2191C x * X ^ k) \u2192 card (support f) = 1", "state_after": "case refine'_2.intro.intro.intro\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nx : R\nhx : x \u2260 0\n\u22a2 card (support (\u2191C x * X ^ k)) = 1"}, {"tactic": "rw [support_C_mul_X_pow k hx, card_singleton]", "state_before": "case refine'_2.intro.intro.intro\nR : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nx : R\nhx : x \u2260 0\n\u22a2 card (support (\u2191C x * X ^ k)) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Control/Traversable/Basic.lean", "full_name": "ApplicativeTransformation.comp_id", "start": [203, 1], "end": [204, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Type.lean", "full_name": "Equiv.Perm.isCycle_of_prime_order'", "start": [356, 1], "end": [358, 87], "traced_tactics": [{"tactic": "classical exact isCycle_of_prime_order h1 (lt_of_le_of_lt \u03c3.support.card_le_univ h2)", "state_before": "\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : Nat.Prime (orderOf \u03c3)\nh2 : Fintype.card \u03b1 < 2 * orderOf \u03c3\n\u22a2 IsCycle \u03c3", "state_after": "no goals"}, {"tactic": "exact isCycle_of_prime_order h1 (lt_of_le_of_lt \u03c3.support.card_le_univ h2)", "state_before": "\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : Nat.Prime (orderOf \u03c3)\nh2 : Fintype.card \u03b1 < 2 * orderOf \u03c3\n\u22a2 IsCycle \u03c3", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Stream/Init.lean", "full_name": "Stream'.interleave_even_odd", "start": [494, 1], "end": [500, 8], "traced_tactics": [{"tactic": "rw [h]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s' s : Stream' \u03b1\nh : s' = even s \u22c8 odd s\n\u22a2 head s' = head s \u2227 (fun s' s => s' = even s \u22c8 odd s) (tail s') (tail s)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s' s : Stream' \u03b1\nh : s' = even s \u22c8 odd s\n\u22a2 head (even s \u22c8 odd s) = head s \u2227 (fun s' s => s' = even s \u22c8 odd s) (tail (even s \u22c8 odd s)) (tail s)"}, {"tactic": "constructor", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s' s : Stream' \u03b1\nh : s' = even s \u22c8 odd s\n\u22a2 head (even s \u22c8 odd s) = head s \u2227 (fun s' s => s' = even s \u22c8 odd s) (tail (even s \u22c8 odd s)) (tail s)", "state_after": "case left\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s' s : Stream' \u03b1\nh : s' = even s \u22c8 odd s\n\u22a2 head (even s \u22c8 odd s) = head s\n\ncase right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s' s : Stream' \u03b1\nh : s' = even s \u22c8 odd s\n\u22a2 (fun s' s => s' = even s \u22c8 odd s) (tail (even s \u22c8 odd s)) (tail s)"}, {"tactic": "rfl", "state_before": "case left\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s' s : Stream' \u03b1\nh : s' = even s \u22c8 odd s\n\u22a2 head (even s \u22c8 odd s) = head s", "state_after": "no goals"}, {"tactic": "simp [odd_eq, odd_eq, tail_interleave, tail_even]", "state_before": "case right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\ns\u2081 s' s : Stream' \u03b1\nh : s' = even s \u22c8 odd s\n\u22a2 (fun s' s => s' = even s \u22c8 odd s) (tail (even s \u22c8 odd s)) (tail s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.snd", "start": [354, 1], "end": [362, 42], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b1\u271d : Type ?u.42117\n\u03c3 : Type ?u.42120\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03c3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03b2\nn : \u2115\n\u22a2 Nat.unpaired\n      (fun z n =>\n        Nat.casesOn n 0 fun y =>\n          Nat.unpaired (fun z n => Nat.casesOn n 0 fun y => Nat.succ (Nat.unpair (Nat.pair z y)).snd)\n            (Nat.pair (Nat.unpair (Nat.pair z y)).snd (encode (decode (Nat.unpair (Nat.pair z y)).fst))))\n      (Nat.pair (Nat.unpair n).snd (encode (decode (Nat.unpair n).fst))) =\n    encode (Option.map Prod.snd (decode n))", "state_after": "\u03b1\u271d : Type ?u.42117\n\u03c3 : Type ?u.42120\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03c3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03b2\nn : \u2115\n\u22a2 Nat.rec 0 (fun n_1 n_ih => Nat.rec 0 (fun n n_ih => Nat.succ n) (encode (decode (Nat.unpair n).snd)))\n      (encode (decode (Nat.unpair n).fst)) =\n    encode\n      (Option.map Prod.snd\n        (Option.bind (decode (Nat.unpair n).fst) fun a => Option.map (Prod.mk a) (decode (Nat.unpair n).snd)))"}, {"tactic": "cases @decode \u03b1 _ n.unpair.1 <;> simp", "state_before": "\u03b1\u271d : Type ?u.42117\n\u03c3 : Type ?u.42120\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03c3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03b2\nn : \u2115\n\u22a2 Nat.rec 0 (fun n_1 n_ih => Nat.rec 0 (fun n n_ih => Nat.succ n) (encode (decode (Nat.unpair n).snd)))\n      (encode (decode (Nat.unpair n).fst)) =\n    encode\n      (Option.map Prod.snd\n        (Option.bind (decode (Nat.unpair n).fst) fun a => Option.map (Prod.mk a) (decode (Nat.unpair n).snd)))", "state_after": "case some\n\u03b1\u271d : Type ?u.42117\n\u03c3 : Type ?u.42120\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03c3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03b2\nn : \u2115\nval\u271d : \u03b1\n\u22a2 Nat.rec 0 (fun n n_ih => Nat.succ n) (encode (decode (Nat.unpair n).snd)) = encode (decode (Nat.unpair n).snd)"}, {"tactic": "cases @decode \u03b2 _ n.unpair.2 <;> simp", "state_before": "case some\n\u03b1\u271d : Type ?u.42117\n\u03c3 : Type ?u.42120\ninst\u271d\u00b3 : Primcodable \u03b1\u271d\ninst\u271d\u00b2 : Primcodable \u03c3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03b2\nn : \u2115\nval\u271d : \u03b1\n\u22a2 Nat.rec 0 (fun n n_ih => Nat.succ n) (encode (decode (Nat.unpair n).snd)) = encode (decode (Nat.unpair n).snd)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/Sqrt.lean", "full_name": "Real.sqrt_zero", "start": [254, 1], "end": [254, 49], "traced_tactics": [{"tactic": "simp [sqrt]", "state_before": "x y : \u211d\n\u22a2 sqrt 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.pairwise_aedisjoint_of_aedisjoint_forall_ne_one", "start": [2602, 1], "end": [2616, 63], "traced_tactics": [{"tactic": "intro g\u2081 g\u2082 hg", "state_before": "\u03b1\u271d : Type ?u.529346\n\u03b2 : Type ?u.529349\n\u03b3 : Type ?u.529352\n\u03b4 : Type ?u.529355\n\u03b9 : Type ?u.529358\nR : Type ?u.529361\nR' : Type ?u.529364\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\n\u22a2 Pairwise (AEDisjoint \u03bc on fun g => g \u2022 s)", "state_after": "\u03b1\u271d : Type ?u.529346\n\u03b2 : Type ?u.529349\n\u03b3 : Type ?u.529352\n\u03b4 : Type ?u.529355\n\u03b9 : Type ?u.529358\nR : Type ?u.529361\nR' : Type ?u.529364\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\ng\u2081 g\u2082 : G\nhg : g\u2081 \u2260 g\u2082\n\u22a2 (AEDisjoint \u03bc on fun g => g \u2022 s) g\u2081 g\u2082"}, {"tactic": "let g := g\u2082\u207b\u00b9 * g\u2081", "state_before": "\u03b1\u271d : Type ?u.529346\n\u03b2 : Type ?u.529349\n\u03b3 : Type ?u.529352\n\u03b4 : Type ?u.529355\n\u03b9 : Type ?u.529358\nR : Type ?u.529361\nR' : Type ?u.529364\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\ng\u2081 g\u2082 : G\nhg : g\u2081 \u2260 g\u2082\n\u22a2 (AEDisjoint \u03bc on fun g => g \u2022 s) g\u2081 g\u2082", "state_after": "\u03b1\u271d : Type ?u.529346\n\u03b2 : Type ?u.529349\n\u03b3 : Type ?u.529352\n\u03b4 : Type ?u.529355\n\u03b9 : Type ?u.529358\nR : Type ?u.529361\nR' : Type ?u.529364\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\ng\u2081 g\u2082 : G\nhg : g\u2081 \u2260 g\u2082\ng : G := g\u2082\u207b\u00b9 * g\u2081\n\u22a2 (AEDisjoint \u03bc on fun g => g \u2022 s) g\u2081 g\u2082"}, {"tactic": "replace hg : g \u2260 1", "state_before": "\u03b1\u271d : Type ?u.529346\n\u03b2 : Type ?u.529349\n\u03b3 : Type ?u.529352\n\u03b4 : Type ?u.529355\n\u03b9 : Type ?u.529358\nR : Type ?u.529361\nR' : Type ?u.529364\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\ng\u2081 g\u2082 : G\nhg : g\u2081 \u2260 g\u2082\ng : G := g\u2082\u207b\u00b9 * g\u2081\n\u22a2 (AEDisjoint \u03bc on fun g => g \u2022 s) g\u2081 g\u2082", "state_after": "case hg\n\u03b1\u271d : Type ?u.529346\n\u03b2 : Type ?u.529349\n\u03b3 : Type ?u.529352\n\u03b4 : Type ?u.529355\n\u03b9 : Type ?u.529358\nR : Type ?u.529361\nR' : Type ?u.529364\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\ng\u2081 g\u2082 : G\nhg : g\u2081 \u2260 g\u2082\ng : G := g\u2082\u207b\u00b9 * g\u2081\n\u22a2 g \u2260 1\n\n\u03b1\u271d : Type ?u.529346\n\u03b2 : Type ?u.529349\n\u03b3 : Type ?u.529352\n\u03b4 : Type ?u.529355\n\u03b9 : Type ?u.529358\nR : Type ?u.529361\nR' : Type ?u.529364\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\ng\u2081 g\u2082 : G\ng : G := g\u2082\u207b\u00b9 * g\u2081\nhg : g \u2260 1\n\u22a2 (AEDisjoint \u03bc on fun g => g \u2022 s) g\u2081 g\u2082"}, {"tactic": "change \u03bc (g\u2081 \u2022 s \u2229 g\u2082 \u2022 s) = 0", "state_before": "\u03b1\u271d : Type ?u.529346\n\u03b2 : Type ?u.529349\n\u03b3 : Type ?u.529352\n\u03b4 : Type ?u.529355\n\u03b9 : Type ?u.529358\nR : Type ?u.529361\nR' : Type ?u.529364\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\ng\u2081 g\u2082 : G\ng : G := g\u2082\u207b\u00b9 * g\u2081\nhg : g \u2260 1\nthis : (fun x x_1 => x \u2022 x_1) g\u2082\u207b\u00b9 \u207b\u00b9' (g \u2022 s \u2229 s) = g\u2081 \u2022 s \u2229 g\u2082 \u2022 s\n\u22a2 (AEDisjoint \u03bc on fun g => g \u2022 s) g\u2081 g\u2082", "state_after": "\u03b1\u271d : Type ?u.529346\n\u03b2 : Type ?u.529349\n\u03b3 : Type ?u.529352\n\u03b4 : Type ?u.529355\n\u03b9 : Type ?u.529358\nR : Type ?u.529361\nR' : Type ?u.529364\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\ng\u2081 g\u2082 : G\ng : G := g\u2082\u207b\u00b9 * g\u2081\nhg : g \u2260 1\nthis : (fun x x_1 => x \u2022 x_1) g\u2082\u207b\u00b9 \u207b\u00b9' (g \u2022 s \u2229 s) = g\u2081 \u2022 s \u2229 g\u2082 \u2022 s\n\u22a2 \u2191\u2191\u03bc (g\u2081 \u2022 s \u2229 g\u2082 \u2022 s) = 0"}, {"tactic": "exact this \u25b8 (h_qmp g\u2082\u207b\u00b9).preimage_null (h_ae_disjoint g hg)", "state_before": "\u03b1\u271d : Type ?u.529346\n\u03b2 : Type ?u.529349\n\u03b3 : Type ?u.529352\n\u03b4 : Type ?u.529355\n\u03b9 : Type ?u.529358\nR : Type ?u.529361\nR' : Type ?u.529364\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\ng\u2081 g\u2082 : G\ng : G := g\u2082\u207b\u00b9 * g\u2081\nhg : g \u2260 1\nthis : (fun x x_1 => x \u2022 x_1) g\u2082\u207b\u00b9 \u207b\u00b9' (g \u2022 s \u2229 s) = g\u2081 \u2022 s \u2229 g\u2082 \u2022 s\n\u22a2 \u2191\u2191\u03bc (g\u2081 \u2022 s \u2229 g\u2082 \u2022 s) = 0", "state_after": "no goals"}, {"tactic": "rw [Ne.def, inv_mul_eq_one]", "state_before": "case hg\n\u03b1\u271d : Type ?u.529346\n\u03b2 : Type ?u.529349\n\u03b3 : Type ?u.529352\n\u03b4 : Type ?u.529355\n\u03b9 : Type ?u.529358\nR : Type ?u.529361\nR' : Type ?u.529364\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\ng\u2081 g\u2082 : G\nhg : g\u2081 \u2260 g\u2082\ng : G := g\u2082\u207b\u00b9 * g\u2081\n\u22a2 g \u2260 1", "state_after": "case hg\n\u03b1\u271d : Type ?u.529346\n\u03b2 : Type ?u.529349\n\u03b3 : Type ?u.529352\n\u03b4 : Type ?u.529355\n\u03b9 : Type ?u.529358\nR : Type ?u.529361\nR' : Type ?u.529364\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\ng\u2081 g\u2082 : G\nhg : g\u2081 \u2260 g\u2082\ng : G := g\u2082\u207b\u00b9 * g\u2081\n\u22a2 \u00acg\u2082 = g\u2081"}, {"tactic": "exact hg.symm", "state_before": "case hg\n\u03b1\u271d : Type ?u.529346\n\u03b2 : Type ?u.529349\n\u03b3 : Type ?u.529352\n\u03b4 : Type ?u.529355\n\u03b9 : Type ?u.529358\nR : Type ?u.529361\nR' : Type ?u.529364\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\ng\u2081 g\u2082 : G\nhg : g\u2081 \u2260 g\u2082\ng : G := g\u2082\u207b\u00b9 * g\u2081\n\u22a2 \u00acg\u2082 = g\u2081", "state_after": "no goals"}, {"tactic": "rw [preimage_eq_iff_eq_image (MulAction.bijective g\u2082\u207b\u00b9), image_smul, smul_set_inter, smul_smul,\n  smul_smul, inv_mul_self, one_smul]", "state_before": "\u03b1\u271d : Type ?u.529346\n\u03b2 : Type ?u.529349\n\u03b3 : Type ?u.529352\n\u03b4 : Type ?u.529355\n\u03b9 : Type ?u.529358\nR : Type ?u.529361\nR' : Type ?u.529364\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\nG : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nh_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s\nh_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g)\ng\u2081 g\u2082 : G\ng : G := g\u2082\u207b\u00b9 * g\u2081\nhg : g \u2260 1\n\u22a2 (fun x x_1 => x \u2022 x_1) g\u2082\u207b\u00b9 \u207b\u00b9' (g \u2022 s \u2229 s) = g\u2081 \u2022 s \u2229 g\u2082 \u2022 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/MonoidLocalization.lean", "full_name": "Submonoid.LocalizationMap.sec_zero_fst", "start": [1854, 1], "end": [1855, 43], "traced_tactics": [{"tactic": "rw [LocalizationMap.sec_spec', mul_zero]", "state_before": "M : Type u_1\ninst\u271d\u00b2 : CommMonoidWithZero M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoidWithZero N\nP : Type ?u.3335552\ninst\u271d : CommMonoidWithZero P\nf : LocalizationMap S N\n\u22a2 \u2191(toMap f) (sec f 0).fst = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Hom/Group.lean", "full_name": "MonoidHom.inv_comp", "start": [1660, 1], "end": [1663, 55], "traced_tactics": [{"tactic": "ext", "state_before": "\u03b1 : Type ?u.240534\n\u03b2 : Type ?u.240537\nM\u271d : Type ?u.240540\nN\u271d : Type ?u.240543\nP : Type ?u.240546\nG : Type ?u.240549\nH : Type ?u.240552\nF : Type ?u.240555\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : CommGroup H\ninst\u271d\u00b3 : MulOneClass M\u271d\nM : Type u_1\nN : Type u_2\nA : Type u_3\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : CommGroup A\n\u03c6 : N \u2192* A\n\u03c8 : M \u2192* N\n\u22a2 comp \u03c6\u207b\u00b9 \u03c8 = (comp \u03c6 \u03c8)\u207b\u00b9", "state_after": "case h\n\u03b1 : Type ?u.240534\n\u03b2 : Type ?u.240537\nM\u271d : Type ?u.240540\nN\u271d : Type ?u.240543\nP : Type ?u.240546\nG : Type ?u.240549\nH : Type ?u.240552\nF : Type ?u.240555\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : CommGroup H\ninst\u271d\u00b3 : MulOneClass M\u271d\nM : Type u_1\nN : Type u_2\nA : Type u_3\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : CommGroup A\n\u03c6 : N \u2192* A\n\u03c8 : M \u2192* N\nx\u271d : M\n\u22a2 \u2191(comp \u03c6\u207b\u00b9 \u03c8) x\u271d = \u2191(comp \u03c6 \u03c8)\u207b\u00b9 x\u271d"}, {"tactic": "simp only [Function.comp_apply, inv_apply, coe_comp]", "state_before": "case h\n\u03b1 : Type ?u.240534\n\u03b2 : Type ?u.240537\nM\u271d : Type ?u.240540\nN\u271d : Type ?u.240543\nP : Type ?u.240546\nG : Type ?u.240549\nH : Type ?u.240552\nF : Type ?u.240555\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : CommGroup H\ninst\u271d\u00b3 : MulOneClass M\u271d\nM : Type u_1\nN : Type u_2\nA : Type u_3\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : MulOneClass N\ninst\u271d : CommGroup A\n\u03c6 : N \u2192* A\n\u03c8 : M \u2192* N\nx\u271d : M\n\u22a2 \u2191(comp \u03c6\u207b\u00b9 \u03c8) x\u271d = \u2191(comp \u03c6 \u03c8)\u207b\u00b9 x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Module/LocalizedModule.lean", "full_name": "IsLocalizedModule.lift_comp", "start": [853, 1], "end": [858, 26], "traced_tactics": [{"tactic": "dsimp only [IsLocalizedModule.lift]", "state_before": "R : Type u_1\ninst\u271d\u2077 : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type u_4\nM'' : Type u_3\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\nf : M \u2192\u2097[R] M'\ng\u271d : M \u2192\u2097[R] M''\ninst\u271d : IsLocalizedModule S f\ng : M \u2192\u2097[R] M''\nh : \u2200 (x : { x // x \u2208 S }), IsUnit (\u2191(algebraMap R (Module.End R M'')) \u2191x)\n\u22a2 LinearMap.comp (lift S f g h) f = g", "state_after": "R : Type u_1\ninst\u271d\u2077 : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type u_4\nM'' : Type u_3\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\nf : M \u2192\u2097[R] M'\ng\u271d : M \u2192\u2097[R] M''\ninst\u271d : IsLocalizedModule S f\ng : M \u2192\u2097[R] M''\nh : \u2200 (x : { x // x \u2208 S }), IsUnit (\u2191(algebraMap R (Module.End R M'')) \u2191x)\n\u22a2 LinearMap.comp (LinearMap.comp (LocalizedModule.lift S g h) \u2191(LinearEquiv.symm (iso S f))) f = g"}, {"tactic": "rw [LinearMap.comp_assoc]", "state_before": "R : Type u_1\ninst\u271d\u2077 : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type u_4\nM'' : Type u_3\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\nf : M \u2192\u2097[R] M'\ng\u271d : M \u2192\u2097[R] M''\ninst\u271d : IsLocalizedModule S f\ng : M \u2192\u2097[R] M''\nh : \u2200 (x : { x // x \u2208 S }), IsUnit (\u2191(algebraMap R (Module.End R M'')) \u2191x)\n\u22a2 LinearMap.comp (LinearMap.comp (LocalizedModule.lift S g h) \u2191(LinearEquiv.symm (iso S f))) f = g", "state_after": "R : Type u_1\ninst\u271d\u2077 : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type u_4\nM'' : Type u_3\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\nf : M \u2192\u2097[R] M'\ng\u271d : M \u2192\u2097[R] M''\ninst\u271d : IsLocalizedModule S f\ng : M \u2192\u2097[R] M''\nh : \u2200 (x : { x // x \u2208 S }), IsUnit (\u2191(algebraMap R (Module.End R M'')) \u2191x)\n\u22a2 LinearMap.comp (LocalizedModule.lift S g h) (LinearMap.comp (\u2191(LinearEquiv.symm (iso S f))) f) = g"}, {"tactic": "convert LocalizedModule.lift_comp S g h", "state_before": "R : Type u_1\ninst\u271d\u2077 : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type u_4\nM'' : Type u_3\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\nf : M \u2192\u2097[R] M'\ng\u271d : M \u2192\u2097[R] M''\ninst\u271d : IsLocalizedModule S f\ng : M \u2192\u2097[R] M''\nh : \u2200 (x : { x // x \u2208 S }), IsUnit (\u2191(algebraMap R (Module.End R M'')) \u2191x)\n\u22a2 LinearMap.comp (LocalizedModule.lift S g h) (LinearMap.comp (\u2191(LinearEquiv.symm (iso S f))) f) = g", "state_after": "case h.e'_2.h.e'_21\nR : Type u_1\ninst\u271d\u2077 : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type u_4\nM'' : Type u_3\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\nf : M \u2192\u2097[R] M'\ng\u271d : M \u2192\u2097[R] M''\ninst\u271d : IsLocalizedModule S f\ng : M \u2192\u2097[R] M''\nh : \u2200 (x : { x // x \u2208 S }), IsUnit (\u2191(algebraMap R (Module.End R M'')) \u2191x)\n\u22a2 LinearMap.comp (\u2191(LinearEquiv.symm (iso S f))) f = LocalizedModule.mkLinearMap S M"}, {"tactic": "exact iso_symm_comp _ _", "state_before": "case h.e'_2.h.e'_21\nR : Type u_1\ninst\u271d\u2077 : CommRing R\nS : Submonoid R\nM : Type u_2\nM' : Type u_4\nM'' : Type u_3\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\nf : M \u2192\u2097[R] M'\ng\u271d : M \u2192\u2097[R] M''\ninst\u271d : IsLocalizedModule S f\ng : M \u2192\u2097[R] M''\nh : \u2200 (x : { x // x \u2208 S }), IsUnit (\u2191(algebraMap R (Module.End R M'')) \u2191x)\n\u22a2 LinearMap.comp (\u2191(LinearEquiv.symm (iso S f))) f = LocalizedModule.mkLinearMap S M", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Pow.lean", "full_name": "Nat.one_lt_pow", "start": [72, 1], "end": [74, 36], "traced_tactics": [{"tactic": "rw [\u2190 one_pow n]", "state_before": "n m : \u2115\nh\u2080 : 0 < n\nh\u2081 : 1 < m\n\u22a2 1 < m ^ n", "state_after": "n m : \u2115\nh\u2080 : 0 < n\nh\u2081 : 1 < m\n\u22a2 1 ^ n < m ^ n"}, {"tactic": "exact pow_lt_pow_of_lt_left h\u2081 h\u2080", "state_before": "n m : \u2115\nh\u2080 : 0 < n\nh\u2081 : 1 < m\n\u22a2 1 ^ n < m ^ n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/CubicDiscriminant.lean", "full_name": "Cubic.zero", "start": [173, 1], "end": [174, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.ae_neBot", "start": [2683, 1], "end": [2684, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/OmegaCompletePartialOrder.lean", "full_name": "Part.mem_chain_of_mem_\u03c9Sup", "start": [344, 1], "end": [350, 23], "traced_tactics": [{"tactic": "simp [Part.\u03c9Sup] at h", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type ?u.14960\nc : Chain (Part \u03b1)\na : \u03b1\nh : a \u2208 Part.\u03c9Sup c\n\u22a2 some a \u2208 c", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type ?u.14960\nc : Chain (Part \u03b1)\na : \u03b1\nh : a \u2208 if h : \u2203 a, some a \u2208 c then some (choose h) else none\n\u22a2 some a \u2208 c"}, {"tactic": "split_ifs at h with h_1", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type ?u.14960\nc : Chain (Part \u03b1)\na : \u03b1\nh : a \u2208 if h : \u2203 a, some a \u2208 c then some (choose h) else none\n\u22a2 some a \u2208 c", "state_after": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type ?u.14960\nc : Chain (Part \u03b1)\na : \u03b1\nh_1 : \u2203 a, some a \u2208 c\nh : a \u2208 some (choose h_1)\n\u22a2 some a \u2208 c\n\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type ?u.14960\nc : Chain (Part \u03b1)\na : \u03b1\nh_1 : \u00ac\u2203 a, some a \u2208 c\nh : a \u2208 none\n\u22a2 some a \u2208 c"}, {"tactic": "have h' := Classical.choose_spec h_1", "state_before": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type ?u.14960\nc : Chain (Part \u03b1)\na : \u03b1\nh_1 : \u2203 a, some a \u2208 c\nh : a \u2208 some (choose h_1)\n\u22a2 some a \u2208 c", "state_after": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type ?u.14960\nc : Chain (Part \u03b1)\na : \u03b1\nh_1 : \u2203 a, some a \u2208 c\nh : a \u2208 some (choose h_1)\nh' : some (choose h_1) \u2208 c\n\u22a2 some a \u2208 c"}, {"tactic": "rw [\u2190 eq_some_iff] at h", "state_before": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type ?u.14960\nc : Chain (Part \u03b1)\na : \u03b1\nh_1 : \u2203 a, some a \u2208 c\nh : a \u2208 some (choose h_1)\nh' : some (choose h_1) \u2208 c\n\u22a2 some a \u2208 c", "state_after": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type ?u.14960\nc : Chain (Part \u03b1)\na : \u03b1\nh_1 : \u2203 a, some a \u2208 c\nh : some (choose h_1) = some a\nh' : some (choose h_1) \u2208 c\n\u22a2 some a \u2208 c"}, {"tactic": "rw [\u2190 h]", "state_before": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type ?u.14960\nc : Chain (Part \u03b1)\na : \u03b1\nh_1 : \u2203 a, some a \u2208 c\nh : some (choose h_1) = some a\nh' : some (choose h_1) \u2208 c\n\u22a2 some a \u2208 c", "state_after": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type ?u.14960\nc : Chain (Part \u03b1)\na : \u03b1\nh_1 : \u2203 a, some a \u2208 c\nh : some (choose h_1) = some a\nh' : some (choose h_1) \u2208 c\n\u22a2 some (choose h_1) \u2208 c"}, {"tactic": "exact h'", "state_before": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type ?u.14960\nc : Chain (Part \u03b1)\na : \u03b1\nh_1 : \u2203 a, some a \u2208 c\nh : some (choose h_1) = some a\nh' : some (choose h_1) \u2208 c\n\u22a2 some (choose h_1) \u2208 c", "state_after": "no goals"}, {"tactic": "rcases h with \u27e8\u27e8\u27e9\u27e9", "state_before": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type ?u.14960\nc : Chain (Part \u03b1)\na : \u03b1\nh_1 : \u00ac\u2203 a, some a \u2208 c\nh : a \u2208 none\n\u22a2 some a \u2208 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fin/Tuple/Basic.lean", "full_name": "Fin.append_elim0'", "start": [317, 1], "end": [319, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean", "full_name": "Real.cos_arctan", "start": [152, 1], "end": [153, 82], "traced_tactics": [{"tactic": "rw_mod_cast [one_div, \u2190 inv_sqrt_one_add_tan_sq (cos_arctan_pos x), tan_arctan]", "state_before": "x : \u211d\n\u22a2 cos (arctan x) = \u21911 / sqrt (\u21911 + x ^ 2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "full_name": "div_eq_iff_mul_eq", "start": [99, 1], "end": [100, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Function.LeftInverse.rightInvOn_range", "start": [1611, 1], "end": [1613, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Pi/Algebra.lean", "full_name": "Sum.elim_mul_mul", "start": [474, 1], "end": [476, 18], "traced_tactics": [{"tactic": "ext x", "state_before": "I : Type u\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_1\nf : I \u2192 Type v\u2081\ng : I \u2192 Type v\u2082\nh : I \u2192 Type v\u2083\nx y : (i : I) \u2192 f i\ni : I\na a' : \u03b1 \u2192 \u03b3\nb b' : \u03b2 \u2192 \u03b3\ninst\u271d : Mul \u03b3\n\u22a2 Sum.elim (a * a') (b * b') = Sum.elim a b * Sum.elim a' b'", "state_after": "case h\nI : Type u\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_1\nf : I \u2192 Type v\u2081\ng : I \u2192 Type v\u2082\nh : I \u2192 Type v\u2083\nx\u271d y : (i : I) \u2192 f i\ni : I\na a' : \u03b1 \u2192 \u03b3\nb b' : \u03b2 \u2192 \u03b3\ninst\u271d : Mul \u03b3\nx : \u03b1 \u2295 \u03b2\n\u22a2 Sum.elim (a * a') (b * b') x = (Sum.elim a b * Sum.elim a' b') x"}, {"tactic": "cases x <;> rfl", "state_before": "case h\nI : Type u\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_1\nf : I \u2192 Type v\u2081\ng : I \u2192 Type v\u2082\nh : I \u2192 Type v\u2083\nx\u271d y : (i : I) \u2192 f i\ni : I\na a' : \u03b1 \u2192 \u03b3\nb b' : \u03b2 \u2192 \u03b3\ninst\u271d : Mul \u03b3\nx : \u03b1 \u2295 \u03b2\n\u22a2 Sum.elim (a * a') (b * b') x = (Sum.elim a b * Sum.elim a' b') x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean", "full_name": "InnerProductGeometry.angle_self", "start": [134, 1], "end": [137, 21], "traced_tactics": [{"tactic": "unfold angle", "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y x : V\nhx : x \u2260 0\n\u22a2 angle x x = 0", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y x : V\nhx : x \u2260 0\n\u22a2 arccos (inner x x / (\u2016x\u2016 * \u2016x\u2016)) = 0"}, {"tactic": "rw [\u2190 real_inner_self_eq_norm_mul_norm, div_self (inner_self_ne_zero.2 hx : \u27eax, x\u27eb \u2260 0),\n  Real.arccos_one]", "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y x : V\nhx : x \u2260 0\n\u22a2 arccos (inner x x / (\u2016x\u2016 * \u2016x\u2016)) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Card.lean", "full_name": "Finset.card_lt_card", "start": [288, 1], "end": [289, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "Filter.Tendsto.min_left", "start": [740, 1], "end": [742, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "full_name": "LinearIsometryEquiv.refl_mul", "start": [918, 1], "end": [919, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Sites/Sheafification.lean", "full_name": "CategoryTheory.Sheaf.Hom.mono_iff_presheaf_mono", "start": [660, 1], "end": [661, 88], "traced_tactics": [{"tactic": "infer_instance", "state_before": "C : Type u\ninst\u271d\u2077 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 :\n  \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nF G : Sheaf J D\nf : F \u27f6 G\nm : Mono f\n\u22a2 Mono f.val", "state_after": "no goals"}, {"tactic": "exact Sheaf.Hom.mono_of_presheaf_mono J D f", "state_before": "C : Type u\ninst\u271d\u2077 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2076 : Category D\ninst\u271d\u2075 : ConcreteCategory D\ninst\u271d\u2074 : PreservesLimits (forget D)\ninst\u271d\u00b3 :\n  \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : GrothendieckTopology.Cover J X), HasMultiequalizer (GrothendieckTopology.Cover.index S P)\ninst\u271d\u00b2 : \u2200 (X : C), HasColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 D\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfShape (GrothendieckTopology.Cover J X)\u1d52\u1d56 (forget D)\ninst\u271d : ReflectsIsomorphisms (forget D)\nF G : Sheaf J D\nf : F \u27f6 G\nm : Mono f.val\n\u22a2 Mono f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Bases.lean", "full_name": "Filter.HasBasis.comp_equiv", "start": [426, 1], "end": [427, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "full_name": "NNReal.inv_rpow", "start": [115, 1], "end": [116, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Order/Floor.lean", "full_name": "ContinuousOn.comp_fract'", "start": [196, 1], "end": [214, 82], "traced_tactics": [{"tactic": "change Continuous (uncurry f \u2218 Prod.map id fract)", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\n\u22a2 Continuous fun st => f st.fst (fract st.snd)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\n\u22a2 Continuous (uncurry f \u2218 Prod.map id fract)"}, {"tactic": "rw [continuous_iff_continuousAt]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\n\u22a2 Continuous (uncurry f \u2218 Prod.map id fract)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\n\u22a2 \u2200 (x : \u03b2 \u00d7 \u03b1), ContinuousAt (uncurry f \u2218 Prod.map id fract) x"}, {"tactic": "rintro \u27e8s, t\u27e9", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\n\u22a2 \u2200 (x : \u03b2 \u00d7 \u03b1), ContinuousAt (uncurry f \u2218 Prod.map id fract) x", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nt : \u03b1\n\u22a2 ContinuousAt (uncurry f \u2218 Prod.map id fract) (s, t)"}, {"tactic": "rcases em (\u2203 n : \u2124, t = n) with (\u27e8n, rfl\u27e9 | ht)", "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nt : \u03b1\n\u22a2 ContinuousAt (uncurry f \u2218 Prod.map id fract) (s, t)", "state_after": "case mk.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 ContinuousAt (uncurry f \u2218 Prod.map id fract) (s, \u2191n)\n\ncase mk.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nt : \u03b1\nht : \u00ac\u2203 n, t = \u2191n\n\u22a2 ContinuousAt (uncurry f \u2218 Prod.map id fract) (s, t)"}, {"tactic": "rw [ContinuousAt, nhds_prod_eq, \u2190 nhds_left'_sup_nhds_right (n : \u03b1), prod_sup, tendsto_sup]", "state_before": "case mk.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 ContinuousAt (uncurry f \u2218 Prod.map id fract) (s, \u2191n)", "state_after": "case mk.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 Tendsto (uncurry f \u2218 Prod.map id fract) (\ud835\udcdd s \u00d7\u02e2 \ud835\udcdd[Iio \u2191n] \u2191n) (\ud835\udcdd ((uncurry f \u2218 Prod.map id fract) (s, \u2191n))) \u2227\n    Tendsto (uncurry f \u2218 Prod.map id fract) (\ud835\udcdd s \u00d7\u02e2 \ud835\udcdd[Ici \u2191n] \u2191n) (\ud835\udcdd ((uncurry f \u2218 Prod.map id fract) (s, \u2191n)))"}, {"tactic": "constructor", "state_before": "case mk.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 Tendsto (uncurry f \u2218 Prod.map id fract) (\ud835\udcdd s \u00d7\u02e2 \ud835\udcdd[Iio \u2191n] \u2191n) (\ud835\udcdd ((uncurry f \u2218 Prod.map id fract) (s, \u2191n))) \u2227\n    Tendsto (uncurry f \u2218 Prod.map id fract) (\ud835\udcdd s \u00d7\u02e2 \ud835\udcdd[Ici \u2191n] \u2191n) (\ud835\udcdd ((uncurry f \u2218 Prod.map id fract) (s, \u2191n)))", "state_after": "case mk.inl.intro.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 Tendsto (uncurry f \u2218 Prod.map id fract) (\ud835\udcdd s \u00d7\u02e2 \ud835\udcdd[Iio \u2191n] \u2191n) (\ud835\udcdd ((uncurry f \u2218 Prod.map id fract) (s, \u2191n)))\n\ncase mk.inl.intro.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 Tendsto (uncurry f \u2218 Prod.map id fract) (\ud835\udcdd s \u00d7\u02e2 \ud835\udcdd[Ici \u2191n] \u2191n) (\ud835\udcdd ((uncurry f \u2218 Prod.map id fract) (s, \u2191n)))"}, {"tactic": "refine (((h (s, 1) \u27e8trivial, zero_le_one, le_rfl\u27e9).tendsto.mono_left ?_).comp\n  (tendsto_id.prod_map (tendsto_fract_left _))).mono_right (le_of_eq ?_)", "state_before": "case mk.inl.intro.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 Tendsto (uncurry f \u2218 Prod.map id fract) (\ud835\udcdd s \u00d7\u02e2 \ud835\udcdd[Iio \u2191n] \u2191n) (\ud835\udcdd ((uncurry f \u2218 Prod.map id fract) (s, \u2191n)))", "state_after": "case mk.inl.intro.left.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 \ud835\udcdd s \u00d7\u02e2 \ud835\udcdd[Iio 1] 1 \u2264 \ud835\udcdd[univ \u00d7\u02e2 Icc 0 1] (s, 1)\n\ncase mk.inl.intro.left.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 \ud835\udcdd (uncurry f (s, 1)) = \ud835\udcdd ((uncurry f \u2218 Prod.map id fract) (s, \u2191n))"}, {"tactic": "rw [nhdsWithin_prod_eq, nhdsWithin_univ, \u2190 nhdsWithin_Ico_eq_nhdsWithin_Iio one_pos]", "state_before": "case mk.inl.intro.left.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 \ud835\udcdd s \u00d7\u02e2 \ud835\udcdd[Iio 1] 1 \u2264 \ud835\udcdd[univ \u00d7\u02e2 Icc 0 1] (s, 1)", "state_after": "case mk.inl.intro.left.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 \ud835\udcdd s \u00d7\u02e2 \ud835\udcdd[Ico 0 1] 1 \u2264 \ud835\udcdd s \u00d7\u02e2 \ud835\udcdd[Icc 0 1] 1"}, {"tactic": "exact Filter.prod_mono le_rfl (nhdsWithin_mono _ Ico_subset_Icc_self)", "state_before": "case mk.inl.intro.left.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 \ud835\udcdd s \u00d7\u02e2 \ud835\udcdd[Ico 0 1] 1 \u2264 \ud835\udcdd s \u00d7\u02e2 \ud835\udcdd[Icc 0 1] 1", "state_after": "no goals"}, {"tactic": "simp [hf]", "state_before": "case mk.inl.intro.left.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 \ud835\udcdd (uncurry f (s, 1)) = \ud835\udcdd ((uncurry f \u2218 Prod.map id fract) (s, \u2191n))", "state_after": "no goals"}, {"tactic": "refine (((h (s, 0) \u27e8trivial, le_rfl, zero_le_one\u27e9).tendsto.mono_left <| le_of_eq ?_).comp\n  (tendsto_id.prod_map (tendsto_fract_right _))).mono_right (le_of_eq ?_) <;>\n  simp [nhdsWithin_prod_eq, nhdsWithin_univ]", "state_before": "case mk.inl.intro.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 Tendsto (uncurry f \u2218 Prod.map id fract) (\ud835\udcdd s \u00d7\u02e2 \ud835\udcdd[Ici \u2191n] \u2191n) (\ud835\udcdd ((uncurry f \u2218 Prod.map id fract) (s, \u2191n)))", "state_after": "no goals"}, {"tactic": "replace ht : t \u2260 \u230at\u230b := fun ht' => ht \u27e8_, ht'\u27e9", "state_before": "case mk.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nt : \u03b1\nht : \u00ac\u2203 n, t = \u2191n\n\u22a2 ContinuousAt (uncurry f \u2218 Prod.map id fract) (s, t)", "state_after": "case mk.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nt : \u03b1\nht : t \u2260 \u2191\u230at\u230b\n\u22a2 ContinuousAt (uncurry f \u2218 Prod.map id fract) (s, t)"}, {"tactic": "refine (h.continuousAt ?_).comp (continuousAt_id.prod_map (continuousAt_fract ht))", "state_before": "case mk.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nt : \u03b1\nht : t \u2260 \u2191\u230at\u230b\n\u22a2 ContinuousAt (uncurry f \u2218 Prod.map id fract) (s, t)", "state_after": "case mk.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nt : \u03b1\nht : t \u2260 \u2191\u230at\u230b\n\u22a2 univ \u00d7\u02e2 Icc 0 1 \u2208 \ud835\udcdd (Prod.map id fract (s, t))"}, {"tactic": "exact prod_mem_nhds univ_mem (Icc_mem_nhds (fract_pos.2 ht) (fract_lt_one _))", "state_before": "case mk.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nt : \u03b1\nht : t \u2260 \u2191\u230at\u230b\n\u22a2 univ \u00d7\u02e2 Icc 0 1 \u2208 \ud835\udcdd (Prod.map id fract (s, t))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "full_name": "Finset.nnnorm_prod_le'", "start": [343, 1], "end": [345, 63], "traced_tactics": [{"tactic": "simp [NNReal.coe_prod]", "state_before": "\u03b1\u271d : Type ?u.119341\n\u03b2 : Type ?u.119344\n\u03b3 : Type ?u.119347\n\u03b9 : Type u_2\ninst\u271d\u00b9 : SeminormedRing \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : NormedCommRing \u03b1\ns : Finset \u03b9\nhs : Finset.Nonempty s\nf : \u03b9 \u2192 \u03b1\n\u22a2 \u220f i in s, \u2016f i\u2016 = (fun a => \u2191a) (\u220f i in s, \u2016f i\u2016\u208a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Sites/Spaces.lean", "full_name": "Opens.pretopology_ofGrothendieck", "start": [83, 1], "end": [91, 47], "traced_tactics": [{"tactic": "apply le_antisymm", "state_before": "T : Type u\ninst\u271d : TopologicalSpace T\n\u22a2 Pretopology.ofGrothendieck (Opens T) (grothendieckTopology T) = pretopology T", "state_after": "case a\nT : Type u\ninst\u271d : TopologicalSpace T\n\u22a2 Pretopology.ofGrothendieck (Opens T) (grothendieckTopology T) \u2264 pretopology T\n\ncase a\nT : Type u\ninst\u271d : TopologicalSpace T\n\u22a2 pretopology T \u2264 Pretopology.ofGrothendieck (Opens T) (grothendieckTopology T)"}, {"tactic": "intro X R hR x hx", "state_before": "case a\nT : Type u\ninst\u271d : TopologicalSpace T\n\u22a2 Pretopology.ofGrothendieck (Opens T) (grothendieckTopology T) \u2264 pretopology T", "state_after": "case a\nT : Type u\ninst\u271d : TopologicalSpace T\nX : Opens T\nR : Presieve X\nhR : R \u2208 Pretopology.coverings (Pretopology.ofGrothendieck (Opens T) (grothendieckTopology T)) X\nx : T\nhx : x \u2208 X\n\u22a2 \u2203 U f, R f \u2227 x \u2208 U"}, {"tactic": "rcases hR x hx with \u27e8U, f, \u27e8V, g\u2081, g\u2082, hg\u2082, _\u27e9, hU\u27e9", "state_before": "case a\nT : Type u\ninst\u271d : TopologicalSpace T\nX : Opens T\nR : Presieve X\nhR : R \u2208 Pretopology.coverings (Pretopology.ofGrothendieck (Opens T) (grothendieckTopology T)) X\nx : T\nhx : x \u2208 X\n\u22a2 \u2203 U f, R f \u2227 x \u2208 U", "state_after": "case a.intro.intro.intro.intro.intro.intro.intro\nT : Type u\ninst\u271d : TopologicalSpace T\nX : Opens T\nR : Presieve X\nhR : R \u2208 Pretopology.coverings (Pretopology.ofGrothendieck (Opens T) (grothendieckTopology T)) X\nx : T\nhx : x \u2208 X\nU : Opens T\nf : U \u27f6 X\nhU : x \u2208 U\nV : Opens T\ng\u2081 : U \u27f6 V\ng\u2082 : V \u27f6 X\nhg\u2082 : R g\u2082\nright\u271d : g\u2081 \u226b g\u2082 = f\n\u22a2 \u2203 U f, R f \u2227 x \u2208 U"}, {"tactic": "exact \u27e8V, g\u2082, hg\u2082, g\u2081.le hU\u27e9", "state_before": "case a.intro.intro.intro.intro.intro.intro.intro\nT : Type u\ninst\u271d : TopologicalSpace T\nX : Opens T\nR : Presieve X\nhR : R \u2208 Pretopology.coverings (Pretopology.ofGrothendieck (Opens T) (grothendieckTopology T)) X\nx : T\nhx : x \u2208 X\nU : Opens T\nf : U \u27f6 X\nhU : x \u2208 U\nV : Opens T\ng\u2081 : U \u27f6 V\ng\u2082 : V \u27f6 X\nhg\u2082 : R g\u2082\nright\u271d : g\u2081 \u226b g\u2082 = f\n\u22a2 \u2203 U f, R f \u2227 x \u2208 U", "state_after": "no goals"}, {"tactic": "intro X R hR x hx", "state_before": "case a\nT : Type u\ninst\u271d : TopologicalSpace T\n\u22a2 pretopology T \u2264 Pretopology.ofGrothendieck (Opens T) (grothendieckTopology T)", "state_after": "case a\nT : Type u\ninst\u271d : TopologicalSpace T\nX : Opens T\nR : Presieve X\nhR : R \u2208 Pretopology.coverings (pretopology T) X\nx : T\nhx : x \u2208 X\n\u22a2 \u2203 U f, (Sieve.generate R).arrows f \u2227 x \u2208 U"}, {"tactic": "rcases hR x hx with \u27e8U, f, hf, hU\u27e9", "state_before": "case a\nT : Type u\ninst\u271d : TopologicalSpace T\nX : Opens T\nR : Presieve X\nhR : R \u2208 Pretopology.coverings (pretopology T) X\nx : T\nhx : x \u2208 X\n\u22a2 \u2203 U f, (Sieve.generate R).arrows f \u2227 x \u2208 U", "state_after": "case a.intro.intro.intro\nT : Type u\ninst\u271d : TopologicalSpace T\nX : Opens T\nR : Presieve X\nhR : R \u2208 Pretopology.coverings (pretopology T) X\nx : T\nhx : x \u2208 X\nU : Opens T\nf : U \u27f6 X\nhf : R f\nhU : x \u2208 U\n\u22a2 \u2203 U f, (Sieve.generate R).arrows f \u2227 x \u2208 U"}, {"tactic": "exact \u27e8U, f, Sieve.le_generate R U hf, hU\u27e9", "state_before": "case a.intro.intro.intro\nT : Type u\ninst\u271d : TopologicalSpace T\nX : Opens T\nR : Presieve X\nhR : R \u2208 Pretopology.coverings (pretopology T) X\nx : T\nhx : x \u2208 X\nU : Opens T\nf : U \u27f6 X\nhf : R f\nhU : x \u2208 U\n\u22a2 \u2203 U f, (Sieve.generate R).arrows f \u2227 x \u2208 U", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.nonempty_compl_range", "start": [1713, 1], "end": [1714, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Instances/ENNReal.lean", "full_name": "ENNReal.summable", "start": [780, 11], "end": [781, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Sym/Sym2.lean", "full_name": "Sym2.eqBool_spec", "start": [666, 1], "end": [667, 55], "traced_tactics": [{"tactic": "aesop (rule_sets [Sym2])", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.77228\n\u03b3 : Type ?u.77231\ninst\u271d : DecidableEq \u03b1\na b : Sym2 \u03b1\n\u22a2 \u2200 (a\u2081 a\u2082 b\u2081 b\u2082 : \u03b1),\n    eqBool (Quotient.mk (Rel.setoid \u03b1) (a\u2081, a\u2082)) (Quotient.mk (Rel.setoid \u03b1) (b\u2081, b\u2082)) = true \u2194\n      Quotient.mk (Rel.setoid \u03b1) (a\u2081, a\u2082) = Quotient.mk (Rel.setoid \u03b1) (b\u2081, b\u2082)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Matrix/Notation.lean", "full_name": "Matrix.cons_mul", "start": [251, 1], "end": [256, 22], "traced_tactics": [{"tactic": "ext (i j)", "state_before": "\u03b1 : Type u\no n m : \u2115\nm' : Type u\u2098\nn' : Type u\u2099\no' : Type u\u2092\na b : \u2115\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Fintype n'\nv : n' \u2192 \u03b1\nA : Fin m \u2192 n' \u2192 \u03b1\nB : Matrix n' o' \u03b1\n\u22a2 \u2191of (vecCons v A) \u2b1d B = \u2191of (vecCons (vecMul v B) (\u2191of.symm (\u2191of A \u2b1d B)))", "state_after": "case a.h\n\u03b1 : Type u\no n m : \u2115\nm' : Type u\u2098\nn' : Type u\u2099\no' : Type u\u2092\na b : \u2115\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Fintype n'\nv : n' \u2192 \u03b1\nA : Fin m \u2192 n' \u2192 \u03b1\nB : Matrix n' o' \u03b1\ni : Fin (Nat.succ m)\nj : o'\n\u22a2 (\u2191of (vecCons v A) \u2b1d B) i j = \u2191of (vecCons (vecMul v B) (\u2191of.symm (\u2191of A \u2b1d B))) i j"}, {"tactic": "refine' Fin.cases _ _ i", "state_before": "case a.h\n\u03b1 : Type u\no n m : \u2115\nm' : Type u\u2098\nn' : Type u\u2099\no' : Type u\u2092\na b : \u2115\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Fintype n'\nv : n' \u2192 \u03b1\nA : Fin m \u2192 n' \u2192 \u03b1\nB : Matrix n' o' \u03b1\ni : Fin (Nat.succ m)\nj : o'\n\u22a2 (\u2191of (vecCons v A) \u2b1d B) i j = \u2191of (vecCons (vecMul v B) (\u2191of.symm (\u2191of A \u2b1d B))) i j", "state_after": "case a.h.refine'_1\n\u03b1 : Type u\no n m : \u2115\nm' : Type u\u2098\nn' : Type u\u2099\no' : Type u\u2092\na b : \u2115\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Fintype n'\nv : n' \u2192 \u03b1\nA : Fin m \u2192 n' \u2192 \u03b1\nB : Matrix n' o' \u03b1\ni : Fin (Nat.succ m)\nj : o'\n\u22a2 (\u2191of (vecCons v A) \u2b1d B) 0 j = \u2191of (vecCons (vecMul v B) (\u2191of.symm (\u2191of A \u2b1d B))) 0 j\n\ncase a.h.refine'_2\n\u03b1 : Type u\no n m : \u2115\nm' : Type u\u2098\nn' : Type u\u2099\no' : Type u\u2092\na b : \u2115\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Fintype n'\nv : n' \u2192 \u03b1\nA : Fin m \u2192 n' \u2192 \u03b1\nB : Matrix n' o' \u03b1\ni : Fin (Nat.succ m)\nj : o'\n\u22a2 \u2200 (i : Fin m),\n    (\u2191of (vecCons v A) \u2b1d B) (Fin.succ i) j = \u2191of (vecCons (vecMul v B) (\u2191of.symm (\u2191of A \u2b1d B))) (Fin.succ i) j"}, {"tactic": "simp [mul_val_succ]", "state_before": "case a.h.refine'_2\n\u03b1 : Type u\no n m : \u2115\nm' : Type u\u2098\nn' : Type u\u2099\no' : Type u\u2092\na b : \u2115\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Fintype n'\nv : n' \u2192 \u03b1\nA : Fin m \u2192 n' \u2192 \u03b1\nB : Matrix n' o' \u03b1\ni : Fin (Nat.succ m)\nj : o'\n\u22a2 \u2200 (i : Fin m),\n    (\u2191of (vecCons v A) \u2b1d B) (Fin.succ i) j = \u2191of (vecCons (vecMul v B) (\u2191of.symm (\u2191of A \u2b1d B))) (Fin.succ i) j", "state_after": "no goals"}, {"tactic": "rfl", "state_before": "case a.h.refine'_1\n\u03b1 : Type u\no n m : \u2115\nm' : Type u\u2098\nn' : Type u\u2099\no' : Type u\u2092\na b : \u2115\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Fintype n'\nv : n' \u2192 \u03b1\nA : Fin m \u2192 n' \u2192 \u03b1\nB : Matrix n' o' \u03b1\ni : Fin (Nat.succ m)\nj : o'\n\u22a2 (\u2191of (vecCons v A) \u2b1d B) 0 j = \u2191of (vecCons (vecMul v B) (\u2191of.symm (\u2191of A \u2b1d B))) 0 j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "full_name": "ContinuousOn.stronglyMeasurableAtFilter", "start": [582, 1], "end": [586, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Sigma.lean", "full_name": "List.kerase_comm", "start": [535, 1], "end": [550, 59], "traced_tactics": [{"tactic": "simp [h]", "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na\u2081 a\u2082 : \u03b1\nl : List (Sigma \u03b2)\nh : a\u2081 = a\u2082\n\u22a2 kerase a\u2082 (kerase a\u2081 l) = kerase a\u2081 (kerase a\u2082 l)", "state_after": "no goals"}, {"tactic": "simp [kerase_append_left h',\n  kerase_append_right (mt (mem_keys_kerase_of_ne h).mp a\u2081_nin_l\u2081)]", "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na\u2081 a\u2082 : \u03b1\nl : List (Sigma \u03b2)\nh : \u00aca\u2081 = a\u2082\nha\u2082 : a\u2082 \u2208 keys l\nb\u2081 : \u03b2 a\u2081\nl\u2081 l\u2082 : List (Sigma \u03b2)\na\u2081_nin_l\u2081 : \u00aca\u2081 \u2208 keys l\u2081\nx\u271d : a\u2082 \u2208 keys (l\u2081 ++ { fst := a\u2081, snd := b\u2081 } :: l\u2082)\nha\u2081 : a\u2081 \u2208 keys (l\u2081 ++ { fst := a\u2081, snd := b\u2081 } :: l\u2082)\nh' : a\u2082 \u2208 keys l\u2081\n\u22a2 kerase a\u2082 (l\u2081 ++ l\u2082) = kerase a\u2081 (kerase a\u2082 (l\u2081 ++ { fst := a\u2081, snd := b\u2081 } :: l\u2082))", "state_after": "no goals"}, {"tactic": "simp [kerase_append_right h', kerase_append_right a\u2081_nin_l\u2081,\n  @kerase_cons_ne _ _ _ a\u2082 \u27e8a\u2081, b\u2081\u27e9 _ (Ne.symm h)]", "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na\u2081 a\u2082 : \u03b1\nl : List (Sigma \u03b2)\nh : \u00aca\u2081 = a\u2082\nha\u2082 : a\u2082 \u2208 keys l\nb\u2081 : \u03b2 a\u2081\nl\u2081 l\u2082 : List (Sigma \u03b2)\na\u2081_nin_l\u2081 : \u00aca\u2081 \u2208 keys l\u2081\nx\u271d : a\u2082 \u2208 keys (l\u2081 ++ { fst := a\u2081, snd := b\u2081 } :: l\u2082)\nha\u2081 : a\u2081 \u2208 keys (l\u2081 ++ { fst := a\u2081, snd := b\u2081 } :: l\u2082)\nh' : \u00aca\u2082 \u2208 keys l\u2081\n\u22a2 kerase a\u2082 (l\u2081 ++ l\u2082) = kerase a\u2081 (kerase a\u2082 (l\u2081 ++ { fst := a\u2081, snd := b\u2081 } :: l\u2082))", "state_after": "no goals"}, {"tactic": "simp [ha\u2082, mt mem_keys_of_mem_keys_kerase ha\u2082]", "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na\u2081 a\u2082 : \u03b1\nl : List (Sigma \u03b2)\nh : \u00aca\u2081 = a\u2082\nha\u2081 : a\u2081 \u2208 keys l\nha\u2082 : \u00aca\u2082 \u2208 keys l\n\u22a2 kerase a\u2082 (kerase a\u2081 l) = kerase a\u2081 (kerase a\u2082 l)", "state_after": "no goals"}, {"tactic": "simp [ha\u2081, mt mem_keys_of_mem_keys_kerase ha\u2081]", "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na\u2081 a\u2082 : \u03b1\nl : List (Sigma \u03b2)\nh : \u00aca\u2081 = a\u2082\nha\u2081 : \u00aca\u2081 \u2208 keys l\n\u22a2 kerase a\u2082 (kerase a\u2081 l) = kerase a\u2081 (kerase a\u2082 l)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Bases.lean", "full_name": 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"https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/FinitePresentation.lean", "full_name": "RingHom.FinitePresentation.of_surjective", "start": [441, 1], "end": [444, 39], "traced_tactics": [{"tactic": "rw [\u2190 f.comp_id]", "state_before": "A : Type u_1\nB : Type u_2\nC : Type ?u.1126473\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : CommRing B\ninst\u271d : CommRing C\nf : A \u2192+* B\nhf : Surjective \u2191f\nhker : Ideal.FG (ker f)\n\u22a2 FinitePresentation f", "state_after": "A : Type u_1\nB : Type u_2\nC : Type ?u.1126473\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : CommRing B\ninst\u271d : CommRing C\nf : A \u2192+* B\nhf : Surjective \u2191f\nhker : Ideal.FG (ker f)\n\u22a2 FinitePresentation (comp f (RingHom.id A))"}, {"tactic": "exact (id A).comp_surjective hf hker", "state_before": "A : Type u_1\nB : Type u_2\nC : Type ?u.1126473\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : CommRing B\ninst\u271d : CommRing C\nf : A \u2192+* B\nhf : Surjective \u2191f\nhker : Ideal.FG (ker f)\n\u22a2 FinitePresentation (comp f (RingHom.id A))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Logic.lean", "full_name": "exists_and_right", "start": [465, 9], "end": [465, 89], "traced_tactics": [{"tactic": "simp [And.comm]", "state_before": "\u03b1 : Sort u_1\np q : \u03b1 \u2192 Prop\nb : Prop\n\u22a2 (\u2203 x, p x \u2227 b) \u2194 (\u2203 x, p x) \u2227 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/TypeVec.lean", "full_name": "TypeVec.appendFun_comp", "start": [260, 1], "end": [265, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Order.lean", "full_name": "Finset.le_prod_nonempty_of_submultiplicative", "start": [62, 1], "end": [65, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.piecewise_mem_set_pi", "start": [2497, 1], "end": [2501, 42], "traced_tactics": [{"tactic": "classical\n  rw [\u2190 piecewise_coe]\n  exact Set.piecewise_mem_pi (\u2191s) hf hg", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.300752\n\u03b3 : Type ?u.300755\n\u03b4\u271d : \u03b1 \u2192 Sort ?u.300760\ns : Finset \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4\u271d i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\n\u03b4 : \u03b1 \u2192 Type u_1\nt : Set \u03b1\nt' : (i : \u03b1) \u2192 Set (\u03b4 i)\nf g : (i : \u03b1) \u2192 \u03b4 i\nhf : f \u2208 Set.pi t t'\nhg : g \u2208 Set.pi t t'\n\u22a2 piecewise s f g \u2208 Set.pi t t'", "state_after": "no goals"}, {"tactic": "rw [\u2190 piecewise_coe]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.300752\n\u03b3 : Type ?u.300755\n\u03b4\u271d : \u03b1 \u2192 Sort ?u.300760\ns : Finset \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4\u271d i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\n\u03b4 : \u03b1 \u2192 Type u_1\nt : Set \u03b1\nt' : (i : \u03b1) \u2192 Set (\u03b4 i)\nf g : (i : \u03b1) \u2192 \u03b4 i\nhf : f \u2208 Set.pi t t'\nhg : g \u2208 Set.pi t t'\n\u22a2 piecewise s f g \u2208 Set.pi t t'", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type ?u.300752\n\u03b3 : Type ?u.300755\n\u03b4\u271d : \u03b1 \u2192 Sort ?u.300760\ns : Finset \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4\u271d i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\n\u03b4 : \u03b1 \u2192 Type u_1\nt : Set \u03b1\nt' : (i : \u03b1) \u2192 Set (\u03b4 i)\nf g : (i : \u03b1) \u2192 \u03b4 i\nhf : f \u2208 Set.pi t t'\nhg : g \u2208 Set.pi t t'\n\u22a2 Set.piecewise (\u2191s) f g \u2208 Set.pi t t'"}, {"tactic": "exact Set.piecewise_mem_pi (\u2191s) hf hg", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.300752\n\u03b3 : Type ?u.300755\n\u03b4\u271d : \u03b1 \u2192 Sort ?u.300760\ns : Finset \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4\u271d i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\n\u03b4 : \u03b1 \u2192 Type u_1\nt : Set \u03b1\nt' : (i : \u03b1) \u2192 Set (\u03b4 i)\nf g : (i : \u03b1) \u2192 \u03b4 i\nhf : f \u2208 Set.pi t t'\nhg : g \u2208 Set.pi t t'\n\u22a2 Set.piecewise (\u2191s) f g \u2208 Set.pi t t'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/GCD/Basic.lean", "full_name": "Nat.lcm_pos", "start": [104, 1], "end": [106, 20], "traced_tactics": [{"tactic": "simp_rw [pos_iff_ne_zero]", "state_before": "m n : \u2115\n\u22a2 0 < m \u2192 0 < n \u2192 0 < lcm m n", "state_after": "m n : \u2115\n\u22a2 m \u2260 0 \u2192 n \u2260 0 \u2192 lcm m n \u2260 0"}, {"tactic": "exact lcm_ne_zero", "state_before": "m n : \u2115\n\u22a2 m \u2260 0 \u2192 n \u2260 0 \u2192 lcm m n \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/AbstractCompletion.lean", "full_name": "AbstractCompletion.continuous_map\u2082", "start": [378, 1], "end": [381, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean", "full_name": "Pmf.support_seq", "start": [133, 1], "end": [134, 65], "traced_tactics": [{"tactic": "simp [-mem_support_iff, seq, @eq_comm \u03b2 b]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.49396\nq : Pmf (\u03b1 \u2192 \u03b2)\np : Pmf \u03b1\nb\u271d b : \u03b2\n\u22a2 b \u2208 support (seq q p) \u2194 b \u2208 \u22c3 (f : \u03b1 \u2192 \u03b2) (_ : f \u2208 support q), f '' support p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/Basic.lean", "full_name": "Function.piCongrLeft'_update", "start": [1930, 1], "end": [1943, 35], "traced_tactics": [{"tactic": "ext b'", "state_before": "\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nP : \u03b1 \u2192 Sort u_3\ne : \u03b1 \u2243 \u03b2\nf : (a : \u03b1) \u2192 P a\nb : \u03b2\nx : P (\u2191e.symm b)\n\u22a2 \u2191(Equiv.piCongrLeft' P e) (update f (\u2191e.symm b) x) = update (\u2191(Equiv.piCongrLeft' P e) f) b x", "state_after": "case h\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nP : \u03b1 \u2192 Sort u_3\ne : \u03b1 \u2243 \u03b2\nf : (a : \u03b1) \u2192 P a\nb : \u03b2\nx : P (\u2191e.symm b)\nb' : \u03b2\n\u22a2 \u2191(Equiv.piCongrLeft' P e) (update f (\u2191e.symm b) x) b' = update (\u2191(Equiv.piCongrLeft' P e) f) b x b'"}, {"tactic": "rcases eq_or_ne b' b with (rfl | h)", "state_before": "case h\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nP : \u03b1 \u2192 Sort u_3\ne : \u03b1 \u2243 \u03b2\nf : (a : \u03b1) \u2192 P a\nb : \u03b2\nx : P (\u2191e.symm b)\nb' : \u03b2\n\u22a2 \u2191(Equiv.piCongrLeft' P e) (update f (\u2191e.symm b) x) b' = update (\u2191(Equiv.piCongrLeft' P e) f) b x b'", "state_after": "case h.inl\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nP : \u03b1 \u2192 Sort u_3\ne : \u03b1 \u2243 \u03b2\nf : (a : \u03b1) \u2192 P a\nb' : \u03b2\nx : P (\u2191e.symm b')\n\u22a2 \u2191(Equiv.piCongrLeft' P e) (update f (\u2191e.symm b') x) b' = update (\u2191(Equiv.piCongrLeft' P e) f) b' x b'\n\ncase h.inr\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nP : \u03b1 \u2192 Sort u_3\ne : \u03b1 \u2243 \u03b2\nf : (a : \u03b1) \u2192 P a\nb : \u03b2\nx : P (\u2191e.symm b)\nb' : \u03b2\nh : b' \u2260 b\n\u22a2 \u2191(Equiv.piCongrLeft' P e) (update f (\u2191e.symm b) x) b' = update (\u2191(Equiv.piCongrLeft' P e) f) b x b'"}, {"tactic": "simp", "state_before": "case h.inl\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nP : \u03b1 \u2192 Sort u_3\ne : \u03b1 \u2243 \u03b2\nf : (a : \u03b1) \u2192 P a\nb' : \u03b2\nx : P (\u2191e.symm b')\n\u22a2 \u2191(Equiv.piCongrLeft' P e) (update f (\u2191e.symm b') x) b' = update (\u2191(Equiv.piCongrLeft' P e) f) b' x b'", "state_after": "no goals"}, {"tactic": "simp only [Equiv.piCongrLeft'_apply, ne_eq, h, not_false_iff, update_noteq]", "state_before": "case h.inr\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nP : \u03b1 \u2192 Sort u_3\ne : \u03b1 \u2243 \u03b2\nf : (a : \u03b1) \u2192 P a\nb : \u03b2\nx : P (\u2191e.symm b)\nb' : \u03b2\nh : b' \u2260 b\n\u22a2 \u2191(Equiv.piCongrLeft' P e) (update f (\u2191e.symm b) x) b' = update (\u2191(Equiv.piCongrLeft' P e) f) b x b'", "state_after": "case h.inr\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nP : \u03b1 \u2192 Sort u_3\ne : \u03b1 \u2243 \u03b2\nf : (a : \u03b1) \u2192 P a\nb : \u03b2\nx : P (\u2191e.symm b)\nb' : \u03b2\nh : b' \u2260 b\n\u22a2 update f (\u2191e.symm b) x (\u2191e.symm b') = f (\u2191e.symm b')"}, {"tactic": "rw [update_noteq _]", "state_before": "case h.inr\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nP : \u03b1 \u2192 Sort u_3\ne : \u03b1 \u2243 \u03b2\nf : (a : \u03b1) \u2192 P a\nb : \u03b2\nx : P (\u2191e.symm b)\nb' : \u03b2\nh : b' \u2260 b\n\u22a2 update f (\u2191e.symm b) x (\u2191e.symm b') = f (\u2191e.symm b')", "state_after": "\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nP : \u03b1 \u2192 Sort u_3\ne : \u03b1 \u2243 \u03b2\nf : (a : \u03b1) \u2192 P a\nb : \u03b2\nx : P (\u2191e.symm b)\nb' : \u03b2\nh : b' \u2260 b\n\u22a2 \u2191e.symm b' \u2260 \u2191e.symm b"}, {"tactic": "rw [ne_eq]", "state_before": "\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nP : \u03b1 \u2192 Sort u_3\ne : \u03b1 \u2243 \u03b2\nf : (a : \u03b1) \u2192 P a\nb : \u03b2\nx : P (\u2191e.symm b)\nb' : \u03b2\nh : b' \u2260 b\n\u22a2 \u2191e.symm b' \u2260 \u2191e.symm b", "state_after": "\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nP : \u03b1 \u2192 Sort u_3\ne : \u03b1 \u2243 \u03b2\nf : (a : \u03b1) \u2192 P a\nb : \u03b2\nx : P (\u2191e.symm b)\nb' : \u03b2\nh : b' \u2260 b\n\u22a2 \u00ac\u2191e.symm b' = \u2191e.symm b"}, {"tactic": "intro h'", "state_before": "\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nP : \u03b1 \u2192 Sort u_3\ne : \u03b1 \u2243 \u03b2\nf : (a : \u03b1) \u2192 P a\nb : \u03b2\nx : P (\u2191e.symm b)\nb' : \u03b2\nh : b' \u2260 b\n\u22a2 \u00ac\u2191e.symm b' = \u2191e.symm b", "state_after": "\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nP : \u03b1 \u2192 Sort u_3\ne : \u03b1 \u2243 \u03b2\nf : (a : \u03b1) \u2192 P a\nb : \u03b2\nx : P (\u2191e.symm b)\nb' : \u03b2\nh : b' \u2260 b\nh' : \u2191e.symm b' = \u2191e.symm b\n\u22a2 False"}, {"tactic": "cases e.symm.injective h' |> h", "state_before": "\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nP : \u03b1 \u2192 Sort u_3\ne : \u03b1 \u2243 \u03b2\nf : (a : \u03b1) \u2192 P a\nb : \u03b2\nx : P (\u2191e.symm b)\nb' : \u03b2\nh : b' \u2260 b\nh' : \u2191e.symm b' = \u2191e.symm b\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Nonarchimedean/Basic.lean", "full_name": "NonarchimedeanGroup.prod_subset", "start": [89, 1], "end": [98, 69], "traced_tactics": [{"tactic": "erw [nhds_prod_eq, Filter.mem_prod_iff] at hU", "state_before": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type ?u.59835\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_2\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nhU : U \u2208 nhds 1\n\u22a2 \u2203 V W, \u2191V \u00d7\u02e2 \u2191W \u2286 U", "state_after": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type ?u.59835\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_2\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nhU : \u2203 t\u2081, t\u2081 \u2208 nhds 1 \u2227 \u2203 t\u2082, t\u2082 \u2208 nhds 1 \u2227 t\u2081 \u00d7\u02e2 t\u2082 \u2286 U\n\u22a2 \u2203 V W, \u2191V \u00d7\u02e2 \u2191W \u2286 U"}, {"tactic": "rcases hU with \u27e8U\u2081, hU\u2081, U\u2082, hU\u2082, h\u27e9", "state_before": "G : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type ?u.59835\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_2\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nhU : \u2203 t\u2081, t\u2081 \u2208 nhds 1 \u2227 \u2203 t\u2082, t\u2082 \u2208 nhds 1 \u2227 t\u2081 \u00d7\u02e2 t\u2082 \u2286 U\n\u22a2 \u2203 V W, \u2191V \u00d7\u02e2 \u2191W \u2286 U", "state_after": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type ?u.59835\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_2\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nU\u2081 : Set G\nhU\u2081 : U\u2081 \u2208 nhds 1\nU\u2082 : Set K\nhU\u2082 : U\u2082 \u2208 nhds 1\nh : U\u2081 \u00d7\u02e2 U\u2082 \u2286 U\n\u22a2 \u2203 V W, \u2191V \u00d7\u02e2 \u2191W \u2286 U"}, {"tactic": "cases' is_nonarchimedean _ hU\u2081 with V hV", "state_before": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type ?u.59835\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_2\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nU\u2081 : Set G\nhU\u2081 : U\u2081 \u2208 nhds 1\nU\u2082 : Set K\nhU\u2082 : U\u2082 \u2208 nhds 1\nh : U\u2081 \u00d7\u02e2 U\u2082 \u2286 U\n\u22a2 \u2203 V W, \u2191V \u00d7\u02e2 \u2191W \u2286 U", "state_after": "case intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type ?u.59835\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_2\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nU\u2081 : Set G\nhU\u2081 : U\u2081 \u2208 nhds 1\nU\u2082 : Set K\nhU\u2082 : U\u2082 \u2208 nhds 1\nh : U\u2081 \u00d7\u02e2 U\u2082 \u2286 U\nV : OpenSubgroup G\nhV : \u2191V \u2286 U\u2081\n\u22a2 \u2203 V W, \u2191V \u00d7\u02e2 \u2191W \u2286 U"}, {"tactic": "cases' is_nonarchimedean _ hU\u2082 with W hW", "state_before": "case intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type ?u.59835\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_2\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nU\u2081 : Set G\nhU\u2081 : U\u2081 \u2208 nhds 1\nU\u2082 : Set K\nhU\u2082 : U\u2082 \u2208 nhds 1\nh : U\u2081 \u00d7\u02e2 U\u2082 \u2286 U\nV : OpenSubgroup G\nhV : \u2191V \u2286 U\u2081\n\u22a2 \u2203 V W, \u2191V \u00d7\u02e2 \u2191W \u2286 U", "state_after": "case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type ?u.59835\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_2\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nU\u2081 : Set G\nhU\u2081 : U\u2081 \u2208 nhds 1\nU\u2082 : Set K\nhU\u2082 : U\u2082 \u2208 nhds 1\nh : U\u2081 \u00d7\u02e2 U\u2082 \u2286 U\nV : OpenSubgroup G\nhV : \u2191V \u2286 U\u2081\nW : OpenSubgroup K\nhW : \u2191W \u2286 U\u2082\n\u22a2 \u2203 V W, \u2191V \u00d7\u02e2 \u2191W \u2286 U"}, {"tactic": "use V", "state_before": "case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type ?u.59835\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_2\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nU\u2081 : Set G\nhU\u2081 : U\u2081 \u2208 nhds 1\nU\u2082 : Set K\nhU\u2082 : U\u2082 \u2208 nhds 1\nh : U\u2081 \u00d7\u02e2 U\u2082 \u2286 U\nV : OpenSubgroup G\nhV : \u2191V \u2286 U\u2081\nW : OpenSubgroup K\nhW : \u2191W \u2286 U\u2082\n\u22a2 \u2203 V W, \u2191V \u00d7\u02e2 \u2191W \u2286 U", "state_after": "case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type ?u.59835\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_2\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nU\u2081 : Set G\nhU\u2081 : U\u2081 \u2208 nhds 1\nU\u2082 : Set K\nhU\u2082 : U\u2082 \u2208 nhds 1\nh : U\u2081 \u00d7\u02e2 U\u2082 \u2286 U\nV : OpenSubgroup G\nhV : \u2191V \u2286 U\u2081\nW : OpenSubgroup K\nhW : \u2191W \u2286 U\u2082\n\u22a2 \u2203 W, \u2191V \u00d7\u02e2 \u2191W \u2286 U"}, {"tactic": "use W", "state_before": "case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type ?u.59835\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_2\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nU\u2081 : Set G\nhU\u2081 : U\u2081 \u2208 nhds 1\nU\u2082 : Set K\nhU\u2082 : U\u2082 \u2208 nhds 1\nh : U\u2081 \u00d7\u02e2 U\u2082 \u2286 U\nV : OpenSubgroup G\nhV : \u2191V \u2286 U\u2081\nW : OpenSubgroup K\nhW : \u2191W \u2286 U\u2082\n\u22a2 \u2203 W, \u2191V \u00d7\u02e2 \u2191W \u2286 U", "state_after": "case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type ?u.59835\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_2\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nU\u2081 : Set G\nhU\u2081 : U\u2081 \u2208 nhds 1\nU\u2082 : Set K\nhU\u2082 : U\u2082 \u2208 nhds 1\nh : U\u2081 \u00d7\u02e2 U\u2082 \u2286 U\nV : OpenSubgroup G\nhV : \u2191V \u2286 U\u2081\nW : OpenSubgroup K\nhW : \u2191W \u2286 U\u2082\n\u22a2 \u2191V \u00d7\u02e2 \u2191W \u2286 U"}, {"tactic": "rw [Set.prod_subset_iff]", "state_before": "case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type ?u.59835\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_2\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nU\u2081 : Set G\nhU\u2081 : U\u2081 \u2208 nhds 1\nU\u2082 : Set K\nhU\u2082 : U\u2082 \u2208 nhds 1\nh : U\u2081 \u00d7\u02e2 U\u2082 \u2286 U\nV : OpenSubgroup G\nhV : \u2191V \u2286 U\u2081\nW : OpenSubgroup K\nhW : \u2191W \u2286 U\u2082\n\u22a2 \u2191V \u00d7\u02e2 \u2191W \u2286 U", "state_after": "case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type ?u.59835\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_2\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nU\u2081 : Set G\nhU\u2081 : U\u2081 \u2208 nhds 1\nU\u2082 : Set K\nhU\u2082 : U\u2082 \u2208 nhds 1\nh : U\u2081 \u00d7\u02e2 U\u2082 \u2286 U\nV : OpenSubgroup G\nhV : \u2191V \u2286 U\u2081\nW : OpenSubgroup K\nhW : \u2191W \u2286 U\u2082\n\u22a2 \u2200 (x : G), x \u2208 \u2191V \u2192 \u2200 (y : K), y \u2208 \u2191W \u2192 (x, y) \u2208 U"}, {"tactic": "intro x hX y hY", "state_before": "case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type ?u.59835\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_2\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nU\u2081 : Set G\nhU\u2081 : U\u2081 \u2208 nhds 1\nU\u2082 : Set K\nhU\u2082 : U\u2082 \u2208 nhds 1\nh : U\u2081 \u00d7\u02e2 U\u2082 \u2286 U\nV : OpenSubgroup G\nhV : \u2191V \u2286 U\u2081\nW : OpenSubgroup K\nhW : \u2191W \u2286 U\u2082\n\u22a2 \u2200 (x : G), x \u2208 \u2191V \u2192 \u2200 (y : K), y \u2208 \u2191W \u2192 (x, y) \u2208 U", "state_after": "case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type ?u.59835\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_2\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nU\u2081 : Set G\nhU\u2081 : U\u2081 \u2208 nhds 1\nU\u2082 : Set K\nhU\u2082 : U\u2082 \u2208 nhds 1\nh : U\u2081 \u00d7\u02e2 U\u2082 \u2286 U\nV : OpenSubgroup G\nhV : \u2191V \u2286 U\u2081\nW : OpenSubgroup K\nhW : \u2191W \u2286 U\u2082\nx : G\nhX : x \u2208 \u2191V\ny : K\nhY : y \u2208 \u2191W\n\u22a2 (x, y) \u2208 U"}, {"tactic": "exact Set.Subset.trans (Set.prod_mono hV hW) h (Set.mem_sep hX hY)", "state_before": "case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : NonarchimedeanGroup G\nH : Type ?u.59835\ninst\u271d\u2075 : Group H\ninst\u271d\u2074 : TopologicalSpace H\ninst\u271d\u00b3 : TopologicalGroup H\nK : Type u_2\ninst\u271d\u00b2 : Group K\ninst\u271d\u00b9 : TopologicalSpace K\ninst\u271d : NonarchimedeanGroup K\nU : Set (G \u00d7 K)\nU\u2081 : Set G\nhU\u2081 : U\u2081 \u2208 nhds 1\nU\u2082 : Set K\nhU\u2082 : U\u2082 \u2208 nhds 1\nh : U\u2081 \u00d7\u02e2 U\u2082 \u2286 U\nV : OpenSubgroup G\nhV : \u2191V \u2286 U\u2081\nW : OpenSubgroup K\nhW : \u2191W \u2286 U\u2082\nx : G\nhX : x \u2208 \u2191V\ny : K\nhY : y \u2208 \u2191W\n\u22a2 (x, y) \u2208 U", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean", "full_name": "EuclideanGeometry.sin_eq_one_iff_angle_eq_pi_div_two", "start": [505, 8], "end": [507, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/NaturalOps.lean", "full_name": "Ordinal.toNatOrdinal_eq_one", "start": [179, 1], "end": [180, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/LinearPMap.lean", "full_name": "LinearPMap.graph_map_snd_eq_range", "start": [751, 1], "end": [752, 80], "traced_tactics": [{"tactic": "ext", "state_before": "R : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type ?u.493697\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\n\u22a2 Submodule.map (LinearMap.snd R E F) (graph f) = LinearMap.range f.toFun", "state_after": "case h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type ?u.493697\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx\u271d : F\n\u22a2 x\u271d \u2208 Submodule.map (LinearMap.snd R E F) (graph f) \u2194 x\u271d \u2208 LinearMap.range f.toFun"}, {"tactic": "simp", "state_before": "case h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type ?u.493697\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nx\u271d : F\n\u22a2 x\u271d \u2208 Submodule.map (LinearMap.snd R E F) (graph f) \u2194 x\u271d \u2208 LinearMap.range f.toFun", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Basic.lean", "full_name": "SimpleGraph.sInf_adj", "start": [314, 1], "end": [315, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.biUnion_singleton", "start": [971, 1], "end": [972, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Caratheodory.lean", "full_name": "eq_pos_convex_span_of_mem_convexHull", "start": [170, 1], "end": [193, 29], "traced_tactics": [{"tactic": "rw [convexHull_eq_union] at hx", "state_before": "\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : x \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom s\n\u22a2 \u2203 \u03b9 x_1 z w x_2 x_3 x_4, \u2211 i : \u03b9, w i = 1 \u2227 \u2211 i : \u03b9, w i \u2022 z i = x", "state_after": "\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : x \u2208 \u22c3 (t : Finset E) (_ : \u2191t \u2286 s) (_ : AffineIndependent \ud835\udd5c Subtype.val), \u2191(convexHull \ud835\udd5c).toOrderHom \u2191t\n\u22a2 \u2203 \u03b9 x_1 z w x_2 x_3 x_4, \u2211 i : \u03b9, w i = 1 \u2227 \u2211 i : \u03b9, w i \u2022 z i = x"}, {"tactic": "simp only [exists_prop, Set.mem_iUnion] at hx", "state_before": "\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : x \u2208 \u22c3 (t : Finset E) (_ : \u2191t \u2286 s) (_ : AffineIndependent \ud835\udd5c Subtype.val), \u2191(convexHull \ud835\udd5c).toOrderHom \u2191t\n\u22a2 \u2203 \u03b9 x_1 z w x_2 x_3 x_4, \u2211 i : \u03b9, w i = 1 \u2227 \u2211 i : \u03b9, w i \u2022 z i = x", "state_after": "\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : \u2203 i, \u2191i \u2286 s \u2227 AffineIndependent \ud835\udd5c Subtype.val \u2227 x \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom \u2191i\n\u22a2 \u2203 \u03b9 x_1 z w x_2 x_3 x_4, \u2211 i : \u03b9, w i = 1 \u2227 \u2211 i : \u03b9, w i \u2022 z i = x"}, {"tactic": "obtain \u27e8t, ht\u2081, ht\u2082, ht\u2083\u27e9 := hx", "state_before": "\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nhx : \u2203 i, \u2191i \u2286 s \u2227 AffineIndependent \ud835\udd5c Subtype.val \u2227 x \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom \u2191i\n\u22a2 \u2203 \u03b9 x_1 z w x_2 x_3 x_4, \u2211 i : \u03b9, w i = 1 \u2227 \u2211 i : \u03b9, w i \u2022 z i = x", "state_after": "case intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nht\u2083 : x \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom \u2191t\n\u22a2 \u2203 \u03b9 x_1 z w x_2 x_3 x_4, \u2211 i : \u03b9, w i = 1 \u2227 \u2211 i : \u03b9, w i \u2022 z i = x"}, {"tactic": "simp only [t.convexHull_eq, exists_prop, Set.mem_setOf_eq] at ht\u2083", "state_before": "case intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nht\u2083 : x \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom \u2191t\n\u22a2 \u2203 \u03b9 x_1 z w x_2 x_3 x_4, \u2211 i : \u03b9, w i = 1 \u2227 \u2211 i : \u03b9, w i \u2022 z i = x", "state_after": "case intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nht\u2083 : \u2203 w, (\u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y) \u2227 \u2211 y in t, w y = 1 \u2227 centerMass t w id = x\n\u22a2 \u2203 \u03b9 x_1 z w x_2 x_3 x_4, \u2211 i : \u03b9, w i = 1 \u2227 \u2211 i : \u03b9, w i \u2022 z i = x"}, {"tactic": "obtain \u27e8w, hw\u2081, hw\u2082, hw\u2083\u27e9 := ht\u2083", "state_before": "case intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nht\u2083 : \u2203 w, (\u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y) \u2227 \u2211 y in t, w y = 1 \u2227 centerMass t w id = x\n\u22a2 \u2203 \u03b9 x_1 z w x_2 x_3 x_4, \u2211 i : \u03b9, w i = 1 \u2227 \u2211 i : \u03b9, w i \u2022 z i = x", "state_after": "case intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\n\u22a2 \u2203 \u03b9 x_1 z w x_2 x_3 x_4, \u2211 i : \u03b9, w i = 1 \u2227 \u2211 i : \u03b9, w i \u2022 z i = x"}, {"tactic": "let t' := t.filter fun i => w i \u2260 0", "state_before": "case intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\n\u22a2 \u2203 \u03b9 x_1 z w x_2 x_3 x_4, \u2211 i : \u03b9, w i = 1 \u2227 \u2211 i : \u03b9, w i \u2022 z i = x", "state_after": "case intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2203 \u03b9 x_1 z w x_2 x_3 x_4, \u2211 i : \u03b9, w i = 1 \u2227 \u2211 i : \u03b9, w i \u2022 z i = x"}, {"tactic": "refine' \u27e8t', t'.fintypeCoeSort, ((\u2191) : t' \u2192 E), w \u2218 ((\u2191) : t' \u2192 E), _, _, _, _, _\u27e9", "state_before": "case intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2203 \u03b9 x_1 z w x_2 x_3 x_4, \u2211 i : \u03b9, w i = 1 \u2227 \u2211 i : \u03b9, w i \u2022 z i = x", "state_after": "case intro.intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 Set.range Subtype.val \u2286 s\n\ncase intro.intro.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 AffineIndependent \ud835\udd5c Subtype.val\n\ncase intro.intro.intro.intro.intro.intro.refine'_3\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2200 (i : { x // x \u2208 t' }), 0 < (w \u2218 Subtype.val) i\n\ncase intro.intro.intro.intro.intro.intro.refine'_4\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2211 i : { x // x \u2208 t' }, (w \u2218 Subtype.val) i = 1\n\ncase intro.intro.intro.intro.intro.intro.refine'_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2211 i : { x // x \u2208 t' }, (w \u2218 Subtype.val) i \u2022 \u2191i = x"}, {"tactic": "rw [Subtype.range_coe_subtype]", "state_before": "case intro.intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 Set.range Subtype.val \u2286 s", "state_after": "case intro.intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 {x | x \u2208 t'} \u2286 s"}, {"tactic": "exact Subset.trans (Finset.filter_subset _ t) ht\u2081", "state_before": "case intro.intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 {x | x \u2208 t'} \u2286 s", "state_after": "no goals"}, {"tactic": "exact ht\u2082.comp_embedding \u27e8_, inclusion_injective (Finset.filter_subset (fun i => w i \u2260 0) t)\u27e9", "state_before": "case intro.intro.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 AffineIndependent \ud835\udd5c Subtype.val", "state_after": "no goals"}, {"tactic": "exact fun i =>\n  (hw\u2081 _ (Finset.mem_filter.mp i.2).1).lt_of_ne (Finset.mem_filter.mp i.property).2.symm", "state_before": "case intro.intro.intro.intro.intro.intro.refine'_3\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2200 (i : { x // x \u2208 t' }), 0 < (w \u2218 Subtype.val) i", "state_after": "no goals"}, {"tactic": "erw [Finset.sum_attach, Finset.sum_filter_ne_zero, hw\u2082]", "state_before": "case intro.intro.intro.intro.intro.intro.refine'_4\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2211 i : { x // x \u2208 t' }, (w \u2218 Subtype.val) i = 1", "state_after": "no goals"}, {"tactic": "change (\u2211 i : t' in t'.attach, (fun e => w e \u2022 e) \u2191i) = x", "state_before": "case intro.intro.intro.intro.intro.intro.refine'_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2211 i : { x // x \u2208 t' }, (w \u2218 Subtype.val) i \u2022 \u2191i = x", "state_after": "case intro.intro.intro.intro.intro.intro.refine'_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2211 i in attach t', (fun e => w e \u2022 e) \u2191i = x"}, {"tactic": "erw [Finset.sum_attach (f := fun e => w e \u2022 e), Finset.sum_filter_of_ne]", "state_before": "case intro.intro.intro.intro.intro.intro.refine'_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2211 i in attach t', (fun e => w e \u2022 e) \u2191i = x", "state_after": "case intro.intro.intro.intro.intro.intro.refine'_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2211 x in t, w x \u2022 x = x\n\ncase intro.intro.intro.intro.intro.intro.refine'_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2200 (x : E), x \u2208 t \u2192 w x \u2022 x \u2260 0 \u2192 w x \u2260 0"}, {"tactic": "rw [t.centerMass_eq_of_sum_1 id hw\u2082] at hw\u2083", "state_before": "case intro.intro.intro.intro.intro.intro.refine'_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2211 x in t, w x \u2022 x = x", "state_after": "case intro.intro.intro.intro.intro.intro.refine'_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : \u2211 i in t, w i \u2022 id i = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2211 x in t, w x \u2022 x = x"}, {"tactic": "exact hw\u2083", "state_before": "case intro.intro.intro.intro.intro.intro.refine'_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : \u2211 i in t, w i \u2022 id i = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2211 x in t, w x \u2022 x = x", "state_after": "no goals"}, {"tactic": "intro e _ hwe contra", "state_before": "case intro.intro.intro.intro.intro.intro.refine'_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\n\u22a2 \u2200 (x : E), x \u2208 t \u2192 w x \u2022 x \u2260 0 \u2192 w x \u2260 0", "state_after": "case intro.intro.intro.intro.intro.intro.refine'_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\ne : E\na\u271d : e \u2208 t\nhwe : w e \u2022 e \u2260 0\ncontra : w e = 0\n\u22a2 False"}, {"tactic": "apply hwe", "state_before": "case intro.intro.intro.intro.intro.intro.refine'_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\ne : E\na\u271d : e \u2208 t\nhwe : w e \u2022 e \u2260 0\ncontra : w e = 0\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro.intro.refine'_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\ne : E\na\u271d : e \u2208 t\nhwe : w e \u2022 e \u2260 0\ncontra : w e = 0\n\u22a2 w e \u2022 e = 0"}, {"tactic": "rw [contra, zero_smul]", "state_before": "case intro.intro.intro.intro.intro.intro.refine'_5\n\ud835\udd5c : Type u_1\nE : Type u\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx : E\nt : Finset E\nht\u2081 : \u2191t \u2286 s\nht\u2082 : AffineIndependent \ud835\udd5c Subtype.val\nw : E \u2192 \ud835\udd5c\nhw\u2081 : \u2200 (y : E), y \u2208 t \u2192 0 \u2264 w y\nhw\u2082 : \u2211 y in t, w y = 1\nhw\u2083 : centerMass t w id = x\nt' : Finset E := filter (fun i => w i \u2260 0) t\ne : E\na\u271d : e \u2208 t\nhwe : w e \u2022 e \u2260 0\ncontra : w e = 0\n\u22a2 w e \u2022 e = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "full_name": "inner_sub_sub_self", "start": [678, 1], "end": [679, 52], "traced_tactics": [{"tactic": "simp only [inner_sub_left, inner_sub_right]", "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.1809059\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 inner (x - y) (x - y) = inner x x - inner x y - inner y x + inner y y", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.1809059\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 inner x x - inner y x - (inner x y - inner y y) = inner x x - inner x y - inner y x + inner y y"}, {"tactic": "ring", "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.1809059\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 inner x x - inner y x - (inner x y - inner y y) = inner x x - inner x y - inner y x + inner y y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.smul_finite", "start": [3899, 1], "end": [3902, 48], "traced_tactics": [{"tactic": "lift c to \u211d\u22650 using hc", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.827912\n\u03b3 : Type ?u.827915\n\u03b4 : Type ?u.827918\n\u03b9 : Type ?u.827921\nR : Type ?u.827924\nR' : Type ?u.827927\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\n\u22a2 IsFiniteMeasure (c \u2022 \u03bc)", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.827912\n\u03b3 : Type ?u.827915\n\u03b4 : Type ?u.827918\n\u03b9 : Type ?u.827921\nR : Type ?u.827924\nR' : Type ?u.827927\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nc : \u211d\u22650\n\u22a2 IsFiniteMeasure (\u2191c \u2022 \u03bc)"}, {"tactic": "exact MeasureTheory.isFiniteMeasureSMulNNReal", "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.827912\n\u03b3 : Type ?u.827915\n\u03b4 : Type ?u.827918\n\u03b9 : Type ?u.827921\nR : Type ?u.827924\nR' : Type ?u.827927\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nc : \u211d\u22650\n\u22a2 IsFiniteMeasure (\u2191c \u2022 \u03bc)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean", "full_name": "PrimeSpectrum.localization_comap_injective", "start": [668, 1], "end": [675, 10], "traced_tactics": [{"tactic": "intro p q h", "state_before": "R : Type u\nS : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\nS' : Type ?u.251612\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R S\nM : Submonoid R\ninst\u271d : IsLocalization M S\n\u22a2 Function.Injective \u2191(comap (algebraMap R S))", "state_after": "R : Type u\nS : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\nS' : Type ?u.251612\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R S\nM : Submonoid R\ninst\u271d : IsLocalization M S\np q : PrimeSpectrum S\nh : \u2191(comap (algebraMap R S)) p = \u2191(comap (algebraMap R S)) q\n\u22a2 p = q"}, {"tactic": "replace h := congr_arg (fun x : PrimeSpectrum R => Ideal.map (algebraMap R S) x.asIdeal) h", "state_before": "R : Type u\nS : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\nS' : Type ?u.251612\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R S\nM : Submonoid R\ninst\u271d : IsLocalization M S\np q : PrimeSpectrum S\nh : \u2191(comap (algebraMap R S)) p = \u2191(comap (algebraMap R S)) q\n\u22a2 p = q", "state_after": "R : Type u\nS : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\nS' : Type ?u.251612\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R S\nM : Submonoid R\ninst\u271d : IsLocalization M S\np q : PrimeSpectrum S\nh :\n  (fun x => Ideal.map (algebraMap R S) x.asIdeal) (\u2191(comap (algebraMap R S)) p) =\n    (fun x => Ideal.map (algebraMap R S) x.asIdeal) (\u2191(comap (algebraMap R S)) q)\n\u22a2 p = q"}, {"tactic": "dsimp only [comap, ContinuousMap.coe_mk] at h", "state_before": "R : Type u\nS : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\nS' : Type ?u.251612\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R S\nM : Submonoid R\ninst\u271d : IsLocalization M S\np q : PrimeSpectrum S\nh :\n  (fun x => Ideal.map (algebraMap R S) x.asIdeal) (\u2191(comap (algebraMap R S)) p) =\n    (fun x => Ideal.map (algebraMap R S) x.asIdeal) (\u2191(comap (algebraMap R S)) q)\n\u22a2 p = q", "state_after": "R : Type u\nS : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\nS' : Type ?u.251612\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R S\nM : Submonoid R\ninst\u271d : IsLocalization M S\np q : PrimeSpectrum S\nh :\n  Ideal.map (algebraMap R S) (Ideal.comap (algebraMap R S) p.asIdeal) =\n    Ideal.map (algebraMap R S) (Ideal.comap (algebraMap R S) q.asIdeal)\n\u22a2 p = q"}, {"tactic": "rw [IsLocalization.map_comap M S, IsLocalization.map_comap M S] at h", "state_before": "R : Type u\nS : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\nS' : Type ?u.251612\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R S\nM : Submonoid R\ninst\u271d : IsLocalization M S\np q : PrimeSpectrum S\nh :\n  Ideal.map (algebraMap R S) (Ideal.comap (algebraMap R S) p.asIdeal) =\n    Ideal.map (algebraMap R S) (Ideal.comap (algebraMap R S) q.asIdeal)\n\u22a2 p = q", "state_after": "R : Type u\nS : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\nS' : Type ?u.251612\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R S\nM : Submonoid R\ninst\u271d : IsLocalization M S\np q : PrimeSpectrum S\nh : p.asIdeal = q.asIdeal\n\u22a2 p = q"}, {"tactic": "ext1", "state_before": "R : Type u\nS : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\nS' : Type ?u.251612\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R S\nM : Submonoid R\ninst\u271d : IsLocalization M S\np q : PrimeSpectrum S\nh : p.asIdeal = q.asIdeal\n\u22a2 p = q", "state_after": "case asIdeal\nR : Type u\nS : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\nS' : Type ?u.251612\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R S\nM : Submonoid R\ninst\u271d : IsLocalization M S\np q : PrimeSpectrum S\nh : p.asIdeal = q.asIdeal\n\u22a2 p.asIdeal = q.asIdeal"}, {"tactic": "exact h", "state_before": "case asIdeal\nR : Type u\nS : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\nS' : Type ?u.251612\ninst\u271d\u00b2 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u00b9 : Algebra R S\nM : Submonoid R\ninst\u271d : IsLocalization M S\np q : PrimeSpectrum S\nh : p.asIdeal = q.asIdeal\n\u22a2 p.asIdeal = q.asIdeal", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Isometry.lean", "full_name": "IsometryEquiv.edist_eq", "start": [349, 11], "end": [350, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Sups.lean", "full_name": "Finset.card_infs_le", "start": [247, 1], "end": [248, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Expand.lean", "full_name": "Polynomial.expand_pow", "start": [91, 1], "end": [93, 64], "traced_tactics": [{"tactic": "rw [pow_zero, expand_one, Function.iterate_zero, id]", "state_before": "R : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np q : \u2115\nf : R[X]\n\u22a2 \u2191(expand R (p ^ Nat.zero)) f = (\u2191(expand R p)^[Nat.zero]) f", "state_after": "no goals"}, {"tactic": "rw [Function.iterate_succ_apply', pow_succ, expand_mul, ih]", "state_before": "R : Type u\ninst\u271d\u00b9 : CommSemiring R\nS : Type v\ninst\u271d : CommSemiring S\np q : \u2115\nf : R[X]\nn : \u2115\nih : \u2191(expand R (p ^ n)) f = (\u2191(expand R p)^[n]) f\n\u22a2 \u2191(expand R (p ^ Nat.succ n)) f = (\u2191(expand R p)^[Nat.succ n]) f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/FreeGroup.lean", "full_name": "FreeGroup.map_one", "start": [1039, 1], "end": [1040, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Hyperoperation.lean", "full_name": "hyperoperation_ge_three_one", "start": [109, 1], "end": [118, 41], "traced_tactics": [{"tactic": "induction' n with nn nih", "state_before": "n : \u2115\n\u22a2 \u2200 (k : \u2115), hyperoperation (n + 3) 1 k = 1", "state_after": "case zero\n\n\u22a2 \u2200 (k : \u2115), hyperoperation (Nat.zero + 3) 1 k = 1\n\ncase succ\nnn : \u2115\nnih : \u2200 (k : \u2115), hyperoperation (nn + 3) 1 k = 1\n\u22a2 \u2200 (k : \u2115), hyperoperation (Nat.succ nn + 3) 1 k = 1"}, {"tactic": "intro k", "state_before": "case zero\n\n\u22a2 \u2200 (k : \u2115), hyperoperation (Nat.zero + 3) 1 k = 1", "state_after": "case zero\nk : \u2115\n\u22a2 hyperoperation (Nat.zero + 3) 1 k = 1"}, {"tactic": "rw [hyperoperation_three]", "state_before": "case zero\nk : \u2115\n\u22a2 hyperoperation (Nat.zero + 3) 1 k = 1", "state_after": "case zero\nk : \u2115\n\u22a2 (fun x x_1 => x ^ x_1) 1 k = 1"}, {"tactic": "dsimp", "state_before": "case zero\nk : \u2115\n\u22a2 (fun x x_1 => x ^ x_1) 1 k = 1", "state_after": "case zero\nk : \u2115\n\u22a2 1 ^ k = 1"}, {"tactic": "rw [one_pow]", "state_before": "case zero\nk : \u2115\n\u22a2 1 ^ k = 1", "state_after": "no goals"}, {"tactic": "intro k", "state_before": "case succ\nnn : \u2115\nnih : \u2200 (k : \u2115), hyperoperation (nn + 3) 1 k = 1\n\u22a2 \u2200 (k : \u2115), hyperoperation (Nat.succ nn + 3) 1 k = 1", "state_after": "case succ\nnn : \u2115\nnih : \u2200 (k : \u2115), hyperoperation (nn + 3) 1 k = 1\nk : \u2115\n\u22a2 hyperoperation (Nat.succ nn + 3) 1 k = 1"}, {"tactic": "cases k", "state_before": "case succ\nnn : \u2115\nnih : \u2200 (k : \u2115), hyperoperation (nn + 3) 1 k = 1\nk : \u2115\n\u22a2 hyperoperation (Nat.succ nn + 3) 1 k = 1", "state_after": "case succ.zero\nnn : \u2115\nnih : \u2200 (k : \u2115), hyperoperation (nn + 3) 1 k = 1\n\u22a2 hyperoperation (Nat.succ nn + 3) 1 Nat.zero = 1\n\ncase succ.succ\nnn : \u2115\nnih : \u2200 (k : \u2115), hyperoperation (nn + 3) 1 k = 1\nn\u271d : \u2115\n\u22a2 hyperoperation (Nat.succ nn + 3) 1 (Nat.succ n\u271d) = 1"}, {"tactic": "rw [hyperoperation_ge_three_eq_one]", "state_before": "case succ.zero\nnn : \u2115\nnih : \u2200 (k : \u2115), hyperoperation (nn + 3) 1 k = 1\n\u22a2 hyperoperation (Nat.succ nn + 3) 1 Nat.zero = 1", "state_after": "no goals"}, {"tactic": "rw [hyperoperation_recursion, nih]", "state_before": "case succ.succ\nnn : \u2115\nnih : \u2200 (k : \u2115), hyperoperation (nn + 3) 1 k = 1\nn\u271d : \u2115\n\u22a2 hyperoperation (Nat.succ nn + 3) 1 (Nat.succ n\u271d) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/RingDivision.lean", "full_name": "Polynomial.pairwise_coprime_X_sub_C", "start": [1059, 1], "end": [1061, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Init/Algebra/Order.lean", "full_name": "compare_iff", "start": [454, 1], "end": [458, 28], "traced_tactics": [{"tactic": "cases o <;> simp only [Ordering.toRel]", "state_before": "\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\no : Ordering\n\u22a2 compare a b = o \u2194 Ordering.toRel o a b", "state_after": "case lt\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 compare a b = Ordering.lt \u2194 a < b\n\ncase eq\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 compare a b = Ordering.eq \u2194 a = b\n\ncase gt\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 compare a b = Ordering.gt \u2194 a > b"}, {"tactic": "exact compare_lt_iff_lt", "state_before": "case lt\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 compare a b = Ordering.lt \u2194 a < b", "state_after": "no goals"}, {"tactic": "exact compare_eq_iff_eq", "state_before": "case eq\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 compare a b = Ordering.eq \u2194 a = b", "state_after": "no goals"}, {"tactic": "exact compare_gt_iff_gt", "state_before": "case gt\n\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 compare a b = Ordering.gt \u2194 a > b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "full_name": "Real.Angle.pi_div_two_ne_zero", "start": [632, 1], "end": [634, 48], "traced_tactics": [{"tactic": "rw [\u2190 toReal_injective.ne_iff, toReal_pi_div_two, toReal_zero]", "state_before": "\u22a2 \u2191(\u03c0 / 2) \u2260 0", "state_after": "\u22a2 \u03c0 / 2 \u2260 0"}, {"tactic": "exact div_ne_zero Real.pi_ne_zero two_ne_zero", "state_before": "\u22a2 \u03c0 / 2 \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/ZPow.lean", "full_name": "Matrix.zpow_bit1'", "start": [323, 1], "end": [324, 52], "traced_tactics": [{"tactic": "rw [zpow_bit1, Commute.mul_zpow (Commute.refl A)]", "state_before": "n' : Type u_1\ninst\u271d\u00b2 : DecidableEq n'\ninst\u271d\u00b9 : Fintype n'\nR : Type u_2\ninst\u271d : CommRing R\nA : M\nn : \u2124\n\u22a2 A ^ bit1 n = (A * A) ^ n * A", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Prod/Basic.lean", "full_name": "Prod.snd_injective", "start": [153, 1], "end": [154, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Data/String/Extra.lean", "full_name": "String.Iterator.sizeOf_next_lt_of_hasNext", "start": [36, 1], "end": [38, 53], "traced_tactics": [{"tactic": "cases i", "state_before": "i : Iterator\nh : hasNext i = true\n\u22a2 sizeOf (next i) < sizeOf i", "state_after": "case mk\ns\u271d : String\ni\u271d : Pos\nh : hasNext { s := s\u271d, i := i\u271d } = true\n\u22a2 sizeOf (next { s := s\u271d, i := i\u271d }) < sizeOf { s := s\u271d, i := i\u271d }"}, {"tactic": "rename_i s pos", "state_before": "case mk\ns\u271d : String\ni\u271d : Pos\nh : hasNext { s := s\u271d, i := i\u271d } = true\n\u22a2 sizeOf (next { s := s\u271d, i := i\u271d }) < sizeOf { s := s\u271d, i := i\u271d }", "state_after": "case mk\ns : String\npos : Pos\nh : hasNext { s := s, i := pos } = true\n\u22a2 sizeOf (next { s := s, i := pos }) < sizeOf { s := s, i := pos }"}, {"tactic": "simp [Iterator.next, Iterator.sizeOf_eq]", "state_before": "case mk\ns : String\npos : Pos\nh : hasNext { s := s, i := pos } = true\n\u22a2 sizeOf (next { s := s, i := pos }) < sizeOf { s := s, i := pos }", "state_after": "case mk\ns : String\npos : Pos\nh : hasNext { s := s, i := pos } = true\n\u22a2 utf8ByteSize s - (String.next s pos).byteIdx < utf8ByteSize s - pos.byteIdx"}, {"tactic": "simp [Iterator.hasNext] at h", "state_before": "case mk\ns : String\npos : Pos\nh : hasNext { s := s, i := pos } = true\n\u22a2 utf8ByteSize s - (String.next s pos).byteIdx < utf8ByteSize s - pos.byteIdx", "state_after": "case mk\ns : String\npos : Pos\nh : pos.byteIdx < (endPos s).byteIdx\n\u22a2 utf8ByteSize s - (String.next s pos).byteIdx < utf8ByteSize s - pos.byteIdx"}, {"tactic": "exact Nat.sub_lt_sub_left h (String.lt_next s pos)", "state_before": "case mk\ns : String\npos : Pos\nh : pos.byteIdx < (endPos s).byteIdx\n\u22a2 utf8ByteSize s - (String.next s pos).byteIdx < utf8ByteSize s - pos.byteIdx", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean", "full_name": "TopCat.range_pullback_map", "start": [189, 1], "end": [215, 15], "traced_tactics": [{"tactic": "ext", "state_before": "J : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\n\u22a2 Set.range ((forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082)) =\n    (forget TopCat).map pullback.fst \u207b\u00b9' Set.range ((forget TopCat).map i\u2081) \u2229\n      (forget TopCat).map pullback.snd \u207b\u00b9' Set.range ((forget TopCat).map i\u2082)", "state_after": "case h\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\n\u22a2 x\u271d \u2208 Set.range ((forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082)) \u2194\n    x\u271d \u2208\n      (forget TopCat).map pullback.fst \u207b\u00b9' Set.range ((forget TopCat).map i\u2081) \u2229\n        (forget TopCat).map pullback.snd \u207b\u00b9' Set.range ((forget TopCat).map i\u2082)"}, {"tactic": "constructor", "state_before": "case h\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\n\u22a2 x\u271d \u2208 Set.range ((forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082)) \u2194\n    x\u271d \u2208\n      (forget TopCat).map pullback.fst \u207b\u00b9' Set.range ((forget TopCat).map i\u2081) \u2229\n        (forget TopCat).map pullback.snd \u207b\u00b9' Set.range ((forget TopCat).map i\u2082)", "state_after": "case h.mp\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\n\u22a2 x\u271d \u2208 Set.range ((forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082)) \u2192\n    x\u271d \u2208\n      (forget TopCat).map pullback.fst \u207b\u00b9' Set.range ((forget TopCat).map i\u2081) \u2229\n        (forget TopCat).map pullback.snd \u207b\u00b9' Set.range ((forget TopCat).map i\u2082)\n\ncase h.mpr\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\n\u22a2 x\u271d \u2208\n      (forget TopCat).map pullback.fst \u207b\u00b9' Set.range ((forget TopCat).map i\u2081) \u2229\n        (forget TopCat).map pullback.snd \u207b\u00b9' Set.range ((forget TopCat).map i\u2082) \u2192\n    x\u271d \u2208 Set.range ((forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082))"}, {"tactic": "rintro \u27e8\u27e8x\u2081, hx\u2081\u27e9, \u27e8x\u2082, hx\u2082\u27e9\u27e9", "state_before": "case h.mpr\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\n\u22a2 x\u271d \u2208\n      (forget TopCat).map pullback.fst \u207b\u00b9' Set.range ((forget TopCat).map i\u2081) \u2229\n        (forget TopCat).map pullback.snd \u207b\u00b9' Set.range ((forget TopCat).map i\u2082) \u2192\n    x\u271d \u2208 Set.range ((forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082))", "state_after": "case h.mpr.intro.intro.intro\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\n\u22a2 x\u271d \u2208 Set.range ((forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082))"}, {"tactic": "have : f\u2081 x\u2081 = f\u2082 x\u2082 := by\n  apply (TopCat.mono_iff_injective _).mp H\u2083\n  simp only [\u2190 comp_apply, eq\u2081, eq\u2082]\n  simp only [comp_apply, hx\u2081, hx\u2082]\n  simp only [\u2190 comp_apply, pullback.condition]", "state_before": "case h.mpr.intro.intro.intro\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\n\u22a2 x\u271d \u2208 Set.range ((forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082))", "state_after": "case h.mpr.intro.intro.intro\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\nthis : (forget TopCat).map f\u2081 x\u2081 = (forget TopCat).map f\u2082 x\u2082\n\u22a2 x\u271d \u2208 Set.range ((forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082))"}, {"tactic": "use (pullbackIsoProdSubtype f\u2081 f\u2082).inv \u27e8\u27e8x\u2081, x\u2082\u27e9, this\u27e9", "state_before": "case h.mpr.intro.intro.intro\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\nthis : (forget TopCat).map f\u2081 x\u2081 = (forget TopCat).map f\u2082 x\u2082\n\u22a2 x\u271d \u2208 Set.range ((forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082))", "state_after": "case h.mpr.intro.intro.intro\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\nthis : (forget TopCat).map f\u2081 x\u2081 = (forget TopCat).map f\u2082 x\u2082\n\u22a2 (forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082)\n      ((forget TopCat).map (pullbackIsoProdSubtype f\u2081 f\u2082).inv { val := (x\u2081, x\u2082), property := this }) =\n    x\u271d"}, {"tactic": "apply Concrete.limit_ext", "state_before": "case h.mpr.intro.intro.intro\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\nthis : (forget TopCat).map f\u2081 x\u2081 = (forget TopCat).map f\u2082 x\u2082\n\u22a2 (forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082)\n      ((forget TopCat).map (pullbackIsoProdSubtype f\u2081 f\u2082).inv { val := (x\u2081, x\u2082), property := this }) =\n    x\u271d", "state_after": "case h.mpr.intro.intro.intro.a\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\nthis : (forget TopCat).map f\u2081 x\u2081 = (forget TopCat).map f\u2082 x\u2082\n\u22a2 \u2200 (j : WalkingCospan),\n    (forget TopCat).map (limit.\u03c0 (cospan g\u2081 g\u2082) j)\n        ((forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082)\n          ((forget TopCat).map (pullbackIsoProdSubtype f\u2081 f\u2082).inv { val := (x\u2081, x\u2082), property := this })) =\n      (forget TopCat).map (limit.\u03c0 (cospan g\u2081 g\u2082) j) x\u271d"}, {"tactic": "rintro (_ | _ | _)", "state_before": "case h.mpr.intro.intro.intro.a\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\nthis : (forget TopCat).map f\u2081 x\u2081 = (forget TopCat).map f\u2082 x\u2082\n\u22a2 \u2200 (j : WalkingCospan),\n    (forget TopCat).map (limit.\u03c0 (cospan g\u2081 g\u2082) j)\n        ((forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082)\n          ((forget TopCat).map (pullbackIsoProdSubtype f\u2081 f\u2082).inv { val := (x\u2081, x\u2082), property := this })) =\n      (forget TopCat).map (limit.\u03c0 (cospan g\u2081 g\u2082) j) x\u271d", "state_after": "case h.mpr.intro.intro.intro.a.none\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\nthis : (forget TopCat).map f\u2081 x\u2081 = (forget TopCat).map f\u2082 x\u2082\n\u22a2 (forget TopCat).map (limit.\u03c0 (cospan g\u2081 g\u2082) none)\n      ((forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082)\n        ((forget TopCat).map (pullbackIsoProdSubtype f\u2081 f\u2082).inv { val := (x\u2081, x\u2082), property := this })) =\n    (forget TopCat).map (limit.\u03c0 (cospan g\u2081 g\u2082) none) x\u271d\n\ncase h.mpr.intro.intro.intro.a.some.left\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\nthis : (forget TopCat).map f\u2081 x\u2081 = (forget TopCat).map f\u2082 x\u2082\n\u22a2 (forget TopCat).map (limit.\u03c0 (cospan g\u2081 g\u2082) (some WalkingPair.left))\n      ((forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082)\n        ((forget TopCat).map (pullbackIsoProdSubtype f\u2081 f\u2082).inv { val := (x\u2081, x\u2082), property := this })) =\n    (forget TopCat).map (limit.\u03c0 (cospan g\u2081 g\u2082) (some WalkingPair.left)) x\u271d\n\ncase h.mpr.intro.intro.intro.a.some.right\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\nthis : (forget TopCat).map f\u2081 x\u2081 = (forget TopCat).map f\u2082 x\u2082\n\u22a2 (forget TopCat).map (limit.\u03c0 (cospan g\u2081 g\u2082) (some WalkingPair.right))\n      ((forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082)\n        ((forget TopCat).map (pullbackIsoProdSubtype f\u2081 f\u2082).inv { val := (x\u2081, x\u2082), property := this })) =\n    (forget TopCat).map (limit.\u03c0 (cospan g\u2081 g\u2082) (some WalkingPair.right)) x\u271d"}, {"tactic": "rintro \u27e8y, rfl\u27e9", "state_before": "case h.mp\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\n\u22a2 x\u271d \u2208 Set.range ((forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082)) \u2192\n    x\u271d \u2208\n      (forget TopCat).map pullback.fst \u207b\u00b9' Set.range ((forget TopCat).map i\u2081) \u2229\n        (forget TopCat).map pullback.snd \u207b\u00b9' Set.range ((forget TopCat).map i\u2082)", "state_after": "case h.mp.intro\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\ny : (forget TopCat).obj (pullback f\u2081 f\u2082)\n\u22a2 (forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082) y \u2208\n    (forget TopCat).map pullback.fst \u207b\u00b9' Set.range ((forget TopCat).map i\u2081) \u2229\n      (forget TopCat).map pullback.snd \u207b\u00b9' Set.range ((forget TopCat).map i\u2082)"}, {"tactic": "simp", "state_before": "case h.mp.intro\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\ny : (forget TopCat).obj (pullback f\u2081 f\u2082)\n\u22a2 (forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082) y \u2208\n    (forget TopCat).map pullback.fst \u207b\u00b9' Set.range ((forget TopCat).map i\u2081) \u2229\n      (forget TopCat).map pullback.snd \u207b\u00b9' Set.range ((forget TopCat).map i\u2082)", "state_after": "no goals"}, {"tactic": "apply (TopCat.mono_iff_injective _).mp H\u2083", "state_before": "J : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\n\u22a2 (forget TopCat).map f\u2081 x\u2081 = (forget TopCat).map f\u2082 x\u2082", "state_after": "case a\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\n\u22a2 (forget TopCat).map i\u2083 ((forget TopCat).map f\u2081 x\u2081) = (forget TopCat).map i\u2083 ((forget TopCat).map f\u2082 x\u2082)"}, {"tactic": "simp only [\u2190 comp_apply, eq\u2081, eq\u2082]", "state_before": "case a\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\n\u22a2 (forget TopCat).map i\u2083 ((forget TopCat).map f\u2081 x\u2081) = (forget TopCat).map i\u2083 ((forget TopCat).map f\u2082 x\u2082)", "state_after": "case a\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\n\u22a2 (forget TopCat).map (i\u2081 \u226b g\u2081) x\u2081 = (forget TopCat).map (i\u2082 \u226b g\u2082) x\u2082"}, {"tactic": "simp only [comp_apply, hx\u2081, hx\u2082]", "state_before": "case a\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\n\u22a2 (forget TopCat).map (i\u2081 \u226b g\u2081) x\u2081 = (forget TopCat).map (i\u2082 \u226b g\u2082) x\u2082", "state_after": "case a\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\n\u22a2 (forget TopCat).map g\u2081 ((forget TopCat).map pullback.fst x\u271d) =\n    (forget TopCat).map g\u2082 ((forget TopCat).map pullback.snd x\u271d)"}, {"tactic": "simp only [\u2190 comp_apply, pullback.condition]", "state_before": "case a\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\n\u22a2 (forget TopCat).map g\u2081 ((forget TopCat).map pullback.fst x\u271d) =\n    (forget TopCat).map g\u2082 ((forget TopCat).map pullback.snd x\u271d)", "state_after": "no goals"}, {"tactic": "simp only [TopCat.comp_app, limit.lift_\u03c0_apply, Category.assoc, PullbackCone.mk_\u03c0_app_one, hx\u2081,\n  pullbackIsoProdSubtype_inv_fst_apply, Subtype.coe_mk]", "state_before": "case h.mpr.intro.intro.intro.a.none\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\nthis : (forget TopCat).map f\u2081 x\u2081 = (forget TopCat).map f\u2082 x\u2082\n\u22a2 (forget TopCat).map (limit.\u03c0 (cospan g\u2081 g\u2082) none)\n      ((forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082)\n        ((forget TopCat).map (pullbackIsoProdSubtype f\u2081 f\u2082).inv { val := (x\u2081, x\u2082), property := this })) =\n    (forget TopCat).map (limit.\u03c0 (cospan g\u2081 g\u2082) none) x\u271d", "state_after": "case h.mpr.intro.intro.intro.a.none\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\nthis : (forget TopCat).map f\u2081 x\u2081 = (forget TopCat).map f\u2082 x\u2082\n\u22a2 (forget TopCat).map g\u2081 ((forget TopCat).map pullback.fst x\u271d) = (forget TopCat).map (limit.\u03c0 (cospan g\u2081 g\u2082) none) x\u271d"}, {"tactic": "simp only [\u2190 comp_apply]", "state_before": "case h.mpr.intro.intro.intro.a.none\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\nthis : (forget TopCat).map f\u2081 x\u2081 = (forget TopCat).map f\u2082 x\u2082\n\u22a2 (forget TopCat).map g\u2081 ((forget TopCat).map pullback.fst x\u271d) = (forget TopCat).map (limit.\u03c0 (cospan g\u2081 g\u2082) none) x\u271d", "state_after": "case h.mpr.intro.intro.intro.a.none\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\nthis : (forget TopCat).map f\u2081 x\u2081 = (forget TopCat).map f\u2082 x\u2082\n\u22a2 (forget TopCat).map (pullback.fst \u226b g\u2081) x\u271d = (forget TopCat).map (limit.\u03c0 (cospan g\u2081 g\u2082) none) x\u271d"}, {"tactic": "have : pullback.fst \u226b g\u2081 = limit.\u03c0 (cospan g\u2081 g\u2082) none := by\n  apply limit.w _ WalkingCospan.Hom.inl", "state_before": "case h.mpr.intro.intro.intro.a.none\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\nthis : (forget TopCat).map f\u2081 x\u2081 = (forget TopCat).map f\u2082 x\u2082\n\u22a2 (forget TopCat).map (pullback.fst \u226b g\u2081) x\u271d = (forget TopCat).map (limit.\u03c0 (cospan g\u2081 g\u2082) none) x\u271d", "state_after": "case h.mpr.intro.intro.intro.a.none\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\nthis\u271d : (forget TopCat).map f\u2081 x\u2081 = (forget TopCat).map f\u2082 x\u2082\nthis : pullback.fst \u226b g\u2081 = limit.\u03c0 (cospan g\u2081 g\u2082) none\n\u22a2 (forget TopCat).map (pullback.fst \u226b g\u2081) x\u271d = (forget TopCat).map (limit.\u03c0 (cospan g\u2081 g\u2082) none) x\u271d"}, {"tactic": "rw [this]", "state_before": "case h.mpr.intro.intro.intro.a.none\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\nthis\u271d : (forget TopCat).map f\u2081 x\u2081 = (forget TopCat).map f\u2082 x\u2082\nthis : pullback.fst \u226b g\u2081 = limit.\u03c0 (cospan g\u2081 g\u2082) none\n\u22a2 (forget TopCat).map (pullback.fst \u226b g\u2081) x\u271d = (forget TopCat).map (limit.\u03c0 (cospan g\u2081 g\u2082) none) x\u271d", "state_after": "no goals"}, {"tactic": "apply limit.w _ WalkingCospan.Hom.inl", "state_before": "J : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\nthis : (forget TopCat).map f\u2081 x\u2081 = (forget TopCat).map f\u2082 x\u2082\n\u22a2 pullback.fst \u226b g\u2081 = limit.\u03c0 (cospan g\u2081 g\u2082) none", "state_after": "no goals"}, {"tactic": "simp [hx\u2081]", "state_before": "case h.mpr.intro.intro.intro.a.some.left\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\nthis : (forget TopCat).map f\u2081 x\u2081 = (forget TopCat).map f\u2082 x\u2082\n\u22a2 (forget TopCat).map (limit.\u03c0 (cospan g\u2081 g\u2082) (some WalkingPair.left))\n      ((forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082)\n        ((forget TopCat).map (pullbackIsoProdSubtype f\u2081 f\u2082).inv { val := (x\u2081, x\u2082), property := this })) =\n    (forget TopCat).map (limit.\u03c0 (cospan g\u2081 g\u2082) (some WalkingPair.left)) x\u271d", "state_after": "no goals"}, {"tactic": "simp [hx\u2082]", "state_before": "case h.mpr.intro.intro.intro.a.some.right\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z\u271d : TopCat\nW X Y Z S T : TopCat\nf\u2081 : W \u27f6 S\nf\u2082 : X \u27f6 S\ng\u2081 : Y \u27f6 T\ng\u2082 : Z \u27f6 T\ni\u2081 : W \u27f6 Y\ni\u2082 : X \u27f6 Z\ni\u2083 : S \u27f6 T\nH\u2083 : Mono i\u2083\neq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081\neq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082\nx\u271d : (forget TopCat).obj (pullback g\u2081 g\u2082)\nx\u2081 : (forget TopCat).obj W\nhx\u2081 : (forget TopCat).map i\u2081 x\u2081 = (forget TopCat).map pullback.fst x\u271d\nx\u2082 : (forget TopCat).obj X\nhx\u2082 : (forget TopCat).map i\u2082 x\u2082 = (forget TopCat).map pullback.snd x\u271d\nthis : (forget TopCat).map f\u2081 x\u2081 = (forget TopCat).map f\u2082 x\u2082\n\u22a2 (forget TopCat).map (limit.\u03c0 (cospan g\u2081 g\u2082) (some WalkingPair.right))\n      ((forget TopCat).map (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082)\n        ((forget TopCat).map (pullbackIsoProdSubtype f\u2081 f\u2082).inv { val := (x\u2081, x\u2082), property := this })) =\n    (forget TopCat).map (limit.\u03c0 (cospan g\u2081 g\u2082) (some WalkingPair.right)) x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Eval.lean", "full_name": "Polynomial.map_id", "start": [810, 1], "end": [810, 85], "traced_tactics": [{"tactic": "simp [Polynomial.ext_iff, coeff_map]", "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\n\u22a2 map (RingHom.id R) p = p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/ContinuousOn.lean", "full_name": "nhdsWithin_pi_eq", "start": [322, 1], "end": [330, 26], "traced_tactics": [{"tactic": "simp only [nhdsWithin, nhds_pi, Filter.pi, pi_def, \u2190 iInf_principal_finite hI, comap_inf,\n  comap_principal, eval]", "state_before": "\u03b1\u271d : Type ?u.51973\n\u03b2 : Type ?u.51976\n\u03b3 : Type ?u.51979\n\u03b4 : Type ?u.51982\ninst\u271d\u00b9 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03b1 i)\nI : Set \u03b9\nhI : Set.Finite I\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nx : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 \ud835\udcdd[Set.pi I s] x =\n    (\u2a05 (i : \u03b9) (_ : i \u2208 I), comap (fun x => x i) (\ud835\udcdd[s i] x i)) \u2293 \u2a05 (i : \u03b9) (_ : \u00aci \u2208 I), comap (fun x => x i) (\ud835\udcdd (x i))", "state_after": "\u03b1\u271d : Type ?u.51973\n\u03b2 : Type ?u.51976\n\u03b3 : Type ?u.51979\n\u03b4 : Type ?u.51982\ninst\u271d\u00b9 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03b1 i)\nI : Set \u03b9\nhI : Set.Finite I\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nx : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 ((\u2a05 (i : \u03b9), comap (fun f => f i) (\ud835\udcdd (x i))) \u2293 \u2a05 (i : \u03b9) (_ : i \u2208 I), \ud835\udcdf ((fun f => f i) \u207b\u00b9' s i)) =\n    (\u2a05 (i : \u03b9) (_ : i \u2208 I), comap (fun x => x i) (\ud835\udcdd (x i)) \u2293 \ud835\udcdf ((fun x => x i) \u207b\u00b9' s i)) \u2293\n      \u2a05 (i : \u03b9) (_ : \u00aci \u2208 I), comap (fun x => x i) (\ud835\udcdd (x i))"}, {"tactic": "rw [iInf_split _ fun i => i \u2208 I, inf_right_comm]", "state_before": "\u03b1\u271d : Type ?u.51973\n\u03b2 : Type ?u.51976\n\u03b3 : Type ?u.51979\n\u03b4 : Type ?u.51982\ninst\u271d\u00b9 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03b1 i)\nI : Set \u03b9\nhI : Set.Finite I\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nx : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 ((\u2a05 (i : \u03b9), comap (fun f => f i) (\ud835\udcdd (x i))) \u2293 \u2a05 (i : \u03b9) (_ : i \u2208 I), \ud835\udcdf ((fun f => f i) \u207b\u00b9' s i)) =\n    (\u2a05 (i : \u03b9) (_ : i \u2208 I), comap (fun x => x i) (\ud835\udcdd (x i)) \u2293 \ud835\udcdf ((fun x => x i) \u207b\u00b9' s i)) \u2293\n      \u2a05 (i : \u03b9) (_ : \u00aci \u2208 I), comap (fun x => x i) (\ud835\udcdd (x i))", "state_after": "\u03b1\u271d : Type ?u.51973\n\u03b2 : Type ?u.51976\n\u03b3 : Type ?u.51979\n\u03b4 : Type ?u.51982\ninst\u271d\u00b9 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03b1 i)\nI : Set \u03b9\nhI : Set.Finite I\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nx : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (((\u2a05 (i : \u03b9) (_ : i \u2208 I), comap (fun f => f i) (\ud835\udcdd (x i))) \u2293 \u2a05 (i : \u03b9) (_ : i \u2208 I), \ud835\udcdf ((fun f => f i) \u207b\u00b9' s i)) \u2293\n      \u2a05 (i : \u03b9) (_ : \u00aci \u2208 I), comap (fun f => f i) (\ud835\udcdd (x i))) =\n    (\u2a05 (i : \u03b9) (_ : i \u2208 I), comap (fun x => x i) (\ud835\udcdd (x i)) \u2293 \ud835\udcdf ((fun x => x i) \u207b\u00b9' s i)) \u2293\n      \u2a05 (i : \u03b9) (_ : \u00aci \u2208 I), comap (fun x => x i) (\ud835\udcdd (x i))"}, {"tactic": "simp only [iInf_inf_eq]", "state_before": "\u03b1\u271d : Type ?u.51973\n\u03b2 : Type ?u.51976\n\u03b3 : Type ?u.51979\n\u03b4 : Type ?u.51982\ninst\u271d\u00b9 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03b1 i)\nI : Set \u03b9\nhI : Set.Finite I\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nx : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (((\u2a05 (i : \u03b9) (_ : i \u2208 I), comap (fun f => f i) (\ud835\udcdd (x i))) \u2293 \u2a05 (i : \u03b9) (_ : i \u2208 I), \ud835\udcdf ((fun f => f i) \u207b\u00b9' s i)) \u2293\n      \u2a05 (i : \u03b9) (_ : \u00aci \u2208 I), comap (fun f => f i) (\ud835\udcdd (x i))) =\n    (\u2a05 (i : \u03b9) (_ : i \u2208 I), comap (fun x => x i) (\ud835\udcdd (x i)) \u2293 \ud835\udcdf ((fun x => x i) \u207b\u00b9' s i)) \u2293\n      \u2a05 (i : \u03b9) (_ : \u00aci \u2208 I), comap (fun x => x i) (\ud835\udcdd (x i))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Star/Subalgebra.lean", "full_name": "StarSubalgebra.subset_adjoin", "start": [443, 1], "end": [444, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Multilinear/Basic.lean", "full_name": "MultilinearMap.toFun_eq_coe", "start": [123, 1], "end": [124, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/CrossProduct.lean", "full_name": "triple_product_permutation", "start": [113, 1], "end": [116, 7], "traced_tactics": [{"tactic": "simp_rw [cross_apply, vec3_dotProduct]", "state_before": "R : Type u_1\ninst\u271d : CommRing R\nu v w : Fin 3 \u2192 R\n\u22a2 u \u2b1d\u1d65 \u2191(\u2191crossProduct v) w = v \u2b1d\u1d65 \u2191(\u2191crossProduct w) u", "state_after": "R : Type u_1\ninst\u271d : CommRing R\nu v w : Fin 3 \u2192 R\n\u22a2 u 0 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0 +\n        u 1 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1 +\n      u 2 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2 =\n    v 0 * vecCons (w 1 * u 2 - w 2 * u 1) ![w 2 * u 0 - w 0 * u 2, w 0 * u 1 - w 1 * u 0] 0 +\n        v 1 * vecCons (w 1 * u 2 - w 2 * u 1) ![w 2 * u 0 - w 0 * u 2, w 0 * u 1 - w 1 * u 0] 1 +\n      v 2 * vecCons (w 1 * u 2 - w 2 * u 1) ![w 2 * u 0 - w 0 * u 2, w 0 * u 1 - w 1 * u 0] 2"}, {"tactic": "norm_num", "state_before": "R : Type u_1\ninst\u271d : CommRing R\nu v w : Fin 3 \u2192 R\n\u22a2 u 0 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 0 +\n        u 1 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 1 +\n      u 2 * vecCons (v 1 * w 2 - v 2 * w 1) ![v 2 * w 0 - v 0 * w 2, v 0 * w 1 - v 1 * w 0] 2 =\n    v 0 * vecCons (w 1 * u 2 - w 2 * u 1) ![w 2 * u 0 - w 0 * u 2, w 0 * u 1 - w 1 * u 0] 0 +\n        v 1 * vecCons (w 1 * u 2 - w 2 * u 1) ![w 2 * u 0 - w 0 * u 2, w 0 * u 1 - w 1 * u 0] 1 +\n      v 2 * vecCons (w 1 * u 2 - w 2 * u 1) ![w 2 * u 0 - w 0 * u 2, w 0 * u 1 - w 1 * u 0] 2", "state_after": "R : Type u_1\ninst\u271d : CommRing R\nu v w : Fin 3 \u2192 R\n\u22a2 u 0 * (v 1 * w 2 - v 2 * w 1) + u 1 * (v 2 * w 0 - v 0 * w 2) + u 2 * (v 0 * w 1 - v 1 * w 0) =\n    v 0 * (w 1 * u 2 - w 2 * u 1) + v 1 * (w 2 * u 0 - w 0 * u 2) + v 2 * (w 0 * u 1 - w 1 * u 0)"}, {"tactic": "ring", "state_before": "R : Type u_1\ninst\u271d : CommRing R\nu v w : Fin 3 \u2192 R\n\u22a2 u 0 * (v 1 * w 2 - v 2 * w 1) + u 1 * (v 2 * w 0 - v 0 * w 2) + u 2 * (v 0 * w 1 - v 1 * w 0) =\n    v 0 * (w 1 * u 2 - w 2 * u 1) + v 1 * (w 2 * u 0 - w 0 * u 2) + v 2 * (w 0 * u 1 - w 1 * u 0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Sups.lean", "full_name": "Set.sups_comm", "start": [198, 1], "end": [199, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "full_name": "orthonormal_subtype_range", "start": [890, 1], 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NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nx : E\n\u22a2 x \u2208 \u22a4\u15ee \u2194 x \u2208 \u22a5"}, {"tactic": "rw [mem_bot, mem_orthogonal]", "state_before": "case h\n\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.84244\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nx : E\n\u22a2 x \u2208 \u22a4\u15ee \u2194 x \u2208 \u22a5", "state_after": "case h\n\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.84244\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nx : E\n\u22a2 (\u2200 (u : E), u \u2208 \u22a4 \u2192 inner u x = 0) \u2194 x = 0"}, {"tactic": "exact\n  \u27e8fun h => inner_self_eq_zero.mp (h x mem_top), by\n    rintro rfl\n    simp\u27e9", "state_before": "case h\n\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.84244\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nx : E\n\u22a2 (\u2200 (u : E), u \u2208 \u22a4 \u2192 inner u x = 0) \u2194 x = 0", "state_after": "no goals"}, {"tactic": "rintro rfl", "state_before": "\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.84244\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nx : E\n\u22a2 x = 0 \u2192 \u2200 (u : E), u \u2208 \u22a4 \u2192 inner u x = 0", "state_after": "\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.84244\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\n\u22a2 \u2200 (u : E), u \u2208 \u22a4 \u2192 inner u 0 = 0"}, {"tactic": "simp", "state_before": "\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.84244\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\n\u22a2 \u2200 (u : E), u \u2208 \u22a4 \u2192 inner u 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/Multilinear.lean", "full_name": "ContinuousMultilinearMap.le_of_op_nnnorm_le", "start": [472, 1], "end": [473, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Int/Basic.lean", "full_name": "Int.associated_natAbs", "start": [365, 1], "end": [366, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/WithZeroTopology.lean", "full_name": "WithZeroTopology.singleton_mem_nhds_of_ne_zero", "start": [109, 1], "end": [109, 95], "traced_tactics": [{"tactic": "simp [h]", "state_before": "\u03b1 : Type ?u.50635\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\nh : \u03b3 \u2260 0\n\u22a2 {\u03b3} \u2208 \ud835\udcdd \u03b3", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Size.lean", "full_name": "Nat.size_le", "start": [142, 1], "end": [155, 69], "traced_tactics": [{"tactic": "decide", "state_before": "m n : \u2115\nh : size m \u2264 n\n\u22a2 2 > 0", "state_after": "no goals"}, {"tactic": "rw [\u2190 one_shiftl]", "state_before": "m n : \u2115\n\u22a2 m < 2 ^ n \u2192 size m \u2264 n", "state_after": "m n : \u2115\n\u22a2 m < shiftl 1 n \u2192 size m \u2264 n"}, {"tactic": "revert n", "state_before": "m n : \u2115\n\u22a2 m < shiftl 1 n \u2192 size m \u2264 n", "state_after": "m : \u2115\n\u22a2 \u2200 {n : \u2115}, m < shiftl 1 n \u2192 size m \u2264 n"}, {"tactic": "apply binaryRec _ _ m", "state_before": "m : \u2115\n\u22a2 \u2200 {n : \u2115}, m < shiftl 1 n \u2192 size m \u2264 n", "state_after": "m : \u2115\n\u22a2 \u2200 {n : \u2115}, 0 < shiftl 1 n \u2192 size 0 \u2264 n\n\nm : \u2115\n\u22a2 \u2200 (b : Bool) (n : \u2115),\n    (\u2200 {n_1 : \u2115}, n < shiftl 1 n_1 \u2192 size n \u2264 n_1) \u2192 \u2200 {n_1 : \u2115}, bit b n < shiftl 1 n_1 \u2192 size (bit b n) \u2264 n_1"}, {"tactic": "intro n", "state_before": "m : \u2115\n\u22a2 \u2200 {n : \u2115}, 0 < shiftl 1 n \u2192 size 0 \u2264 n", "state_after": "m n : \u2115\n\u22a2 0 < shiftl 1 n \u2192 size 0 \u2264 n"}, {"tactic": "simp", "state_before": "m n : \u2115\n\u22a2 0 < shiftl 1 n \u2192 size 0 \u2264 n", "state_after": "no goals"}, {"tactic": "intro b m IH n h", "state_before": "m : \u2115\n\u22a2 \u2200 (b : Bool) (n : \u2115),\n    (\u2200 {n_1 : \u2115}, n < shiftl 1 n_1 \u2192 size n \u2264 n_1) \u2192 \u2200 {n_1 : \u2115}, bit b n < shiftl 1 n_1 \u2192 size (bit b n) \u2264 n_1", "state_after": "m\u271d : \u2115\nb : Bool\nm : \u2115\nIH : \u2200 {n : \u2115}, m < shiftl 1 n \u2192 size m \u2264 n\nn : \u2115\nh : bit b m < shiftl 1 n\n\u22a2 size (bit b m) \u2264 n"}, {"tactic": "by_cases e : bit b m = 0", "state_before": "m\u271d : \u2115\nb : Bool\nm : \u2115\nIH : \u2200 {n : \u2115}, m < shiftl 1 n \u2192 size m \u2264 n\nn : \u2115\nh : bit b m < shiftl 1 n\n\u22a2 size (bit b m) \u2264 n", "state_after": "case pos\nm\u271d : \u2115\nb : Bool\nm : \u2115\nIH : \u2200 {n : \u2115}, m < shiftl 1 n \u2192 size m \u2264 n\nn : \u2115\nh : bit b m < shiftl 1 n\ne : bit b m = 0\n\u22a2 size (bit b m) \u2264 n\n\ncase neg\nm\u271d : \u2115\nb : Bool\nm : \u2115\nIH : \u2200 {n : \u2115}, m < shiftl 1 n \u2192 size m \u2264 n\nn : \u2115\nh : bit b m < shiftl 1 n\ne : \u00acbit b m = 0\n\u22a2 size (bit b m) \u2264 n"}, {"tactic": "rw [size_bit e]", "state_before": "case neg\nm\u271d : \u2115\nb : Bool\nm : \u2115\nIH : \u2200 {n : \u2115}, m < shiftl 1 n \u2192 size m \u2264 n\nn : \u2115\nh : bit b m < shiftl 1 n\ne : \u00acbit b m = 0\n\u22a2 size (bit b m) \u2264 n", "state_after": "case neg\nm\u271d : \u2115\nb : Bool\nm : \u2115\nIH : \u2200 {n : \u2115}, m < shiftl 1 n \u2192 size m \u2264 n\nn : \u2115\nh : bit b m < shiftl 1 n\ne : \u00acbit b m = 0\n\u22a2 succ (size m) \u2264 n"}, {"tactic": "cases' n with n", "state_before": "case neg\nm\u271d : \u2115\nb : Bool\nm : \u2115\nIH : \u2200 {n : \u2115}, m < shiftl 1 n \u2192 size m \u2264 n\nn : \u2115\nh : bit b m < shiftl 1 n\ne : \u00acbit b m = 0\n\u22a2 succ (size m) \u2264 n", "state_after": "case neg.zero\nm\u271d : \u2115\nb : Bool\nm : \u2115\nIH : \u2200 {n : \u2115}, m < shiftl 1 n \u2192 size m \u2264 n\ne : \u00acbit b m = 0\nh : bit b m < shiftl 1 zero\n\u22a2 succ (size m) \u2264 zero\n\ncase neg.succ\nm\u271d : \u2115\nb : Bool\nm : \u2115\nIH : \u2200 {n : \u2115}, m < shiftl 1 n \u2192 size m \u2264 n\ne : \u00acbit b m = 0\nn : \u2115\nh : bit b m < shiftl 1 (succ n)\n\u22a2 succ (size m) \u2264 succ n"}, {"tactic": "simp [e]", "state_before": "case pos\nm\u271d : \u2115\nb : Bool\nm : \u2115\nIH : \u2200 {n : \u2115}, m < shiftl 1 n \u2192 size m \u2264 n\nn : \u2115\nh : bit b m < shiftl 1 n\ne : bit b m = 0\n\u22a2 size (bit b m) \u2264 n", "state_after": "no goals"}, {"tactic": "exact e.elim (Nat.eq_zero_of_le_zero (le_of_lt_succ h))", "state_before": "case neg.zero\nm\u271d : \u2115\nb : Bool\nm : \u2115\nIH : \u2200 {n : \u2115}, m < shiftl 1 n \u2192 size m \u2264 n\ne : \u00acbit b m = 0\nh : bit b m < shiftl 1 zero\n\u22a2 succ (size m) \u2264 zero", "state_after": "no goals"}, {"tactic": "apply succ_le_succ (IH _)", "state_before": "case neg.succ\nm\u271d : \u2115\nb : Bool\nm : \u2115\nIH : \u2200 {n : \u2115}, m < shiftl 1 n \u2192 size m \u2264 n\ne : \u00acbit b m = 0\nn : \u2115\nh : bit b m < shiftl 1 (succ n)\n\u22a2 succ (size m) \u2264 succ n", "state_after": "m\u271d : \u2115\nb : Bool\nm : \u2115\nIH : \u2200 {n : \u2115}, m < shiftl 1 n \u2192 size m \u2264 n\ne : \u00acbit b m = 0\nn : \u2115\nh : bit b m < shiftl 1 (succ n)\n\u22a2 m < shiftl 1 n"}, {"tactic": "apply lt_imp_lt_of_le_imp_le (fun h' => bit0_le_bit _ h') h", "state_before": "m\u271d : \u2115\nb : Bool\nm : \u2115\nIH : \u2200 {n : \u2115}, m < shiftl 1 n \u2192 size m \u2264 n\ne : \u00acbit b m = 0\nn : \u2115\nh : bit b m < shiftl 1 (succ n)\n\u22a2 m < shiftl 1 n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/BoxIntegral/Partition/SubboxInduction.lean", "full_name": "BoxIntegral.Prepartition.isPartition_splitCenter", "start": [61, 1], "end": [62, 12], "traced_tactics": [{"tactic": "simp [hx]", "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J I : Box \u03b9\nx : \u03b9 \u2192 \u211d\nhx : x \u2208 I\n\u22a2 \u2203 J, J \u2208 splitCenter I \u2227 x \u2208 J", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean", "full_name": "CategoryTheory.Limits.diagonalObjPullbackFstIso_hom_fst_snd", "start": [313, 1], "end": [317, 7], "traced_tactics": [{"tactic": "delta diagonalObjPullbackFstIso", "state_before": "C : Type u_2\ninst\u271d\u2075 : Category C\nX\u271d Y\u271d Z\u271d : C\ninst\u271d\u2074 : HasPullbacks C\nS T : C\nf\u271d : X\u271d \u27f6 T\ng\u271d : Y\u271d \u27f6 T\ni : T \u27f6 S\ninst\u271d\u00b3 : HasPullback i i\ninst\u271d\u00b2 : HasPullback f\u271d g\u271d\ninst\u271d\u00b9 : HasPullback (f\u271d \u226b i) (g\u271d \u226b i)\ninst\u271d :\n  HasPullback (diagonal i)\n    (map (f\u271d \u226b i) (g\u271d \u226b i) i i f\u271d g\u271d (\ud835\udfd9 S) (_ : (f\u271d \u226b i) \u226b \ud835\udfd9 S = f\u271d \u226b i) (_ : (g\u271d \u226b i) \u226b \ud835\udfd9 S = g\u271d \u226b i))\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 (diagonalObjPullbackFstIso f g).hom \u226b fst \u226b snd = snd \u226b snd", "state_after": "C : Type u_2\ninst\u271d\u2075 : Category C\nX\u271d Y\u271d Z\u271d : C\ninst\u271d\u2074 : HasPullbacks C\nS T : C\nf\u271d : X\u271d \u27f6 T\ng\u271d : Y\u271d \u27f6 T\ni : T \u27f6 S\ninst\u271d\u00b3 : HasPullback i i\ninst\u271d\u00b2 : HasPullback f\u271d g\u271d\ninst\u271d\u00b9 : HasPullback (f\u271d \u226b i) (g\u271d \u226b i)\ninst\u271d :\n  HasPullback (diagonal i)\n    (map (f\u271d \u226b i) (g\u271d \u226b i) i i f\u271d g\u271d (\ud835\udfd9 S) (_ : (f\u271d \u226b i) \u226b \ud835\udfd9 S = f\u271d \u226b i) (_ : (g\u271d \u226b i) \u226b \ud835\udfd9 S = g\u271d \u226b i))\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 (pullbackRightPullbackFstIso f g fst \u226a\u226b\n          congrHom (_ : fst \u226b f = snd \u226b g) (_ : g = g) \u226a\u226b\n            pullbackAssoc f g g g \u226a\u226b pullbackSymmetry f (fst \u226b g) \u226a\u226b congrHom (_ : fst \u226b g = snd \u226b g) (_ : f = f)).hom \u226b\n      fst \u226b snd =\n    snd \u226b snd"}, {"tactic": "simp", "state_before": "C : Type u_2\ninst\u271d\u2075 : Category C\nX\u271d Y\u271d Z\u271d : C\ninst\u271d\u2074 : HasPullbacks C\nS T : C\nf\u271d : X\u271d \u27f6 T\ng\u271d : Y\u271d \u27f6 T\ni : T \u27f6 S\ninst\u271d\u00b3 : HasPullback i i\ninst\u271d\u00b2 : HasPullback f\u271d g\u271d\ninst\u271d\u00b9 : HasPullback (f\u271d \u226b i) (g\u271d \u226b i)\ninst\u271d :\n  HasPullback (diagonal i)\n    (map (f\u271d \u226b i) (g\u271d \u226b i) i i f\u271d g\u271d (\ud835\udfd9 S) (_ : (f\u271d \u226b i) \u226b \ud835\udfd9 S = f\u271d \u226b i) (_ : (g\u271d \u226b i) \u226b \ud835\udfd9 S = g\u271d \u226b i))\nX Y Z : C\nf : X \u27f6 Z\ng : Y \u27f6 Z\n\u22a2 (pullbackRightPullbackFstIso f g fst \u226a\u226b\n          congrHom (_ : fst \u226b f = snd \u226b g) (_ : g = g) \u226a\u226b\n            pullbackAssoc f g g g \u226a\u226b pullbackSymmetry f (fst \u226b g) \u226a\u226b congrHom (_ : fst \u226b g = snd \u226b g) (_ : f = f)).hom \u226b\n      fst \u226b snd =\n    snd \u226b snd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Dfinsupp/Basic.lean", "full_name": "Dfinsupp.mapRange_apply", "start": [158, 1], "end": [160, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Yoneda.lean", "full_name": "CategoryTheory.Coyoneda.isIso", "start": [138, 1], "end": [139, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Preadditive/Basic.lean", "full_name": "CategoryTheory.Preadditive.isColimitCoforkOfCokernelCofork_desc", "start": [404, 1], "end": [407, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Seq/Seq.lean", "full_name": "Stream'.Seq.ofList_cons", "start": [459, 1], "end": [460, 23], "traced_tactics": [{"tactic": "ext1 (_ | n) <;> rfl", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nl : List \u03b1\n\u22a2 \u2191(a :: l) = cons a \u2191l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Complex/Arg.lean", "full_name": "Complex.abs_sub_eq", "start": [68, 1], "end": [69, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/Basic.lean", "full_name": "UniformSpace.mem_nhds_iff", "start": [740, 1], "end": [742, 16], "traced_tactics": [{"tactic": "rw [nhds_eq_comap_uniformity, mem_comap]", "state_before": "\u03b1 : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort ?u.59770\ninst\u271d : UniformSpace \u03b1\nx : \u03b1\ns : Set \u03b1\n\u22a2 s \u2208 \ud835\udcdd x \u2194 \u2203 V, V \u2208 \ud835\udce4 \u03b1 \u2227 ball x V \u2286 s", "state_after": "\u03b1 : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort ?u.59770\ninst\u271d : UniformSpace \u03b1\nx : \u03b1\ns : Set \u03b1\n\u22a2 (\u2203 t, t \u2208 \ud835\udce4 \u03b1 \u2227 Prod.mk x \u207b\u00b9' t \u2286 s) \u2194 \u2203 V, V \u2208 \ud835\udce4 \u03b1 \u2227 ball x V \u2286 s"}, {"tactic": "exact Iff.rfl", "state_before": "\u03b1 : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort ?u.59770\ninst\u271d : UniformSpace \u03b1\nx : \u03b1\ns : Set \u03b1\n\u22a2 (\u2203 t, t \u2208 \ud835\udce4 \u03b1 \u2227 Prod.mk x \u207b\u00b9' t \u2286 s) \u2194 \u2203 V, V \u2208 \ud835\udce4 \u03b1 \u2227 ball x V \u2286 s", "state_after": "no goals"}]}, {"url": 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"file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Int.floor_sub_nat", "start": [766, 1], "end": [766, 101], "traced_tactics": [{"tactic": "rw [\u2190 Int.cast_ofNat, floor_sub_int]", "state_before": "F : Type ?u.136124\n\u03b1 : Type u_1\n\u03b2 : Type ?u.136130\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nn : \u2115\n\u22a2 \u230aa - \u2191n\u230b = \u230aa\u230b - \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Int/Lemmas.lean", "full_name": "Int.lt_of_le_of_lt", "start": [674, 11], "end": [675, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.card_map", "start": [1233, 1], "end": [1234, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Topology.lean", "full_name": "Convex.openSegment_interior_closure_subset_interior", "start": [182, 1], "end": [185, 66], "traced_tactics": [{"tactic": "rintro _ \u27e8a, b, ha, hb, hab, rfl\u27e9", "state_before": "\u03b9 : Type ?u.99733\n\ud835\udd5c : Type u_2\nE : Type u_1\ninst\u271d\u2075 : LinearOrderedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousConstSMul \ud835\udd5c E\ns : Set E\nhs : Convex \ud835\udd5c s\nx y : E\nhx : x \u2208 interior s\nhy : y \u2208 closure s\n\u22a2 openSegment \ud835\udd5c x y \u2286 interior s", "state_after": "case intro.intro.intro.intro.intro\n\u03b9 : Type ?u.99733\n\ud835\udd5c : Type u_2\nE : Type u_1\ninst\u271d\u2075 : LinearOrderedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousConstSMul \ud835\udd5c E\ns : Set E\nhs : Convex \ud835\udd5c s\nx y : E\nhx : x \u2208 interior s\nhy : y \u2208 closure s\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 x + b \u2022 y \u2208 interior s"}, {"tactic": "exact hs.combo_interior_closure_mem_interior hx hy ha hb.le hab", "state_before": "case intro.intro.intro.intro.intro\n\u03b9 : Type ?u.99733\n\ud835\udd5c : Type u_2\nE : Type u_1\ninst\u271d\u2075 : LinearOrderedField \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousConstSMul \ud835\udd5c E\ns : Set E\nhs : Convex \ud835\udd5c s\nx y : E\nhx : x \u2208 interior s\nhy : y \u2208 closure s\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 x + b \u2022 y \u2208 interior s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "full_name": "Matrix.det_fromBlocks_zero\u2082\u2081", "start": [650, 1], "end": [700, 48], "traced_tactics": [{"tactic": "simp_rw [det_apply']", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 det (fromBlocks A B 0 D) = det A * det D", "state_after": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2211 x : Perm (m \u2295 n), \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1 =\n    (\u2211 x : Perm m, \u2191\u2191(\u2191sign x) * \u220f x_1 : m, A (\u2191x x_1) x_1) * \u2211 x : Perm n, \u2191\u2191(\u2191sign x) * \u220f x_1 : n, D (\u2191x x_1) x_1"}, {"tactic": "convert Eq.symm <|\n  sum_subset (\u03b2 := R) (subset_univ ((sumCongrHom m n).range : Set (Perm (Sum m n))).toFinset) ?_", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2211 x : Perm (m \u2295 n), \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1 =\n    (\u2211 x : Perm m, \u2191\u2191(\u2191sign x) * \u220f x_1 : m, A (\u2191x x_1) x_1) * \u2211 x : Perm n, \u2191\u2191(\u2191sign x) * \u220f x_1 : n, D (\u2191x x_1) x_1", "state_after": "case h.e'_3\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 (\u2211 x : Perm m, \u2191\u2191(\u2191sign x) * \u220f x_1 : m, A (\u2191x x_1) x_1) * \u2211 x : Perm n, \u2191\u2191(\u2191sign x) * \u220f x_1 : n, D (\u2191x x_1) x_1 =\n    \u2211 x in Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)),\n      \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1\n\ncase convert_2\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (x : Perm (m \u2295 n)),\n    x \u2208 univ \u2192\n      \u00acx \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)) \u2192\n        \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1 = 0"}, {"tactic": "rw [sum_mul_sum]", "state_before": "case h.e'_3\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 (\u2211 x : Perm m, \u2191\u2191(\u2191sign x) * \u220f x_1 : m, A (\u2191x x_1) x_1) * \u2211 x : Perm n, \u2191\u2191(\u2191sign x) * \u220f x_1 : n, D (\u2191x x_1) x_1 =\n    \u2211 x in Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)),\n      \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1\n\ncase convert_2\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (x : Perm (m \u2295 n)),\n    x \u2208 univ \u2192\n      \u00acx \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)) \u2192\n        \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1 = 0", "state_after": "case h.e'_3\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2211 p in univ \u00d7\u02e2 univ, (\u2191\u2191(\u2191sign p.fst) * \u220f x : m, A (\u2191p.fst x) x) * (\u2191\u2191(\u2191sign p.snd) * \u220f x : n, D (\u2191p.snd x) x) =\n    \u2211 x in Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)),\n      \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1\n\ncase convert_2\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (x : Perm (m \u2295 n)),\n    x \u2208 univ \u2192\n      \u00acx \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)) \u2192\n        \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1 = 0"}, {"tactic": "simp_rw [univ_product_univ]", "state_before": "case h.e'_3\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2211 p in univ \u00d7\u02e2 univ, (\u2191\u2191(\u2191sign p.fst) * \u220f x : m, A (\u2191p.fst x) x) * (\u2191\u2191(\u2191sign p.snd) * \u220f x : n, D (\u2191p.snd x) x) =\n    \u2211 x in Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)),\n      \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1\n\ncase convert_2\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (x : Perm (m \u2295 n)),\n    x \u2208 univ \u2192\n      \u00acx \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)) \u2192\n        \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1 = 0", "state_after": "case h.e'_3\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2211 p : Perm m \u00d7 Perm n, (\u2191\u2191(\u2191sign p.fst) * \u220f x : m, A (\u2191p.fst x) x) * (\u2191\u2191(\u2191sign p.snd) * \u220f x : n, D (\u2191p.snd x) x) =\n    \u2211 x in Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)),\n      \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1\n\ncase convert_2\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (x : Perm (m \u2295 n)),\n    x \u2208 univ \u2192\n      \u00acx \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)) \u2192\n        \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1 = 0"}, {"tactic": "rw [(sum_bij (fun (\u03c3 : Perm m \u00d7 Perm n) _ => Equiv.sumCongr \u03c3.fst \u03c3.snd) _ _ _ _).symm]", "state_before": "case h.e'_3\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2211 p : Perm m \u00d7 Perm n, (\u2191\u2191(\u2191sign p.fst) * \u220f x : m, A (\u2191p.fst x) x) * (\u2191\u2191(\u2191sign p.snd) * \u220f x : n, D (\u2191p.snd x) x) =\n    \u2211 x in Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)),\n      \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1\n\ncase convert_2\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (x : Perm (m \u2295 n)),\n    x \u2208 univ \u2192\n      \u00acx \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)) \u2192\n        \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1 = 0", "state_after": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (a : Perm m \u00d7 Perm n) (ha : a \u2208 univ),\n    (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a ha \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\n\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (a : Perm m \u00d7 Perm n) (ha : a \u2208 univ),\n    (\u2191\u2191(\u2191sign a.fst) * \u220f x : m, A (\u2191a.fst x) x) * (\u2191\u2191(\u2191sign a.snd) * \u220f x : n, D (\u2191a.snd x) x) =\n      \u2191\u2191(\u2191sign ((fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a ha)) *\n        \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191((fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a ha) x) x\n\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (a\u2081 a\u2082 : Perm m \u00d7 Perm n) (ha\u2081 : a\u2081 \u2208 univ) (ha\u2082 : a\u2082 \u2208 univ),\n    (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a\u2081 ha\u2081 = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\n\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (b : m \u2295 n \u2243 m \u2295 n),\n    b \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)) \u2192 \u2203 a ha, b = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a ha\n\ncase convert_2\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (x : Perm (m \u2295 n)),\n    x \u2208 univ \u2192\n      \u00acx \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)) \u2192\n        \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1 = 0"}, {"tactic": "intro \u03c3\u2081\u2082 h", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (a : Perm m \u00d7 Perm n) (ha : a \u2208 univ),\n    (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a ha \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))", "state_after": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081\u2082 : Perm m \u00d7 Perm n\nh : \u03c3\u2081\u2082 \u2208 univ\n\u22a2 (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) \u03c3\u2081\u2082 h \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))"}, {"tactic": "simp only", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081\u2082 : Perm m \u00d7 Perm n\nh : \u03c3\u2081\u2082 \u2208 univ\n\u22a2 (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) \u03c3\u2081\u2082 h \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))", "state_after": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081\u2082 : Perm m \u00d7 Perm n\nh : \u03c3\u2081\u2082 \u2208 univ\n\u22a2 Equiv.sumCongr \u03c3\u2081\u2082.fst \u03c3\u2081\u2082.snd \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))"}, {"tactic": "erw [Set.mem_toFinset, MonoidHom.mem_range]", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081\u2082 : Perm m \u00d7 Perm n\nh : \u03c3\u2081\u2082 \u2208 univ\n\u22a2 Equiv.sumCongr \u03c3\u2081\u2082.fst \u03c3\u2081\u2082.snd \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))", "state_after": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081\u2082 : Perm m \u00d7 Perm n\nh : \u03c3\u2081\u2082 \u2208 univ\n\u22a2 \u2203 x, \u2191(sumCongrHom m n) x = Equiv.sumCongr \u03c3\u2081\u2082.fst \u03c3\u2081\u2082.snd"}, {"tactic": "use \u03c3\u2081\u2082", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081\u2082 : Perm m \u00d7 Perm n\nh : \u03c3\u2081\u2082 \u2208 univ\n\u22a2 \u2203 x, \u2191(sumCongrHom m n) x = Equiv.sumCongr \u03c3\u2081\u2082.fst \u03c3\u2081\u2082.snd", "state_after": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081\u2082 : Perm m \u00d7 Perm n\nh : \u03c3\u2081\u2082 \u2208 univ\n\u22a2 \u2191(sumCongrHom m n) \u03c3\u2081\u2082 = Equiv.sumCongr \u03c3\u2081\u2082.fst \u03c3\u2081\u2082.snd"}, {"tactic": "simp only [sumCongrHom_apply]", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081\u2082 : Perm m \u00d7 Perm n\nh : \u03c3\u2081\u2082 \u2208 univ\n\u22a2 \u2191(sumCongrHom m n) \u03c3\u2081\u2082 = Equiv.sumCongr \u03c3\u2081\u2082.fst \u03c3\u2081\u2082.snd", "state_after": "no goals"}, {"tactic": "simp only [forall_prop_of_true, Prod.forall, mem_univ]", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (a : Perm m \u00d7 Perm n) (ha : a \u2208 univ),\n    (\u2191\u2191(\u2191sign a.fst) * \u220f x : m, A (\u2191a.fst x) x) * (\u2191\u2191(\u2191sign a.snd) * \u220f x : n, D (\u2191a.snd x) x) =\n      \u2191\u2191(\u2191sign ((fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a ha)) *\n        \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191((fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a ha) x) x", "state_after": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (a : Perm m) (b : Perm n),\n    (\u2191\u2191(\u2191sign a) * \u220f x : m, A (\u2191a x) x) * (\u2191\u2191(\u2191sign b) * \u220f x : n, D (\u2191b x) x) =\n      \u2191\u2191(\u2191sign (Equiv.sumCongr a b)) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191(Equiv.sumCongr a b) x) x"}, {"tactic": "intro \u03c3\u2081 \u03c3\u2082", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (a : Perm m) (b : Perm n),\n    (\u2191\u2191(\u2191sign a) * \u220f x : m, A (\u2191a x) x) * (\u2191\u2191(\u2191sign b) * \u220f x : n, D (\u2191b x) x) =\n      \u2191\u2191(\u2191sign (Equiv.sumCongr a b)) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191(Equiv.sumCongr a b) x) x", "state_after": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 : Perm m\n\u03c3\u2082 : Perm n\n\u22a2 (\u2191\u2191(\u2191sign \u03c3\u2081) * \u220f x : m, A (\u2191\u03c3\u2081 x) x) * (\u2191\u2191(\u2191sign \u03c3\u2082) * \u220f x : n, D (\u2191\u03c3\u2082 x) x) =\n    \u2191\u2191(\u2191sign (Equiv.sumCongr \u03c3\u2081 \u03c3\u2082)) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191(Equiv.sumCongr \u03c3\u2081 \u03c3\u2082) x) x"}, {"tactic": "rw [Fintype.prod_sum_type]", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 : Perm m\n\u03c3\u2082 : Perm n\n\u22a2 (\u2191\u2191(\u2191sign \u03c3\u2081) * \u220f x : m, A (\u2191\u03c3\u2081 x) x) * (\u2191\u2191(\u2191sign \u03c3\u2082) * \u220f x : n, D (\u2191\u03c3\u2082 x) x) =\n    \u2191\u2191(\u2191sign (Equiv.sumCongr \u03c3\u2081 \u03c3\u2082)) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191(Equiv.sumCongr \u03c3\u2081 \u03c3\u2082) x) x", "state_after": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 : Perm m\n\u03c3\u2082 : Perm n\n\u22a2 (\u2191\u2191(\u2191sign \u03c3\u2081) * \u220f x : m, A (\u2191\u03c3\u2081 x) x) * (\u2191\u2191(\u2191sign \u03c3\u2082) * \u220f x : n, D (\u2191\u03c3\u2082 x) x) =\n    \u2191\u2191(\u2191sign (Equiv.sumCongr \u03c3\u2081 \u03c3\u2082)) *\n      ((\u220f a\u2081 : m, fromBlocks A B 0 D (\u2191(Equiv.sumCongr \u03c3\u2081 \u03c3\u2082) (Sum.inl a\u2081)) (Sum.inl a\u2081)) *\n        \u220f a\u2082 : n, fromBlocks A B 0 D (\u2191(Equiv.sumCongr \u03c3\u2081 \u03c3\u2082) (Sum.inr a\u2082)) (Sum.inr a\u2082))"}, {"tactic": "simp_rw [Equiv.sumCongr_apply, Sum.map_inr, Sum.map_inl, fromBlocks_apply\u2081\u2081,\n  fromBlocks_apply\u2082\u2082]", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 : Perm m\n\u03c3\u2082 : Perm n\n\u22a2 (\u2191\u2191(\u2191sign \u03c3\u2081) * \u220f x : m, A (\u2191\u03c3\u2081 x) x) * (\u2191\u2191(\u2191sign \u03c3\u2082) * \u220f x : n, D (\u2191\u03c3\u2082 x) x) =\n    \u2191\u2191(\u2191sign (Equiv.sumCongr \u03c3\u2081 \u03c3\u2082)) *\n      ((\u220f a\u2081 : m, fromBlocks A B 0 D (\u2191(Equiv.sumCongr \u03c3\u2081 \u03c3\u2082) (Sum.inl a\u2081)) (Sum.inl a\u2081)) *\n        \u220f a\u2082 : n, fromBlocks A B 0 D (\u2191(Equiv.sumCongr \u03c3\u2081 \u03c3\u2082) (Sum.inr a\u2082)) (Sum.inr a\u2082))", "state_after": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 : Perm m\n\u03c3\u2082 : Perm n\n\u22a2 (\u2191\u2191(\u2191sign \u03c3\u2081) * \u220f x : m, A (\u2191\u03c3\u2081 x) x) * (\u2191\u2191(\u2191sign \u03c3\u2082) * \u220f x : n, D (\u2191\u03c3\u2082 x) x) =\n    \u2191\u2191(\u2191sign (Equiv.sumCongr \u03c3\u2081 \u03c3\u2082)) * ((\u220f x : m, A (\u2191\u03c3\u2081 x) x) * \u220f x : n, D (\u2191\u03c3\u2082 x) x)"}, {"tactic": "rw [mul_mul_mul_comm]", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 : Perm m\n\u03c3\u2082 : Perm n\n\u22a2 (\u2191\u2191(\u2191sign \u03c3\u2081) * \u220f x : m, A (\u2191\u03c3\u2081 x) x) * (\u2191\u2191(\u2191sign \u03c3\u2082) * \u220f x : n, D (\u2191\u03c3\u2082 x) x) =\n    \u2191\u2191(\u2191sign (Equiv.sumCongr \u03c3\u2081 \u03c3\u2082)) * ((\u220f x : m, A (\u2191\u03c3\u2081 x) x) * \u220f x : n, D (\u2191\u03c3\u2082 x) x)", "state_after": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 : Perm m\n\u03c3\u2082 : Perm n\n\u22a2 \u2191\u2191(\u2191sign \u03c3\u2081) * \u2191\u2191(\u2191sign \u03c3\u2082) * ((\u220f x : m, A (\u2191\u03c3\u2081 x) x) * \u220f x : n, D (\u2191\u03c3\u2082 x) x) =\n    \u2191\u2191(\u2191sign (Equiv.sumCongr \u03c3\u2081 \u03c3\u2082)) * ((\u220f x : m, A (\u2191\u03c3\u2081 x) x) * \u220f x : n, D (\u2191\u03c3\u2082 x) x)"}, {"tactic": "congr", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 : Perm m\n\u03c3\u2082 : Perm n\n\u22a2 \u2191\u2191(\u2191sign \u03c3\u2081) * \u2191\u2191(\u2191sign \u03c3\u2082) * ((\u220f x : m, A (\u2191\u03c3\u2081 x) x) * \u220f x : n, D (\u2191\u03c3\u2082 x) x) =\n    \u2191\u2191(\u2191sign (Equiv.sumCongr \u03c3\u2081 \u03c3\u2082)) * ((\u220f x : m, A (\u2191\u03c3\u2081 x) x) * \u220f x : n, D (\u2191\u03c3\u2082 x) x)", "state_after": "case e_a\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 : Perm m\n\u03c3\u2082 : Perm n\n\u22a2 \u2191\u2191(\u2191sign \u03c3\u2081) * \u2191\u2191(\u2191sign \u03c3\u2082) = \u2191\u2191(\u2191sign (Equiv.sumCongr \u03c3\u2081 \u03c3\u2082))"}, {"tactic": "rw [sign_sumCongr, Units.val_mul, Int.cast_mul]", "state_before": "case e_a\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 : Perm m\n\u03c3\u2082 : Perm n\n\u22a2 \u2191\u2191(\u2191sign \u03c3\u2081) * \u2191\u2191(\u2191sign \u03c3\u2082) = \u2191\u2191(\u2191sign (Equiv.sumCongr \u03c3\u2081 \u03c3\u2082))", "state_after": "no goals"}, {"tactic": "intro \u03c3\u2081 \u03c3\u2082 h\u2081 h\u2082", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (a\u2081 a\u2082 : Perm m \u00d7 Perm n) (ha\u2081 : a\u2081 \u2208 univ) (ha\u2082 : a\u2082 \u2208 univ),\n    (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a\u2081 ha\u2081 = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082", "state_after": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\n\u22a2 (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) \u03c3\u2081 h\u2081 = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) \u03c3\u2082 h\u2082 \u2192 \u03c3\u2081 = \u03c3\u2082"}, {"tactic": "dsimp only", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\n\u22a2 (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) \u03c3\u2081 h\u2081 = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) \u03c3\u2082 h\u2082 \u2192 \u03c3\u2081 = \u03c3\u2082", "state_after": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\n\u22a2 Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd \u2192 \u03c3\u2081 = \u03c3\u2082"}, {"tactic": "intro h", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\n\u22a2 Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd \u2192 \u03c3\u2081 = \u03c3\u2082", "state_after": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\nh : Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd\n\u22a2 \u03c3\u2081 = \u03c3\u2082"}, {"tactic": "have h2 : \u2200 x, Perm.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd x = Perm.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd x := by\n  intro x\n  exact congr_fun (congr_arg toFun h) x", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\nh : Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd\n\u22a2 \u03c3\u2081 = \u03c3\u2082", "state_after": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\nh : Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd\nh2 : \u2200 (x : m \u2295 n), \u2191(Perm.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd) x = \u2191(Perm.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd) x\n\u22a2 \u03c3\u2081 = \u03c3\u2082"}, {"tactic": "simp only [Sum.map_inr, Sum.map_inl, Perm.sumCongr_apply, Sum.forall, Sum.inl.injEq,\n  Sum.inr.injEq] at h2", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\nh : Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd\nh2 : \u2200 (x : m \u2295 n), \u2191(Perm.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd) x = \u2191(Perm.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd) x\n\u22a2 \u03c3\u2081 = \u03c3\u2082", "state_after": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\nh : Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd\nh2 : (\u2200 (a : m), \u2191\u03c3\u2081.fst a = \u2191\u03c3\u2082.fst a) \u2227 \u2200 (b : n), \u2191\u03c3\u2081.snd b = \u2191\u03c3\u2082.snd b\n\u22a2 \u03c3\u2081 = \u03c3\u2082"}, {"tactic": "ext x", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\nh : Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd\nh2 : (\u2200 (a : m), \u2191\u03c3\u2081.fst a = \u2191\u03c3\u2082.fst a) \u2227 \u2200 (b : n), \u2191\u03c3\u2081.snd b = \u2191\u03c3\u2082.snd b\n\u22a2 \u03c3\u2081 = \u03c3\u2082", "state_after": "case h\u2081.H\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\nh : Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd\nh2 : (\u2200 (a : m), \u2191\u03c3\u2081.fst a = \u2191\u03c3\u2082.fst a) \u2227 \u2200 (b : n), \u2191\u03c3\u2081.snd b = \u2191\u03c3\u2082.snd b\nx : m\n\u22a2 \u2191\u03c3\u2081.fst x = \u2191\u03c3\u2082.fst x\n\ncase h\u2082.H\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\nh : Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd\nh2 : (\u2200 (a : m), \u2191\u03c3\u2081.fst a = \u2191\u03c3\u2082.fst a) \u2227 \u2200 (b : n), \u2191\u03c3\u2081.snd b = \u2191\u03c3\u2082.snd b\nx : n\n\u22a2 \u2191\u03c3\u2081.snd x = \u2191\u03c3\u2082.snd x"}, {"tactic": "intro x", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\nh : Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd\n\u22a2 \u2200 (x : m \u2295 n), \u2191(Perm.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd) x = \u2191(Perm.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd) x", "state_after": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\nh : Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd\nx : m \u2295 n\n\u22a2 \u2191(Perm.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd) x = \u2191(Perm.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd) x"}, {"tactic": "exact congr_fun (congr_arg toFun h) x", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\nh : Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd\nx : m \u2295 n\n\u22a2 \u2191(Perm.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd) x = \u2191(Perm.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd) x", "state_after": "no goals"}, {"tactic": "exact h2.left x", "state_before": "case h\u2081.H\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\nh : Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd\nh2 : (\u2200 (a : m), \u2191\u03c3\u2081.fst a = \u2191\u03c3\u2082.fst a) \u2227 \u2200 (b : n), \u2191\u03c3\u2081.snd b = \u2191\u03c3\u2082.snd b\nx : m\n\u22a2 \u2191\u03c3\u2081.fst x = \u2191\u03c3\u2082.fst x", "state_after": "no goals"}, {"tactic": "exact h2.right x", "state_before": "case h\u2082.H\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\nh : Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd\nh2 : (\u2200 (a : m), \u2191\u03c3\u2081.fst a = \u2191\u03c3\u2082.fst a) \u2227 \u2200 (b : n), \u2191\u03c3\u2081.snd b = \u2191\u03c3\u2082.snd b\nx : n\n\u22a2 \u2191\u03c3\u2081.snd x = \u2191\u03c3\u2082.snd x", "state_after": "no goals"}, {"tactic": "intro \u03c3 h\u03c3", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (b : m \u2295 n \u2243 m \u2295 n),\n    b \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)) \u2192 \u2203 a ha, b = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a ha", "state_after": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : m \u2295 n \u2243 m \u2295 n\nh\u03c3 : \u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\n\u22a2 \u2203 a ha, \u03c3 = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a ha"}, {"tactic": "erw [Set.mem_toFinset, MonoidHom.mem_range] at h\u03c3", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : m \u2295 n \u2243 m \u2295 n\nh\u03c3 : \u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\n\u22a2 \u2203 a ha, \u03c3 = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a ha", "state_after": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : m \u2295 n \u2243 m \u2295 n\nh\u03c3 : \u2203 x, \u2191(sumCongrHom m n) x = \u03c3\n\u22a2 \u2203 a ha, \u03c3 = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a ha"}, {"tactic": "obtain \u27e8\u03c3\u2081\u2082, h\u03c3\u2081\u2082\u27e9 := h\u03c3", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : m \u2295 n \u2243 m \u2295 n\nh\u03c3 : \u2203 x, \u2191(sumCongrHom m n) x = \u03c3\n\u22a2 \u2203 a ha, \u03c3 = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a ha", "state_after": "case intro\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : m \u2295 n \u2243 m \u2295 n\n\u03c3\u2081\u2082 : Perm m \u00d7 Perm n\nh\u03c3\u2081\u2082 : \u2191(sumCongrHom m n) \u03c3\u2081\u2082 = \u03c3\n\u22a2 \u2203 a ha, \u03c3 = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a ha"}, {"tactic": "use \u03c3\u2081\u2082", "state_before": "case intro\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : m \u2295 n \u2243 m \u2295 n\n\u03c3\u2081\u2082 : Perm m \u00d7 Perm n\nh\u03c3\u2081\u2082 : \u2191(sumCongrHom m n) \u03c3\u2081\u2082 = \u03c3\n\u22a2 \u2203 a ha, \u03c3 = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a ha", "state_after": "case intro\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : m \u2295 n \u2243 m \u2295 n\n\u03c3\u2081\u2082 : Perm m \u00d7 Perm n\nh\u03c3\u2081\u2082 : \u2191(sumCongrHom m n) \u03c3\u2081\u2082 = \u03c3\n\u22a2 \u2203 ha, \u03c3 = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) \u03c3\u2081\u2082 ha"}, {"tactic": "rw [\u2190 h\u03c3\u2081\u2082]", "state_before": "case intro\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : m \u2295 n \u2243 m \u2295 n\n\u03c3\u2081\u2082 : Perm m \u00d7 Perm n\nh\u03c3\u2081\u2082 : \u2191(sumCongrHom m n) \u03c3\u2081\u2082 = \u03c3\n\u22a2 \u2203 ha, \u03c3 = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) \u03c3\u2081\u2082 ha", "state_after": "case intro\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : m \u2295 n \u2243 m \u2295 n\n\u03c3\u2081\u2082 : Perm m \u00d7 Perm n\nh\u03c3\u2081\u2082 : \u2191(sumCongrHom m n) \u03c3\u2081\u2082 = \u03c3\n\u22a2 \u2203 ha, \u2191(sumCongrHom m n) \u03c3\u2081\u2082 = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) \u03c3\u2081\u2082 ha"}, {"tactic": "simp", "state_before": "case intro\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : m \u2295 n \u2243 m \u2295 n\n\u03c3\u2081\u2082 : Perm m \u00d7 Perm n\nh\u03c3\u2081\u2082 : \u2191(sumCongrHom m n) \u03c3\u2081\u2082 = \u03c3\n\u22a2 \u2203 ha, \u2191(sumCongrHom m n) \u03c3\u2081\u2082 = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) \u03c3\u2081\u2082 ha", "state_after": "no goals"}, {"tactic": "rintro \u03c3 - h\u03c3n", "state_before": "case convert_2\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (x : Perm (m \u2295 n)),\n    x \u2208 univ \u2192\n      \u00acx \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)) \u2192\n        \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1 = 0", "state_after": "case convert_2\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191\u03c3 x) x = 0"}, {"tactic": "have h1 : \u00ac\u2200 x, \u2203 y, Sum.inl y = \u03c3 (Sum.inl x) := by\n  rw [Set.mem_toFinset] at h\u03c3n\n  simpa only [Set.MapsTo, Set.mem_range, forall_exists_index, forall_apply_eq_imp_iff'] using\n    mt mem_sumCongrHom_range_of_perm_mapsTo_inl h\u03c3n", "state_before": "case convert_2\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191\u03c3 x) x = 0", "state_after": "case convert_2\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\nh1 : \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191\u03c3 x) x = 0"}, {"tactic": "obtain \u27e8a, ha\u27e9 := not_forall.mp h1", "state_before": "case convert_2\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\nh1 : \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191\u03c3 x) x = 0", "state_after": "case convert_2.intro\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\nh1 : \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)\na : m\nha : \u00ac\u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl a)\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191\u03c3 x) x = 0"}, {"tactic": "cases' hx : \u03c3 (Sum.inl a) with a2 b", "state_before": "case convert_2.intro\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\nh1 : \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)\na : m\nha : \u00ac\u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl a)\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191\u03c3 x) x = 0", "state_after": "case convert_2.intro.inl\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\nh1 : \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)\na : m\nha : \u00ac\u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl a)\na2 : m\nhx : \u2191\u03c3 (Sum.inl a) = Sum.inl a2\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191\u03c3 x) x = 0\n\ncase convert_2.intro.inr\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\nh1 : \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)\na : m\nha : \u00ac\u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl a)\nb : n\nhx : \u2191\u03c3 (Sum.inl a) = Sum.inr b\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191\u03c3 x) x = 0"}, {"tactic": "rw [Set.mem_toFinset] at h\u03c3n", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\n\u22a2 \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)", "state_after": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 \u2191(MonoidHom.range (sumCongrHom m n))\n\u22a2 \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)"}, {"tactic": "simpa only [Set.MapsTo, Set.mem_range, forall_exists_index, forall_apply_eq_imp_iff'] using\n  mt mem_sumCongrHom_range_of_perm_mapsTo_inl h\u03c3n", "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 \u2191(MonoidHom.range (sumCongrHom m n))\n\u22a2 \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)", "state_after": "no goals"}, {"tactic": "have hn := (not_exists.mp ha) a2", "state_before": "case convert_2.intro.inl\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\nh1 : \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)\na : m\nha : \u00ac\u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl a)\na2 : m\nhx : \u2191\u03c3 (Sum.inl a) = Sum.inl a2\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191\u03c3 x) x = 0", "state_after": "case convert_2.intro.inl\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\nh1 : \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)\na : m\nha : \u00ac\u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl a)\na2 : m\nhx : \u2191\u03c3 (Sum.inl a) = Sum.inl a2\nhn : \u00acSum.inl a2 = \u2191\u03c3 (Sum.inl a)\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191\u03c3 x) x = 0"}, {"tactic": "exact absurd hx.symm hn", "state_before": "case convert_2.intro.inl\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\nh1 : \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)\na : m\nha : \u00ac\u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl a)\na2 : m\nhx : \u2191\u03c3 (Sum.inl a) = Sum.inl a2\nhn : \u00acSum.inl a2 = \u2191\u03c3 (Sum.inl a)\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191\u03c3 x) x = 0", "state_after": "no goals"}, {"tactic": "rw [Finset.prod_eq_zero (Finset.mem_univ (Sum.inl a)), MulZeroClass.mul_zero]", "state_before": "case convert_2.intro.inr\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\nh1 : \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)\na : m\nha : \u00ac\u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl a)\nb : n\nhx : \u2191\u03c3 (Sum.inl a) = Sum.inr b\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191\u03c3 x) x = 0", "state_after": "case convert_2.intro.inr\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\nh1 : \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)\na : m\nha : \u00ac\u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl a)\nb : n\nhx : \u2191\u03c3 (Sum.inl a) = Sum.inr b\n\u22a2 fromBlocks A B 0 D (\u2191\u03c3 (Sum.inl a)) (Sum.inl a) = 0"}, {"tactic": "rw [hx, fromBlocks_apply\u2082\u2081, zero_apply]", "state_before": "case convert_2.intro.inr\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\nh1 : \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)\na : m\nha : \u00ac\u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl a)\nb : n\nhx : \u2191\u03c3 (Sum.inl a) = Sum.inr b\n\u22a2 fromBlocks A B 0 D (\u2191\u03c3 (Sum.inl a)) (Sum.inl a) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Prod.range_snd", "start": [872, 1], "end": [873, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Localization/LocalizationLocalization.lean", "full_name": "IsLocalization.localization_localization_isLocalization_of_has_all_units", "start": [136, 1], "end": [144, 49], "traced_tactics": [{"tactic": "convert localization_localization_isLocalization M N T using 1", "state_before": "R : Type u_3\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_1\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type ?u.238314\ninst\u271d\u2076 : CommRing P\nN : Submonoid S\nT : Type u_2\ninst\u271d\u2075 : CommRing T\ninst\u271d\u2074 : Algebra R T\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\ninst\u271d\u00b9 : IsLocalization M S\ninst\u271d : IsLocalization N T\nH : \u2200 (x : S), IsUnit x \u2192 x \u2208 N\n\u22a2 IsLocalization (Submonoid.comap (algebraMap R S) N) T", "state_after": "case h.e'_3\nR : Type u_3\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_1\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type ?u.238314\ninst\u271d\u2076 : CommRing P\nN : Submonoid S\nT : Type u_2\ninst\u271d\u2075 : CommRing T\ninst\u271d\u2074 : Algebra R T\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\ninst\u271d\u00b9 : IsLocalization M S\ninst\u271d : IsLocalization N T\nH : \u2200 (x : S), IsUnit x \u2192 x \u2208 N\n\u22a2 Submonoid.comap (algebraMap R S) N = localizationLocalizationSubmodule M N"}, {"tactic": "dsimp [localizationLocalizationSubmodule]", "state_before": "case h.e'_3\nR : Type u_3\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_1\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type ?u.238314\ninst\u271d\u2076 : CommRing P\nN : Submonoid S\nT : Type u_2\ninst\u271d\u2075 : CommRing T\ninst\u271d\u2074 : Algebra R T\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\ninst\u271d\u00b9 : IsLocalization M S\ninst\u271d : IsLocalization N T\nH : \u2200 (x : S), IsUnit x \u2192 x \u2208 N\n\u22a2 Submonoid.comap (algebraMap R S) N = localizationLocalizationSubmodule M N", "state_after": "case h.e'_3\nR : Type u_3\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_1\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type ?u.238314\ninst\u271d\u2076 : CommRing P\nN : Submonoid S\nT : Type u_2\ninst\u271d\u2075 : CommRing T\ninst\u271d\u2074 : Algebra R T\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\ninst\u271d\u00b9 : IsLocalization M S\ninst\u271d : IsLocalization N T\nH : \u2200 (x : S), IsUnit x \u2192 x \u2208 N\n\u22a2 Submonoid.comap (algebraMap R S) N = Submonoid.comap (algebraMap R S) (N \u2294 Submonoid.map (algebraMap R S) M)"}, {"tactic": "congr", "state_before": "case h.e'_3\nR : Type u_3\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_1\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type ?u.238314\ninst\u271d\u2076 : CommRing P\nN : Submonoid S\nT : Type u_2\ninst\u271d\u2075 : CommRing T\ninst\u271d\u2074 : Algebra R T\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\ninst\u271d\u00b9 : IsLocalization M S\ninst\u271d : IsLocalization N T\nH : \u2200 (x : S), IsUnit x \u2192 x \u2208 N\n\u22a2 Submonoid.comap (algebraMap R S) N = Submonoid.comap (algebraMap R S) (N \u2294 Submonoid.map (algebraMap R S) M)", "state_after": "case h.e'_3.e_S\nR : Type u_3\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_1\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type ?u.238314\ninst\u271d\u2076 : CommRing P\nN : Submonoid S\nT : Type u_2\ninst\u271d\u2075 : CommRing T\ninst\u271d\u2074 : Algebra R T\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\ninst\u271d\u00b9 : IsLocalization M S\ninst\u271d : IsLocalization N T\nH : \u2200 (x : S), IsUnit x \u2192 x \u2208 N\n\u22a2 N = N \u2294 Submonoid.map (algebraMap R S) M"}, {"tactic": "symm", "state_before": "case h.e'_3.e_S\nR : Type u_3\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_1\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type ?u.238314\ninst\u271d\u2076 : CommRing P\nN : Submonoid S\nT : Type u_2\ninst\u271d\u2075 : CommRing T\ninst\u271d\u2074 : Algebra R T\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\ninst\u271d\u00b9 : IsLocalization M S\ninst\u271d : IsLocalization N T\nH : \u2200 (x : S), IsUnit x \u2192 x \u2208 N\n\u22a2 N = N \u2294 Submonoid.map (algebraMap R S) M", "state_after": "case h.e'_3.e_S\nR : Type u_3\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_1\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type ?u.238314\ninst\u271d\u2076 : CommRing P\nN : Submonoid S\nT : Type u_2\ninst\u271d\u2075 : CommRing T\ninst\u271d\u2074 : Algebra R T\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\ninst\u271d\u00b9 : IsLocalization M S\ninst\u271d : IsLocalization N T\nH : \u2200 (x : S), IsUnit x \u2192 x \u2208 N\n\u22a2 N \u2294 Submonoid.map (algebraMap R S) M = N"}, {"tactic": "rw [sup_eq_left]", "state_before": "case h.e'_3.e_S\nR : Type u_3\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_1\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type ?u.238314\ninst\u271d\u2076 : CommRing P\nN : Submonoid S\nT : Type u_2\ninst\u271d\u2075 : CommRing T\ninst\u271d\u2074 : Algebra R T\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\ninst\u271d\u00b9 : IsLocalization M S\ninst\u271d : IsLocalization N T\nH : \u2200 (x : S), IsUnit x \u2192 x \u2208 N\n\u22a2 N \u2294 Submonoid.map (algebraMap R S) M = N", "state_after": "case h.e'_3.e_S\nR : Type u_3\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_1\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type ?u.238314\ninst\u271d\u2076 : CommRing P\nN : Submonoid S\nT : Type u_2\ninst\u271d\u2075 : CommRing T\ninst\u271d\u2074 : Algebra R T\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\ninst\u271d\u00b9 : IsLocalization M S\ninst\u271d : IsLocalization N T\nH : \u2200 (x : S), IsUnit x \u2192 x \u2208 N\n\u22a2 Submonoid.map (algebraMap R S) M \u2264 N"}, {"tactic": "rintro _ \u27e8x, hx, rfl\u27e9", "state_before": "case h.e'_3.e_S\nR : Type u_3\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_1\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type ?u.238314\ninst\u271d\u2076 : CommRing P\nN : Submonoid S\nT : Type u_2\ninst\u271d\u2075 : CommRing T\ninst\u271d\u2074 : Algebra R T\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\ninst\u271d\u00b9 : IsLocalization M S\ninst\u271d : IsLocalization N T\nH : \u2200 (x : S), IsUnit x \u2192 x \u2208 N\n\u22a2 Submonoid.map (algebraMap R S) M \u2264 N", "state_after": "case h.e'_3.e_S.intro.intro\nR : Type u_3\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_1\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type ?u.238314\ninst\u271d\u2076 : CommRing P\nN : Submonoid S\nT : Type u_2\ninst\u271d\u2075 : CommRing T\ninst\u271d\u2074 : Algebra R T\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\ninst\u271d\u00b9 : IsLocalization M S\ninst\u271d : IsLocalization N T\nH : \u2200 (x : S), IsUnit x \u2192 x \u2208 N\nx : R\nhx : x \u2208 \u2191M\n\u22a2 \u2191(algebraMap R S) x \u2208 N"}, {"tactic": "exact H _ (IsLocalization.map_units _ \u27e8x, hx\u27e9)", "state_before": "case h.e'_3.e_S.intro.intro\nR : Type u_3\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_1\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type ?u.238314\ninst\u271d\u2076 : CommRing P\nN : Submonoid S\nT : Type u_2\ninst\u271d\u2075 : CommRing T\ninst\u271d\u2074 : Algebra R T\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\ninst\u271d\u00b9 : IsLocalization M S\ninst\u271d : IsLocalization N T\nH : \u2200 (x : S), IsUnit x \u2192 x \u2208 N\nx : R\nhx : x \u2208 \u2191M\n\u22a2 \u2191(algebraMap R S) x \u2208 N", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Rat/Lemmas.lean", "full_name": "Rat.divInt_ofNat", "start": [116, 9], "end": [117, 36], "traced_tactics": [{"tactic": "simp [divInt, normalize_eq_mkRat]", "state_before": "num : Int\nden : Nat\n\u22a2 num /. \u2191den = mkRat num den", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/UniformRing.lean", "full_name": "UniformSpace.ring_sep_rel", "start": [242, 1], "end": [245, 98], "traced_tactics": [{"tactic": "rfl", "state_before": "\u03b1\u271d : Type ?u.320437\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CommRing \u03b1\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformAddGroup \u03b1\ninst\u271d : TopologicalRing \u03b1\nx y : \u03b1\n\u22a2 x - y \u2208 closure {0} \u2194 x - y \u2208 Ideal.closure \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UrysohnsLemma.lean", "full_name": "Urysohns.CU.approx_le_approx_of_U_sub_C", "start": [178, 1], "end": [186, 59], "traced_tactics": 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"IsCompactOperator.add", "start": [219, 1], "end": [224, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Adjunction/Evaluation.lean", "full_name": "CategoryTheory.NatTrans.mono_iff_mono_app", "start": [82, 1], "end": [87, 36], "traced_tactics": [{"tactic": "constructor", "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category C\nD : Type u\u2082\ninst\u271d\u00b9 : Category D\ninst\u271d : \u2200 (a b : C), HasCoproductsOfShape (a \u27f6 b) D\nF G : C \u2964 D\n\u03b7 : F \u27f6 G\n\u22a2 Mono \u03b7 \u2194 \u2200 (c : C), Mono (\u03b7.app c)", "state_after": "case mp\nC : Type u\u2081\ninst\u271d\u00b2 : Category C\nD : Type u\u2082\ninst\u271d\u00b9 : Category D\ninst\u271d : \u2200 (a b : C), HasCoproductsOfShape (a \u27f6 b) D\nF G : C \u2964 D\n\u03b7 : F \u27f6 G\n\u22a2 Mono \u03b7 \u2192 \u2200 (c : C), Mono (\u03b7.app c)\n\ncase mpr\nC : Type u\u2081\ninst\u271d\u00b2 : Category C\nD : Type u\u2082\ninst\u271d\u00b9 : Category D\ninst\u271d : \u2200 (a b : C), HasCoproductsOfShape (a \u27f6 b) D\nF G : C \u2964 D\n\u03b7 : F \u27f6 G\n\u22a2 (\u2200 (c : C), Mono (\u03b7.app c)) \u2192 Mono \u03b7"}, {"tactic": "intro h c", "state_before": "case mp\nC : Type u\u2081\ninst\u271d\u00b2 : Category C\nD : Type u\u2082\ninst\u271d\u00b9 : Category D\ninst\u271d : \u2200 (a b : C), HasCoproductsOfShape (a \u27f6 b) D\nF G : C \u2964 D\n\u03b7 : F \u27f6 G\n\u22a2 Mono \u03b7 \u2192 \u2200 (c : C), Mono (\u03b7.app c)", "state_after": "case mp\nC : Type u\u2081\ninst\u271d\u00b2 : Category C\nD : Type u\u2082\ninst\u271d\u00b9 : Category D\ninst\u271d : \u2200 (a b : C), HasCoproductsOfShape (a \u27f6 b) D\nF G : C \u2964 D\n\u03b7 : F \u27f6 G\nh : Mono \u03b7\nc : C\n\u22a2 Mono (\u03b7.app c)"}, {"tactic": "exact (inferInstance : Mono (((evaluation _ _).obj c).map \u03b7))", "state_before": "case mp\nC : Type u\u2081\ninst\u271d\u00b2 : Category C\nD : Type u\u2082\ninst\u271d\u00b9 : Category D\ninst\u271d : \u2200 (a b : C), HasCoproductsOfShape (a \u27f6 b) D\nF G : C \u2964 D\n\u03b7 : F \u27f6 G\nh : Mono \u03b7\nc : C\n\u22a2 Mono (\u03b7.app c)", "state_after": "no goals"}, {"tactic": "intro _", "state_before": "case mpr\nC : Type u\u2081\ninst\u271d\u00b2 : Category C\nD : Type u\u2082\ninst\u271d\u00b9 : Category D\ninst\u271d : \u2200 (a b : C), HasCoproductsOfShape (a \u27f6 b) D\nF G : C \u2964 D\n\u03b7 : F \u27f6 G\n\u22a2 (\u2200 (c : C), Mono (\u03b7.app c)) \u2192 Mono \u03b7", "state_after": "case mpr\nC : Type u\u2081\ninst\u271d\u00b2 : Category C\nD : Type u\u2082\ninst\u271d\u00b9 : Category D\ninst\u271d : \u2200 (a b : C), HasCoproductsOfShape (a \u27f6 b) D\nF G : C \u2964 D\n\u03b7 : F \u27f6 G\na\u271d : \u2200 (c : C), Mono (\u03b7.app c)\n\u22a2 Mono \u03b7"}, {"tactic": "apply NatTrans.mono_of_mono_app", "state_before": "case mpr\nC : Type u\u2081\ninst\u271d\u00b2 : Category C\nD : Type u\u2082\ninst\u271d\u00b9 : Category D\ninst\u271d : \u2200 (a b : C), HasCoproductsOfShape (a \u27f6 b) D\nF G : C \u2964 D\n\u03b7 : F \u27f6 G\na\u271d : \u2200 (c : C), Mono (\u03b7.app c)\n\u22a2 Mono \u03b7", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/List/Lemmas.lean", "full_name": "List.range'_subset_right", "start": [1861, 1], "end": [1865, 82], "traced_tactics": [{"tactic": "refine \u27e8fun h => Nat.le_of_not_lt fun hn => ?_, fun h => (range'_sublist_right.2 h).subset\u27e9", "state_before": "step s m n : Nat\nstep0 : 0 < step\n\u22a2 range' s m step \u2286 range' s n step \u2194 m \u2264 n", "state_after": "step s m n : Nat\nstep0 : 0 < step\nh : range' s m step \u2286 range' s n step\nhn : n < m\n\u22a2 False"}, {"tactic": "have \u27e8i, h', e\u27e9 := mem_range'.1 <| h <| mem_range'.2 \u27e8_, hn, rfl\u27e9", "state_before": "step s m n : Nat\nstep0 : 0 < step\nh : range' s m step \u2286 range' s n step\nhn : n < m\n\u22a2 False", "state_after": "step s m n : Nat\nstep0 : 0 < step\nh : range' s m step \u2286 range' s n step\nhn : n < m\ni : Nat\nh' : i < n\ne : s + step * n = s + step * i\n\u22a2 False"}, {"tactic": "exact Nat.ne_of_gt h' (Nat.eq_of_mul_eq_mul_left step0 (Nat.add_left_cancel e))", "state_before": "step s m n : Nat\nstep0 : 0 < step\nh : range' s m step \u2286 range' s n step\nhn : n < m\ni : Nat\nh' : i < n\ne : s + step * n = s + step * i\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Defs.lean", "full_name": "zpow_zero", "start": [919, 42], "end": [920, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.Lp.simpleFunc.stronglyMeasurable", "start": [605, 11], "end": [607, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/String/Lemmas.lean", "full_name": "Substring.Valid.data_take", "start": [1028, 1], "end": [1029, 89], "traced_tactics": [{"tactic": "simp [(h.take _).toString, h.toString]", "state_before": "x\u271d\u00b9 : Substring\nh\u271d : Valid x\u271d\u00b9\nx\u271d : Nat\nw\u271d\u00b2 w\u271d\u00b9 w\u271d : List Char\nh : ValidFor w\u271d\u00b2 w\u271d\u00b9 w\u271d x\u271d\u00b9\n\u22a2 (toString (Substring.take x\u271d\u00b9 x\u271d)).data = List.take x\u271d (toString x\u271d\u00b9).data", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "full_name": "Orthonormal.equiv_refl", "start": [1414, 1], "end": [1417, 94], "traced_tactics": [{"tactic": "simp only [Orthonormal.equiv_apply, Equiv.coe_refl, id.def, LinearIsometryEquiv.coe_refl]", "state_before": "\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type ?u.3082509\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_1\n\u03b9' : Type ?u.3082561\n\u03b9'' : Type ?u.3082564\nE' : Type ?u.3082567\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nE'' : Type ?u.3082585\ninst\u271d\u00b9 : NormedAddCommGroup E''\ninst\u271d : InnerProductSpace \ud835\udd5c E''\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\ni : \u03b9\n\u22a2 \u2191(equiv hv hv (Equiv.refl \u03b9)) (\u2191v i) = \u2191(LinearIsometryEquiv.refl \ud835\udd5c E) (\u2191v i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean", "full_name": "HasFDerivWithinAt.const_cpow", "start": [111, 1], "end": [113, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/DualNumber.lean", "full_name": "DualNumber.eps_mul_eps", "start": [84, 1], "end": [85, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.one_lt_succ_succ", "start": [946, 1], "end": [950, 21], "traced_tactics": [{"tactic": "cases n", "state_before": "n m : \u2115\na : Fin n\n\u22a2 1 < succ (succ a)", "state_after": "case zero\nm : \u2115\na : Fin zero\n\u22a2 1 < succ (succ a)\n\ncase succ\nm n\u271d : \u2115\na : Fin (Nat.succ n\u271d)\n\u22a2 1 < succ (succ a)"}, {"tactic": "exact Fin.elim0 a", "state_before": "case zero\nm : \u2115\na : Fin zero\n\u22a2 1 < succ (succ a)", "state_after": "no goals"}, {"tactic": "rw [\u2190 succ_zero_eq_one, succ_lt_succ_iff]", "state_before": "case succ\nm n\u271d : \u2115\na : Fin (Nat.succ n\u271d)\n\u22a2 1 < succ (succ a)", "state_after": "case succ\nm n\u271d : \u2115\na : Fin (Nat.succ n\u271d)\n\u22a2 0 < succ a"}, {"tactic": "exact succ_pos a", "state_before": "case succ\nm n\u271d : \u2115\na : Fin (Nat.succ n\u271d)\n\u22a2 0 < succ a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/SuccPred/Limit.lean", "full_name": "Order.not_isPredLimit_pred_of_not_isMin", "start": [295, 1], "end": [297, 17], "traced_tactics": [{"tactic": "contrapose! ha", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\na : \u03b1\ninst\u271d : PredOrder \u03b1\nha : \u00acIsMin a\n\u22a2 \u00acIsPredLimit (pred a)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\na : \u03b1\ninst\u271d : PredOrder \u03b1\nha : IsPredLimit (pred a)\n\u22a2 IsMin a"}, {"tactic": "exact ha.isMin", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\na : \u03b1\ninst\u271d : PredOrder \u03b1\nha : IsPredLimit (pred a)\n\u22a2 IsMin a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/LiminfLimsup.lean", "full_name": "Filter.liminf_le_of_frequently_le'", "start": [817, 1], "end": [825, 31], "traced_tactics": [{"tactic": "rw [liminf_eq]", "state_before": "\u03b1\u271d : Type ?u.122316\n\u03b2\u271d : Type ?u.122319\n\u03b3 : Type ?u.122322\n\u03b9 : Type ?u.122325\ninst\u271d\u00b9 : CompleteLattice \u03b1\u271d\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CompleteLattice \u03b2\nf : Filter \u03b1\nu : \u03b1 \u2192 \u03b2\nx : \u03b2\nh : \u2203\u1da0 (a : \u03b1) in f, u a \u2264 x\n\u22a2 liminf u f \u2264 x", "state_after": "\u03b1\u271d : Type ?u.122316\n\u03b2\u271d : Type ?u.122319\n\u03b3 : Type ?u.122322\n\u03b9 : Type ?u.122325\ninst\u271d\u00b9 : CompleteLattice \u03b1\u271d\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CompleteLattice \u03b2\nf : Filter \u03b1\nu : \u03b1 \u2192 \u03b2\nx : \u03b2\nh : \u2203\u1da0 (a : \u03b1) in f, u a \u2264 x\n\u22a2 sSup {a | \u2200\u1da0 (n : \u03b1) in f, a \u2264 u n} \u2264 x"}, {"tactic": "refine' sSup_le fun b hb => _", "state_before": "\u03b1\u271d : Type ?u.122316\n\u03b2\u271d : Type ?u.122319\n\u03b3 : Type ?u.122322\n\u03b9 : Type ?u.122325\ninst\u271d\u00b9 : CompleteLattice \u03b1\u271d\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CompleteLattice \u03b2\nf : Filter \u03b1\nu : \u03b1 \u2192 \u03b2\nx : \u03b2\nh : \u2203\u1da0 (a : \u03b1) in f, u a \u2264 x\n\u22a2 sSup {a | \u2200\u1da0 (n : \u03b1) in f, a \u2264 u n} \u2264 x", "state_after": "\u03b1\u271d : Type ?u.122316\n\u03b2\u271d : Type ?u.122319\n\u03b3 : Type ?u.122322\n\u03b9 : Type ?u.122325\ninst\u271d\u00b9 : CompleteLattice \u03b1\u271d\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CompleteLattice \u03b2\nf : Filter \u03b1\nu : \u03b1 \u2192 \u03b2\nx : \u03b2\nh : \u2203\u1da0 (a : \u03b1) in f, u a \u2264 x\nb : \u03b2\nhb : b \u2208 {a | \u2200\u1da0 (n : \u03b1) in f, a \u2264 u n}\n\u22a2 b \u2264 x"}, {"tactic": "have hbx : \u2203\u1da0 _ in f, b \u2264 x := by\n  revert h\n  rw [\u2190 not_imp_not, not_frequently, not_frequently]\n  exact fun h => hb.mp (h.mono fun a hbx hba hax => hbx (hba.trans hax))", "state_before": "\u03b1\u271d : Type ?u.122316\n\u03b2\u271d : Type ?u.122319\n\u03b3 : Type ?u.122322\n\u03b9 : Type ?u.122325\ninst\u271d\u00b9 : CompleteLattice \u03b1\u271d\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CompleteLattice \u03b2\nf : Filter \u03b1\nu : \u03b1 \u2192 \u03b2\nx : \u03b2\nh : \u2203\u1da0 (a : \u03b1) in f, u a \u2264 x\nb : \u03b2\nhb : b \u2208 {a | \u2200\u1da0 (n : \u03b1) in f, a \u2264 u n}\n\u22a2 b \u2264 x", "state_after": "\u03b1\u271d : Type ?u.122316\n\u03b2\u271d : Type ?u.122319\n\u03b3 : Type ?u.122322\n\u03b9 : Type ?u.122325\ninst\u271d\u00b9 : CompleteLattice \u03b1\u271d\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CompleteLattice \u03b2\nf : Filter \u03b1\nu : \u03b1 \u2192 \u03b2\nx : \u03b2\nh : \u2203\u1da0 (a : \u03b1) in f, u a \u2264 x\nb : \u03b2\nhb : b \u2208 {a | \u2200\u1da0 (n : \u03b1) in f, a \u2264 u n}\nhbx : \u2203\u1da0 (x_1 : \u03b1) in f, b \u2264 x\n\u22a2 b \u2264 x"}, {"tactic": "exact hbx.exists.choose_spec", "state_before": "\u03b1\u271d : Type ?u.122316\n\u03b2\u271d : Type ?u.122319\n\u03b3 : Type ?u.122322\n\u03b9 : Type ?u.122325\ninst\u271d\u00b9 : CompleteLattice \u03b1\u271d\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CompleteLattice \u03b2\nf : Filter \u03b1\nu : \u03b1 \u2192 \u03b2\nx : \u03b2\nh : \u2203\u1da0 (a : \u03b1) in f, u a \u2264 x\nb : \u03b2\nhb : b \u2208 {a | \u2200\u1da0 (n : \u03b1) in f, a \u2264 u n}\nhbx : \u2203\u1da0 (x_1 : \u03b1) in f, b \u2264 x\n\u22a2 b \u2264 x", "state_after": "no goals"}, {"tactic": "revert h", "state_before": "\u03b1\u271d : Type ?u.122316\n\u03b2\u271d : Type ?u.122319\n\u03b3 : Type ?u.122322\n\u03b9 : Type ?u.122325\ninst\u271d\u00b9 : CompleteLattice \u03b1\u271d\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CompleteLattice \u03b2\nf : Filter \u03b1\nu : \u03b1 \u2192 \u03b2\nx : \u03b2\nh : \u2203\u1da0 (a : \u03b1) in f, u a \u2264 x\nb : \u03b2\nhb : b \u2208 {a | \u2200\u1da0 (n : \u03b1) in f, a \u2264 u n}\n\u22a2 \u2203\u1da0 (x_1 : \u03b1) in f, b \u2264 x", "state_after": "\u03b1\u271d : Type ?u.122316\n\u03b2\u271d : Type ?u.122319\n\u03b3 : Type ?u.122322\n\u03b9 : Type ?u.122325\ninst\u271d\u00b9 : CompleteLattice \u03b1\u271d\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CompleteLattice \u03b2\nf : Filter \u03b1\nu : \u03b1 \u2192 \u03b2\nx b : \u03b2\nhb : b \u2208 {a | \u2200\u1da0 (n : \u03b1) in f, a \u2264 u n}\n\u22a2 (\u2203\u1da0 (a : \u03b1) in f, u a \u2264 x) \u2192 \u2203\u1da0 (x_1 : \u03b1) in f, b \u2264 x"}, {"tactic": "rw [\u2190 not_imp_not, not_frequently, not_frequently]", "state_before": "\u03b1\u271d : Type ?u.122316\n\u03b2\u271d : Type ?u.122319\n\u03b3 : Type ?u.122322\n\u03b9 : Type ?u.122325\ninst\u271d\u00b9 : CompleteLattice \u03b1\u271d\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CompleteLattice \u03b2\nf : Filter \u03b1\nu : \u03b1 \u2192 \u03b2\nx b : \u03b2\nhb : b \u2208 {a | \u2200\u1da0 (n : \u03b1) in f, a \u2264 u n}\n\u22a2 (\u2203\u1da0 (a : \u03b1) in f, u a \u2264 x) \u2192 \u2203\u1da0 (x_1 : \u03b1) in f, b \u2264 x", "state_after": "\u03b1\u271d : Type ?u.122316\n\u03b2\u271d : Type ?u.122319\n\u03b3 : Type ?u.122322\n\u03b9 : Type ?u.122325\ninst\u271d\u00b9 : CompleteLattice \u03b1\u271d\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CompleteLattice \u03b2\nf : Filter \u03b1\nu : \u03b1 \u2192 \u03b2\nx b : \u03b2\nhb : b \u2208 {a | \u2200\u1da0 (n : \u03b1) in f, a \u2264 u n}\n\u22a2 (\u2200\u1da0 (x_1 : \u03b1) in f, \u00acb \u2264 x) \u2192 \u2200\u1da0 (x_1 : \u03b1) in f, \u00acu x_1 \u2264 x"}, {"tactic": "exact fun h => hb.mp (h.mono fun a hbx hba hax => hbx (hba.trans hax))", "state_before": "\u03b1\u271d : Type ?u.122316\n\u03b2\u271d : Type ?u.122319\n\u03b3 : Type ?u.122322\n\u03b9 : Type ?u.122325\ninst\u271d\u00b9 : CompleteLattice \u03b1\u271d\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CompleteLattice \u03b2\nf : Filter \u03b1\nu : \u03b1 \u2192 \u03b2\nx b : \u03b2\nhb : b \u2208 {a | \u2200\u1da0 (n : \u03b1) in f, a \u2264 u n}\n\u22a2 (\u2200\u1da0 (x_1 : \u03b1) in f, \u00acb \u2264 x) \u2192 \u2200\u1da0 (x_1 : \u03b1) in f, \u00acu x_1 \u2264 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/ArithmeticFunction.lean", "full_name": "Nat.ArithmeticFunction.coe_zeta_mul_coe_moebius", "start": [1055, 1], "end": [1056, 65], "traced_tactics": [{"tactic": "rw [\u2190 coe_coe, \u2190 intCoe_mul, coe_zeta_mul_moebius, intCoe_one]", "state_before": "R : Type u_1\ninst\u271d : Ring R\n\u22a2 \u2191\u03b6 * \u2191\u03bc = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/Defs.lean", "full_name": "Equiv.symm_apply_apply", "start": [286, 9], "end": [286, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/ContDiff.lean", "full_name": "norm_iteratedFDerivWithin_clm_apply", "start": [2714, 1], "end": [2725, 79], "traced_tactics": [{"tactic": "let B : (F \u2192L[\ud835\udd5c] G) \u2192L[\ud835\udd5c] F \u2192L[\ud835\udd5c] G := ContinuousLinearMap.flip (ContinuousLinearMap.apply \ud835\udd5c G)", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.4352265\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nf : E \u2192 F \u2192L[\ud835\udd5c] G\ng : E \u2192 F\ns : Set E\nx : E\nN : \u2115\u221e\nn : \u2115\nhf : ContDiffOn \ud835\udd5c N f s\nhg : ContDiffOn \ud835\udd5c N g s\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\nhn : \u2191n \u2264 N\n\u22a2 \u2016iteratedFDerivWithin \ud835\udd5c n (fun y => \u2191(f y) (g y)) s x\u2016 \u2264\n    \u2211 i in Finset.range (n + 1),\n      \u2191(Nat.choose n i) * \u2016iteratedFDerivWithin \ud835\udd5c i f s x\u2016 * \u2016iteratedFDerivWithin \ud835\udd5c (n - i) g s x\u2016", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.4352265\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nf : E \u2192 F \u2192L[\ud835\udd5c] G\ng : E \u2192 F\ns : Set E\nx : E\nN : \u2115\u221e\nn : \u2115\nhf : ContDiffOn \ud835\udd5c N f s\nhg : ContDiffOn \ud835\udd5c N g s\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\nhn : \u2191n \u2264 N\nB : (F \u2192L[\ud835\udd5c] G) \u2192L[\ud835\udd5c] F \u2192L[\ud835\udd5c] G := ContinuousLinearMap.flip (ContinuousLinearMap.apply \ud835\udd5c G)\n\u22a2 \u2016iteratedFDerivWithin \ud835\udd5c n (fun y => \u2191(f y) (g y)) s x\u2016 \u2264\n    \u2211 i in Finset.range (n + 1),\n      \u2191(Nat.choose n i) * \u2016iteratedFDerivWithin \ud835\udd5c i f s x\u2016 * \u2016iteratedFDerivWithin \ud835\udd5c (n - i) g s x\u2016"}, {"tactic": "have hB : \u2016B\u2016 \u2264 1 := by\n  simp only [ContinuousLinearMap.op_norm_flip, ContinuousLinearMap.apply]\n  refine' ContinuousLinearMap.op_norm_le_bound _ zero_le_one fun f => _\n  simp only [ContinuousLinearMap.coe_id', id.def, one_mul]\n  rfl", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.4352265\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nf : E \u2192 F \u2192L[\ud835\udd5c] G\ng : E \u2192 F\ns : Set E\nx : E\nN : \u2115\u221e\nn : \u2115\nhf : ContDiffOn \ud835\udd5c N f s\nhg : ContDiffOn \ud835\udd5c N g s\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\nhn : \u2191n \u2264 N\nB : (F \u2192L[\ud835\udd5c] G) \u2192L[\ud835\udd5c] F \u2192L[\ud835\udd5c] G := ContinuousLinearMap.flip (ContinuousLinearMap.apply \ud835\udd5c G)\n\u22a2 \u2016iteratedFDerivWithin \ud835\udd5c n (fun y => \u2191(f y) (g y)) s x\u2016 \u2264\n    \u2211 i in Finset.range (n + 1),\n      \u2191(Nat.choose n i) * \u2016iteratedFDerivWithin \ud835\udd5c i f s x\u2016 * \u2016iteratedFDerivWithin \ud835\udd5c (n - i) g s x\u2016", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.4352265\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nf : E \u2192 F \u2192L[\ud835\udd5c] G\ng : E \u2192 F\ns : Set E\nx : E\nN : \u2115\u221e\nn : \u2115\nhf : ContDiffOn \ud835\udd5c N f s\nhg : ContDiffOn \ud835\udd5c N g s\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\nhn : \u2191n \u2264 N\nB : (F \u2192L[\ud835\udd5c] G) \u2192L[\ud835\udd5c] F \u2192L[\ud835\udd5c] G := ContinuousLinearMap.flip (ContinuousLinearMap.apply \ud835\udd5c G)\nhB : \u2016B\u2016 \u2264 1\n\u22a2 \u2016iteratedFDerivWithin \ud835\udd5c n (fun y => \u2191(f y) (g y)) s x\u2016 \u2264\n    \u2211 i in Finset.range (n + 1),\n      \u2191(Nat.choose n i) * \u2016iteratedFDerivWithin \ud835\udd5c i f s x\u2016 * \u2016iteratedFDerivWithin \ud835\udd5c (n - i) g s x\u2016"}, {"tactic": "exact B.norm_iteratedFDerivWithin_le_of_bilinear_of_le_one hf hg hs hx hn hB", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.4352265\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nf : E \u2192 F \u2192L[\ud835\udd5c] G\ng : E \u2192 F\ns : Set E\nx : E\nN : \u2115\u221e\nn : \u2115\nhf : ContDiffOn \ud835\udd5c N f s\nhg : ContDiffOn \ud835\udd5c N g s\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\nhn : \u2191n \u2264 N\nB : (F \u2192L[\ud835\udd5c] G) \u2192L[\ud835\udd5c] F \u2192L[\ud835\udd5c] G := ContinuousLinearMap.flip (ContinuousLinearMap.apply \ud835\udd5c G)\nhB : \u2016B\u2016 \u2264 1\n\u22a2 \u2016iteratedFDerivWithin \ud835\udd5c n (fun y => \u2191(f y) (g y)) s x\u2016 \u2264\n    \u2211 i in Finset.range (n + 1),\n      \u2191(Nat.choose n i) * \u2016iteratedFDerivWithin \ud835\udd5c i f s x\u2016 * \u2016iteratedFDerivWithin \ud835\udd5c (n - i) g s x\u2016", "state_after": "no goals"}, {"tactic": "simp only [ContinuousLinearMap.op_norm_flip, ContinuousLinearMap.apply]", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.4352265\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nf : E \u2192 F \u2192L[\ud835\udd5c] G\ng : E \u2192 F\ns : Set E\nx : E\nN : \u2115\u221e\nn : \u2115\nhf : ContDiffOn \ud835\udd5c N f s\nhg : ContDiffOn \ud835\udd5c N g s\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\nhn : \u2191n \u2264 N\nB : (F \u2192L[\ud835\udd5c] G) \u2192L[\ud835\udd5c] F \u2192L[\ud835\udd5c] G := ContinuousLinearMap.flip (ContinuousLinearMap.apply \ud835\udd5c G)\n\u22a2 \u2016B\u2016 \u2264 1", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.4352265\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nf : E \u2192 F \u2192L[\ud835\udd5c] G\ng : E \u2192 F\ns : Set E\nx : E\nN : \u2115\u221e\nn : \u2115\nhf : ContDiffOn \ud835\udd5c N f s\nhg : ContDiffOn \ud835\udd5c N g s\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\nhn : \u2191n \u2264 N\nB : (F \u2192L[\ud835\udd5c] G) \u2192L[\ud835\udd5c] F \u2192L[\ud835\udd5c] G := ContinuousLinearMap.flip (ContinuousLinearMap.apply \ud835\udd5c G)\n\u22a2 \u2016ContinuousLinearMap.id \ud835\udd5c (F \u2192L[\ud835\udd5c] G)\u2016 \u2264 1"}, {"tactic": "refine' ContinuousLinearMap.op_norm_le_bound _ zero_le_one fun f => _", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.4352265\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t u : Set E\nf\u271d f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nf : E \u2192 F \u2192L[\ud835\udd5c] G\ng : E \u2192 F\ns : Set E\nx : E\nN : \u2115\u221e\nn : \u2115\nhf : ContDiffOn \ud835\udd5c N f s\nhg : ContDiffOn \ud835\udd5c N g s\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\nhn : \u2191n \u2264 N\nB : (F \u2192L[\ud835\udd5c] G) \u2192L[\ud835\udd5c] F \u2192L[\ud835\udd5c] G := ContinuousLinearMap.flip (ContinuousLinearMap.apply \ud835\udd5c G)\n\u22a2 \u2016ContinuousLinearMap.id \ud835\udd5c (F \u2192L[\ud835\udd5c] G)\u2016 \u2264 1", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.4352265\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t u : Set E\nf\u271d\u00b9 f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nf\u271d : E \u2192 F \u2192L[\ud835\udd5c] G\ng : E \u2192 F\ns : Set E\nx : E\nN : \u2115\u221e\nn : \u2115\nhf : ContDiffOn \ud835\udd5c N f\u271d s\nhg : ContDiffOn \ud835\udd5c N g s\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\nhn : \u2191n \u2264 N\nB : (F \u2192L[\ud835\udd5c] G) \u2192L[\ud835\udd5c] F \u2192L[\ud835\udd5c] G := ContinuousLinearMap.flip (ContinuousLinearMap.apply \ud835\udd5c G)\nf : F \u2192L[\ud835\udd5c] G\n\u22a2 \u2016\u2191(ContinuousLinearMap.id \ud835\udd5c (F \u2192L[\ud835\udd5c] G)) f\u2016 \u2264 1 * \u2016f\u2016"}, {"tactic": "simp only [ContinuousLinearMap.coe_id', id.def, one_mul]", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.4352265\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t u : Set E\nf\u271d\u00b9 f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nf\u271d : E \u2192 F \u2192L[\ud835\udd5c] G\ng : E \u2192 F\ns : Set E\nx : E\nN : \u2115\u221e\nn : \u2115\nhf : ContDiffOn \ud835\udd5c N f\u271d s\nhg : ContDiffOn \ud835\udd5c N g s\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\nhn : \u2191n \u2264 N\nB : (F \u2192L[\ud835\udd5c] G) \u2192L[\ud835\udd5c] F \u2192L[\ud835\udd5c] G := ContinuousLinearMap.flip (ContinuousLinearMap.apply \ud835\udd5c G)\nf : F \u2192L[\ud835\udd5c] G\n\u22a2 \u2016\u2191(ContinuousLinearMap.id \ud835\udd5c (F \u2192L[\ud835\udd5c] G)) f\u2016 \u2264 1 * \u2016f\u2016", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.4352265\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t u : Set E\nf\u271d\u00b9 f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nf\u271d : E \u2192 F \u2192L[\ud835\udd5c] G\ng : E \u2192 F\ns : Set E\nx : E\nN : \u2115\u221e\nn : \u2115\nhf : ContDiffOn \ud835\udd5c N f\u271d s\nhg : ContDiffOn \ud835\udd5c N g s\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\nhn : \u2191n \u2264 N\nB : (F \u2192L[\ud835\udd5c] G) \u2192L[\ud835\udd5c] F \u2192L[\ud835\udd5c] G := ContinuousLinearMap.flip (ContinuousLinearMap.apply \ud835\udd5c G)\nf : F \u2192L[\ud835\udd5c] G\n\u22a2 \u2016f\u2016 \u2264 \u2016f\u2016"}, {"tactic": "rfl", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type ?u.4352265\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns\u271d s\u2081 t u : Set E\nf\u271d\u00b9 f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nf\u271d : E \u2192 F \u2192L[\ud835\udd5c] G\ng : E \u2192 F\ns : Set E\nx : E\nN : \u2115\u221e\nn : \u2115\nhf : ContDiffOn \ud835\udd5c N f\u271d s\nhg : ContDiffOn \ud835\udd5c N g s\nhs : UniqueDiffOn \ud835\udd5c s\nhx : x \u2208 s\nhn : \u2191n \u2264 N\nB : (F \u2192L[\ud835\udd5c] G) \u2192L[\ud835\udd5c] F \u2192L[\ud835\udd5c] G := ContinuousLinearMap.flip (ContinuousLinearMap.apply \ud835\udd5c G)\nf : F \u2192L[\ud835\udd5c] G\n\u22a2 \u2016f\u2016 \u2264 \u2016f\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "full_name": "LinearIsometryEquiv.coe_toIsometryEquiv", "start": [648, 1], "end": [649, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.succAbove_left_injective", "start": [2195, 1], "end": [2196, 91], "traced_tactics": [{"tactic": "simpa [range_succAbove] using congr_arg (fun f : Fin n \u21aao Fin (n + 1) => Set.range f\u1d9c) h", "state_before": "n m : \u2115\nx\u271d\u00b9 x\u271d : Fin (n + 1)\nh : succAbove x\u271d\u00b9 = succAbove x\u271d\n\u22a2 x\u271d\u00b9 = x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Function/Iterate.lean", "full_name": "Function.Commute.self_iterate", "start": [158, 1], "end": [159, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Join.lean", "full_name": "segment_subset_convexJoin", "start": [108, 1], "end": [109, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/ENNReal.lean", "full_name": "ENNReal.pow_ne_top", "start": [634, 1], "end": [635, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Tropical/Basic.lean", "full_name": "Tropical.trop_top", "start": [239, 1], "end": [240, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Monoid/TypeTags.lean", "full_name": "Additive.ofMul_lt", "start": [128, 1], "end": [129, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/MvPolynomial/Counit.lean", "full_name": "MvPolynomial.counit_C", "start": [93, 1], "end": [94, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Derivation/Basic.lean", "full_name": "Derivation.coe_mk'_linearMap", "start": [368, 1], "end": [369, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Homotopy/Equiv.lean", "full_name": "ContinuousMap.HomotopyEquiv.continuous", "start": [71, 1], "end": [72, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "MeasureTheory.NullMeasurableSet.insert", "start": [225, 11], "end": [227, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Init/Data/List/Basic.lean", "full_name": "List.le_eq_not_gt", "start": [79, 1], "end": [79, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "full_name": "MonoidHom.eqOn_closure", "start": [2905, 1], "end": [2906, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/UnionLift.lean", "full_name": "Set.iUnionLift_inclusion", "start": [73, 1], "end": [75, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Nontrivial.lean", "full_name": "exists_pair_ne", "start": [42, 1], "end": [43, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/ContDiffDef.lean", "full_name": "norm_iteratedFDeriv_zero", "start": [1522, 1], "end": [1524, 77], "traced_tactics": [{"tactic": "rw [iteratedFDeriv_zero_eq_comp, comp_apply, LinearIsometryEquiv.norm_map]", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 \u2016iteratedFDeriv \ud835\udd5c 0 f x\u2016 = \u2016f x\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Lie/Nilpotent.lean", "full_name": "LieAlgebra.isNilpotent_range_ad_iff", "start": [655, 1], "end": [662, 37], "traced_tactics": [{"tactic": "refine' \u27e8fun h => _, _\u27e9", "state_before": "R : Type u\nL : Type v\nL' : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\ninst\u271d\u00b9 : LieRing L'\ninst\u271d : LieAlgebra R L'\n\u22a2 IsNilpotent R { x // x \u2208 LieHom.range (ad R L) } \u2194 IsNilpotent R L", "state_after": "case refine'_1\nR : Type u\nL : Type v\nL' : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\ninst\u271d\u00b9 : LieRing L'\ninst\u271d : LieAlgebra R L'\nh : IsNilpotent R { x // x \u2208 LieHom.range (ad R L) }\n\u22a2 IsNilpotent R L\n\ncase refine'_2\nR : Type u\nL : Type v\nL' : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\ninst\u271d\u00b9 : LieRing L'\ninst\u271d : LieAlgebra R L'\n\u22a2 IsNilpotent R L \u2192 IsNilpotent R { x // x \u2208 LieHom.range (ad R L) }"}, {"tactic": "have : (ad R L).ker = center R L := by simp", "state_before": "case refine'_1\nR : Type u\nL : Type v\nL' : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\ninst\u271d\u00b9 : LieRing L'\ninst\u271d : LieAlgebra R L'\nh : IsNilpotent R { x // x \u2208 LieHom.range (ad R L) }\n\u22a2 IsNilpotent R L", "state_after": "case refine'_1\nR : Type u\nL : Type v\nL' : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\ninst\u271d\u00b9 : LieRing L'\ninst\u271d : LieAlgebra R L'\nh : IsNilpotent R { x // x \u2208 LieHom.range (ad R L) }\nthis : LieHom.ker (ad R L) = center R L\n\u22a2 IsNilpotent R L"}, {"tactic": "exact\n  LieAlgebra.nilpotent_of_nilpotent_quotient (le_of_eq this)\n    ((ad R L).quotKerEquivRange.nilpotent_iff_equiv_nilpotent.mpr h)", "state_before": "case refine'_1\nR : Type u\nL : Type v\nL' : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\ninst\u271d\u00b9 : LieRing L'\ninst\u271d : LieAlgebra R L'\nh : IsNilpotent R { x // x \u2208 LieHom.range (ad R L) }\nthis : LieHom.ker (ad R L) = center R L\n\u22a2 IsNilpotent R L", "state_after": "no goals"}, {"tactic": "simp", "state_before": "R : Type u\nL : Type v\nL' : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\ninst\u271d\u00b9 : LieRing L'\ninst\u271d : LieAlgebra R L'\nh : IsNilpotent R { x // x \u2208 LieHom.range (ad R L) }\n\u22a2 LieHom.ker (ad R L) = center R L", "state_after": "no goals"}, {"tactic": "intro h", "state_before": "case refine'_2\nR : Type u\nL : Type v\nL' : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\ninst\u271d\u00b9 : LieRing L'\ninst\u271d : LieAlgebra R L'\n\u22a2 IsNilpotent R L \u2192 IsNilpotent R { x // x \u2208 LieHom.range (ad R L) }", "state_after": "case refine'_2\nR : Type u\nL : Type v\nL' : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\ninst\u271d\u00b9 : LieRing L'\ninst\u271d : LieAlgebra R L'\nh : IsNilpotent R L\n\u22a2 IsNilpotent R { x // x \u2208 LieHom.range (ad R L) }"}, {"tactic": "exact (ad R L).isNilpotent_range", "state_before": "case refine'_2\nR : Type u\nL : Type v\nL' : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\ninst\u271d\u00b9 : LieRing L'\ninst\u271d : LieAlgebra R L'\nh : IsNilpotent R L\n\u22a2 IsNilpotent R { x // x \u2208 LieHom.range (ad R L) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/Finite/Basic.lean", "full_name": "ZMod.card_units", "start": [423, 1], "end": [424, 32], "traced_tactics": [{"tactic": "rw [Fintype.card_units, card]", "state_before": "K : Type ?u.1003941\nR : Type ?u.1003944\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 Fintype.card (ZMod p)\u02e3 = p - 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Complex/Basic.lean", "full_name": "Complex.abs_apply", "start": [949, 1], "end": [950, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/IsROrC/Basic.lean", "full_name": "IsROrC.I_im'", "start": [323, 1], "end": [323, 75], "traced_tactics": [{"tactic": "rw [mul_comm, I_im]", "state_before": "K : Type u_1\nE : Type ?u.2325884\ninst\u271d : IsROrC K\nz : K\n\u22a2 \u2191im I * \u2191im z = \u2191im z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/MatrixAlgebra.lean", "full_name": "MatrixEquivTensor.invFun_zero", "start": [99, 1], "end": [99, 61], "traced_tactics": [{"tactic": "simp [invFun]", "state_before": "R : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nn : Type w\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\n\u22a2 invFun R A n 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Bounds/Basic.lean", "full_name": "IsLeast.isLeast_image2_of_isGreatest", "start": [1445, 1], "end": [1448, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean", "full_name": "PrimeSpectrum.subset_vanishingIdeal_zeroLocus", "start": [213, 1], "end": [214, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/ClassGroup.lean", "full_name": "card_classGroup_eq_one_iff", "start": [410, 1], "end": [419, 93], "traced_tactics": [{"tactic": "constructor", "state_before": "R : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\n\u22a2 Fintype.card (ClassGroup R) = 1 \u2194 IsPrincipalIdealRing R", "state_after": "case mp\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\n\u22a2 Fintype.card (ClassGroup R) = 1 \u2192 IsPrincipalIdealRing R\n\ncase mpr\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\n\u22a2 IsPrincipalIdealRing R \u2192 Fintype.card (ClassGroup R) = 1"}, {"tactic": "swap", "state_before": "case mp\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\n\u22a2 Fintype.card (ClassGroup R) = 1 \u2192 IsPrincipalIdealRing R\n\ncase mpr\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\n\u22a2 IsPrincipalIdealRing R \u2192 Fintype.card (ClassGroup R) = 1", "state_after": "case mpr\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\n\u22a2 IsPrincipalIdealRing R \u2192 Fintype.card (ClassGroup R) = 1\n\ncase mp\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\n\u22a2 Fintype.card (ClassGroup R) = 1 \u2192 IsPrincipalIdealRing R"}, {"tactic": "rw [Fintype.card_eq_one_iff]", "state_before": "case mp\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\n\u22a2 Fintype.card (ClassGroup R) = 1 \u2192 IsPrincipalIdealRing R", "state_after": "case mp\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\n\u22a2 (\u2203 x, \u2200 (y : ClassGroup R), y = x) \u2192 IsPrincipalIdealRing R"}, {"tactic": "rintro \u27e8I, hI\u27e9", "state_before": "case mp\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\n\u22a2 (\u2203 x, \u2200 (y : ClassGroup R), y = x) \u2192 IsPrincipalIdealRing R", "state_after": "case mp.intro\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\nI : ClassGroup R\nhI : \u2200 (y : ClassGroup R), y = I\n\u22a2 IsPrincipalIdealRing R"}, {"tactic": "have eq_one : \u2200 J : ClassGroup R, J = 1 := fun J => (hI J).trans (hI 1).symm", "state_before": "case mp.intro\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\nI : ClassGroup R\nhI : \u2200 (y : ClassGroup R), y = I\n\u22a2 IsPrincipalIdealRing R", "state_after": "case mp.intro\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\nI : ClassGroup R\nhI : \u2200 (y : ClassGroup R), y = I\neq_one : \u2200 (J : ClassGroup R), J = 1\n\u22a2 IsPrincipalIdealRing R"}, {"tactic": "refine \u27e8fun I => ?_\u27e9", "state_before": "case mp.intro\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\nI : ClassGroup R\nhI : \u2200 (y : ClassGroup R), y = I\neq_one : \u2200 (J : ClassGroup R), J = 1\n\u22a2 IsPrincipalIdealRing R", "state_after": "case mp.intro\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\nI\u271d : ClassGroup R\nhI : \u2200 (y : ClassGroup R), y = I\u271d\neq_one : \u2200 (J : ClassGroup R), J = 1\nI : Ideal R\n\u22a2 Submodule.IsPrincipal I"}, {"tactic": "by_cases hI : I = \u22a5", "state_before": "case mp.intro\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\nI\u271d : ClassGroup R\nhI : \u2200 (y : ClassGroup R), y = I\u271d\neq_one : \u2200 (J : ClassGroup R), J = 1\nI : Ideal R\n\u22a2 Submodule.IsPrincipal I", "state_after": "case pos\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\nI\u271d : ClassGroup R\nhI\u271d : \u2200 (y : ClassGroup R), y = I\u271d\neq_one : \u2200 (J : ClassGroup R), J = 1\nI : Ideal R\nhI : I = \u22a5\n\u22a2 Submodule.IsPrincipal I\n\ncase neg\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\nI\u271d : ClassGroup R\nhI\u271d : \u2200 (y : ClassGroup R), y = I\u271d\neq_one : \u2200 (J : ClassGroup R), J = 1\nI : Ideal R\nhI : \u00acI = \u22a5\n\u22a2 Submodule.IsPrincipal I"}, {"tactic": "intros", "state_before": "case mpr\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\n\u22a2 IsPrincipalIdealRing R \u2192 Fintype.card (ClassGroup R) = 1", "state_after": "case mpr\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\na\u271d : IsPrincipalIdealRing R\n\u22a2 Fintype.card (ClassGroup R) = 1"}, {"tactic": "convert card_classGroup_eq_one (R := R)", "state_before": "case mpr\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\na\u271d : IsPrincipalIdealRing R\n\u22a2 Fintype.card (ClassGroup R) = 1", "state_after": "no goals"}, {"tactic": "rw [hI]", "state_before": "case pos\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\nI\u271d : ClassGroup R\nhI\u271d : \u2200 (y : ClassGroup R), y = I\u271d\neq_one : \u2200 (J : ClassGroup R), J = 1\nI : Ideal R\nhI : I = \u22a5\n\u22a2 Submodule.IsPrincipal I", "state_after": "case pos\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\nI\u271d : ClassGroup R\nhI\u271d : \u2200 (y : ClassGroup R), y = I\u271d\neq_one : \u2200 (J : ClassGroup R), J = 1\nI : Ideal R\nhI : I = \u22a5\n\u22a2 Submodule.IsPrincipal \u22a5"}, {"tactic": "exact bot_isPrincipal", "state_before": "case pos\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\nI\u271d : ClassGroup R\nhI\u271d : \u2200 (y : ClassGroup R), y = I\u271d\neq_one : \u2200 (J : ClassGroup R), J = 1\nI : Ideal R\nhI : I = \u22a5\n\u22a2 Submodule.IsPrincipal \u22a5", "state_after": "no goals"}, {"tactic": "exact (ClassGroup.mk0_eq_one_iff (mem_nonZeroDivisors_iff_ne_zero.mpr hI)).mp (eq_one _)", "state_before": "case neg\nR : Type u_1\nK : Type ?u.1978209\nL : Type ?u.1978212\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : DecidableEq L\ninst\u271d\u2078 : Algebra R K\ninst\u271d\u2077 : IsFractionRing R K\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : FiniteDimensional K L\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsDedekindDomain R\ninst\u271d : Fintype (ClassGroup R)\nI\u271d : ClassGroup R\nhI\u271d : \u2200 (y : ClassGroup R), y = I\u271d\neq_one : \u2200 (J : ClassGroup R), J = 1\nI : Ideal R\nhI : \u00acI = \u22a5\n\u22a2 Submodule.IsPrincipal I", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Quiver/Basic.lean", "full_name": "Prefunctor.id_comp", "start": [125, 1], "end": [126, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Prod/Basic.lean", "full_name": "Function.LeftInverse.Prod_map", "start": [329, 1], "end": [331, 73], "traced_tactics": [{"tactic": "rw [Prod.map_map, hf.comp_eq_id, hg.comp_eq_id, map_id, id]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type u_4\n\u03b4 : Type u_3\nf : \u03b1 \u2192 \u03b3\ng : \u03b2 \u2192 \u03b4\nf\u2081 : \u03b1 \u2192 \u03b2\ng\u2081 : \u03b3 \u2192 \u03b4\nf\u2082 : \u03b2 \u2192 \u03b1\ng\u2082 : \u03b4 \u2192 \u03b3\nhf : LeftInverse f\u2081 f\u2082\nhg : LeftInverse g\u2081 g\u2082\na : \u03b2 \u00d7 \u03b4\n\u22a2 map f\u2081 g\u2081 (map f\u2082 g\u2082 a) = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Partition/Finpartition.lean", "full_name": "Finpartition.card_mono", "start": [343, 1], "end": [352, 97], "traced_tactics": [{"tactic": "classical\n  have : \u2200 b \u2208 Q.parts, \u2203 c \u2208 P.parts, c \u2264 b := fun b \u21a6 exists_le_of_le h\n  choose f hP hf using this\n  rw [\u2190 card_attach]\n  refine' card_le_card_of_inj_on (fun b \u21a6 f _ b.2) (fun b _ \u21a6 hP _ b.2) fun b _ c _ h \u21a6 _\n  exact\n    Subtype.coe_injective\n      (Q.disjoint.elim b.2 c.2 fun H \u21a6\n        P.ne_bot (hP _ b.2) <| disjoint_self.1 <| H.mono (hf _ b.2) <| h.le.trans <| hf _ c.2)", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : OrderBot \u03b1\na : \u03b1\nP Q : Finpartition a\nh : P \u2264 Q\n\u22a2 card Q.parts \u2264 card P.parts", "state_after": "no goals"}, {"tactic": "have : \u2200 b \u2208 Q.parts, \u2203 c \u2208 P.parts, c \u2264 b := fun b \u21a6 exists_le_of_le h", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : OrderBot \u03b1\na : \u03b1\nP Q : Finpartition a\nh : P \u2264 Q\n\u22a2 card Q.parts \u2264 card P.parts", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : OrderBot \u03b1\na : \u03b1\nP Q : Finpartition a\nh : P \u2264 Q\nthis : \u2200 (b : \u03b1), b \u2208 Q.parts \u2192 \u2203 c, c \u2208 P.parts \u2227 c \u2264 b\n\u22a2 card Q.parts \u2264 card P.parts"}, {"tactic": "choose f hP hf using this", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : OrderBot \u03b1\na : \u03b1\nP Q : Finpartition a\nh : P \u2264 Q\nthis : \u2200 (b : \u03b1), b \u2208 Q.parts \u2192 \u2203 c, c \u2208 P.parts \u2227 c \u2264 b\n\u22a2 card Q.parts \u2264 card P.parts", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : OrderBot \u03b1\na : \u03b1\nP Q : Finpartition a\nh : P \u2264 Q\nf : (b : \u03b1) \u2192 b \u2208 Q.parts \u2192 \u03b1\nhP : \u2200 (b : \u03b1) (a_1 : b \u2208 Q.parts), f b a_1 \u2208 P.parts\nhf : \u2200 (b : \u03b1) (a : b \u2208 Q.parts), f b a \u2264 b\n\u22a2 card Q.parts \u2264 card P.parts"}, {"tactic": "rw [\u2190 card_attach]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : OrderBot \u03b1\na : \u03b1\nP Q : Finpartition a\nh : P \u2264 Q\nf : (b : \u03b1) \u2192 b \u2208 Q.parts \u2192 \u03b1\nhP : \u2200 (b : \u03b1) (a_1 : b \u2208 Q.parts), f b a_1 \u2208 P.parts\nhf : \u2200 (b : \u03b1) (a : b \u2208 Q.parts), f b a \u2264 b\n\u22a2 card Q.parts \u2264 card P.parts", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : OrderBot \u03b1\na : \u03b1\nP Q : Finpartition a\nh : P \u2264 Q\nf : (b : \u03b1) \u2192 b \u2208 Q.parts \u2192 \u03b1\nhP : \u2200 (b : \u03b1) (a_1 : b \u2208 Q.parts), f b a_1 \u2208 P.parts\nhf : \u2200 (b : \u03b1) (a : b \u2208 Q.parts), f b a \u2264 b\n\u22a2 card (attach Q.parts) \u2264 card P.parts"}, {"tactic": "refine' card_le_card_of_inj_on (fun b \u21a6 f _ b.2) (fun b _ \u21a6 hP _ b.2) fun b _ c _ h \u21a6 _", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : OrderBot \u03b1\na : \u03b1\nP Q : Finpartition a\nh : P \u2264 Q\nf : (b : \u03b1) \u2192 b \u2208 Q.parts \u2192 \u03b1\nhP : \u2200 (b : \u03b1) (a_1 : b \u2208 Q.parts), f b a_1 \u2208 P.parts\nhf : \u2200 (b : \u03b1) (a : b \u2208 Q.parts), f b a \u2264 b\n\u22a2 card (attach Q.parts) \u2264 card P.parts", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : OrderBot \u03b1\na : \u03b1\nP Q : Finpartition a\nh\u271d : P \u2264 Q\nf : (b : \u03b1) \u2192 b \u2208 Q.parts \u2192 \u03b1\nhP : \u2200 (b : \u03b1) (a_1 : b \u2208 Q.parts), f b a_1 \u2208 P.parts\nhf : \u2200 (b : \u03b1) (a : b \u2208 Q.parts), f b a \u2264 b\nb : { x // x \u2208 Q.parts }\nx\u271d\u00b9 : b \u2208 attach Q.parts\nc : { x // x \u2208 Q.parts }\nx\u271d : c \u2208 attach Q.parts\nh : (fun b => f \u2191b (_ : \u2191b \u2208 Q.parts)) b = (fun b => f \u2191b (_ : \u2191b \u2208 Q.parts)) c\n\u22a2 b = c"}, {"tactic": "exact\n  Subtype.coe_injective\n    (Q.disjoint.elim b.2 c.2 fun H \u21a6\n      P.ne_bot (hP _ b.2) <| disjoint_self.1 <| H.mono (hf _ b.2) <| h.le.trans <| hf _ c.2)", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : OrderBot \u03b1\na : \u03b1\nP Q : Finpartition a\nh\u271d : P \u2264 Q\nf : (b : \u03b1) \u2192 b \u2208 Q.parts \u2192 \u03b1\nhP : \u2200 (b : \u03b1) (a_1 : b \u2208 Q.parts), f b a_1 \u2208 P.parts\nhf : \u2200 (b : \u03b1) (a : b \u2208 Q.parts), f b a \u2264 b\nb : { x // x \u2208 Q.parts }\nx\u271d\u00b9 : b \u2208 attach Q.parts\nc : { x // x \u2208 Q.parts }\nx\u271d : c \u2208 attach Q.parts\nh : (fun b => f \u2191b (_ : \u2191b \u2208 Q.parts)) b = (fun b => f \u2191b (_ : \u2191b \u2208 Q.parts)) c\n\u22a2 b = c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Filtration.lean", "full_name": "Ideal.Filtration.Stable.of_le", "start": [408, 1], "end": [415, 61], "traced_tactics": [{"tactic": "haveI := isNoetherian_of_fg_of_noetherian' h.1", "state_before": "R M : Type u\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh\u271d : Stable F\ninst\u271d : IsNoetherianRing R\nh : Module.Finite R M\nhF : Stable F\nF' : Filtration I M\nhf : F' \u2264 F\n\u22a2 Stable F'", "state_after": "R M : Type u\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh\u271d : Stable F\ninst\u271d : IsNoetherianRing R\nh : Module.Finite R M\nhF : Stable F\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\n\u22a2 Stable F'"}, {"tactic": "rw [\u2190 submodule_fg_iff_stable] at hF\u22a2", "state_before": "R M : Type u\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh\u271d : Stable F\ninst\u271d : IsNoetherianRing R\nh : Module.Finite R M\nhF : Stable F\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\n\u22a2 Stable F'", "state_after": "R M : Type u\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh\u271d : Stable F\ninst\u271d : IsNoetherianRing R\nh : Module.Finite R M\nhF : Submodule.FG (Filtration.submodule F)\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\n\u22a2 Submodule.FG (Filtration.submodule F')\n\ncase hF'\nR M : Type u\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh\u271d : Stable F\ninst\u271d : IsNoetherianRing R\nh : Module.Finite R M\nhF : Submodule.FG (Filtration.submodule F)\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\n\u22a2 \u2200 (i : \u2115), Submodule.FG (N F' i)\n\ncase hF'\nR M : Type u\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh\u271d : Stable F\ninst\u271d : IsNoetherianRing R\nh : Module.Finite R M\nhF : Stable F\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\n\u22a2 \u2200 (i : \u2115), Submodule.FG (N F i)"}, {"tactic": "any_goals intro i; exact IsNoetherian.noetherian _", "state_before": "R M : Type u\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh\u271d : Stable F\ninst\u271d : IsNoetherianRing R\nh : Module.Finite R M\nhF : Submodule.FG (Filtration.submodule F)\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\n\u22a2 Submodule.FG (Filtration.submodule F')\n\ncase hF'\nR M : Type u\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh\u271d : Stable F\ninst\u271d : IsNoetherianRing R\nh : Module.Finite R M\nhF : Submodule.FG (Filtration.submodule F)\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\n\u22a2 \u2200 (i : \u2115), Submodule.FG (N F' i)\n\ncase hF'\nR M : Type u\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh\u271d : Stable F\ninst\u271d : IsNoetherianRing R\nh : Module.Finite R M\nhF : Stable F\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\n\u22a2 \u2200 (i : \u2115), Submodule.FG (N F i)", "state_after": "R M : Type u\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh\u271d : Stable F\ninst\u271d : IsNoetherianRing R\nh : Module.Finite R M\nhF : Submodule.FG (Filtration.submodule F)\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\n\u22a2 Submodule.FG (Filtration.submodule F')"}, {"tactic": "have := isNoetherian_of_fg_of_noetherian _ hF", "state_before": "R M : Type u\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh\u271d : Stable F\ninst\u271d : IsNoetherianRing R\nh : Module.Finite R M\nhF : Submodule.FG (Filtration.submodule F)\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\n\u22a2 Submodule.FG (Filtration.submodule F')", "state_after": "R M : Type u\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh\u271d : Stable F\ninst\u271d : IsNoetherianRing R\nh : Module.Finite R M\nhF : Submodule.FG (Filtration.submodule F)\nF' : Filtration I M\nhf : F' \u2264 F\nthis\u271d : IsNoetherian R M\nthis : IsNoetherian { x // x \u2208 reesAlgebra I } { x // x \u2208 Filtration.submodule F }\n\u22a2 Submodule.FG (Filtration.submodule F')"}, {"tactic": "rw [isNoetherian_submodule] at this", "state_before": "R M : Type u\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh\u271d : Stable F\ninst\u271d : IsNoetherianRing R\nh : Module.Finite R M\nhF : Submodule.FG (Filtration.submodule F)\nF' : Filtration I M\nhf : F' \u2264 F\nthis\u271d : IsNoetherian R M\nthis : IsNoetherian { x // x \u2208 reesAlgebra I } { x // x \u2208 Filtration.submodule F }\n\u22a2 Submodule.FG (Filtration.submodule F')", "state_after": "R M : Type u\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh\u271d : Stable F\ninst\u271d : IsNoetherianRing R\nh : Module.Finite R M\nhF : Submodule.FG (Filtration.submodule F)\nF' : Filtration I M\nhf : F' \u2264 F\nthis\u271d : IsNoetherian R M\nthis : \u2200 (s : Submodule { x // x \u2208 reesAlgebra I } (PolynomialModule R M)), s \u2264 Filtration.submodule F \u2192 Submodule.FG s\n\u22a2 Submodule.FG (Filtration.submodule F')"}, {"tactic": "exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)", "state_before": "R M : Type u\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh\u271d : Stable F\ninst\u271d : IsNoetherianRing R\nh : Module.Finite R M\nhF : Submodule.FG (Filtration.submodule F)\nF' : Filtration I M\nhf : F' \u2264 F\nthis\u271d : IsNoetherian R M\nthis : \u2200 (s : Submodule { x // x \u2208 reesAlgebra I } (PolynomialModule R M)), s \u2264 Filtration.submodule F \u2192 Submodule.FG s\n\u22a2 Submodule.FG (Filtration.submodule F')", "state_after": "no goals"}, {"tactic": "intro i", "state_before": "case hF'\nR M : Type u\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh\u271d : Stable F\ninst\u271d : IsNoetherianRing R\nh : Module.Finite R M\nhF : Stable F\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\n\u22a2 \u2200 (i : \u2115), Submodule.FG (N F i)", "state_after": "case hF'\nR M : Type u\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh\u271d : Stable F\ninst\u271d : IsNoetherianRing R\nh : Module.Finite R M\nhF : Stable F\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\ni : \u2115\n\u22a2 Submodule.FG (N F i)"}, {"tactic": "exact IsNoetherian.noetherian _", "state_before": "case hF'\nR M : Type u\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh\u271d : Stable F\ninst\u271d : IsNoetherianRing R\nh : Module.Finite R M\nhF : Stable F\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\ni : \u2115\n\u22a2 Submodule.FG (N F i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Localization/Basic.lean", "full_name": "Field.localization_map_bijective", "start": [1283, 1], "end": [1286, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "full_name": "Filter.Tendsto.mass", "start": [488, 1], "end": [490, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/Basic.lean", "full_name": "Equiv.prodCongr_refl_left", "start": [736, 1], "end": [739, 7], "traced_tactics": [{"tactic": "ext \u27e8a, b\u27e9 : 1", "state_before": "\u03b1\u2081 : Type u_3\n\u03b2\u2081 : Type u_1\n\u03b2\u2082 : Type u_2\ne\u271d : \u03b1\u2081 \u2192 \u03b2\u2081 \u2243 \u03b2\u2082\ne : \u03b2\u2081 \u2243 \u03b2\u2082\n\u22a2 prodCongr (Equiv.refl \u03b1\u2081) e = prodCongrRight fun x => e", "state_after": "case H.mk\n\u03b1\u2081 : Type u_3\n\u03b2\u2081 : Type u_1\n\u03b2\u2082 : Type u_2\ne\u271d : \u03b1\u2081 \u2192 \u03b2\u2081 \u2243 \u03b2\u2082\ne : \u03b2\u2081 \u2243 \u03b2\u2082\na : \u03b1\u2081\nb : \u03b2\u2081\n\u22a2 \u2191(prodCongr (Equiv.refl \u03b1\u2081) e) (a, b) = \u2191(prodCongrRight fun x => e) (a, b)"}, {"tactic": "simp", "state_before": "case H.mk\n\u03b1\u2081 : Type u_3\n\u03b2\u2081 : Type u_1\n\u03b2\u2082 : Type u_2\ne\u271d : \u03b1\u2081 \u2192 \u03b2\u2081 \u2243 \u03b2\u2082\ne : \u03b2\u2081 \u2243 \u03b2\u2082\na : \u03b1\u2081\nb : \u03b2\u2081\n\u22a2 \u2191(prodCongr (Equiv.refl \u03b1\u2081) e) (a, b) = \u2191(prodCongrRight fun x => e) (a, b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Heyting/Basic.lean", "full_name": "himp_le_himp_right", "start": [418, 1], "end": [419, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/EpiMono.lean", "full_name": "CategoryTheory.IsIso.of_mono_retraction'", "start": [185, 1], "end": [187, 79], "traced_tactics": [{"tactic": "simp", "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category C\nX Y : C\nf : X \u27f6 Y\nhf : SplitMono f\ninst\u271d : Mono hf.retraction\n\u22a2 f \u226b hf.retraction = \ud835\udfd9 X", "state_after": "no goals"}, {"tactic": "simp", "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category C\nX Y : C\nf : X \u27f6 Y\nhf : SplitMono f\ninst\u271d : Mono hf.retraction\n\u22a2 (hf.retraction \u226b f) \u226b hf.retraction = hf.retraction", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/LocalEquiv.lean", "full_name": "LocalEquiv.EqOnSource.source_eq", "start": [833, 1], "end": [834, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Nat/Gcd.lean", "full_name": "Nat.gcd_one_right", "start": [81, 9], "end": [81, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.nthLe_enum", "start": [3965, 1], "end": [3967, 58], "traced_tactics": [{"tactic": "simpa [length_enum] using hi'", "state_before": "\u03b9 : Type ?u.461588\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 l : List \u03b1\ni : \u2115\nhi' : i < length (enum l)\n\u22a2 i < length l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SetFamily/HarrisKleitman.lean", "full_name": "IsLowerSet.memberSubfamily", "start": [45, 1], "end": [49, 65], "traced_tactics": [{"tactic": "rintro s t hts", "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\ns : Finset \u03b1\na : \u03b1\nh : IsLowerSet \u2191\ud835\udc9c\n\u22a2 IsLowerSet \u2191(Finset.memberSubfamily a \ud835\udc9c)", "state_after": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na : \u03b1\nh : IsLowerSet \u2191\ud835\udc9c\ns t : Finset \u03b1\nhts : t \u2264 s\n\u22a2 s \u2208 \u2191(Finset.memberSubfamily a \ud835\udc9c) \u2192 t \u2208 \u2191(Finset.memberSubfamily a \ud835\udc9c)"}, {"tactic": "simp_rw [mem_coe, mem_memberSubfamily]", "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na : \u03b1\nh : IsLowerSet \u2191\ud835\udc9c\ns t : Finset \u03b1\nhts : t \u2264 s\n\u22a2 s \u2208 \u2191(Finset.memberSubfamily a \ud835\udc9c) \u2192 t \u2208 \u2191(Finset.memberSubfamily a \ud835\udc9c)", "state_after": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na : \u03b1\nh : IsLowerSet \u2191\ud835\udc9c\ns t : Finset \u03b1\nhts : t \u2264 s\n\u22a2 insert a s \u2208 \ud835\udc9c \u2227 \u00aca \u2208 s \u2192 insert a t \u2208 \ud835\udc9c \u2227 \u00aca \u2208 t"}, {"tactic": "exact And.imp (h <| insert_subset_insert _ hts) (mt <| @hts _)", "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na : \u03b1\nh : IsLowerSet \u2191\ud835\udc9c\ns t : Finset \u03b1\nhts : t \u2264 s\n\u22a2 insert a s \u2208 \ud835\udc9c \u2227 \u00aca \u2208 s \u2192 insert a t \u2208 \ud835\udc9c \u2227 \u00aca \u2208 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.image_smul_prod", "start": [93, 1], "end": [94, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.Integrable.apply_continuousLinearMap", "start": [1439, 1], "end": [1442, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean", "full_name": "FormalMultilinearSeries.norm_apply_eq_norm_coef", "start": [294, 1], "end": [295, 71], "traced_tactics": [{"tactic": "rw [\u2190 mkPiField_coeff_eq p, ContinuousMultilinearMap.norm_mkPiField]", "state_before": "\ud835\udd5c : Type u\n\ud835\udd5c' : Type u'\nE : Type v\nF : Type w\nG : Type x\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : E\np : FormalMultilinearSeries \ud835\udd5c \ud835\udd5c E\nf : \ud835\udd5c \u2192 E\nn : \u2115\nz z\u2080 : \ud835\udd5c\ny : Fin n \u2192 \ud835\udd5c\n\u22a2 \u2016p n\u2016 = \u2016coeff p n\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "full_name": "mulPosMonoRev_iff_mulPosReflectLT", "start": [1018, 1], "end": [1019, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/Egorov.lean", "full_name": "MeasureTheory.Egorov.tendstoUniformlyOn_diff_iUnionNotConvergentSeq", "start": [172, 1], "end": [188, 36], "traced_tactics": [{"tactic": "rw [Metric.tendstoUniformlyOn_iff]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 TendstoUniformlyOn f g atTop (s \\ iUnionNotConvergentSeq h\u03b5 hf hg hsm hs hfg)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 \u2200 (\u03b5_1 : \u211d),\n    \u03b5_1 > 0 \u2192\n      \u2200\u1da0 (n : \u03b9) in atTop, \u2200 (x : \u03b1), x \u2208 s \\ iUnionNotConvergentSeq h\u03b5 hf hg hsm hs hfg \u2192 dist (g x) (f n x) < \u03b5_1"}, {"tactic": "intro \u03b4 h\u03b4", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 \u2200 (\u03b5_1 : \u211d),\n    \u03b5_1 > 0 \u2192\n      \u2200\u1da0 (n : \u03b9) in atTop, \u2200 (x : \u03b1), x \u2208 s \\ iUnionNotConvergentSeq h\u03b5 hf hg hsm hs hfg \u2192 dist (g x) (f n x) < \u03b5_1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\n\u22a2 \u2200\u1da0 (n : \u03b9) in atTop, \u2200 (x : \u03b1), x \u2208 s \\ iUnionNotConvergentSeq h\u03b5 hf hg hsm hs hfg \u2192 dist (g x) (f n x) < \u03b4"}, {"tactic": "obtain \u27e8N, hN\u27e9 := exists_nat_one_div_lt h\u03b4", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\n\u22a2 \u2200\u1da0 (n : \u03b9) in atTop, \u2200 (x : \u03b1), x \u2208 s \\ iUnionNotConvergentSeq h\u03b5 hf hg hsm hs hfg \u2192 dist (g x) (f n x) < \u03b4", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\n\u22a2 \u2200\u1da0 (n : \u03b9) in atTop, \u2200 (x : \u03b1), x \u2208 s \\ iUnionNotConvergentSeq h\u03b5 hf hg hsm hs hfg \u2192 dist (g x) (f n x) < \u03b4"}, {"tactic": "rw [eventually_atTop]", "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\n\u22a2 \u2200\u1da0 (n : \u03b9) in atTop, \u2200 (x : \u03b1), x \u2208 s \\ iUnionNotConvergentSeq h\u03b5 hf hg hsm hs hfg \u2192 dist (g x) (f n x) < \u03b4", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\n\u22a2 \u2203 a, \u2200 (b : \u03b9), b \u2265 a \u2192 \u2200 (x : \u03b1), x \u2208 s \\ iUnionNotConvergentSeq h\u03b5 hf hg hsm hs hfg \u2192 dist (g x) (f b x) < \u03b4"}, {"tactic": "refine' \u27e8Egorov.notConvergentSeqLtIndex (half_pos h\u03b5) hf hg hsm hs hfg N, fun n hn x hx => _\u27e9", "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\n\u22a2 \u2203 a, \u2200 (b : \u03b9), b \u2265 a \u2192 \u2200 (x : \u03b1), x \u2208 s \\ iUnionNotConvergentSeq h\u03b5 hf hg hsm hs hfg \u2192 dist (g x) (f b x) < \u03b4", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\nn : \u03b9\nhn : n \u2265 notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N\nx : \u03b1\nhx : x \u2208 s \\ iUnionNotConvergentSeq h\u03b5 hf hg hsm hs hfg\n\u22a2 dist (g x) (f n x) < \u03b4"}, {"tactic": "simp only [Set.mem_diff, Egorov.iUnionNotConvergentSeq, not_exists, Set.mem_iUnion,\n  Set.mem_inter_iff, not_and, exists_and_left] at hx", "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\nn : \u03b9\nhn : n \u2265 notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N\nx : \u03b1\nhx : x \u2208 s \\ iUnionNotConvergentSeq h\u03b5 hf hg hsm hs hfg\n\u22a2 dist (g x) (f n x) < \u03b4", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\nn : \u03b9\nhn : n \u2265 notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N\nx : \u03b1\nhx :\n  x \u2208 s \u2227\n    (x \u2208 s \u2192\n      \u2200 (x_1 : \u2115),\n        \u00acx \u2208 notConvergentSeq (fun n => f n) g x_1 (notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg x_1))\n\u22a2 dist (g x) (f n x) < \u03b4"}, {"tactic": "obtain \u27e8hxs, hx\u27e9 := hx", "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\nn : \u03b9\nhn : n \u2265 notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N\nx : \u03b1\nhx :\n  x \u2208 s \u2227\n    (x \u2208 s \u2192\n      \u2200 (x_1 : \u2115),\n        \u00acx \u2208 notConvergentSeq (fun n => f n) g x_1 (notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg x_1))\n\u22a2 dist (g x) (f n x) < \u03b4", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\nn : \u03b9\nhn : n \u2265 notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N\nx : \u03b1\nhxs : x \u2208 s\nhx :\n  x \u2208 s \u2192\n    \u2200 (x_1 : \u2115),\n      \u00acx \u2208 notConvergentSeq (fun n => f n) g x_1 (notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg x_1)\n\u22a2 dist (g x) (f n x) < \u03b4"}, {"tactic": "specialize hx hxs N", "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\nn : \u03b9\nhn : n \u2265 notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N\nx : \u03b1\nhxs : x \u2208 s\nhx :\n  x \u2208 s \u2192\n    \u2200 (x_1 : \u2115),\n      \u00acx \u2208 notConvergentSeq (fun n => f n) g x_1 (notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg x_1)\n\u22a2 dist (g x) (f n x) < \u03b4", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\nn : \u03b9\nhn : n \u2265 notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N\nx : \u03b1\nhxs : x \u2208 s\nhx : \u00acx \u2208 notConvergentSeq (fun n => f n) g N (notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N)\n\u22a2 dist (g x) (f n x) < \u03b4"}, {"tactic": "rw [Egorov.mem_notConvergentSeq_iff] at hx", "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\nn : \u03b9\nhn : n \u2265 notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N\nx : \u03b1\nhxs : x \u2208 s\nhx : \u00acx \u2208 notConvergentSeq (fun n => f n) g N (notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N)\n\u22a2 dist (g x) (f n x) < \u03b4", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\nn : \u03b9\nhn : n \u2265 notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N\nx : \u03b1\nhxs : x \u2208 s\nhx : \u00ac\u2203 k x_1, 1 / (\u2191N + 1) < dist (f k x) (g x)\n\u22a2 dist (g x) (f n x) < \u03b4"}, {"tactic": "push_neg  at hx", "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\nn : \u03b9\nhn : n \u2265 notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N\nx : \u03b1\nhxs : x \u2208 s\nhx : \u00ac\u2203 k x_1, 1 / (\u2191N + 1) < dist (f k x) (g x)\n\u22a2 dist (g x) (f n x) < \u03b4", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\nn : \u03b9\nhn : n \u2265 notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N\nx : \u03b1\nhxs : x \u2208 s\nhx : \u2200 (k : \u03b9), notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N \u2264 k \u2192 dist (f k x) (g x) \u2264 1 / (\u2191N + 1)\n\u22a2 dist (g x) (f n x) < \u03b4"}, {"tactic": "rw [dist_comm]", "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\nn : \u03b9\nhn : n \u2265 notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N\nx : \u03b1\nhxs : x \u2208 s\nhx : \u2200 (k : \u03b9), notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N \u2264 k \u2192 dist (f k x) (g x) \u2264 1 / (\u2191N + 1)\n\u22a2 dist (g x) (f n x) < \u03b4", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\nn : \u03b9\nhn : n \u2265 notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N\nx : \u03b1\nhxs : x \u2208 s\nhx : \u2200 (k : \u03b9), notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N \u2264 k \u2192 dist (f k x) (g x) \u2264 1 / (\u2191N + 1)\n\u22a2 dist (f n x) (g x) < \u03b4"}, {"tactic": "exact lt_of_le_of_lt (hx n hn) hN", "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : SemilatticeSup \u03b9\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : Countable \u03b9\nh\u03b5 : 0 < \u03b5\nhf : \u2200 (n : \u03b9), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhsm : MeasurableSet s\nhs : \u2191\u2191\u03bc s \u2260 \u22a4\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nN : \u2115\nhN : 1 / (\u2191N + 1) < \u03b4\nn : \u03b9\nhn : n \u2265 notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N\nx : \u03b1\nhxs : x \u2208 s\nhx : \u2200 (k : \u03b9), notConvergentSeqLtIndex (_ : 0 < \u03b5 / 2) hf hg hsm hs hfg N \u2264 k \u2192 dist (f k x) (g x) \u2264 1 / (\u2191N + 1)\n\u22a2 dist (f n x) (g x) < \u03b4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.mem_ofOption", "start": [332, 1], "end": [334, 99], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Semiquot.lean", "full_name": "Semiquot.get_mem", "start": [220, 1], "end": [222, 53], "traced_tactics": [{"tactic": "let \u27e8a, h\u27e9 := exists_mem q", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.15301\nq : Semiquot \u03b1\np : IsPure q\n\u22a2 get q p \u2208 q", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.15301\nq : Semiquot \u03b1\np : IsPure q\na : \u03b1\nh : a \u2208 q\n\u22a2 get q p \u2208 q"}, {"tactic": "unfold get", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.15301\nq : Semiquot \u03b1\np : IsPure q\na : \u03b1\nh : a \u2208 q\n\u22a2 get q p \u2208 q", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.15301\nq : Semiquot \u03b1\np : IsPure q\na : \u03b1\nh : a \u2208 q\n\u22a2 liftOn q id p \u2208 q"}, {"tactic": "rw [liftOn_ofMem q _ _ a h]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.15301\nq : Semiquot \u03b1\np : IsPure q\na : \u03b1\nh : a \u2208 q\n\u22a2 liftOn q id p \u2208 q", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.15301\nq : Semiquot \u03b1\np : IsPure q\na : \u03b1\nh : a \u2208 q\n\u22a2 id a \u2208 q"}, {"tactic": "exact h", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.15301\nq : Semiquot \u03b1\np : IsPure q\na : \u03b1\nh : a \u2208 q\n\u22a2 id a \u2208 q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm.lean", "full_name": "SemilinearMapClass.bound_of_continuous", "start": [80, 1], "end": [92, 40], "traced_tactics": [{"tactic": "rcases NormedAddCommGroup.tendsto_nhds_nhds.1 (hf.tendsto 0) 1 zero_lt_one with \u27e8\u03b5, \u03b5_pos, h\u03b5\u27e9", "state_before": "\ud835\udd5c : Type u_2\n\ud835\udd5c\u2082 : Type u_3\n\ud835\udd5c\u2083 : Type ?u.156253\nE : Type u_4\nE\u2097 : Type ?u.156259\nF : Type u_5\nF\u2097 : Type ?u.156265\nG : Type ?u.156268\nG\u2097 : Type ?u.156271\n\ud835\udcd5 : Type u_1\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d : SemilinearMapClass \ud835\udcd5 \u03c3\u2081\u2082 E F\nf : \ud835\udcd5\nhf : Continuous \u2191f\n\u22a2 \u2203 C, 0 < C \u2227 \u2200 (x : E), \u2016\u2191f x\u2016 \u2264 C * \u2016x\u2016", "state_after": "case intro.intro\n\ud835\udd5c : Type u_2\n\ud835\udd5c\u2082 : Type u_3\n\ud835\udd5c\u2083 : Type ?u.156253\nE : Type u_4\nE\u2097 : Type ?u.156259\nF : Type u_5\nF\u2097 : Type ?u.156265\nG : Type ?u.156268\nG\u2097 : Type ?u.156271\n\ud835\udcd5 : Type u_1\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d : SemilinearMapClass \ud835\udcd5 \u03c3\u2081\u2082 E F\nf : \ud835\udcd5\nhf : Continuous \u2191f\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nh\u03b5 : \u2200 (x' : E), \u2016x' - 0\u2016 < \u03b5 \u2192 \u2016\u2191f x' - \u2191f 0\u2016 < 1\n\u22a2 \u2203 C, 0 < C \u2227 \u2200 (x : E), \u2016\u2191f x\u2016 \u2264 C * \u2016x\u2016"}, {"tactic": "simp only [sub_zero, map_zero] at h\u03b5", "state_before": "case intro.intro\n\ud835\udd5c : Type u_2\n\ud835\udd5c\u2082 : Type u_3\n\ud835\udd5c\u2083 : Type ?u.156253\nE : Type u_4\nE\u2097 : Type ?u.156259\nF : Type u_5\nF\u2097 : Type ?u.156265\nG : Type ?u.156268\nG\u2097 : Type ?u.156271\n\ud835\udcd5 : Type u_1\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d : SemilinearMapClass \ud835\udcd5 \u03c3\u2081\u2082 E F\nf : \ud835\udcd5\nhf : Continuous \u2191f\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nh\u03b5 : \u2200 (x' : E), \u2016x' - 0\u2016 < \u03b5 \u2192 \u2016\u2191f x' - \u2191f 0\u2016 < 1\n\u22a2 \u2203 C, 0 < C \u2227 \u2200 (x : E), \u2016\u2191f x\u2016 \u2264 C * \u2016x\u2016", "state_after": "case intro.intro\n\ud835\udd5c : Type u_2\n\ud835\udd5c\u2082 : Type u_3\n\ud835\udd5c\u2083 : Type ?u.156253\nE : Type u_4\nE\u2097 : Type ?u.156259\nF : Type u_5\nF\u2097 : Type ?u.156265\nG : Type ?u.156268\nG\u2097 : Type ?u.156271\n\ud835\udcd5 : Type u_1\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d : SemilinearMapClass \ud835\udcd5 \u03c3\u2081\u2082 E F\nf : \ud835\udcd5\nhf : Continuous \u2191f\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nh\u03b5 : \u2200 (x' : E), \u2016x'\u2016 < \u03b5 \u2192 \u2016\u2191f x'\u2016 < 1\n\u22a2 \u2203 C, 0 < C \u2227 \u2200 (x : E), \u2016\u2191f x\u2016 \u2264 C * \u2016x\u2016"}, {"tactic": "rcases NormedField.exists_one_lt_norm \ud835\udd5c with \u27e8c, hc\u27e9", "state_before": "case intro.intro\n\ud835\udd5c : Type u_2\n\ud835\udd5c\u2082 : Type u_3\n\ud835\udd5c\u2083 : Type ?u.156253\nE : Type u_4\nE\u2097 : Type ?u.156259\nF : Type u_5\nF\u2097 : Type ?u.156265\nG : Type ?u.156268\nG\u2097 : Type ?u.156271\n\ud835\udcd5 : Type u_1\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d : SemilinearMapClass \ud835\udcd5 \u03c3\u2081\u2082 E F\nf : \ud835\udcd5\nhf : Continuous \u2191f\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nh\u03b5 : \u2200 (x' : E), \u2016x'\u2016 < \u03b5 \u2192 \u2016\u2191f x'\u2016 < 1\n\u22a2 \u2203 C, 0 < C \u2227 \u2200 (x : E), \u2016\u2191f x\u2016 \u2264 C * \u2016x\u2016", "state_after": "case intro.intro.intro\n\ud835\udd5c : Type u_2\n\ud835\udd5c\u2082 : Type u_3\n\ud835\udd5c\u2083 : Type ?u.156253\nE : Type u_4\nE\u2097 : Type ?u.156259\nF : Type u_5\nF\u2097 : Type ?u.156265\nG : Type ?u.156268\nG\u2097 : Type ?u.156271\n\ud835\udcd5 : Type u_1\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d : SemilinearMapClass \ud835\udcd5 \u03c3\u2081\u2082 E F\nf : \ud835\udcd5\nhf : Continuous \u2191f\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nh\u03b5 : \u2200 (x' : E), \u2016x'\u2016 < \u03b5 \u2192 \u2016\u2191f x'\u2016 < 1\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\n\u22a2 \u2203 C, 0 < C \u2227 \u2200 (x : E), \u2016\u2191f x\u2016 \u2264 C * \u2016x\u2016"}, {"tactic": "have : 0 < \u2016c\u2016 / \u03b5 := div_pos (zero_lt_one.trans hc) \u03b5_pos", "state_before": "case intro.intro.intro\n\ud835\udd5c : Type u_2\n\ud835\udd5c\u2082 : Type u_3\n\ud835\udd5c\u2083 : Type ?u.156253\nE : Type u_4\nE\u2097 : Type ?u.156259\nF : Type u_5\nF\u2097 : Type ?u.156265\nG : Type ?u.156268\nG\u2097 : Type ?u.156271\n\ud835\udcd5 : Type u_1\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d : SemilinearMapClass \ud835\udcd5 \u03c3\u2081\u2082 E F\nf : \ud835\udcd5\nhf : Continuous \u2191f\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nh\u03b5 : \u2200 (x' : E), \u2016x'\u2016 < \u03b5 \u2192 \u2016\u2191f x'\u2016 < 1\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\n\u22a2 \u2203 C, 0 < C \u2227 \u2200 (x : E), \u2016\u2191f x\u2016 \u2264 C * \u2016x\u2016", "state_after": "case intro.intro.intro\n\ud835\udd5c : Type u_2\n\ud835\udd5c\u2082 : Type u_3\n\ud835\udd5c\u2083 : Type ?u.156253\nE : Type u_4\nE\u2097 : Type ?u.156259\nF : Type u_5\nF\u2097 : Type ?u.156265\nG : Type ?u.156268\nG\u2097 : Type ?u.156271\n\ud835\udcd5 : Type u_1\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d : SemilinearMapClass \ud835\udcd5 \u03c3\u2081\u2082 E F\nf : \ud835\udcd5\nhf : Continuous \u2191f\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nh\u03b5 : \u2200 (x' : E), \u2016x'\u2016 < \u03b5 \u2192 \u2016\u2191f x'\u2016 < 1\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nthis : 0 < \u2016c\u2016 / \u03b5\n\u22a2 \u2203 C, 0 < C \u2227 \u2200 (x : E), \u2016\u2191f x\u2016 \u2264 C * \u2016x\u2016"}, {"tactic": "refine' \u27e8\u2016c\u2016 / \u03b5, this, fun x => _\u27e9", "state_before": "case intro.intro.intro\n\ud835\udd5c : Type u_2\n\ud835\udd5c\u2082 : Type u_3\n\ud835\udd5c\u2083 : Type ?u.156253\nE : Type u_4\nE\u2097 : Type ?u.156259\nF : Type u_5\nF\u2097 : Type ?u.156265\nG : Type ?u.156268\nG\u2097 : Type ?u.156271\n\ud835\udcd5 : Type u_1\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d : SemilinearMapClass \ud835\udcd5 \u03c3\u2081\u2082 E F\nf : \ud835\udcd5\nhf : Continuous \u2191f\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nh\u03b5 : \u2200 (x' : E), \u2016x'\u2016 < \u03b5 \u2192 \u2016\u2191f x'\u2016 < 1\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nthis : 0 < \u2016c\u2016 / \u03b5\n\u22a2 \u2203 C, 0 < C \u2227 \u2200 (x : E), \u2016\u2191f x\u2016 \u2264 C * \u2016x\u2016", "state_after": "case intro.intro.intro\n\ud835\udd5c : Type u_2\n\ud835\udd5c\u2082 : Type u_3\n\ud835\udd5c\u2083 : Type ?u.156253\nE : Type u_4\nE\u2097 : Type ?u.156259\nF : Type u_5\nF\u2097 : Type ?u.156265\nG : Type ?u.156268\nG\u2097 : Type ?u.156271\n\ud835\udcd5 : Type u_1\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d : SemilinearMapClass \ud835\udcd5 \u03c3\u2081\u2082 E F\nf : \ud835\udcd5\nhf : Continuous \u2191f\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nh\u03b5 : \u2200 (x' : E), \u2016x'\u2016 < \u03b5 \u2192 \u2016\u2191f x'\u2016 < 1\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nthis : 0 < \u2016c\u2016 / \u03b5\nx : E\n\u22a2 \u2016\u2191f x\u2016 \u2264 \u2016c\u2016 / \u03b5 * \u2016x\u2016"}, {"tactic": "by_cases hx : \u2016x\u2016 = 0", "state_before": "case intro.intro.intro\n\ud835\udd5c : Type u_2\n\ud835\udd5c\u2082 : Type u_3\n\ud835\udd5c\u2083 : Type ?u.156253\nE : Type u_4\nE\u2097 : Type ?u.156259\nF : Type u_5\nF\u2097 : Type ?u.156265\nG : Type ?u.156268\nG\u2097 : Type ?u.156271\n\ud835\udcd5 : Type u_1\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d : SemilinearMapClass \ud835\udcd5 \u03c3\u2081\u2082 E F\nf : \ud835\udcd5\nhf : Continuous \u2191f\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nh\u03b5 : \u2200 (x' : E), \u2016x'\u2016 < \u03b5 \u2192 \u2016\u2191f x'\u2016 < 1\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nthis : 0 < \u2016c\u2016 / \u03b5\nx : E\n\u22a2 \u2016\u2191f x\u2016 \u2264 \u2016c\u2016 / \u03b5 * \u2016x\u2016", "state_after": "case pos\n\ud835\udd5c : Type u_2\n\ud835\udd5c\u2082 : Type u_3\n\ud835\udd5c\u2083 : Type ?u.156253\nE : Type u_4\nE\u2097 : Type ?u.156259\nF : Type u_5\nF\u2097 : Type ?u.156265\nG : Type ?u.156268\nG\u2097 : Type ?u.156271\n\ud835\udcd5 : Type u_1\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d : SemilinearMapClass \ud835\udcd5 \u03c3\u2081\u2082 E F\nf : \ud835\udcd5\nhf : Continuous \u2191f\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nh\u03b5 : \u2200 (x' : E), \u2016x'\u2016 < \u03b5 \u2192 \u2016\u2191f x'\u2016 < 1\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nthis : 0 < \u2016c\u2016 / \u03b5\nx : E\nhx : \u2016x\u2016 = 0\n\u22a2 \u2016\u2191f x\u2016 \u2264 \u2016c\u2016 / \u03b5 * \u2016x\u2016\n\ncase neg\n\ud835\udd5c : Type u_2\n\ud835\udd5c\u2082 : Type u_3\n\ud835\udd5c\u2083 : Type ?u.156253\nE : Type u_4\nE\u2097 : Type ?u.156259\nF : Type u_5\nF\u2097 : Type ?u.156265\nG : Type ?u.156268\nG\u2097 : Type ?u.156271\n\ud835\udcd5 : Type u_1\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d : SemilinearMapClass \ud835\udcd5 \u03c3\u2081\u2082 E F\nf : \ud835\udcd5\nhf : Continuous \u2191f\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nh\u03b5 : \u2200 (x' : E), \u2016x'\u2016 < \u03b5 \u2192 \u2016\u2191f x'\u2016 < 1\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nthis : 0 < \u2016c\u2016 / \u03b5\nx : E\nhx : \u00ac\u2016x\u2016 = 0\n\u22a2 \u2016\u2191f x\u2016 \u2264 \u2016c\u2016 / \u03b5 * \u2016x\u2016"}, {"tactic": "refine' SemilinearMapClass.bound_of_shell_semi_normed f \u03b5_pos hc (fun x hle hlt => _) hx", "state_before": "case neg\n\ud835\udd5c : Type u_2\n\ud835\udd5c\u2082 : Type u_3\n\ud835\udd5c\u2083 : Type ?u.156253\nE : Type u_4\nE\u2097 : Type ?u.156259\nF : Type u_5\nF\u2097 : Type ?u.156265\nG : Type ?u.156268\nG\u2097 : Type ?u.156271\n\ud835\udcd5 : Type u_1\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d : SemilinearMapClass \ud835\udcd5 \u03c3\u2081\u2082 E F\nf : \ud835\udcd5\nhf : Continuous \u2191f\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nh\u03b5 : \u2200 (x' : E), \u2016x'\u2016 < \u03b5 \u2192 \u2016\u2191f x'\u2016 < 1\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nthis : 0 < \u2016c\u2016 / \u03b5\nx : E\nhx : \u00ac\u2016x\u2016 = 0\n\u22a2 \u2016\u2191f x\u2016 \u2264 \u2016c\u2016 / \u03b5 * \u2016x\u2016", "state_after": "case neg\n\ud835\udd5c : Type u_2\n\ud835\udd5c\u2082 : Type u_3\n\ud835\udd5c\u2083 : Type ?u.156253\nE : Type u_4\nE\u2097 : Type ?u.156259\nF : Type u_5\nF\u2097 : Type ?u.156265\nG : Type ?u.156268\nG\u2097 : Type ?u.156271\n\ud835\udcd5 : Type u_1\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d : SemilinearMapClass \ud835\udcd5 \u03c3\u2081\u2082 E F\nf : \ud835\udcd5\nhf : Continuous \u2191f\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nh\u03b5 : \u2200 (x' : E), \u2016x'\u2016 < \u03b5 \u2192 \u2016\u2191f x'\u2016 < 1\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nthis : 0 < \u2016c\u2016 / \u03b5\nx\u271d : E\nhx : \u00ac\u2016x\u271d\u2016 = 0\nx : E\nhle : \u03b5 / \u2016c\u2016 \u2264 \u2016x\u2016\nhlt : \u2016x\u2016 < \u03b5\n\u22a2 \u2016\u2191f x\u2016 \u2264 \u2016c\u2016 / \u03b5 * \u2016x\u2016"}, {"tactic": "refine' (h\u03b5 _ hlt).le.trans _", "state_before": "case neg\n\ud835\udd5c : Type u_2\n\ud835\udd5c\u2082 : Type u_3\n\ud835\udd5c\u2083 : Type ?u.156253\nE : Type u_4\nE\u2097 : Type ?u.156259\nF : Type u_5\nF\u2097 : Type ?u.156265\nG : Type ?u.156268\nG\u2097 : Type ?u.156271\n\ud835\udcd5 : Type u_1\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d : SemilinearMapClass \ud835\udcd5 \u03c3\u2081\u2082 E F\nf : \ud835\udcd5\nhf : Continuous \u2191f\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nh\u03b5 : \u2200 (x' : E), \u2016x'\u2016 < \u03b5 \u2192 \u2016\u2191f x'\u2016 < 1\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nthis : 0 < \u2016c\u2016 / \u03b5\nx\u271d : E\nhx : \u00ac\u2016x\u271d\u2016 = 0\nx : E\nhle : \u03b5 / \u2016c\u2016 \u2264 \u2016x\u2016\nhlt : \u2016x\u2016 < \u03b5\n\u22a2 \u2016\u2191f x\u2016 \u2264 \u2016c\u2016 / \u03b5 * \u2016x\u2016", "state_after": "case neg\n\ud835\udd5c : Type u_2\n\ud835\udd5c\u2082 : Type u_3\n\ud835\udd5c\u2083 : Type ?u.156253\nE : Type u_4\nE\u2097 : Type ?u.156259\nF : Type u_5\nF\u2097 : Type ?u.156265\nG : Type ?u.156268\nG\u2097 : Type ?u.156271\n\ud835\udcd5 : Type u_1\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d : SemilinearMapClass \ud835\udcd5 \u03c3\u2081\u2082 E F\nf : \ud835\udcd5\nhf : Continuous \u2191f\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nh\u03b5 : \u2200 (x' : E), \u2016x'\u2016 < \u03b5 \u2192 \u2016\u2191f x'\u2016 < 1\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nthis : 0 < \u2016c\u2016 / \u03b5\nx\u271d : E\nhx : \u00ac\u2016x\u271d\u2016 = 0\nx : E\nhle : \u03b5 / \u2016c\u2016 \u2264 \u2016x\u2016\nhlt : \u2016x\u2016 < \u03b5\n\u22a2 1 \u2264 \u2016c\u2016 / \u03b5 * \u2016x\u2016"}, {"tactic": "rwa [\u2190 div_le_iff' this, one_div_div]", "state_before": "case neg\n\ud835\udd5c : Type u_2\n\ud835\udd5c\u2082 : Type u_3\n\ud835\udd5c\u2083 : Type ?u.156253\nE : Type u_4\nE\u2097 : Type ?u.156259\nF : Type u_5\nF\u2097 : Type ?u.156265\nG : Type ?u.156268\nG\u2097 : Type ?u.156271\n\ud835\udcd5 : Type u_1\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d : SemilinearMapClass \ud835\udcd5 \u03c3\u2081\u2082 E F\nf : \ud835\udcd5\nhf : Continuous \u2191f\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nh\u03b5 : \u2200 (x' : E), \u2016x'\u2016 < \u03b5 \u2192 \u2016\u2191f x'\u2016 < 1\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nthis : 0 < \u2016c\u2016 / \u03b5\nx\u271d : E\nhx : \u00ac\u2016x\u271d\u2016 = 0\nx : E\nhle : \u03b5 / \u2016c\u2016 \u2264 \u2016x\u2016\nhlt : \u2016x\u2016 < \u03b5\n\u22a2 1 \u2264 \u2016c\u2016 / \u03b5 * \u2016x\u2016", "state_after": "no goals"}, {"tactic": "rw [hx, MulZeroClass.mul_zero]", "state_before": "case pos\n\ud835\udd5c : Type u_2\n\ud835\udd5c\u2082 : Type u_3\n\ud835\udd5c\u2083 : Type ?u.156253\nE : Type u_4\nE\u2097 : Type ?u.156259\nF : Type u_5\nF\u2097 : Type ?u.156265\nG : Type ?u.156268\nG\u2097 : Type ?u.156271\n\ud835\udcd5 : Type u_1\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d : SemilinearMapClass \ud835\udcd5 \u03c3\u2081\u2082 E F\nf : \ud835\udcd5\nhf : Continuous \u2191f\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nh\u03b5 : \u2200 (x' : E), \u2016x'\u2016 < \u03b5 \u2192 \u2016\u2191f x'\u2016 < 1\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nthis : 0 < \u2016c\u2016 / \u03b5\nx : E\nhx : \u2016x\u2016 = 0\n\u22a2 \u2016\u2191f x\u2016 \u2264 \u2016c\u2016 / \u03b5 * \u2016x\u2016", "state_after": "case pos\n\ud835\udd5c : Type u_2\n\ud835\udd5c\u2082 : Type u_3\n\ud835\udd5c\u2083 : Type ?u.156253\nE : Type u_4\nE\u2097 : Type ?u.156259\nF : Type u_5\nF\u2097 : Type ?u.156265\nG : Type ?u.156268\nG\u2097 : Type ?u.156271\n\ud835\udcd5 : Type u_1\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d : SemilinearMapClass \ud835\udcd5 \u03c3\u2081\u2082 E F\nf : \ud835\udcd5\nhf : Continuous \u2191f\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nh\u03b5 : \u2200 (x' : E), \u2016x'\u2016 < \u03b5 \u2192 \u2016\u2191f x'\u2016 < 1\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nthis : 0 < \u2016c\u2016 / \u03b5\nx : E\nhx : \u2016x\u2016 = 0\n\u22a2 \u2016\u2191f x\u2016 \u2264 0"}, {"tactic": "exact le_of_eq (norm_image_of_norm_zero f hf hx)", "state_before": "case pos\n\ud835\udd5c : Type u_2\n\ud835\udd5c\u2082 : Type u_3\n\ud835\udd5c\u2083 : Type ?u.156253\nE : Type u_4\nE\u2097 : Type ?u.156259\nF : Type u_5\nF\u2097 : Type ?u.156265\nG : Type ?u.156268\nG\u2097 : Type ?u.156271\n\ud835\udcd5 : Type u_1\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2077 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2075 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomIsometric \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomIsometric \u03c3\u2082\u2083\ninst\u271d : SemilinearMapClass \ud835\udcd5 \u03c3\u2081\u2082 E F\nf : \ud835\udcd5\nhf : Continuous \u2191f\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nh\u03b5 : \u2200 (x' : E), \u2016x'\u2016 < \u03b5 \u2192 \u2016\u2191f x'\u2016 < 1\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nthis : 0 < \u2016c\u2016 / \u03b5\nx : E\nhx : \u2016x\u2016 = 0\n\u22a2 \u2016\u2191f x\u2016 \u2264 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Init/Data/Bool/Lemmas.lean", "full_name": "Bool.decide_false_iff", "start": [143, 1], "end": [144, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/Equicontinuity.lean", "full_name": "UniformEquicontinuous.uniformContinuous", "start": [177, 1], "end": [179, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Localization/Integer.lean", "full_name": "IsLocalization.isInteger_smul", "start": [69, 1], "end": [72, 56], "traced_tactics": [{"tactic": "rcases hb with \u27e8b', hb\u27e9", "state_before": "R : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type ?u.5566\ninst\u271d : CommRing P\na : R\nb : S\nhb : IsInteger R b\n\u22a2 IsInteger R (a \u2022 b)", "state_after": "case intro\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type ?u.5566\ninst\u271d : CommRing P\na : R\nb : S\nb' : R\nhb : \u2191(algebraMap R S) b' = b\n\u22a2 IsInteger R (a \u2022 b)"}, {"tactic": "use a * b'", "state_before": "case intro\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type ?u.5566\ninst\u271d : CommRing P\na : R\nb : S\nb' : R\nhb : \u2191(algebraMap R S) b' = b\n\u22a2 IsInteger R (a \u2022 b)", "state_after": "case intro\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type ?u.5566\ninst\u271d : CommRing P\na : R\nb : S\nb' : R\nhb : \u2191(algebraMap R S) b' = b\n\u22a2 \u2191(algebraMap R S) (a * b') = a \u2022 b"}, {"tactic": "rw [\u2190 hb, (algebraMap R S).map_mul, Algebra.smul_def]", "state_before": "case intro\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nP : Type ?u.5566\ninst\u271d : CommRing P\na : R\nb : S\nb' : R\nhb : \u2191(algebraMap R S) b' = b\n\u22a2 \u2191(algebraMap R S) (a * b') = a \u2022 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "full_name": "ContinuousLinearMap.neg_comp", "start": [1405, 1], "end": [1408, 7], "traced_tactics": [{"tactic": "ext", "state_before": "R : Type u_1\ninst\u271d\u00b9\u2075 : Ring R\nR\u2082 : Type u_2\ninst\u271d\u00b9\u2074 : Ring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b9\u00b3 : Ring R\u2083\nM : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : AddCommGroup M\nM\u2082 : Type u_5\ninst\u271d\u00b9\u2070 : TopologicalSpace M\u2082\ninst\u271d\u2079 : AddCommGroup M\u2082\nM\u2083 : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\u2083\ninst\u271d\u2077 : AddCommGroup M\u2083\nM\u2084 : Type ?u.848558\ninst\u271d\u2076 : TopologicalSpace M\u2084\ninst\u271d\u2075 : AddCommGroup M\u2084\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : TopologicalAddGroup M\u2083\ng : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083\nf : M \u2192SL[\u03c3\u2081\u2082] M\u2082\n\u22a2 comp (-g) f = -comp g f", "state_after": "case h\nR : Type u_1\ninst\u271d\u00b9\u2075 : Ring R\nR\u2082 : Type u_2\ninst\u271d\u00b9\u2074 : Ring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b9\u00b3 : Ring R\u2083\nM : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : AddCommGroup M\nM\u2082 : Type u_5\ninst\u271d\u00b9\u2070 : TopologicalSpace M\u2082\ninst\u271d\u2079 : AddCommGroup M\u2082\nM\u2083 : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\u2083\ninst\u271d\u2077 : AddCommGroup M\u2083\nM\u2084 : Type ?u.848558\ninst\u271d\u2076 : TopologicalSpace M\u2084\ninst\u271d\u2075 : AddCommGroup M\u2084\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : TopologicalAddGroup M\u2083\ng : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083\nf : M \u2192SL[\u03c3\u2081\u2082] M\u2082\nx\u271d : M\n\u22a2 \u2191(comp (-g) f) x\u271d = \u2191(-comp g f) x\u271d"}, {"tactic": "simp", "state_before": "case h\nR : Type u_1\ninst\u271d\u00b9\u2075 : Ring R\nR\u2082 : Type u_2\ninst\u271d\u00b9\u2074 : Ring R\u2082\nR\u2083 : Type u_3\ninst\u271d\u00b9\u00b3 : Ring R\u2083\nM : Type u_6\ninst\u271d\u00b9\u00b2 : TopologicalSpace M\ninst\u271d\u00b9\u00b9 : AddCommGroup M\nM\u2082 : Type u_5\ninst\u271d\u00b9\u2070 : TopologicalSpace M\u2082\ninst\u271d\u2079 : AddCommGroup M\u2082\nM\u2083 : Type u_4\ninst\u271d\u2078 : TopologicalSpace M\u2083\ninst\u271d\u2077 : AddCommGroup M\u2083\nM\u2084 : Type ?u.848558\ninst\u271d\u2076 : TopologicalSpace M\u2084\ninst\u271d\u2075 : AddCommGroup M\u2084\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d : TopologicalAddGroup M\u2083\ng : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083\nf : M \u2192SL[\u03c3\u2081\u2082] M\u2082\nx\u271d : M\n\u22a2 \u2191(comp (-g) f) x\u271d = \u2191(-comp g f) x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/ENNReal.lean", "full_name": "ENNReal.one_lt_two", "start": [399, 8], "end": [400, 61], "traced_tactics": [{"tactic": "exact_mod_cast (one_lt_two : 1 < 2)", "state_before": "\u03b1 : Type ?u.33456\n\u03b2 : Type ?u.33459\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u22a2 \u21911 < \u21912", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Normed/Group/Hom.lean", "full_name": "NormedAddGroupHom.Equalizer.\u03b9_comp_map", "start": [957, 1], "end": [959, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Sheaves/SheafCondition/UniqueGluing.lean", "full_name": "TopCat.Sheaf.eq_of_locally_eq'", "start": [297, 1], "end": [307, 14], "traced_tactics": [{"tactic": "have V_eq_supr_U : V = iSup U := le_antisymm hcover (iSup_le fun i => (iUV i).le)", "state_before": "C : Type u\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : ReflectsIsomorphisms ConcreteCategory.forget\ninst\u271d : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nV : Opens \u2191X\niUV : (i : \u03b9) \u2192 U i \u27f6 V\nhcover : V \u2264 iSup U\ns t : (CategoryTheory.forget C).obj (F.val.obj V.op)\nh :\n  \u2200 (i : \u03b9),\n    (CategoryTheory.forget C).map (F.val.map (iUV i).op) s = (CategoryTheory.forget C).map (F.val.map (iUV i).op) t\n\u22a2 s = t", "state_after": "C : Type u\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : ReflectsIsomorphisms ConcreteCategory.forget\ninst\u271d : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nV : Opens \u2191X\niUV : (i : \u03b9) \u2192 U i \u27f6 V\nhcover : V \u2264 iSup U\ns t : (CategoryTheory.forget C).obj (F.val.obj V.op)\nh :\n  \u2200 (i : \u03b9),\n    (CategoryTheory.forget C).map (F.val.map (iUV i).op) s = (CategoryTheory.forget C).map (F.val.map (iUV i).op) t\nV_eq_supr_U : V = iSup U\n\u22a2 s = t"}, {"tactic": "suffices F.1.map (eqToHom V_eq_supr_U.symm).op s = F.1.map (eqToHom V_eq_supr_U.symm).op t by\n  convert congr_arg (F.1.map (eqToHom V_eq_supr_U).op) this <;>\n  rw [\u2190 comp_apply, \u2190 F.1.map_comp, eqToHom_op, eqToHom_op, eqToHom_trans, eqToHom_refl,\n    F.1.map_id, id_apply]", "state_before": "C : Type u\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : ReflectsIsomorphisms ConcreteCategory.forget\ninst\u271d : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nV : Opens \u2191X\niUV : (i : \u03b9) \u2192 U i \u27f6 V\nhcover : V \u2264 iSup U\ns t : (CategoryTheory.forget C).obj (F.val.obj V.op)\nh :\n  \u2200 (i : \u03b9),\n    (CategoryTheory.forget C).map (F.val.map (iUV i).op) s = (CategoryTheory.forget C).map (F.val.map (iUV i).op) t\nV_eq_supr_U : V = iSup U\n\u22a2 s = t", "state_after": "C : Type u\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : ReflectsIsomorphisms ConcreteCategory.forget\ninst\u271d : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nV : Opens \u2191X\niUV : (i : \u03b9) \u2192 U i \u27f6 V\nhcover : V \u2264 iSup U\ns t : (CategoryTheory.forget C).obj (F.val.obj V.op)\nh :\n  \u2200 (i : \u03b9),\n    (CategoryTheory.forget C).map (F.val.map (iUV i).op) s = (CategoryTheory.forget C).map (F.val.map (iUV i).op) t\nV_eq_supr_U : V = iSup U\n\u22a2 (CategoryTheory.forget C).map (F.val.map (eqToHom (_ : iSup U = V)).op) s =\n    (CategoryTheory.forget C).map (F.val.map (eqToHom (_ : iSup U = V)).op) t"}, {"tactic": "apply eq_of_locally_eq", "state_before": "C : Type u\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : ReflectsIsomorphisms ConcreteCategory.forget\ninst\u271d : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nV : Opens \u2191X\niUV : (i : \u03b9) \u2192 U i \u27f6 V\nhcover : V \u2264 iSup U\ns t : (CategoryTheory.forget C).obj (F.val.obj V.op)\nh :\n  \u2200 (i : \u03b9),\n    (CategoryTheory.forget C).map (F.val.map (iUV i).op) s = (CategoryTheory.forget C).map (F.val.map (iUV i).op) t\nV_eq_supr_U : V = iSup U\n\u22a2 (CategoryTheory.forget C).map (F.val.map (eqToHom (_ : iSup U = V)).op) s =\n    (CategoryTheory.forget C).map (F.val.map (eqToHom (_ : iSup U = V)).op) t", "state_after": "case h\nC : Type u\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : ReflectsIsomorphisms ConcreteCategory.forget\ninst\u271d : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nV : Opens \u2191X\niUV : (i : \u03b9) \u2192 U i \u27f6 V\nhcover : V \u2264 iSup U\ns t : (CategoryTheory.forget C).obj (F.val.obj V.op)\nh :\n  \u2200 (i : \u03b9),\n    (CategoryTheory.forget C).map (F.val.map (iUV i).op) s = (CategoryTheory.forget C).map (F.val.map (iUV i).op) t\nV_eq_supr_U : V = iSup U\n\u22a2 \u2200 (i : \u03b9),\n    (CategoryTheory.forget C).map (F.val.map (leSupr U i).op)\n        ((CategoryTheory.forget C).map (F.val.map (eqToHom (_ : iSup U = V)).op) s) =\n      (CategoryTheory.forget C).map (F.val.map (leSupr U i).op)\n        ((CategoryTheory.forget C).map (F.val.map (eqToHom (_ : iSup U = V)).op) t)"}, {"tactic": "intro i", "state_before": "case h\nC : Type u\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : ReflectsIsomorphisms ConcreteCategory.forget\ninst\u271d : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nV : Opens \u2191X\niUV : (i : \u03b9) \u2192 U i \u27f6 V\nhcover : V \u2264 iSup U\ns t : (CategoryTheory.forget C).obj (F.val.obj V.op)\nh :\n  \u2200 (i : \u03b9),\n    (CategoryTheory.forget C).map (F.val.map (iUV i).op) s = (CategoryTheory.forget C).map (F.val.map (iUV i).op) t\nV_eq_supr_U : V = iSup U\n\u22a2 \u2200 (i : \u03b9),\n    (CategoryTheory.forget C).map (F.val.map (leSupr U i).op)\n        ((CategoryTheory.forget C).map (F.val.map (eqToHom (_ : iSup U = V)).op) s) =\n      (CategoryTheory.forget C).map (F.val.map (leSupr U i).op)\n        ((CategoryTheory.forget C).map (F.val.map (eqToHom (_ : iSup U = V)).op) t)", "state_after": "case h\nC : Type u\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : ReflectsIsomorphisms ConcreteCategory.forget\ninst\u271d : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nV : Opens \u2191X\niUV : (i : \u03b9) \u2192 U i \u27f6 V\nhcover : V \u2264 iSup U\ns t : (CategoryTheory.forget C).obj (F.val.obj V.op)\nh :\n  \u2200 (i : \u03b9),\n    (CategoryTheory.forget C).map (F.val.map (iUV i).op) s = (CategoryTheory.forget C).map (F.val.map (iUV i).op) t\nV_eq_supr_U : V = iSup U\ni : \u03b9\n\u22a2 (CategoryTheory.forget C).map (F.val.map (leSupr U i).op)\n      ((CategoryTheory.forget C).map (F.val.map (eqToHom (_ : iSup U = V)).op) s) =\n    (CategoryTheory.forget C).map (F.val.map (leSupr U i).op)\n      ((CategoryTheory.forget C).map (F.val.map (eqToHom (_ : iSup U = V)).op) t)"}, {"tactic": "rw [\u2190 comp_apply, \u2190 comp_apply, \u2190 F.1.map_comp]", "state_before": "case h\nC : Type u\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : ReflectsIsomorphisms ConcreteCategory.forget\ninst\u271d : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nV : Opens \u2191X\niUV : (i : \u03b9) \u2192 U i \u27f6 V\nhcover : V \u2264 iSup U\ns t : (CategoryTheory.forget C).obj (F.val.obj V.op)\nh :\n  \u2200 (i : \u03b9),\n    (CategoryTheory.forget C).map (F.val.map (iUV i).op) s = (CategoryTheory.forget C).map (F.val.map (iUV i).op) t\nV_eq_supr_U : V = iSup U\ni : \u03b9\n\u22a2 (CategoryTheory.forget C).map (F.val.map (leSupr U i).op)\n      ((CategoryTheory.forget C).map (F.val.map (eqToHom (_ : iSup U = V)).op) s) =\n    (CategoryTheory.forget C).map (F.val.map (leSupr U i).op)\n      ((CategoryTheory.forget C).map (F.val.map (eqToHom (_ : iSup U = V)).op) t)", "state_after": "case h\nC : Type u\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : ReflectsIsomorphisms ConcreteCategory.forget\ninst\u271d : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nV : Opens \u2191X\niUV : (i : \u03b9) \u2192 U i \u27f6 V\nhcover : V \u2264 iSup U\ns t : (CategoryTheory.forget C).obj (F.val.obj V.op)\nh :\n  \u2200 (i : \u03b9),\n    (CategoryTheory.forget C).map (F.val.map (iUV i).op) s = (CategoryTheory.forget C).map (F.val.map (iUV i).op) t\nV_eq_supr_U : V = iSup U\ni : \u03b9\n\u22a2 (CategoryTheory.forget C).map (F.val.map ((eqToHom (_ : iSup U = V)).op \u226b (leSupr U i).op)) s =\n    (CategoryTheory.forget C).map (F.val.map ((eqToHom (_ : iSup U = V)).op \u226b (leSupr U i).op)) t"}, {"tactic": "convert h i", "state_before": "case h\nC : Type u\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : ReflectsIsomorphisms ConcreteCategory.forget\ninst\u271d : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nV : Opens \u2191X\niUV : (i : \u03b9) \u2192 U i \u27f6 V\nhcover : V \u2264 iSup U\ns t : (CategoryTheory.forget C).obj (F.val.obj V.op)\nh :\n  \u2200 (i : \u03b9),\n    (CategoryTheory.forget C).map (F.val.map (iUV i).op) s = (CategoryTheory.forget C).map (F.val.map (iUV i).op) t\nV_eq_supr_U : V = iSup U\ni : \u03b9\n\u22a2 (CategoryTheory.forget C).map (F.val.map ((eqToHom (_ : iSup U = V)).op \u226b (leSupr U i).op)) s =\n    (CategoryTheory.forget C).map (F.val.map ((eqToHom (_ : iSup U = V)).op \u226b (leSupr U i).op)) t", "state_after": "no goals"}, {"tactic": "convert congr_arg (F.1.map (eqToHom V_eq_supr_U).op) this <;>\nrw [\u2190 comp_apply, \u2190 F.1.map_comp, eqToHom_op, eqToHom_op, eqToHom_trans, eqToHom_refl,\n  F.1.map_id, id_apply]", "state_before": "C : Type u\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : HasLimits C\ninst\u271d\u00b9 : ReflectsIsomorphisms ConcreteCategory.forget\ninst\u271d : PreservesLimits ConcreteCategory.forget\nX : TopCat\nF : Sheaf C X\n\u03b9 : Type v\nU : \u03b9 \u2192 Opens \u2191X\nV : Opens \u2191X\niUV : (i : \u03b9) \u2192 U i \u27f6 V\nhcover : V \u2264 iSup U\ns t : (CategoryTheory.forget C).obj (F.val.obj V.op)\nh :\n  \u2200 (i : \u03b9),\n    (CategoryTheory.forget C).map (F.val.map (iUV i).op) s = (CategoryTheory.forget C).map (F.val.map (iUV i).op) t\nV_eq_supr_U : V = iSup U\nthis :\n  (CategoryTheory.forget C).map (F.val.map (eqToHom (_ : iSup U = V)).op) s =\n    (CategoryTheory.forget C).map (F.val.map (eqToHom (_ : iSup U = V)).op) t\n\u22a2 s = t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/ContinuousOn.lean", "full_name": "tendsto_nhdsWithin_mono_right", "start": [371, 1], "end": [373, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Quaternion.lean", "full_name": "Quaternion.one_im", "start": [880, 9], "end": [880, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/UpperLower/Basic.lean", "full_name": "isLowerSet_iUnion", "start": [133, 1], "end": [134, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean", "full_name": "Real.differentiable_sinh", "start": [643, 1], "end": [643, 101], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/ENNReal.lean", "full_name": "ENNReal.toReal_nonneg", "start": [218, 9], "end": [218, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Localization/FractionRing.lean", "full_name": "IsFractionRing.isFractionRing_iff_of_base_ringEquiv", "start": [250, 1], "end": [266, 47], "traced_tactics": [{"tactic": "delta IsFractionRing", "state_before": "R : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_3\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_2\ninst\u271d\u2078 : CommRing P\nA : Type ?u.377174\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type ?u.377348\nB : Type ?u.377351\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type ?u.377528\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\n\u22a2 IsFractionRing R S \u2194 IsFractionRing P S", "state_after": "R : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_3\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_2\ninst\u271d\u2078 : CommRing P\nA : Type ?u.377174\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type ?u.377348\nB : Type ?u.377351\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type ?u.377528\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\n\u22a2 IsLocalization (nonZeroDivisors R) S \u2194 IsLocalization (nonZeroDivisors P) S"}, {"tactic": "convert isLocalization_iff_of_base_ringEquiv (nonZeroDivisors R) S h", "state_before": "R : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_3\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_2\ninst\u271d\u2078 : CommRing P\nA : Type ?u.377174\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type ?u.377348\nB : Type ?u.377351\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type ?u.377528\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\n\u22a2 IsLocalization (nonZeroDivisors R) S \u2194 IsLocalization (nonZeroDivisors P) S", "state_after": "case h.e'_2.h.e'_3\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_3\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_2\ninst\u271d\u2078 : CommRing P\nA : Type ?u.377174\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type ?u.377348\nB : Type ?u.377351\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type ?u.377528\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\n\u22a2 nonZeroDivisors P = Submonoid.map (RingEquiv.toMonoidHom h) (nonZeroDivisors R)"}, {"tactic": "ext x", "state_before": "case h.e'_2.h.e'_3\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_3\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_2\ninst\u271d\u2078 : CommRing P\nA : Type ?u.377174\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type ?u.377348\nB : Type ?u.377351\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type ?u.377528\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\n\u22a2 nonZeroDivisors P = Submonoid.map (RingEquiv.toMonoidHom h) (nonZeroDivisors R)", "state_after": "case h.e'_2.h.e'_3.h\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_3\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_2\ninst\u271d\u2078 : CommRing P\nA : Type ?u.377174\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type ?u.377348\nB : Type ?u.377351\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type ?u.377528\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\nx : P\n\u22a2 x \u2208 nonZeroDivisors P \u2194 x \u2208 Submonoid.map (RingEquiv.toMonoidHom h) (nonZeroDivisors R)"}, {"tactic": "erw [Submonoid.map_equiv_eq_comap_symm]", "state_before": "case h.e'_2.h.e'_3.h\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_3\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_2\ninst\u271d\u2078 : CommRing P\nA : Type ?u.377174\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type ?u.377348\nB : Type ?u.377351\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type ?u.377528\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\nx : P\n\u22a2 x \u2208 nonZeroDivisors P \u2194 x \u2208 Submonoid.map (RingEquiv.toMonoidHom h) (nonZeroDivisors R)", "state_after": "case h.e'_2.h.e'_3.h\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_3\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_2\ninst\u271d\u2078 : CommRing P\nA : Type ?u.377174\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type ?u.377348\nB : Type ?u.377351\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type ?u.377528\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\nx : P\n\u22a2 x \u2208 nonZeroDivisors P \u2194\n    x \u2208 Submonoid.comap (MulEquiv.toMonoidHom (MulEquiv.symm (RingEquiv.toMulEquiv h))) (nonZeroDivisors R)"}, {"tactic": "simp only [MulEquiv.coe_toMonoidHom, RingEquiv.toMulEquiv_eq_coe, Submonoid.mem_comap]", "state_before": "case h.e'_2.h.e'_3.h\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_3\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_2\ninst\u271d\u2078 : CommRing P\nA : Type ?u.377174\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type ?u.377348\nB : Type ?u.377351\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type ?u.377528\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\nx : P\n\u22a2 x \u2208 nonZeroDivisors P \u2194\n    x \u2208 Submonoid.comap (MulEquiv.toMonoidHom (MulEquiv.symm (RingEquiv.toMulEquiv h))) (nonZeroDivisors R)", "state_after": "case h.e'_2.h.e'_3.h\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_3\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_2\ninst\u271d\u2078 : CommRing P\nA : Type ?u.377174\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type ?u.377348\nB : Type ?u.377351\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : IsDomain B\ninst\u271d\u00b3 : Field K\nL : Type ?u.377528\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\nx : P\n\u22a2 x \u2208 nonZeroDivisors P \u2194 \u2191(MulEquiv.symm \u2191h) x \u2208 nonZeroDivisors R"}, {"tactic": "constructor", "state_before": "case h.e'_2.h.e'_3.h\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nM : Submonoid R\nS : Type u_3\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra R S\nP : Type u_2\ninst\u271d\u2078 : CommRing P\nA : Type ?u.377174\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : IsDomain A\nK : Type ?u.377348\nB : Type 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B\ninst\u271d\u00b3 : Field K\nL : Type ?u.377528\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\ng : A \u2192+* L\nh : R \u2243+* P\nx : P\nhx : \u2191(RingEquiv.symm h) x \u2208 nonZeroDivisors R\nz : P\nhz : z * x = 0\n\u22a2 \u2191(RingEquiv.symm h) z * \u2191(RingEquiv.symm h) x = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Seq/Computation.lean", "full_name": "Computation.thinkN_equiv", "start": [1024, 1], "end": [1024, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Hom/Bounded.lean", "full_name": "TopHom.comp_id", "start": [277, 1], "end": [278, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Subring/Basic.lean", "full_name": "Subring.mem_mk'", "start": [310, 1], "end": [312, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Sort.lean", "full_name": "Multiset.coe_sort", "start": [38, 1], "end": [39, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/BinomialHeap.lean", "full_name": "Std.BinomialHeapImp.Heap.WellFormed.merge'", "start": [362, 1], "end": [399, 38], "traced_tactics": [{"tactic": "unfold merge", "state_before": "\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\ns\u2081 s\u2082 : Heap \u03b1\u271d\nn : Nat\nh\u2081 : WellFormed le n s\u2081\nh\u2082 : WellFormed le n s\u2082\n\u22a2 WellFormed le n (merge le s\u2081 s\u2082) \u2227 ((Heap.rankGT s\u2081 n \u2194 Heap.rankGT s\u2082 n) \u2192 Heap.rankGT (merge le s\u2081 s\u2082) n)", "state_after": "\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\ns\u2081 s\u2082 : Heap \u03b1\u271d\nn : Nat\nh\u2081 : WellFormed le n s\u2081\nh\u2082 : WellFormed le n s\u2082\n\u22a2 WellFormed le n\n      (match s\u2081, s\u2082 with\n      | Heap.nil, h => h\n      | h, Heap.nil => h\n      | s\u2081@h:(cons r\u2081 a\u2081 n\u2081 t\u2081), s\u2082@h_1:(cons r\u2082 a\u2082 n\u2082 t\u2082) =>\n        if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 s\u2082)\n        else\n          if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le s\u2081 t\u2082)\n          else\n            match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n            | (a, n) =>\n              let r := r\u2081 + 1;\n              if Heap.rankGT t\u2081 r then\n                if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n              else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT s\u2081 n \u2194 Heap.rankGT s\u2082 n) \u2192\n      Heap.rankGT\n        (match s\u2081, s\u2082 with\n        | Heap.nil, h => h\n        | h, Heap.nil => h\n        | s\u2081@h:(cons r\u2081 a\u2081 n\u2081 t\u2081), s\u2082@h_1:(cons r\u2082 a\u2082 n\u2082 t\u2082) =>\n          if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 s\u2082)\n          else\n            if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le s\u2081 t\u2082)\n            else\n              match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n              | (a, n) =>\n                let r := r\u2081 + 1;\n                if Heap.rankGT t\u2081 r then\n                  if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n                else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082))\n        n)"}, {"tactic": "split", "state_before": "\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\ns\u2081 s\u2082 : Heap \u03b1\u271d\nn : Nat\nh\u2081 : WellFormed le n s\u2081\nh\u2082 : WellFormed le n s\u2082\n\u22a2 WellFormed le n\n      (match s\u2081, s\u2082 with\n      | Heap.nil, h => h\n      | h, Heap.nil => h\n      | s\u2081@h:(cons r\u2081 a\u2081 n\u2081 t\u2081), s\u2082@h_1:(cons r\u2082 a\u2082 n\u2082 t\u2082) =>\n        if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 s\u2082)\n        else\n          if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le s\u2081 t\u2082)\n          else\n            match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n            | (a, n) =>\n              let r := r\u2081 + 1;\n              if Heap.rankGT t\u2081 r then\n                if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n              else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT s\u2081 n \u2194 Heap.rankGT s\u2082 n) \u2192\n      Heap.rankGT\n        (match s\u2081, s\u2082 with\n        | Heap.nil, h => h\n        | h, Heap.nil => h\n        | s\u2081@h:(cons r\u2081 a\u2081 n\u2081 t\u2081), s\u2082@h_1:(cons r\u2082 a\u2082 n\u2082 t\u2082) =>\n          if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 s\u2082)\n          else\n            if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le s\u2081 t\u2082)\n            else\n              match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n              | (a, n) =>\n                let r := r\u2081 + 1;\n                if Heap.rankGT t\u2081 r then\n                  if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n                else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082))\n        n)", "state_after": "case h_1\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\ns\u2082 : Heap \u03b1\u271d\nn : Nat\nh\u2082 : WellFormed le n s\u2082\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nh\u2081 : WellFormed le n Heap.nil\n\u22a2 WellFormed le n s\u2082 \u2227 ((Heap.rankGT Heap.nil n \u2194 Heap.rankGT s\u2082 n) \u2192 Heap.rankGT s\u2082 n)\n\ncase h_2\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\ns\u2081 : Heap \u03b1\u271d\nn : Nat\nh\u2081 : WellFormed le n s\u2081\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nx\u271d : s\u2081 = Heap.nil \u2192 False\nh\u2082 : WellFormed le n Heap.nil\n\u22a2 WellFormed le n s\u2081 \u2227 ((Heap.rankGT s\u2081 n \u2194 Heap.rankGT Heap.nil n) \u2192 Heap.rankGT s\u2081 n)\n\ncase h_3\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081\u271d : Nat\na\u2081\u271d : \u03b1\u271d\nn\u2081\u271d : HeapNode \u03b1\u271d\nt\u2081\u271d : Heap \u03b1\u271d\nr\u2082\u271d : Nat\na\u2082\u271d : \u03b1\u271d\nn\u2082\u271d : HeapNode \u03b1\u271d\nt\u2082\u271d : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081\u271d a\u2081\u271d n\u2081\u271d t\u2081\u271d)\nh\u2082 : WellFormed le n (cons r\u2082\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d)\n\u22a2 WellFormed le n\n      (if r\u2081\u271d < r\u2082\u271d then cons r\u2081\u271d a\u2081\u271d n\u2081\u271d (merge le t\u2081\u271d (cons r\u2082\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d))\n      else\n        if r\u2082\u271d < r\u2081\u271d then cons r\u2082\u271d a\u2082\u271d n\u2082\u271d (merge le (cons r\u2081\u271d a\u2081\u271d n\u2081\u271d t\u2081\u271d) t\u2082\u271d)\n        else\n          match combine le a\u2081\u271d a\u2082\u271d n\u2081\u271d n\u2082\u271d with\n          | (a, n) =>\n            let r := r\u2081\u271d + 1;\n            if Heap.rankGT t\u2081\u271d r then\n              if Heap.rankGT t\u2082\u271d r then cons r a n (merge le t\u2081\u271d t\u2082\u271d) else merge le (cons r a n t\u2081\u271d) t\u2082\u271d\n            else if Heap.rankGT t\u2082\u271d r then merge le t\u2081\u271d (cons r a n t\u2082\u271d) else cons r a n (merge le t\u2081\u271d t\u2082\u271d)) \u2227\n    ((Heap.rankGT (cons r\u2081\u271d a\u2081\u271d n\u2081\u271d t\u2081\u271d) n \u2194 Heap.rankGT (cons r\u2082\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d) n) \u2192\n      Heap.rankGT\n        (if r\u2081\u271d < r\u2082\u271d then cons r\u2081\u271d a\u2081\u271d n\u2081\u271d (merge le t\u2081\u271d (cons r\u2082\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d))\n        else\n          if r\u2082\u271d < r\u2081\u271d then cons r\u2082\u271d a\u2082\u271d n\u2082\u271d (merge le (cons r\u2081\u271d a\u2081\u271d n\u2081\u271d t\u2081\u271d) t\u2082\u271d)\n          else\n            match combine le a\u2081\u271d a\u2082\u271d n\u2081\u271d n\u2082\u271d with\n            | (a, n) =>\n              let r := r\u2081\u271d + 1;\n              if Heap.rankGT t\u2081\u271d r then\n                if Heap.rankGT t\u2082\u271d r then cons r a n (merge le t\u2081\u271d t\u2082\u271d) else merge le (cons r a n t\u2081\u271d) t\u2082\u271d\n              else if Heap.rankGT t\u2082\u271d r then merge le t\u2081\u271d (cons r a n t\u2082\u271d) else cons r a n (merge le t\u2081\u271d t\u2082\u271d))\n        n)"}, {"tactic": "exact \u27e8h\u2082, fun h => h.1 h\u2081\u27e9", "state_before": "case h_1\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\ns\u2082 : Heap \u03b1\u271d\nn : Nat\nh\u2082 : WellFormed le n s\u2082\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nh\u2081 : WellFormed le n Heap.nil\n\u22a2 WellFormed le n s\u2082 \u2227 ((Heap.rankGT Heap.nil n \u2194 Heap.rankGT s\u2082 n) \u2192 Heap.rankGT s\u2082 n)", "state_after": "no goals"}, {"tactic": "exact \u27e8h\u2081, fun h => h.2 h\u2082\u27e9", "state_before": "case h_2\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\ns\u2081 : Heap \u03b1\u271d\nn : Nat\nh\u2081 : WellFormed le n s\u2081\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nx\u271d : s\u2081 = Heap.nil \u2192 False\nh\u2082 : WellFormed le n Heap.nil\n\u22a2 WellFormed le n s\u2081 \u2227 ((Heap.rankGT s\u2081 n \u2194 Heap.rankGT Heap.nil n) \u2192 Heap.rankGT s\u2081 n)", "state_after": "no goals"}, {"tactic": "rename_i r\u2081 a\u2081 n\u2081 t\u2081 r\u2082 a\u2082 n\u2082 t\u2082", "state_before": "case h_3\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081\u271d : Nat\na\u2081\u271d : \u03b1\u271d\nn\u2081\u271d : HeapNode \u03b1\u271d\nt\u2081\u271d : Heap \u03b1\u271d\nr\u2082\u271d : Nat\na\u2082\u271d : \u03b1\u271d\nn\u2082\u271d : HeapNode \u03b1\u271d\nt\u2082\u271d : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081\u271d a\u2081\u271d n\u2081\u271d t\u2081\u271d)\nh\u2082 : WellFormed le n (cons r\u2082\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d)\n\u22a2 WellFormed le n\n      (if r\u2081\u271d < r\u2082\u271d then cons r\u2081\u271d a\u2081\u271d n\u2081\u271d (merge le t\u2081\u271d (cons r\u2082\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d))\n      else\n        if r\u2082\u271d < r\u2081\u271d then cons r\u2082\u271d a\u2082\u271d n\u2082\u271d (merge le (cons r\u2081\u271d a\u2081\u271d n\u2081\u271d t\u2081\u271d) t\u2082\u271d)\n        else\n          match combine le a\u2081\u271d a\u2082\u271d n\u2081\u271d n\u2082\u271d with\n          | (a, n) =>\n            let r := r\u2081\u271d + 1;\n            if Heap.rankGT t\u2081\u271d r then\n              if Heap.rankGT t\u2082\u271d r then cons r a n (merge le t\u2081\u271d t\u2082\u271d) else merge le (cons r a n t\u2081\u271d) t\u2082\u271d\n            else if Heap.rankGT t\u2082\u271d r then merge le t\u2081\u271d (cons r a n t\u2082\u271d) else cons r a n (merge le t\u2081\u271d t\u2082\u271d)) \u2227\n    ((Heap.rankGT (cons r\u2081\u271d a\u2081\u271d n\u2081\u271d t\u2081\u271d) n \u2194 Heap.rankGT (cons r\u2082\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d) n) \u2192\n      Heap.rankGT\n        (if r\u2081\u271d < r\u2082\u271d then cons r\u2081\u271d a\u2081\u271d n\u2081\u271d (merge le t\u2081\u271d (cons r\u2082\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d))\n        else\n          if r\u2082\u271d < r\u2081\u271d then cons r\u2082\u271d a\u2082\u271d n\u2082\u271d (merge le (cons r\u2081\u271d a\u2081\u271d n\u2081\u271d t\u2081\u271d) t\u2082\u271d)\n          else\n            match combine le a\u2081\u271d a\u2082\u271d n\u2081\u271d n\u2082\u271d with\n            | (a, n) =>\n              let r := r\u2081\u271d + 1;\n              if Heap.rankGT t\u2081\u271d r then\n                if Heap.rankGT t\u2082\u271d r then cons r a n (merge le t\u2081\u271d t\u2082\u271d) else merge le (cons r a n t\u2081\u271d) t\u2082\u271d\n              else if Heap.rankGT t\u2082\u271d r then merge le t\u2081\u271d (cons r a n t\u2082\u271d) else cons r a n (merge le t\u2081\u271d t\u2082\u271d))\n        n)", "state_after": "case h_3\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WellFormed le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\n\u22a2 WellFormed le n\n      (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n      else\n        if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n        else\n          match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n          | (a, n) =>\n            let r := r\u2081 + 1;\n            if Heap.rankGT t\u2081 r then\n              if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n            else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n \u2194 Heap.rankGT (cons r\u2082 a\u2082 n\u2082 t\u2082) n) \u2192\n      Heap.rankGT\n        (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n        else\n          if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n          else\n            match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n            | (a, n) =>\n              let r := r\u2081 + 1;\n              if Heap.rankGT t\u2081 r then\n                if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n              else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082))\n        n)"}, {"tactic": "let \u27e8hr\u2081, hn\u2081, ht\u2081\u27e9 := h\u2081", "state_before": "case h_3\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WellFormed le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\n\u22a2 WellFormed le n\n      (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n      else\n        if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n        else\n          match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n          | (a, n) =>\n            let r := r\u2081 + 1;\n            if Heap.rankGT t\u2081 r then\n              if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n            else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n \u2194 Heap.rankGT (cons r\u2082 a\u2082 n\u2082 t\u2082) n) \u2192\n      Heap.rankGT\n        (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n        else\n          if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n          else\n            match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n            | (a, n) =>\n              let r := r\u2081 + 1;\n              if Heap.rankGT t\u2081 r then\n                if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n              else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082))\n        n)", "state_after": "case h_3\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WellFormed le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\n\u22a2 WellFormed le n\n      (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n      else\n        if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n        else\n          match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n          | (a, n) =>\n            let r := r\u2081 + 1;\n            if Heap.rankGT t\u2081 r then\n              if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n            else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n \u2194 Heap.rankGT (cons r\u2082 a\u2082 n\u2082 t\u2082) n) \u2192\n      Heap.rankGT\n        (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n        else\n          if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n          else\n            match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n            | (a, n) =>\n              let r := r\u2081 + 1;\n              if Heap.rankGT t\u2081 r then\n                if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n              else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082))\n        n)"}, {"tactic": "let \u27e8hr\u2082, hn\u2082, ht\u2082\u27e9 := h\u2082", "state_before": "case h_3\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WellFormed le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\n\u22a2 WellFormed le n\n      (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n      else\n        if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n        else\n          match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n          | (a, n) =>\n            let r := r\u2081 + 1;\n            if Heap.rankGT t\u2081 r then\n              if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n            else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n \u2194 Heap.rankGT (cons r\u2082 a\u2082 n\u2082 t\u2082) n) \u2192\n      Heap.rankGT\n        (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n        else\n          if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n          else\n            match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n            | (a, n) =>\n              let r := r\u2081 + 1;\n              if Heap.rankGT t\u2081 r then\n                if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n              else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082))\n        n)", "state_after": "case h_3\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WellFormed le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2082\nht\u2082 : WellFormed le (r\u2082 + 1) t\u2082\n\u22a2 WellFormed le n\n      (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n      else\n        if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n        else\n          match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n          | (a, n) =>\n            let r := r\u2081 + 1;\n            if Heap.rankGT t\u2081 r then\n              if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n            else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n \u2194 Heap.rankGT (cons r\u2082 a\u2082 n\u2082 t\u2082) n) \u2192\n      Heap.rankGT\n        (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n        else\n          if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n          else\n            match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n            | (a, n) =>\n              let r := r\u2081 + 1;\n              if Heap.rankGT t\u2081 r then\n                if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n              else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082))\n        n)"}, {"tactic": "split <;> rename_i lt\u2081", "state_before": "case h_3\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WellFormed le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2082\nht\u2082 : WellFormed le (r\u2082 + 1) t\u2082\n\u22a2 WellFormed le n\n      (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n      else\n        if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n        else\n          match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n          | (a, n) =>\n            let r := r\u2081 + 1;\n            if Heap.rankGT t\u2081 r then\n              if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n            else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n \u2194 Heap.rankGT (cons r\u2082 a\u2082 n\u2082 t\u2082) n) \u2192\n      Heap.rankGT\n        (if r\u2081 < r\u2082 then cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))\n        else\n          if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n          else\n            match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n            | (a, n) =>\n              let r := r\u2081 + 1;\n              if Heap.rankGT t\u2081 r then\n                if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n              else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082))\n        n)", "state_after": "case h_3.inl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WellFormed le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2082\nht\u2082 : WellFormed le (r\u2082 + 1) t\u2082\nlt\u2081 : r\u2081 < r\u2082\n\u22a2 WellFormed le n (cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n \u2194 Heap.rankGT (cons r\u2082 a\u2082 n\u2082 t\u2082) n) \u2192\n      Heap.rankGT (cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))) n)\n\ncase h_3.inr\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WellFormed le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2082\nht\u2082 : WellFormed le (r\u2082 + 1) t\u2082\nlt\u2081 : \u00acr\u2081 < r\u2082\n\u22a2 WellFormed le n\n      (if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n      else\n        match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n        | (a, n) =>\n          let r := r\u2081 + 1;\n          if Heap.rankGT t\u2081 r then if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n          else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n \u2194 Heap.rankGT (cons r\u2082 a\u2082 n\u2082 t\u2082) n) \u2192\n      Heap.rankGT\n        (if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n        else\n          match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n          | (a, n) =>\n            let r := r\u2081 + 1;\n            if Heap.rankGT t\u2081 r then\n              if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n            else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082))\n        n)"}, {"tactic": "split <;> rename_i lt\u2082", "state_before": "case h_3.inr\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WellFormed le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2082\nht\u2082 : WellFormed le (r\u2082 + 1) t\u2082\nlt\u2081 : \u00acr\u2081 < r\u2082\n\u22a2 WellFormed le n\n      (if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n      else\n        match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n        | (a, n) =>\n          let r := r\u2081 + 1;\n          if Heap.rankGT t\u2081 r then if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n          else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n \u2194 Heap.rankGT (cons r\u2082 a\u2082 n\u2082 t\u2082) n) \u2192\n      Heap.rankGT\n        (if r\u2082 < r\u2081 then cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)\n        else\n          match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n          | (a, n) =>\n            let r := r\u2081 + 1;\n            if Heap.rankGT t\u2081 r then\n              if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n            else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082))\n        n)", "state_after": "case h_3.inr.inl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WellFormed le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2082\nht\u2082 : WellFormed le (r\u2082 + 1) t\u2082\nlt\u2081 : \u00acr\u2081 < r\u2082\nlt\u2082 : r\u2082 < r\u2081\n\u22a2 WellFormed le n (cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n \u2194 Heap.rankGT (cons r\u2082 a\u2082 n\u2082 t\u2082) n) \u2192\n      Heap.rankGT (cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)) n)\n\ncase h_3.inr.inr\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WellFormed le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2082\nht\u2082 : WellFormed le (r\u2082 + 1) t\u2082\nlt\u2081 : \u00acr\u2081 < r\u2082\nlt\u2082 : \u00acr\u2082 < r\u2081\n\u22a2 WellFormed le n\n      (match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n      | (a, n) =>\n        let r := r\u2081 + 1;\n        if Heap.rankGT t\u2081 r then if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n        else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n \u2194 Heap.rankGT (cons r\u2082 a\u2082 n\u2082 t\u2082) n) \u2192\n      Heap.rankGT\n        (match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n        | (a, n) =>\n          let r := r\u2081 + 1;\n          if Heap.rankGT t\u2081 r then if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n          else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082))\n        n)"}, {"tactic": "cases Nat.le_antisymm (Nat.ge_of_not_lt lt\u2082) (Nat.ge_of_not_lt lt\u2081)", "state_before": "case h_3.inr.inr\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WellFormed le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2082\nht\u2082 : WellFormed le (r\u2082 + 1) t\u2082\nlt\u2081 : \u00acr\u2081 < r\u2082\nlt\u2082 : \u00acr\u2082 < r\u2081\n\u22a2 WellFormed le n\n      (match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n      | (a, n) =>\n        let r := r\u2081 + 1;\n        if Heap.rankGT t\u2081 r then if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n        else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n \u2194 Heap.rankGT (cons r\u2082 a\u2082 n\u2082 t\u2082) n) \u2192\n      Heap.rankGT\n        (match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n        | (a, n) =>\n          let r := r\u2081 + 1;\n          if Heap.rankGT t\u2081 r then if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n          else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082))\n        n)", "state_after": "case h_3.inr.inr.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\n\u22a2 WellFormed le n\n      (match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n      | (a, n) =>\n        let r := r\u2081 + 1;\n        if Heap.rankGT t\u2081 r then if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n        else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n \u2194 Heap.rankGT (cons r\u2081 a\u2082 n\u2082 t\u2082) n) \u2192\n      Heap.rankGT\n        (match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n        | (a, n) =>\n          let r := r\u2081 + 1;\n          if Heap.rankGT t\u2081 r then if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n          else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082))\n        n)"}, {"tactic": "split", "state_before": "case h_3.inr.inr.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\n\u22a2 WellFormed le n\n      (match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n      | (a, n) =>\n        let r := r\u2081 + 1;\n        if Heap.rankGT t\u2081 r then if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n        else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n \u2194 Heap.rankGT (cons r\u2081 a\u2082 n\u2082 t\u2082) n) \u2192\n      Heap.rankGT\n        (match combine le a\u2081 a\u2082 n\u2081 n\u2082 with\n        | (a, n) =>\n          let r := r\u2081 + 1;\n          if Heap.rankGT t\u2081 r then if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n          else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082))\n        n)", "state_after": "case h_3.inr.inr.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na\u271d : \u03b1\u271d\nn\u271d : HeapNode \u03b1\u271d\nheq\u271d : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a\u271d, n\u271d)\n\u22a2 WellFormed le n\n      (let r := r\u2081 + 1;\n      if Heap.rankGT t\u2081 r then if Heap.rankGT t\u2082 r then cons r a\u271d n\u271d (merge le t\u2081 t\u2082) else merge le (cons r a\u271d n\u271d t\u2081) t\u2082\n      else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a\u271d n\u271d t\u2082) else cons r a\u271d n\u271d (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n \u2194 Heap.rankGT (cons r\u2081 a\u2082 n\u2082 t\u2082) n) \u2192\n      Heap.rankGT\n        (let r := r\u2081 + 1;\n        if Heap.rankGT t\u2081 r then\n          if Heap.rankGT t\u2082 r then cons r a\u271d n\u271d (merge le t\u2081 t\u2082) else merge le (cons r a\u271d n\u271d t\u2081) t\u2082\n        else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a\u271d n\u271d t\u2082) else cons r a\u271d n\u271d (merge le t\u2081 t\u2082))\n        n)"}, {"tactic": "rename_i a n eq", "state_before": "case h_3.inr.inr.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na\u271d : \u03b1\u271d\nn\u271d : HeapNode \u03b1\u271d\nheq\u271d : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a\u271d, n\u271d)\n\u22a2 WellFormed le n\n      (let r := r\u2081 + 1;\n      if Heap.rankGT t\u2081 r then if Heap.rankGT t\u2082 r then cons r a\u271d n\u271d (merge le t\u2081 t\u2082) else merge le (cons r a\u271d n\u271d t\u2081) t\u2082\n      else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a\u271d n\u271d t\u2082) else cons r a\u271d n\u271d (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n \u2194 Heap.rankGT (cons r\u2081 a\u2082 n\u2082 t\u2082) n) \u2192\n      Heap.rankGT\n        (let r := r\u2081 + 1;\n        if Heap.rankGT t\u2081 r then\n          if Heap.rankGT t\u2082 r then cons r a\u271d n\u271d (merge le t\u2081 t\u2082) else merge le (cons r a\u271d n\u271d t\u2081) t\u2082\n        else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a\u271d n\u271d t\u2082) else cons r a\u271d n\u271d (merge le t\u2081 t\u2082))\n        n)", "state_after": "case h_3.inr.inr.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\n\u22a2 WellFormed le n\u271d\n      (let r := r\u2081 + 1;\n      if Heap.rankGT t\u2081 r then if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n      else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n\u271d \u2194 Heap.rankGT (cons r\u2081 a\u2082 n\u2082 t\u2082) n\u271d) \u2192\n      Heap.rankGT\n        (let r := r\u2081 + 1;\n        if Heap.rankGT t\u2081 r then if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n        else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082))\n        n\u271d)"}, {"tactic": "simp only", "state_before": "case h_3.inr.inr.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WellFormed le a n (r\u2081 + 1)\n\u22a2 WellFormed le n\u271d\n      (let r := r\u2081 + 1;\n      if Heap.rankGT t\u2081 r then if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n      else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n\u271d \u2194 Heap.rankGT (cons r\u2081 a\u2082 n\u2082 t\u2082) n\u271d) \u2192\n      Heap.rankGT\n        (let r := r\u2081 + 1;\n        if Heap.rankGT t\u2081 r then if Heap.rankGT t\u2082 r then cons r a n (merge le t\u2081 t\u2082) else merge le (cons r a n t\u2081) t\u2082\n        else if Heap.rankGT t\u2082 r then merge le t\u2081 (cons r a n t\u2082) else cons r a n (merge le t\u2081 t\u2082))\n        n\u271d)", "state_after": "case h_3.inr.inr.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WellFormed le a n (r\u2081 + 1)\n\u22a2 WellFormed le n\u271d\n      (if Heap.rankGT t\u2081 (r\u2081 + 1) then\n        if Heap.rankGT t\u2082 (r\u2081 + 1) then cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082) else merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082\n      else if Heap.rankGT t\u2082 (r\u2081 + 1) then merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082) else cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n\u271d \u2194 Heap.rankGT (cons r\u2081 a\u2082 n\u2082 t\u2082) n\u271d) \u2192\n      Heap.rankGT\n        (if Heap.rankGT t\u2081 (r\u2081 + 1) then\n          if Heap.rankGT t\u2082 (r\u2081 + 1) then cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082) else merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082\n        else if Heap.rankGT t\u2082 (r\u2081 + 1) then merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082) else cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082))\n        n\u271d)"}, {"tactic": "split <;> split <;> rename_i hl\u2081 hl\u2082", "state_before": "case h_3.inr.inr.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WellFormed le a n (r\u2081 + 1)\n\u22a2 WellFormed le n\u271d\n      (if Heap.rankGT t\u2081 (r\u2081 + 1) then\n        if Heap.rankGT t\u2082 (r\u2081 + 1) then cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082) else merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082\n      else if Heap.rankGT t\u2082 (r\u2081 + 1) then merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082) else cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n\u271d \u2194 Heap.rankGT (cons r\u2081 a\u2082 n\u2082 t\u2082) n\u271d) \u2192\n      Heap.rankGT\n        (if Heap.rankGT t\u2081 (r\u2081 + 1) then\n          if Heap.rankGT t\u2082 (r\u2081 + 1) then cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082) else merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082\n        else if Heap.rankGT t\u2082 (r\u2081 + 1) then merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082) else cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082))\n        n\u271d)", "state_after": "case h_3.inr.inr.refl.inl.inl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WellFormed le a n (r\u2081 + 1)\nhl\u2081 : Heap.rankGT t\u2081 (r\u2081 + 1)\nhl\u2082 : Heap.rankGT t\u2082 (r\u2081 + 1)\n\u22a2 WellFormed le n\u271d (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n\u271d \u2194 Heap.rankGT (cons r\u2081 a\u2082 n\u2082 t\u2082) n\u271d) \u2192\n      Heap.rankGT (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)) n\u271d)\n\ncase h_3.inr.inr.refl.inl.inr\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WellFormed le a n (r\u2081 + 1)\nhl\u2081 : Heap.rankGT t\u2081 (r\u2081 + 1)\nhl\u2082 : \u00acHeap.rankGT t\u2082 (r\u2081 + 1)\n\u22a2 WellFormed le n\u271d (merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n\u271d \u2194 Heap.rankGT (cons r\u2081 a\u2082 n\u2082 t\u2082) n\u271d) \u2192\n      Heap.rankGT (merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082) n\u271d)\n\ncase h_3.inr.inr.refl.inr.inl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WellFormed le a n (r\u2081 + 1)\nhl\u2081 : \u00acHeap.rankGT t\u2081 (r\u2081 + 1)\nhl\u2082 : Heap.rankGT t\u2082 (r\u2081 + 1)\n\u22a2 WellFormed le n\u271d (merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n\u271d \u2194 Heap.rankGT (cons r\u2081 a\u2082 n\u2082 t\u2082) n\u271d) \u2192\n      Heap.rankGT (merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082)) n\u271d)\n\ncase h_3.inr.inr.refl.inr.inr\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WellFormed le a n (r\u2081 + 1)\nhl\u2081 : \u00acHeap.rankGT t\u2081 (r\u2081 + 1)\nhl\u2082 : \u00acHeap.rankGT t\u2082 (r\u2081 + 1)\n\u22a2 WellFormed le n\u271d (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n\u271d \u2194 Heap.rankGT (cons r\u2081 a\u2082 n\u2082 t\u2082) n\u271d) \u2192\n      Heap.rankGT (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)) n\u271d)"}, {"tactic": "refine \u27e8\u27e8hr\u2081, hn\u2081, And.left (merge' ht\u2081 \u27e8lt\u2081, hn\u2082, ht\u2082\u27e9)\u27e9, fun h => ?_\u27e9", "state_before": "case h_3.inl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WellFormed le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2082\nht\u2082 : WellFormed le (r\u2082 + 1) t\u2082\nlt\u2081 : r\u2081 < r\u2082\n\u22a2 WellFormed le n (cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n \u2194 Heap.rankGT (cons r\u2082 a\u2082 n\u2082 t\u2082) n) \u2192\n      Heap.rankGT (cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))) n)", "state_after": "case h_3.inl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WellFormed le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2082\nht\u2082 : WellFormed le (r\u2082 + 1) t\u2082\nlt\u2081 : r\u2081 < r\u2082\nh : Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n \u2194 Heap.rankGT (cons r\u2082 a\u2082 n\u2082 t\u2082) n\n\u22a2 Heap.rankGT (cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))) n"}, {"tactic": "exact h.2 <| Nat.lt_of_le_of_lt hr\u2081 lt\u2081", "state_before": "case h_3.inl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WellFormed le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2082\nht\u2082 : WellFormed le (r\u2082 + 1) t\u2082\nlt\u2081 : r\u2081 < r\u2082\nh : Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n \u2194 Heap.rankGT (cons r\u2082 a\u2082 n\u2082 t\u2082) n\n\u22a2 Heap.rankGT (cons r\u2081 a\u2081 n\u2081 (merge le t\u2081 (cons r\u2082 a\u2082 n\u2082 t\u2082))) n", "state_after": "no goals"}, {"tactic": "refine \u27e8\u27e8hr\u2082, hn\u2082, And.left (merge' \u27e8lt\u2082, hn\u2081, ht\u2081\u27e9 ht\u2082)\u27e9, fun h => ?_\u27e9", "state_before": "case h_3.inr.inl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WellFormed le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2082\nht\u2082 : WellFormed le (r\u2082 + 1) t\u2082\nlt\u2081 : \u00acr\u2081 < r\u2082\nlt\u2082 : r\u2082 < r\u2081\n\u22a2 WellFormed le n (cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n \u2194 Heap.rankGT (cons r\u2082 a\u2082 n\u2082 t\u2082) n) \u2192\n      Heap.rankGT (cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)) n)", "state_after": "case h_3.inr.inl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WellFormed le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2082\nht\u2082 : WellFormed le (r\u2082 + 1) t\u2082\nlt\u2081 : \u00acr\u2081 < r\u2082\nlt\u2082 : r\u2082 < r\u2081\nh : Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n \u2194 Heap.rankGT (cons r\u2082 a\u2082 n\u2082 t\u2082) n\n\u22a2 Heap.rankGT (cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)) n"}, {"tactic": "exact h.1 <| Nat.lt_of_le_of_lt hr\u2082 lt\u2082", "state_before": "case h_3.inr.inl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b9 x\u271d : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\nr\u2082 : Nat\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nh\u2082 : WellFormed le n (cons r\u2082 a\u2082 n\u2082 t\u2082)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nhr\u2082 : n \u2264 r\u2082\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2082\nht\u2082 : WellFormed le (r\u2082 + 1) t\u2082\nlt\u2081 : \u00acr\u2081 < r\u2082\nlt\u2082 : r\u2082 < r\u2081\nh : Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n \u2194 Heap.rankGT (cons r\u2082 a\u2082 n\u2082 t\u2082) n\n\u22a2 Heap.rankGT (cons r\u2082 a\u2082 n\u2082 (merge le (cons r\u2081 a\u2081 n\u2081 t\u2081) t\u2082)) n", "state_after": "no goals"}, {"tactic": "unfold combine at eq", "state_before": "\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\n\u22a2 HeapNode.WellFormed le a n (r\u2081 + 1)", "state_after": "\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : (if le a\u2081 a\u2082 = true then (a\u2081, HeapNode.node a\u2082 n\u2082 n\u2081) else (a\u2082, HeapNode.node a\u2081 n\u2081 n\u2082)) = (a, n)\n\u22a2 HeapNode.WellFormed le a n (r\u2081 + 1)"}, {"tactic": "split at eq <;> cases eq <;> rename_i h", "state_before": "\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : (if le a\u2081 a\u2082 = true then (a\u2081, HeapNode.node a\u2082 n\u2082 n\u2081) else (a\u2082, HeapNode.node a\u2081 n\u2081 n\u2082)) = (a, n)\n\u22a2 HeapNode.WellFormed le a n (r\u2081 + 1)", "state_after": "case inl.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\nh : le a\u2081 a\u2082 = true\n\u22a2 HeapNode.WellFormed le a\u2081 (HeapNode.node a\u2082 n\u2082 n\u2081) (r\u2081 + 1)\n\ncase inr.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\nh : \u00acle a\u2081 a\u2082 = true\n\u22a2 HeapNode.WellFormed le a\u2082 (HeapNode.node a\u2081 n\u2081 n\u2082) (r\u2081 + 1)"}, {"tactic": "exact \u27e8r\u2081, rfl, h, hn\u2082, hn\u2081\u27e9", "state_before": "case inl.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\nh : le a\u2081 a\u2082 = true\n\u22a2 HeapNode.WellFormed le a\u2081 (HeapNode.node a\u2082 n\u2082 n\u2081) (r\u2081 + 1)", "state_after": "no goals"}, {"tactic": "exact \u27e8r\u2081, rfl, TotalBLE.total.resolve_left h, hn\u2081, hn\u2082\u27e9", "state_before": "case inr.refl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\nh : \u00acle a\u2081 a\u2082 = true\n\u22a2 HeapNode.WellFormed le a\u2082 (HeapNode.node a\u2081 n\u2081 n\u2082) (r\u2081 + 1)", "state_after": "no goals"}, {"tactic": "exact \u27e8\u27e8Nat.le_succ_of_le hr\u2081, this,\n  (merge' (ht\u2081.of_rankGT hl\u2081) (ht\u2082.of_rankGT hl\u2082)).1\u27e9,\n  fun _ => Nat.lt_succ_of_le hr\u2081\u27e9", "state_before": "case h_3.inr.inr.refl.inl.inl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WellFormed le a n (r\u2081 + 1)\nhl\u2081 : Heap.rankGT t\u2081 (r\u2081 + 1)\nhl\u2082 : Heap.rankGT t\u2082 (r\u2081 + 1)\n\u22a2 WellFormed le n\u271d (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n\u271d \u2194 Heap.rankGT (cons r\u2081 a\u2082 n\u2082 t\u2082) n\u271d) \u2192\n      Heap.rankGT (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)) n\u271d)", "state_after": "no goals"}, {"tactic": "let \u27e8ih\u2081, ih\u2082\u27e9 := merge' (s\u2081 := .cons ..)\n  \u27e8Nat.le_succ_of_le hr\u2081, this, ht\u2081.of_rankGT hl\u2081\u27e9\n  (ht\u2082.of_le (Nat.le_succ_of_le hr\u2081))", "state_before": "case h_3.inr.inr.refl.inl.inr\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WellFormed le a n (r\u2081 + 1)\nhl\u2081 : Heap.rankGT t\u2081 (r\u2081 + 1)\nhl\u2082 : \u00acHeap.rankGT t\u2082 (r\u2081 + 1)\n\u22a2 WellFormed le n\u271d (merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n\u271d \u2194 Heap.rankGT (cons r\u2081 a\u2082 n\u2082 t\u2082) n\u271d) \u2192\n      Heap.rankGT (merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082) n\u271d)", "state_after": "case h_3.inr.inr.refl.inl.inr\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WellFormed le a n (r\u2081 + 1)\nhl\u2081 : Heap.rankGT t\u2081 (r\u2081 + 1)\nhl\u2082 : \u00acHeap.rankGT t\u2082 (r\u2081 + 1)\nih\u2081 : WellFormed le n\u271d (merge le (cons (Nat.succ r\u2081) a n t\u2081) t\u2082)\nih\u2082 :\n  (Heap.rankGT (cons (Nat.succ r\u2081) a n t\u2081) n\u271d \u2194 Heap.rankGT t\u2082 n\u271d) \u2192\n    Heap.rankGT (merge le (cons (Nat.succ r\u2081) a n t\u2081) t\u2082) n\u271d\n\u22a2 WellFormed le n\u271d (merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n\u271d \u2194 Heap.rankGT (cons r\u2081 a\u2082 n\u2082 t\u2082) n\u271d) \u2192\n      Heap.rankGT (merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082) n\u271d)"}, {"tactic": "exact \u27e8ih\u2081, fun _ => ih\u2082 \u27e8fun _ => ht\u2082.rankGT.le_trans hr\u2081, fun h => Nat.lt_succ_of_le hr\u2081\u27e9\u27e9", "state_before": "case h_3.inr.inr.refl.inl.inr\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WellFormed le a n (r\u2081 + 1)\nhl\u2081 : Heap.rankGT t\u2081 (r\u2081 + 1)\nhl\u2082 : \u00acHeap.rankGT t\u2082 (r\u2081 + 1)\nih\u2081 : WellFormed le n\u271d (merge le (cons (Nat.succ r\u2081) a n t\u2081) t\u2082)\nih\u2082 :\n  (Heap.rankGT (cons (Nat.succ r\u2081) a n t\u2081) n\u271d \u2194 Heap.rankGT t\u2082 n\u271d) \u2192\n    Heap.rankGT (merge le (cons (Nat.succ r\u2081) a n t\u2081) t\u2082) n\u271d\n\u22a2 WellFormed le n\u271d (merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n\u271d \u2194 Heap.rankGT (cons r\u2081 a\u2082 n\u2082 t\u2082) n\u271d) \u2192\n      Heap.rankGT (merge le (cons (r\u2081 + 1) a n t\u2081) t\u2082) n\u271d)", "state_after": "no goals"}, {"tactic": "let \u27e8ih\u2081, ih\u2082\u27e9 := merge' (s\u2082 := .cons ..) (ht\u2081.of_le (Nat.le_succ_of_le hr\u2081))\n  \u27e8Nat.le_succ_of_le hr\u2081, this, ht\u2082.of_rankGT hl\u2082\u27e9", "state_before": "case h_3.inr.inr.refl.inr.inl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WellFormed le a n (r\u2081 + 1)\nhl\u2081 : \u00acHeap.rankGT t\u2081 (r\u2081 + 1)\nhl\u2082 : Heap.rankGT t\u2082 (r\u2081 + 1)\n\u22a2 WellFormed le n\u271d (merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n\u271d \u2194 Heap.rankGT (cons r\u2081 a\u2082 n\u2082 t\u2082) n\u271d) \u2192\n      Heap.rankGT (merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082)) n\u271d)", "state_after": "case h_3.inr.inr.refl.inr.inl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WellFormed le a n (r\u2081 + 1)\nhl\u2081 : \u00acHeap.rankGT t\u2081 (r\u2081 + 1)\nhl\u2082 : Heap.rankGT t\u2082 (r\u2081 + 1)\nih\u2081 : WellFormed le n\u271d (merge le t\u2081 (cons (Nat.succ r\u2081) a n t\u2082))\nih\u2082 :\n  (Heap.rankGT t\u2081 n\u271d \u2194 Heap.rankGT (cons (Nat.succ r\u2081) a n t\u2082) n\u271d) \u2192\n    Heap.rankGT (merge le t\u2081 (cons (Nat.succ r\u2081) a n t\u2082)) n\u271d\n\u22a2 WellFormed le n\u271d (merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n\u271d \u2194 Heap.rankGT (cons r\u2081 a\u2082 n\u2082 t\u2082) n\u271d) \u2192\n      Heap.rankGT (merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082)) n\u271d)"}, {"tactic": "exact \u27e8ih\u2081, fun _ => ih\u2082 \u27e8fun h => Nat.lt_succ_of_le hr\u2081, fun _ => ht\u2081.rankGT.le_trans hr\u2081\u27e9\u27e9", "state_before": "case h_3.inr.inr.refl.inr.inl\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WellFormed le a n (r\u2081 + 1)\nhl\u2081 : \u00acHeap.rankGT t\u2081 (r\u2081 + 1)\nhl\u2082 : Heap.rankGT t\u2082 (r\u2081 + 1)\nih\u2081 : WellFormed le n\u271d (merge le t\u2081 (cons (Nat.succ r\u2081) a n t\u2082))\nih\u2082 :\n  (Heap.rankGT t\u2081 n\u271d \u2194 Heap.rankGT (cons (Nat.succ r\u2081) a n t\u2082) n\u271d) \u2192\n    Heap.rankGT (merge le t\u2081 (cons (Nat.succ r\u2081) a n t\u2082)) n\u271d\n\u22a2 WellFormed le n\u271d (merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n\u271d \u2194 Heap.rankGT (cons r\u2081 a\u2082 n\u2082 t\u2082) n\u271d) \u2192\n      Heap.rankGT (merge le t\u2081 (cons (r\u2081 + 1) a n t\u2082)) n\u271d)", "state_after": "no goals"}, {"tactic": "let \u27e8ih\u2081, ih\u2082\u27e9 := merge' ht\u2081 ht\u2082", "state_before": "case h_3.inr.inr.refl.inr.inr\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WellFormed le a n (r\u2081 + 1)\nhl\u2081 : \u00acHeap.rankGT t\u2081 (r\u2081 + 1)\nhl\u2082 : \u00acHeap.rankGT t\u2082 (r\u2081 + 1)\n\u22a2 WellFormed le n\u271d (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n\u271d \u2194 Heap.rankGT (cons r\u2081 a\u2082 n\u2082 t\u2082) n\u271d) \u2192\n      Heap.rankGT (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)) n\u271d)", "state_after": "case h_3.inr.inr.refl.inr.inr\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WellFormed le a n (r\u2081 + 1)\nhl\u2081 : \u00acHeap.rankGT t\u2081 (r\u2081 + 1)\nhl\u2082 : \u00acHeap.rankGT t\u2082 (r\u2081 + 1)\nih\u2081 : WellFormed le (r\u2081 + 1) (merge le t\u2081 t\u2082)\nih\u2082 : (Heap.rankGT t\u2081 (r\u2081 + 1) \u2194 Heap.rankGT t\u2082 (r\u2081 + 1)) \u2192 Heap.rankGT (merge le t\u2081 t\u2082) (r\u2081 + 1)\n\u22a2 WellFormed le n\u271d (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n\u271d \u2194 Heap.rankGT (cons r\u2081 a\u2082 n\u2082 t\u2082) n\u271d) \u2192\n      Heap.rankGT (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)) n\u271d)"}, {"tactic": "exact \u27e8\u27e8Nat.le_succ_of_le hr\u2081, this, ih\u2081.of_rankGT (ih\u2082 (iff_of_false hl\u2081 hl\u2082))\u27e9,\n  fun _ => Nat.lt_succ_of_le hr\u2081\u27e9", "state_before": "case h_3.inr.inr.refl.inr.inr\n\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\nn\u271d : Nat\nx\u271d\u00b2 x\u271d\u00b9 : Heap \u03b1\u271d\nr\u2081 : Nat\na\u2081 : \u03b1\u271d\nn\u2081 : HeapNode \u03b1\u271d\nt\u2081 : Heap \u03b1\u271d\na\u2082 : \u03b1\u271d\nn\u2082 : HeapNode \u03b1\u271d\nt\u2082 : Heap \u03b1\u271d\nh\u2081 : WellFormed le n\u271d (cons r\u2081 a\u2081 n\u2081 t\u2081)\nhr\u2081 : n\u271d \u2264 r\u2081\nhn\u2081 : HeapNode.WellFormed le a\u2081 n\u2081 r\u2081\nht\u2081 : WellFormed le (r\u2081 + 1) t\u2081\nh\u2082 : WellFormed le n\u271d (cons r\u2081 a\u2082 n\u2082 t\u2082)\nhr\u2082 : n\u271d \u2264 r\u2081\nhn\u2082 : HeapNode.WellFormed le a\u2082 n\u2082 r\u2081\nht\u2082 : WellFormed le (r\u2081 + 1) t\u2082\nlt\u2081 lt\u2082 : \u00acr\u2081 < r\u2081\nx\u271d : \u03b1\u271d \u00d7 HeapNode \u03b1\u271d\na : \u03b1\u271d\nn : HeapNode \u03b1\u271d\neq : combine le a\u2081 a\u2082 n\u2081 n\u2082 = (a, n)\nthis : HeapNode.WellFormed le a n (r\u2081 + 1)\nhl\u2081 : \u00acHeap.rankGT t\u2081 (r\u2081 + 1)\nhl\u2082 : \u00acHeap.rankGT t\u2082 (r\u2081 + 1)\nih\u2081 : WellFormed le (r\u2081 + 1) (merge le t\u2081 t\u2082)\nih\u2082 : (Heap.rankGT t\u2081 (r\u2081 + 1) \u2194 Heap.rankGT t\u2082 (r\u2081 + 1)) \u2192 Heap.rankGT (merge le t\u2081 t\u2082) (r\u2081 + 1)\n\u22a2 WellFormed le n\u271d (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)) \u2227\n    ((Heap.rankGT (cons r\u2081 a\u2081 n\u2081 t\u2081) n\u271d \u2194 Heap.rankGT (cons r\u2081 a\u2082 n\u2082 t\u2082) n\u271d) \u2192\n      Heap.rankGT (cons (r\u2081 + 1) a n (merge le t\u2081 t\u2082)) n\u271d)", "state_after": "no goals"}, {"tactic": "simp_wf", "state_before": "\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\n_x\u271d : (s\u2081 : Heap \u03b1\u271d) \u00d7' (s\u2082 : Heap \u03b1\u271d) \u00d7' (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082\na\u271d\u2075 :\n  \u2200 (y : (s\u2081 : Heap \u03b1\u271d) \u00d7' (s\u2082 : Heap \u03b1\u271d) \u00d7' (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082),\n    (invImage\n            (fun a =>\n              PSigma.casesOn a fun s\u2081 snd =>\n                PSigma.casesOn snd fun s\u2082 snd =>\n                  PSigma.casesOn snd fun n snd => PSigma.casesOn snd fun h\u2081 snd => length s\u2081 + length s\u2082)\n            instWellFoundedRelation).1\n        y _x\u271d \u2192\n      WellFormed le y.2.2.1 (merge le y.1 y.2.1) \u2227\n        ((Heap.rankGT y.1 y.2.2.1 \u2194 Heap.rankGT y.2.1 y.2.2.1) \u2192 Heap.rankGT (merge le y.1 y.2.1) y.2.2.1)\ns\u2081 : Heap \u03b1\u271d\ns\u2082\u271d : (s\u2082 : Heap \u03b1\u271d) \u00d7' (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082\na\u271d\u2074 :\n  \u2200 (y : (s\u2081 : Heap \u03b1\u271d) \u00d7' (s\u2082 : Heap \u03b1\u271d) \u00d7' (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082),\n    (invImage\n            (fun a =>\n              PSigma.casesOn a fun s\u2081 snd =>\n                PSigma.casesOn snd fun s\u2082 snd =>\n                  PSigma.casesOn snd fun n snd => PSigma.casesOn snd fun h\u2081 snd => length s\u2081 + length s\u2082)\n            instWellFoundedRelation).1\n        y { fst := s\u2081, snd := s\u2082\u271d } \u2192\n      WellFormed le y.2.2.1 (merge le y.1 y.2.1) \u2227\n        ((Heap.rankGT y.1 y.2.2.1 \u2194 Heap.rankGT y.2.1 y.2.2.1) \u2192 Heap.rankGT (merge le y.1 y.2.1) y.2.2.1)\ns\u2082 : Heap \u03b1\u271d\nn\u271d\u00b9 : (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082\na\u271d\u00b3 :\n  \u2200 (y : (s\u2081 : Heap \u03b1\u271d) \u00d7' (s\u2082 : Heap \u03b1\u271d) \u00d7' (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082),\n    (invImage\n            (fun a =>\n              PSigma.casesOn a fun s\u2081 snd =>\n                PSigma.casesOn snd fun s\u2082 snd =>\n                  PSigma.casesOn snd fun n snd => PSigma.casesOn snd fun h\u2081 snd => length s\u2081 + length s\u2082)\n            instWellFoundedRelation).1\n        y { fst := s\u2081, snd := { fst := s\u2082, snd := n\u271d\u00b9 } } \u2192\n      WellFormed le y.2.2.1 (merge le y.1 y.2.1) \u2227\n        ((Heap.rankGT y.1 y.2.2.1 \u2194 Heap.rankGT y.2.1 y.2.2.1) \u2192 Heap.rankGT (merge le y.1 y.2.1) y.2.2.1)\nn : Nat\nh\u2081\u271d\u00b9 : (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082\na\u271d\u00b2 :\n  \u2200 (y : (s\u2081 : Heap \u03b1\u271d) \u00d7' (s\u2082 : Heap \u03b1\u271d) \u00d7' (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082),\n    (invImage\n            (fun a =>\n              PSigma.casesOn a fun s\u2081 snd =>\n                PSigma.casesOn snd fun s\u2082 snd =>\n                  PSigma.casesOn snd fun n snd => PSigma.casesOn snd fun h\u2081 snd => length s\u2081 + length s\u2082)\n            instWellFoundedRelation).1\n        y { fst := s\u2081, snd := { fst := s\u2082, snd := { fst := n, snd := h\u2081\u271d\u00b9 } } } \u2192\n      WellFormed le y.2.2.1 (merge le y.1 y.2.1) \u2227\n        ((Heap.rankGT y.1 y.2.2.1 \u2194 Heap.rankGT y.2.1 y.2.2.1) \u2192 Heap.rankGT (merge le y.1 y.2.1) y.2.2.1)\nh\u2081\u271d : WellFormed le n s\u2081\nh\u2082\u271d\u00b9 : WellFormed le n s\u2082\na\u271d\u00b9 :\n  \u2200 (y : (s\u2081 : Heap \u03b1\u271d) \u00d7' (s\u2082 : Heap \u03b1\u271d) \u00d7' (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082),\n    (invImage\n            (fun a =>\n              PSigma.casesOn a fun s\u2081 snd =>\n                PSigma.casesOn snd fun s\u2082 snd =>\n                  PSigma.casesOn snd fun n snd => PSigma.casesOn snd fun h\u2081 snd => length s\u2081 + length s\u2082)\n            instWellFoundedRelation).1\n        y { fst := s\u2081, snd := { fst := s\u2082, snd := { fst := n, snd := { fst := h\u2081\u271d, snd := h\u2082\u271d\u00b9 } } } } \u2192\n      WellFormed le y.2.2.1 (merge le y.1 y.2.1) \u2227\n        ((Heap.rankGT y.1 y.2.2.1 \u2194 Heap.rankGT y.2.1 y.2.2.1) \u2192 Heap.rankGT (merge le y.1 y.2.1) y.2.2.1)\nr\u2081\u271d : Nat\na\u2081\u271d : \u03b1\u271d\nn\u2081\u271d : HeapNode \u03b1\u271d\nt\u2081\u271d : Heap \u03b1\u271d\nr\u2082\u271d : Nat\na\u2082\u271d : \u03b1\u271d\nn\u2082\u271d : HeapNode \u03b1\u271d\nt\u2082\u271d : Heap \u03b1\u271d\nh\u271d\u2078 : s\u2081 = cons r\u2081\u271d a\u2081\u271d n\u2081\u271d t\u2081\u271d\nh\u2081 : WellFormed le n (cons r\u2081\u271d a\u2081\u271d n\u2081\u271d t\u2081\u271d)\nh\u271d\u2077 : s\u2082 = cons r\u2082\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d\nh\u2082\u271d : WellFormed le n (cons r\u2082\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d)\nhr\u2081 : n \u2264 r\u2081\u271d\nhn\u2081 : HeapNode.WellFormed le a\u2081\u271d n\u2081\u271d r\u2081\u271d\nht\u2081 : WellFormed le (r\u2081\u271d + 1) t\u2081\u271d\nhr\u2082\u271d : n \u2264 r\u2082\u271d\nhn\u2082\u271d : HeapNode.WellFormed le a\u2082\u271d n\u2082\u271d r\u2082\u271d\nht\u2082\u271d : WellFormed le (r\u2082\u271d + 1) t\u2082\u271d\nh\u271d\u2076 : \u00acr\u2081\u271d < r\u2082\u271d\nh\u271d\u2075 : \u00acr\u2082\u271d < r\u2081\u271d\nh\u271d\u2074 : r\u2082\u271d = r\u2081\u271d\nh\u2082 : WellFormed le n (cons r\u2081\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d)\nhr\u2082 : n \u2264 r\u2081\u271d\nhn\u2082 : HeapNode.WellFormed le a\u2082\u271d n\u2082\u271d r\u2081\u271d\nht\u2082 : WellFormed le (r\u2081\u271d + 1) t\u2082\u271d\nlt\u2081 lt\u2082 : \u00acr\u2081\u271d < r\u2081\u271d\nx\u271d : r\u2081\u271d = r\u2081\u271d\nh\u271d\u00b3 : HEq x\u271d (_ : r\u2081\u271d = r\u2081\u271d)\na\u271d : \u03b1\u271d\nn\u271d : HeapNode \u03b1\u271d\nh\u271d\u00b2 heq\u271d : combine le a\u2081\u271d a\u2082\u271d n\u2081\u271d n\u2082\u271d = (a\u271d, n\u271d)\nthis : HeapNode.WellFormed le a\u271d n\u271d (r\u2081\u271d + 1)\nh\u271d\u00b9 : \u00acHeap.rankGT t\u2081\u271d (r\u2081\u271d + 1)\nh\u271d : \u00acHeap.rankGT t\u2082\u271d (r\u2081\u271d + 1)\n\u22a2 (invImage\n        (fun a =>\n          PSigma.casesOn a fun s\u2081 snd =>\n            PSigma.casesOn snd fun s\u2082 snd =>\n              PSigma.casesOn snd fun n snd => PSigma.casesOn snd fun h\u2081 snd => length s\u2081 + length s\u2082)\n        instWellFoundedRelation).1\n    { fst := t\u2081\u271d, snd := { fst := t\u2082\u271d, snd := { fst := r\u2081\u271d + 1, snd := { fst := ht\u2081, snd := ht\u2082 } } } }\n    { fst := s\u2081, snd := { fst := s\u2082, snd := { fst := n, snd := { fst := h\u2081\u271d, snd := h\u2082\u271d\u00b9 } } } }", "state_after": "\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\n_x\u271d : (s\u2081 : Heap \u03b1\u271d) \u00d7' (s\u2082 : Heap \u03b1\u271d) \u00d7' (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082\na\u271d\u2075 :\n  \u2200 (y : (s\u2081 : Heap \u03b1\u271d) \u00d7' (s\u2082 : Heap \u03b1\u271d) \u00d7' (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082),\n    (invImage\n            (fun a =>\n              PSigma.casesOn a fun s\u2081 snd =>\n                PSigma.casesOn snd fun s\u2082 snd =>\n                  PSigma.casesOn snd fun n snd => PSigma.casesOn snd fun h\u2081 snd => length s\u2081 + length s\u2082)\n            instWellFoundedRelation).1\n        y _x\u271d \u2192\n      WellFormed le y.2.2.1 (merge le y.1 y.2.1) \u2227\n        ((Heap.rankGT y.1 y.2.2.1 \u2194 Heap.rankGT y.2.1 y.2.2.1) \u2192 Heap.rankGT (merge le y.1 y.2.1) y.2.2.1)\ns\u2081 : Heap \u03b1\u271d\ns\u2082\u271d : (s\u2082 : Heap \u03b1\u271d) \u00d7' (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082\na\u271d\u2074 :\n  \u2200 (y : (s\u2081 : Heap \u03b1\u271d) \u00d7' (s\u2082 : Heap \u03b1\u271d) \u00d7' (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082),\n    (invImage\n            (fun a =>\n              PSigma.casesOn a fun s\u2081 snd =>\n                PSigma.casesOn snd fun s\u2082 snd =>\n                  PSigma.casesOn snd fun n snd => PSigma.casesOn snd fun h\u2081 snd => length s\u2081 + length s\u2082)\n            instWellFoundedRelation).1\n        y { fst := s\u2081, snd := s\u2082\u271d } \u2192\n      WellFormed le y.2.2.1 (merge le y.1 y.2.1) \u2227\n        ((Heap.rankGT y.1 y.2.2.1 \u2194 Heap.rankGT y.2.1 y.2.2.1) \u2192 Heap.rankGT (merge le y.1 y.2.1) y.2.2.1)\ns\u2082 : Heap \u03b1\u271d\nn\u271d\u00b9 : (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082\na\u271d\u00b3 :\n  \u2200 (y : (s\u2081 : Heap \u03b1\u271d) \u00d7' (s\u2082 : Heap \u03b1\u271d) \u00d7' (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082),\n    (invImage\n            (fun a =>\n              PSigma.casesOn a fun s\u2081 snd =>\n                PSigma.casesOn snd fun s\u2082 snd =>\n                  PSigma.casesOn snd fun n snd => PSigma.casesOn snd fun h\u2081 snd => length s\u2081 + length s\u2082)\n            instWellFoundedRelation).1\n        y { fst := s\u2081, snd := { fst := s\u2082, snd := n\u271d\u00b9 } } \u2192\n      WellFormed le y.2.2.1 (merge le y.1 y.2.1) \u2227\n        ((Heap.rankGT y.1 y.2.2.1 \u2194 Heap.rankGT y.2.1 y.2.2.1) \u2192 Heap.rankGT (merge le y.1 y.2.1) y.2.2.1)\nn : Nat\nh\u2081\u271d\u00b9 : (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082\na\u271d\u00b2 :\n  \u2200 (y : (s\u2081 : Heap \u03b1\u271d) \u00d7' (s\u2082 : Heap \u03b1\u271d) \u00d7' (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082),\n    (invImage\n            (fun a =>\n              PSigma.casesOn a fun s\u2081 snd =>\n                PSigma.casesOn snd fun s\u2082 snd =>\n                  PSigma.casesOn snd fun n snd => PSigma.casesOn snd fun h\u2081 snd => length s\u2081 + length s\u2082)\n            instWellFoundedRelation).1\n        y { fst := s\u2081, snd := { fst := s\u2082, snd := { fst := n, snd := h\u2081\u271d\u00b9 } } } \u2192\n      WellFormed le y.2.2.1 (merge le y.1 y.2.1) \u2227\n        ((Heap.rankGT y.1 y.2.2.1 \u2194 Heap.rankGT y.2.1 y.2.2.1) \u2192 Heap.rankGT (merge le y.1 y.2.1) y.2.2.1)\nh\u2081\u271d : WellFormed le n s\u2081\nh\u2082\u271d\u00b9 : WellFormed le n s\u2082\na\u271d\u00b9 :\n  \u2200 (y : (s\u2081 : Heap \u03b1\u271d) \u00d7' (s\u2082 : Heap \u03b1\u271d) \u00d7' (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082),\n    (invImage\n            (fun a =>\n              PSigma.casesOn a fun s\u2081 snd =>\n                PSigma.casesOn snd fun s\u2082 snd =>\n                  PSigma.casesOn snd fun n snd => PSigma.casesOn snd fun h\u2081 snd => length s\u2081 + length s\u2082)\n            instWellFoundedRelation).1\n        y { fst := s\u2081, snd := { fst := s\u2082, snd := { fst := n, snd := { fst := h\u2081\u271d, snd := h\u2082\u271d\u00b9 } } } } \u2192\n      WellFormed le y.2.2.1 (merge le y.1 y.2.1) \u2227\n        ((Heap.rankGT y.1 y.2.2.1 \u2194 Heap.rankGT y.2.1 y.2.2.1) \u2192 Heap.rankGT (merge le y.1 y.2.1) y.2.2.1)\nr\u2081\u271d : Nat\na\u2081\u271d : \u03b1\u271d\nn\u2081\u271d : HeapNode \u03b1\u271d\nt\u2081\u271d : Heap \u03b1\u271d\nr\u2082\u271d : Nat\na\u2082\u271d : \u03b1\u271d\nn\u2082\u271d : HeapNode \u03b1\u271d\nt\u2082\u271d : Heap \u03b1\u271d\nh\u271d\u2078 : s\u2081 = cons r\u2081\u271d a\u2081\u271d n\u2081\u271d t\u2081\u271d\nh\u2081 : WellFormed le n (cons r\u2081\u271d a\u2081\u271d n\u2081\u271d t\u2081\u271d)\nh\u271d\u2077 : s\u2082 = cons r\u2082\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d\nh\u2082\u271d : WellFormed le n (cons r\u2082\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d)\nhr\u2081 : n \u2264 r\u2081\u271d\nhn\u2081 : HeapNode.WellFormed le a\u2081\u271d n\u2081\u271d r\u2081\u271d\nht\u2081 : WellFormed le (r\u2081\u271d + 1) t\u2081\u271d\nhr\u2082\u271d : n \u2264 r\u2082\u271d\nhn\u2082\u271d : HeapNode.WellFormed le a\u2082\u271d n\u2082\u271d r\u2082\u271d\nht\u2082\u271d : WellFormed le (r\u2082\u271d + 1) t\u2082\u271d\nh\u271d\u2076 : \u00acr\u2081\u271d < r\u2082\u271d\nh\u271d\u2075 : \u00acr\u2082\u271d < r\u2081\u271d\nh\u271d\u2074 : r\u2082\u271d = r\u2081\u271d\nh\u2082 : WellFormed le n (cons r\u2081\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d)\nhr\u2082 : n \u2264 r\u2081\u271d\nhn\u2082 : HeapNode.WellFormed le a\u2082\u271d n\u2082\u271d r\u2081\u271d\nht\u2082 : WellFormed le (r\u2081\u271d + 1) t\u2082\u271d\nlt\u2081 lt\u2082 : \u00acr\u2081\u271d < r\u2081\u271d\nx\u271d : r\u2081\u271d = r\u2081\u271d\nh\u271d\u00b3 : HEq x\u271d (_ : r\u2081\u271d = r\u2081\u271d)\na\u271d : \u03b1\u271d\nn\u271d : HeapNode \u03b1\u271d\nh\u271d\u00b2 heq\u271d : combine le a\u2081\u271d a\u2082\u271d n\u2081\u271d n\u2082\u271d = (a\u271d, n\u271d)\nthis : HeapNode.WellFormed le a\u271d n\u271d (r\u2081\u271d + 1)\nh\u271d\u00b9 : \u00acHeap.rankGT t\u2081\u271d (r\u2081\u271d + 1)\nh\u271d : \u00acHeap.rankGT t\u2082\u271d (r\u2081\u271d + 1)\n\u22a2 length t\u2081\u271d + length t\u2082\u271d < length s\u2081 + length s\u2082"}, {"tactic": "simp_arith [*]", "state_before": "\u03b1\u271d : Type u_1\nle : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Bool\n_x\u271d : (s\u2081 : Heap \u03b1\u271d) \u00d7' (s\u2082 : Heap \u03b1\u271d) \u00d7' (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082\na\u271d\u2075 :\n  \u2200 (y : (s\u2081 : Heap \u03b1\u271d) \u00d7' (s\u2082 : Heap \u03b1\u271d) \u00d7' (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082),\n    (invImage\n            (fun a =>\n              PSigma.casesOn a fun s\u2081 snd =>\n                PSigma.casesOn snd fun s\u2082 snd =>\n                  PSigma.casesOn snd fun n snd => PSigma.casesOn snd fun h\u2081 snd => length s\u2081 + length s\u2082)\n            instWellFoundedRelation).1\n        y _x\u271d \u2192\n      WellFormed le y.2.2.1 (merge le y.1 y.2.1) \u2227\n        ((Heap.rankGT y.1 y.2.2.1 \u2194 Heap.rankGT y.2.1 y.2.2.1) \u2192 Heap.rankGT (merge le y.1 y.2.1) y.2.2.1)\ns\u2081 : Heap \u03b1\u271d\ns\u2082\u271d : (s\u2082 : Heap \u03b1\u271d) \u00d7' (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082\na\u271d\u2074 :\n  \u2200 (y : (s\u2081 : Heap \u03b1\u271d) \u00d7' (s\u2082 : Heap \u03b1\u271d) \u00d7' (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082),\n    (invImage\n            (fun a =>\n              PSigma.casesOn a fun s\u2081 snd =>\n                PSigma.casesOn snd fun s\u2082 snd =>\n                  PSigma.casesOn snd fun n snd => PSigma.casesOn snd fun h\u2081 snd => length s\u2081 + length s\u2082)\n            instWellFoundedRelation).1\n        y { fst := s\u2081, snd := s\u2082\u271d } \u2192\n      WellFormed le y.2.2.1 (merge le y.1 y.2.1) \u2227\n        ((Heap.rankGT y.1 y.2.2.1 \u2194 Heap.rankGT y.2.1 y.2.2.1) \u2192 Heap.rankGT (merge le y.1 y.2.1) y.2.2.1)\ns\u2082 : Heap \u03b1\u271d\nn\u271d\u00b9 : (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082\na\u271d\u00b3 :\n  \u2200 (y : (s\u2081 : Heap \u03b1\u271d) \u00d7' (s\u2082 : Heap \u03b1\u271d) \u00d7' (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082),\n    (invImage\n            (fun a =>\n              PSigma.casesOn a fun s\u2081 snd =>\n                PSigma.casesOn snd fun s\u2082 snd =>\n                  PSigma.casesOn snd fun n snd => PSigma.casesOn snd fun h\u2081 snd => length s\u2081 + length s\u2082)\n            instWellFoundedRelation).1\n        y { fst := s\u2081, snd := { fst := s\u2082, snd := n\u271d\u00b9 } } \u2192\n      WellFormed le y.2.2.1 (merge le y.1 y.2.1) \u2227\n        ((Heap.rankGT y.1 y.2.2.1 \u2194 Heap.rankGT y.2.1 y.2.2.1) \u2192 Heap.rankGT (merge le y.1 y.2.1) y.2.2.1)\nn : Nat\nh\u2081\u271d\u00b9 : (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082\na\u271d\u00b2 :\n  \u2200 (y : (s\u2081 : Heap \u03b1\u271d) \u00d7' (s\u2082 : Heap \u03b1\u271d) \u00d7' (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082),\n    (invImage\n            (fun a =>\n              PSigma.casesOn a fun s\u2081 snd =>\n                PSigma.casesOn snd fun s\u2082 snd =>\n                  PSigma.casesOn snd fun n snd => PSigma.casesOn snd fun h\u2081 snd => length s\u2081 + length s\u2082)\n            instWellFoundedRelation).1\n        y { fst := s\u2081, snd := { fst := s\u2082, snd := { fst := n, snd := h\u2081\u271d\u00b9 } } } \u2192\n      WellFormed le y.2.2.1 (merge le y.1 y.2.1) \u2227\n        ((Heap.rankGT y.1 y.2.2.1 \u2194 Heap.rankGT y.2.1 y.2.2.1) \u2192 Heap.rankGT (merge le y.1 y.2.1) y.2.2.1)\nh\u2081\u271d : WellFormed le n s\u2081\nh\u2082\u271d\u00b9 : WellFormed le n s\u2082\na\u271d\u00b9 :\n  \u2200 (y : (s\u2081 : Heap \u03b1\u271d) \u00d7' (s\u2082 : Heap \u03b1\u271d) \u00d7' (n : Nat) \u00d7' (_ : WellFormed le n s\u2081) \u00d7' WellFormed le n s\u2082),\n    (invImage\n            (fun a =>\n              PSigma.casesOn a fun s\u2081 snd =>\n                PSigma.casesOn snd fun s\u2082 snd =>\n                  PSigma.casesOn snd fun n snd => PSigma.casesOn snd fun h\u2081 snd => length s\u2081 + length s\u2082)\n            instWellFoundedRelation).1\n        y { fst := s\u2081, snd := { fst := s\u2082, snd := { fst := n, snd := { fst := h\u2081\u271d, snd := h\u2082\u271d\u00b9 } } } } \u2192\n      WellFormed le y.2.2.1 (merge le y.1 y.2.1) \u2227\n        ((Heap.rankGT y.1 y.2.2.1 \u2194 Heap.rankGT y.2.1 y.2.2.1) \u2192 Heap.rankGT (merge le y.1 y.2.1) y.2.2.1)\nr\u2081\u271d : Nat\na\u2081\u271d : \u03b1\u271d\nn\u2081\u271d : HeapNode \u03b1\u271d\nt\u2081\u271d : Heap \u03b1\u271d\nr\u2082\u271d : Nat\na\u2082\u271d : \u03b1\u271d\nn\u2082\u271d : HeapNode \u03b1\u271d\nt\u2082\u271d : Heap \u03b1\u271d\nh\u271d\u2078 : s\u2081 = cons r\u2081\u271d a\u2081\u271d n\u2081\u271d t\u2081\u271d\nh\u2081 : WellFormed le n (cons r\u2081\u271d a\u2081\u271d n\u2081\u271d t\u2081\u271d)\nh\u271d\u2077 : s\u2082 = cons r\u2082\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d\nh\u2082\u271d : WellFormed le n (cons r\u2082\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d)\nhr\u2081 : n \u2264 r\u2081\u271d\nhn\u2081 : HeapNode.WellFormed le a\u2081\u271d n\u2081\u271d r\u2081\u271d\nht\u2081 : WellFormed le (r\u2081\u271d + 1) t\u2081\u271d\nhr\u2082\u271d : n \u2264 r\u2082\u271d\nhn\u2082\u271d : HeapNode.WellFormed le a\u2082\u271d n\u2082\u271d r\u2082\u271d\nht\u2082\u271d : WellFormed le (r\u2082\u271d + 1) t\u2082\u271d\nh\u271d\u2076 : \u00acr\u2081\u271d < r\u2082\u271d\nh\u271d\u2075 : \u00acr\u2082\u271d < r\u2081\u271d\nh\u271d\u2074 : r\u2082\u271d = r\u2081\u271d\nh\u2082 : WellFormed le n (cons r\u2081\u271d a\u2082\u271d n\u2082\u271d t\u2082\u271d)\nhr\u2082 : n \u2264 r\u2081\u271d\nhn\u2082 : HeapNode.WellFormed le a\u2082\u271d n\u2082\u271d r\u2081\u271d\nht\u2082 : WellFormed le (r\u2081\u271d + 1) t\u2082\u271d\nlt\u2081 lt\u2082 : \u00acr\u2081\u271d < r\u2081\u271d\nx\u271d : r\u2081\u271d = r\u2081\u271d\nh\u271d\u00b3 : HEq x\u271d (_ : r\u2081\u271d = r\u2081\u271d)\na\u271d : \u03b1\u271d\nn\u271d : HeapNode \u03b1\u271d\nh\u271d\u00b2 heq\u271d : combine le a\u2081\u271d a\u2082\u271d n\u2081\u271d n\u2082\u271d = (a\u271d, n\u271d)\nthis : HeapNode.WellFormed le a\u271d n\u271d (r\u2081\u271d + 1)\nh\u271d\u00b9 : \u00acHeap.rankGT t\u2081\u271d (r\u2081\u271d + 1)\nh\u271d : \u00acHeap.rankGT t\u2082\u271d (r\u2081\u271d + 1)\n\u22a2 length t\u2081\u271d + length t\u2082\u271d < length s\u2081 + length s\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fintype/Perm.lean", "full_name": "Fintype.card_perm", "start": [165, 1], "end": [166, 94], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Configuration.lean", "full_name": "Configuration.HasPoints.card_le", "start": [260, 1], "end": [262, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "full_name": "LinearIsometry.toLinearMap_inj", "start": [144, 1], "end": [145, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Tactic/Linarith/Lemmas.lean", "full_name": "lt_zero_of_zero_gt", "start": [77, 1], "end": [77, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Bool/Basic.lean", "full_name": "Bool.not_iff_not", "start": [236, 1], "end": [236, 55], "traced_tactics": [{"tactic": "simp", "state_before": "\u22a2 \u2200 {b : Bool}, (!b) = true \u2194 \u00acb = true", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/ModelTheory/Basic.lean", "full_name": "FirstOrder.Sequence\u2082.sum_card", "start": [104, 1], "end": [106, 30], "traced_tactics": [{"tactic": "rw [sum_nat_eq_add_sum_succ, sum_nat_eq_add_sum_succ, sum_nat_eq_add_sum_succ]", "state_before": "a\u2080 a\u2081 a\u2082 : Type u\n\u22a2 (sum fun i => #Sequence\u2082 a\u2080 a\u2081 a\u2082 i) = (#a\u2080) + (#a\u2081) + (#a\u2082)", "state_after": "a\u2080 a\u2081 a\u2082 : Type u\n\u22a2 (#Sequence\u2082 a\u2080 a\u2081 a\u2082 0) +\n      ((#Sequence\u2082 a\u2080 a\u2081 a\u2082 (0 + 1)) +\n        ((#Sequence\u2082 a\u2080 a\u2081 a\u2082 (0 + 1 + 1)) + sum fun i => #Sequence\u2082 a\u2080 a\u2081 a\u2082 (i + 1 + 1 + 1))) =\n    (#a\u2080) + (#a\u2081) + (#a\u2082)"}, {"tactic": "simp [add_assoc, Sequence\u2082]", "state_before": "a\u2080 a\u2081 a\u2082 : Type u\n\u22a2 (#Sequence\u2082 a\u2080 a\u2081 a\u2082 0) +\n      ((#Sequence\u2082 a\u2080 a\u2081 a\u2082 (0 + 1)) +\n        ((#Sequence\u2082 a\u2080 a\u2081 a\u2082 (0 + 1 + 1)) + sum fun i => #Sequence\u2082 a\u2080 a\u2081 a\u2082 (i + 1 + 1 + 1))) =\n    (#a\u2080) + (#a\u2081) + (#a\u2082)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Pi.lean", "full_name": "Filter.coprod\u1d62_neBot_iff", "start": [219, 1], "end": [220, 31], "traced_tactics": [{"tactic": "simp [coprod\u1d62_neBot_iff', *]", "state_before": "\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_1\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\n\u22a2 NeBot (Filter.coprod\u1d62 f) \u2194 \u2203 d, NeBot (f d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Instances/ENNReal.lean", "full_name": "ENNReal.tsum_iUnion_le_tsum", "start": [999, 1], "end": [1003, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "full_name": "ContinuousLinearEquiv.equivOfRightInverse_symm_apply", "start": [2419, 1], "end": [2422, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.measure_toMeasurable_add_inter_left", "start": [934, 1], "end": [942, 15], "traced_tactics": [{"tactic": "refine' (measure_inter_eq_of_measure_eq hs _ (subset_toMeasurable _ _) _).symm", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.140079\n\u03b3 : Type ?u.140082\n\u03b4 : Type ?u.140085\n\u03b9 : Type ?u.140088\nR : Type ?u.140091\nR' : Type ?u.140094\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nhs : MeasurableSet s\nht : \u2191\u2191(\u03bc + \u03bd) t \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (toMeasurable (\u03bc + \u03bd) t \u2229 s) = \u2191\u2191\u03bc (t \u2229 s)", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type ?u.140079\n\u03b3 : Type ?u.140082\n\u03b4 : Type ?u.140085\n\u03b9 : Type ?u.140088\nR : Type ?u.140091\nR' : Type ?u.140094\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nhs : MeasurableSet s\nht : \u2191\u2191(\u03bc + \u03bd) t \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc t = \u2191\u2191\u03bc (toMeasurable (\u03bc + \u03bd) t)\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type ?u.140079\n\u03b3 : Type ?u.140082\n\u03b4 : Type ?u.140085\n\u03b9 : Type ?u.140088\nR : Type ?u.140091\nR' : Type ?u.140094\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nhs : MeasurableSet s\nht : \u2191\u2191(\u03bc + \u03bd) t \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc t \u2260 \u22a4"}, {"tactic": "refine'\n  measure_eq_left_of_subset_of_measure_add_eq _ (subset_toMeasurable _ _)\n    (measure_toMeasurable t).symm", "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type ?u.140079\n\u03b3 : Type ?u.140082\n\u03b4 : Type ?u.140085\n\u03b9 : Type ?u.140088\nR : Type ?u.140091\nR' : Type ?u.140094\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nhs : MeasurableSet s\nht : \u2191\u2191(\u03bc + \u03bd) t \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc t = \u2191\u2191\u03bc (toMeasurable (\u03bc + \u03bd) t)", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type ?u.140079\n\u03b3 : Type ?u.140082\n\u03b4 : Type ?u.140085\n\u03b9 : Type ?u.140088\nR : Type ?u.140091\nR' : Type ?u.140094\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nhs : MeasurableSet s\nht : \u2191\u2191(\u03bc + \u03bd) t \u2260 \u22a4\n\u22a2 \u2191\u2191(\u03bc + \u03bd) (toMeasurable (\u03bc + \u03bd) t) \u2260 \u22a4"}, {"tactic": "rwa [measure_toMeasurable t]", "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type ?u.140079\n\u03b3 : Type ?u.140082\n\u03b4 : Type ?u.140085\n\u03b9 : Type ?u.140088\nR : Type ?u.140091\nR' : Type ?u.140094\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nhs : MeasurableSet s\nht : \u2191\u2191(\u03bc + \u03bd) t \u2260 \u22a4\n\u22a2 \u2191\u2191(\u03bc + \u03bd) (toMeasurable (\u03bc + \u03bd) t) \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "simp only [not_or, ENNReal.add_eq_top, Pi.add_apply, Ne.def, coe_add] at ht", "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type ?u.140079\n\u03b3 : Type ?u.140082\n\u03b4 : Type ?u.140085\n\u03b9 : Type ?u.140088\nR : Type ?u.140091\nR' : Type ?u.140094\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nhs : MeasurableSet s\nht : \u2191\u2191(\u03bc + \u03bd) t \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc t \u2260 \u22a4", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type ?u.140079\n\u03b3 : Type ?u.140082\n\u03b4 : Type ?u.140085\n\u03b9 : Type ?u.140088\nR : Type ?u.140091\nR' : Type ?u.140094\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nhs : MeasurableSet s\nht : \u00ac\u2191\u2191\u03bc t = \u22a4 \u2227 \u00ac\u2191\u2191\u03bd t = \u22a4\n\u22a2 \u2191\u2191\u03bc t \u2260 \u22a4"}, {"tactic": "exact ht.1", "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type ?u.140079\n\u03b3 : Type ?u.140082\n\u03b4 : Type ?u.140085\n\u03b9 : Type ?u.140088\nR : Type ?u.140091\nR' : Type ?u.140094\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d s t : Set \u03b1\nhs : MeasurableSet s\nht : \u00ac\u2191\u2191\u03bc t = \u22a4 \u2227 \u00ac\u2191\u2191\u03bd t = \u22a4\n\u22a2 \u2191\u2191\u03bc t \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Quotient.ind\u2082'", "start": [659, 11], "end": [662, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Matrix/Basic.lean", "full_name": "Matrix.one_dotProduct_one", "start": [851, 1], "end": [852, 34], "traced_tactics": [{"tactic": "simp [dotProduct, Fintype.card]", "state_before": "l : Type ?u.170404\nm : Type ?u.170407\nn : Type u_1\no : Type ?u.170413\nm' : o \u2192 Type ?u.170418\nn' : o \u2192 Type ?u.170423\nR : Type ?u.170426\nS : Type ?u.170429\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type ?u.170436\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : NonAssocSemiring \u03b1\n\u22a2 1 \u2b1d\u1d65 1 = \u2191(Fintype.card n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Group/Prod.lean", "full_name": "MeasureTheory.measure_lintegral_div_measure", "start": [325, 1], "end": [334, 78], "traced_tactics": [{"tactic": "set g := fun y => f y\u207b\u00b9 / \u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)", "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\n\u22a2 (\u2191\u2191\u03bc s * \u222b\u207b (y : G), f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s) \u2202\u03bd) = \u222b\u207b (x : G), f x \u2202\u03bc", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\n\u22a2 \u2191\u2191\u03bc s * lintegral \u03bd g = \u222b\u207b (x : G), f x \u2202\u03bc"}, {"tactic": "have hg : Measurable g :=\n  (hf.comp measurable_inv).div ((measurable_measure_mul_right \u03bd sm).comp measurable_inv)", "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\n\u22a2 \u2191\u2191\u03bc s * lintegral \u03bd g = \u222b\u207b (x : G), f x \u2202\u03bc", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\nhg : Measurable g\n\u22a2 \u2191\u2191\u03bc s * lintegral \u03bd g = \u222b\u207b (x : G), f x \u2202\u03bc"}, {"tactic": "simp_rw [measure_mul_lintegral_eq \u03bc \u03bd sm g hg, inv_inv]", "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\nhg : Measurable g\n\u22a2 \u2191\u2191\u03bc s * lintegral \u03bd g = \u222b\u207b (x : G), f x \u2202\u03bc", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\nhg : Measurable g\n\u22a2 (\u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * (f x / \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s)) \u2202\u03bc) = \u222b\u207b (x : G), f x \u2202\u03bc"}, {"tactic": "refine' lintegral_congr_ae _", "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\nhg : Measurable g\n\u22a2 (\u222b\u207b (x : G), \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * (f x / \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s)) \u2202\u03bc) = \u222b\u207b (x : G), f x \u2202\u03bc", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\nhg : Measurable g\n\u22a2 (fun x => \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * (f x / \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s))) =\u1da0[ae \u03bc] fun x => f x"}, {"tactic": "refine' (ae_measure_preimage_mul_right_lt_top_of_ne_zero \u03bc \u03bd sm h2s h3s).mono fun x hx => _", "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\nhg : Measurable g\n\u22a2 (fun x => \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * (f x / \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s))) =\u1da0[ae \u03bc] fun x => f x", "state_after": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\nhg : Measurable g\nx : G\nhx : \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4\n\u22a2 (fun x => \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * (f x / \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s))) x = (fun x => f x) x"}, {"tactic": "simp_rw [ENNReal.mul_div_cancel' (measure_mul_right_ne_zero \u03bd h2s _) hx.ne]", "state_before": "G : Type u_1\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : Group G\ninst\u271d\u2075 : MeasurableMul\u2082 G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : SigmaFinite \u03bc\ns : Set G\ninst\u271d\u00b2 : MeasurableInv G\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulLeftInvariant \u03bd\nsm : MeasurableSet s\nh2s : \u2191\u2191\u03bd s \u2260 0\nh3s : \u2191\u2191\u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\u22650\u221e\nhf : Measurable f\ng : G \u2192 \u211d\u22650\u221e := fun y => f y\u207b\u00b9 / \u2191\u2191\u03bd ((fun x => x * y\u207b\u00b9) \u207b\u00b9' s)\nhg : Measurable g\nx : G\nhx : \u2191\u2191\u03bd ((fun y => y * x) \u207b\u00b9' s) < \u22a4\n\u22a2 (fun x => \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s) * (f x / \u2191\u2191\u03bd ((fun z => z * x) \u207b\u00b9' s))) x = (fun x => f x) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean", "full_name": "InnerProductGeometry.inner_eq_zero_iff_angle_eq_pi_div_two", "start": [234, 1], "end": [235, 73], "traced_tactics": [{"tactic": "simp (config := { contextual := true }) [angle, or_imp]", "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\n\u22a2 angle x y = \u03c0 / 2 \u2194 inner x y = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/LinearIndependent.lean", "full_name": "surjective_of_linearIndependent_of_span", "start": [934, 1], "end": [952, 14], "traced_tactics": [{"tactic": "intro i", "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\n\u22a2 Surjective \u2191f", "state_after": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\n\u22a2 \u2203 a, \u2191f a = i"}, {"tactic": "let repr : (span R (range (v \u2218 f)) : Type _) \u2192 \u03b9' \u2192\u2080 R := (hv.comp f f.injective).repr", "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\n\u22a2 \u2203 a, \u2191f a = i", "state_after": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\n\u22a2 \u2203 a, \u2191f a = i"}, {"tactic": "let l := (repr \u27e8v i, hss (mem_range_self i)\u27e9).mapDomain f", "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\n\u22a2 \u2203 a, \u2191f a = i", "state_after": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\n\u22a2 \u2203 a, \u2191f a = i"}, {"tactic": "have h_total_l : Finsupp.total \u03b9 M R v l = v i := by\n  dsimp only []\n  rw [Finsupp.total_mapDomain]\n  rw [(hv.comp f f.injective).total_repr]", "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\n\u22a2 \u2203 a, \u2191f a = i", "state_after": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\n\u22a2 \u2203 a, \u2191f a = i"}, {"tactic": "have h_total_eq : (Finsupp.total \u03b9 M R v) l = (Finsupp.total \u03b9 M R v) (Finsupp.single i 1) := by\n  rw [h_total_l, Finsupp.total_single, one_smul]", "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\n\u22a2 \u2203 a, \u2191f a = i", "state_after": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\nh_total_eq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i 1)\n\u22a2 \u2203 a, \u2191f a = i"}, {"tactic": "have l_eq : l = _ := LinearMap.ker_eq_bot.1 hv h_total_eq", "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\nh_total_eq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i 1)\n\u22a2 \u2203 a, \u2191f a = i", "state_after": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\nh_total_eq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i 1)\nl_eq : l = Finsupp.single i 1\n\u22a2 \u2203 a, \u2191f a = i"}, {"tactic": "dsimp only [] at l_eq", "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\nh_total_eq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i 1)\nl_eq : l = Finsupp.single i 1\n\u22a2 \u2203 a, \u2191f a = i", "state_after": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\nh_total_eq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i 1)\nl_eq :\n  Finsupp.mapDomain (\u2191f)\n      (\u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\n        { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) }) =\n    Finsupp.single i 1\n\u22a2 \u2203 a, \u2191f a = i"}, {"tactic": "rw [\u2190 Finsupp.embDomain_eq_mapDomain] at l_eq", "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\nh_total_eq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i 1)\nl_eq :\n  Finsupp.mapDomain (\u2191f)\n      (\u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\n        { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) }) =\n    Finsupp.single i 1\n\u22a2 \u2203 a, \u2191f a = i", "state_after": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\nh_total_eq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i 1)\nl_eq :\n  Finsupp.embDomain f\n      (\u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\n        { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) }) =\n    Finsupp.single i 1\n\u22a2 \u2203 a, \u2191f a = i"}, {"tactic": "rcases Finsupp.single_of_embDomain_single (repr \u27e8v i, _\u27e9) f i (1 : R) zero_ne_one.symm l_eq with\n  \u27e8i', hi'\u27e9", "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\nh_total_eq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i 1)\nl_eq :\n  Finsupp.embDomain f\n      (\u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\n        { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) }) =\n    Finsupp.single i 1\n\u22a2 \u2203 a, \u2191f a = i", "state_after": "case intro\n\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\nh_total_eq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i 1)\nl_eq :\n  Finsupp.embDomain f\n      (\u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\n        { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) }) =\n    Finsupp.single i 1\ni' : \u03b9'\nhi' : repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) } = Finsupp.single i' 1 \u2227 \u2191f i' = i\n\u22a2 \u2203 a, \u2191f a = i"}, {"tactic": "use i'", "state_before": "case intro\n\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\nh_total_eq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i 1)\nl_eq :\n  Finsupp.embDomain f\n      (\u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\n        { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) }) =\n    Finsupp.single i 1\ni' : \u03b9'\nhi' : repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) } = Finsupp.single i' 1 \u2227 \u2191f i' = i\n\u22a2 \u2203 a, \u2191f a = i", "state_after": "case intro\n\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\nh_total_eq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i 1)\nl_eq :\n  Finsupp.embDomain f\n      (\u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\n        { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) }) =\n    Finsupp.single i 1\ni' : \u03b9'\nhi' : repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) } = Finsupp.single i' 1 \u2227 \u2191f i' = i\n\u22a2 \u2191f i' = i"}, {"tactic": "exact hi'.2", "state_before": "case intro\n\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\nh_total_eq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i 1)\nl_eq :\n  Finsupp.embDomain f\n      (\u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\n        { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) }) =\n    Finsupp.single i 1\ni' : \u03b9'\nhi' : repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) } = Finsupp.single i' 1 \u2227 \u2191f i' = i\n\u22a2 \u2191f i' = i", "state_after": "no goals"}, {"tactic": "dsimp only []", "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\n\u22a2 \u2191(Finsupp.total \u03b9 M R v) l = v i", "state_after": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\n\u22a2 \u2191(Finsupp.total \u03b9 M R v)\n      (Finsupp.mapDomain (\u2191f)\n        (\u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\n          { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })) =\n    v i"}, {"tactic": "rw [Finsupp.total_mapDomain]", "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\n\u22a2 \u2191(Finsupp.total \u03b9 M R v)\n      (Finsupp.mapDomain (\u2191f)\n        (\u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\n          { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })) =\n    v i", "state_after": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\n\u22a2 \u2191(Finsupp.total \u03b9' M R (v \u2218 \u2191f))\n      (\u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\n        { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) }) =\n    v i"}, {"tactic": "rw [(hv.comp f f.injective).total_repr]", "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\n\u22a2 \u2191(Finsupp.total \u03b9' M R (v \u2218 \u2191f))\n      (\u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\n        { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) }) =\n    v i", "state_after": "no goals"}, {"tactic": "rw [h_total_l, Finsupp.total_single, one_smul]", "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_3\nR : Type u_1\nK : Type ?u.522813\nM : Type u_2\nM' : Type ?u.522819\nM'' : Type ?u.522822\nV : Type u\nV' : Type ?u.522827\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\n\u22a2 \u2191(Finsupp.total \u03b9 M R v) l = \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.le_map_iff", "start": [2560, 1], "end": [2561, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/IntermediateField.lean", "full_name": "IntermediateField.coe_val", "start": [498, 1], "end": [499, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/BilinearForm.lean", "full_name": "BilinForm.nondegenerate_toBilin'_of_det_ne_zero'", "start": [603, 1], "end": [605, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Basic.lean", "full_name": "div_left_inj", "start": [796, 1], "end": [798, 23], "traced_tactics": [{"tactic": "rw [div_eq_mul_inv, div_eq_mul_inv]", "state_before": "\u03b1 : Type ?u.60285\n\u03b2 : Type ?u.60288\nG : Type u_1\ninst\u271d : Group G\na b c d : G\n\u22a2 b / a = c / a \u2194 b = c", "state_after": "\u03b1 : Type ?u.60285\n\u03b2 : Type ?u.60288\nG : Type u_1\ninst\u271d : Group G\na b c d : G\n\u22a2 b * a\u207b\u00b9 = c * a\u207b\u00b9 \u2194 b = c"}, {"tactic": "exact mul_left_inj _", "state_before": "\u03b1 : Type ?u.60285\n\u03b2 : Type ?u.60288\nG : Type u_1\ninst\u271d : Group G\na b c d : G\n\u22a2 b * a\u207b\u00b9 = c * a\u207b\u00b9 \u2194 b = c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Monotone/Basic.lean", "full_name": "List.foldl_strictMono", "start": [729, 1], "end": [731, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/LiminfLimsup.lean", "full_name": "Filter.liminf_eq_iSup_iInf", "start": [754, 1], "end": [755, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Lipschitz.lean", "full_name": "LipschitzWith.iterate", "start": [272, 11], "end": [274, 75], "traced_tactics": [{"tactic": "simpa only [pow_zero] using LipschitzWith.id", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b1\nhf : LipschitzWith K f\n\u22a2 LipschitzWith (K ^ 0) (f^[0])", "state_after": "no goals"}, {"tactic": "rw [pow_succ']", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b1\nhf : LipschitzWith K f\nn : \u2115\n\u22a2 LipschitzWith (K ^ (n + 1)) (f^[n + 1])", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b1\nhf : LipschitzWith K f\nn : \u2115\n\u22a2 LipschitzWith (K ^ n * K) (f^[n + 1])"}, {"tactic": "exact (LipschitzWith.iterate hf n).comp hf", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b1\nhf : LipschitzWith K f\nn : \u2115\n\u22a2 LipschitzWith (K ^ n * K) (f^[n + 1])", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "full_name": "MeasureTheory.Measure.snd_univ", "start": [846, 1], "end": [846, 94], "traced_tactics": [{"tactic": "rw [snd_apply MeasurableSet.univ, preimage_univ]", "state_before": "\u03b1 : Type u_2\n\u03b1' : Type ?u.5373035\n\u03b2 : Type u_1\n\u03b2' : Type ?u.5373041\n\u03b3 : Type ?u.5373044\nE : Type ?u.5373047\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SigmaFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\n\u22a2 \u2191\u2191(snd \u03c1) univ = \u2191\u2191\u03c1 univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/DList/Basic.lean", "full_name": "Std.DList.ofList_toList", "start": [60, 1], "end": [64, 16], "traced_tactics": [{"tactic": "cases' l with app inv", "state_before": "\u03b1 : Type u_1\nl : DList \u03b1\n\u22a2 ofList (toList l) = l", "state_after": "case mk\n\u03b1 : Type u_1\napp : List \u03b1 \u2192 List \u03b1\ninv : \u2200 (l : List \u03b1), app l = app [] ++ l\n\u22a2 ofList (toList { apply := app, invariant := inv }) = { apply := app, invariant := inv }"}, {"tactic": "simp only [ofList, toList, mk.injEq]", "state_before": "case mk\n\u03b1 : Type u_1\napp : List \u03b1 \u2192 List \u03b1\ninv : \u2200 (l : List \u03b1), app l = app [] ++ l\n\u22a2 ofList (toList { apply := app, invariant := inv }) = { apply := app, invariant := inv }", "state_after": "case mk\n\u03b1 : Type u_1\napp : List \u03b1 \u2192 List \u03b1\ninv : \u2200 (l : List \u03b1), app l = app [] ++ l\n\u22a2 (fun x => app [] ++ x) = app"}, {"tactic": "funext x", "state_before": "case mk\n\u03b1 : Type u_1\napp : List \u03b1 \u2192 List \u03b1\ninv : \u2200 (l : List \u03b1), app l = app [] ++ l\n\u22a2 (fun x => app [] ++ x) = app", "state_after": "case mk.h\n\u03b1 : Type u_1\napp : List \u03b1 \u2192 List \u03b1\ninv : \u2200 (l : List \u03b1), app l = app [] ++ l\nx : List \u03b1\n\u22a2 app [] ++ x = app x"}, {"tactic": "rw [(inv x)]", "state_before": "case mk.h\n\u03b1 : Type u_1\napp : List \u03b1 \u2192 List \u03b1\ninv : \u2200 (l : List \u03b1), app l = app [] ++ l\nx : List \u03b1\n\u22a2 app [] ++ x = app x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.erase_diff_erase_sublist_of_sublist", "start": [3796, 1], "end": [3806, 70], "traced_tactics": [{"tactic": "simp only [heq, erase_cons_head, diff_cons]", "state_before": "\u03b9 : Type ?u.428945\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : b :: l\u2081 <+ l\u2082\nheq : b = a\n\u22a2 List.diff (List.erase l\u2082 a) (List.erase (b :: l\u2081) a) <+ List.diff l\u2082 (b :: l\u2081)", "state_after": "\u03b9 : Type ?u.428945\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : b :: l\u2081 <+ l\u2082\nheq : b = a\n\u22a2 List.diff (List.erase l\u2082 a) l\u2081 <+ List.diff (List.erase l\u2082 a) l\u2081"}, {"tactic": "rfl", "state_before": "\u03b9 : Type ?u.428945\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : b :: l\u2081 <+ l\u2082\nheq : b = a\n\u22a2 List.diff (List.erase l\u2082 a) l\u2081 <+ List.diff (List.erase l\u2082 a) l\u2081", "state_after": "no goals"}, {"tactic": "simp only [erase_cons_head b l\u2081, erase_cons_tail l\u2081 heq,\n  diff_cons ((List.erase l\u2082 a)) (List.erase l\u2081 a) b, diff_cons l\u2082 l\u2081 b, erase_comm a b l\u2082]", "state_before": "\u03b9 : Type ?u.428945\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : b :: l\u2081 <+ l\u2082\nheq : \u00acb = a\n\u22a2 List.diff (List.erase l\u2082 a) (List.erase (b :: l\u2081) a) <+ List.diff l\u2082 (b :: l\u2081)", "state_after": "\u03b9 : Type ?u.428945\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : b :: l\u2081 <+ l\u2082\nheq : \u00acb = a\n\u22a2 List.diff (List.erase (List.erase l\u2082 b) a) (List.erase l\u2081 a) <+ List.diff (List.erase l\u2082 b) l\u2081"}, {"tactic": "have h' := h.erase b", "state_before": "\u03b9 : Type ?u.428945\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : b :: l\u2081 <+ l\u2082\nheq : \u00acb = a\n\u22a2 List.diff (List.erase (List.erase l\u2082 b) a) (List.erase l\u2081 a) <+ List.diff (List.erase l\u2082 b) l\u2081", "state_after": "\u03b9 : Type ?u.428945\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : b :: l\u2081 <+ l\u2082\nheq : \u00acb = a\nh' : List.erase (b :: l\u2081) b <+ List.erase l\u2082 b\n\u22a2 List.diff (List.erase (List.erase l\u2082 b) a) (List.erase l\u2081 a) <+ List.diff (List.erase l\u2082 b) l\u2081"}, {"tactic": "rw [erase_cons_head] at h'", "state_before": "\u03b9 : Type ?u.428945\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : b :: l\u2081 <+ l\u2082\nheq : \u00acb = a\nh' : List.erase (b :: l\u2081) b <+ List.erase l\u2082 b\n\u22a2 List.diff (List.erase (List.erase l\u2082 b) a) (List.erase l\u2081 a) <+ List.diff (List.erase l\u2082 b) l\u2081", "state_after": "\u03b9 : Type ?u.428945\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : b :: l\u2081 <+ l\u2082\nheq : \u00acb = a\nh' : l\u2081 <+ List.erase l\u2082 b\n\u22a2 List.diff (List.erase (List.erase l\u2082 b) a) (List.erase l\u2081 a) <+ List.diff (List.erase l\u2082 b) l\u2081"}, {"tactic": "exact @erase_diff_erase_sublist_of_sublist _ l\u2081 (l\u2082.erase b) h'", "state_before": "\u03b9 : Type ?u.428945\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : b :: l\u2081 <+ l\u2082\nheq : \u00acb = a\nh' : l\u2081 <+ List.erase l\u2082 b\n\u22a2 List.diff (List.erase (List.erase l\u2082 b) a) (List.erase l\u2081 a) <+ List.diff (List.erase l\u2082 b) l\u2081", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Segment.lean", "full_name": "segment_eq_Icc'", "start": [517, 1], "end": [520, 71], "traced_tactics": [{"tactic": "cases' le_total x y with h h", "state_before": "\ud835\udd5c : Type u_1\nE : Type ?u.289591\nF : Type ?u.289594\nG : Type ?u.289597\n\u03b9 : Type ?u.289600\n\u03c0 : \u03b9 \u2192 Type ?u.289605\ninst\u271d : LinearOrderedField \ud835\udd5c\nx\u271d y\u271d z x y : \ud835\udd5c\n\u22a2 [x-[\ud835\udd5c]y] = Icc (min x y) (max x y)", "state_after": "case inl\n\ud835\udd5c : Type u_1\nE : Type ?u.289591\nF : Type ?u.289594\nG : Type ?u.289597\n\u03b9 : Type ?u.289600\n\u03c0 : \u03b9 \u2192 Type ?u.289605\ninst\u271d : LinearOrderedField \ud835\udd5c\nx\u271d y\u271d z x y : \ud835\udd5c\nh : x \u2264 y\n\u22a2 [x-[\ud835\udd5c]y] = Icc (min x y) (max x y)\n\ncase inr\n\ud835\udd5c : Type u_1\nE : Type ?u.289591\nF : Type ?u.289594\nG : Type ?u.289597\n\u03b9 : Type ?u.289600\n\u03c0 : \u03b9 \u2192 Type ?u.289605\ninst\u271d : LinearOrderedField \ud835\udd5c\nx\u271d y\u271d z x y : \ud835\udd5c\nh : y \u2264 x\n\u22a2 [x-[\ud835\udd5c]y] = Icc (min x y) (max x y)"}, {"tactic": "rw [segment_eq_Icc h, max_eq_right h, min_eq_left h]", "state_before": "case inl\n\ud835\udd5c : Type u_1\nE : Type ?u.289591\nF : Type ?u.289594\nG : Type ?u.289597\n\u03b9 : Type ?u.289600\n\u03c0 : \u03b9 \u2192 Type ?u.289605\ninst\u271d : LinearOrderedField \ud835\udd5c\nx\u271d y\u271d z x y : \ud835\udd5c\nh : x \u2264 y\n\u22a2 [x-[\ud835\udd5c]y] = Icc (min x y) (max x y)", "state_after": "no goals"}, {"tactic": "rw [segment_symm, segment_eq_Icc h, max_eq_left h, min_eq_right h]", "state_before": "case inr\n\ud835\udd5c : Type u_1\nE : Type ?u.289591\nF : Type ?u.289594\nG : Type ?u.289597\n\u03b9 : Type ?u.289600\n\u03c0 : \u03b9 \u2192 Type ?u.289605\ninst\u271d : LinearOrderedField \ud835\udd5c\nx\u271d y\u271d z x y : \ud835\udd5c\nh : y \u2264 x\n\u22a2 [x-[\ud835\udd5c]y] = Icc (min x y) (max x y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/QuotientOperations.lean", "full_name": "Ideal.comp_quotientMap_eq_of_comp_eq", "start": [422, 1], "end": [432, 31], "traced_tactics": [{"tactic": "refine RingHom.ext fun a => ?_", "state_before": "R : Type u\nS : Type v\nF : Type w\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\nR\u2081 : Type ?u.1497719\nR\u2082 : Type ?u.1497722\nA : Type ?u.1497725\nB : Type ?u.1497728\ninst\u271d\u2078 : CommSemiring R\u2081\ninst\u271d\u2077 : CommSemiring R\u2082\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R\u2081 A\ninst\u271d\u00b3 : Algebra R\u2082 A\ninst\u271d\u00b2 : Algebra R\u2081 B\nR' : Type u_1\nS' : Type u_2\ninst\u271d\u00b9 : CommRing R'\ninst\u271d : CommRing S'\nf : R \u2192+* S\nf' : R' \u2192+* S'\ng : R \u2192+* R'\ng' : S \u2192+* S'\nhfg : RingHom.comp f' g = RingHom.comp g' f\nI : Ideal S'\n\u22a2 let leq := (_ : comap f (comap g' I) \u2264 comap g (comap f' I));\n  RingHom.comp (quotientMap I g' (_ : comap g' I \u2264 comap g' I))\n      (quotientMap (comap g' I) f (_ : comap f (comap g' I) \u2264 comap f (comap g' I))) =\n    RingHom.comp (quotientMap I f' (_ : comap f' I \u2264 comap f' I)) (quotientMap (comap f' I) g leq)", "state_after": "R : Type u\nS : Type v\nF : Type w\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\nR\u2081 : Type ?u.1497719\nR\u2082 : Type ?u.1497722\nA : Type ?u.1497725\nB : Type ?u.1497728\ninst\u271d\u2078 : CommSemiring R\u2081\ninst\u271d\u2077 : CommSemiring R\u2082\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R\u2081 A\ninst\u271d\u00b3 : Algebra R\u2082 A\ninst\u271d\u00b2 : Algebra R\u2081 B\nR' : Type u_1\nS' : Type u_2\ninst\u271d\u00b9 : CommRing R'\ninst\u271d : CommRing S'\nf : R \u2192+* S\nf' : R' \u2192+* S'\ng : R \u2192+* R'\ng' : S \u2192+* S'\nhfg : RingHom.comp f' g = RingHom.comp g' f\nI : Ideal S'\na : R \u29f8 comap f (comap g' I)\n\u22a2 \u2191(RingHom.comp (quotientMap I g' (_ : comap g' I \u2264 comap g' I))\n          (quotientMap (comap g' I) f (_ : comap f (comap g' I) \u2264 comap f (comap g' I))))\n      a =\n    \u2191(RingHom.comp (quotientMap I f' (_ : comap f' I \u2264 comap f' I))\n          (quotientMap (comap f' I) g (_ : comap f (comap g' I) \u2264 comap g (comap f' I))))\n      a"}, {"tactic": "obtain \u27e8r, rfl\u27e9 := Quotient.mk_surjective a", "state_before": "R : Type u\nS : Type v\nF : Type w\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\nR\u2081 : Type ?u.1497719\nR\u2082 : Type ?u.1497722\nA : Type ?u.1497725\nB : Type ?u.1497728\ninst\u271d\u2078 : CommSemiring R\u2081\ninst\u271d\u2077 : CommSemiring R\u2082\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R\u2081 A\ninst\u271d\u00b3 : Algebra R\u2082 A\ninst\u271d\u00b2 : Algebra R\u2081 B\nR' : Type u_1\nS' : Type u_2\ninst\u271d\u00b9 : CommRing R'\ninst\u271d : CommRing S'\nf : R \u2192+* S\nf' : R' \u2192+* S'\ng : R \u2192+* R'\ng' : S \u2192+* S'\nhfg : RingHom.comp f' g = RingHom.comp g' f\nI : Ideal S'\na : R \u29f8 comap f (comap g' I)\n\u22a2 \u2191(RingHom.comp (quotientMap I g' (_ : comap g' I \u2264 comap g' I))\n          (quotientMap (comap g' I) f (_ : comap f (comap g' I) \u2264 comap f (comap g' I))))\n      a =\n    \u2191(RingHom.comp (quotientMap I f' (_ : comap f' I \u2264 comap f' I))\n          (quotientMap (comap f' I) g (_ : comap f (comap g' I) \u2264 comap g (comap f' I))))\n      a", "state_after": "case intro\nR : Type u\nS : Type v\nF : Type w\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\nR\u2081 : Type ?u.1497719\nR\u2082 : Type ?u.1497722\nA : Type ?u.1497725\nB : Type ?u.1497728\ninst\u271d\u2078 : CommSemiring R\u2081\ninst\u271d\u2077 : CommSemiring R\u2082\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R\u2081 A\ninst\u271d\u00b3 : Algebra R\u2082 A\ninst\u271d\u00b2 : Algebra R\u2081 B\nR' : Type u_1\nS' : Type u_2\ninst\u271d\u00b9 : CommRing R'\ninst\u271d : CommRing S'\nf : R \u2192+* S\nf' : R' \u2192+* S'\ng : R \u2192+* R'\ng' : S \u2192+* S'\nhfg : RingHom.comp f' g = RingHom.comp g' f\nI : Ideal S'\nr : R\n\u22a2 \u2191(RingHom.comp (quotientMap I g' (_ : comap g' I \u2264 comap g' I))\n          (quotientMap (comap g' I) f (_ : comap f (comap g' I) \u2264 comap f (comap g' I))))\n      (\u2191(Quotient.mk (comap f (comap g' I))) r) =\n    \u2191(RingHom.comp (quotientMap I f' (_ : comap f' I \u2264 comap f' I))\n          (quotientMap (comap f' I) g (_ : comap f (comap g' I) \u2264 comap g (comap f' I))))\n      (\u2191(Quotient.mk (comap f (comap g' I))) r)"}, {"tactic": "simp only [RingHom.comp_apply, quotientMap_mk]", "state_before": "case intro\nR : Type u\nS : Type v\nF : Type w\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\nR\u2081 : Type ?u.1497719\nR\u2082 : Type ?u.1497722\nA : Type ?u.1497725\nB : Type ?u.1497728\ninst\u271d\u2078 : CommSemiring R\u2081\ninst\u271d\u2077 : CommSemiring R\u2082\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R\u2081 A\ninst\u271d\u00b3 : Algebra R\u2082 A\ninst\u271d\u00b2 : Algebra R\u2081 B\nR' : Type u_1\nS' : Type u_2\ninst\u271d\u00b9 : CommRing R'\ninst\u271d : CommRing S'\nf : R \u2192+* S\nf' : R' \u2192+* S'\ng : R \u2192+* R'\ng' : S \u2192+* S'\nhfg : RingHom.comp f' g = RingHom.comp g' f\nI : Ideal S'\nr : R\n\u22a2 \u2191(RingHom.comp (quotientMap I g' (_ : comap g' I \u2264 comap g' I))\n          (quotientMap (comap g' I) f (_ : comap f (comap g' I) \u2264 comap f (comap g' I))))\n      (\u2191(Quotient.mk (comap f (comap g' I))) r) =\n    \u2191(RingHom.comp (quotientMap I f' (_ : comap f' I \u2264 comap f' I))\n          (quotientMap (comap f' I) g (_ : comap f (comap g' I) \u2264 comap g (comap f' I))))\n      (\u2191(Quotient.mk (comap f (comap g' I))) r)", "state_after": "case intro\nR : Type u\nS : Type v\nF : Type w\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\nR\u2081 : Type ?u.1497719\nR\u2082 : Type ?u.1497722\nA : Type ?u.1497725\nB : Type ?u.1497728\ninst\u271d\u2078 : CommSemiring R\u2081\ninst\u271d\u2077 : CommSemiring R\u2082\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R\u2081 A\ninst\u271d\u00b3 : Algebra R\u2082 A\ninst\u271d\u00b2 : Algebra R\u2081 B\nR' : Type u_1\nS' : Type u_2\ninst\u271d\u00b9 : CommRing R'\ninst\u271d : CommRing S'\nf : R \u2192+* S\nf' : R' \u2192+* S'\ng : R \u2192+* R'\ng' : S \u2192+* S'\nhfg : RingHom.comp f' g = RingHom.comp g' f\nI : Ideal S'\nr : R\n\u22a2 \u2191(Quotient.mk I) (\u2191g' (\u2191f r)) = \u2191(Quotient.mk I) (\u2191f' (\u2191g r))"}, {"tactic": "exact congr_arg (Ideal.Quotient.mk I) (_root_.trans (g'.comp_apply f r).symm\n  (hfg \u25b8 f'.comp_apply g r))", "state_before": "case intro\nR : Type u\nS : Type v\nF : Type w\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\nR\u2081 : Type ?u.1497719\nR\u2082 : Type ?u.1497722\nA : Type ?u.1497725\nB : Type ?u.1497728\ninst\u271d\u2078 : CommSemiring R\u2081\ninst\u271d\u2077 : CommSemiring R\u2082\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra R\u2081 A\ninst\u271d\u00b3 : Algebra R\u2082 A\ninst\u271d\u00b2 : Algebra R\u2081 B\nR' : Type u_1\nS' : Type u_2\ninst\u271d\u00b9 : CommRing R'\ninst\u271d : CommRing S'\nf : R \u2192+* S\nf' : R' \u2192+* S'\ng : R \u2192+* R'\ng' : S \u2192+* S'\nhfg : RingHom.comp f' g = RingHom.comp g' f\nI : Ideal S'\nr : R\n\u22a2 \u2191(Quotient.mk I) (\u2191g' (\u2191f r)) = \u2191(Quotient.mk I) (\u2191f' (\u2191g r))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean", "full_name": "aeSeq.aeSeq_eq_mk_of_mem_aeSeqSet", "start": [62, 1], "end": [64, 33], "traced_tactics": [{"tactic": "simp only [aeSeq, hx, if_true]", "state_before": "\u03b9 : Sort u_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.207057\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\n\u03bc : MeasureTheory.Measure \u03b1\np : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop\nhf : \u2200 (i : \u03b9), AEMeasurable (f i)\nx : \u03b1\nhx : x \u2208 aeSeqSet hf p\ni : \u03b9\n\u22a2 aeSeq hf p i x = AEMeasurable.mk (f i) (_ : AEMeasurable (f i)) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "full_name": "ContinuousLinearEquiv.symm_toHomeomorph", "start": [1955, 1], "end": [1956, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Closure.lean", "full_name": "ClosureOperator.ext", "start": [96, 1], "end": [98, 10], "traced_tactics": [{"tactic": "congr", "state_before": "\u03b1 : Type u_1\n\u03b9 : Sort ?u.1113\n\u03ba : \u03b9 \u2192 Sort ?u.1118\ninst\u271d\u00b9 inst\u271d : PartialOrder \u03b1\nc : ClosureOperator \u03b1\nc\u2081 : \u03b1 \u2192 \u03b1\nmonotone'\u271d\u00b9 : Monotone c\u2081\nle_closure'\u271d\u00b9 : \u2200 (x : \u03b1), x \u2264 \u2191{ toFun := c\u2081, monotone' := monotone'\u271d\u00b9 } x\nidempotent'\u271d\u00b9 :\n  \u2200 (x : \u03b1),\n    \u2191{ toFun := c\u2081, monotone' := monotone'\u271d\u00b9 } (\u2191{ toFun := c\u2081, monotone' := monotone'\u271d\u00b9 } x) =\n      \u2191{ toFun := c\u2081, monotone' := monotone'\u271d\u00b9 } x\nc\u2082 : \u03b1 \u2192 \u03b1\nmonotone'\u271d : Monotone c\u2082\nle_closure'\u271d : \u2200 (x : \u03b1), x \u2264 \u2191{ toFun := c\u2082, monotone' := monotone'\u271d } x\nidempotent'\u271d :\n  \u2200 (x : \u03b1),\n    \u2191{ toFun := c\u2082, monotone' := monotone'\u271d } (\u2191{ toFun := c\u2082, monotone' := monotone'\u271d } x) =\n      \u2191{ toFun := c\u2082, monotone' := monotone'\u271d } x\nh :\n  \u2191{ toOrderHom := { toFun := c\u2081, monotone' := monotone'\u271d\u00b9 }, le_closure' := le_closure'\u271d\u00b9,\n          idempotent' := idempotent'\u271d\u00b9 }.toOrderHom =\n    \u2191{ toOrderHom := { toFun := c\u2082, monotone' := monotone'\u271d }, le_closure' := le_closure'\u271d,\n          idempotent' := idempotent'\u271d }.toOrderHom\n\u22a2 { toOrderHom := { toFun := c\u2081, monotone' := monotone'\u271d\u00b9 }, le_closure' := le_closure'\u271d\u00b9,\n      idempotent' := idempotent'\u271d\u00b9 } =\n    { toOrderHom := { toFun := c\u2082, monotone' := monotone'\u271d }, le_closure' := le_closure'\u271d, idempotent' := idempotent'\u271d }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "full_name": "Real.sin_two_pi_sub", "start": [263, 1], "end": [264, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Hom/NonUnitalAlg.lean", "full_name": "NonUnitalAlgHom.to_distribMulActionHom_injective", "start": [207, 1], "end": [210, 42], "traced_tactics": [{"tactic": "ext a", "state_before": "R : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nf g : A \u2192\u2099\u2090[R] B\nh : \u2191f = \u2191g\n\u22a2 f = g", "state_after": "case h\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nf g : A \u2192\u2099\u2090[R] B\nh : \u2191f = \u2191g\na : A\n\u22a2 \u2191f a = \u2191g a"}, {"tactic": "exact DistribMulActionHom.congr_fun h a", "state_before": "case h\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nf g : A \u2192\u2099\u2090[R] B\nh : \u2191f = \u2191g\na : A\n\u22a2 \u2191f a = \u2191g a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Sites/Sheafification.lean", "full_name": "CategoryTheory.GrothendieckTopology.Plus.toPlus_eq_mk", "start": [220, 1], "end": [229, 6], "traced_tactics": [{"tactic": "dsimp [mk, toPlus]", "state_before": "C : Type u\ninst\u271d\u2075 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj X.op)\n\u22a2 (forget D).map ((toPlus J P).app X.op) x = mk (Meq.mk \u22a4 x)", "state_after": "C : Type u\ninst\u271d\u2075 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj X.op)\n\u22a2 (forget D).map (Cover.toMultiequalizer \u22a4 P \u226b colimit.\u03b9 (diagram J P X) \u22a4.op) x =\n    (forget D).map (colimit.\u03b9 (diagram J P X) \u22a4.op) (\u2191(Meq.equiv P \u22a4).symm (Meq.mk \u22a4 x))"}, {"tactic": "delta Cover.toMultiequalizer", "state_before": "C : Type u\ninst\u271d\u2075 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj X.op)\n\u22a2 (forget D).map (Cover.toMultiequalizer \u22a4 P \u226b colimit.\u03b9 (diagram J P X) \u22a4.op) x =\n    (forget D).map (colimit.\u03b9 (diagram J P X) \u22a4.op) (\u2191(Meq.equiv P \u22a4).symm (Meq.mk \u22a4 x))", "state_after": "C : Type u\ninst\u271d\u2075 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj X.op)\n\u22a2 (forget D).map\n      (Multiequalizer.lift (Cover.index \u22a4 P) (P.obj X.op) (fun I => P.map I.f.op)\n          (_ :\n            \u2200 (I : (Cover.index \u22a4 P).R),\n              (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n                  MulticospanIndex.fst (Cover.index \u22a4 P) I =\n                (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n                  MulticospanIndex.snd (Cover.index \u22a4 P) I) \u226b\n        colimit.\u03b9 (diagram J P X) \u22a4.op)\n      x =\n    (forget D).map (colimit.\u03b9 (diagram J P X) \u22a4.op) (\u2191(Meq.equiv P \u22a4).symm (Meq.mk \u22a4 x))"}, {"tactic": "simp only [comp_apply]", "state_before": "C : Type u\ninst\u271d\u2075 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj X.op)\n\u22a2 (forget D).map\n      (Multiequalizer.lift (Cover.index \u22a4 P) (P.obj X.op) (fun I => P.map I.f.op)\n          (_ :\n            \u2200 (I : (Cover.index \u22a4 P).R),\n              (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n                  MulticospanIndex.fst (Cover.index \u22a4 P) I =\n                (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n                  MulticospanIndex.snd (Cover.index \u22a4 P) I) \u226b\n        colimit.\u03b9 (diagram J P X) \u22a4.op)\n      x =\n    (forget D).map (colimit.\u03b9 (diagram J P X) \u22a4.op) (\u2191(Meq.equiv P \u22a4).symm (Meq.mk \u22a4 x))", "state_after": "C : Type u\ninst\u271d\u2075 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj X.op)\n\u22a2 (forget D).map (colimit.\u03b9 (diagram J P X) \u22a4.op)\n      ((forget D).map\n        (Multiequalizer.lift (Cover.index \u22a4 P) (P.obj X.op) (fun I => P.map I.f.op)\n          (_ :\n            \u2200 (I : (Cover.index \u22a4 P).R),\n              (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n                  MulticospanIndex.fst (Cover.index \u22a4 P) I =\n                (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n                  MulticospanIndex.snd (Cover.index \u22a4 P) I))\n        x) =\n    (forget D).map (colimit.\u03b9 (diagram J P X) \u22a4.op) (\u2191(Meq.equiv P \u22a4).symm (Meq.mk \u22a4 x))"}, {"tactic": "apply congr_arg", "state_before": "C : Type u\ninst\u271d\u2075 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj X.op)\n\u22a2 (forget D).map (colimit.\u03b9 (diagram J P X) \u22a4.op)\n      ((forget D).map\n        (Multiequalizer.lift (Cover.index \u22a4 P) (P.obj X.op) (fun I => P.map I.f.op)\n          (_ :\n            \u2200 (I : (Cover.index \u22a4 P).R),\n              (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n                  MulticospanIndex.fst (Cover.index \u22a4 P) I =\n                (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n                  MulticospanIndex.snd (Cover.index \u22a4 P) I))\n        x) =\n    (forget D).map (colimit.\u03b9 (diagram J P X) \u22a4.op) (\u2191(Meq.equiv P \u22a4).symm (Meq.mk \u22a4 x))", "state_after": "case h\nC : Type u\ninst\u271d\u2075 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj X.op)\n\u22a2 (forget D).map\n      (Multiequalizer.lift (Cover.index \u22a4 P) (P.obj X.op) (fun I => P.map I.f.op)\n        (_ :\n          \u2200 (I : (Cover.index \u22a4 P).R),\n            (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n                MulticospanIndex.fst (Cover.index \u22a4 P) I =\n              (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n                MulticospanIndex.snd (Cover.index \u22a4 P) I))\n      x =\n    \u2191(Meq.equiv P \u22a4).symm (Meq.mk \u22a4 x)"}, {"tactic": "apply (Meq.equiv P \u22a4).injective", "state_before": "case h\nC : Type u\ninst\u271d\u2075 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj X.op)\n\u22a2 (forget D).map\n      (Multiequalizer.lift (Cover.index \u22a4 P) (P.obj X.op) (fun I => P.map I.f.op)\n        (_ :\n          \u2200 (I : (Cover.index \u22a4 P).R),\n            (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n                MulticospanIndex.fst (Cover.index \u22a4 P) I =\n              (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n                MulticospanIndex.snd (Cover.index \u22a4 P) I))\n      x =\n    \u2191(Meq.equiv P \u22a4).symm (Meq.mk \u22a4 x)", "state_after": "case h.a\nC : Type u\ninst\u271d\u2075 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj X.op)\n\u22a2 \u2191(Meq.equiv P \u22a4)\n      ((forget D).map\n        (Multiequalizer.lift (Cover.index \u22a4 P) (P.obj X.op) (fun I => P.map I.f.op)\n          (_ :\n            \u2200 (I : (Cover.index \u22a4 P).R),\n              (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n                  MulticospanIndex.fst (Cover.index \u22a4 P) I =\n                (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n                  MulticospanIndex.snd (Cover.index \u22a4 P) I))\n        x) =\n    \u2191(Meq.equiv P \u22a4) (\u2191(Meq.equiv P \u22a4).symm (Meq.mk \u22a4 x))"}, {"tactic": "ext i", "state_before": "case h.a\nC : Type u\ninst\u271d\u2075 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj X.op)\n\u22a2 \u2191(Meq.equiv P \u22a4)\n      ((forget D).map\n        (Multiequalizer.lift (Cover.index \u22a4 P) (P.obj X.op) (fun I => P.map I.f.op)\n          (_ :\n            \u2200 (I : (Cover.index \u22a4 P).R),\n              (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n                  MulticospanIndex.fst (Cover.index \u22a4 P) I =\n                (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n                  MulticospanIndex.snd (Cover.index \u22a4 P) I))\n        x) =\n    \u2191(Meq.equiv P \u22a4) (\u2191(Meq.equiv P \u22a4).symm (Meq.mk \u22a4 x))", "state_after": "case h.a.h\nC : Type u\ninst\u271d\u2075 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj X.op)\ni : Cover.Arrow \u22a4\n\u22a2 \u2191(\u2191(Meq.equiv P \u22a4)\n          ((forget D).map\n            (Multiequalizer.lift (Cover.index \u22a4 P) (P.obj X.op) (fun I => P.map I.f.op)\n              (_ :\n                \u2200 (I : (Cover.index \u22a4 P).R),\n                  (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n                      MulticospanIndex.fst (Cover.index \u22a4 P) I =\n                    (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n                      MulticospanIndex.snd (Cover.index \u22a4 P) I))\n            x))\n      i =\n    \u2191(\u2191(Meq.equiv P \u22a4) (\u2191(Meq.equiv P \u22a4).symm (Meq.mk \u22a4 x))) i"}, {"tactic": "simp", "state_before": "case h.a.h\nC : Type u\ninst\u271d\u2075 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj X.op)\ni : Cover.Arrow \u22a4\n\u22a2 \u2191(\u2191(Meq.equiv P \u22a4)\n          ((forget D).map\n            (Multiequalizer.lift (Cover.index \u22a4 P) (P.obj X.op) (fun I => P.map I.f.op)\n              (_ :\n                \u2200 (I : (Cover.index \u22a4 P).R),\n                  (fun I => P.map I.f.op) (MulticospanIndex.fstTo (Cover.index \u22a4 P) I) \u226b\n                      MulticospanIndex.fst (Cover.index \u22a4 P) I =\n                    (fun I => P.map I.f.op) (MulticospanIndex.sndTo (Cover.index \u22a4 P) I) \u226b\n                      MulticospanIndex.snd (Cover.index \u22a4 P) I))\n            x))\n      i =\n    \u2191(\u2191(Meq.equiv P \u22a4) (\u2191(Meq.equiv P \u22a4).symm (Meq.mk \u22a4 x))) i", "state_after": "case h.a.h\nC : Type u\ninst\u271d\u2075 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj X.op)\ni : Cover.Arrow \u22a4\n\u22a2 (forget D).map (P.map i.f.op) x = \u2191(Meq.mk \u22a4 x) i"}, {"tactic": "rfl", "state_before": "case h.a.h\nC : Type u\ninst\u271d\u2075 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u2074 : Category D\ninst\u271d\u00b3 : ConcreteCategory D\ninst\u271d\u00b2 : PreservesLimits (forget D)\ninst\u271d\u00b9 : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\ninst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nX : C\nP : C\u1d52\u1d56 \u2964 D\nx : (forget D).obj (P.obj X.op)\ni : Cover.Arrow \u22a4\n\u22a2 (forget D).map (P.map i.f.op) x = \u2191(Meq.mk \u22a4 x) i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "List.toFinset_reverse", "start": [3310, 1], "end": [3311, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Module/LocalizedModule.lean", "full_name": "IsLocalizedModule.mk'_cancel_left", "start": [961, 1], "end": [963, 45], "traced_tactics": [{"tactic": "delta mk'", "state_before": "R : Type u_1\ninst\u271d\u2077 : CommRing R\nS : Submonoid R\nM : Type u_3\nM' : Type u_2\nM'' : Type ?u.841263\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\nf : M \u2192\u2097[R] M'\ng : M \u2192\u2097[R] M''\ninst\u271d : IsLocalizedModule S f\nm : M\ns\u2081 s\u2082 : { x // x \u2208 S }\n\u22a2 mk' f (s\u2081 \u2022 m) (s\u2081 * s\u2082) = mk' f m s\u2082", "state_after": "R : Type u_1\ninst\u271d\u2077 : CommRing R\nS : Submonoid R\nM : Type u_3\nM' : Type u_2\nM'' : Type ?u.841263\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\nf : M \u2192\u2097[R] M'\ng : M \u2192\u2097[R] M''\ninst\u271d : IsLocalizedModule S f\nm : M\ns\u2081 s\u2082 : { x // x \u2208 S }\n\u22a2 \u2191(fromLocalizedModule S f) (LocalizedModule.mk (s\u2081 \u2022 m) (s\u2081 * s\u2082)) =\n    \u2191(fromLocalizedModule S f) (LocalizedModule.mk m s\u2082)"}, {"tactic": "rw [LocalizedModule.mk_cancel_common_left]", "state_before": "R : Type u_1\ninst\u271d\u2077 : CommRing R\nS : Submonoid R\nM : Type u_3\nM' : Type u_2\nM'' : Type ?u.841263\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\nf : M \u2192\u2097[R] M'\ng : M \u2192\u2097[R] M''\ninst\u271d : IsLocalizedModule S f\nm : M\ns\u2081 s\u2082 : { x // x \u2208 S }\n\u22a2 \u2191(fromLocalizedModule S f) (LocalizedModule.mk (s\u2081 \u2022 m) (s\u2081 * s\u2082)) =\n    \u2191(fromLocalizedModule S f) (LocalizedModule.mk m s\u2082)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "IsPiSystem.pi", "start": [75, 1], "end": [79, 97], "traced_tactics": [{"tactic": "rintro _ \u27e8s\u2081, hs\u2081, rfl\u27e9 _ \u27e8s\u2082, hs\u2082, rfl\u27e9 hst", "state_before": "\u03b9 : Type u_2\n\u03b9' : Type ?u.16\n\u03b1 : \u03b9 \u2192 Type u_1\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), IsPiSystem (C i)\n\u22a2 IsPiSystem (Set.pi univ '' Set.pi univ C)", "state_after": "case intro.intro.intro.intro\n\u03b9 : Type u_2\n\u03b9' : Type ?u.16\n\u03b1 : \u03b9 \u2192 Type u_1\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), IsPiSystem (C i)\ns\u2081 : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs\u2081 : s\u2081 \u2208 Set.pi univ C\ns\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs\u2082 : s\u2082 \u2208 Set.pi univ C\nhst : Set.Nonempty (Set.pi univ s\u2081 \u2229 Set.pi univ s\u2082)\n\u22a2 Set.pi univ s\u2081 \u2229 Set.pi univ s\u2082 \u2208 Set.pi univ '' Set.pi univ C"}, {"tactic": "rw [\u2190 pi_inter_distrib] at hst\u22a2", "state_before": "case intro.intro.intro.intro\n\u03b9 : Type u_2\n\u03b9' : Type ?u.16\n\u03b1 : \u03b9 \u2192 Type u_1\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), IsPiSystem (C i)\ns\u2081 : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs\u2081 : s\u2081 \u2208 Set.pi univ C\ns\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs\u2082 : s\u2082 \u2208 Set.pi univ C\nhst : Set.Nonempty (Set.pi univ s\u2081 \u2229 Set.pi univ s\u2082)\n\u22a2 Set.pi univ s\u2081 \u2229 Set.pi univ s\u2082 \u2208 Set.pi univ '' Set.pi univ C", "state_after": "case intro.intro.intro.intro\n\u03b9 : Type u_2\n\u03b9' : Type ?u.16\n\u03b1 : \u03b9 \u2192 Type u_1\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), IsPiSystem (C i)\ns\u2081 : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs\u2081 : s\u2081 \u2208 Set.pi univ C\ns\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs\u2082 : s\u2082 \u2208 Set.pi univ C\nhst : Set.Nonempty (Set.pi univ fun i => s\u2081 i \u2229 s\u2082 i)\n\u22a2 (Set.pi univ fun i => s\u2081 i \u2229 s\u2082 i) \u2208 Set.pi univ '' Set.pi univ C"}, {"tactic": "rw [univ_pi_nonempty_iff] at hst", "state_before": "case intro.intro.intro.intro\n\u03b9 : Type u_2\n\u03b9' : Type ?u.16\n\u03b1 : \u03b9 \u2192 Type u_1\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), IsPiSystem (C i)\ns\u2081 : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs\u2081 : s\u2081 \u2208 Set.pi univ C\ns\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs\u2082 : s\u2082 \u2208 Set.pi univ C\nhst : Set.Nonempty (Set.pi univ fun i => s\u2081 i \u2229 s\u2082 i)\n\u22a2 (Set.pi univ fun i => s\u2081 i \u2229 s\u2082 i) \u2208 Set.pi univ '' Set.pi univ C", "state_after": "case intro.intro.intro.intro\n\u03b9 : Type u_2\n\u03b9' : Type ?u.16\n\u03b1 : \u03b9 \u2192 Type u_1\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), IsPiSystem (C i)\ns\u2081 : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs\u2081 : s\u2081 \u2208 Set.pi univ C\ns\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs\u2082 : s\u2082 \u2208 Set.pi univ C\nhst : \u2200 (i : \u03b9), Set.Nonempty (s\u2081 i \u2229 s\u2082 i)\n\u22a2 (Set.pi univ fun i => s\u2081 i \u2229 s\u2082 i) \u2208 Set.pi univ '' Set.pi univ C"}, {"tactic": "exact mem_image_of_mem _ fun i _ => hC i _ (hs\u2081 i (mem_univ 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\u211a\n\u22a2 0 \u2264 \u2191n \u2194 0 \u2264 n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/SeparableDegree.lean", "full_name": "Polynomial.IsSeparableContraction.dvd_degree'", "start": [76, 1], "end": [80, 24], "traced_tactics": [{"tactic": "obtain \u27e8m, rfl\u27e9 := hf.2", "state_before": "F : Type u_1\ninst\u271d : CommSemiring F\nq : \u2115\nf : F[X]\nhf\u271d : HasSeparableContraction q f\ng : F[X]\nhf : IsSeparableContraction q f g\n\u22a2 \u2203 m, natDegree g * q ^ m = natDegree f", "state_after": "case intro\nF : Type u_1\ninst\u271d : CommSemiring F\nq : \u2115\ng : F[X]\nm : \u2115\nhf\u271d : HasSeparableContraction q (\u2191(expand F (q ^ m)) g)\nhf : IsSeparableContraction q (\u2191(expand F (q ^ m)) g) g\n\u22a2 \u2203 m_1, natDegree g * q ^ m_1 = natDegree (\u2191(expand F (q ^ m)) g)"}, {"tactic": "use m", "state_before": "case intro\nF : Type u_1\ninst\u271d : CommSemiring F\nq : \u2115\ng : F[X]\nm : \u2115\nhf\u271d : HasSeparableContraction q (\u2191(expand F (q ^ m)) g)\nhf : IsSeparableContraction q (\u2191(expand F (q ^ m)) g) g\n\u22a2 \u2203 m_1, natDegree g * q ^ m_1 = natDegree (\u2191(expand F (q ^ m)) g)", "state_after": "case intro\nF : Type u_1\ninst\u271d : CommSemiring F\nq : \u2115\ng : F[X]\nm : \u2115\nhf\u271d : HasSeparableContraction q (\u2191(expand F (q ^ m)) g)\nhf : IsSeparableContraction q (\u2191(expand F (q ^ m)) g) g\n\u22a2 natDegree g * q ^ m = natDegree (\u2191(expand F (q ^ m)) g)"}, {"tactic": "rw [natDegree_expand]", "state_before": "case intro\nF : Type u_1\ninst\u271d : CommSemiring F\nq : \u2115\ng : F[X]\nm : \u2115\nhf\u271d : HasSeparableContraction q (\u2191(expand F (q ^ m)) g)\nhf : IsSeparableContraction q (\u2191(expand F (q ^ m)) g) g\n\u22a2 natDegree g * q ^ m = natDegree (\u2191(expand F (q ^ m)) g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.choose_mem", "start": [3711, 1], "end": [3712, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Nat.ceil_mono", "start": [319, 1], "end": [320, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "full_name": "PiNat.mem_cylinder_iff_eq", "start": [136, 1], "end": [149, 32], "traced_tactics": [{"tactic": "constructor", "state_before": "E : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\n\u22a2 y \u2208 cylinder x n \u2194 cylinder y n = cylinder x n", "state_after": "case mp\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\n\u22a2 y \u2208 cylinder x n \u2192 cylinder y n = cylinder x n\n\ncase mpr\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\n\u22a2 cylinder y n = cylinder x n \u2192 y \u2208 cylinder x n"}, {"tactic": "intro hy", "state_before": "case mp\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\n\u22a2 y \u2208 cylinder x n \u2192 cylinder y n = cylinder x n", "state_after": "case mp\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhy : y \u2208 cylinder x n\n\u22a2 cylinder y n = cylinder x n"}, {"tactic": "apply Subset.antisymm", "state_before": "case mp\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhy : y \u2208 cylinder x n\n\u22a2 cylinder y n = cylinder x n", "state_after": "case mp.h\u2081\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhy : y \u2208 cylinder x n\n\u22a2 cylinder y n \u2286 cylinder x n\n\ncase mp.h\u2082\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhy : y \u2208 cylinder x n\n\u22a2 cylinder x n \u2286 cylinder y n"}, {"tactic": "intro z hz i hi", "state_before": "case mp.h\u2081\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhy : y \u2208 cylinder x n\n\u22a2 cylinder y n \u2286 cylinder x n", "state_after": "case mp.h\u2081\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhy : y \u2208 cylinder x n\nz : (n : \u2115) \u2192 E n\nhz : z \u2208 cylinder y n\ni : \u2115\nhi : i < n\n\u22a2 z i = x i"}, {"tactic": "rw [\u2190 hy i hi]", "state_before": "case mp.h\u2081\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhy : y \u2208 cylinder x n\nz : (n : \u2115) \u2192 E n\nhz : z \u2208 cylinder y n\ni : \u2115\nhi : i < n\n\u22a2 z i = x i", "state_after": "case mp.h\u2081\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhy : y \u2208 cylinder x n\nz : (n : \u2115) \u2192 E n\nhz : z \u2208 cylinder y n\ni : \u2115\nhi : i < n\n\u22a2 z i = y i"}, {"tactic": "exact hz i hi", "state_before": "case mp.h\u2081\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhy : y \u2208 cylinder x n\nz : (n : \u2115) \u2192 E n\nhz : z \u2208 cylinder y n\ni : \u2115\nhi : i < n\n\u22a2 z i = y i", "state_after": "no goals"}, {"tactic": "intro z hz i hi", "state_before": "case mp.h\u2082\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhy : y \u2208 cylinder x n\n\u22a2 cylinder x n \u2286 cylinder y n", "state_after": "case mp.h\u2082\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhy : y \u2208 cylinder x n\nz : (n : \u2115) \u2192 E n\nhz : z \u2208 cylinder x n\ni : \u2115\nhi : i < n\n\u22a2 z i = y i"}, {"tactic": "rw [hy i hi]", "state_before": "case mp.h\u2082\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhy : y \u2208 cylinder x n\nz : (n : \u2115) \u2192 E n\nhz : z \u2208 cylinder x n\ni : \u2115\nhi : i < n\n\u22a2 z i = y i", "state_after": "case mp.h\u2082\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhy : y \u2208 cylinder x n\nz : (n : \u2115) \u2192 E n\nhz : z \u2208 cylinder x n\ni : \u2115\nhi : i < n\n\u22a2 z i = x i"}, {"tactic": "exact hz i hi", "state_before": "case mp.h\u2082\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nhy : y \u2208 cylinder x n\nz : (n : \u2115) \u2192 E n\nhz : z \u2208 cylinder x n\ni : \u2115\nhi : i < n\n\u22a2 z i = x i", "state_after": "no goals"}, {"tactic": "intro h", "state_before": "case mpr\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\n\u22a2 cylinder y n = cylinder x n \u2192 y \u2208 cylinder x n", "state_after": "case mpr\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nh : cylinder y n = cylinder x n\n\u22a2 y \u2208 cylinder x n"}, {"tactic": "rw [\u2190 h]", "state_before": "case mpr\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nh : cylinder y n = cylinder x n\n\u22a2 y \u2208 cylinder x n", "state_after": "case mpr\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nh : cylinder y n = cylinder x n\n\u22a2 y \u2208 cylinder y n"}, {"tactic": "exact self_mem_cylinder _ _", "state_before": "case mpr\nE : \u2115 \u2192 Type u_1\nx y : (n : \u2115) \u2192 E n\nn : \u2115\nh : cylinder y n = cylinder x n\n\u22a2 y \u2208 cylinder y n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "fst_integral", "start": [1192, 1], "end": [1193, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subsemigroup/Operations.lean", "full_name": "Subsemigroup.monotone_map", "start": [305, 1], "end": [306, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/LocalHomeomorph.lean", "full_name": "LocalHomeomorph.IsImage.inter", "start": [554, 11], "end": [555, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/NAry.lean", "full_name": "Filter.map\u2082_bot_left", "start": [106, 1], "end": [107, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Init/Logic.lean", "full_name": "trans_rel_right", "start": [45, 15], "end": [46, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Iso.lean", "full_name": "CategoryTheory.IsIso.inv_eq_of_hom_inv_id", "start": [345, 1], "end": [348, 20], "traced_tactics": [{"tactic": "apply (cancel_epi f).mp", "state_before": "C : Type u\ninst\u271d\u00b9 : Category C\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d : IsIso f\ng : Y \u27f6 X\nhom_inv_id : f \u226b g = \ud835\udfd9 X\n\u22a2 inv f = g", "state_after": "C : Type u\ninst\u271d\u00b9 : Category C\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d : IsIso f\ng : Y \u27f6 X\nhom_inv_id : f \u226b g = \ud835\udfd9 X\n\u22a2 f \u226b inv f = f \u226b g"}, {"tactic": "simp [hom_inv_id]", "state_before": "C : Type u\ninst\u271d\u00b9 : Category C\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d : IsIso f\ng : Y \u27f6 X\nhom_inv_id : f \u226b g = \ud835\udfd9 X\n\u22a2 f \u226b inv f = f \u226b g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Monic.lean", "full_name": "Polynomial.ne_zero_of_ne_zero_of_monic", "start": [61, 1], "end": [65, 15], "traced_tactics": [{"tactic": "rintro rfl", "state_before": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np q r : R[X]\nhp : p \u2260 0\nhq : Monic q\n\u22a2 q \u2260 0", "state_after": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np r : R[X]\nhp : p \u2260 0\nhq : Monic 0\n\u22a2 False"}, {"tactic": "rw [Monic.def, leadingCoeff_zero] at hq", "state_before": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np r : R[X]\nhp : p \u2260 0\nhq : Monic 0\n\u22a2 False", "state_after": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np r : R[X]\nhp : p \u2260 0\nhq : 0 = 1\n\u22a2 False"}, {"tactic": "rw [\u2190 mul_one p, \u2190 C_1, \u2190 hq, C_0, mul_zero] at hp", "state_before": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np r : R[X]\nhp : p \u2260 0\nhq : 0 = 1\n\u22a2 False", "state_after": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np r : R[X]\nhp : 0 \u2260 0\nhq : 0 = 1\n\u22a2 False"}, {"tactic": "exact hp rfl", "state_before": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : Semiring R\np r : R[X]\nhp : 0 \u2260 0\nhq : 0 = 1\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/RamificationInertia.lean", "full_name": "Ideal.IsDedekindDomain.ramificationIdx_eq_factors_count", "start": [159, 1], "end": [162, 34], "traced_tactics": [{"tactic": "rw [IsDedekindDomain.ramificationIdx_eq_normalizedFactors_count hp0 hP hP0,\n  factors_eq_normalizedFactors]", "state_before": "R : Type u\ninst\u271d\u00b3 : CommRing R\nS : Type v\ninst\u271d\u00b2 : CommRing S\nf : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : IsDedekindDomain S\nhp0 : map f p \u2260 \u22a5\nhP : IsPrime P\nhP0 : P \u2260 \u22a5\n\u22a2 ramificationIdx f p P = Multiset.count P (factors (map f p))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "full_name": "Matrix.det_units_conj'", "start": [224, 1], "end": [226, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/Operations.lean", "full_name": "Ideal.finset_inf_span_singleton", "start": [627, 1], "end": [632, 93], "traced_tactics": [{"tactic": "ext x", "state_before": "R : Type u\n\u03b9\u271d : Type ?u.253392\ninst\u271d : CommSemiring R\nI\u271d J K L : Ideal R\n\u03b9 : Type u_1\ns : Finset \u03b9\nI : \u03b9 \u2192 R\nhI : Set.Pairwise (\u2191s) (IsCoprime on I)\n\u22a2 (Finset.inf s fun i => span {I i}) = span {\u220f i in s, I i}", "state_after": "case h\nR : Type u\n\u03b9\u271d : Type ?u.253392\ninst\u271d : CommSemiring R\nI\u271d J K L : Ideal R\n\u03b9 : Type u_1\ns : Finset \u03b9\nI : \u03b9 \u2192 R\nhI : Set.Pairwise (\u2191s) (IsCoprime on I)\nx : R\n\u22a2 (x \u2208 Finset.inf s fun i => span {I i}) \u2194 x \u2208 span {\u220f i in s, I i}"}, {"tactic": "simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton]", "state_before": "case h\nR : Type u\n\u03b9\u271d : Type ?u.253392\ninst\u271d : CommSemiring R\nI\u271d J K L : Ideal R\n\u03b9 : Type u_1\ns : Finset \u03b9\nI : \u03b9 \u2192 R\nhI : Set.Pairwise (\u2191s) (IsCoprime on I)\nx : R\n\u22a2 (x \u2208 Finset.inf s fun i => span {I i}) \u2194 x \u2208 span {\u220f i in s, I i}", "state_after": "case h\nR : Type u\n\u03b9\u271d : Type ?u.253392\ninst\u271d : CommSemiring R\nI\u271d J K L : Ideal R\n\u03b9 : Type u_1\ns : Finset \u03b9\nI : \u03b9 \u2192 R\nhI : Set.Pairwise (\u2191s) (IsCoprime on I)\nx : R\n\u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 I i \u2223 x) \u2194 \u220f i in s, I i \u2223 x"}, {"tactic": "exact \u27e8Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h\u27e9", "state_before": "case h\nR : Type u\n\u03b9\u271d : Type ?u.253392\ninst\u271d : CommSemiring R\nI\u271d J K L : Ideal R\n\u03b9 : Type u_1\ns : Finset \u03b9\nI : \u03b9 \u2192 R\nhI : Set.Pairwise (\u2191s) (IsCoprime on I)\nx : R\n\u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 I i \u2223 x) \u2194 \u220f i in s, I i \u2223 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.toMeasure_smul", "start": [229, 1], "end": [230, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/MeasurableSpace.lean", "full_name": "measurable_prod_mk_right", "start": [690, 1], "end": [691, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Normed/Group/Seminorm.lean", "full_name": "AddGroupNorm.apply_one", "start": [864, 1], "end": [865, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/MvPolynomial/Cardinal.lean", "full_name": "MvPolynomial.cardinal_mk_eq_max", "start": [61, 1], "end": [62, 60], "traced_tactics": [{"tactic": "simp", "state_before": "\u03c3 R : Type u\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Nonempty \u03c3\ninst\u271d : Nontrivial R\n\u22a2 (#MvPolynomial \u03c3 R) = max (max (#R) (#\u03c3)) \u2135\u2080", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Reflexive.lean", "full_name": "CategoryTheory.IsReflexivePair.mk'", "start": [66, 1], "end": [68, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Cone/Dual.lean", "full_name": "ConvexCone.hyperplane_separation_of_nonempty_of_isClosed_of_nmem", "start": [166, 1], "end": [196, 49], "traced_tactics": [{"tactic": "obtain \u27e8z, hzK, infi\u27e9 := exists_norm_eq_iInf_of_complete_convex ne hc.isComplete K.convex b", "state_before": "\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\n\u22a2 \u2203 y, (\u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x y) \u2227 inner y b < 0", "state_after": "case intro.intro\n\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\n\u22a2 \u2203 y, (\u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x y) \u2227 inner y b < 0"}, {"tactic": "have hinner := (norm_eq_iInf_iff_real_inner_le_zero K.convex hzK).1 infi", "state_before": "case intro.intro\n\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\n\u22a2 \u2203 y, (\u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x y) \u2227 inner y b < 0", "state_after": "case intro.intro\n\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\n\u22a2 \u2203 y, (\u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x y) \u2227 inner y b < 0"}, {"tactic": "use z - b", "state_before": "case intro.intro\n\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\n\u22a2 \u2203 y, (\u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x y) \u2227 inner y b < 0", "state_after": "case intro.intro\n\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\n\u22a2 (\u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x (z - b)) \u2227 inner (z - b) b < 0"}, {"tactic": "constructor", "state_before": "case intro.intro\n\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\n\u22a2 (\u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x (z - b)) \u2227 inner (z - b) b < 0", "state_after": "case intro.intro.left\n\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\n\u22a2 \u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x (z - b)\n\ncase intro.intro.right\n\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\n\u22a2 inner (z - b) b < 0"}, {"tactic": "rintro x hxK", "state_before": "case intro.intro.left\n\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\n\u22a2 \u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x (z - b)", "state_after": "case intro.intro.left\n\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nx : H\nhxK : x \u2208 K\n\u22a2 0 \u2264 inner x (z - b)"}, {"tactic": "specialize hinner _ (K.add_mem hxK hzK)", "state_before": "case intro.intro.left\n\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nx : H\nhxK : x \u2208 K\n\u22a2 0 \u2264 inner x (z - b)", "state_after": "case intro.intro.left\n\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nx : H\nhxK : x \u2208 K\nhinner : inner (b - z) (x + z - z) \u2264 0\n\u22a2 0 \u2264 inner x (z - b)"}, {"tactic": "rwa [add_sub_cancel, real_inner_comm, \u2190 neg_nonneg, neg_eq_neg_one_mul, \u2190 real_inner_smul_right,\n  neg_smul, one_smul, neg_sub] at hinner", "state_before": "case intro.intro.left\n\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nx : H\nhxK : x \u2208 K\nhinner : inner (b - z) (x + z - z) \u2264 0\n\u22a2 0 \u2264 inner x (z - b)", "state_after": "no goals"}, {"tactic": "have hinner\u2080 := hinner 0 (K.pointed_of_nonempty_of_isClosed ne hc)", "state_before": "case intro.intro.right\n\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\n\u22a2 inner (z - b) b < 0", "state_after": "case intro.intro.right\n\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : inner (b - z) (0 - z) \u2264 0\n\u22a2 inner (z - b) b < 0"}, {"tactic": "rw [zero_sub, inner_neg_right, Right.neg_nonpos_iff] at hinner\u2080", "state_before": "case intro.intro.right\n\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : inner (b - z) (0 - z) \u2264 0\n\u22a2 inner (z - b) b < 0", "state_after": "case intro.intro.right\n\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\n\u22a2 inner (z - b) b < 0"}, {"tactic": "have hbz : b - z \u2260 0 := by\n  rw [sub_ne_zero]\n  contrapose! hzK\n  rwa [\u2190 hzK]", "state_before": "case intro.intro.right\n\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\n\u22a2 inner (z - b) b < 0", "state_after": "case intro.intro.right\n\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\nhbz : b - z \u2260 0\n\u22a2 inner (z - b) b < 0"}, {"tactic": "rw [\u2190 neg_zero, lt_neg, \u2190 neg_one_mul, \u2190 real_inner_smul_left, smul_sub, neg_smul, one_smul,\n  neg_smul, neg_sub_neg, one_smul]", "state_before": "case intro.intro.right\n\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\nhbz : b - z \u2260 0\n\u22a2 inner (z - b) b < 0", "state_after": "case intro.intro.right\n\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\nhbz : b - z \u2260 0\n\u22a2 0 < inner (b - z) b"}, {"tactic": "calc\n  0 < \u27eab - z, b - z\u27eb_\u211d := lt_of_not_le ((Iff.not real_inner_self_nonpos).2 hbz)\n  _ = \u27eab - z, b - z\u27eb_\u211d + 0 := (add_zero _).symm\n  _ \u2264 \u27eab - z, b - z\u27eb_\u211d + \u27eab - z, z\u27eb_\u211d := (add_le_add rfl.ge hinner\u2080)\n  _ = \u27eab - z, b - z + z\u27eb_\u211d := (inner_add_right _ _ _).symm\n  _ = \u27eab - z, b\u27eb_\u211d := by rw [sub_add_cancel]", "state_before": "case intro.intro.right\n\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\nhbz : b - z \u2260 0\n\u22a2 0 < inner (b - z) b", "state_after": "no goals"}, {"tactic": "rw [sub_ne_zero]", "state_before": "\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\n\u22a2 b - z \u2260 0", "state_after": "\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\n\u22a2 b \u2260 z"}, {"tactic": "contrapose! hzK", "state_before": "\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\n\u22a2 b \u2260 z", "state_after": "\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\nhzK : b = z\n\u22a2 \u00acz \u2208 \u2191K"}, {"tactic": "rwa [\u2190 hzK]", "state_before": "\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type ?u.105001\nH : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\ninfi : \u2016b - z\u2016 = \u2a05 (w : \u2191\u2191K), \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\nhzK : b = z\n\u22a2 \u00acz \u2208 \u2191K", "state_after": "no goals"}, {"tactic": "rw [sub_add_cancel]", "state_before": "\ud835\udd5c : Type ?u.104992\nE : Type ?u.104995\nF : Type ?u.104998\nG : Type 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(ucs k \u22a5) = ucs k \u22a5", "state_after": "case zero\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nk : \u2115\nN N\u2081 N\u2082 : LieSubmodule R L M\n\u22a2 comap (incl N) (ucs Nat.zero \u22a5) = ucs Nat.zero \u22a5\n\ncase succ\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nk\u271d : \u2115\nN N\u2081 N\u2082 : LieSubmodule R L M\nk : \u2115\nih : comap (incl N) (ucs k \u22a5) = ucs k \u22a5\n\u22a2 comap (incl N) (ucs (Nat.succ k) \u22a5) = ucs (Nat.succ k) \u22a5"}, {"tactic": "exact N.ker_incl", "state_before": "case zero\nR : Type u\nL : Type 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lt_trichotomy 0 a with (h | h | h)", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\n\u22a2 0 \u2264 \u2191sign a \u2194 0 \u2264 a", "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\nh : 0 < a\n\u22a2 0 \u2264 \u2191sign a \u2194 0 \u2264 a\n\ncase inr.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\nh : 0 = a\n\u22a2 0 \u2264 \u2191sign a \u2194 0 \u2264 a\n\ncase inr.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\nh : a < 0\n\u22a2 0 \u2264 \u2191sign a \u2194 0 \u2264 a"}, {"tactic": "simp [h, h.le]", "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\nh : 0 < a\n\u22a2 0 \u2264 \u2191sign a \u2194 0 \u2264 a", "state_after": "no goals"}, {"tactic": "simp [\u2190h]", 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maximalGeneralizedEigenspace f \u03bc = \u2191(generalizedEigenspace f \u03bc) (maximalGeneralizedEigenspaceIndex f \u03bc)"}, {"tactic": "exact (WellFounded.iSup_eq_monotonicSequenceLimit h (f.generalizedEigenspace \u03bc) : _)", "state_before": "K R : Type v\nV M : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nh : WellFounded fun x x_1 => x > x_1\nf : End R M\n\u03bc : R\n\u22a2 maximalGeneralizedEigenspace f \u03bc = \u2191(generalizedEigenspace f \u03bc) (maximalGeneralizedEigenspaceIndex f \u03bc)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Semicontinuous.lean", "full_name": "ContinuousOn.lowerSemicontinuousOn", "start": [293, 1], "end": [294, 82], "traced_tactics": []}, {"url": 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Type u_1\ninst\u271d\u00b9 : LinearOrderedSemifield R\ninst\u271d : FloorSemiring R\nb : \u2115\nhb : 1 < b\nz : \u2124\n\u22a2 clog b (\u2191b ^ z) = z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "full_name": "SimpleGraph.Walk.darts_nil", "start": [685, 1], "end": [685, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/UnionFind.lean", "full_name": "UFModel.Models.parent_eq'", "start": [129, 1], "end": [130, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/LocalEquiv.lean", "full_name": "LocalEquiv.IsImage.leftInvOn_piecewise", "start": [452, 1], "end": [458, 22], "traced_tactics": [{"tactic": "rintro x (\u27e8he, hs\u27e9 | \u27e8he, hs : x \u2209 s\u27e9)", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.29796\n\u03b4 : Type ?u.29799\ne : LocalEquiv \u03b1 \u03b2\ne'\u271d : LocalEquiv \u03b2 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nx : \u03b1\ny : \u03b2\ne' : LocalEquiv \u03b1 \u03b2\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Decidable (i \u2208 s)\ninst\u271d : (i : \u03b2) \u2192 Decidable (i \u2208 t)\nh : IsImage e s t\nh' : IsImage e' s t\n\u22a2 LeftInvOn (piecewise t \u2191(LocalEquiv.symm e) \u2191(LocalEquiv.symm e')) (piecewise s \u2191e \u2191e') (Set.ite s e.source e'.source)", "state_after": "case inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.29796\n\u03b4 : Type ?u.29799\ne : LocalEquiv \u03b1 \u03b2\ne'\u271d : LocalEquiv \u03b2 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nx\u271d : \u03b1\ny : \u03b2\ne' : LocalEquiv \u03b1 \u03b2\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Decidable (i \u2208 s)\ninst\u271d : (i : \u03b2) \u2192 Decidable (i \u2208 t)\nh : IsImage e s t\nh' : IsImage e' s t\nx : \u03b1\nhe : x \u2208 e.source\nhs : x \u2208 s\n\u22a2 piecewise t (\u2191(LocalEquiv.symm e)) (\u2191(LocalEquiv.symm e')) (piecewise s (\u2191e) (\u2191e') x) = x\n\ncase inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.29796\n\u03b4 : Type ?u.29799\ne : LocalEquiv \u03b1 \u03b2\ne'\u271d : LocalEquiv \u03b2 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nx\u271d : \u03b1\ny : \u03b2\ne' : LocalEquiv \u03b1 \u03b2\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Decidable (i \u2208 s)\ninst\u271d : (i : \u03b2) \u2192 Decidable (i \u2208 t)\nh : IsImage e s t\nh' : IsImage e' s t\nx : \u03b1\nhe : x \u2208 e'.source\nhs : \u00acx \u2208 s\n\u22a2 piecewise t (\u2191(LocalEquiv.symm e)) (\u2191(LocalEquiv.symm e')) (piecewise s (\u2191e) (\u2191e') x) = x"}, {"tactic": "rw [piecewise_eq_of_mem _ _ _ hs, piecewise_eq_of_mem _ _ _ ((h he).2 hs), e.left_inv he]", "state_before": "case inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.29796\n\u03b4 : Type ?u.29799\ne : LocalEquiv \u03b1 \u03b2\ne'\u271d : LocalEquiv \u03b2 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nx\u271d : \u03b1\ny : \u03b2\ne' : LocalEquiv \u03b1 \u03b2\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Decidable (i \u2208 s)\ninst\u271d : (i : \u03b2) \u2192 Decidable (i \u2208 t)\nh : IsImage e s t\nh' : IsImage e' s t\nx : \u03b1\nhe : x \u2208 e.source\nhs : x \u2208 s\n\u22a2 piecewise t (\u2191(LocalEquiv.symm e)) (\u2191(LocalEquiv.symm e')) (piecewise s (\u2191e) (\u2191e') x) = x", "state_after": "no goals"}, {"tactic": "rw [piecewise_eq_of_not_mem _ _ _ hs, piecewise_eq_of_not_mem _ _ _ ((h'.compl he).2 hs),\n  e'.left_inv he]", "state_before": "case inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.29796\n\u03b4 : Type ?u.29799\ne : LocalEquiv \u03b1 \u03b2\ne'\u271d : LocalEquiv \u03b2 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nx\u271d : \u03b1\ny : \u03b2\ne' : LocalEquiv \u03b1 \u03b2\ninst\u271d\u00b9 : (i : \u03b1) \u2192 Decidable (i \u2208 s)\ninst\u271d : (i : \u03b2) \u2192 Decidable (i \u2208 t)\nh : IsImage e s t\nh' : IsImage e' s t\nx : \u03b1\nhe : x \u2208 e'.source\nhs : \u00acx \u2208 s\n\u22a2 piecewise t (\u2191(LocalEquiv.symm e)) (\u2191(LocalEquiv.symm e')) (piecewise s (\u2191e) (\u2191e') x) = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Func.lean", "full_name": "List.Func.get_map'", "start": [208, 1], "end": [215, 13], "traced_tactics": [{"tactic": "intro h1", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nas : List \u03b1\n\u22a2 f default = default \u2192 get n (map f as) = f (get n as)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nas : List \u03b1\nh1 : f default = default\n\u22a2 get n (map f as) = f (get n as)"}, {"tactic": "by_cases h2 : n < as.length", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nas : List \u03b1\nh1 : f default = default\n\u22a2 get n (map f as) = f (get n as)", "state_after": "case pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nas : List \u03b1\nh1 : f default = default\nh2 : n < length as\n\u22a2 get n (map f as) = f (get n as)\n\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nas : List \u03b1\nh1 : f default = default\nh2 : \u00acn < length as\n\u22a2 get n (map f as) = f (get n as)"}, {"tactic": "apply get_map h2", "state_before": "case pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nas : List \u03b1\nh1 : f default = default\nh2 : n < length as\n\u22a2 get n (map f as) = f (get n as)", "state_after": "no goals"}, {"tactic": "rw [not_lt] at h2", "state_before": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nas : List \u03b1\nh1 : f default = default\nh2 : \u00acn < length as\n\u22a2 get n (map f as) = f (get n as)", "state_after": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nas : List \u03b1\nh1 : f default = default\nh2 : length as \u2264 n\n\u22a2 get n (map f as) = f (get n as)"}, {"tactic": "rw [get_eq_default_of_le _ h2, get_eq_default_of_le, h1]", "state_before": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nas : List \u03b1\nh1 : f default = default\nh2 : length as \u2264 n\n\u22a2 get n (map f as) = f (get n as)", "state_after": "case neg.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nas : List \u03b1\nh1 : f default = default\nh2 : length as \u2264 n\n\u22a2 length (map f as) \u2264 n"}, {"tactic": "rw [length_map]", "state_before": "case neg.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nas : List \u03b1\nh1 : f default = default\nh2 : length as \u2264 n\n\u22a2 length (map f as) \u2264 n", "state_after": "case neg.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nas\u271d as1 as2 as3 : List \u03b1\ninst\u271d\u00b9 : Inhabited \u03b1\ninst\u271d : Inhabited \u03b2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nas : List \u03b1\nh1 : f default = default\nh2 : length as \u2264 n\n\u22a2 length as \u2264 n"}, {"tactic": "apply h2", "state_before": "case neg.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : 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Function.Injective ((forget Profinite).map f)", "state_after": "case mp\nX\u271d Y\u271d : Profinite\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Profinite\nf : X \u27f6 Y\n\u22a2 Mono f \u2192 Function.Injective ((forget Profinite).map f)\n\ncase mpr\nX\u271d Y\u271d : Profinite\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Profinite\nf : X \u27f6 Y\n\u22a2 Function.Injective ((forget Profinite).map f) \u2192 Mono f"}, {"tactic": "intro h", "state_before": "case mp\nX\u271d Y\u271d : Profinite\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Profinite\nf : X \u27f6 Y\n\u22a2 Mono f \u2192 Function.Injective ((forget Profinite).map f)", "state_after": "case mp\nX\u271d Y\u271d : Profinite\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Profinite\nf : X \u27f6 Y\nh : Mono f\n\u22a2 Function.Injective ((forget Profinite).map f)"}, {"tactic": "haveI : Limits.PreservesLimits profiniteToCompHaus := inferInstance", "state_before": "case mp\nX\u271d Y\u271d : Profinite\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Profinite\nf : X \u27f6 Y\nh : Mono f\n\u22a2 Function.Injective ((forget Profinite).map f)", "state_after": "case mp\nX\u271d Y\u271d : Profinite\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Profinite\nf : X \u27f6 Y\nh : Mono f\nthis : Limits.PreservesLimits profiniteToCompHaus\n\u22a2 Function.Injective ((forget Profinite).map f)"}, {"tactic": "haveI : Mono (profiniteToCompHaus.map f) := inferInstance", "state_before": "case mp\nX\u271d Y\u271d : Profinite\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Profinite\nf : X \u27f6 Y\nh : Mono f\nthis : Limits.PreservesLimits profiniteToCompHaus\n\u22a2 Function.Injective ((forget Profinite).map f)", "state_after": "case mp\nX\u271d Y\u271d : Profinite\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Profinite\nf : X \u27f6 Y\nh : Mono f\nthis\u271d : Limits.PreservesLimits profiniteToCompHaus\nthis : Mono (profiniteToCompHaus.map f)\n\u22a2 Function.Injective ((forget Profinite).map f)"}, {"tactic": "erw [\u2190 CompHaus.mono_iff_injective]", "state_before": "case mp\nX\u271d Y\u271d : Profinite\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Profinite\nf : X \u27f6 Y\nh : Mono f\nthis\u271d : Limits.PreservesLimits profiniteToCompHaus\nthis : Mono (profiniteToCompHaus.map f)\n\u22a2 Function.Injective ((forget Profinite).map f)", "state_after": "case mp\nX\u271d Y\u271d : Profinite\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Profinite\nf : X \u27f6 Y\nh : Mono f\nthis\u271d : Limits.PreservesLimits profiniteToCompHaus\nthis : Mono (profiniteToCompHaus.map f)\n\u22a2 Mono ((inducedFunctor toCompHaus).map f)"}, {"tactic": "assumption", "state_before": "case mp\nX\u271d Y\u271d : Profinite\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Profinite\nf : X \u27f6 Y\nh : Mono f\nthis\u271d : Limits.PreservesLimits profiniteToCompHaus\nthis : Mono (profiniteToCompHaus.map f)\n\u22a2 Mono ((inducedFunctor toCompHaus).map f)", "state_after": "no goals"}, {"tactic": "rw [\u2190 CategoryTheory.mono_iff_injective]", "state_before": "case mpr\nX\u271d Y\u271d : Profinite\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Profinite\nf : X \u27f6 Y\n\u22a2 Function.Injective ((forget Profinite).map f) \u2192 Mono f", "state_after": "case mpr\nX\u271d Y\u271d : Profinite\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Profinite\nf : X \u27f6 Y\n\u22a2 Mono ((forget Profinite).map f) \u2192 Mono f"}, {"tactic": "apply (forget Profinite).mono_of_mono_map", "state_before": "case mpr\nX\u271d Y\u271d : Profinite\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Profinite\nf : X \u27f6 Y\n\u22a2 Mono ((forget Profinite).map f) \u2192 Mono f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean", "full_name": "InnerProductGeometry.angle_neg_neg", "start": [92, 1], "end": [94, 41], "traced_tactics": [{"tactic": "unfold angle", "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\n\u22a2 angle (-x) (-y) = angle x y", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\n\u22a2 arccos (inner (-x) (-y) / (\u2016-x\u2016 * \u2016-y\u2016)) = arccos (inner x y / (\u2016x\u2016 * \u2016y\u2016))"}, {"tactic": "rw [inner_neg_neg, norm_neg, norm_neg]", "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\n\u22a2 arccos (inner (-x) (-y) / (\u2016-x\u2016 * \u2016-y\u2016)) = arccos (inner x y / (\u2016x\u2016 * \u2016y\u2016))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/MonoidLocalization.lean", "full_name": "Submonoid.LocalizationMap.lift_surjective_iff", "start": [1084, 1], "end": [1096, 51], "traced_tactics": [{"tactic": "constructor", "state_before": "M : Type u_3\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_2\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\n\u22a2 Surjective \u2191(lift f hg) \u2194 \u2200 (v : P), \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst", "state_after": "case mp\nM : Type u_3\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_2\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\n\u22a2 Surjective \u2191(lift f hg) \u2192 \u2200 (v : P), \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst\n\ncase mpr\nM : Type u_3\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_2\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\n\u22a2 (\u2200 (v : P), \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst) \u2192 Surjective \u2191(lift f hg)"}, {"tactic": "intro H v", "state_before": "case mp\nM : Type u_3\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_2\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\n\u22a2 Surjective \u2191(lift f hg) \u2192 \u2200 (v : P), \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst", "state_after": "case mp\nM : Type u_3\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_2\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : Surjective \u2191(lift f hg)\nv : P\n\u22a2 \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst"}, {"tactic": "obtain \u27e8z, hz\u27e9 := H v", "state_before": "case mp\nM : Type u_3\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_2\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : Surjective \u2191(lift f hg)\nv : P\n\u22a2 \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst", "state_after": "case mp.intro\nM : Type u_3\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_2\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : Surjective \u2191(lift f hg)\nv : P\nz : N\nhz : \u2191(lift f hg) z = v\n\u22a2 \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst"}, {"tactic": "obtain \u27e8x, hx\u27e9 := f.surj z", "state_before": "case mp.intro\nM : Type u_3\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_2\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : Surjective \u2191(lift f hg)\nv : P\nz : N\nhz : \u2191(lift f hg) z = v\n\u22a2 \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst", "state_after": "case mp.intro.intro\nM : Type u_3\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_2\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : Surjective \u2191(lift f hg)\nv : P\nz : N\nhz : \u2191(lift f hg) z = v\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst"}, {"tactic": "use x", "state_before": "case mp.intro.intro\nM : Type u_3\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_2\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : Surjective \u2191(lift f hg)\nv : P\nz : N\nhz : \u2191(lift f hg) z = v\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst", "state_after": "case mp.intro.intro\nM : Type u_3\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_2\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : Surjective \u2191(lift f hg)\nv : P\nz : N\nhz : \u2191(lift f hg) z = v\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 v * \u2191g \u2191x.snd = \u2191g x.fst"}, {"tactic": "rw [\u2190 hz, f.eq_mk'_iff_mul_eq.2 hx, lift_mk', mul_assoc, mul_comm _ (g \u2191x.2)]", "state_before": "case mp.intro.intro\nM : Type u_3\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_2\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : Surjective \u2191(lift f hg)\nv : P\nz : N\nhz : \u2191(lift f hg) z = v\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 v * \u2191g \u2191x.snd = \u2191g x.fst", "state_after": "case mp.intro.intro\nM : Type u_3\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_2\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : Surjective \u2191(lift f hg)\nv : P\nz : N\nhz : \u2191(lift f hg) z = v\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191g x.fst * (\u2191g \u2191x.snd * \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict g S) hg) x.snd)\u207b\u00b9) = \u2191g x.fst"}, {"tactic": "erw [IsUnit.mul_liftRight_inv (g.restrict S) hg, mul_one]", "state_before": "case mp.intro.intro\nM : Type u_3\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_2\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : Surjective \u2191(lift f hg)\nv : P\nz : N\nhz : \u2191(lift f hg) z = v\nx : M \u00d7 { x // x \u2208 S }\nhx : z * \u2191(toMap f) \u2191x.snd = \u2191(toMap f) x.fst\n\u22a2 \u2191g x.fst * (\u2191g \u2191x.snd * \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict g S) hg) x.snd)\u207b\u00b9) = \u2191g x.fst", "state_after": "no goals"}, {"tactic": "intro H v", "state_before": "case mpr\nM : Type u_3\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_2\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\n\u22a2 (\u2200 (v : P), \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst) \u2192 Surjective \u2191(lift f hg)", "state_after": "case mpr\nM : Type u_3\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_2\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : \u2200 (v : P), \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst\nv : P\n\u22a2 \u2203 a, \u2191(lift f hg) a = v"}, {"tactic": "obtain \u27e8x, hx\u27e9 := H v", "state_before": "case mpr\nM : Type u_3\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_2\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : \u2200 (v : P), \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst\nv : P\n\u22a2 \u2203 a, \u2191(lift f hg) a = v", "state_after": "case mpr.intro\nM : Type u_3\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_2\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : \u2200 (v : P), \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst\nv : P\nx : M \u00d7 { x // x \u2208 S }\nhx : v * \u2191g \u2191x.snd = \u2191g x.fst\n\u22a2 \u2203 a, \u2191(lift f hg) a = v"}, {"tactic": "use f.mk' x.1 x.2", "state_before": "case mpr.intro\nM : Type u_3\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_2\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : \u2200 (v : P), \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst\nv : P\nx : M \u00d7 { x // x \u2208 S }\nhx : v * \u2191g \u2191x.snd = \u2191g x.fst\n\u22a2 \u2203 a, \u2191(lift f hg) a = v", "state_after": "case mpr.intro\nM : Type u_3\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_2\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : \u2200 (v : P), \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst\nv : P\nx : M \u00d7 { x // x \u2208 S }\nhx : v * \u2191g \u2191x.snd = \u2191g x.fst\n\u22a2 \u2191(lift f hg) (mk' f x.fst x.snd) = v"}, {"tactic": "rw [lift_mk', mul_inv_left hg, mul_comm, \u2190 hx]", "state_before": "case mpr.intro\nM : Type u_3\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_2\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nH : \u2200 (v : P), \u2203 x, v * \u2191g \u2191x.snd = \u2191g x.fst\nv : P\nx : M \u00d7 { x // x \u2208 S }\nhx : v * \u2191g \u2191x.snd = \u2191g x.fst\n\u22a2 \u2191(lift f hg) (mk' f x.fst x.snd) = v", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/Liouville/LiouvilleWith.lean", "full_name": "LiouvilleWith.add_int", "start": [232, 1], "end": [233, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "full_name": "Subgroup.toSubmonoid_eq", "start": [460, 1], "end": [461, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/IteratedDeriv.lean", "full_name": "ContDiffOn.differentiableOn_iteratedDerivWithin", "start": [172, 1], "end": [175, 90], "traced_tactics": [{"tactic": "rwa [insert_eq_of_mem hx]", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type ?u.58253\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nn\u271d : \u2115\nf : \ud835\udd5c \u2192 F\ns : Set \ud835\udd5c\nx\u271d : \ud835\udd5c\nn : \u2115\u221e\nm : \u2115\nh : ContDiffOn \ud835\udd5c n f s\nhmn : \u2191m < n\nhs : UniqueDiffOn \ud835\udd5c s\nx : \ud835\udd5c\nhx : x \u2208 s\n\u22a2 UniqueDiffOn \ud835\udd5c (insert x s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.sub_inter", "start": [1903, 1], "end": [1904, 91], "traced_tactics": [{"tactic": "rw [sub_add_inter s t, tsub_add_cancel_of_le (inter_le_left s t)]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.195832\n\u03b3 : Type ?u.195835\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u : Multiset \u03b1\na b : \u03b1\ns t : Multiset \u03b1\n\u22a2 s - s \u2229 t + ?m.196298 s t = s - t + ?m.196298 s t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Clique.lean", "full_name": "SimpleGraph.IsClique.subset", "start": [81, 1], "end": [83, 28], "traced_tactics": [{"tactic": "simp_rw [isClique_iff]", "state_before": "\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ns t : Set \u03b1\nh : t \u2286 s\n\u22a2 IsClique G s \u2192 IsClique G t", "state_after": "\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ns t : Set \u03b1\nh : t \u2286 s\n\u22a2 Set.Pairwise s G.Adj \u2192 Set.Pairwise t G.Adj"}, {"tactic": "exact Set.Pairwise.mono h", "state_before": "\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\ns t : Set \u03b1\nh : t \u2286 s\n\u22a2 Set.Pairwise s G.Adj \u2192 Set.Pairwise t G.Adj", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "full_name": "MeasureTheory.ProbabilityMeasure.toMeasure_comp_toFiniteMeasure_eq_toMeasure", "start": [157, 1], "end": [159, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Monotone/Basic.lean", "full_name": "StrictAnti.ite'", "start": [614, 11], "end": [618, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.coe_inv", "start": [272, 1], "end": [273, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/Basic.lean", "full_name": "Equiv.Perm.extendDomain_refl", "start": [1454, 1], "end": [1455, 27], "traced_tactics": [{"tactic": "simp [Perm.extendDomain]", "state_before": "\u03b1' : Type u_2\n\u03b2' : Type u_1\ne : Perm \u03b1'\np : \u03b2' \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1' \u2243 Subtype p\n\u22a2 extendDomain (Equiv.refl \u03b1') f = Equiv.refl \u03b2'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/Basic.lean", "full_name": "Ideal.zero_mem", "start": [57, 11], "end": [58, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/Adjoin.lean", "full_name": "IntermediateField.isSplittingField_iff", "start": [531, 1], "end": [538, 30], "traced_tactics": [{"tactic": "suffices _ \u2192 (Algebra.adjoin F (p.rootSet K) = \u22a4 \u2194 K = adjoin F (p.rootSet E)) by\n  exact \u27e8fun h => \u27e8h.1, (this h.1).mp h.2\u27e9, fun h => \u27e8h.1, (this h.1).mpr h.2\u27e9\u27e9", "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nS : Set E\n\u03b1 : E\np : F[X]\nK : IntermediateField F E\n\u22a2 IsSplittingField F { x // x \u2208 K } p \u2194 Splits (algebraMap F { x // x \u2208 K }) p \u2227 K = adjoin F (rootSet p E)", "state_after": "F : Type u_1\ninst\u271d\u00b2 : Field F\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nS : Set E\n\u03b1 : E\np : F[X]\nK : IntermediateField F E\n\u22a2 Splits (algebraMap F { x // x \u2208 K }) p \u2192\n    (Algebra.adjoin F (rootSet p { x // x \u2208 K }) = \u22a4 \u2194 K = adjoin F (rootSet p E))"}, {"tactic": "simp_rw [SetLike.ext_iff, \u2190 mem_toSubalgebra, \u2190 SetLike.ext_iff]", "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nS : Set E\n\u03b1 : E\np : F[X]\nK : IntermediateField F E\n\u22a2 Splits (algebraMap F { x // x \u2208 K }) p \u2192\n    (Algebra.adjoin F (rootSet p { x // x \u2208 K }) = \u22a4 \u2194 K = adjoin F (rootSet p E))", "state_after": "F : Type u_1\ninst\u271d\u00b2 : Field F\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nS : Set E\n\u03b1 : E\np : F[X]\nK : IntermediateField F E\n\u22a2 Splits (algebraMap F { x // x \u2208 K }) p \u2192\n    (Algebra.adjoin F (rootSet p { x // x \u2208 K }) = \u22a4 \u2194 K.toSubalgebra = (adjoin F (rootSet p E)).toSubalgebra)"}, {"tactic": "rw [adjoin_algebraic_toSubalgebra fun x => isAlgebraic_of_mem_rootSet]", "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nS : Set E\n\u03b1 : E\np : F[X]\nK : IntermediateField F E\n\u22a2 Splits (algebraMap F { x // x \u2208 K }) p \u2192\n    (Algebra.adjoin F (rootSet p { x // x \u2208 K }) = \u22a4 \u2194 K.toSubalgebra = (adjoin F (rootSet p E)).toSubalgebra)", "state_after": "F : Type u_1\ninst\u271d\u00b2 : Field F\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nS : Set E\n\u03b1 : E\np : F[X]\nK : IntermediateField F E\n\u22a2 Splits (algebraMap F { x // x \u2208 K }) p \u2192\n    (Algebra.adjoin F (rootSet p { x // x \u2208 K }) = \u22a4 \u2194 K.toSubalgebra = Algebra.adjoin F (rootSet p E))"}, {"tactic": "refine' fun hp => (adjoin_rootSet_eq_range hp K.val).symm.trans _", "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nS : Set E\n\u03b1 : E\np : F[X]\nK : IntermediateField F E\n\u22a2 Splits (algebraMap F { x // x \u2208 K }) p \u2192\n    (Algebra.adjoin F (rootSet p { x // x \u2208 K }) = \u22a4 \u2194 K.toSubalgebra = Algebra.adjoin F (rootSet p E))", "state_after": "F : Type u_1\ninst\u271d\u00b2 : Field F\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nS : Set E\n\u03b1 : E\np : F[X]\nK : IntermediateField F E\nhp : Splits (algebraMap F { x // x \u2208 K }) p\n\u22a2 Algebra.adjoin F (rootSet p E) = AlgHom.range (val K) \u2194 K.toSubalgebra = Algebra.adjoin F (rootSet p E)"}, {"tactic": "rw [\u2190 K.range_val, eq_comm]", "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nS : Set E\n\u03b1 : E\np : F[X]\nK : IntermediateField F E\nhp : Splits (algebraMap F { x // x \u2208 K }) p\n\u22a2 Algebra.adjoin F (rootSet p E) = AlgHom.range (val K) \u2194 K.toSubalgebra = 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"5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean", "full_name": "midpoint_sub_add", "start": [229, 1], "end": [230, 84], "traced_tactics": [{"tactic": "rw [sub_eq_add_neg, \u2190 vadd_eq_add, \u2190 vadd_eq_add, \u2190 midpoint_vadd_midpoint]", "state_before": "R : Type u_2\nV : Type u_1\nV' : Type ?u.138949\nP : Type ?u.138952\nP' : Type ?u.138955\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y\u271d z : P\nx y : V\n\u22a2 midpoint R (x - y) (x + y) = x", "state_after": "R : Type u_2\nV : Type u_1\nV' : Type ?u.138949\nP : Type ?u.138952\nP' : Type ?u.138955\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y\u271d z : P\nx y : V\n\u22a2 midpoint R x x +\u1d65 midpoint R (-y) y = x"}, {"tactic": "simp", "state_before": "R : Type u_2\nV : Type u_1\nV' : Type ?u.138949\nP : Type ?u.138952\nP' : Type ?u.138955\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y\u271d z : P\nx y : V\n\u22a2 midpoint R x x +\u1d65 midpoint R (-y) y = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/PNat/Prime.lean", "full_name": "PNat.gcd_eq_left", "start": [293, 1], "end": [298, 26], "traced_tactics": [{"tactic": "rw [dvd_iff]", "state_before": "m n : \u2115+\n\u22a2 m \u2223 n \u2192 gcd m n = m", "state_after": "m n : \u2115+\n\u22a2 \u2191m \u2223 \u2191n \u2192 gcd m n = m"}, {"tactic": "intro h", "state_before": "m n : \u2115+\n\u22a2 \u2191m \u2223 \u2191n \u2192 gcd m n = m", "state_after": "m n : \u2115+\nh : \u2191m \u2223 \u2191n\n\u22a2 gcd m n = m"}, {"tactic": "apply eq", "state_before": "m n : \u2115+\nh : \u2191m \u2223 \u2191n\n\u22a2 gcd m n = m", "state_after": "case a\nm n : \u2115+\nh : \u2191m \u2223 \u2191n\n\u22a2 \u2191(gcd m n) = \u2191m"}, {"tactic": "simp only [gcd_coe]", "state_before": "case a\nm n : \u2115+\nh : \u2191m \u2223 \u2191n\n\u22a2 \u2191(gcd m n) = \u2191m", "state_after": "case a\nm n : \u2115+\nh : \u2191m \u2223 \u2191n\n\u22a2 Nat.gcd \u2191m \u2191n = \u2191m"}, {"tactic": "apply Nat.gcd_eq_left h", "state_before": "case a\nm n : \u2115+\nh : \u2191m \u2223 \u2191n\n\u22a2 Nat.gcd \u2191m \u2191n = \u2191m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Sigma/Interval.lean", "full_name": "Sigma.card_Ioc", "start": [71, 1], "end": [72, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/MonoidAlgebra/Basic.lean", "full_name": "MonoidAlgebra.mul_apply_right", "start": [1070, 1], "end": [1075, 58], "traced_tactics": [{"tactic": "rw [\u2190 Finsupp.sum_apply, \u2190 Finsupp.mul_sum f g, g.sum_single]", "state_before": "k : Type u\u2081\nG : Type u\u2082\nR : Type ?u.1518387\ninst\u271d\u00b9 : Semiring k\ninst\u271d : Group G\nf g : MonoidAlgebra k G\nx : G\n\u22a2 \u2191(f * g) x = sum g fun a b => \u2191(f * single a b) x", "state_after": "no goals"}, {"tactic": "simp only [mul_single_apply, Finsupp.sum]", "state_before": "k : Type u\u2081\nG : Type u\u2082\nR : Type ?u.1518387\ninst\u271d\u00b9 : Semiring k\ninst\u271d : Group G\nf g : MonoidAlgebra k G\nx : G\n\u22a2 (sum g fun a b => \u2191(f * single a b) x) = sum g fun a b => \u2191f (x * a\u207b\u00b9) * b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicTopology/SimplexCategory.lean", "full_name": "SimplexCategory.Hom.mk_toOrderHom_apply", "start": [132, 1], "end": [134, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/ZPow.lean", "full_name": "Matrix.zpow_neg", "start": [139, 1], "end": [144, 18], "traced_tactics": [{"tactic": "rw [zpow_negSucc, neg_negSucc, zpow_ofNat, nonsing_inv_nonsing_inv]", "state_before": "n' : Type u_1\ninst\u271d\u00b2 : DecidableEq n'\ninst\u271d\u00b9 : Fintype n'\nR : Type u_2\ninst\u271d : CommRing R\nA : M\nh : IsUnit (det A)\nn : \u2115\n\u22a2 A ^ (- -[n+1]) = (A ^ -[n+1])\u207b\u00b9", "state_after": "case h\nn' : Type u_1\ninst\u271d\u00b2 : DecidableEq n'\ninst\u271d\u00b9 : Fintype n'\nR : Type u_2\ninst\u271d : CommRing R\nA : M\nh : IsUnit (det A)\nn : \u2115\n\u22a2 IsUnit (det (A ^ (n + 1)))"}, {"tactic": "rw [det_pow]", "state_before": "case h\nn' : Type u_1\ninst\u271d\u00b2 : DecidableEq n'\ninst\u271d\u00b9 : Fintype n'\nR : Type u_2\ninst\u271d : CommRing R\nA : M\nh : IsUnit (det A)\nn : \u2115\n\u22a2 IsUnit (det (A ^ (n + 1)))", "state_after": "case h\nn' : Type u_1\ninst\u271d\u00b2 : DecidableEq n'\ninst\u271d\u00b9 : Fintype n'\nR : Type u_2\ninst\u271d : CommRing R\nA : M\nh : IsUnit (det A)\nn : \u2115\n\u22a2 IsUnit (det A ^ (n + 1))"}, {"tactic": "exact h.pow _", "state_before": "case h\nn' : Type u_1\ninst\u271d\u00b2 : DecidableEq n'\ninst\u271d\u00b9 : Fintype n'\nR : Type u_2\ninst\u271d : CommRing R\nA : M\nh : IsUnit (det A)\nn : \u2115\n\u22a2 IsUnit (det A ^ (n + 1))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/ContinuousOn.lean", "full_name": "Inducing.continuousWithinAt_iff", "start": [1070, 1], "end": [1072, 66], "traced_tactics": [{"tactic": "simp_rw [ContinuousWithinAt, Inducing.tendsto_nhds_iff hg]", "state_before": "\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.350125\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nhg : Inducing g\ns : Set \u03b1\nx : \u03b1\n\u22a2 ContinuousWithinAt f s x \u2194 ContinuousWithinAt (g \u2218 f) s x", "state_after": "\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.350125\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nhg : Inducing g\ns : Set \u03b1\nx : \u03b1\n\u22a2 Tendsto (g \u2218 f) (\ud835\udcdd[s] x) (\ud835\udcdd (g (f x))) \u2194 Tendsto (g \u2218 f) (\ud835\udcdd[s] x) (\ud835\udcdd ((g \u2218 f) x))"}, {"tactic": "rfl", "state_before": "\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.350125\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nhg : Inducing g\ns : Set \u03b1\nx : \u03b1\n\u22a2 Tendsto (g \u2218 f) (\ud835\udcdd[s] x) (\ud835\udcdd (g (f x))) \u2194 Tendsto (g \u2218 f) (\ud835\udcdd[s] x) (\ud835\udcdd ((g \u2218 f) x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean", "full_name": "CategoryTheory.IsPullback.of_hasPullback", "start": [239, 1], "end": [241, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Subsemiring/Basic.lean", "full_name": "Subsemiring.list_sum_mem", "start": [333, 11], "end": [334, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Congruence.lean", "full_name": "RingCon.coe_smul", "start": [242, 1], "end": [243, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Dfinsupp/Basic.lean", "full_name": "Dfinsupp.filter_single_pos", "start": [710, 1], "end": [711, 74], "traced_tactics": [{"tactic": "rw [filter_single, if_pos h]", "state_before": "\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d : \u03b9\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\ni : \u03b9\nx : \u03b2 i\nh : p i\n\u22a2 filter p (single i x) = single i x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GroupPower/Order.lean", "full_name": "pow_lt_pow", "start": [501, 1], "end": [502, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/Language.lean", "full_name": "Language.mem_iSup", "start": [213, 1], "end": [214, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Rat/Lemmas.lean", "full_name": "Rat.neg_divInt", "start": [215, 1], "end": [216, 83], "traced_tactics": [{"tactic": "rcases Int.eq_nat_or_neg d with \u27e8_, rfl | rfl\u27e9 <;> simp [divInt_neg', neg_mkRat]", "state_before": "n d : Int\n\u22a2 -(n /. d) = -n /. d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Stream/Init.lean", "full_name": "Stream'.corec_id_f_eq_iterate", "start": [386, 1], "end": [387, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "full_name": "ModelWithCorners.secondCountableTopology", "start": [364, 11], "end": [366, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean", "full_name": "Polynomial.derivWithin", "start": [145, 11], "end": [148, 16], "traced_tactics": [{"tactic": "rw [DifferentiableAt.derivWithin p.differentiableAt hxs]", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR : Type ?u.85451\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Algebra R \ud835\udd5c\np : \ud835\udd5c[X]\nq : R[X]\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\n\u22a2 derivWithin (fun x => eval x p) s x = eval x (\u2191derivative p)", "state_after": "\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR : Type ?u.85451\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Algebra R \ud835\udd5c\np : \ud835\udd5c[X]\nq : R[X]\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\n\u22a2 deriv (fun x => eval x p) x = eval x (\u2191derivative p)"}, {"tactic": "exact p.deriv", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR : Type ?u.85451\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Algebra R \ud835\udd5c\np : \ud835\udd5c[X]\nq : R[X]\nhxs : UniqueDiffWithinAt \ud835\udd5c s x\n\u22a2 deriv (fun x => eval x p) x = eval x (\u2191derivative p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Order/Lemmas.lean", "full_name": "Nat.div_div_div_eq_div", "start": [195, 1], "end": [208, 57], "traced_tactics": [{"tactic": "simp", "state_before": "a\u271d b\u271d m n k a b c x\u271d\u00b9 x\u271d : \u2115\ndvd : x\u271d\u00b9 \u2223 0\ndvd2 : 0 \u2223 x\u271d\n\u22a2 x\u271d / (0 / x\u271d\u00b9) / x\u271d\u00b9 = x\u271d / 0", "state_after": "no goals"}, {"tactic": "simp at dvd", "state_before": "a\u271d\u00b9 b\u271d m n k a\u271d b c a x\u271d : \u2115\ndvd : 0 \u2223 a + 1\ndvd2 : a + 1 \u2223 x\u271d\n\u22a2 x\u271d / ((a + 1) / 0) / 0 = x\u271d / (a + 1)", "state_after": "no goals"}, {"tactic": "have a_split : a + 1 \u2260 0 := succ_ne_zero a", "state_before": "a\u271d\u00b9 b\u271d m n k a\u271d b c\u271d a c x\u271d : \u2115\ndvd : c + 1 \u2223 a + 1\ndvd2 : a + 1 \u2223 x\u271d\n\u22a2 x\u271d / ((a + 1) / (c + 1)) / (c + 1) = x\u271d / (a + 1)", "state_after": "a\u271d\u00b9 b\u271d m n k a\u271d b c\u271d a c x\u271d : \u2115\ndvd : c + 1 \u2223 a + 1\ndvd2 : a + 1 \u2223 x\u271d\na_split : a + 1 \u2260 0\n\u22a2 x\u271d / ((a + 1) / (c + 1)) / (c + 1) = x\u271d / (a + 1)"}, {"tactic": "have c_split : c + 1 \u2260 0 := succ_ne_zero c", "state_before": "a\u271d\u00b9 b\u271d m n k a\u271d b c\u271d a c x\u271d : \u2115\ndvd : c + 1 \u2223 a + 1\ndvd2 : a + 1 \u2223 x\u271d\na_split : a + 1 \u2260 0\n\u22a2 x\u271d / ((a + 1) / (c + 1)) / (c + 1) = x\u271d / (a + 1)", "state_after": "a\u271d\u00b9 b\u271d m n k a\u271d b c\u271d a c x\u271d : \u2115\ndvd : c + 1 \u2223 a + 1\ndvd2 : a + 1 \u2223 x\u271d\na_split : a + 1 \u2260 0\nc_split : c + 1 \u2260 0\n\u22a2 x\u271d / ((a + 1) / (c + 1)) / (c + 1) = x\u271d / (a + 1)"}, {"tactic": "rcases dvd2 with \u27e8k, rfl\u27e9", "state_before": "a\u271d\u00b9 b\u271d m n k a\u271d b c\u271d a c x\u271d : \u2115\ndvd : c + 1 \u2223 a + 1\ndvd2 : a + 1 \u2223 x\u271d\na_split : a + 1 \u2260 0\nc_split : c + 1 \u2260 0\n\u22a2 x\u271d / ((a + 1) / (c + 1)) / (c + 1) = x\u271d / (a + 1)", "state_after": "case intro\na\u271d\u00b9 b\u271d m n k\u271d a\u271d b c\u271d a c : \u2115\ndvd : c + 1 \u2223 a + 1\na_split : a + 1 \u2260 0\nc_split : c + 1 \u2260 0\nk : \u2115\n\u22a2 (a + 1) * k / ((a + 1) / (c + 1)) / (c + 1) = (a + 1) * k / (a + 1)"}, {"tactic": "rcases dvd with \u27e8k2, pr\u27e9", "state_before": "case intro\na\u271d\u00b9 b\u271d m n k\u271d a\u271d b c\u271d a c : \u2115\ndvd : c + 1 \u2223 a + 1\na_split : a + 1 \u2260 0\nc_split : c + 1 \u2260 0\nk : \u2115\n\u22a2 (a + 1) * k / ((a + 1) / (c + 1)) / (c + 1) = (a + 1) * k / (a + 1)", "state_after": "case intro.intro\na\u271d\u00b9 b\u271d m n k\u271d a\u271d b c\u271d a c : \u2115\na_split : a + 1 \u2260 0\nc_split : c + 1 \u2260 0\nk k2 : \u2115\npr : a + 1 = (c + 1) * k2\n\u22a2 (a + 1) * k / ((a + 1) / (c + 1)) / (c + 1) = (a + 1) * k / (a + 1)"}, {"tactic": "have k2_nonzero : k2 \u2260 0 := fun k2_zero => by simp [k2_zero] at pr", "state_before": "case intro.intro\na\u271d\u00b9 b\u271d m n k\u271d a\u271d b c\u271d a c : \u2115\na_split : a + 1 \u2260 0\nc_split : c + 1 \u2260 0\nk k2 : \u2115\npr : a + 1 = (c + 1) * k2\n\u22a2 (a + 1) * k / ((a + 1) / (c + 1)) / (c + 1) = (a + 1) * k / (a + 1)", "state_after": "case intro.intro\na\u271d\u00b9 b\u271d m n k\u271d a\u271d b c\u271d a c : \u2115\na_split : a + 1 \u2260 0\nc_split : c + 1 \u2260 0\nk k2 : \u2115\npr : a + 1 = (c + 1) * k2\nk2_nonzero : k2 \u2260 0\n\u22a2 (a + 1) * k / ((a + 1) / (c + 1)) / (c + 1) = (a + 1) * k / (a + 1)"}, {"tactic": "rw [Nat.mul_div_cancel_left k (Nat.pos_of_ne_zero a_split), pr,\n  Nat.mul_div_cancel_left k2 (Nat.pos_of_ne_zero c_split), Nat.mul_comm ((c + 1) * k2) k, \u2190\n  Nat.mul_assoc k (c + 1) k2, Nat.mul_div_cancel _ (Nat.pos_of_ne_zero k2_nonzero),\n  Nat.mul_div_cancel _ (Nat.pos_of_ne_zero c_split)]", "state_before": "case intro.intro\na\u271d\u00b9 b\u271d m n k\u271d a\u271d b c\u271d a c : \u2115\na_split : a + 1 \u2260 0\nc_split : c + 1 \u2260 0\nk k2 : \u2115\npr : a + 1 = (c + 1) * k2\nk2_nonzero : k2 \u2260 0\n\u22a2 (a + 1) * k / ((a + 1) / (c + 1)) / (c + 1) = (a + 1) * k / (a + 1)", "state_after": "no goals"}, {"tactic": "simp [k2_zero] at pr", "state_before": "a\u271d\u00b9 b\u271d m n k\u271d a\u271d b c\u271d a c : \u2115\na_split : a + 1 \u2260 0\nc_split : c + 1 \u2260 0\nk k2 : \u2115\npr : a + 1 = (c + 1) * k2\nk2_zero : k2 = 0\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finsupp/Defs.lean", "full_name": "Finsupp.support_eq_empty", "start": [217, 1], "end": [218, 56], "traced_tactics": [{"tactic": "exact_mod_cast @Function.support_eq_empty_iff _ _ _ f", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.15456\n\u03b3 : Type ?u.15459\n\u03b9 : Type ?u.15462\nM : Type u_2\nM' : Type ?u.15468\nN : Type ?u.15471\nP : Type ?u.15474\nG : Type ?u.15477\nH : Type ?u.15480\nR : Type ?u.15483\nS : Type ?u.15486\ninst\u271d : Zero M\nf : \u03b1 \u2192\u2080 M\n\u22a2 f.support = \u2205 \u2194 f = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Probability/ConditionalProbability.lean", "full_name": "ProbabilityTheory.inter_pos_of_cond_ne_zero", "start": [113, 1], "end": [117, 35], "traced_tactics": [{"tactic": "refine' pos_iff_ne_zero.mpr (right_ne_zero_of_mul _)", "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\ns t : Set \u03a9\nhms : MeasurableSet s\nhcst : \u2191\u2191(\u03bc[|s]) t \u2260 0\n\u22a2 0 < \u2191\u2191\u03bc (s \u2229 t)", "state_after": "case refine'_1\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\ns t : Set \u03a9\nhms : MeasurableSet s\nhcst : \u2191\u2191(\u03bc[|s]) t \u2260 0\n\u22a2 \u211d\u22650\u221e\n\ncase refine'_2\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\ns t : Set \u03a9\nhms : MeasurableSet s\nhcst : \u2191\u2191(\u03bc[|s]) t \u2260 0\n\u22a2 ?refine'_1 * \u2191\u2191\u03bc (s \u2229 t) \u2260 0"}, {"tactic": "convert hcst", "state_before": "case refine'_2\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\ns t : Set \u03a9\nhms : MeasurableSet s\nhcst : \u2191\u2191(\u03bc[|s]) t \u2260 0\n\u22a2 (\u2191\u2191\u03bc s)\u207b\u00b9 * \u2191\u2191\u03bc (s \u2229 t) \u2260 0", "state_after": "case h.e'_2\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\ns t : Set \u03a9\nhms : MeasurableSet s\nhcst : \u2191\u2191(\u03bc[|s]) t \u2260 0\n\u22a2 (\u2191\u2191\u03bc s)\u207b\u00b9 * \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191(\u03bc[|s]) t"}, {"tactic": "simp [hms, Set.inter_comm, cond]", "state_before": "case h.e'_2\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\ns t : Set \u03a9\nhms : MeasurableSet s\nhcst : \u2191\u2191(\u03bc[|s]) t \u2260 0\n\u22a2 (\u2191\u2191\u03bc s)\u207b\u00b9 * \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191(\u03bc[|s]) t", "state_after": "no goals"}, {"tactic": "exact (\u03bc s)\u207b\u00b9", "state_before": "case refine'_1\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\n\u03bc : MeasureTheory.Measure \u03a9\ns t : Set \u03a9\nhms : MeasurableSet s\nhcst : \u2191\u2191(\u03bc[|s]) t \u2260 0\n\u22a2 \u211d\u22650\u221e", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Laurent.lean", "full_name": "LaurentPolynomial.leftInverse_trunc_toLaurent", "start": [364, 1], "end": [371, 15], "traced_tactics": [{"tactic": "refine' fun f => f.induction_on' _ _", "state_before": "R : Type u_1\ninst\u271d : Semiring R\n\u22a2 Function.LeftInverse \u2191trunc \u2191toLaurent", "state_after": "case refine'_1\nR : Type u_1\ninst\u271d : Semiring R\nf : R[X]\n\u22a2 \u2200 (p q : R[X]), \u2191trunc (\u2191toLaurent p) = p \u2192 \u2191trunc (\u2191toLaurent q) = q \u2192 \u2191trunc (\u2191toLaurent (p + q)) = p + q\n\ncase refine'_2\nR : Type u_1\ninst\u271d : Semiring R\nf : R[X]\n\u22a2 \u2200 (n : \u2115) (a : R), \u2191trunc (\u2191toLaurent (\u2191(monomial n) a)) = \u2191(monomial n) a"}, {"tactic": "intro f g hf hg", "state_before": "case refine'_1\nR : Type u_1\ninst\u271d : Semiring R\nf : R[X]\n\u22a2 \u2200 (p q : R[X]), \u2191trunc (\u2191toLaurent p) = p \u2192 \u2191trunc (\u2191toLaurent q) = q \u2192 \u2191trunc (\u2191toLaurent (p + q)) = p + q", "state_after": "case refine'_1\nR : Type u_1\ninst\u271d : Semiring R\nf\u271d f g : R[X]\nhf : \u2191trunc (\u2191toLaurent f) = f\nhg : \u2191trunc (\u2191toLaurent g) = g\n\u22a2 \u2191trunc (\u2191toLaurent (f + g)) = f + g"}, {"tactic": "simp only [hf, hg, _root_.map_add]", "state_before": "case refine'_1\nR : Type u_1\ninst\u271d : Semiring R\nf\u271d f g : R[X]\nhf : \u2191trunc (\u2191toLaurent f) = f\nhg : \u2191trunc (\u2191toLaurent g) = g\n\u22a2 \u2191trunc (\u2191toLaurent (f + g)) = f + g", "state_after": "no goals"}, {"tactic": "intro n r", "state_before": "case refine'_2\nR : Type u_1\ninst\u271d : Semiring R\nf : R[X]\n\u22a2 \u2200 (n : \u2115) (a : R), \u2191trunc (\u2191toLaurent (\u2191(monomial n) a)) = \u2191(monomial n) a", "state_after": "case refine'_2\nR : Type u_1\ninst\u271d : Semiring R\nf : R[X]\nn : \u2115\nr : R\n\u22a2 \u2191trunc (\u2191toLaurent (\u2191(monomial n) r)) = \u2191(monomial n) r"}, {"tactic": "simp only [Polynomial.toLaurent_C_mul_T, trunc_C_mul_T, Int.coe_nat_nonneg, Int.toNat_coe_nat,\n  if_true]", "state_before": "case refine'_2\nR : Type u_1\ninst\u271d : Semiring R\nf : R[X]\nn : \u2115\nr : R\n\u22a2 \u2191trunc (\u2191toLaurent (\u2191(monomial n) r)) = \u2191(monomial n) r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fin/Tuple/Basic.lean", "full_name": "Fin.update_snoc_last", "start": [533, 1], "end": [539, 9], "traced_tactics": [{"tactic": "ext j", "state_before": "m n : \u2115\n\u03b1 : Fin (n + 1) \u2192 Type u\nx : \u03b1 (last n)\nq : (i : Fin (n + 1)) \u2192 \u03b1 i\np : (i : Fin n) \u2192 \u03b1 (\u2191castSucc i)\ni : Fin n\ny : \u03b1 (\u2191castSucc i)\nz : \u03b1 (last n)\n\u22a2 update (snoc p x) (last n) z = snoc p z", "state_after": "case h\nm n : \u2115\n\u03b1 : Fin (n + 1) \u2192 Type u\nx : \u03b1 (last n)\nq : (i : Fin (n + 1)) \u2192 \u03b1 i\np : (i : Fin n) \u2192 \u03b1 (\u2191castSucc i)\ni : Fin n\ny : \u03b1 (\u2191castSucc i)\nz : \u03b1 (last n)\nj : Fin (n + 1)\n\u22a2 update (snoc p x) (last n) z j = snoc p z j"}, {"tactic": "by_cases h : j.val < n", "state_before": "case h\nm n : \u2115\n\u03b1 : Fin (n + 1) \u2192 Type u\nx : \u03b1 (last n)\nq : (i : Fin (n + 1)) \u2192 \u03b1 i\np : (i : Fin n) \u2192 \u03b1 (\u2191castSucc i)\ni : Fin n\ny : \u03b1 (\u2191castSucc i)\nz : \u03b1 (last n)\nj : Fin (n + 1)\n\u22a2 update (snoc p x) (last n) z j = snoc p z j", "state_after": "case pos\nm n : \u2115\n\u03b1 : Fin (n + 1) \u2192 Type u\nx : \u03b1 (last n)\nq : (i : Fin (n + 1)) \u2192 \u03b1 i\np : (i : Fin n) \u2192 \u03b1 (\u2191castSucc i)\ni : Fin n\ny : \u03b1 (\u2191castSucc i)\nz : \u03b1 (last n)\nj : Fin (n + 1)\nh : \u2191j < n\n\u22a2 update (snoc p x) (last n) z j = snoc p z j\n\ncase neg\nm n : \u2115\n\u03b1 : Fin (n + 1) \u2192 Type u\nx : \u03b1 (last n)\nq : (i : Fin (n + 1)) \u2192 \u03b1 i\np : (i : Fin n) \u2192 \u03b1 (\u2191castSucc i)\ni : Fin n\ny : \u03b1 (\u2191castSucc i)\nz : \u03b1 (last n)\nj : Fin (n + 1)\nh : \u00ac\u2191j < n\n\u22a2 update (snoc p x) (last n) z j = snoc p z j"}, {"tactic": "have : j \u2260 last n := ne_of_lt h", "state_before": "case pos\nm n : \u2115\n\u03b1 : Fin (n + 1) \u2192 Type u\nx : \u03b1 (last n)\nq : (i : Fin (n + 1)) \u2192 \u03b1 i\np : (i : Fin n) \u2192 \u03b1 (\u2191castSucc i)\ni : Fin n\ny : \u03b1 (\u2191castSucc i)\nz : \u03b1 (last n)\nj : Fin (n + 1)\nh : \u2191j < n\n\u22a2 update (snoc p x) (last n) z j = snoc p z j", "state_after": "case pos\nm n : \u2115\n\u03b1 : Fin (n + 1) \u2192 Type u\nx : \u03b1 (last n)\nq : (i : Fin (n + 1)) \u2192 \u03b1 i\np : (i : Fin n) \u2192 \u03b1 (\u2191castSucc i)\ni : Fin n\ny : \u03b1 (\u2191castSucc i)\nz : \u03b1 (last n)\nj : Fin (n + 1)\nh : \u2191j < n\nthis : j \u2260 last n\n\u22a2 update (snoc p x) (last n) z j = snoc p z j"}, {"tactic": "simp [h, update_noteq, this, snoc]", "state_before": "case pos\nm n : \u2115\n\u03b1 : Fin (n + 1) \u2192 Type u\nx : \u03b1 (last n)\nq : (i : Fin (n + 1)) \u2192 \u03b1 i\np : (i : Fin n) \u2192 \u03b1 (\u2191castSucc i)\ni : Fin n\ny : \u03b1 (\u2191castSucc i)\nz : \u03b1 (last n)\nj : Fin (n + 1)\nh : \u2191j < n\nthis : j \u2260 last n\n\u22a2 update (snoc p x) (last n) z j = snoc p z j", "state_after": "no goals"}, {"tactic": "rw [eq_last_of_not_lt h]", "state_before": "case neg\nm n : \u2115\n\u03b1 : Fin (n + 1) \u2192 Type u\nx : \u03b1 (last n)\nq : (i : Fin (n + 1)) \u2192 \u03b1 i\np : (i : Fin n) \u2192 \u03b1 (\u2191castSucc i)\ni : Fin n\ny : \u03b1 (\u2191castSucc i)\nz : \u03b1 (last n)\nj : Fin (n + 1)\nh : \u00ac\u2191j < n\n\u22a2 update (snoc p x) (last n) z j = snoc p z j", "state_after": "case neg\nm n : \u2115\n\u03b1 : Fin (n + 1) \u2192 Type u\nx : \u03b1 (last n)\nq : (i : Fin (n + 1)) \u2192 \u03b1 i\np : (i : Fin n) \u2192 \u03b1 (\u2191castSucc i)\ni : Fin n\ny : \u03b1 (\u2191castSucc i)\nz : \u03b1 (last n)\nj : Fin (n + 1)\nh : \u00ac\u2191j < n\n\u22a2 update (snoc p x) (last n) z (last n) = snoc p z (last n)"}, {"tactic": "simp", "state_before": "case neg\nm n : \u2115\n\u03b1 : Fin (n + 1) \u2192 Type u\nx : \u03b1 (last n)\nq : (i : Fin (n + 1)) \u2192 \u03b1 i\np : (i : Fin n) \u2192 \u03b1 (\u2191castSucc i)\ni : Fin n\ny : \u03b1 (\u2191castSucc i)\nz : \u03b1 (last n)\nj : Fin (n + 1)\nh : \u00ac\u2191j < n\n\u22a2 update (snoc p x) (last n) z (last n) = snoc p z (last n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.coe_max'", "start": [1515, 1], "end": [1516, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Sigma.lean", "full_name": "List.Perm.kunion_left", "start": [734, 1], "end": [737, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/SesquilinearForm.lean", "full_name": "LinearMap.isOrtho_flip", "start": [78, 1], "end": [79, 36], "traced_tactics": [{"tactic": "simp_rw [isOrtho_def, flip_apply]", "state_before": "R : Type u_2\nR\u2081 : Type u_1\nR\u2082 : Type ?u.19748\nR\u2083 : Type ?u.19751\nM : Type ?u.19754\nM\u2081 : Type u_3\nM\u2082 : Type ?u.19760\nM\u2097\u2081 : Type ?u.19763\nM\u2097\u2081' : Type ?u.19766\nM\u2097\u2082 : Type ?u.19769\nM\u2097\u2082' : Type ?u.19772\nK : Type ?u.19775\nK\u2081 : Type ?u.19778\nK\u2082 : Type ?u.19781\nV : Type ?u.19784\nV\u2081 : Type ?u.19787\nV\u2082 : Type ?u.19790\nn : Type ?u.19793\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring R\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b3 : Module R\u2081 M\u2081\ninst\u271d\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9 : AddCommMonoid M\u2082\ninst\u271d : Module R\u2082 M\u2082\nI\u2081 : R\u2081 \u2192+* R\nI\u2082 : R\u2082 \u2192+* R\nI\u2081' : R\u2081 \u2192+* R\nB : M\u2081 \u2192\u209b\u2097[I\u2081] M\u2081 \u2192\u209b\u2097[I\u2081'] R\nx y : M\u2081\n\u22a2 IsOrtho B x y \u2194 IsOrtho (flip B) y x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Complex/Basic.lean", "full_name": "Complex.lim_conj", "start": [1318, 1], "end": [1320, 78], "traced_tactics": [{"tactic": "simp [cauSeqConj, (lim_re _).symm, cauSeqRe]", "state_before": "f : CauSeq \u2102 \u2191abs\n\u22a2 (CauSeq.lim (cauSeqConj f)).re = (\u2191(starRingEnd \u2102) (CauSeq.lim f)).re", "state_after": "no goals"}, {"tactic": "simp [cauSeqConj, (lim_im _).symm, cauSeqIm, (lim_neg _).symm]", "state_before": "f : CauSeq \u2102 \u2191abs\n\u22a2 (CauSeq.lim (cauSeqConj f)).im = (\u2191(starRingEnd \u2102) (CauSeq.lim f)).im", "state_after": "f : CauSeq \u2102 \u2191abs\n\u22a2 CauSeq.lim { val := fun n => -(\u2191f n).im, property := (_ : (fun f => IsCauSeq abs' f) fun n => -(\u2191f n).im) } =\n    CauSeq.lim (-{ val := fun n => (\u2191f n).im, property := (_ : IsCauSeq abs' fun n => (\u2191f n).im) })"}, {"tactic": "rfl", "state_before": "f : CauSeq \u2102 \u2191abs\n\u22a2 CauSeq.lim { val := fun n => -(\u2191f n).im, property := (_ : (fun f => IsCauSeq abs' f) fun n => -(\u2191f n).im) } =\n    CauSeq.lim (-{ val := fun n => (\u2191f n).im, property := (_ : IsCauSeq abs' fun n => (\u2191f n).im) })", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Group/Abs.lean", "full_name": "neg_lt_of_abs_lt", "start": [203, 1], "end": [204, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Pointwise.lean", "full_name": "Filter.map_div", "start": [903, 11], "end": [904, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean", "full_name": "NonUnitalSubsemiring.closure_univ", "start": [723, 1], "end": [724, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.EventuallyEq.le", "start": [1647, 1], "end": [1648, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Constructions.lean", "full_name": "ContinuousAt.update", "start": [1237, 1], "end": [1239, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.volume_preserving_finTwoArrow", "start": [821, 1], "end": [824, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/SubsetProperties.lean", "full_name": "Filter.hasBasis_cocompact", "start": [521, 1], "end": [526, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Cofinite.lean", "full_name": "Filter.atTop_le_cofinite", "start": [108, 1], "end": [109, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GroupWithZero/Commute.lean", "full_name": "Commute.zero_left", "start": [60, 1], "end": [61, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.disjoint_union_left", "start": [1502, 1], "end": [1503, 59], "traced_tactics": [{"tactic": "simp only [disjoint_left, mem_union, or_imp, forall_and]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.153416\n\u03b3 : Type ?u.153419\ninst\u271d : DecidableEq \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\n\u22a2 _root_.Disjoint (s \u222a t) u \u2194 _root_.Disjoint s u \u2227 _root_.Disjoint t u", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "full_name": "CategoryTheory.Presieve.ofArrows_pUnit", "start": [136, 1], "end": [143, 33], "traced_tactics": [{"tactic": "funext Y", "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category C\nD : Type u\u2082\ninst\u271d : Category D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\n\u22a2 (ofArrows (fun x => Y) fun x => f) = singleton f", "state_after": "case h\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\nD : Type u\u2082\ninst\u271d : Category D\nF : C \u2964 D\nX Y\u271d Z : C\nf : Y\u271d \u27f6 X\nY : C\n\u22a2 (ofArrows (fun x => Y\u271d) fun x => f) = singleton f"}, {"tactic": "ext g", "state_before": "case h\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\nD : Type u\u2082\ninst\u271d : Category D\nF : C \u2964 D\nX Y\u271d Z : C\nf : Y\u271d \u27f6 X\nY : C\n\u22a2 (ofArrows (fun x => Y\u271d) fun x => f) = singleton f", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\nD : Type u\u2082\ninst\u271d : Category D\nF : C \u2964 D\nX Y\u271d Z : C\nf : Y\u271d \u27f6 X\nY : C\ng : Y \u27f6 X\n\u22a2 (g \u2208 ofArrows (fun x => Y\u271d) fun x => f) \u2194 g \u2208 singleton f"}, {"tactic": "constructor", "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\nD : Type u\u2082\ninst\u271d : Category D\nF : C \u2964 D\nX Y\u271d Z : C\nf : Y\u271d \u27f6 X\nY : C\ng : Y \u27f6 X\n\u22a2 (g \u2208 ofArrows (fun x => Y\u271d) fun x => f) \u2194 g \u2208 singleton f", "state_after": "case h.h.mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\nD : Type u\u2082\ninst\u271d : Category D\nF : C \u2964 D\nX Y\u271d Z : C\nf : Y\u271d \u27f6 X\nY : C\ng : Y \u27f6 X\n\u22a2 (g \u2208 ofArrows (fun x => Y\u271d) fun x => f) \u2192 g \u2208 singleton f\n\ncase h.h.mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\nD : Type u\u2082\ninst\u271d : Category D\nF : C \u2964 D\nX Y\u271d Z : C\nf : Y\u271d \u27f6 X\nY : C\ng : Y \u27f6 X\n\u22a2 g \u2208 singleton f \u2192 g \u2208 ofArrows (fun x => Y\u271d) fun x => f"}, {"tactic": "rintro \u27e8_\u27e9", "state_before": "case h.h.mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\nD : Type u\u2082\ninst\u271d : Category D\nF : C \u2964 D\nX Y\u271d Z : C\nf : Y\u271d \u27f6 X\nY : C\ng : Y \u27f6 X\n\u22a2 (g \u2208 ofArrows (fun x => Y\u271d) fun x => f) \u2192 g \u2208 singleton f", "state_after": "case h.h.mp.mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\nD : Type u\u2082\ninst\u271d : Category D\nF : C \u2964 D\nX Y\u271d Z : C\nf : Y\u271d \u27f6 X\nY : C\ni\u271d : PUnit\n\u22a2 f \u2208 singleton f"}, {"tactic": "apply singleton.mk", "state_before": "case h.h.mp.mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\nD : Type u\u2082\ninst\u271d : Category D\nF : C \u2964 D\nX Y\u271d Z : C\nf : Y\u271d \u27f6 X\nY : C\ni\u271d : PUnit\n\u22a2 f \u2208 singleton f", "state_after": "no goals"}, {"tactic": "rintro \u27e8_\u27e9", "state_before": "case h.h.mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\nD : Type u\u2082\ninst\u271d : Category D\nF : C \u2964 D\nX Y\u271d Z : C\nf : Y\u271d \u27f6 X\nY : C\ng : Y \u27f6 X\n\u22a2 g \u2208 singleton f \u2192 g \u2208 ofArrows (fun x => Y\u271d) fun x => f", "state_after": "case h.h.mpr.mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\nD : Type u\u2082\ninst\u271d : Category D\nF : C \u2964 D\nX Y\u271d Z : C\nf : Y\u271d \u27f6 X\nY : C\n\u22a2 f \u2208 ofArrows (fun x => Y\u271d) fun x => f"}, {"tactic": "exact ofArrows.mk PUnit.unit", "state_before": "case h.h.mpr.mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\nD : Type u\u2082\ninst\u271d : Category D\nF : C \u2964 D\nX Y\u271d Z : C\nf : Y\u271d \u27f6 X\nY : C\n\u22a2 f \u2208 ofArrows (fun x => Y\u271d) fun x => f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/MStructure.lean", "full_name": "IsLprojection.commute", "start": [108, 1], "end": [147, 35], "traced_tactics": [{"tactic": "have PR_eq_RPR : \u2200 R : M, IsLprojection X R \u2192 P * R = R * P * R := fun R h\u2083 => by\n  refine @eq_of_smul_eq_smul _ X _ _ _ _ fun x => by\n    rw [\u2190 norm_sub_eq_zero_iff]\n    have e1 : \u2016R \u2022 x\u2016 \u2265 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 :=\n      calc\n        \u2016R \u2022 x\u2016 = \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 +\n            (\u2016(R * R) \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 (1 - P) \u2022 R \u2022 x\u2016) := by\n          rw [h\u2081.Lnorm, h\u2083.Lnorm, h\u2083.Lnorm ((1 - P) \u2022 R \u2022 x), sub_smul 1 P, one_smul, smul_sub,\n            mul_smul]\n        _ = \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 +\n            (\u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016((1 - R) * R) \u2022 x - (1 - R) \u2022 P \u2022 R \u2022 x\u2016) := by\n          rw [h\u2083.proj.eq, sub_smul 1 P, one_smul, smul_sub, mul_smul]\n        _ = \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 +\n            (\u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016) := by\n          rw [sub_mul, h\u2083.proj.eq, one_mul, sub_self, zero_smul, zero_sub, norm_neg]\n        _ = \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 := by abel\n        _ \u2265 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 := by\n          rw [GE.ge]\n          have :=\n            add_le_add_right (norm_le_insert' (R \u2022 x) (R \u2022 P \u2022 R \u2022 x)) (2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016)\n          simpa only [mul_smul, sub_smul, one_smul] using this\n    rw [GE.ge] at e1\n    nth_rewrite 2 [\u2190 add_zero \u2016R \u2022 x\u2016]  at e1\n    rw [add_le_add_iff_left, two_smul, \u2190 two_mul] at e1\n    rw [le_antisymm_iff]\n    refine' \u27e8_, norm_nonneg _\u27e9\n    rwa [\u2190 MulZeroClass.mul_zero (2 : \u211d), mul_le_mul_left (show (0 : \u211d) < 2 by norm_num)] at e1", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\n\u22a2 Commute P Q", "state_after": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nPR_eq_RPR : \u2200 (R : M), IsLprojection X R \u2192 P * R = R * P * R\n\u22a2 Commute P Q"}, {"tactic": "have QP_eq_QPQ : Q * P = Q * P * Q := by\n  have e1 : P * (1 - Q) = P * (1 - Q) - (Q * P - Q * P * Q) :=\n    calc\n      P * (1 - Q) = (1 - Q) * P * (1 - Q) := by rw [PR_eq_RPR (1 - Q) h\u2082.Lcomplement]\n      _ = P * (1 - Q) - (Q * P - Q * P * Q) := by noncomm_ring\n  rwa [eq_sub_iff_add_eq, add_right_eq_self, sub_eq_zero] at e1", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nPR_eq_RPR : \u2200 (R : M), IsLprojection X R \u2192 P * R = R * P * R\n\u22a2 Commute P Q", "state_after": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nPR_eq_RPR : \u2200 (R : M), IsLprojection X R \u2192 P * R = R * P * R\nQP_eq_QPQ : Q * P = Q * P * Q\n\u22a2 Commute P Q"}, {"tactic": "show P * Q = Q * P", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nPR_eq_RPR : \u2200 (R : M), IsLprojection X R \u2192 P * R = R * P * R\nQP_eq_QPQ : Q * P = Q * P * Q\n\u22a2 Commute P Q", "state_after": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nPR_eq_RPR : \u2200 (R : M), IsLprojection X R \u2192 P * R = R * P * R\nQP_eq_QPQ : Q * P = Q * P * Q\n\u22a2 P * Q = Q * P"}, {"tactic": "refine @eq_of_smul_eq_smul _ X _ _ _ _ fun x => by\n  rw [\u2190 norm_sub_eq_zero_iff]\n  have e1 : \u2016R \u2022 x\u2016 \u2265 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 :=\n    calc\n      \u2016R \u2022 x\u2016 = \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 +\n          (\u2016(R * R) \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 (1 - P) \u2022 R \u2022 x\u2016) := by\n        rw [h\u2081.Lnorm, h\u2083.Lnorm, h\u2083.Lnorm ((1 - P) \u2022 R \u2022 x), sub_smul 1 P, one_smul, smul_sub,\n          mul_smul]\n      _ = \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 +\n          (\u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016((1 - R) * R) \u2022 x - (1 - R) \u2022 P \u2022 R \u2022 x\u2016) := by\n        rw [h\u2083.proj.eq, sub_smul 1 P, one_smul, smul_sub, mul_smul]\n      _ = \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 +\n          (\u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016) := by\n        rw [sub_mul, h\u2083.proj.eq, one_mul, sub_self, zero_smul, zero_sub, norm_neg]\n      _ = \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 := by abel\n      _ \u2265 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 := by\n        rw [GE.ge]\n        have :=\n          add_le_add_right (norm_le_insert' (R \u2022 x) (R \u2022 P \u2022 R \u2022 x)) (2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016)\n        simpa only [mul_smul, sub_smul, one_smul] using this\n  rw [GE.ge] at e1\n  nth_rewrite 2 [\u2190 add_zero \u2016R \u2022 x\u2016]  at e1\n  rw [add_le_add_iff_left, two_smul, \u2190 two_mul] at e1\n  rw [le_antisymm_iff]\n  refine' \u27e8_, norm_nonneg _\u27e9\n  rwa [\u2190 MulZeroClass.mul_zero (2 : \u211d), mul_le_mul_left (show (0 : \u211d) < 2 by norm_num)] at e1", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\n\u22a2 P * R = R * P * R", "state_after": "no goals"}, {"tactic": "rw [\u2190 norm_sub_eq_zero_iff]", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\n\u22a2 (P * R) \u2022 x = (R * P * R) \u2022 x", "state_after": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\n\u22a2 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 = 0"}, {"tactic": "have e1 : \u2016R \u2022 x\u2016 \u2265 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 :=\n  calc\n    \u2016R \u2022 x\u2016 = \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 +\n        (\u2016(R * R) \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 (1 - P) \u2022 R \u2022 x\u2016) := by\n      rw [h\u2081.Lnorm, h\u2083.Lnorm, h\u2083.Lnorm ((1 - P) \u2022 R \u2022 x), sub_smul 1 P, one_smul, smul_sub,\n        mul_smul]\n    _ = \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 +\n        (\u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016((1 - R) * R) \u2022 x - (1 - R) \u2022 P \u2022 R \u2022 x\u2016) := by\n      rw [h\u2083.proj.eq, sub_smul 1 P, one_smul, smul_sub, mul_smul]\n    _ = \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 +\n        (\u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016) := by\n      rw [sub_mul, h\u2083.proj.eq, one_mul, sub_self, zero_smul, zero_sub, norm_neg]\n    _ = \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 := by abel\n    _ \u2265 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 := by\n      rw [GE.ge]\n      have :=\n        add_le_add_right (norm_le_insert' (R \u2022 x) (R \u2022 P \u2022 R \u2022 x)) (2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016)\n      simpa only [mul_smul, sub_smul, one_smul] using this", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\n\u22a2 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 = 0", "state_after": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\ne1 : \u2016R \u2022 x\u2016 \u2265 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016\n\u22a2 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 = 0"}, {"tactic": "rw [GE.ge] at e1", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\ne1 : \u2016R \u2022 x\u2016 \u2265 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016\n\u22a2 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 = 0", "state_after": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\ne1 : \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 \u2016R \u2022 x\u2016\n\u22a2 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 = 0"}, {"tactic": "nth_rewrite 2 [\u2190 add_zero \u2016R \u2022 x\u2016]  at e1", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\ne1 : \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 \u2016R \u2022 x\u2016\n\u22a2 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 = 0", "state_after": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\ne1 : \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 \u2016R \u2022 x\u2016 + 0\n\u22a2 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 = 0"}, {"tactic": "rw [add_le_add_iff_left, two_smul, \u2190 two_mul] at e1", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\ne1 : \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 \u2016R \u2022 x\u2016 + 0\n\u22a2 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 = 0", "state_after": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\ne1 : 2 * \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 0\n\u22a2 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 = 0"}, {"tactic": "rw [le_antisymm_iff]", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\ne1 : 2 * \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 0\n\u22a2 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 = 0", "state_after": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\ne1 : 2 * \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 0\n\u22a2 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 0 \u2227 0 \u2264 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016"}, {"tactic": "refine' \u27e8_, norm_nonneg _\u27e9", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\ne1 : 2 * \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 0\n\u22a2 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 0 \u2227 0 \u2264 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016", "state_after": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\ne1 : 2 * \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 0\n\u22a2 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 0"}, {"tactic": "rwa [\u2190 MulZeroClass.mul_zero (2 : \u211d), mul_le_mul_left (show (0 : \u211d) < 2 by norm_num)] at e1", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\ne1 : 2 * \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 0\n\u22a2 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 0", "state_after": "no goals"}, {"tactic": "rw [h\u2081.Lnorm, h\u2083.Lnorm, h\u2083.Lnorm ((1 - P) \u2022 R \u2022 x), sub_smul 1 P, one_smul, smul_sub,\n  mul_smul]", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\n\u22a2 \u2016R \u2022 x\u2016 = \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 + (\u2016(R * R) \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 (1 - P) \u2022 R \u2022 x\u2016)", "state_after": "no goals"}, {"tactic": "rw [h\u2083.proj.eq, sub_smul 1 P, one_smul, smul_sub, mul_smul]", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\n\u22a2 \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 + (\u2016(R * R) \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 (1 - P) \u2022 R \u2022 x\u2016) =\n    \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 + (\u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016((1 - R) * R) \u2022 x - (1 - R) \u2022 P \u2022 R \u2022 x\u2016)", "state_after": "no goals"}, {"tactic": "rw [sub_mul, h\u2083.proj.eq, one_mul, sub_self, zero_smul, zero_sub, norm_neg]", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\n\u22a2 \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 + (\u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016((1 - R) * R) \u2022 x - (1 - R) \u2022 P \u2022 R \u2022 x\u2016) =\n    \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 + (\u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016)", "state_after": "no goals"}, {"tactic": "abel", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\n\u22a2 \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 + (\u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016) =\n    \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016", "state_after": "no goals"}, {"tactic": "rw [GE.ge]", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\n\u22a2 \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 \u2265 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016", "state_after": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\n\u22a2 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016"}, {"tactic": "have :=\n  add_le_add_right (norm_le_insert' (R \u2022 x) (R \u2022 P \u2022 R \u2022 x)) (2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016)", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\n\u22a2 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016", "state_after": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\nthis : \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 \u2264 \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016\n\u22a2 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016"}, {"tactic": "simpa only [mul_smul, sub_smul, one_smul] using this", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\nthis : \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016 \u2264 \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016\n\u22a2 \u2016R \u2022 x\u2016 + 2 \u2022 \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 \u2016R \u2022 P \u2022 R \u2022 x\u2016 + \u2016R \u2022 x - R \u2022 P \u2022 R \u2022 x\u2016 + 2 \u2022 \u2016(1 - R) \u2022 P \u2022 R \u2022 x\u2016", "state_after": "no goals"}, {"tactic": "norm_num", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nR : M\nh\u2083 : IsLprojection X R\nx : X\ne1 : 2 * \u2016(P * R) \u2022 x - (R * P * R) \u2022 x\u2016 \u2264 2 * 0\n\u22a2 0 < 2", "state_after": "no goals"}, {"tactic": "have e1 : P * (1 - Q) = P * (1 - Q) - (Q * P - Q * P * Q) :=\n  calc\n    P * (1 - Q) = (1 - Q) * P * (1 - Q) := by rw [PR_eq_RPR (1 - Q) h\u2082.Lcomplement]\n    _ = P * (1 - Q) - (Q * P - Q * P * Q) := by noncomm_ring", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nPR_eq_RPR : \u2200 (R : M), IsLprojection X R \u2192 P * R = R * P * R\n\u22a2 Q * P = Q * P * Q", "state_after": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nPR_eq_RPR : \u2200 (R : M), IsLprojection X R \u2192 P * R = R * P * R\ne1 : P * (1 - Q) = P * (1 - Q) - (Q * P - Q * P * Q)\n\u22a2 Q * P = Q * P * Q"}, {"tactic": "rwa [eq_sub_iff_add_eq, add_right_eq_self, sub_eq_zero] at e1", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nPR_eq_RPR : \u2200 (R : M), IsLprojection X R \u2192 P * R = R * P * R\ne1 : P * (1 - Q) = P * (1 - Q) - (Q * P - Q * P * Q)\n\u22a2 Q * P = Q * P * Q", "state_after": "no goals"}, {"tactic": "rw [PR_eq_RPR (1 - Q) h\u2082.Lcomplement]", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nPR_eq_RPR : \u2200 (R : M), IsLprojection X R \u2192 P * R = R * P * R\n\u22a2 P * (1 - Q) = (1 - Q) * P * (1 - Q)", "state_after": "no goals"}, {"tactic": "noncomm_ring", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nPR_eq_RPR : \u2200 (R : M), IsLprojection X R \u2192 P * R = R * P * R\n\u22a2 (1 - Q) * P * (1 - Q) = P * (1 - Q) - (Q * P - Q * P * Q)", "state_after": "no goals"}, {"tactic": "rw [QP_eq_QPQ, PR_eq_RPR Q h\u2082]", "state_before": "X : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\nM : Type u_1\ninst\u271d\u00b2 : Ring M\ninst\u271d\u00b9 : Module M X\ninst\u271d : FaithfulSMul M X\nP Q : M\nh\u2081 : IsLprojection X P\nh\u2082 : IsLprojection X Q\nPR_eq_RPR : \u2200 (R : M), IsLprojection X R \u2192 P * R = R * P * R\nQP_eq_QPQ : Q * P = Q * P * Q\n\u22a2 P * Q = Q * P", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Nat.mem_range_succ", "start": [707, 1], "end": [710, 67], "traced_tactics": [{"tactic": "rintro \u27e8n, rfl\u27e9", "state_before": "\u03b1 : Type ?u.70793\n\u03b2 : Type ?u.70796\n\u03b3 : Type ?u.70799\n\u03b9 : Sort ?u.70802\n\u03b9' : Sort ?u.70805\nf : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\ni : \u2115\n\u22a2 i \u2208 range Nat.succ \u2192 0 < i", "state_after": "case intro\n\u03b1 : Type ?u.70793\n\u03b2 : Type ?u.70796\n\u03b3 : Type ?u.70799\n\u03b9 : Sort ?u.70802\n\u03b9' : Sort ?u.70805\nf : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\nn : \u2115\n\u22a2 0 < Nat.succ n"}, {"tactic": "exact Nat.succ_pos n", "state_before": "case intro\n\u03b1 : Type ?u.70793\n\u03b2 : Type ?u.70796\n\u03b3 : Type ?u.70799\n\u03b9 : Sort ?u.70802\n\u03b9' : Sort ?u.70805\nf : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\nn : \u2115\n\u22a2 0 < Nat.succ n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "full_name": "Real.sin_pi_div_thirty_two", "start": [837, 1], "end": [841, 7], "traced_tactics": [{"tactic": "trans sin (\u03c0 / 2 ^ 5)", "state_before": "x : \u211d\n\u22a2 sin (\u03c0 / 32) = sqrt (2 - sqrt (2 + sqrt (2 + sqrt 2))) / 2", "state_after": "x : \u211d\n\u22a2 sin (\u03c0 / 32) = sin (\u03c0 / 2 ^ 5)\n\nx : \u211d\n\u22a2 sin (\u03c0 / 2 ^ 5) = sqrt (2 - sqrt (2 + sqrt (2 + sqrt 2))) / 2"}, {"tactic": "congr", "state_before": "x : \u211d\n\u22a2 sin (\u03c0 / 32) = sin (\u03c0 / 2 ^ 5)\n\nx : \u211d\n\u22a2 sin (\u03c0 / 2 ^ 5) = 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"sdiff_sup_cancel", "start": [584, 1], "end": [584, 99], "traced_tactics": [{"tactic": "rw [sup_comm, sup_sdiff_cancel_right h]", "state_before": "\u03b9 : Type ?u.92974\n\u03b1 : Type u_1\n\u03b2 : Type ?u.92980\ninst\u271d : GeneralizedCoheytingAlgebra \u03b1\na b c d : \u03b1\nh : b \u2264 a\n\u22a2 a \\ b \u2294 b = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/ModelTheory/FinitelyGenerated.lean", "full_name": "FirstOrder.Language.Substructure.fg_def", "start": [48, 1], "end": [52, 20], "traced_tactics": [{"tactic": "rintro \u27e8t', h, rfl\u27e9", "state_before": "L : Language\nM : Type u_3\ninst\u271d : Structure L M\nN : Substructure L M\n\u22a2 (\u2203 S, Set.Finite S \u2227 LowerAdjoint.toFun (closure L) S = N) \u2192 FG N", "state_after": "case intro.intro\nL : Language\nM : Type u_3\ninst\u271d : Structure L M\nt' : Set M\nh : Set.Finite t'\n\u22a2 FG 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[]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Homology/HomologicalComplex.lean", "full_name": "HomologicalComplex.eqToHom_f", "start": [237, 1], "end": [241, 6], "traced_tactics": [{"tactic": "subst h", "state_before": "\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category V\ninst\u271d : HasZeroMorphisms V\nc : ComplexShape \u03b9\nC C\u2081 C\u2082 : HomologicalComplex V c\nh : C\u2081 = C\u2082\nn : \u03b9\n\u22a2 Hom.f (eqToHom h) n = eqToHom (_ : X C\u2081 n = X C\u2082 n)", "state_after": "\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category V\ninst\u271d : HasZeroMorphisms V\nc : ComplexShape \u03b9\nC C\u2081 : HomologicalComplex V c\nn : \u03b9\n\u22a2 Hom.f (eqToHom (_ : C\u2081 = C\u2081)) n = eqToHom (_ : X C\u2081 n = X C\u2081 n)"}, {"tactic": "rfl", "state_before": "\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category V\ninst\u271d : HasZeroMorphisms V\nc : ComplexShape \u03b9\nC C\u2081 : HomologicalComplex V c\nn : \u03b9\n\u22a2 Hom.f (eqToHom (_ : C\u2081 = C\u2081)) n = eqToHom (_ : X C\u2081 n = X C\u2081 n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "full_name": "Summable.star", "start": [1403, 1], "end": [1404, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean", "full_name": "Polynomial.differentiable", "start": [120, 11], "end": [120, 101], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/PartENat.lean", "full_name": "PartENat.get_le_get", "start": [285, 1], "end": [288, 48], 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"full_name": "List.subperm_cons", "start": [671, 1], "end": [675, 81], "traced_tactics": [{"tactic": "cases' s with _ _ _ s' u _ _ s'", "state_before": "\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nx\u271d : a :: l\u2081 <+~ a :: l\u2082\nl : List \u03b1\np : l ~ a :: l\u2081\ns : l <+ a :: l\u2082\n\u22a2 l\u2081 <+~ l\u2082", "state_after": "case cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nx\u271d : a :: l\u2081 <+~ a :: l\u2082\nl : List \u03b1\np : l ~ a :: l\u2081\ns' : l <+ l\u2082\n\u22a2 l\u2081 <+~ l\u2082\n\ncase cons\u2082\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nx\u271d : a :: l\u2081 <+~ a :: l\u2082\nu : List \u03b1\np : a :: u ~ a :: l\u2081\ns' : u <+ l\u2082\n\u22a2 l\u2081 <+~ l\u2082"}, {"tactic": "exact (p.subperm_left.2 <| (sublist_cons _ _).subperm).trans s'.subperm", "state_before": "case cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nx\u271d : a :: l\u2081 <+~ a :: l\u2082\nl : List \u03b1\np : l ~ a :: l\u2081\ns' : l <+ l\u2082\n\u22a2 l\u2081 <+~ l\u2082", "state_after": "no goals"}, {"tactic": "exact \u27e8u, p.cons_inv, s'\u27e9", "state_before": "case cons\u2082\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\nx\u271d : a :: l\u2081 <+~ a :: l\u2082\nu : List \u03b1\np : a :: u ~ a :: l\u2081\ns' : u <+ l\u2082\n\u22a2 l\u2081 <+~ l\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "full_name": "Cardinal.aleph0_le_aleph'", "start": [289, 1], "end": [289, 100], "traced_tactics": [{"tactic": "rw [\u2190 aleph'_omega, aleph'_le]", "state_before": "o : Ordinal\n\u22a2 \u2135\u2080 \u2264 aleph' o \u2194 \u03c9 \u2264 o", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Sym/Sym2.lean", "full_name": "Sym2.fromRel_toRel", "start": [548, 1], "end": [549, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/ModelTheory/Syntax.lean", "full_name": "FirstOrder.Language.directed_distinctConstantsTheory", "start": [1131, 1], "end": [1133, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Ring/Equiv.lean", "full_name": "RingEquiv.apply_symm_apply", "start": [339, 1], "end": [340, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Sets/Compacts.lean", "full_name": "TopologicalSpace.CompactOpens.map_id", "start": [588, 1], "end": [589, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Int/Parity.lean", "full_name": "Int.even_or_odd'", "start": [71, 1], "end": [72, 65], "traced_tactics": [{"tactic": "simpa only [two_mul, exists_or, Odd, Even] using even_or_odd n", "state_before": "m n\u271d n : \u2124\n\u22a2 \u2203 k, n = 2 * k \u2228 n = 2 * k + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/IsLimit.lean", "full_name": "CategoryTheory.Limits.IsLimit.conePointUniqueUpToIso_hom_comp", "start": [159, 1], "end": [161, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Game/PGame.lean", "full_name": "PGame.IsOption.mk_right", "start": [231, 1], "end": [233, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.smul_neg", "start": [1083, 11], "end": [1085, 41], "traced_tactics": [{"tactic": "simp_rw [\u2190 image_neg]", "state_before": "F : Type ?u.209091\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.209100\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : DistribMulAction \u03b1 \u03b2\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 s \u2022 -t = -(s \u2022 t)", "state_after": "F : Type ?u.209091\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.209100\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : DistribMulAction \u03b1 \u03b2\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 s \u2022 Neg.neg '' t = Neg.neg '' (s \u2022 t)"}, {"tactic": "exact image_image2_right_comm smul_neg", "state_before": "F : Type ?u.209091\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.209100\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : DistribMulAction \u03b1 \u03b2\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 s \u2022 Neg.neg '' t = Neg.neg '' (s \u2022 t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Cover.lean", "full_name": "Covby.ge_of_gt", "start": [442, 1], "end": [443, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Disjointed.lean", "full_name": "disjoint_disjointed", "start": [77, 1], "end": [83, 84], "traced_tactics": [{"tactic": "refine' (Symmetric.pairwise_on Disjoint.symm _).2 fun m n h => _", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.2549\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nf : \u2115 \u2192 \u03b1\n\u22a2 Pairwise (Disjoint on disjointed f)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.2549\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nf : \u2115 \u2192 \u03b1\nm n : \u2115\nh : m < n\n\u22a2 Disjoint (disjointed f m) (disjointed f n)"}, {"tactic": "cases n", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.2549\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nf : \u2115 \u2192 \u03b1\nm n : \u2115\nh : m < n\n\u22a2 Disjoint (disjointed f m) (disjointed f n)", "state_after": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type ?u.2549\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nf : \u2115 \u2192 \u03b1\nm : \u2115\nh : m < Nat.zero\n\u22a2 Disjoint (disjointed f m) (disjointed f Nat.zero)\n\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type ?u.2549\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nf : \u2115 \u2192 \u03b1\nm n\u271d : \u2115\nh : m < Nat.succ n\u271d\n\u22a2 Disjoint (disjointed f m) (disjointed f (Nat.succ n\u271d))"}, {"tactic": "exact\n  disjoint_sdiff_self_right.mono_left\n    ((disjointed_le f m).trans (le_partialSups_of_le f (Nat.lt_add_one_iff.1 h)))", "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type ?u.2549\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nf : \u2115 \u2192 \u03b1\nm n\u271d : \u2115\nh : m < Nat.succ n\u271d\n\u22a2 Disjoint (disjointed f m) (disjointed f (Nat.succ n\u271d))", "state_after": "no goals"}, {"tactic": "exact (Nat.not_lt_zero _ h).elim", "state_before": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type ?u.2549\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nf : \u2115 \u2192 \u03b1\nm : \u2115\nh : m < Nat.zero\n\u22a2 Disjoint (disjointed f m) (disjointed f Nat.zero)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Heyting/Boundary.lean", "full_name": "Coheyting.boundary_hnot_hnot", "start": [74, 1], "end": [75, 47], "traced_tactics": [{"tactic": "simp_rw [boundary, hnot_hnot_hnot, inf_comm]", "state_before": "\u03b1 : Type u_1\ninst\u271d : CoheytingAlgebra \u03b1\na\u271d b a : \u03b1\n\u22a2 \u2202 (\uffe2\uffe2a) = \u2202 (\uffe2a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Basic.lean", "full_name": "Metric.bounded_iff_subset_ball", "start": [2334, 1], "end": [2341, 37], "traced_tactics": [{"tactic": "constructor <;> rintro \u27e8C, hC\u27e9", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.488478\n\u03b9 : Type ?u.488481\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr : \u211d\nc : \u03b1\n\u22a2 Bounded s \u2194 \u2203 r, s \u2286 closedBall c r", "state_after": "case mp.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.488478\n\u03b9 : Type ?u.488481\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr : \u211d\nc : \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist x y \u2264 C\n\u22a2 \u2203 r, s \u2286 closedBall c r\n\ncase mpr.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.488478\n\u03b9 : Type ?u.488481\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr : \u211d\nc : \u03b1\nC : \u211d\nhC : s \u2286 closedBall c C\n\u22a2 Bounded s"}, {"tactic": "rcases s.eq_empty_or_nonempty with (rfl | \u27e8x, hx\u27e9)", "state_before": "case mp.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.488478\n\u03b9 : Type ?u.488481\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr : \u211d\nc : \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist x y \u2264 C\n\u22a2 \u2203 r, s \u2286 closedBall c r", "state_after": "case mp.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.488478\n\u03b9 : Type ?u.488481\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\nt : Set \u03b1\nr : \u211d\nc : \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), x \u2208 \u2205 \u2192 \u2200 (y : \u03b1), y \u2208 \u2205 \u2192 dist x y \u2264 C\n\u22a2 \u2203 r, \u2205 \u2286 closedBall c r\n\ncase mp.intro.inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.488478\n\u03b9 : Type ?u.488481\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d : \u03b1\ns t : Set \u03b1\nr : \u211d\nc : \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist x y \u2264 C\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2203 r, s \u2286 closedBall c r"}, {"tactic": "exact \u27e80, by simp\u27e9", "state_before": "case mp.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.488478\n\u03b9 : Type ?u.488481\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\nt : Set \u03b1\nr : \u211d\nc : \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), x \u2208 \u2205 \u2192 \u2200 (y : \u03b1), y \u2208 \u2205 \u2192 dist x y \u2264 C\n\u22a2 \u2203 r, \u2205 \u2286 closedBall c r", "state_after": "no goals"}, {"tactic": "simp", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.488478\n\u03b9 : Type ?u.488481\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\nt : Set \u03b1\nr : \u211d\nc : \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), x \u2208 \u2205 \u2192 \u2200 (y : \u03b1), y \u2208 \u2205 \u2192 dist x y \u2264 C\n\u22a2 \u2205 \u2286 closedBall c 0", "state_after": "no goals"}, {"tactic": "exact \u27e8C + dist x c, fun y hy =>\n  calc dist y c \u2264 dist y x + dist x c := dist_triangle _ _ _\n  _ \u2264 C + dist x c := add_le_add_right (hC y hy x hx) _\u27e9", "state_before": "case mp.intro.inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.488478\n\u03b9 : Type ?u.488481\ninst\u271d : PseudoMetricSpace \u03b1\nx\u271d : \u03b1\ns t : Set \u03b1\nr : \u211d\nc : \u03b1\nC : \u211d\nhC : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist x y \u2264 C\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2203 r, s \u2286 closedBall c r", "state_after": "no goals"}, {"tactic": "exact bounded_closedBall.mono hC", "state_before": "case mpr.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.488478\n\u03b9 : Type ?u.488481\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr : \u211d\nc : \u03b1\nC : \u211d\nhC : s \u2286 closedBall c C\n\u22a2 Bounded s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/Set.lean", "full_name": "Set.image_equiv_eq_preimage_symm", "start": [53, 1], "end": [55, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Pointwise/BigOperators.lean", "full_name": "Set.multiset_prod_mem_multiset_prod", "start": [130, 1], "end": [134, 41], "traced_tactics": [{"tactic": "induction t using Quotient.inductionOn", "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type ?u.78098\nF : Type ?u.78101\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nt : Multiset \u03b9\nf : \u03b9 \u2192 Set \u03b1\ng : \u03b9 \u2192 \u03b1\nhg : \u2200 (i : \u03b9), i \u2208 t \u2192 g i \u2208 f i\n\u22a2 Multiset.prod (Multiset.map g t) \u2208 Multiset.prod (Multiset.map f t)", "state_after": "case h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type ?u.78098\nF : Type ?u.78101\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\ng : \u03b9 \u2192 \u03b1\na\u271d : List \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 Quotient.mk (List.isSetoid \u03b9) a\u271d \u2192 g i \u2208 f i\n\u22a2 Multiset.prod (Multiset.map g (Quotient.mk (List.isSetoid \u03b9) a\u271d)) \u2208\n    Multiset.prod (Multiset.map f (Quotient.mk (List.isSetoid \u03b9) a\u271d))"}, {"tactic": "simp_rw [Multiset.quot_mk_to_coe, Multiset.coe_map, Multiset.coe_prod]", "state_before": "case h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type ?u.78098\nF : Type ?u.78101\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\ng : \u03b9 \u2192 \u03b1\na\u271d : List \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 Quotient.mk (List.isSetoid \u03b9) a\u271d \u2192 g i \u2208 f i\n\u22a2 Multiset.prod (Multiset.map g (Quotient.mk (List.isSetoid \u03b9) a\u271d)) \u2208\n    Multiset.prod (Multiset.map f (Quotient.mk (List.isSetoid \u03b9) a\u271d))", "state_after": "case h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type ?u.78098\nF : Type ?u.78101\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\ng : \u03b9 \u2192 \u03b1\na\u271d : List \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 Quotient.mk (List.isSetoid \u03b9) a\u271d \u2192 g i \u2208 f i\n\u22a2 List.prod (List.map g a\u271d) \u2208 List.prod (List.map f a\u271d)"}, {"tactic": "exact list_prod_mem_list_prod _ _ _ hg", "state_before": "case h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type ?u.78098\nF : Type ?u.78101\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : \u03b9 \u2192 Set \u03b1\ng : \u03b9 \u2192 \u03b1\na\u271d : List \u03b9\nhg : \u2200 (i : \u03b9), i \u2208 Quotient.mk (List.isSetoid \u03b9) a\u271d \u2192 g i \u2208 f i\n\u22a2 List.prod (List.map g a\u271d) \u2208 List.prod (List.map f a\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Monoidal/Functor.lean", "full_name": "CategoryTheory.LaxMonoidalFunctor.prod'_\u03bc", "start": [415, 1], "end": [417, 7], "traced_tactics": [{"tactic": "dsimp [prod']", "state_before": "C : Type u\u2081\ninst\u271d\u2075 : Category C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor C E\nX Y : C\n\u22a2 \u03bc (prod' F G) X Y = (\u03bc F X Y, \u03bc G X Y)", "state_after": "C : Type u\u2081\ninst\u271d\u2075 : Category C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor C E\nX Y : C\n\u22a2 (\u03bc F X Y \u226b F.map (\ud835\udfd9 (X \u2297 Y)), \u03bc G X Y \u226b G.map (\ud835\udfd9 (X \u2297 Y))) = (\u03bc F X Y, \u03bc G X Y)"}, {"tactic": "simp", "state_before": "C : Type u\u2081\ninst\u271d\u2075 : Category C\ninst\u271d\u2074 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b3 : Category D\ninst\u271d\u00b2 : MonoidalCategory D\nE : Type u\u2083\ninst\u271d\u00b9 : Category E\ninst\u271d : MonoidalCategory E\nF : LaxMonoidalFunctor C D\nG : LaxMonoidalFunctor C E\nX Y : C\n\u22a2 (\u03bc F X Y \u226b F.map (\ud835\udfd9 (X \u2297 Y)), \u03bc G X Y \u226b G.map (\ud835\udfd9 (X \u2297 Y))) = (\u03bc F X Y, \u03bc G X Y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "ClusterPt.of_le_nhds", "start": [1114, 1], "end": [1115, 38], "traced_tactics": [{"tactic": "rwa [ClusterPt, inf_eq_right.mpr H]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\na : \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\np p\u2081 p\u2082 : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nx : \u03b1\nf : Filter \u03b1\nH : f \u2264 \ud835\udcdd x\ninst\u271d : NeBot f\n\u22a2 ClusterPt x f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/RootsOfUnity/Basic.lean", "full_name": "mem_rootsOfUnity_iff_mem_nthRoots", "start": [190, 1], "end": [193, 30], "traced_tactics": [{"tactic": "simp only [mem_rootsOfUnity, mem_nthRoots k.pos, Units.ext_iff, Units.val_one,\n  Units.val_pow_eq_pow_val]", "state_before": "M : Type ?u.1044729\nN : Type ?u.1044732\nG : Type ?u.1044735\nR : Type u_1\nS : Type ?u.1044741\nF : Type ?u.1044744\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115+\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\u02e3\n\u22a2 \u03b6 \u2208 rootsOfUnity k R \u2194 \u2191\u03b6 \u2208 nthRoots (\u2191k) 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Prelude.lean", "full_name": "Nat.le_of_succ_le_succ", "start": [1628, 1], "end": [1629, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Basic.lean", "full_name": "div_self'", "start": [748, 1], "end": [748, 81], "traced_tactics": [{"tactic": "rw [div_eq_mul_inv, mul_right_inv a]", "state_before": "\u03b1 : Type ?u.57205\n\u03b2 : Type ?u.57208\nG : Type u_1\ninst\u271d : Group G\na\u271d b c d a : G\n\u22a2 a / a = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Monoidal/End.lean", "full_name": "CategoryTheory.\u03b5_naturality", "start": [119, 1], "end": [120, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "full_name": "Finset.tsum_subtype'", "start": [595, 1], "end": [597, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/UnionFind.lean", "full_name": "UFModel.Agrees.size_eq", "start": [80, 1], "end": [81, 34], "traced_tactics": [{"tactic": "cases H", "state_before": "\u03b1 : Type u_1\nn : \u2115\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\nm : Fin n \u2192 \u03b2\nH : Agrees arr f m\n\u22a2 n = Array.size arr", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\n\u22a2 Array.size arr = Array.size arr"}, {"tactic": "rfl", "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2192 \u03b2\narr : Array \u03b1\n\u22a2 Array.size arr = Array.size arr", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Constructions.lean", "full_name": "continuous_uLift_up", "start": [1585, 1], "end": [1586, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/LinearPMap.lean", "full_name": "LinearPMap.smul_domain", "start": [413, 1], "end": [414, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "full_name": "ENNReal.lintegral_rpow_add_le_add_snorm_mul_lintegral_rpow_add", "start": [277, 1], "end": [312, 94], "traced_tactics": [{"tactic": "refine' lintegral_mono fun a => _", "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2264 \u222b\u207b (a : \u03b1), (f + g) a * (f + g) a ^ (p - 1) \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\na : \u03b1\n\u22a2 (f + g) a ^ p \u2264 (f + g) a * (f + g) a ^ (p - 1)"}, {"tactic": "dsimp only", "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\na : \u03b1\n\u22a2 (f + g) a ^ p \u2264 (f + g) a * (f + g) a ^ (p - 1)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\na : \u03b1\n\u22a2 (f + g) a ^ p \u2264 (f + g) a * (f + g) a ^ (p - 1)"}, {"tactic": "by_cases h_zero : (f + g) a = 0", "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\na : \u03b1\n\u22a2 (f + g) a ^ p \u2264 (f + g) a * (f + g) a ^ (p - 1)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\na : \u03b1\nh_zero : (f + g) a = 0\n\u22a2 (f + g) a ^ p \u2264 (f + g) a * (f + g) a ^ (p - 1)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\na : \u03b1\nh_zero : \u00ac(f + g) a = 0\n\u22a2 (f + g) a ^ p \u2264 (f + g) a * (f + g) a ^ (p - 1)"}, {"tactic": "by_cases h_top : (f + g) a = \u22a4", "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\na : \u03b1\nh_zero : \u00ac(f + g) a = 0\n\u22a2 (f + g) a ^ p \u2264 (f + g) a * (f + g) a ^ (p - 1)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\na : \u03b1\nh_zero : \u00ac(f + g) a = 0\nh_top : (f + g) a = \u22a4\n\u22a2 (f + g) a ^ p \u2264 (f + g) a * (f + g) a ^ (p - 1)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\na : \u03b1\nh_zero : \u00ac(f + g) a = 0\nh_top : \u00ac(f + g) a = \u22a4\n\u22a2 (f + g) a ^ p \u2264 (f + g) a * (f + g) a ^ (p - 1)"}, {"tactic": "refine' le_of_eq _", "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\na : \u03b1\nh_zero : \u00ac(f + g) a = 0\nh_top : \u00ac(f + g) a = \u22a4\n\u22a2 (f + g) a ^ p \u2264 (f + g) a * (f + g) a ^ (p - 1)", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\na : \u03b1\nh_zero : \u00ac(f + g) a = 0\nh_top : \u00ac(f + g) a = \u22a4\n\u22a2 (f + g) a ^ p = (f + g) a * (f + g) a ^ (p - 1)"}, {"tactic": "nth_rw 2 [\u2190 ENNReal.rpow_one ((f + g) a)]", "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\na : \u03b1\nh_zero : \u00ac(f + g) a = 0\nh_top : \u00ac(f + g) a = \u22a4\n\u22a2 (f + g) a ^ p = (f + g) a * (f + g) a ^ (p - 1)", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\na : \u03b1\nh_zero : \u00ac(f + g) a = 0\nh_top : \u00ac(f + g) a = \u22a4\n\u22a2 (f + g) a ^ p = (f + g) a ^ 1 * (f + g) a ^ (p - 1)"}, {"tactic": "rw [\u2190 ENNReal.rpow_add _ _ h_zero h_top, add_sub_cancel'_right]", "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\na : \u03b1\nh_zero : \u00ac(f + g) a = 0\nh_top : \u00ac(f + g) a = \u22a4\n\u22a2 (f + g) a ^ p = (f + g) a ^ 1 * (f + g) a ^ (p - 1)", "state_after": "no goals"}, {"tactic": "rw [h_zero, ENNReal.zero_rpow_of_pos hpq.pos]", "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\na : \u03b1\nh_zero : (f + g) a = 0\n\u22a2 (f + g) a ^ p \u2264 (f + g) a * (f + g) a ^ (p - 1)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\na : \u03b1\nh_zero : (f + g) a = 0\n\u22a2 0 \u2264 0 * 0 ^ (p - 1)"}, {"tactic": "exact zero_le _", "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\na : \u03b1\nh_zero : (f + g) a = 0\n\u22a2 0 \u2264 0 * 0 ^ (p - 1)", "state_after": "no goals"}, {"tactic": "rw [h_top, ENNReal.top_rpow_of_pos hpq.sub_one_pos, ENNReal.top_mul_top]", "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\na : \u03b1\nh_zero : \u00ac(f + g) a = 0\nh_top : (f + g) a = \u22a4\n\u22a2 (f + g) a ^ p \u2264 (f + g) a * (f + g) a ^ (p - 1)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\na : \u03b1\nh_zero : \u00ac(f + g) a = 0\nh_top : (f + g) a = \u22a4\n\u22a2 \u22a4 ^ p \u2264 \u22a4"}, {"tactic": "exact le_top", "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\na : \u03b1\nh_zero : \u00ac(f + g) a = 0\nh_top : (f + g) a = \u22a4\n\u22a2 \u22a4 ^ p \u2264 \u22a4", "state_after": "no goals"}, {"tactic": "have h_add_m : AEMeasurable (fun a : \u03b1 => (f + g) a ^ (p - 1 : \u211d)) \u03bc :=\n  (hf.add hg).pow_const _", "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a * (f + g) a ^ (p - 1) \u2202\u03bc) =\n    (\u222b\u207b (a : \u03b1), f a * (f + g) a ^ (p - 1) \u2202\u03bc) + \u222b\u207b (a : \u03b1), g a * (f + g) a ^ (p - 1) \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\nh_add_m : AEMeasurable fun a => (f + g) a ^ (p - 1)\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a * (f + g) a ^ (p - 1) \u2202\u03bc) =\n    (\u222b\u207b (a : \u03b1), f a * (f + g) a ^ (p - 1) \u2202\u03bc) + \u222b\u207b (a : \u03b1), g a * (f + g) a ^ (p - 1) \u2202\u03bc"}, {"tactic": "have h_add_apply :\n  (\u222b\u207b a : \u03b1, (f + g) a * (f + g) a ^ (p - 1) \u2202\u03bc) =\n    \u222b\u207b a : \u03b1, (f a + g a) * (f + g) a ^ (p - 1) \u2202\u03bc :=\n  rfl", "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\nh_add_m : AEMeasurable fun a => (f + g) a ^ (p - 1)\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a * (f + g) a ^ (p - 1) \u2202\u03bc) =\n    (\u222b\u207b (a : \u03b1), f a * (f + g) a ^ (p - 1) \u2202\u03bc) + \u222b\u207b (a : \u03b1), g a * (f + g) a ^ (p - 1) \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\nh_add_m : AEMeasurable fun a => (f + g) a ^ (p - 1)\nh_add_apply : (\u222b\u207b (a : \u03b1), (f + g) a * (f + g) a ^ (p - 1) \u2202\u03bc) = \u222b\u207b (a : \u03b1), (f a + g a) * (f + g) a ^ (p - 1) \u2202\u03bc\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a * (f + g) a ^ (p - 1) \u2202\u03bc) =\n    (\u222b\u207b (a : \u03b1), f a * (f + g) a ^ (p - 1) \u2202\u03bc) + \u222b\u207b (a : \u03b1), g a * (f + g) a ^ (p - 1) \u2202\u03bc"}, {"tactic": "simp_rw [h_add_apply, add_mul]", "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\nh_add_m : AEMeasurable fun a => (f + g) a ^ (p - 1)\nh_add_apply : (\u222b\u207b (a : \u03b1), (f + g) a * (f + g) a ^ (p - 1) \u2202\u03bc) = \u222b\u207b (a : \u03b1), (f a + g a) * (f + g) a ^ (p - 1) \u2202\u03bc\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a * (f + g) a ^ (p - 1) \u2202\u03bc) =\n    (\u222b\u207b (a : \u03b1), f a * (f + g) a ^ (p - 1) \u2202\u03bc) + \u222b\u207b (a : \u03b1), g a * (f + g) a ^ (p - 1) \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\nh_add_m : AEMeasurable fun a => (f + g) a ^ (p - 1)\nh_add_apply : (\u222b\u207b (a : \u03b1), (f + g) a * (f + g) a ^ (p - 1) \u2202\u03bc) = \u222b\u207b (a : \u03b1), (f a + g a) * (f + g) a ^ (p - 1) \u2202\u03bc\n\u22a2 (\u222b\u207b (a : \u03b1), f a * (f + g) a ^ (p - 1) + g a * (f + g) a ^ (p - 1) \u2202\u03bc) =\n    (\u222b\u207b (a : \u03b1), f a * (f + g) a ^ (p - 1) \u2202\u03bc) + \u222b\u207b (a : \u03b1), g a * (f + g) a ^ (p - 1) \u2202\u03bc"}, {"tactic": "rw [lintegral_add_left' (hf.mul h_add_m)]", "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\nh_add_m : AEMeasurable fun a => (f + g) a ^ (p - 1)\nh_add_apply : (\u222b\u207b (a : \u03b1), (f + g) a * (f + g) a ^ (p - 1) \u2202\u03bc) = \u222b\u207b (a : \u03b1), (f a + g a) * (f + g) a ^ (p - 1) \u2202\u03bc\n\u22a2 (\u222b\u207b (a : \u03b1), f a * (f + g) a ^ (p - 1) + g a * (f + g) a ^ (p - 1) \u2202\u03bc) =\n    (\u222b\u207b (a : \u03b1), f a * (f + g) a ^ (p - 1) \u2202\u03bc) + \u222b\u207b (a : \u03b1), g a * (f + g) a ^ (p - 1) \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [add_mul]", "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\n\u22a2 ((\u222b\u207b (a : \u03b1), f a * (f + g) a ^ (p - 1) \u2202\u03bc) + \u222b\u207b (a : \u03b1), g a * (f + g) a ^ (p - 1) \u2202\u03bc) \u2264\n    ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)) *\n      (\u222b\u207b (a : \u03b1), (f a + g a) ^ p \u2202\u03bc) ^ (1 / q)", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\n\u22a2 ((\u222b\u207b (a : \u03b1), f a * (f + g) a ^ (p - 1) \u2202\u03bc) + \u222b\u207b (a : \u03b1), g a * (f + g) a ^ (p - 1) \u2202\u03bc) \u2264\n    (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), (f a + g a) ^ p \u2202\u03bc) ^ (1 / q) +\n      (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), (f a + g a) ^ p \u2202\u03bc) ^ (1 / q)"}, {"tactic": "exact\n  add_le_add\n    (lintegral_mul_rpow_le_lintegral_rpow_mul_lintegral_rpow hpq hf (hf.add hg) hf_top)\n    (lintegral_mul_rpow_le_lintegral_rpow_mul_lintegral_rpow hpq hg (hf.add hg) hg_top)", "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhf_top : (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) \u2260 \u22a4\nhg : AEMeasurable g\nhg_top : (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) \u2260 \u22a4\n\u22a2 ((\u222b\u207b (a : \u03b1), f a * (f + g) a ^ (p - 1) \u2202\u03bc) + \u222b\u207b (a : \u03b1), g a * (f + g) a ^ (p - 1) \u2202\u03bc) \u2264\n    (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), (f a + g a) ^ p \u2202\u03bc) ^ (1 / q) +\n      (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), (f a + g a) ^ p \u2202\u03bc) ^ (1 / q)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "full_name": "MeasureTheory.LocallyIntegrableOn.aestronglyMeasurable", "start": [79, 1], "end": [96, 49], "traced_tactics": [{"tactic": "have : \u2200 x : s, \u2203 u, IsOpen u \u2227 x.1 \u2208 u \u2227 IntegrableOn f (u \u2229 s) \u03bc := by\n  rintro \u27e8x, hx\u27e9\n  rcases hf x hx with \u27e8t, ht, h't\u27e9\n  rcases mem_nhdsWithin.1 ht with \u27e8u, u_open, x_mem, u_sub\u27e9\n  refine' \u27e8u, u_open, x_mem, h't.mono_set u_sub\u27e9", "state_before": "X : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc s)", "state_after": "X : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nthis : \u2200 (x : \u2191s), \u2203 u, IsOpen u \u2227 \u2191x \u2208 u \u2227 IntegrableOn f (u \u2229 s)\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc s)"}, {"tactic": "choose u u_open xu hu using this", "state_before": "X : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nthis : \u2200 (x : \u2191s), \u2203 u, IsOpen u \u2227 \u2191x \u2208 u \u2227 IntegrableOn f (u \u2229 s)\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc s)", "state_after": "X : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc s)"}, {"tactic": "obtain \u27e8T, T_count, hT\u27e9 : \u2203 T : Set s, T.Countable \u2227 s = \u22c3 i : T, u i \u2229 s := by\n  have : s \u2286 \u22c3 x : s, u x := fun y hy => mem_iUnion_of_mem \u27e8y, hy\u27e9 (xu \u27e8y, hy\u27e9)\n  obtain \u27e8T, hT_count, hT_un\u27e9 := isOpen_iUnion_countable u u_open\n  refine' \u27e8T, hT_count, _\u27e9\n  rw [\u2190 hT_un, biUnion_eq_iUnion] at this\n  rw [\u2190 iUnion_inter, eq_comm, inter_eq_right_iff_subset]\n  exact this", "state_before": "X : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc s)", "state_after": "case intro.intro\nX : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nT : Set \u2191s\nT_count : Set.Countable T\nhT : s = \u22c3 (i : \u2191T), u \u2191i \u2229 s\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc s)"}, {"tactic": "have : Countable T := countable_coe_iff.mpr T_count", "state_before": "case intro.intro\nX : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nT : Set \u2191s\nT_count : Set.Countable T\nhT : s = \u22c3 (i : \u2191T), u \u2191i \u2229 s\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc s)", "state_after": "case intro.intro\nX : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nT : Set \u2191s\nT_count : Set.Countable T\nhT : s = \u22c3 (i : \u2191T), u \u2191i \u2229 s\nthis : Countable \u2191T\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc s)"}, {"tactic": "rw [hT, aestronglyMeasurable_iUnion_iff]", "state_before": "case intro.intro\nX : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nT : Set \u2191s\nT_count : Set.Countable T\nhT : s = \u22c3 (i : \u2191T), u \u2191i \u2229 s\nthis : Countable \u2191T\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc s)", "state_after": "case intro.intro\nX : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nT : Set \u2191s\nT_count : Set.Countable T\nhT : s = \u22c3 (i : \u2191T), u \u2191i \u2229 s\nthis : Countable \u2191T\n\u22a2 \u2200 (i : \u2191T), AEStronglyMeasurable f (Measure.restrict \u03bc (u \u2191i \u2229 s))"}, {"tactic": "exact fun i : T => (hu i).aestronglyMeasurable", "state_before": "case intro.intro\nX : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nT : Set \u2191s\nT_count : Set.Countable T\nhT : s = \u22c3 (i : \u2191T), u \u2191i \u2229 s\nthis : Countable \u2191T\n\u22a2 \u2200 (i : \u2191T), AEStronglyMeasurable f (Measure.restrict \u03bc (u \u2191i \u2229 s))", "state_after": "no goals"}, {"tactic": "rintro \u27e8x, hx\u27e9", "state_before": "X : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\n\u22a2 \u2200 (x : \u2191s), \u2203 u, IsOpen u \u2227 \u2191x \u2208 u \u2227 IntegrableOn f (u \u2229 s)", "state_after": "case mk\nX : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nx : X\nhx : x \u2208 s\n\u22a2 \u2203 u, IsOpen u \u2227 \u2191{ val := x, property := hx } \u2208 u \u2227 IntegrableOn f (u \u2229 s)"}, {"tactic": "rcases hf x hx with \u27e8t, ht, h't\u27e9", "state_before": "case mk\nX : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nx : X\nhx : x \u2208 s\n\u22a2 \u2203 u, IsOpen u \u2227 \u2191{ val := x, property := hx } \u2208 u \u2227 IntegrableOn f (u \u2229 s)", "state_after": "case mk.intro.intro\nX : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nx : X\nhx : x \u2208 s\nt : Set X\nht : t \u2208 \ud835\udcdd[s] x\nh't : IntegrableOn f t\n\u22a2 \u2203 u, IsOpen u \u2227 \u2191{ val := x, property := hx } \u2208 u \u2227 IntegrableOn f (u \u2229 s)"}, {"tactic": "rcases mem_nhdsWithin.1 ht with \u27e8u, u_open, x_mem, u_sub\u27e9", "state_before": "case mk.intro.intro\nX : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nx : X\nhx : x \u2208 s\nt : Set X\nht : t \u2208 \ud835\udcdd[s] x\nh't : IntegrableOn f t\n\u22a2 \u2203 u, IsOpen u \u2227 \u2191{ val := x, property := hx } \u2208 u \u2227 IntegrableOn f (u \u2229 s)", "state_after": "case mk.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nx : X\nhx : x \u2208 s\nt : Set X\nht : t \u2208 \ud835\udcdd[s] x\nh't : IntegrableOn f t\nu : Set X\nu_open : IsOpen u\nx_mem : x \u2208 u\nu_sub : u \u2229 s \u2286 t\n\u22a2 \u2203 u, IsOpen u \u2227 \u2191{ val := x, property := hx } \u2208 u \u2227 IntegrableOn f (u \u2229 s)"}, {"tactic": "refine' \u27e8u, u_open, x_mem, h't.mono_set u_sub\u27e9", "state_before": "case mk.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nx : X\nhx : x \u2208 s\nt : Set X\nht : t \u2208 \ud835\udcdd[s] x\nh't : IntegrableOn f t\nu : Set X\nu_open : IsOpen u\nx_mem : x \u2208 u\nu_sub : u \u2229 s \u2286 t\n\u22a2 \u2203 u, IsOpen u \u2227 \u2191{ val := x, property := hx } \u2208 u \u2227 IntegrableOn f (u \u2229 s)", "state_after": "no goals"}, {"tactic": "have : s \u2286 \u22c3 x : s, u x := fun y hy => mem_iUnion_of_mem \u27e8y, hy\u27e9 (xu \u27e8y, hy\u27e9)", "state_before": "X : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\n\u22a2 \u2203 T, Set.Countable T \u2227 s = \u22c3 (i : \u2191T), u \u2191i \u2229 s", "state_after": "X : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nthis : s \u2286 \u22c3 (x : \u2191s), u x\n\u22a2 \u2203 T, Set.Countable T \u2227 s = \u22c3 (i : \u2191T), u \u2191i \u2229 s"}, {"tactic": "obtain \u27e8T, hT_count, hT_un\u27e9 := isOpen_iUnion_countable u u_open", "state_before": "X : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nthis : s \u2286 \u22c3 (x : \u2191s), u x\n\u22a2 \u2203 T, Set.Countable T \u2227 s = \u22c3 (i : \u2191T), u \u2191i \u2229 s", "state_after": "case intro.intro\nX : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nthis : s \u2286 \u22c3 (x : \u2191s), u x\nT : Set \u2191s\nhT_count : Set.Countable T\nhT_un : (\u22c3 (i : \u2191s) (_ : i \u2208 T), u i) = \u22c3 (i : \u2191s), u i\n\u22a2 \u2203 T, Set.Countable T \u2227 s = \u22c3 (i : \u2191T), u \u2191i \u2229 s"}, {"tactic": "refine' \u27e8T, hT_count, _\u27e9", "state_before": "case intro.intro\nX : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nthis : s \u2286 \u22c3 (x : \u2191s), u x\nT : Set \u2191s\nhT_count : Set.Countable T\nhT_un : (\u22c3 (i : \u2191s) (_ : i \u2208 T), u i) = \u22c3 (i : \u2191s), u i\n\u22a2 \u2203 T, Set.Countable T \u2227 s = \u22c3 (i : \u2191T), u \u2191i \u2229 s", "state_after": "case intro.intro\nX : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nthis : s \u2286 \u22c3 (x : \u2191s), u x\nT : Set \u2191s\nhT_count : Set.Countable T\nhT_un : (\u22c3 (i : \u2191s) (_ : i \u2208 T), u i) = \u22c3 (i : \u2191s), u i\n\u22a2 s = \u22c3 (i : \u2191T), u \u2191i \u2229 s"}, {"tactic": "rw [\u2190 hT_un, biUnion_eq_iUnion] at this", "state_before": "case intro.intro\nX : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nthis : s \u2286 \u22c3 (x : \u2191s), u x\nT : Set \u2191s\nhT_count : Set.Countable T\nhT_un : (\u22c3 (i : \u2191s) (_ : i \u2208 T), u i) = \u22c3 (i : \u2191s), u i\n\u22a2 s = \u22c3 (i : \u2191T), u \u2191i \u2229 s", "state_after": "case intro.intro\nX : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nT : Set \u2191s\nthis : s \u2286 \u22c3 (x : \u2191T), u \u2191x\nhT_count : Set.Countable T\nhT_un : (\u22c3 (i : \u2191s) (_ : i \u2208 T), u i) = \u22c3 (i : \u2191s), u i\n\u22a2 s = \u22c3 (i : \u2191T), u \u2191i \u2229 s"}, {"tactic": "rw [\u2190 iUnion_inter, eq_comm, inter_eq_right_iff_subset]", "state_before": "case intro.intro\nX : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nT : Set \u2191s\nthis : s \u2286 \u22c3 (x : \u2191T), u \u2191x\nhT_count : Set.Countable T\nhT_un : (\u22c3 (i : \u2191s) (_ : i \u2208 T), u i) = \u22c3 (i : \u2191s), u i\n\u22a2 s = \u22c3 (i : \u2191T), u \u2191i \u2229 s", "state_after": "case intro.intro\nX : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nT : Set \u2191s\nthis : s \u2286 \u22c3 (x : \u2191T), u \u2191x\nhT_count : Set.Countable T\nhT_un : (\u22c3 (i : \u2191s) (_ : i \u2208 T), u i) = \u22c3 (i : \u2191s), u i\n\u22a2 s \u2286 \u22c3 (i : \u2191T), u \u2191i"}, {"tactic": "exact this", "state_before": "case intro.intro\nX : Type u_1\nY : Type ?u.117872\nE : Type u_2\nR : Type ?u.117878\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormedAddCommGroup E\nf : X \u2192 E\n\u03bc : Measure X\ns : Set X\ninst\u271d : SecondCountableTopology X\nhf : LocallyIntegrableOn f s\nu : \u2191s \u2192 Set X\nu_open : \u2200 (x : \u2191s), IsOpen (u x)\nxu : \u2200 (x : \u2191s), \u2191x \u2208 u x\nhu : \u2200 (x : \u2191s), IntegrableOn f (u x \u2229 s)\nT : Set \u2191s\nthis : s \u2286 \u22c3 (x : \u2191T), u \u2191x\nhT_count : Set.Countable T\nhT_un : (\u22c3 (i : \u2191s) (_ : i \u2208 T), u i) = \u22c3 (i : \u2191s), u i\n\u22a2 s \u2286 \u22c3 (i : \u2191T), u \u2191i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/Set.lean", "full_name": "Equiv.self_comp_ofInjective_symm", "start": [632, 1], "end": [634, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Derivative.lean", "full_name": "Polynomial.iterate_derivative_sub", "start": [612, 1], "end": [614, 55], "traced_tactics": [{"tactic": "induction' k with k ih generalizing f g <;> simp [*]", "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Ring R\nk : \u2115\nf g : R[X]\n\u22a2 (\u2191derivative^[k]) (f - g) = (\u2191derivative^[k]) f - (\u2191derivative^[k]) g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Tactic/IntervalCases.lean", "full_name": "Mathlib.Tactic.IntervalCases.of_not_lt_right", "start": [133, 1], "end": [133, 99], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Stream/Init.lean", "full_name": "Stream'.take_zero", "start": [570, 1], "end": [571, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Sites/Grothendieck.lean", "full_name": "CategoryTheory.GrothendieckTopology.superset_covering", "start": [147, 1], "end": [152, 38], "traced_tactics": [{"tactic": "apply J.transitive sjx R fun Y f hf => _", "state_before": "C : Type u\ninst\u271d : Category C\nX Y : C\nS R : Sieve X\nJ : GrothendieckTopology C\nHss : S \u2264 R\nsjx : S \u2208 sieves J X\n\u22a2 R \u2208 sieves J X", "state_after": "C : Type u\ninst\u271d : Category C\nX Y : C\nS R : Sieve X\nJ : GrothendieckTopology C\nHss : S \u2264 R\nsjx : S \u2208 sieves J X\n\u22a2 \u2200 (Y : C) (f : Y \u27f6 X), S.arrows f \u2192 Sieve.pullback f R \u2208 sieves J Y"}, {"tactic": "intros Y f hf", "state_before": "C : Type u\ninst\u271d : Category C\nX Y : C\nS R : Sieve X\nJ : GrothendieckTopology C\nHss : S \u2264 R\nsjx : S \u2208 sieves J X\n\u22a2 \u2200 (Y : C) (f : Y \u27f6 X), S.arrows f \u2192 Sieve.pullback f R \u2208 sieves J Y", "state_after": "C : Type u\ninst\u271d : Category C\nX Y\u271d : C\nS R : Sieve X\nJ : GrothendieckTopology C\nHss : S \u2264 R\nsjx : S \u2208 sieves J X\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\n\u22a2 Sieve.pullback f R \u2208 sieves J Y"}, {"tactic": "apply covering_of_eq_top", "state_before": "C : Type u\ninst\u271d : Category C\nX Y\u271d : C\nS R : Sieve X\nJ : GrothendieckTopology C\nHss : S \u2264 R\nsjx : S \u2208 sieves J X\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\n\u22a2 Sieve.pullback f R \u2208 sieves J Y", "state_after": "case a\nC : Type u\ninst\u271d : Category C\nX Y\u271d : C\nS R : Sieve X\nJ : GrothendieckTopology C\nHss : S \u2264 R\nsjx : S \u2208 sieves J X\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\n\u22a2 Sieve.pullback f R = \u22a4"}, {"tactic": "rw [\u2190 top_le_iff, \u2190 S.pullback_eq_top_of_mem hf]", "state_before": "case a\nC : Type u\ninst\u271d : Category C\nX Y\u271d : C\nS R : Sieve X\nJ : GrothendieckTopology C\nHss : S \u2264 R\nsjx : S \u2208 sieves J X\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\n\u22a2 Sieve.pullback f R = \u22a4", "state_after": "case a\nC : Type u\ninst\u271d : Category C\nX Y\u271d : C\nS R : Sieve X\nJ : GrothendieckTopology C\nHss : S \u2264 R\nsjx : S \u2208 sieves J X\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\n\u22a2 Sieve.pullback f S \u2264 Sieve.pullback f R"}, {"tactic": "apply Sieve.pullback_monotone _ Hss", "state_before": "case a\nC : Type u\ninst\u271d : Category C\nX Y\u271d : C\nS R : Sieve X\nJ : GrothendieckTopology C\nHss : S \u2264 R\nsjx : S \u2208 sieves J X\nY : C\nf : Y \u27f6 X\nhf : S.arrows f\n\u22a2 Sieve.pullback f S \u2264 Sieve.pullback f R", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Star/Subalgebra.lean", "full_name": "StarSubalgebra.toSubalgebra_le_iff", "start": [109, 1], "end": [111, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/OreLocalization/Basic.lean", "full_name": "OreLocalization.mul_one", "start": [300, 11], "end": [302, 72], "traced_tactics": [{"tactic": "induction' x using OreLocalization.ind with r s", "state_before": "R : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nx : OreLocalization R S\n\u22a2 x * 1 = x", "state_after": "case c\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\n\u22a2 r /\u2092 s * 1 = r /\u2092 s"}, {"tactic": "simp [OreLocalization.one_def, oreDiv_mul_char r 1 s 1 1 s (by simp)]", "state_before": "case c\nR : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\n\u22a2 r /\u2092 s * 1 = r /\u2092 s", "state_after": "no goals"}, {"tactic": "simp", "state_before": "R : Type u_1\ninst\u271d\u00b9 : Monoid R\nS : Submonoid R\ninst\u271d : OreSet S\nr : R\ns : { x // x \u2208 S }\n\u22a2 1 * \u2191s = \u2191s * 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Nodup.lean", "full_name": "List.Nodup.concat", "start": [371, 11], "end": [372, 86], "traced_tactics": [{"tactic": "rw [concat_eq_append]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nl l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nh : \u00aca \u2208 l\nh' : Nodup l\n\u22a2 Nodup (concat l a)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nl l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nh : \u00aca \u2208 l\nh' : Nodup l\n\u22a2 Nodup (l ++ [a])"}, {"tactic": "exact h'.append (nodup_singleton _) (disjoint_singleton.2 h)", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nl l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nh : \u00aca \u2208 l\nh' : Nodup l\n\u22a2 Nodup (l ++ [a])", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean", "full_name": "PseudoEpimorphism.toFun_eq_coe", "start": [128, 1], "end": [128, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Prod.lean", "full_name": "Prod.swap_inv", "start": [155, 1], "end": [156, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "subset_interior_div_right", "start": [1355, 1], "end": [1356, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/IntegralClosure.lean", "full_name": "RingHom.isIntegral_of_surjective", "start": [1027, 1], "end": [1028, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/ModelTheory/Basic.lean", "full_name": "FirstOrder.Language.constants_mk\u2082", "start": [143, 1], "end": [145, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/LocalAtTarget.lean", "full_name": "Set.restrictPreimage_isClosedMap", "start": [70, 1], "end": [79, 81], "traced_tactics": [{"tactic": "rintro t \u27e8u, hu, e\u27e9", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type ?u.4199\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : IsClosedMap f\n\u22a2 IsClosedMap (restrictPreimage s f)", "state_after": "case mk.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type ?u.4199\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : IsClosedMap f\nt : Set \u2191(f \u207b\u00b9' s)\nu : Set \u03b1\nhu : IsOpen u\ne : Subtype.val \u207b\u00b9' u = t\u1d9c\n\u22a2 IsClosed (restrictPreimage s f '' t)"}, {"tactic": "refine' \u27e8\u27e8_, (H _ (IsOpen.isClosed_compl hu)).1, _\u27e9\u27e9", "state_before": "case mk.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type ?u.4199\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : IsClosedMap f\nt : Set \u2191(f \u207b\u00b9' s)\nu : Set \u03b1\nhu : IsOpen u\ne : Subtype.val \u207b\u00b9' u = t\u1d9c\n\u22a2 IsClosed (restrictPreimage s f '' t)", "state_after": "case mk.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type ?u.4199\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : IsClosedMap f\nt : Set \u2191(f \u207b\u00b9' s)\nu : Set \u03b1\nhu : IsOpen u\ne : Subtype.val \u207b\u00b9' u = t\u1d9c\n\u22a2 Subtype.val \u207b\u00b9' (f '' u\u1d9c)\u1d9c = (restrictPreimage s f '' t)\u1d9c"}, {"tactic": "rw [\u2190 (congr_arg HasCompl.compl e).trans (compl_compl t)]", "state_before": "case mk.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type ?u.4199\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : IsClosedMap f\nt : Set \u2191(f \u207b\u00b9' s)\nu : Set \u03b1\nhu : IsOpen u\ne : Subtype.val \u207b\u00b9' u = t\u1d9c\n\u22a2 Subtype.val \u207b\u00b9' (f '' u\u1d9c)\u1d9c = (restrictPreimage s f '' t)\u1d9c", "state_after": "case mk.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type ?u.4199\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : IsClosedMap f\nt : Set \u2191(f \u207b\u00b9' s)\nu : Set \u03b1\nhu : IsOpen u\ne : Subtype.val \u207b\u00b9' u = t\u1d9c\n\u22a2 Subtype.val \u207b\u00b9' (f '' u\u1d9c)\u1d9c = (restrictPreimage s f '' (Subtype.val \u207b\u00b9' u)\u1d9c)\u1d9c"}, {"tactic": "simp only [Set.preimage_compl, compl_inj_iff]", "state_before": "case mk.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type ?u.4199\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : IsClosedMap f\nt : Set \u2191(f \u207b\u00b9' s)\nu : Set \u03b1\nhu : IsOpen u\ne : Subtype.val \u207b\u00b9' u = t\u1d9c\n\u22a2 Subtype.val \u207b\u00b9' (f '' u\u1d9c)\u1d9c = (restrictPreimage s f '' (Subtype.val \u207b\u00b9' u)\u1d9c)\u1d9c", "state_after": "case mk.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type ?u.4199\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : IsClosedMap f\nt : Set \u2191(f \u207b\u00b9' s)\nu : Set \u03b1\nhu : IsOpen u\ne : Subtype.val \u207b\u00b9' u = t\u1d9c\n\u22a2 Subtype.val \u207b\u00b9' (f '' u\u1d9c) = restrictPreimage s f '' (Subtype.val \u207b\u00b9' u)\u1d9c"}, {"tactic": "ext \u27e8x, hx\u27e9", "state_before": "case mk.intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type ?u.4199\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : IsClosedMap f\nt : Set \u2191(f \u207b\u00b9' s)\nu : Set \u03b1\nhu : IsOpen u\ne : Subtype.val \u207b\u00b9' u = t\u1d9c\n\u22a2 Subtype.val \u207b\u00b9' (f '' u\u1d9c) = restrictPreimage s f '' (Subtype.val \u207b\u00b9' u)\u1d9c", "state_after": "case mk.intro.intro.h.mk\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type ?u.4199\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : IsClosedMap f\nt : Set \u2191(f \u207b\u00b9' s)\nu : Set \u03b1\nhu : IsOpen u\ne : Subtype.val \u207b\u00b9' u = t\u1d9c\nx : \u03b2\nhx : x \u2208 s\n\u22a2 { val := x, property := hx } \u2208 Subtype.val \u207b\u00b9' (f '' u\u1d9c) \u2194\n    { val := x, property := hx } \u2208 restrictPreimage s f '' (Subtype.val \u207b\u00b9' u)\u1d9c"}, {"tactic": "suffices (\u2203 y, y \u2209 u \u2227 f y = x) \u2194 \u2203 y, y \u2209 u \u2227 f y \u2208 s \u2227 f y = x by\n  simpa [Set.restrictPreimage, \u2190 Subtype.coe_inj]", "state_before": "case mk.intro.intro.h.mk\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type ?u.4199\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : IsClosedMap f\nt : Set \u2191(f \u207b\u00b9' s)\nu : Set \u03b1\nhu : IsOpen u\ne : Subtype.val \u207b\u00b9' u = t\u1d9c\nx : \u03b2\nhx : x \u2208 s\n\u22a2 { val := x, property := hx } \u2208 Subtype.val \u207b\u00b9' (f '' u\u1d9c) \u2194\n    { val := x, property := hx } \u2208 restrictPreimage s f '' (Subtype.val \u207b\u00b9' u)\u1d9c", "state_after": "case mk.intro.intro.h.mk\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type ?u.4199\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : IsClosedMap f\nt : Set \u2191(f \u207b\u00b9' s)\nu : Set \u03b1\nhu : IsOpen u\ne : Subtype.val \u207b\u00b9' u = t\u1d9c\nx : \u03b2\nhx : x \u2208 s\n\u22a2 (\u2203 y, \u00acy \u2208 u \u2227 f y = x) \u2194 \u2203 y, \u00acy \u2208 u \u2227 f y \u2208 s \u2227 f y = x"}, {"tactic": "exact \u27e8fun \u27e8a, b, c\u27e9 => \u27e8a, b, c.symm \u25b8 hx, c\u27e9, fun \u27e8a, b, _, c\u27e9 => \u27e8a, b, c\u27e9\u27e9", "state_before": "case mk.intro.intro.h.mk\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type ?u.4199\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : IsClosedMap f\nt : Set \u2191(f \u207b\u00b9' s)\nu : Set \u03b1\nhu : IsOpen u\ne : Subtype.val \u207b\u00b9' u = t\u1d9c\nx : \u03b2\nhx : x \u2208 s\n\u22a2 (\u2203 y, \u00acy \u2208 u \u2227 f y = x) \u2194 \u2203 y, \u00acy \u2208 u \u2227 f y \u2208 s \u2227 f y = x", "state_after": "no goals"}, {"tactic": "simpa [Set.restrictPreimage, \u2190 Subtype.coe_inj]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns\u271d : Set \u03b2\n\u03b9 : Type ?u.4199\nU : \u03b9 \u2192 Opens \u03b2\nhU : iSup U = \u22a4\ns : Set \u03b2\nH : IsClosedMap f\nt : Set \u2191(f \u207b\u00b9' s)\nu : Set \u03b1\nhu : IsOpen u\ne : Subtype.val \u207b\u00b9' u = t\u1d9c\nx : \u03b2\nhx : x \u2208 s\nthis : (\u2203 y, \u00acy \u2208 u \u2227 f y = x) \u2194 \u2203 y, \u00acy \u2208 u \u2227 f y \u2208 s \u2227 f y = x\n\u22a2 { val := x, property := hx } \u2208 Subtype.val \u207b\u00b9' (f '' u\u1d9c) \u2194\n    { val := x, property := hx } \u2208 restrictPreimage s f '' (Subtype.val \u207b\u00b9' u)\u1d9c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.count_filter_of_neg", "start": [2493, 1], "end": [2495, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Monoidal/Mon_.lean", "full_name": "Mon_.mul_one_hom", "start": [81, 1], "end": [82, 85], "traced_tactics": [{"tactic": "rw [\u2190 tensor_id_comp_id_tensor, Category.assoc, M.mul_one, rightUnitor_naturality]", "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category C\ninst\u271d : MonoidalCategory C\nM : Mon_ C\nZ : C\nf : Z \u27f6 M.X\n\u22a2 (f \u2297 M.one) \u226b M.mul = (\u03c1_ Z).hom \u226b f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Core.lean", "full_name": "Quotient.inductionOn\u2082", "start": [1455, 11], "end": [1463, 10], "traced_tactics": [{"tactic": "induction q\u2081 using Quotient.ind", "state_before": "a b c d : Prop\n\u03b1 : Sort uA\n\u03b2 : Sort uB\n\u03c6 : Sort uC\ns\u2081 : Setoid \u03b1\ns\u2082 : Setoid \u03b2\nmotive : Quotient s\u2081 \u2192 Quotient s\u2082 \u2192 Prop\nq\u2081 : Quotient s\u2081\nq\u2082 : Quotient s\u2082\nh : \u2200 (a : \u03b1) (b : \u03b2), motive (Quotient.mk s\u2081 a) (Quotient.mk s\u2082 b)\n\u22a2 motive q\u2081 q\u2082", "state_after": "case a\na b c d : Prop\n\u03b1 : Sort uA\n\u03b2 : Sort uB\n\u03c6 : Sort uC\ns\u2081 : Setoid \u03b1\ns\u2082 : Setoid \u03b2\nmotive : Quotient s\u2081 \u2192 Quotient s\u2082 \u2192 Prop\nq\u2082 : Quotient s\u2082\nh : \u2200 (a : \u03b1) (b : \u03b2), motive (Quotient.mk s\u2081 a) (Quotient.mk s\u2082 b)\na\u271d : \u03b1\n\u22a2 motive (Quotient.mk s\u2081 a\u271d) q\u2082"}, {"tactic": "induction q\u2082 using Quotient.ind", "state_before": "case a\na b c d : Prop\n\u03b1 : Sort uA\n\u03b2 : Sort uB\n\u03c6 : Sort uC\ns\u2081 : Setoid \u03b1\ns\u2082 : Setoid \u03b2\nmotive : Quotient s\u2081 \u2192 Quotient s\u2082 \u2192 Prop\nq\u2082 : Quotient s\u2082\nh : \u2200 (a : \u03b1) (b : \u03b2), motive (Quotient.mk s\u2081 a) (Quotient.mk s\u2082 b)\na\u271d : \u03b1\n\u22a2 motive (Quotient.mk s\u2081 a\u271d) q\u2082", "state_after": "case a.a\na b c d : Prop\n\u03b1 : Sort uA\n\u03b2 : Sort uB\n\u03c6 : Sort uC\ns\u2081 : Setoid \u03b1\ns\u2082 : Setoid \u03b2\nmotive : Quotient s\u2081 \u2192 Quotient s\u2082 \u2192 Prop\nh : \u2200 (a : \u03b1) (b : \u03b2), motive (Quotient.mk s\u2081 a) (Quotient.mk s\u2082 b)\na\u271d\u00b9 : \u03b1\na\u271d : \u03b2\n\u22a2 motive (Quotient.mk s\u2081 a\u271d\u00b9) (Quotient.mk s\u2082 a\u271d)"}, {"tactic": "apply h", "state_before": "case a.a\na b c d : Prop\n\u03b1 : Sort uA\n\u03b2 : Sort uB\n\u03c6 : Sort uC\ns\u2081 : Setoid \u03b1\ns\u2082 : Setoid \u03b2\nmotive : Quotient s\u2081 \u2192 Quotient s\u2082 \u2192 Prop\nh : \u2200 (a : \u03b1) (b : \u03b2), motive (Quotient.mk s\u2081 a) (Quotient.mk s\u2082 b)\na\u271d\u00b9 : \u03b1\na\u271d : \u03b2\n\u22a2 motive (Quotient.mk s\u2081 a\u271d\u00b9) (Quotient.mk s\u2082 a\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Degree/TrailingDegree.lean", "full_name": "Polynomial.natTrailingDegree_eq_support_min'", "start": [322, 1], "end": [329, 72], "traced_tactics": [{"tactic": "apply le_antisymm", "state_before": "R : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : p \u2260 0\n\u22a2 natTrailingDegree p = min' (support p) (_ : Finset.Nonempty (support p))", "state_after": "case a\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : p \u2260 0\n\u22a2 natTrailingDegree p \u2264 min' (support p) (_ : Finset.Nonempty (support p))\n\ncase a\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : p \u2260 0\n\u22a2 min' (support p) (_ : Finset.Nonempty (support p)) \u2264 natTrailingDegree p"}, {"tactic": "apply le_min'", "state_before": "case a\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : p \u2260 0\n\u22a2 natTrailingDegree p \u2264 min' (support p) (_ : Finset.Nonempty (support p))", "state_after": "case a.H2\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : p \u2260 0\n\u22a2 \u2200 (y : \u2115), y \u2208 support p \u2192 natTrailingDegree p \u2264 y"}, {"tactic": "intro y hy", "state_before": "case a.H2\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : p \u2260 0\n\u22a2 \u2200 (y : \u2115), y \u2208 support p \u2192 natTrailingDegree p \u2264 y", "state_after": "case a.H2\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : p \u2260 0\ny : \u2115\nhy : y \u2208 support p\n\u22a2 natTrailingDegree p \u2264 y"}, {"tactic": "exact natTrailingDegree_le_of_mem_supp y hy", "state_before": "case a.H2\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : p \u2260 0\ny : \u2115\nhy : y \u2208 support p\n\u22a2 natTrailingDegree p \u2264 y", "state_after": "no goals"}, {"tactic": "apply Finset.min'_le", "state_before": "case a\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : p \u2260 0\n\u22a2 min' (support p) (_ : Finset.Nonempty (support p)) \u2264 natTrailingDegree p", "state_after": "case a.H2\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : p \u2260 0\n\u22a2 natTrailingDegree p \u2208 support p"}, {"tactic": "exact mem_support_iff.mpr (trailingCoeff_nonzero_iff_nonzero.mpr h)", "state_before": "case a.H2\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : p \u2260 0\n\u22a2 natTrailingDegree p \u2208 support p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finite/Defs.lean", "full_name": "Finite.exists_equiv_fin", "start": [69, 1], "end": [70, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Dual.lean", "full_name": "Subspace.finrank_add_finrank_dualCoannihilator_eq", "start": [1140, 1], "end": [1147, 91], "traced_tactics": [{"tactic": "rw [finrank_dualCoannihilator_eq]", "state_before": "K : Type u\nV : Type v\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nW\u271d : Subspace K V\nV\u2081 : Type ?u.1350893\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : FiniteDimensional K V\ninst\u271d : FiniteDimensional K V\u2081\nW : Subspace K (Module.Dual K V)\n\u22a2 finrank K { x // x \u2208 W } + finrank K { x // x \u2208 dualCoannihilator W } = finrank K V", "state_after": "K : Type u\nV : Type v\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nW\u271d : Subspace K V\nV\u2081 : Type ?u.1350893\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : FiniteDimensional K V\ninst\u271d : FiniteDimensional K V\u2081\nW : Subspace K (Module.Dual K V)\n\u22a2 finrank K { x // x \u2208 W } + finrank K { x // x \u2208 dualAnnihilator W } = finrank K V"}, {"tactic": "let equiv := W.quotEquivAnnihilator", "state_before": "K : Type u\nV : Type v\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nW\u271d : Subspace K V\nV\u2081 : Type ?u.1350893\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : FiniteDimensional K V\ninst\u271d : FiniteDimensional K V\u2081\nW : Subspace K (Module.Dual K V)\n\u22a2 finrank K { x // x \u2208 W } + finrank K { x // x \u2208 dualAnnihilator W } = finrank K V", "state_after": "K : Type u\nV : Type v\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nW\u271d : Subspace K V\nV\u2081 : Type ?u.1350893\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : FiniteDimensional K V\ninst\u271d : FiniteDimensional K V\u2081\nW : Subspace K (Module.Dual K V)\nequiv : (Module.Dual K V \u29f8 W) \u2243\u2097[K] { x // x \u2208 dualAnnihilator W } := quotEquivAnnihilator W\n\u22a2 finrank K { x // x \u2208 W } + finrank K { x // x \u2208 dualAnnihilator W } = finrank K V"}, {"tactic": "have eq := LinearEquiv.finrank_eq (R := K) (M := (Module.Dual K V) \u29f8 W)\n  (M\u2082 := { x // x \u2208 dualAnnihilator W }) equiv", "state_before": "K : Type u\nV : Type v\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nW\u271d : Subspace K V\nV\u2081 : Type ?u.1350893\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : FiniteDimensional K V\ninst\u271d : FiniteDimensional K V\u2081\nW : Subspace K (Module.Dual K V)\nequiv : (Module.Dual K V \u29f8 W) \u2243\u2097[K] { x // x \u2208 dualAnnihilator W } := quotEquivAnnihilator W\n\u22a2 finrank K { x // x \u2208 W } + finrank K { x // x \u2208 dualAnnihilator W } = finrank K V", "state_after": "K : Type u\nV : Type v\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nW\u271d : Subspace K V\nV\u2081 : Type ?u.1350893\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : FiniteDimensional K V\ninst\u271d : FiniteDimensional K V\u2081\nW : Subspace K (Module.Dual K V)\nequiv : (Module.Dual K V \u29f8 W) \u2243\u2097[K] { x // x \u2208 dualAnnihilator W } := quotEquivAnnihilator W\neq : finrank K (Module.Dual K V \u29f8 W) = finrank K { x // x \u2208 dualAnnihilator W }\n\u22a2 finrank K { x // x \u2208 W } + finrank K { x // x \u2208 dualAnnihilator W } = finrank K V"}, {"tactic": "rw [eq.symm, add_comm, Submodule.finrank_quotient_add_finrank, Subspace.dual_finrank_eq]", "state_before": "K : Type u\nV : Type v\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nW\u271d : Subspace K V\nV\u2081 : Type ?u.1350893\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : FiniteDimensional K V\ninst\u271d : FiniteDimensional K V\u2081\nW : Subspace K (Module.Dual K V)\nequiv : (Module.Dual K V \u29f8 W) \u2243\u2097[K] { x // x \u2208 dualAnnihilator W } := quotEquivAnnihilator W\neq : finrank K (Module.Dual K V \u29f8 W) = finrank K { x // x \u2208 dualAnnihilator W }\n\u22a2 finrank K { x // x \u2208 W } + finrank K { x // x \u2208 dualAnnihilator W } = finrank K V", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SumIntegralComparisons.lean", "full_name": "AntitoneOn.integral_le_sum_Ico", "start": [79, 1], "end": [101, 59], "traced_tactics": [{"tactic": "rw [(Nat.sub_add_cancel hab).symm, Nat.cast_add]", "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n\u22a2 (\u222b (x : \u211d) in \u2191a..\u2191b, f x) \u2264 \u2211 x in Finset.Ico a b, f \u2191x", "state_after": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n\u22a2 (\u222b (x : \u211d) in \u2191a..\u2191(b - a) + \u2191a, f x) \u2264 \u2211 x in Finset.Ico a (b - a + a), f \u2191x"}, {"tactic": "rw [\u2190 Finset.sum_Ico_add, Nat.Ico_zero_eq_range]", "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n\u22a2 (\u222b (x : \u211d) in \u2191a..\u2191a + \u2191(b - a), f x) \u2264 \u2211 x in Finset.Ico (0 + a) (b - a + a), f \u2191x", "state_after": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n\u22a2 (\u222b (x : \u211d) in \u2191a..\u2191a + \u2191(b - a), f x) \u2264 \u2211 x in Finset.range (b - a), f \u2191(a + x)"}, {"tactic": "apply AntitoneOn.integral_le_sum", "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n\u22a2 (\u222b (x : \u211d) in \u2191a..\u2191a + \u2191(b - a), f x) \u2264 \u2211 x in Finset.range (b - a), f (\u2191a + \u2191x)", "state_after": "case hf\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n\u22a2 AntitoneOn (fun x => f x) (Icc (\u2191a) (\u2191a + \u2191(b - a)))"}, {"tactic": "simp only [hf, hab, Nat.cast_sub, add_sub_cancel'_right]", "state_before": "case hf\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : AntitoneOn f (Icc \u2191a \u2191b)\n\u22a2 AntitoneOn (fun x => f x) (Icc (\u2191a) (\u2191a + \u2191(b - a)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Eval.lean", "full_name": "Polynomial.map_smul", "start": [734, 11], "end": [735, 51], "traced_tactics": [{"tactic": "rw [map, eval\u2082_smul, RingHom.comp_apply, C_mul']", "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r\u271d : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nr : R\n\u22a2 map f (r \u2022 p) = \u2191f r \u2022 map f p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Nat/Lemmas.lean", "full_name": "Nat.lt_iff_le_and_ne", "start": [44, 11], "end": [46, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/ModEq.lean", "full_name": "Nat.frequently_mod_eq", "start": [31, 1], "end": [32, 75], "traced_tactics": [{"tactic": "simpa only [Nat.ModEq, mod_eq_of_lt h] using frequently_modEq h.ne_bot d", "state_before": "d n : \u2115\nh : d < n\n\u22a2 \u2203\u1da0 (m : \u2115) in atTop, m % n = d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/IndicatorFunction.lean", "full_name": "Antitone.tendsto_indicator", "start": [76, 1], "end": [89, 72], "traced_tactics": [{"tactic": "by_cases h : \u2203 i, a \u2209 s i", "state_before": "\u03b1 : Type u_3\n\u03b2 : Type u_2\nM : Type ?u.4482\nE : Type ?u.4485\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : Zero \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\n\u22a2 Tendsto (fun i => indicator (s i) f a) atTop (pure (indicator (\u22c2 (i : \u03b9), s i) f a))", "state_after": "case pos\n\u03b1 : Type u_3\n\u03b2 : Type u_2\nM : Type ?u.4482\nE : Type ?u.4485\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : Zero \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nh : \u2203 i, \u00aca \u2208 s i\n\u22a2 Tendsto (fun i => indicator (s i) f a) atTop (pure (indicator (\u22c2 (i : \u03b9), s i) f a))\n\ncase neg\n\u03b1 : Type u_3\n\u03b2 : Type u_2\nM : Type ?u.4482\nE : Type ?u.4485\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : Zero \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nh : \u00ac\u2203 i, \u00aca \u2208 s i\n\u22a2 Tendsto (fun i => indicator (s i) f a) atTop (pure (indicator (\u22c2 (i : \u03b9), s i) f a))"}, {"tactic": "rcases h with \u27e8i, hi\u27e9", "state_before": "case pos\n\u03b1 : Type u_3\n\u03b2 : Type u_2\nM : Type ?u.4482\nE : Type ?u.4485\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : Zero \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nh : \u2203 i, \u00aca \u2208 s i\n\u22a2 Tendsto (fun i => indicator (s i) f a) atTop (pure (indicator (\u22c2 (i : \u03b9), s i) f a))", "state_after": "case pos.intro\n\u03b1 : Type u_3\n\u03b2 : Type u_2\nM : Type ?u.4482\nE : Type ?u.4485\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : Zero \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ni : \u03b9\nhi : \u00aca \u2208 s i\n\u22a2 Tendsto (fun i => indicator (s i) f a) atTop (pure (indicator (\u22c2 (i : \u03b9), s i) f a))"}, {"tactic": "refine' tendsto_pure.2 ((eventually_ge_atTop i).mono fun n hn => _)", "state_before": "case pos.intro\n\u03b1 : Type u_3\n\u03b2 : Type u_2\nM : Type ?u.4482\nE : Type ?u.4485\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : Zero \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ni : \u03b9\nhi : \u00aca \u2208 s i\n\u22a2 Tendsto (fun i => indicator (s i) f a) atTop (pure (indicator (\u22c2 (i : \u03b9), s i) f a))", "state_after": "case pos.intro\n\u03b1 : Type u_3\n\u03b2 : Type u_2\nM : Type ?u.4482\nE : Type ?u.4485\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : Zero \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ni : \u03b9\nhi : \u00aca \u2208 s i\nn : \u03b9\nhn : i \u2264 n\n\u22a2 indicator (s n) f a = indicator (\u22c2 (i : \u03b9), s i) f a"}, {"tactic": "rw [indicator_of_not_mem _ _, indicator_of_not_mem _ _]", "state_before": "case pos.intro\n\u03b1 : Type u_3\n\u03b2 : Type u_2\nM : Type ?u.4482\nE : Type ?u.4485\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : Zero \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ni : \u03b9\nhi : \u00aca \u2208 s i\nn : \u03b9\nhn : i \u2264 n\n\u22a2 indicator (s n) f a = indicator (\u22c2 (i : \u03b9), s i) f a", "state_after": "\u03b1 : Type u_3\n\u03b2 : Type u_2\nM : Type ?u.4482\nE : Type ?u.4485\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : Zero \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ni : \u03b9\nhi : \u00aca \u2208 s i\nn : \u03b9\nhn : i \u2264 n\n\u22a2 \u00aca \u2208 \u22c2 (i : \u03b9), s i\n\n\u03b1 : Type u_3\n\u03b2 : Type u_2\nM : Type ?u.4482\nE : Type ?u.4485\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : Zero \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ni : \u03b9\nhi : \u00aca \u2208 s i\nn : \u03b9\nhn : i \u2264 n\n\u22a2 \u00aca \u2208 s n"}, {"tactic": "simp only [mem_iInter, not_forall]", "state_before": "\u03b1 : Type u_3\n\u03b2 : Type u_2\nM : Type ?u.4482\nE : Type ?u.4485\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : Zero \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ni : \u03b9\nhi : \u00aca \u2208 s i\nn : \u03b9\nhn : i \u2264 n\n\u22a2 \u00aca \u2208 \u22c2 (i : \u03b9), s i", "state_after": "\u03b1 : Type u_3\n\u03b2 : Type u_2\nM : Type ?u.4482\nE : Type ?u.4485\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : Zero \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ni : \u03b9\nhi : \u00aca \u2208 s i\nn : \u03b9\nhn : i \u2264 n\n\u22a2 \u2203 x, \u00aca \u2208 s x"}, {"tactic": "exact \u27e8i, hi\u27e9", "state_before": "\u03b1 : Type u_3\n\u03b2 : Type u_2\nM : Type ?u.4482\nE : Type ?u.4485\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : Zero \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ni : \u03b9\nhi : \u00aca \u2208 s i\nn : \u03b9\nhn : i \u2264 n\n\u22a2 \u2203 x, \u00aca \u2208 s x", "state_after": "no goals"}, {"tactic": "intro h", "state_before": "\u03b1 : Type u_3\n\u03b2 : Type u_2\nM : Type ?u.4482\nE : Type ?u.4485\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : Zero \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ni : \u03b9\nhi : \u00aca \u2208 s i\nn : \u03b9\nhn : i \u2264 n\n\u22a2 \u00aca \u2208 s n", "state_after": "\u03b1 : Type u_3\n\u03b2 : Type u_2\nM : Type ?u.4482\nE : Type ?u.4485\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : Zero \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ni : \u03b9\nhi : \u00aca \u2208 s i\nn : \u03b9\nhn : i \u2264 n\nh : a \u2208 s n\n\u22a2 False"}, {"tactic": "have := hs hn h", "state_before": "\u03b1 : Type u_3\n\u03b2 : Type u_2\nM : Type ?u.4482\nE : Type ?u.4485\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : Zero \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ni : \u03b9\nhi : \u00aca \u2208 s i\nn : \u03b9\nhn : i \u2264 n\nh : a \u2208 s n\n\u22a2 False", "state_after": "\u03b1 : Type u_3\n\u03b2 : Type u_2\nM : Type ?u.4482\nE : Type ?u.4485\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : Zero \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ni : \u03b9\nhi : \u00aca \u2208 s i\nn : \u03b9\nhn : i \u2264 n\nh : a \u2208 s n\nthis : a \u2208 s i\n\u22a2 False"}, {"tactic": "contradiction", "state_before": "\u03b1 : Type u_3\n\u03b2 : Type u_2\nM : Type ?u.4482\nE : Type ?u.4485\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : Zero \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ni : \u03b9\nhi : \u00aca \u2208 s i\nn : \u03b9\nhn : i \u2264 n\nh : a \u2208 s n\nthis : a \u2208 s i\n\u22a2 False", "state_after": "no goals"}, {"tactic": "push_neg  at h", "state_before": "case neg\n\u03b1 : Type u_3\n\u03b2 : Type u_2\nM : Type ?u.4482\nE : Type ?u.4485\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : Zero \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nh : \u00ac\u2203 i, \u00aca \u2208 s i\n\u22a2 Tendsto (fun i => indicator (s i) f a) atTop (pure (indicator (\u22c2 (i : \u03b9), s i) f a))", "state_after": "case neg\n\u03b1 : Type u_3\n\u03b2 : Type u_2\nM : Type ?u.4482\nE : Type ?u.4485\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : Zero \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nh : \u2200 (i : \u03b9), a \u2208 s i\n\u22a2 Tendsto (fun i => indicator (s i) f a) atTop (pure (indicator (\u22c2 (i : \u03b9), s i) f a))"}, {"tactic": "simp only [indicator_of_mem, h, mem_iInter.2 h, tendsto_const_pure]", "state_before": "case neg\n\u03b1 : Type u_3\n\u03b2 : Type u_2\nM : Type ?u.4482\nE : Type ?u.4485\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : Zero \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nh : \u2200 (i : \u03b9), a \u2208 s i\n\u22a2 Tendsto (fun i => indicator (s i) f a) atTop (pure (indicator (\u22c2 (i : \u03b9), s i) f a))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Prod.lean", "full_name": "SimpleGraph.boxProd_degree", "start": [243, 1], "end": [247, 73], "traced_tactics": [{"tactic": "rw [degree, degree, degree, boxProd_neighborFinset, Finset.card_disjUnion]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.89124\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nx : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b2 : Fintype \u2191(neighborSet G x.fst)\ninst\u271d\u00b9 : Fintype \u2191(neighborSet H x.snd)\ninst\u271d : Fintype \u2191(neighborSet (G \u25a1 H) x)\n\u22a2 degree (G \u25a1 H) x = degree G x.fst + degree H x.snd", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.89124\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nx : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b2 : Fintype \u2191(neighborSet G x.fst)\ninst\u271d\u00b9 : Fintype \u2191(neighborSet H x.snd)\ninst\u271d : Fintype \u2191(neighborSet (G \u25a1 H) x)\n\u22a2 Finset.card (neighborFinset G x.fst \u00d7\u02e2 {x.snd}) + Finset.card ({x.fst} \u00d7\u02e2 neighborFinset H x.snd) =\n    Finset.card (neighborFinset G x.fst) + Finset.card (neighborFinset H x.snd)"}, {"tactic": "simp_rw [Finset.card_product, Finset.card_singleton, mul_one, one_mul]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.89124\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nx : \u03b1 \u00d7 \u03b2\ninst\u271d\u00b2 : Fintype \u2191(neighborSet G x.fst)\ninst\u271d\u00b9 : Fintype \u2191(neighborSet H x.snd)\ninst\u271d : Fintype \u2191(neighborSet (G \u25a1 H) x)\n\u22a2 Finset.card (neighborFinset G x.fst \u00d7\u02e2 {x.snd}) + Finset.card ({x.fst} \u00d7\u02e2 neighborFinset H x.snd) =\n    Finset.card (neighborFinset G x.fst) + Finset.card (neighborFinset H x.snd)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Rat/NNRat.lean", "full_name": "NNRat.bddAbove_coe", "start": [301, 1], "end": [305, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Bounds.lean", "full_name": "mul_mem_lowerBounds_mul", "start": [116, 1], "end": [118, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Sets/Order.lean", "full_name": "ClopenUpperSet.coe_mk", "start": [67, 1], "end": [68, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/Adjoin.lean", "full_name": "IntermediateField.finrank_eq_one_iff", "start": [718, 1], "end": [720, 22], "traced_tactics": [{"tactic": "rw [\u2190 toSubalgebra_eq_iff, \u2190 finrank_eq_finrank_subalgebra, Subalgebra.finrank_eq_one_iff,\n  bot_toSubalgebra]", "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\n\u03b1 : E\nS : Set E\nK L : IntermediateField F E\n\u22a2 finrank F { x // x \u2208 K } = 1 \u2194 K = \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.AEEqFun.Integrable.add", "start": [1242, 1], "end": [1245, 20], "traced_tactics": [{"tactic": "refine' induction_on\u2082 f g fun f hf g hg hfi hgi => _", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.1266441\n\u03b4 : Type ?u.1266444\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf g : \u03b1 \u2192\u2098[\u03bc] \u03b2\n\u22a2 Integrable f \u2192 Integrable g \u2192 Integrable (f + g)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.1266441\n\u03b4 : Type ?u.1266444\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf\u271d g\u271d : \u03b1 \u2192\u2098[\u03bc] \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 \u03b2\nhg : AEStronglyMeasurable g \u03bc\nhfi : Integrable (mk f hf)\nhgi : Integrable (mk g hg)\n\u22a2 Integrable (mk f hf + mk g hg)"}, {"tactic": "simp only [integrable_mk, mk_add_mk] at hfi hgi \u22a2", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.1266441\n\u03b4 : Type ?u.1266444\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf\u271d g\u271d : \u03b1 \u2192\u2098[\u03bc] \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 \u03b2\nhg : AEStronglyMeasurable g \u03bc\nhfi : Integrable (mk f hf)\nhgi : Integrable (mk g hg)\n\u22a2 Integrable (mk f hf + mk g hg)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.1266441\n\u03b4 : Type ?u.1266444\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf\u271d g\u271d : \u03b1 \u2192\u2098[\u03bc] \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 \u03b2\nhg : AEStronglyMeasurable g \u03bc\nhfi : MeasureTheory.Integrable f\nhgi : MeasureTheory.Integrable g\n\u22a2 MeasureTheory.Integrable (f + g)"}, {"tactic": "exact hfi.add hgi", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.1266441\n\u03b4 : Type ?u.1266444\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf\u271d g\u271d : \u03b1 \u2192\u2098[\u03bc] \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : AEStronglyMeasurable f \u03bc\ng : \u03b1 \u2192 \u03b2\nhg : AEStronglyMeasurable g \u03bc\nhfi : MeasureTheory.Integrable f\nhgi : MeasureTheory.Integrable g\n\u22a2 MeasureTheory.Integrable (f + g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "full_name": "LinearIsometryEquiv.trans_refl", "start": [817, 1], "end": [818, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Data/Nat/Basic.lean", "full_name": "Nat.not_beq_eq_true_eq", "start": [105, 9], "end": [110, 22], "traced_tactics": [{"tactic": "subst h\u2082", "state_before": "a b : Nat\nh\u2081 : (!a == b) = true\nh\u2082 : a = b\n\u22a2 False", "state_after": "a : Nat\nh\u2081 : (!a == a) = true\n\u22a2 False"}, {"tactic": "rw [LawfulBEq.rfl] at h\u2081", "state_before": "a : Nat\nh\u2081 : (!a == a) = true\n\u22a2 False", "state_after": "a : Nat\nh\u2081 : (!true) = true\n\u22a2 False"}, {"tactic": "contradiction", "state_before": "a : Nat\nh\u2081 : (!true) = true\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simp [this]", "state_before": "a b : Nat\nh : \u00aca = b\nthis : \u00ac(a == b) = true\n\u22a2 (!a == b) = true", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Homology/HomologicalComplex.lean", "full_name": "HomologicalComplex.dFrom_eq", "start": [420, 1], "end": [422, 34], "traced_tactics": [{"tactic": "obtain rfl := c.next_eq' r", "state_before": "\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category V\ninst\u271d : HasZeroMorphisms V\nc : ComplexShape \u03b9\nC : HomologicalComplex V c\ni j : \u03b9\nr : ComplexShape.Rel c i j\n\u22a2 dFrom C i = d C i j \u226b (xNextIso C r).inv", "state_after": "\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category V\ninst\u271d : HasZeroMorphisms V\nc : ComplexShape \u03b9\nC : HomologicalComplex V c\ni : \u03b9\nr : ComplexShape.Rel c i (ComplexShape.next c i)\n\u22a2 dFrom C i = d C i (ComplexShape.next c i) \u226b (xNextIso C r).inv"}, {"tactic": "exact (Category.comp_id _).symm", "state_before": "\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b9 : Category V\ninst\u271d : HasZeroMorphisms V\nc : ComplexShape \u03b9\nC : HomologicalComplex V c\ni : \u03b9\nr : ComplexShape.Rel c i (ComplexShape.next c i)\n\u22a2 dFrom C i = d C i (ComplexShape.next c i) \u226b (xNextIso C r).inv", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Eval.lean", "full_name": "Polynomial.eval\u2082_ofFinsupp", "start": [173, 1], "end": [176, 6], "traced_tactics": [{"tactic": "simp only [eval\u2082_eq_sum, sum, toFinsupp_sum, support, coeff]", "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np\u271d q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf\u271d : R \u2192+* S\nx\u271d : S\ninst\u271d : Semiring T\nf : R \u2192+* S\nx : S\np : AddMonoidAlgebra R \u2115\n\u22a2 eval\u2082 f x { toFinsupp := p } = \u2191(liftNC \u2191f \u2191(\u2191(powersHom S) x)) p", "state_after": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np\u271d q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf\u271d : R \u2192+* S\nx\u271d : S\ninst\u271d : Semiring T\nf : R \u2192+* S\nx : S\np : AddMonoidAlgebra R \u2115\n\u22a2 \u2211 x_1 in p.support, \u2191f (\u2191p x_1) * x ^ x_1 = \u2191(liftNC \u2191f \u2191(\u2191(powersHom S) x)) p"}, {"tactic": "rfl", "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np\u271d q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf\u271d : R \u2192+* S\nx\u271d : S\ninst\u271d : Semiring T\nf : R \u2192+* S\nx : S\np : AddMonoidAlgebra R \u2115\n\u22a2 \u2211 x_1 in p.support, \u2191f (\u2191p x_1) * x ^ x_1 = \u2191(liftNC \u2191f \u2191(\u2191(powersHom S) x)) p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Module/LinearPMap.lean", "full_name": "LinearPMap.closureHasCore", "start": [178, 1], "end": [189, 42], "traced_tactics": [{"tactic": "refine' \u27e8f.le_closure.1, _\u27e9", "state_before": "R : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : AddCommGroup E\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module R E\ninst\u271d\u2077 : Module R F\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : ContinuousAdd E\ninst\u271d\u00b3 : ContinuousAdd F\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ContinuousSMul R E\ninst\u271d : ContinuousSMul R F\nf : E \u2192\u2097.[R] F\n\u22a2 HasCore (closure f) f.domain", "state_after": "R : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : AddCommGroup E\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module R E\ninst\u271d\u2077 : Module R F\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : ContinuousAdd E\ninst\u271d\u00b3 : ContinuousAdd F\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ContinuousSMul R E\ninst\u271d : ContinuousSMul R F\nf : E \u2192\u2097.[R] F\n\u22a2 closure (domRestrict (closure f) f.domain) = closure f"}, {"tactic": "congr", "state_before": "R : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : AddCommGroup E\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module R E\ninst\u271d\u2077 : Module R F\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : ContinuousAdd E\ninst\u271d\u00b3 : ContinuousAdd F\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ContinuousSMul R E\ninst\u271d : ContinuousSMul R F\nf : E \u2192\u2097.[R] F\n\u22a2 closure (domRestrict (closure f) f.domain) = closure f", "state_after": "case e_f\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : AddCommGroup E\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module R E\ninst\u271d\u2077 : Module R F\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : ContinuousAdd E\ninst\u271d\u00b3 : ContinuousAdd F\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ContinuousSMul R E\ninst\u271d : ContinuousSMul R F\nf : E \u2192\u2097.[R] F\n\u22a2 domRestrict (closure f) f.domain = f"}, {"tactic": "ext x", "state_before": "case e_f\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : AddCommGroup E\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module R E\ninst\u271d\u2077 : Module R F\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : ContinuousAdd E\ninst\u271d\u00b3 : ContinuousAdd F\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ContinuousSMul R E\ninst\u271d : ContinuousSMul R F\nf : E \u2192\u2097.[R] F\n\u22a2 domRestrict (closure f) f.domain = f", "state_after": "case e_f.h.h\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : AddCommGroup E\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module R E\ninst\u271d\u2077 : Module R F\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : ContinuousAdd E\ninst\u271d\u00b3 : ContinuousAdd F\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ContinuousSMul R E\ninst\u271d : ContinuousSMul R F\nf : E \u2192\u2097.[R] F\nx : E\n\u22a2 x \u2208 (domRestrict (closure f) f.domain).domain \u2194 x \u2208 f.domain\n\ncase e_f.h'\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : AddCommGroup E\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module R E\ninst\u271d\u2077 : Module R F\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : ContinuousAdd E\ninst\u271d\u00b3 : ContinuousAdd F\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ContinuousSMul R E\ninst\u271d : ContinuousSMul R F\nf : E \u2192\u2097.[R] F\nx : { x // x \u2208 (domRestrict (closure f) f.domain).domain }\n\u22a2 \u2200 \u2983y : { x // x \u2208 f.domain }\u2984, \u2191x = \u2191y \u2192 \u2191(domRestrict (closure f) f.domain) x = \u2191f y"}, {"tactic": "intro y hxy", "state_before": "case e_f.h'\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : AddCommGroup E\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module R E\ninst\u271d\u2077 : Module R F\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : ContinuousAdd E\ninst\u271d\u00b3 : ContinuousAdd F\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ContinuousSMul R E\ninst\u271d : ContinuousSMul R F\nf : E \u2192\u2097.[R] F\nx : { x // x \u2208 (domRestrict (closure f) f.domain).domain }\n\u22a2 \u2200 \u2983y : { x // x \u2208 f.domain }\u2984, \u2191x = \u2191y \u2192 \u2191(domRestrict (closure f) f.domain) x = \u2191f y", "state_after": "case e_f.h'\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : AddCommGroup E\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module R E\ninst\u271d\u2077 : Module R F\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : ContinuousAdd E\ninst\u271d\u00b3 : ContinuousAdd F\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ContinuousSMul R E\ninst\u271d : ContinuousSMul R F\nf : E \u2192\u2097.[R] F\nx : { x // x \u2208 (domRestrict (closure f) f.domain).domain }\ny : { x // x \u2208 f.domain }\nhxy : \u2191x = \u2191y\n\u22a2 \u2191(domRestrict (closure f) f.domain) x = \u2191f y"}, {"tactic": "let z : f.closure.domain := \u27e8y.1, f.le_closure.1 y.2\u27e9", "state_before": "case e_f.h'\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : AddCommGroup E\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module R E\ninst\u271d\u2077 : Module R F\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : ContinuousAdd E\ninst\u271d\u00b3 : ContinuousAdd F\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ContinuousSMul R E\ninst\u271d : ContinuousSMul R F\nf : E \u2192\u2097.[R] F\nx : { x // x \u2208 (domRestrict (closure f) f.domain).domain }\ny : { x // x \u2208 f.domain }\nhxy : \u2191x = \u2191y\n\u22a2 \u2191(domRestrict (closure f) f.domain) x = \u2191f y", "state_after": "case e_f.h'\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : AddCommGroup E\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module R E\ninst\u271d\u2077 : Module R F\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : ContinuousAdd E\ninst\u271d\u00b3 : ContinuousAdd F\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ContinuousSMul R E\ninst\u271d : ContinuousSMul R F\nf : E \u2192\u2097.[R] F\nx : { x // x \u2208 (domRestrict (closure f) f.domain).domain }\ny : { x // x \u2208 f.domain }\nhxy : \u2191x = \u2191y\nz : { x // x \u2208 (closure f).domain } := { val := \u2191y, property := (_ : \u2191y \u2208 (closure f).domain) }\n\u22a2 \u2191(domRestrict (closure f) f.domain) x = \u2191f y"}, {"tactic": "have hyz : (y : E) = z := by simp", "state_before": "case e_f.h'\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : AddCommGroup E\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module R E\ninst\u271d\u2077 : Module R F\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : ContinuousAdd E\ninst\u271d\u00b3 : ContinuousAdd F\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ContinuousSMul R E\ninst\u271d : ContinuousSMul R F\nf : E \u2192\u2097.[R] F\nx : { x // x \u2208 (domRestrict (closure f) f.domain).domain }\ny : { x // x \u2208 f.domain }\nhxy : \u2191x = \u2191y\nz : { x // x \u2208 (closure f).domain } := { val := \u2191y, property := (_ : \u2191y \u2208 (closure f).domain) }\n\u22a2 \u2191(domRestrict (closure f) f.domain) x = \u2191f y", "state_after": "case e_f.h'\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : AddCommGroup E\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module R E\ninst\u271d\u2077 : Module R F\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : ContinuousAdd E\ninst\u271d\u00b3 : ContinuousAdd F\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ContinuousSMul R E\ninst\u271d : ContinuousSMul R F\nf : E \u2192\u2097.[R] F\nx : { x // x \u2208 (domRestrict (closure f) f.domain).domain }\ny : { x // x \u2208 f.domain }\nhxy : \u2191x = \u2191y\nz : { x // x \u2208 (closure f).domain } := { val := \u2191y, property := (_ : \u2191y \u2208 (closure f).domain) }\nhyz : \u2191y = \u2191z\n\u22a2 \u2191(domRestrict (closure f) f.domain) x = \u2191f y"}, {"tactic": "rw [f.le_closure.2 hyz]", "state_before": "case e_f.h'\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : AddCommGroup E\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module R E\ninst\u271d\u2077 : Module R F\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : ContinuousAdd E\ninst\u271d\u00b3 : ContinuousAdd F\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ContinuousSMul R E\ninst\u271d : ContinuousSMul R F\nf : E \u2192\u2097.[R] F\nx : { x // x \u2208 (domRestrict (closure f) f.domain).domain }\ny : { x // x \u2208 f.domain }\nhxy : \u2191x = \u2191y\nz : { x // x \u2208 (closure f).domain } := { val := \u2191y, property := (_ : \u2191y \u2208 (closure f).domain) }\nhyz : \u2191y = \u2191z\n\u22a2 \u2191(domRestrict (closure f) f.domain) x = \u2191f y", "state_after": "case e_f.h'\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : AddCommGroup E\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module R E\ninst\u271d\u2077 : Module R F\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : ContinuousAdd E\ninst\u271d\u00b3 : ContinuousAdd F\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ContinuousSMul R E\ninst\u271d : ContinuousSMul R F\nf : E \u2192\u2097.[R] F\nx : { x // x \u2208 (domRestrict (closure f) f.domain).domain }\ny : { x // x \u2208 f.domain }\nhxy : \u2191x = \u2191y\nz : { x // x \u2208 (closure f).domain } := { val := \u2191y, property := (_ : \u2191y \u2208 (closure f).domain) }\nhyz : \u2191y = \u2191z\n\u22a2 \u2191(domRestrict (closure f) f.domain) x = \u2191(closure f) z"}, {"tactic": "exact domRestrict_apply (hxy.trans hyz)", "state_before": "case e_f.h'\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : AddCommGroup E\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module R E\ninst\u271d\u2077 : Module R F\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : ContinuousAdd E\ninst\u271d\u00b3 : ContinuousAdd F\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ContinuousSMul R E\ninst\u271d : ContinuousSMul R F\nf : E \u2192\u2097.[R] F\nx : { x // x \u2208 (domRestrict (closure f) f.domain).domain }\ny : { x // x \u2208 f.domain }\nhxy : \u2191x = \u2191y\nz : { x // x \u2208 (closure f).domain } := { val := \u2191y, property := (_ : \u2191y \u2208 (closure f).domain) }\nhyz : \u2191y = \u2191z\n\u22a2 \u2191(domRestrict (closure f) f.domain) x = \u2191(closure f) z", "state_after": "no goals"}, {"tactic": "simp only [domRestrict_domain, Submodule.mem_inf, and_iff_left_iff_imp]", "state_before": "case e_f.h.h\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : AddCommGroup E\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module R E\ninst\u271d\u2077 : Module R F\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : ContinuousAdd E\ninst\u271d\u00b3 : ContinuousAdd F\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ContinuousSMul R E\ninst\u271d : ContinuousSMul R F\nf : E \u2192\u2097.[R] F\nx : E\n\u22a2 x \u2208 (domRestrict (closure f) f.domain).domain \u2194 x \u2208 f.domain", "state_after": "case e_f.h.h\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : AddCommGroup E\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module R E\ninst\u271d\u2077 : Module R F\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : ContinuousAdd E\ninst\u271d\u00b3 : ContinuousAdd F\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ContinuousSMul R E\ninst\u271d : ContinuousSMul R F\nf : E \u2192\u2097.[R] F\nx : E\n\u22a2 x \u2208 f.domain \u2192 x \u2208 (closure f).domain"}, {"tactic": "intro hx", "state_before": "case e_f.h.h\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : AddCommGroup E\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module R E\ninst\u271d\u2077 : Module R F\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : ContinuousAdd E\ninst\u271d\u00b3 : ContinuousAdd F\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ContinuousSMul R E\ninst\u271d : ContinuousSMul R F\nf : E \u2192\u2097.[R] F\nx : E\n\u22a2 x \u2208 f.domain \u2192 x \u2208 (closure f).domain", "state_after": "case e_f.h.h\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : AddCommGroup E\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module R E\ninst\u271d\u2077 : Module R F\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : ContinuousAdd E\ninst\u271d\u00b3 : ContinuousAdd F\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ContinuousSMul R E\ninst\u271d : ContinuousSMul R F\nf : E \u2192\u2097.[R] F\nx : E\nhx : x \u2208 f.domain\n\u22a2 x \u2208 (closure f).domain"}, {"tactic": "exact f.le_closure.1 hx", "state_before": "case e_f.h.h\nR : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : AddCommGroup E\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module R E\ninst\u271d\u2077 : Module R F\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : ContinuousAdd E\ninst\u271d\u00b3 : ContinuousAdd F\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ContinuousSMul R E\ninst\u271d : ContinuousSMul R F\nf : E \u2192\u2097.[R] F\nx : E\nhx : x \u2208 f.domain\n\u22a2 x \u2208 (closure f).domain", "state_after": "no goals"}, {"tactic": "simp", "state_before": "R : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : AddCommGroup E\ninst\u271d\u2079 : AddCommGroup F\ninst\u271d\u2078 : Module R E\ninst\u271d\u2077 : Module R F\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : TopologicalSpace F\ninst\u271d\u2074 : ContinuousAdd E\ninst\u271d\u00b3 : ContinuousAdd F\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ContinuousSMul R E\ninst\u271d : ContinuousSMul R F\nf : E \u2192\u2097.[R] F\nx : { x // x \u2208 (domRestrict (closure f) f.domain).domain }\ny : { x // x \u2208 f.domain }\nhxy : \u2191x = \u2191y\nz : { x // x \u2208 (closure f).domain } := { val := \u2191y, property := (_ : \u2191y \u2208 (closure f).domain) }\n\u22a2 \u2191y = \u2191z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "iSup_iInf_ge_nat_add", "start": [1651, 1], "end": [1655, 55], "traced_tactics": [{"tactic": "have hf : Monotone fun n => \u2a05 i \u2265 n, f i := fun n m h => biInf_mono fun i => h.trans", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.178268\n\u03b2\u2082 : Type ?u.178271\n\u03b3 : Type ?u.178274\n\u03b9 : Sort ?u.178277\n\u03b9' : Sort ?u.178280\n\u03ba : \u03b9 \u2192 Sort ?u.178285\n\u03ba' : \u03b9' \u2192 Sort ?u.178290\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u2115 \u2192 \u03b1\nk : \u2115\n\u22a2 (\u2a06 (n : \u2115), \u2a05 (i : \u2115) (_ : i \u2265 n), f (i + k)) = \u2a06 (n : \u2115), \u2a05 (i : \u2115) (_ : i \u2265 n), f i", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.178268\n\u03b2\u2082 : Type ?u.178271\n\u03b3 : Type ?u.178274\n\u03b9 : Sort ?u.178277\n\u03b9' : Sort ?u.178280\n\u03ba : \u03b9 \u2192 Sort ?u.178285\n\u03ba' : \u03b9' \u2192 Sort ?u.178290\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u2115 \u2192 \u03b1\nk : \u2115\nhf : Monotone fun n => \u2a05 (i : \u2115) (_ : i \u2265 n), f i\n\u22a2 (\u2a06 (n : \u2115), \u2a05 (i : \u2115) (_ : i \u2265 n), f (i + k)) = \u2a06 (n : \u2115), \u2a05 (i : \u2115) (_ : i \u2265 n), f i"}, {"tactic": "rw [\u2190 Monotone.iSup_nat_add hf k]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.178268\n\u03b2\u2082 : Type ?u.178271\n\u03b3 : Type ?u.178274\n\u03b9 : Sort ?u.178277\n\u03b9' : Sort ?u.178280\n\u03ba : \u03b9 \u2192 Sort ?u.178285\n\u03ba' : \u03b9' \u2192 Sort ?u.178290\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u2115 \u2192 \u03b1\nk : \u2115\nhf : Monotone fun n => \u2a05 (i : \u2115) (_ : i \u2265 n), f i\n\u22a2 (\u2a06 (n : \u2115), \u2a05 (i : \u2115) (_ : i \u2265 n), f (i + k)) = \u2a06 (n : \u2115), \u2a05 (i : \u2115) (_ : i \u2265 n), f i", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.178268\n\u03b2\u2082 : Type ?u.178271\n\u03b3 : Type ?u.178274\n\u03b9 : Sort ?u.178277\n\u03b9' : Sort ?u.178280\n\u03ba : \u03b9 \u2192 Sort ?u.178285\n\u03ba' : \u03b9' \u2192 Sort ?u.178290\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u2115 \u2192 \u03b1\nk : \u2115\nhf : Monotone fun n => \u2a05 (i : \u2115) (_ : i \u2265 n), f i\n\u22a2 (\u2a06 (n : \u2115), \u2a05 (i : \u2115) (_ : i \u2265 n), f (i + k)) = \u2a06 (n : \u2115), \u2a05 (i : \u2115) (_ : i \u2265 n + k), f i"}, {"tactic": "simp_rw [iInf_ge_eq_iInf_nat_add, \u2190 Nat.add_assoc]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.178268\n\u03b2\u2082 : Type ?u.178271\n\u03b3 : Type ?u.178274\n\u03b9 : Sort ?u.178277\n\u03b9' : Sort ?u.178280\n\u03ba : \u03b9 \u2192 Sort ?u.178285\n\u03ba' : \u03b9' \u2192 Sort ?u.178290\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u2115 \u2192 \u03b1\nk : \u2115\nhf : Monotone fun n => \u2a05 (i : \u2115) (_ : i \u2265 n), f i\n\u22a2 (\u2a06 (n : \u2115), \u2a05 (i : \u2115) (_ : i \u2265 n), f (i + k)) = \u2a06 (n : \u2115), \u2a05 (i : \u2115) (_ : i \u2265 n + k), f i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/GCD/Basic.lean", "full_name": "Nat.coprime_add_mul_right_right", "start": [153, 1], "end": [154, 49], "traced_tactics": [{"tactic": "rw [coprime, coprime, gcd_add_mul_right_right]", "state_before": "m n k : \u2115\n\u22a2 coprime m (n + k * m) \u2194 coprime m n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/AbstractCompletion.lean", "full_name": "AbstractCompletion.induction_on", "start": [112, 1], "end": [114, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Associated.lean", "full_name": "Associates.dvdNotUnit_of_lt", "start": [1066, 1], "end": [1075, 7], "traced_tactics": [{"tactic": "constructor", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.325963\n\u03b3 : Type ?u.325966\n\u03b4 : Type ?u.325969\ninst\u271d : CommMonoidWithZero \u03b1\na b : Associates \u03b1\nhlt : a < b\n\u22a2 DvdNotUnit a b", "state_after": "case left\n\u03b1 : Type u_1\n\u03b2 : Type ?u.325963\n\u03b3 : Type ?u.325966\n\u03b4 : Type ?u.325969\ninst\u271d : CommMonoidWithZero \u03b1\na b : Associates \u03b1\nhlt : a < b\n\u22a2 a \u2260 0\n\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type ?u.325963\n\u03b3 : Type ?u.325966\n\u03b4 : Type ?u.325969\ninst\u271d : CommMonoidWithZero \u03b1\na b : Associates \u03b1\nhlt : a < b\n\u22a2 \u2203 x, \u00acIsUnit x \u2227 b = a * x"}, {"tactic": "rcases hlt with \u27e8\u27e8x, rfl\u27e9, ndvd\u27e9", "state_before": "case right\n\u03b1 : Type u_1\n\u03b2 : Type ?u.325963\n\u03b3 : Type ?u.325966\n\u03b4 : Type ?u.325969\ninst\u271d : CommMonoidWithZero \u03b1\na b : Associates \u03b1\nhlt : a < b\n\u22a2 \u2203 x, \u00acIsUnit x \u2227 b = a * x", "state_after": "case right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.325963\n\u03b3 : Type ?u.325966\n\u03b4 : Type ?u.325969\ninst\u271d : CommMonoidWithZero \u03b1\na : Associates \u03b1\nx : Associates \u03b1\nndvd : \u00aca * x \u2223 a\n\u22a2 \u2203 x_1, \u00acIsUnit x_1 \u2227 a * x = a * x_1"}, {"tactic": "refine' \u27e8x, _, rfl\u27e9", "state_before": "case right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.325963\n\u03b3 : Type ?u.325966\n\u03b4 : Type ?u.325969\ninst\u271d : CommMonoidWithZero \u03b1\na : Associates \u03b1\nx : Associates \u03b1\nndvd : \u00aca * x \u2223 a\n\u22a2 \u2203 x_1, \u00acIsUnit x_1 \u2227 a * x = a * x_1", "state_after": "case right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.325963\n\u03b3 : Type ?u.325966\n\u03b4 : Type ?u.325969\ninst\u271d : CommMonoidWithZero \u03b1\na : Associates \u03b1\nx : Associates \u03b1\nndvd : \u00aca * x \u2223 a\n\u22a2 \u00acIsUnit x"}, {"tactic": "contrapose! ndvd", "state_before": "case right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.325963\n\u03b3 : Type ?u.325966\n\u03b4 : Type ?u.325969\ninst\u271d : CommMonoidWithZero \u03b1\na : Associates \u03b1\nx : Associates \u03b1\nndvd : \u00aca * x \u2223 a\n\u22a2 \u00acIsUnit x", "state_after": "case right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.325963\n\u03b3 : Type ?u.325966\n\u03b4 : Type ?u.325969\ninst\u271d : CommMonoidWithZero \u03b1\na : Associates \u03b1\nx : Associates \u03b1\nndvd : IsUnit x\n\u22a2 a * x \u2223 a"}, {"tactic": "rcases ndvd with \u27e8u, rfl\u27e9", "state_before": "case right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.325963\n\u03b3 : Type ?u.325966\n\u03b4 : Type ?u.325969\ninst\u271d : CommMonoidWithZero \u03b1\na : Associates \u03b1\nx : Associates \u03b1\nndvd : IsUnit x\n\u22a2 a * x \u2223 a", "state_after": "case right.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.325963\n\u03b3 : Type ?u.325966\n\u03b4 : Type ?u.325969\ninst\u271d : CommMonoidWithZero \u03b1\na : Associates \u03b1\nu : (Associates \u03b1)\u02e3\n\u22a2 a * \u2191u \u2223 a"}, {"tactic": "simp", "state_before": "case right.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.325963\n\u03b3 : Type ?u.325966\n\u03b4 : Type ?u.325969\ninst\u271d : CommMonoidWithZero \u03b1\na : Associates \u03b1\nu : (Associates \u03b1)\u02e3\n\u22a2 a * \u2191u \u2223 a", "state_after": "no goals"}, {"tactic": "rintro rfl", "state_before": "case left\n\u03b1 : Type u_1\n\u03b2 : Type ?u.325963\n\u03b3 : Type ?u.325966\n\u03b4 : Type ?u.325969\ninst\u271d : CommMonoidWithZero \u03b1\na b : Associates \u03b1\nhlt : a < b\n\u22a2 a \u2260 0", "state_after": "case left\n\u03b1 : Type u_1\n\u03b2 : Type ?u.325963\n\u03b3 : Type ?u.325966\n\u03b4 : Type ?u.325969\ninst\u271d : CommMonoidWithZero \u03b1\nb : Associates \u03b1\nhlt : 0 < b\n\u22a2 False"}, {"tactic": "apply not_lt_of_le _ hlt", "state_before": "case left\n\u03b1 : Type u_1\n\u03b2 : Type ?u.325963\n\u03b3 : Type ?u.325966\n\u03b4 : Type ?u.325969\ninst\u271d : CommMonoidWithZero \u03b1\nb : Associates \u03b1\nhlt : 0 < b\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.325963\n\u03b3 : Type ?u.325966\n\u03b4 : Type ?u.325969\ninst\u271d : CommMonoidWithZero \u03b1\nb : Associates \u03b1\nhlt : 0 < b\n\u22a2 b \u2264 0"}, {"tactic": "apply dvd_zero", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.325963\n\u03b3 : Type ?u.325966\n\u03b4 : Type ?u.325969\ninst\u271d : CommMonoidWithZero \u03b1\nb : Associates \u03b1\nhlt : 0 < b\n\u22a2 b \u2264 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finsupp/ToDfinsupp.lean", "full_name": "Finsupp.toDfinsupp_add", "start": [159, 1], "end": [161, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.induction_on", "start": [447, 1], "end": [449, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/ModelTheory/Encoding.lean", "full_name": "FirstOrder.Language.Term.card_sigma", "start": [127, 1], "end": [156, 44], "traced_tactics": [{"tactic": "refine' le_antisymm _ _", "state_before": "L : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 (#(n : \u2115) \u00d7 Term L (\u03b1 \u2295 Fin n)) = max \u2135\u2080 (#\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i)", "state_after": "case refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 (#(n : \u2115) \u00d7 Term L (\u03b1 \u2295 Fin n)) \u2264 max \u2135\u2080 (#\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i)\n\ncase refine'_2\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 max \u2135\u2080 (#\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i) \u2264 (#(n : \u2115) \u00d7 Term L (\u03b1 \u2295 Fin n))"}, {"tactic": "rw [mk_sigma]", "state_before": "case refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 (#(n : \u2115) \u00d7 Term L (\u03b1 \u2295 Fin n)) \u2264 max \u2135\u2080 (#\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i)", "state_after": "case refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 (Cardinal.sum fun i => #Term L (\u03b1 \u2295 Fin i)) \u2264 max \u2135\u2080 (#\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i)"}, {"tactic": "refine' (sum_le_iSup_lift _).trans _", "state_before": "case refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 (Cardinal.sum fun i => #Term L (\u03b1 \u2295 Fin i)) \u2264 max \u2135\u2080 (#\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i)", "state_after": "case refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 (lift (#\u2115) * \u2a06 (i : \u2115), #Term L (\u03b1 \u2295 Fin i)) \u2264 max \u2135\u2080 (#\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i)"}, {"tactic": "rw [mk_nat, lift_aleph0, mul_eq_max_of_aleph0_le_left le_rfl, max_le_iff,\n  ciSup_le_iff' (bddAbove_range _)]", "state_before": "case refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 (lift (#\u2115) * \u2a06 (i : \u2115), #Term L (\u03b1 \u2295 Fin i)) \u2264 max \u2135\u2080 (#\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i)", "state_after": "case refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 \u2135\u2080 \u2264 max \u2135\u2080 (#\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i) \u2227 \u2200 (i : \u2115), (#Term L (\u03b1 \u2295 Fin i)) \u2264 max \u2135\u2080 (#\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i)\n\ncase refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 (\u2a06 (i : \u2115), #Term L (\u03b1 \u2295 Fin i)) \u2260 0"}, {"tactic": "refine' \u27e8le_max_left _ _, fun i => card_le.trans _\u27e9", "state_before": "case refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 \u2135\u2080 \u2264 max \u2135\u2080 (#\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i) \u2227 \u2200 (i : \u2115), (#Term L (\u03b1 \u2295 Fin i)) \u2264 max \u2135\u2080 (#\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i)", "state_after": "case refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ni : \u2115\n\u22a2 max \u2135\u2080 (#(\u03b1 \u2295 Fin i) \u2295 (i : \u2115) \u00d7 Functions L i) \u2264 max \u2135\u2080 (#\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i)"}, {"tactic": "refine' max_le (le_max_left _ _) _", "state_before": "case refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ni : \u2115\n\u22a2 max \u2135\u2080 (#(\u03b1 \u2295 Fin i) \u2295 (i : \u2115) \u00d7 Functions L i) \u2264 max \u2135\u2080 (#\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i)", "state_after": "case refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ni : \u2115\n\u22a2 (#(\u03b1 \u2295 Fin i) \u2295 (i : \u2115) \u00d7 Functions L i) \u2264 max \u2135\u2080 (#\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i)"}, {"tactic": "rw [\u2190 add_eq_max le_rfl, mk_sum, mk_sum, mk_sum, add_comm (Cardinal.lift (#\u03b1)), lift_add,\n  add_assoc, lift_lift, lift_lift, mk_fin, lift_natCast]", "state_before": "case refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ni : \u2115\n\u22a2 (#(\u03b1 \u2295 Fin i) \u2295 (i : \u2115) \u00d7 Functions L i) \u2264 max \u2135\u2080 (#\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i)", "state_after": "case refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ni : \u2115\n\u22a2 \u2191i + (lift (#\u03b1) + lift (#(i : \u2115) \u00d7 Functions L i)) \u2264 \u2135\u2080 + (lift (#\u03b1) + lift (#(i : \u2115) \u00d7 Functions L i))"}, {"tactic": "exact add_le_add_right (nat_lt_aleph0 _).le _", "state_before": "case refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ni : \u2115\n\u22a2 \u2191i + (lift (#\u03b1) + lift (#(i : \u2115) \u00d7 Functions L i)) \u2264 \u2135\u2080 + (lift (#\u03b1) + lift (#(i : \u2115) \u00d7 Functions L i))", "state_after": "no goals"}, {"tactic": "rw [\u2190 one_le_iff_ne_zero]", "state_before": "case refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 (\u2a06 (i : \u2115), #Term L (\u03b1 \u2295 Fin i)) \u2260 0", "state_after": "case refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 1 \u2264 \u2a06 (i : \u2115), #Term L (\u03b1 \u2295 Fin i)"}, {"tactic": "refine' _root_.trans _ (le_ciSup (bddAbove_range _) 1)", "state_before": "case refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 1 \u2264 \u2a06 (i : \u2115), #Term L (\u03b1 \u2295 Fin i)", "state_after": "case refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 1 \u2264 (#Term L (\u03b1 \u2295 Fin 1))"}, {"tactic": "rw [one_le_iff_ne_zero, mk_ne_zero_iff]", "state_before": "case refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 1 \u2264 (#Term L (\u03b1 \u2295 Fin 1))", "state_after": "case refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 Nonempty (Term L (\u03b1 \u2295 Fin 1))"}, {"tactic": "exact \u27e8var (Sum.inr 0)\u27e9", "state_before": "case refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 Nonempty (Term L (\u03b1 \u2295 Fin 1))", "state_after": "no goals"}, {"tactic": "rw [max_le_iff, \u2190 infinite_iff]", "state_before": "case refine'_2\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 max \u2135\u2080 (#\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i) \u2264 (#(n : \u2115) \u00d7 Term L (\u03b1 \u2295 Fin n))", "state_after": "case refine'_2\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 Infinite ((n : \u2115) \u00d7 Term L (\u03b1 \u2295 Fin n)) \u2227 (#\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i) \u2264 (#(n : \u2115) \u00d7 Term L (\u03b1 \u2295 Fin n))"}, {"tactic": "refine' \u27e8Infinite.of_injective (fun i => \u27e8i + 1, var (Sum.inr i)\u27e9) fun i j ij => _, _\u27e9", "state_before": "case refine'_2\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 Infinite ((n : \u2115) \u00d7 Term L (\u03b1 \u2295 Fin n)) \u2227 (#\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i) \u2264 (#(n : \u2115) \u00d7 Term L (\u03b1 \u2295 Fin n))", "state_after": "case refine'_2.refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ni j : \u2115\nij : (fun i => { fst := i + 1, snd := var (Sum.inr \u2191i) }) i = (fun i => { fst := i + 1, snd := var (Sum.inr \u2191i) }) j\n\u22a2 i = j\n\ncase refine'_2.refine'_2\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 (#\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i) \u2264 (#(n : \u2115) \u00d7 Term L (\u03b1 \u2295 Fin n))"}, {"tactic": "cases ij", "state_before": "case refine'_2.refine'_1\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ni j : \u2115\nij : (fun i => { fst := i + 1, snd := var (Sum.inr \u2191i) }) i = (fun i => { fst := i + 1, snd := var (Sum.inr \u2191i) }) j\n\u22a2 i = j", "state_after": "case refine'_2.refine'_1.refl\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ni : \u2115\n\u22a2 i = i"}, {"tactic": "rfl", "state_before": "case refine'_2.refine'_1.refl\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ni : \u2115\n\u22a2 i = i", "state_after": "no goals"}, {"tactic": "rw [Cardinal.le_def]", "state_before": "case refine'_2.refine'_2\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 (#\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i) \u2264 (#(n : \u2115) \u00d7 Term L (\u03b1 \u2295 Fin n))", "state_after": "case refine'_2.refine'_2\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 Nonempty (\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i \u21aa (n : \u2115) \u00d7 Term L (\u03b1 \u2295 Fin n))"}, {"tactic": "refine' \u27e8\u27e8Sum.elim (fun i => \u27e80, var (Sum.inl i)\u27e9)\n  fun F => \u27e81, func F.2 fun _ => var (Sum.inr 0)\u27e9, _\u27e9\u27e9", "state_before": "case refine'_2.refine'_2\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 Nonempty (\u03b1 \u2295 (i : \u2115) \u00d7 Functions L i \u21aa (n : \u2115) \u00d7 Term L (\u03b1 \u2295 Fin n))", "state_after": "case refine'_2.refine'_2\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 Function.Injective\n    (Sum.elim (fun i => { fst := 0, snd := var (Sum.inl i) }) fun F =>\n      { fst := 1, snd := func F.snd fun x => var (Sum.inr 0) })"}, {"tactic": "rintro (a | a) (b | b) h", "state_before": "case refine'_2.refine'_2\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u22a2 Function.Injective\n    (Sum.elim (fun i => { fst := 0, snd := var (Sum.inl i) }) fun F =>\n      { fst := 1, snd := func F.snd fun x => var (Sum.inr 0) })", "state_after": "case refine'_2.refine'_2.inl.inl\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\na b : \u03b1\nh :\n  Sum.elim (fun i => { fst := 0, snd := var (Sum.inl i) })\n      (fun F => { fst := 1, snd := func F.snd fun x => var (Sum.inr 0) }) (Sum.inl a) =\n    Sum.elim (fun i => { fst := 0, snd := var (Sum.inl i) })\n      (fun F => { fst := 1, snd := func F.snd fun x => var (Sum.inr 0) }) (Sum.inl b)\n\u22a2 Sum.inl a = Sum.inl b\n\ncase refine'_2.refine'_2.inl.inr\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\na : \u03b1\nb : (i : \u2115) \u00d7 Functions L i\nh :\n  Sum.elim (fun i => { fst := 0, snd := var (Sum.inl i) })\n      (fun F => { fst := 1, snd := func F.snd fun x => var (Sum.inr 0) }) (Sum.inl a) =\n    Sum.elim (fun i => { fst := 0, snd := var (Sum.inl i) })\n      (fun F => { fst := 1, snd := func F.snd fun x => var (Sum.inr 0) }) (Sum.inr b)\n\u22a2 Sum.inl a = Sum.inr b\n\ncase refine'_2.refine'_2.inr.inl\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\na : (i : \u2115) \u00d7 Functions L i\nb : \u03b1\nh :\n  Sum.elim (fun i => { fst := 0, snd := var (Sum.inl i) })\n      (fun F => { fst := 1, snd := func F.snd fun x => var (Sum.inr 0) }) (Sum.inr a) =\n    Sum.elim (fun i => { fst := 0, snd := var (Sum.inl i) })\n      (fun F => { fst := 1, snd := func F.snd fun x => var (Sum.inr 0) }) (Sum.inl b)\n\u22a2 Sum.inr a = Sum.inl b\n\ncase refine'_2.refine'_2.inr.inr\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\na b : (i : \u2115) \u00d7 Functions L i\nh :\n  Sum.elim (fun i => { fst := 0, snd := var (Sum.inl i) })\n      (fun F => { fst := 1, snd := func F.snd fun x => var (Sum.inr 0) }) (Sum.inr a) =\n    Sum.elim (fun i => { fst := 0, snd := var (Sum.inl i) })\n      (fun F => { fst := 1, snd := func F.snd fun x => var (Sum.inr 0) }) (Sum.inr b)\n\u22a2 Sum.inr a = Sum.inr b"}, {"tactic": "simp only [Sum.elim_inl, Sigma.mk.inj_iff, heq_eq_eq, var.injEq, Sum.inl.injEq, true_and]\n  at h", "state_before": "case refine'_2.refine'_2.inl.inl\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\na b : \u03b1\nh :\n  Sum.elim (fun i => { fst := 0, snd := var (Sum.inl i) })\n      (fun F => { fst := 1, snd := func F.snd fun x => var (Sum.inr 0) }) (Sum.inl a) =\n    Sum.elim (fun i => { fst := 0, snd := var (Sum.inl i) })\n      (fun F => { fst := 1, snd := func F.snd fun x => var (Sum.inr 0) }) (Sum.inl b)\n\u22a2 Sum.inl a = Sum.inl b", "state_after": "case refine'_2.refine'_2.inl.inl\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\na b : \u03b1\nh : a = b\n\u22a2 Sum.inl a = Sum.inl b"}, {"tactic": "rw [h]", "state_before": "case refine'_2.refine'_2.inl.inl\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\na b : \u03b1\nh : a = b\n\u22a2 Sum.inl a = Sum.inl b", "state_after": "no goals"}, {"tactic": "simp only [Sum.elim_inl, Sum.elim_inr, Sigma.mk.inj_iff, false_and] at h", "state_before": "case refine'_2.refine'_2.inl.inr\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\na : \u03b1\nb : (i : \u2115) \u00d7 Functions L i\nh :\n  Sum.elim (fun i => { fst := 0, snd := var (Sum.inl i) })\n      (fun F => { fst := 1, snd := func F.snd fun x => var (Sum.inr 0) }) (Sum.inl a) =\n    Sum.elim (fun i => { fst := 0, snd := var (Sum.inl i) })\n      (fun F => { fst := 1, snd := func F.snd fun x => var (Sum.inr 0) }) (Sum.inr b)\n\u22a2 Sum.inl a = Sum.inr b", "state_after": "no goals"}, {"tactic": "simp only [Sum.elim_inr, Sum.elim_inl, Sigma.mk.inj_iff, false_and] at h", "state_before": "case refine'_2.refine'_2.inr.inl\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\na : (i : \u2115) \u00d7 Functions L i\nb : \u03b1\nh :\n  Sum.elim (fun i => { fst := 0, snd := var (Sum.inl i) })\n      (fun F => { fst := 1, snd := func F.snd fun x => var (Sum.inr 0) }) (Sum.inr a) =\n    Sum.elim (fun i => { fst := 0, snd := var (Sum.inl i) })\n      (fun F => { fst := 1, snd := func F.snd fun x => var (Sum.inr 0) }) (Sum.inl b)\n\u22a2 Sum.inr a = Sum.inl b", "state_after": "no goals"}, {"tactic": "simp only [Sum.elim_inr, Sigma.mk.inj_iff, heq_eq_eq, func.injEq, true_and] at h", "state_before": "case refine'_2.refine'_2.inr.inr\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\na b : (i : \u2115) \u00d7 Functions L i\nh :\n  Sum.elim (fun i => { fst := 0, snd := var (Sum.inl i) })\n      (fun F => { fst := 1, snd := func F.snd fun x => var (Sum.inr 0) }) (Sum.inr a) =\n    Sum.elim (fun i => { fst := 0, snd := var (Sum.inl i) })\n      (fun F => { fst := 1, snd := func F.snd fun x => var (Sum.inr 0) }) (Sum.inr b)\n\u22a2 Sum.inr a = Sum.inr b", "state_after": "case refine'_2.refine'_2.inr.inr\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\na b : (i : \u2115) \u00d7 Functions L i\nh : a.fst = b.fst \u2227 HEq a.snd b.snd \u2227 HEq (fun x => var (Sum.inr 0)) fun x => var (Sum.inr 0)\n\u22a2 Sum.inr a = Sum.inr b"}, {"tactic": "rw [Sigma.ext_iff.2 \u27e8h.1, h.2.1\u27e9]", "state_before": "case refine'_2.refine'_2.inr.inr\nL : Language\nM : Type w\nN : Type ?u.10289\nP : Type ?u.10292\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\na b : (i : \u2115) \u00d7 Functions L i\nh : a.fst = b.fst \u2227 HEq a.snd b.snd \u2227 HEq (fun x => var (Sum.inr 0)) fun x => var (Sum.inr 0)\n\u22a2 Sum.inr a = Sum.inr b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Submonoid/Inverses.lean", "full_name": "Submonoid.fromLeftInv_one", "start": [113, 1], "end": [114, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/Operations.lean", "full_name": "RingHom.injective_iff_ker_eq_bot", "start": [2032, 1], "end": [2034, 37], "traced_tactics": [{"tactic": "rw [SetLike.ext'_iff, ker_eq, Set.ext_iff]", "state_before": "R : Type u\nS : Type v\nT : Type w\nF : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\n\u22a2 Function.Injective \u2191f \u2194 ker f = \u22a5", "state_after": "R : Type u\nS : Type v\nT : Type w\nF : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\n\u22a2 Function.Injective \u2191f \u2194 \u2200 (x : R), x \u2208 \u2191f \u207b\u00b9' {0} \u2194 x \u2208 \u2191\u22a5"}, {"tactic": "exact injective_iff_map_eq_zero' f", "state_before": "R : Type u\nS : Type v\nT : Type w\nF : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\n\u22a2 Function.Injective \u2191f \u2194 \u2200 (x : R), x \u2208 \u2191f \u207b\u00b9' {0} \u2194 x \u2208 \u2191\u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/AlgebraMap.lean", "full_name": "Polynomial.aeval_algHom", "start": [262, 1], "end": [263, 58], "traced_tactics": [{"tactic": "simp only [aeval_X, AlgHom.comp_apply]", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.831458\nB' : Type ?u.831461\na b : R\nn : \u2115\ninst\u271d\u2076 : CommSemiring A'\ninst\u271d\u2075 : Semiring B'\ninst\u271d\u2074 : CommSemiring R\np q : R[X]\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type u_1\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nx\u271d : A\nf : A \u2192\u2090[R] B\nx : A\n\u22a2 \u2191(aeval (\u2191f x)) X = \u2191(AlgHom.comp f (aeval x)) X", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Lie/IdealOperations.lean", "full_name": "LieSubmodule.lie_le_inf", "start": [134, 1], "end": [134, 101], "traced_tactics": [{"tactic": "rw [le_inf_iff]", "state_before": "R : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\n\u22a2 \u2045I, J\u2046 \u2264 I \u2293 J", "state_after": "R : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\n\u22a2 \u2045I, J\u2046 \u2264 I \u2227 \u2045I, J\u2046 \u2264 J"}, {"tactic": "exact \u27e8lie_le_left I J, lie_le_right J I\u27e9", "state_before": "R : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\n\u22a2 \u2045I, J\u2046 \u2264 I \u2227 \u2045I, J\u2046 \u2264 J", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Monotone/Basic.lean", "full_name": "StrictAnti.minimal_of_maximal_image", "start": [863, 1], "end": [865, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Functor/EpiMono.lean", "full_name": "CategoryTheory.Functor.preservesEpimorphisms_of_preserves_of_reflects", "start": [104, 1], "end": [106, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean", "full_name": "CircleDeg1Lift.le_floor_map_map_zero", "start": [500, 1], "end": [503, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/BooleanAlgebra.lean", "full_name": "Disjoint.sdiff_unique", "start": [236, 11], "end": [243, 47], "traced_tactics": [{"tactic": "rw [\u2190 inf_eq_right] at hs", "state_before": "\u03b1 : Type u\n\u03b2 : Type ?u.12767\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhd : Disjoint x z\nhz : z \u2264 y\nhs : y \u2264 x \u2294 z\n\u22a2 y \u2293 x \u2294 z = y", "state_after": "\u03b1 : Type u\n\u03b2 : Type ?u.12767\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhd : Disjoint x z\nhz : z \u2264 y\nhs : (x \u2294 z) \u2293 y = y\n\u22a2 y \u2293 x \u2294 z = y"}, {"tactic": "rwa [sup_inf_right, inf_sup_right, @sup_comm _ _ x, inf_sup_self, inf_comm, @sup_comm _ _ z,\n  hs, sup_eq_left]", "state_before": "\u03b1 : Type u\n\u03b2 : Type ?u.12767\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhd : Disjoint x z\nhz : z \u2264 y\nhs : (x \u2294 z) \u2293 y = y\n\u22a2 y \u2293 x \u2294 z = y", "state_after": "no goals"}, {"tactic": "rw [inf_assoc, hd.eq_bot, inf_bot_eq]", "state_before": "\u03b1 : Type u\n\u03b2 : Type ?u.12767\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhd : Disjoint x z\nhz : z \u2264 y\nhs : y \u2264 x \u2294 z\n\u22a2 y \u2293 x \u2293 z = \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/AbstractCompletion.lean", "full_name": "AbstractCompletion.map\u2082_coe_coe", "start": [384, 1], "end": [386, 94], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.range_const_subset", "start": [165, 1], "end": [166, 33], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b1\u271d : Type ?u.16047\n\u03b2 : Type u_2\n\u03b3 : Type ?u.16053\n\u03b4 : Type ?u.16056\ninst\u271d\u00b9 : MeasurableSpace \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nb : \u03b2\n\u22a2 \u2191(SimpleFunc.range (const \u03b1 b)) \u2286 \u2191{b}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finsupp/Basic.lean", "full_name": "Finsupp.split_apply", "start": [1793, 1], "end": [1795, 25], "traced_tactics": [{"tactic": "dsimp only [split]", "state_before": "\u03b1 : Type ?u.737860\n\u03b2 : Type ?u.737863\n\u03b3 : Type ?u.737866\n\u03b9 : Type u_3\nM : Type u_1\nM' : Type ?u.737875\nN : Type ?u.737878\nP : Type ?u.737881\nG : Type ?u.737884\nH : Type ?u.737887\nR : Type ?u.737890\nS : Type ?u.737893\n\u03b1s : \u03b9 \u2192 Type u_2\ninst\u271d : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\ni : \u03b9\nx : \u03b1s i\n\u22a2 \u2191(split l i) x = \u2191l { fst := i, snd := x }", "state_after": "\u03b1 : Type ?u.737860\n\u03b2 : Type ?u.737863\n\u03b3 : Type ?u.737866\n\u03b9 : Type u_3\nM : Type u_1\nM' : Type ?u.737875\nN : Type ?u.737878\nP : Type ?u.737881\nG : Type ?u.737884\nH : Type ?u.737887\nR : Type ?u.737890\nS : Type ?u.737893\n\u03b1s : \u03b9 \u2192 Type u_2\ninst\u271d : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\ni : \u03b9\nx : \u03b1s i\n\u22a2 \u2191(comapDomain (Sigma.mk i) l\n          (_ :\n            \u2200 (x : \u03b1s i),\n              x \u2208 Sigma.mk i \u207b\u00b9' \u2191l.support \u2192\n                \u2200 (x_2 : \u03b1s i),\n                  x_2 \u2208 Sigma.mk i \u207b\u00b9' \u2191l.support \u2192 { fst := i, snd := x } = { fst := i, snd := x_2 } \u2192 x = x_2))\n      x =\n    \u2191l { fst := i, snd := x }"}, {"tactic": "rw [comapDomain_apply]", "state_before": "\u03b1 : Type ?u.737860\n\u03b2 : Type ?u.737863\n\u03b3 : Type ?u.737866\n\u03b9 : Type u_3\nM : Type u_1\nM' : Type ?u.737875\nN : Type ?u.737878\nP : Type ?u.737881\nG : Type ?u.737884\nH : Type ?u.737887\nR : Type ?u.737890\nS : Type ?u.737893\n\u03b1s : \u03b9 \u2192 Type u_2\ninst\u271d : Zero M\nl : (i : \u03b9) \u00d7 \u03b1s i \u2192\u2080 M\ni : \u03b9\nx : \u03b1s i\n\u22a2 \u2191(comapDomain (Sigma.mk i) l\n          (_ :\n            \u2200 (x : \u03b1s i),\n              x \u2208 Sigma.mk i \u207b\u00b9' \u2191l.support \u2192\n                \u2200 (x_2 : \u03b1s i),\n                  x_2 \u2208 Sigma.mk i \u207b\u00b9' \u2191l.support \u2192 { fst := i, snd := x } = { fst := i, snd := x_2 } \u2192 x = x_2))\n      x =\n    \u2191l { fst := i, snd := x }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Hom.lean", "full_name": "AlgHom.commutes", "start": [237, 1], "end": [238, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Span.lean", "full_name": "Submodule.span_singleton_group_smul_eq", "start": [441, 1], "end": [445, 33], "traced_tactics": [{"tactic": "refine' le_antisymm (span_singleton_smul_le R g x) _", "state_before": "R : Type u_2\nR\u2082 : Type ?u.140038\nK : Type ?u.140041\nM : Type u_3\nM\u2082 : Type ?u.140047\nV : Type ?u.140050\nS : Type ?u.140053\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2076 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\ns t : Set M\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : SMul G R\ninst\u271d\u00b9 : MulAction G M\ninst\u271d : IsScalarTower G R M\ng : G\nx : M\n\u22a2 span R {g \u2022 x} = span R {x}", "state_after": "R : Type u_2\nR\u2082 : Type ?u.140038\nK : Type ?u.140041\nM : Type u_3\nM\u2082 : Type ?u.140047\nV : Type ?u.140050\nS : Type ?u.140053\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2076 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\ns t : Set M\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : SMul G R\ninst\u271d\u00b9 : MulAction G M\ninst\u271d : IsScalarTower G R M\ng : G\nx : M\n\u22a2 span R {x} \u2264 span R {g \u2022 x}"}, {"tactic": "convert span_singleton_smul_le R g\u207b\u00b9 (g \u2022 x)", "state_before": "R : Type u_2\nR\u2082 : Type ?u.140038\nK : Type ?u.140041\nM : Type u_3\nM\u2082 : Type ?u.140047\nV : Type ?u.140050\nS : Type ?u.140053\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2076 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\ns t : Set M\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : SMul G R\ninst\u271d\u00b9 : MulAction G M\ninst\u271d : IsScalarTower G R M\ng : G\nx : M\n\u22a2 span R {x} \u2264 span R {g \u2022 x}", "state_after": "case h.e'_3.h.e'_6.h.e'_4\nR : Type u_2\nR\u2082 : Type ?u.140038\nK : Type ?u.140041\nM : Type u_3\nM\u2082 : Type ?u.140047\nV : Type ?u.140050\nS : Type ?u.140053\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2076 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\ns t : Set M\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : SMul G R\ninst\u271d\u00b9 : MulAction G M\ninst\u271d : IsScalarTower G R M\ng : G\nx : M\n\u22a2 x = g\u207b\u00b9 \u2022 g \u2022 x"}, {"tactic": "exact (inv_smul_smul g x).symm", "state_before": "case h.e'_3.h.e'_6.h.e'_4\nR : Type u_2\nR\u2082 : Type ?u.140038\nK : Type ?u.140041\nM : Type u_3\nM\u2082 : Type ?u.140047\nV : Type ?u.140050\nS : Type ?u.140053\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : Module R M\nx\u271d : M\np p' : Submodule R M\ninst\u271d\u2076 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\ns t : Set M\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : SMul G R\ninst\u271d\u00b9 : MulAction G M\ninst\u271d : IsScalarTower G R M\ng : G\nx : M\n\u22a2 x = g\u207b\u00b9 \u2022 g \u2022 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/UpperLower/Basic.lean", "full_name": "LowerSet.lower", "start": [467, 11], "end": [468, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "full_name": "Real.Angle.cos_neg", "start": [406, 1], "end": [408, 23], "traced_tactics": [{"tactic": "induction \u03b8 using Real.Angle.induction_on", "state_before": "\u03b8 : Angle\n\u22a2 cos (-\u03b8) = cos \u03b8", "state_after": "case h\nx\u271d : \u211d\n\u22a2 cos (-\u2191x\u271d) = cos \u2191x\u271d"}, {"tactic": "exact Real.cos_neg _", "state_before": "case h\nx\u271d : \u211d\n\u22a2 cos (-\u2191x\u271d) = cos \u2191x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/RBMap/Alter.lean", "full_name": "Std.RBNode.Path.zoom_ins", "start": [57, 1], "end": [66, 25], "traced_tactics": [{"tactic": "unfold RBNode.ins", "state_before": "\u03b1 : Type u_1\nv : \u03b1\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nt : RBNode \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\n\u22a2 zoom (cmp v) t path = (t', path') \u2192 ins path (RBNode.ins cmp v t) = ins path' (setRoot v t')", "state_after": "\u03b1 : Type u_1\nv : \u03b1\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nt : RBNode \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\n\u22a2 zoom (cmp v) t path = (t', path') \u2192\n    ins path\n        (match t with\n        | nil => node red nil v nil\n        | node red a y b =>\n          match cmp v y with\n          | Ordering.lt => node red (RBNode.ins cmp v a) y b\n          | Ordering.gt => node red a y (RBNode.ins cmp v b)\n          | Ordering.eq => node red a v b\n        | node black a y b =>\n          match cmp v y with\n          | Ordering.lt => balance1 (RBNode.ins cmp v a) y b\n          | Ordering.gt => balance2 a y (RBNode.ins cmp v b)\n          | Ordering.eq => node black a v b) =\n      ins path' (setRoot v t')"}, {"tactic": "split <;> simp [zoom]", "state_before": "\u03b1 : Type u_1\nv : \u03b1\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nt : RBNode \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\n\u22a2 zoom (cmp v) t path = (t', path') \u2192\n    ins path\n        (match t with\n        | nil => node red nil v nil\n        | node red a y b =>\n          match cmp v y with\n          | Ordering.lt => node red (RBNode.ins cmp v a) y b\n          | Ordering.gt => node red a y (RBNode.ins cmp v b)\n          | Ordering.eq => node red a v b\n        | node black a y b =>\n          match cmp v y with\n          | Ordering.lt => balance1 (RBNode.ins cmp v a) y b\n          | Ordering.gt => balance2 a y (RBNode.ins cmp v b)\n          | Ordering.eq => node black a v b) =\n      ins path' (setRoot v t')", "state_after": "case h_1\n\u03b1 : Type u_1\nv : \u03b1\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nx\u271d : RBNode \u03b1\n\u22a2 nil = t' \u2192 path = path' \u2192 ins path (node red nil v nil) = ins path' (setRoot v t')\n\ncase h_2\n\u03b1 : Type u_1\nv : \u03b1\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nx\u271d a\u271d : RBNode \u03b1\ny\u271d : \u03b1\nb\u271d : RBNode \u03b1\n\u22a2 (match cmp v y\u271d with\n      | Ordering.lt => zoom (cmp v) a\u271d (left red path y\u271d b\u271d)\n      | Ordering.gt => zoom (cmp v) b\u271d (right red a\u271d y\u271d path)\n      | Ordering.eq => (node red a\u271d y\u271d b\u271d, path)) =\n      (t', path') \u2192\n    ins path\n        (match cmp v y\u271d with\n        | Ordering.lt => node red (RBNode.ins cmp v a\u271d) y\u271d b\u271d\n        | Ordering.gt => node red a\u271d y\u271d (RBNode.ins cmp v b\u271d)\n        | Ordering.eq => node red a\u271d v b\u271d) =\n      ins path' (setRoot v t')\n\ncase h_3\n\u03b1 : Type u_1\nv : \u03b1\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nx\u271d a\u271d : RBNode \u03b1\ny\u271d : \u03b1\nb\u271d : RBNode \u03b1\n\u22a2 (match cmp v y\u271d with\n      | Ordering.lt => zoom (cmp v) a\u271d (left black path y\u271d b\u271d)\n      | Ordering.gt => zoom (cmp v) b\u271d (right black a\u271d y\u271d path)\n      | Ordering.eq => (node black a\u271d y\u271d b\u271d, path)) =\n      (t', path') \u2192\n    ins path\n        (match cmp v y\u271d with\n        | Ordering.lt => balance1 (RBNode.ins cmp v a\u271d) y\u271d b\u271d\n        | Ordering.gt => balance2 a\u271d y\u271d (RBNode.ins cmp v b\u271d)\n        | Ordering.eq => node black a\u271d v b\u271d) =\n      ins path' (setRoot v t')"}, {"tactic": "intro | rfl, rfl => rfl", "state_before": "case h_1\n\u03b1 : Type u_1\nv : \u03b1\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nx\u271d : RBNode \u03b1\n\u22a2 nil = t' \u2192 path = path' \u2192 ins path (node red nil v nil) = ins path' (setRoot v t')", "state_after": "no goals"}, {"tactic": "rfl", "state_before": "\u03b1 : Type u_1\nv : \u03b1\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nx\u271d\u00b2 : RBNode \u03b1\nx\u271d\u00b9 : nil = t'\nx\u271d : path = path'\n\u22a2 ins path (node red nil v nil) = ins path (setRoot v nil)", "state_after": "no goals"}, {"tactic": "split", "state_before": "case h_3\n\u03b1 : Type u_1\nv : \u03b1\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nx\u271d a\u271d : RBNode \u03b1\ny\u271d : \u03b1\nb\u271d : RBNode \u03b1\n\u22a2 (match cmp v y\u271d with\n      | Ordering.lt => zoom (cmp v) a\u271d (left black path y\u271d b\u271d)\n      | Ordering.gt => zoom (cmp v) b\u271d (right black a\u271d y\u271d path)\n      | Ordering.eq => (node black a\u271d y\u271d b\u271d, path)) =\n      (t', path') \u2192\n    ins path\n        (match cmp v y\u271d with\n        | Ordering.lt => balance1 (RBNode.ins cmp v a\u271d) y\u271d b\u271d\n        | Ordering.gt => balance2 a\u271d y\u271d (RBNode.ins cmp v b\u271d)\n        | Ordering.eq => node black a\u271d v b\u271d) =\n      ins path' (setRoot v t')", "state_after": "case h_3.h_1\n\u03b1 : Type u_1\nv : \u03b1\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nx\u271d\u00b9 a\u271d : RBNode \u03b1\ny\u271d : \u03b1\nb\u271d : RBNode \u03b1\nx\u271d : Ordering\nheq\u271d : cmp v y\u271d = Ordering.lt\n\u22a2 zoom (cmp v) a\u271d (left black path y\u271d b\u271d) = (t', path') \u2192\n    ins path (balance1 (RBNode.ins cmp v a\u271d) y\u271d b\u271d) = ins path' (setRoot v t')\n\ncase h_3.h_2\n\u03b1 : Type u_1\nv : \u03b1\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nx\u271d\u00b9 a\u271d : RBNode \u03b1\ny\u271d : \u03b1\nb\u271d : RBNode \u03b1\nx\u271d : Ordering\nheq\u271d : cmp v y\u271d = Ordering.gt\n\u22a2 zoom (cmp v) b\u271d (right black a\u271d y\u271d path) = (t', path') \u2192\n    ins path (balance2 a\u271d y\u271d (RBNode.ins cmp v b\u271d)) = ins path' (setRoot v t')\n\ncase h_3.h_3\n\u03b1 : Type u_1\nv : \u03b1\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nx\u271d\u00b9 a\u271d : RBNode \u03b1\ny\u271d : \u03b1\nb\u271d : RBNode \u03b1\nx\u271d : Ordering\nheq\u271d : cmp v y\u271d = Ordering.eq\n\u22a2 (node black a\u271d y\u271d b\u271d, path) = (t', path') \u2192 ins path (node black a\u271d v b\u271d) = ins path' (setRoot v t')"}, {"tactic": "exact zoom_ins", "state_before": "case h_3.h_1\n\u03b1 : Type u_1\nv : \u03b1\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nx\u271d\u00b9 a\u271d : RBNode \u03b1\ny\u271d : \u03b1\nb\u271d : RBNode \u03b1\nx\u271d : Ordering\nheq\u271d : cmp v y\u271d = Ordering.lt\n\u22a2 zoom (cmp v) a\u271d (left black path y\u271d b\u271d) = (t', path') \u2192\n    ins path (balance1 (RBNode.ins cmp v a\u271d) y\u271d b\u271d) = ins path' (setRoot v t')", "state_after": "no goals"}, {"tactic": "exact zoom_ins", "state_before": "case h_3.h_2\n\u03b1 : Type u_1\nv : \u03b1\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nx\u271d\u00b9 a\u271d : RBNode \u03b1\ny\u271d : \u03b1\nb\u271d : RBNode \u03b1\nx\u271d : Ordering\nheq\u271d : cmp v y\u271d = Ordering.gt\n\u22a2 zoom (cmp v) b\u271d (right black a\u271d y\u271d path) = (t', path') \u2192\n    ins path (balance2 a\u271d y\u271d (RBNode.ins cmp v b\u271d)) = ins path' (setRoot v t')", "state_after": "no goals"}, {"tactic": "intro | rfl => rfl", "state_before": "case h_3.h_3\n\u03b1 : Type u_1\nv : \u03b1\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nx\u271d\u00b9 a\u271d : RBNode \u03b1\ny\u271d : \u03b1\nb\u271d : RBNode \u03b1\nx\u271d : Ordering\nheq\u271d : cmp v y\u271d = Ordering.eq\n\u22a2 (node black a\u271d y\u271d b\u271d, path) = (t', path') \u2192 ins path (node black a\u271d v b\u271d) = ins path' (setRoot v t')", "state_after": "no goals"}, {"tactic": "rfl", "state_before": "\u03b1 : Type u_1\nv : \u03b1\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nx\u271d\u00b2 a\u271d : RBNode \u03b1\ny\u271d : \u03b1\nb\u271d : RBNode \u03b1\nx\u271d\u00b9 : Ordering\nheq\u271d : cmp v y\u271d = Ordering.eq\nx\u271d : (node black a\u271d y\u271d b\u271d, path) = (t', path')\n\u22a2 ins path (node black a\u271d v b\u271d) = ins path (setRoot v (node black a\u271d y\u271d b\u271d))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/SuccPred/Limit.lean", "full_name": "Order.isPredLimit_of_pred_lt", "start": [355, 1], "end": [356, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/MinMax.lean", "full_name": "max_commutative", "start": [287, 1], "end": [288, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Option/Basic.lean", "full_name": "Option.casesOn'_coe", "start": [354, 1], "end": [355, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "full_name": "Submodule.coe_zero", "start": [313, 1], "end": [314, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Int.abs_sub_lt_one_of_floor_eq_floor", "start": [769, 1], "end": [776, 52], "traced_tactics": [{"tactic": "have : a < \u230aa\u230b + 1 := lt_floor_add_one a", "state_before": "F : Type ?u.136653\n\u03b1\u271d : Type ?u.136656\n\u03b2 : Type ?u.136659\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\u271d\ninst\u271d\u00b2 : FloorRing \u03b1\u271d\nz : \u2124\na\u271d : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedCommRing \u03b1\ninst\u271d : FloorRing \u03b1\na b : \u03b1\nh : \u230aa\u230b = \u230ab\u230b\n\u22a2 abs (a - b) < 1", "state_after": "F : Type ?u.136653\n\u03b1\u271d : Type ?u.136656\n\u03b2 : Type ?u.136659\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\u271d\ninst\u271d\u00b2 : FloorRing \u03b1\u271d\nz : \u2124\na\u271d : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedCommRing \u03b1\ninst\u271d : FloorRing \u03b1\na b : \u03b1\nh : \u230aa\u230b = \u230ab\u230b\nthis : a < \u2191\u230aa\u230b + 1\n\u22a2 abs (a - b) < 1"}, {"tactic": "have : b < \u230ab\u230b + 1 := lt_floor_add_one b", "state_before": "F : Type ?u.136653\n\u03b1\u271d : Type ?u.136656\n\u03b2 : Type ?u.136659\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\u271d\ninst\u271d\u00b2 : FloorRing \u03b1\u271d\nz : \u2124\na\u271d : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedCommRing \u03b1\ninst\u271d : FloorRing \u03b1\na b : \u03b1\nh : \u230aa\u230b = \u230ab\u230b\nthis : a < \u2191\u230aa\u230b + 1\n\u22a2 abs (a - b) < 1", "state_after": "F : Type ?u.136653\n\u03b1\u271d : Type ?u.136656\n\u03b2 : Type ?u.136659\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\u271d\ninst\u271d\u00b2 : FloorRing \u03b1\u271d\nz : \u2124\na\u271d : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedCommRing \u03b1\ninst\u271d : FloorRing \u03b1\na b : \u03b1\nh : \u230aa\u230b = \u230ab\u230b\nthis\u271d : a < \u2191\u230aa\u230b + 1\nthis : b < \u2191\u230ab\u230b + 1\n\u22a2 abs (a - b) < 1"}, {"tactic": "have : (\u230aa\u230b : \u03b1) = \u230ab\u230b := Int.cast_inj.2 h", "state_before": "F : Type ?u.136653\n\u03b1\u271d : Type ?u.136656\n\u03b2 : Type ?u.136659\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\u271d\ninst\u271d\u00b2 : FloorRing \u03b1\u271d\nz : \u2124\na\u271d : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedCommRing \u03b1\ninst\u271d : FloorRing \u03b1\na b : \u03b1\nh : \u230aa\u230b = \u230ab\u230b\nthis\u271d : a < \u2191\u230aa\u230b + 1\nthis : b < \u2191\u230ab\u230b + 1\n\u22a2 abs (a - b) < 1", "state_after": "F : Type ?u.136653\n\u03b1\u271d : Type ?u.136656\n\u03b2 : Type ?u.136659\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\u271d\ninst\u271d\u00b2 : FloorRing \u03b1\u271d\nz : \u2124\na\u271d : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedCommRing \u03b1\ninst\u271d : FloorRing \u03b1\na b : \u03b1\nh : \u230aa\u230b = \u230ab\u230b\nthis\u271d\u00b9 : a < \u2191\u230aa\u230b + 1\nthis\u271d : b < \u2191\u230ab\u230b + 1\nthis : \u2191\u230aa\u230b = \u2191\u230ab\u230b\n\u22a2 abs (a - b) < 1"}, {"tactic": "have : (\u230aa\u230b : \u03b1) \u2264 a := floor_le a", "state_before": "F : Type ?u.136653\n\u03b1\u271d : Type ?u.136656\n\u03b2 : Type ?u.136659\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\u271d\ninst\u271d\u00b2 : FloorRing \u03b1\u271d\nz : \u2124\na\u271d : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedCommRing \u03b1\ninst\u271d : FloorRing \u03b1\na b : \u03b1\nh : \u230aa\u230b = \u230ab\u230b\nthis\u271d\u00b9 : a < \u2191\u230aa\u230b + 1\nthis\u271d : b < \u2191\u230ab\u230b + 1\nthis : \u2191\u230aa\u230b = \u2191\u230ab\u230b\n\u22a2 abs (a - b) < 1", "state_after": "F : Type ?u.136653\n\u03b1\u271d : Type ?u.136656\n\u03b2 : Type ?u.136659\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\u271d\ninst\u271d\u00b2 : FloorRing \u03b1\u271d\nz : \u2124\na\u271d : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedCommRing \u03b1\ninst\u271d : FloorRing \u03b1\na b : \u03b1\nh : \u230aa\u230b = \u230ab\u230b\nthis\u271d\u00b2 : a < \u2191\u230aa\u230b + 1\nthis\u271d\u00b9 : b < \u2191\u230ab\u230b + 1\nthis\u271d : \u2191\u230aa\u230b = \u2191\u230ab\u230b\nthis : \u2191\u230aa\u230b \u2264 a\n\u22a2 abs (a - b) < 1"}, {"tactic": "have : (\u230ab\u230b : \u03b1) \u2264 b := floor_le b", "state_before": "F : Type ?u.136653\n\u03b1\u271d : Type ?u.136656\n\u03b2 : Type ?u.136659\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\u271d\ninst\u271d\u00b2 : FloorRing \u03b1\u271d\nz : \u2124\na\u271d : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedCommRing \u03b1\ninst\u271d : FloorRing \u03b1\na b : \u03b1\nh : \u230aa\u230b = \u230ab\u230b\nthis\u271d\u00b2 : a < \u2191\u230aa\u230b + 1\nthis\u271d\u00b9 : b < \u2191\u230ab\u230b + 1\nthis\u271d : \u2191\u230aa\u230b = \u2191\u230ab\u230b\nthis : \u2191\u230aa\u230b \u2264 a\n\u22a2 abs (a - b) < 1", "state_after": "F : Type ?u.136653\n\u03b1\u271d : Type ?u.136656\n\u03b2 : Type ?u.136659\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\u271d\ninst\u271d\u00b2 : FloorRing \u03b1\u271d\nz : \u2124\na\u271d : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedCommRing \u03b1\ninst\u271d : FloorRing \u03b1\na b : \u03b1\nh : \u230aa\u230b = \u230ab\u230b\nthis\u271d\u00b3 : a < \u2191\u230aa\u230b + 1\nthis\u271d\u00b2 : b < \u2191\u230ab\u230b + 1\nthis\u271d\u00b9 : \u2191\u230aa\u230b = \u2191\u230ab\u230b\nthis\u271d : \u2191\u230aa\u230b \u2264 a\nthis : \u2191\u230ab\u230b \u2264 b\n\u22a2 abs (a - b) < 1"}, {"tactic": "exact abs_sub_lt_iff.2 \u27e8by linarith, by linarith\u27e9", "state_before": "F : Type ?u.136653\n\u03b1\u271d : Type ?u.136656\n\u03b2 : Type ?u.136659\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\u271d\ninst\u271d\u00b2 : FloorRing \u03b1\u271d\nz : \u2124\na\u271d : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedCommRing \u03b1\ninst\u271d : FloorRing \u03b1\na b : \u03b1\nh : \u230aa\u230b = \u230ab\u230b\nthis\u271d\u00b3 : a < \u2191\u230aa\u230b + 1\nthis\u271d\u00b2 : b < \u2191\u230ab\u230b + 1\nthis\u271d\u00b9 : \u2191\u230aa\u230b = \u2191\u230ab\u230b\nthis\u271d : \u2191\u230aa\u230b \u2264 a\nthis : \u2191\u230ab\u230b \u2264 b\n\u22a2 abs (a - b) < 1", "state_after": "no goals"}, {"tactic": "linarith", "state_before": "F : Type ?u.136653\n\u03b1\u271d : Type ?u.136656\n\u03b2 : Type ?u.136659\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\u271d\ninst\u271d\u00b2 : FloorRing \u03b1\u271d\nz : \u2124\na\u271d : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedCommRing \u03b1\ninst\u271d : FloorRing \u03b1\na b : \u03b1\nh : \u230aa\u230b = \u230ab\u230b\nthis\u271d\u00b3 : a < \u2191\u230aa\u230b + 1\nthis\u271d\u00b2 : b < \u2191\u230ab\u230b + 1\nthis\u271d\u00b9 : \u2191\u230aa\u230b = \u2191\u230ab\u230b\nthis\u271d : \u2191\u230aa\u230b \u2264 a\nthis : \u2191\u230ab\u230b \u2264 b\n\u22a2 a - b < 1", "state_after": "no goals"}, {"tactic": "linarith", "state_before": "F : Type ?u.136653\n\u03b1\u271d : Type ?u.136656\n\u03b2 : Type ?u.136659\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\u271d\ninst\u271d\u00b2 : FloorRing \u03b1\u271d\nz : \u2124\na\u271d : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedCommRing \u03b1\ninst\u271d : FloorRing \u03b1\na b : \u03b1\nh : \u230aa\u230b = \u230ab\u230b\nthis\u271d\u00b3 : a < \u2191\u230aa\u230b + 1\nthis\u271d\u00b2 : b < \u2191\u230ab\u230b + 1\nthis\u271d\u00b9 : \u2191\u230aa\u230b = \u2191\u230ab\u230b\nthis\u271d : \u2191\u230aa\u230b \u2264 a\nthis : \u2191\u230ab\u230b \u2264 b\n\u22a2 b - a < 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Basis.lean", "full_name": "Basis.ofVectorSpace_apply_self", "start": [1475, 1], "end": [1477, 29], "traced_tactics": [{"tactic": "unfold ofVectorSpace", "state_before": "\u03b9 : Type ?u.1145319\n\u03b9' : Type ?u.1145322\nR : Type ?u.1145325\nR\u2082 : Type ?u.1145328\nK : Type u_1\nM : Type ?u.1145334\nM' : Type ?u.1145337\nM'' : Type ?u.1145340\nV : Type u\nV' : Type ?u.1145345\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx\u271d y z : V\nx : \u2191(ofVectorSpaceIndex K V)\n\u22a2 \u2191(ofVectorSpace K V) x = \u2191x", "state_after": "\u03b9 : Type ?u.1145319\n\u03b9' : Type ?u.1145322\nR : Type ?u.1145325\nR\u2082 : Type ?u.1145328\nK : Type u_1\nM : Type ?u.1145334\nM' : Type ?u.1145337\nM'' : Type ?u.1145340\nV : Type u\nV' : Type ?u.1145345\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx\u271d y z : V\nx : \u2191(ofVectorSpaceIndex K V)\n\u22a2 \u2191(extend (_ : LinearIndependent K fun x => \u2191x)) x = \u2191x"}, {"tactic": "exact Basis.mk_apply _ _ _", "state_before": "\u03b9 : Type ?u.1145319\n\u03b9' : Type ?u.1145322\nR : Type ?u.1145325\nR\u2082 : Type ?u.1145328\nK : Type u_1\nM : Type ?u.1145334\nM' : Type ?u.1145337\nM'' : Type ?u.1145340\nV : Type u\nV' : Type ?u.1145345\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx\u271d y z : V\nx : \u2191(ofVectorSpaceIndex K V)\n\u22a2 \u2191(extend (_ : LinearIndependent K fun x => \u2191x)) x = \u2191x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Hom/Group.lean", "full_name": "NeZero.of_map", "start": [112, 1], "end": [114, 66], "traced_tactics": [{"tactic": "rw [h]", "state_before": "\u03b1 : Type ?u.1009\n\u03b2 : Type ?u.1012\nM\u271d : Type ?u.1015\nN : Type ?u.1018\nP : Type ?u.1021\nG : Type ?u.1024\nH : Type ?u.1027\nF : Type u_3\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Zero R\ninst\u271d\u00b9 : Zero M\ninst\u271d : ZeroHomClass F R M\nf : F\nr : R\nneZero : NeZero (\u2191f r)\nh : r = 0\n\u22a2 \u2191f r = 0", "state_after": "\u03b1 : Type ?u.1009\n\u03b2 : Type ?u.1012\nM\u271d : Type ?u.1015\nN : Type ?u.1018\nP : Type ?u.1021\nG : Type ?u.1024\nH : Type ?u.1027\nF : Type u_3\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Zero R\ninst\u271d\u00b9 : Zero M\ninst\u271d : ZeroHomClass F R M\nf : F\nr : R\nneZero : NeZero (\u2191f r)\nh : r = 0\n\u22a2 \u2191f 0 = 0"}, {"tactic": "exact ZeroHomClass.map_zero f", "state_before": "\u03b1 : Type ?u.1009\n\u03b2 : Type ?u.1012\nM\u271d : Type ?u.1015\nN : Type ?u.1018\nP : Type ?u.1021\nG : Type ?u.1024\nH : Type ?u.1027\nF : Type u_3\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Zero R\ninst\u271d\u00b9 : Zero M\ninst\u271d : ZeroHomClass F R M\nf : F\nr : R\nneZero : NeZero (\u2191f r)\nh : r = 0\n\u22a2 \u2191f 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/OrderSynonym.lean", "full_name": "toLex_inv", "start": [264, 1], "end": [264, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "full_name": "lt_one_of_mul_lt_left", "start": [497, 1], "end": [500, 53], "traced_tactics": [{"tactic": "simpa only [one_mul]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.24391\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b : \u03b1\nh : a * b < b\n\u22a2 a * ?m.24757 h < 1 * ?m.24757 h", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Arsinh.lean", "full_name": "Real.exp_arsinh", "start": [60, 1], "end": [64, 7], "traced_tactics": [{"tactic": "apply exp_log", "state_before": "x\u271d y x : \u211d\n\u22a2 exp (arsinh x) = x + sqrt (1 + x ^ 2)", "state_after": "case hx\nx\u271d y x : \u211d\n\u22a2 0 < x + sqrt (1 + x ^ 2)"}, {"tactic": "rw [\u2190 neg_lt_iff_pos_add']", "state_before": "case hx\nx\u271d y x : \u211d\n\u22a2 0 < x + sqrt (1 + x ^ 2)", "state_after": "case hx\nx\u271d y x : \u211d\n\u22a2 -x < sqrt (1 + x ^ 2)"}, {"tactic": "apply lt_sqrt_of_sq_lt", "state_before": "case hx\nx\u271d y x : \u211d\n\u22a2 -x < sqrt (1 + x ^ 2)", "state_after": "case hx.h\nx\u271d y x : \u211d\n\u22a2 (-x) ^ 2 < 1 + x ^ 2"}, {"tactic": "simp", "state_before": "case hx.h\nx\u271d y x : \u211d\n\u22a2 (-x) ^ 2 < 1 + x ^ 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Archimedean.lean", "full_name": "exists_rat_btwn", "start": [263, 1], "end": [281, 22], "traced_tactics": [{"tactic": "cases' exists_nat_gt (y - x)\u207b\u00b9 with n nh", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\n\u22a2 \u2203 q, x < \u2191q \u2227 \u2191q < y", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\n\u22a2 \u2203 q, x < \u2191q \u2227 \u2191q < y"}, {"tactic": "cases' exists_floor (x * n) with z zh", "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\n\u22a2 \u2203 q, x < \u2191q \u2227 \u2191q < y", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\n\u22a2 \u2203 q, x < \u2191q \u2227 \u2191q < y"}, {"tactic": "refine' \u27e8(z + 1 : \u2124) / n, _\u27e9", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\n\u22a2 \u2203 q, x < \u2191q \u2227 \u2191q < y", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\n\u22a2 x < \u2191(\u2191(z + 1) / \u2191n) \u2227 \u2191(\u2191(z + 1) / \u2191n) < y"}, {"tactic": "have n0' := (inv_pos.2 (sub_pos.2 h)).trans nh", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\n\u22a2 x < \u2191(\u2191(z + 1) / \u2191n) \u2227 \u2191(\u2191(z + 1) / \u2191n) < y", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\n\u22a2 x < \u2191(\u2191(z + 1) / \u2191n) \u2227 \u2191(\u2191(z + 1) / \u2191n) < y"}, {"tactic": "have n0 := Nat.cast_pos.1 n0'", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\n\u22a2 x < \u2191(\u2191(z + 1) / \u2191n) \u2227 \u2191(\u2191(z + 1) / \u2191n) < y", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 x < \u2191(\u2191(z + 1) / \u2191n) \u2227 \u2191(\u2191(z + 1) / \u2191n) < y"}, {"tactic": "rw [Rat.cast_div_of_ne_zero, Rat.cast_coe_nat, Rat.cast_coe_int, div_lt_iff n0']", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 x < \u2191(\u2191(z + 1) / \u2191n) \u2227 \u2191(\u2191(z + 1) / \u2191n) < y", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 x < \u2191(z + 1) / \u2191n \u2227 \u2191(z + 1) < y * \u2191n\n\ncase intro.intro.md\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191(z + 1)).den \u2260 0\n\ncase intro.intro.nn\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191n).num \u2260 0\n\ncase intro.intro.nd\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191n).den \u2260 0"}, {"tactic": "refine' \u27e8(lt_div_iff n0').2 <| (lt_iff_lt_of_le_iff_le (zh _)).1 (lt_add_one _), _\u27e9", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 x < \u2191(z + 1) / \u2191n \u2227 \u2191(z + 1) < y * \u2191n\n\ncase intro.intro.md\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191(z + 1)).den \u2260 0\n\ncase intro.intro.nn\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191n).num \u2260 0\n\ncase intro.intro.nd\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191n).den \u2260 0", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(z + 1) < y * \u2191n\n\ncase intro.intro.md\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191(z + 1)).den \u2260 0\n\ncase intro.intro.nn\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191n).num \u2260 0\n\ncase intro.intro.nd\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191n).den \u2260 0"}, {"tactic": "rw [Int.cast_add, Int.cast_one]", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(z + 1) < y * \u2191n\n\ncase intro.intro.md\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191(z + 1)).den \u2260 0\n\ncase intro.intro.nn\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191n).num \u2260 0\n\ncase intro.intro.nd\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191n).den \u2260 0", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191z + 1 < y * \u2191n\n\ncase intro.intro.md\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191(z + 1)).den \u2260 0\n\ncase intro.intro.nn\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191n).num \u2260 0\n\ncase intro.intro.nd\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191n).den \u2260 0"}, {"tactic": "refine' lt_of_le_of_lt (add_le_add_right ((zh _).1 le_rfl) _) _", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191z + 1 < y * \u2191n\n\ncase intro.intro.md\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191(z + 1)).den \u2260 0\n\ncase intro.intro.nn\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191n).num \u2260 0\n\ncase intro.intro.nd\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191n).den \u2260 0", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 x * \u2191n + 1 < y * \u2191n\n\ncase intro.intro.md\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191(z + 1)).den \u2260 0\n\ncase intro.intro.nn\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191n).num \u2260 0\n\ncase intro.intro.nd\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191n).den \u2260 0"}, {"tactic": "rwa [\u2190 lt_sub_iff_add_lt', \u2190 sub_mul, \u2190 div_lt_iff' (sub_pos.2 h), one_div]", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 x * \u2191n + 1 < y * \u2191n\n\ncase intro.intro.md\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191(z + 1)).den \u2260 0\n\ncase intro.intro.nn\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191n).num \u2260 0\n\ncase intro.intro.nd\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191n).den \u2260 0", "state_after": "case intro.intro.md\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191(z + 1)).den \u2260 0\n\ncase intro.intro.nn\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191n).num \u2260 0\n\ncase intro.intro.nd\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191n).den \u2260 0"}, {"tactic": "rw [Rat.coe_int_den, Nat.cast_one]", "state_before": "case intro.intro.md\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191(z + 1)).den \u2260 0", "state_after": "case intro.intro.md\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 1 \u2260 0"}, {"tactic": "exact one_ne_zero", "state_before": "case intro.intro.md\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 1 \u2260 0", "state_after": "no goals"}, {"tactic": "intro H", "state_before": "case intro.intro.nn\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191n).num \u2260 0", "state_after": "case intro.intro.nn\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\nH : \u2191(\u2191n).num = 0\n\u22a2 False"}, {"tactic": "rw [Rat.coe_nat_num, Int.cast_ofNat, Nat.cast_eq_zero] at H", "state_before": "case intro.intro.nn\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\nH : \u2191(\u2191n).num = 0\n\u22a2 False", "state_after": "case intro.intro.nn\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\nH : n = 0\n\u22a2 False"}, {"tactic": "subst H", "state_before": "case intro.intro.nn\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\nH : n = 0\n\u22a2 False", "state_after": "case intro.intro.nn\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nz : \u2124\nnh : (y - x)\u207b\u00b9 < \u21910\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u21910\nn0' : 0 < \u21910\nn0 : 0 < 0\n\u22a2 False"}, {"tactic": "cases n0", "state_before": "case intro.intro.nn\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nz : \u2124\nnh : (y - x)\u207b\u00b9 < \u21910\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u21910\nn0' : 0 < \u21910\nn0 : 0 < 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rw [Rat.coe_nat_den, Nat.cast_one]", "state_before": "case intro.intro.nd\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 \u2191(\u2191n).den \u2260 0", "state_after": "case intro.intro.nd\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 1 \u2260 0"}, {"tactic": "exact one_ne_zero", "state_before": "case intro.intro.nd\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : Archimedean \u03b1\nx\u271d y\u271d \u03b5 x y : \u03b1\nh : x < y\nn : \u2115\nnh : (y - x)\u207b\u00b9 < \u2191n\nz : \u2124\nzh : \u2200 (z_1 : \u2124), z_1 \u2264 z \u2194 \u2191z_1 \u2264 x * \u2191n\nn0' : 0 < \u2191n\nn0 : 0 < n\n\u22a2 1 \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/StarSubalgebra.lean", "full_name": "elementalStarAlgebra.closedEmbedding_coe", "start": [218, 1], "end": [227, 54], "traced_tactics": [{"tactic": "convert elementalStarAlgebra.isClosed R x", "state_before": "R : Type u_2\nA : Type u_1\nB : Type ?u.308604\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : StarRing R\ninst\u271d\u00b9\u2070 : TopologicalSpace A\ninst\u271d\u2079 : Semiring A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : TopologicalSemiring A\ninst\u271d\u2076 : ContinuousStar A\ninst\u271d\u2075 : Algebra R A\ninst\u271d\u2074 : StarModule R A\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : Algebra R B\nx : A\n\u22a2 IsClosed (range Subtype.val)", "state_after": "case h.e'_3\nR : Type u_2\nA : Type u_1\nB : Type ?u.308604\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : StarRing R\ninst\u271d\u00b9\u2070 : TopologicalSpace A\ninst\u271d\u2079 : Semiring A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : TopologicalSemiring A\ninst\u271d\u2076 : ContinuousStar A\ninst\u271d\u2075 : Algebra R A\ninst\u271d\u2074 : StarModule R A\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : Algebra R B\nx : A\n\u22a2 range Subtype.val = \u2191(elementalStarAlgebra R x)"}, {"tactic": "exact\n  Set.ext fun y =>\n    \u27e8by\n      rintro \u27e8y, rfl\u27e9\n      exact y.prop, fun hy => \u27e8\u27e8y, hy\u27e9, rfl\u27e9\u27e9", "state_before": "case h.e'_3\nR : Type u_2\nA : Type u_1\nB : Type ?u.308604\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : StarRing R\ninst\u271d\u00b9\u2070 : TopologicalSpace A\ninst\u271d\u2079 : Semiring A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : TopologicalSemiring A\ninst\u271d\u2076 : ContinuousStar A\ninst\u271d\u2075 : Algebra R A\ninst\u271d\u2074 : StarModule R A\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : Algebra R B\nx : A\n\u22a2 range Subtype.val = \u2191(elementalStarAlgebra R x)", "state_after": "no goals"}, {"tactic": "rintro \u27e8y, rfl\u27e9", "state_before": "R : Type u_2\nA : Type u_1\nB : Type ?u.308604\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : StarRing R\ninst\u271d\u00b9\u2070 : TopologicalSpace A\ninst\u271d\u2079 : Semiring A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : TopologicalSemiring A\ninst\u271d\u2076 : ContinuousStar A\ninst\u271d\u2075 : Algebra R A\ninst\u271d\u2074 : StarModule R A\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : Algebra R B\nx y : A\n\u22a2 y \u2208 range Subtype.val \u2192 y \u2208 \u2191(elementalStarAlgebra R x)", "state_after": "case intro\nR : Type u_2\nA : Type u_1\nB : Type ?u.308604\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : StarRing R\ninst\u271d\u00b9\u2070 : TopologicalSpace A\ninst\u271d\u2079 : Semiring A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : TopologicalSemiring A\ninst\u271d\u2076 : ContinuousStar A\ninst\u271d\u2075 : Algebra R A\ninst\u271d\u2074 : StarModule R A\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : Algebra R B\nx : A\ny : { x_1 // x_1 \u2208 \u2191(elementalStarAlgebra R x) }\n\u22a2 \u2191y \u2208 \u2191(elementalStarAlgebra R x)"}, {"tactic": "exact y.prop", "state_before": "case intro\nR : Type u_2\nA : Type u_1\nB : Type ?u.308604\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : StarRing R\ninst\u271d\u00b9\u2070 : TopologicalSpace A\ninst\u271d\u2079 : Semiring A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : TopologicalSemiring A\ninst\u271d\u2076 : ContinuousStar A\ninst\u271d\u2075 : Algebra R A\ninst\u271d\u2074 : StarModule R A\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : Algebra R B\nx : A\ny : { x_1 // x_1 \u2208 \u2191(elementalStarAlgebra R x) }\n\u22a2 \u2191y \u2208 \u2191(elementalStarAlgebra R x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Basic.lean", "full_name": "Set.Icc_inter_Icc", "start": [1756, 1], "end": [1757, 81], "traced_tactics": [{"tactic": "simp only [Ici_inter_Iic.symm, Ici_inter_Ici.symm, Iic_inter_Iic.symm]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.182781\ninst\u271d : Lattice \u03b1\na b c a\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\n\u22a2 Icc a\u2081 b\u2081 \u2229 Icc a\u2082 b\u2082 = Icc (a\u2081 \u2294 a\u2082) (b\u2081 \u2293 b\u2082)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.182781\ninst\u271d : Lattice \u03b1\na b c a\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\n\u22a2 Ici a\u2081 \u2229 Iic b\u2081 \u2229 (Ici a\u2082 \u2229 Iic b\u2082) = Ici a\u2081 \u2229 Ici a\u2082 \u2229 (Iic b\u2081 \u2229 Iic b\u2082)"}, {"tactic": "ac_rfl", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.182781\ninst\u271d : Lattice \u03b1\na b c a\u2081 a\u2082 b\u2081 b\u2082 : \u03b1\n\u22a2 Ici a\u2081 \u2229 Iic b\u2081 \u2229 (Ici a\u2082 \u2229 Iic b\u2082) = Ici a\u2081 \u2229 Ici a\u2082 \u2229 (Iic b\u2081 \u2229 Iic b\u2082)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Module/LocalizedModule.lean", "full_name": "IsLocalizedModule.eq_zero_iff", "start": [1044, 1], "end": [1045, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "full_name": "left_ne_zero_of_mul_eq_one", "start": [150, 1], "end": [151, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Int/Lemmas.lean", "full_name": "Int.negSucc_mul_ofNat", "start": [375, 9], "end": [375, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "full_name": "hasStrictFDerivAt_iff_hasStrictDerivAt", "start": [202, 1], "end": [204, 45], "traced_tactics": [{"tactic": "simp [HasStrictDerivAt, HasStrictFDerivAt]", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf'\u271d f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nf' : \ud835\udd5c \u2192L[\ud835\udd5c] F\n\u22a2 HasStrictFDerivAt f f' x \u2194 HasStrictDerivAt f (\u2191f' 1) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Bind.lean", "full_name": "Multiset.attach_bind_coe", "start": [224, 1], "end": [226, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/IsLimit.lean", "full_name": "CategoryTheory.Limits.IsLimit.conePointUniqueUpToIso_inv_comp", "start": [165, 1], "end": [167, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Parity.lean", "full_name": "Odd.neg_one_pow", "start": [392, 1], "end": [392, 86], "traced_tactics": [{"tactic": "rw [h.neg_pow, one_pow]", "state_before": "F : Type ?u.96735\n\u03b1 : Type u_1\n\u03b2 : Type ?u.96741\nR : Type ?u.96744\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : HasDistribNeg \u03b1\na : \u03b1\nn : \u2115\nh : Odd n\n\u22a2 (-1) ^ n = -1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Index.lean", "full_name": "Subgroup.nat_card_dvd_of_surjective", "start": [313, 1], "end": [316, 61], "traced_tactics": [{"tactic": "rw [\u2190 Nat.card_congr (QuotientGroup.quotientKerEquivOfSurjective f hf).toEquiv]", "state_before": "G\u271d : Type ?u.108101\ninst\u271d\u00b2 : Group G\u271d\nH\u271d K L : Subgroup G\u271d\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192* H\nhf : Function.Surjective \u2191f\n\u22a2 Nat.card H \u2223 Nat.card G", "state_after": "G\u271d : Type ?u.108101\ninst\u271d\u00b2 : Group G\u271d\nH\u271d K L : Subgroup G\u271d\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192* H\nhf : Function.Surjective \u2191f\n\u22a2 Nat.card (G \u29f8 MonoidHom.ker f) \u2223 Nat.card G"}, {"tactic": "exact Dvd.intro_left (Nat.card f.ker) f.ker.card_mul_index", "state_before": "G\u271d : Type ?u.108101\ninst\u271d\u00b2 : Group G\u271d\nH\u271d K L : Subgroup G\u271d\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192* H\nhf : Function.Surjective \u2191f\n\u22a2 Nat.card (G \u29f8 MonoidHom.ker f) \u2223 Nat.card G", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Instances/ENNReal.lean", "full_name": "ENNReal.exists_frequently_lt_of_liminf_ne_top'", "start": [720, 1], "end": [726, 80], "traced_tactics": [{"tactic": "by_contra h", "state_before": "\u03b1 : Type ?u.186400\n\u03b2 : Type ?u.186403\n\u03b3 : Type ?u.186406\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Type u_1\nl : Filter \u03b9\nx : \u03b9 \u2192 \u211d\nhx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4\n\u22a2 \u2203 R, \u2203\u1da0 (n : \u03b9) in l, R < x n", "state_after": "\u03b1 : Type ?u.186400\n\u03b2 : Type ?u.186403\n\u03b3 : Type ?u.186406\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Type u_1\nl : Filter \u03b9\nx : \u03b9 \u2192 \u211d\nhx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4\nh : \u00ac\u2203 R, \u2203\u1da0 (n : \u03b9) in l, R < x n\n\u22a2 False"}, {"tactic": "simp_rw [not_exists, not_frequently, not_lt] at h", "state_before": "\u03b1 : Type ?u.186400\n\u03b2 : Type ?u.186403\n\u03b3 : Type ?u.186406\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Type u_1\nl : Filter \u03b9\nx : \u03b9 \u2192 \u211d\nhx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4\nh : \u00ac\u2203 R, \u2203\u1da0 (n : \u03b9) in l, R < x n\n\u22a2 False", "state_after": "\u03b1 : Type ?u.186400\n\u03b2 : Type ?u.186403\n\u03b3 : Type ?u.186406\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Type u_1\nl : Filter \u03b9\nx : \u03b9 \u2192 \u211d\nhx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4\nh : \u2200 (x_1 : \u211d), \u2200\u1da0 (x_2 : \u03b9) in l, x x_2 \u2264 x_1\n\u22a2 False"}, {"tactic": "refine hx (ENNReal.eq_top_of_forall_nnreal_le fun r => le_limsInf_of_le (by isBoundedDefault) ?_)", "state_before": "\u03b1 : Type ?u.186400\n\u03b2 : Type ?u.186403\n\u03b3 : Type ?u.186406\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Type u_1\nl : Filter \u03b9\nx : \u03b9 \u2192 \u211d\nhx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4\nh : \u2200 (x_1 : \u211d), \u2200\u1da0 (x_2 : \u03b9) in l, x x_2 \u2264 x_1\n\u22a2 False", "state_after": "\u03b1 : Type ?u.186400\n\u03b2 : Type ?u.186403\n\u03b3 : Type ?u.186406\na b c d : \u211d\u22650\u221e\nr\u271d p q : \u211d\u22650\nx\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Type u_1\nl : Filter \u03b9\nx : \u03b9 \u2192 \u211d\nhx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4\nh : \u2200 (x_1 : \u211d), \u2200\u1da0 (x_2 : \u03b9) in l, x x_2 \u2264 x_1\nr : \u211d\u22650\n\u22a2 \u2200\u1da0 (n : \u211d\u22650\u221e) in map (fun n => \u2191(\u2191Real.nnabs (x n))) l, \u2191r \u2264 n"}, {"tactic": "simp only [eventually_map, ENNReal.coe_le_coe]", "state_before": "\u03b1 : Type ?u.186400\n\u03b2 : Type ?u.186403\n\u03b3 : Type ?u.186406\na b c d : \u211d\u22650\u221e\nr\u271d p q : \u211d\u22650\nx\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Type u_1\nl : Filter \u03b9\nx : \u03b9 \u2192 \u211d\nhx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4\nh : \u2200 (x_1 : \u211d), \u2200\u1da0 (x_2 : \u03b9) in l, x x_2 \u2264 x_1\nr : \u211d\u22650\n\u22a2 \u2200\u1da0 (n : \u211d\u22650\u221e) in map (fun n => \u2191(\u2191Real.nnabs (x n))) l, \u2191r \u2264 n", "state_after": "\u03b1 : Type ?u.186400\n\u03b2 : Type ?u.186403\n\u03b3 : Type ?u.186406\na b c d : \u211d\u22650\u221e\nr\u271d p q : \u211d\u22650\nx\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Type u_1\nl : Filter \u03b9\nx : \u03b9 \u2192 \u211d\nhx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4\nh : \u2200 (x_1 : \u211d), \u2200\u1da0 (x_2 : \u03b9) in l, x x_2 \u2264 x_1\nr : \u211d\u22650\n\u22a2 \u2200\u1da0 (a : \u03b9) in l, r \u2264 \u2191Real.nnabs (x a)"}, {"tactic": "filter_upwards [h (-r)]with i hi using(le_neg.1 hi).trans (neg_le_abs_self _)", "state_before": "\u03b1 : Type ?u.186400\n\u03b2 : Type ?u.186403\n\u03b3 : Type ?u.186406\na b c d : \u211d\u22650\u221e\nr\u271d p q : \u211d\u22650\nx\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Type u_1\nl : Filter \u03b9\nx : \u03b9 \u2192 \u211d\nhx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4\nh : \u2200 (x_1 : \u211d), \u2200\u1da0 (x_2 : \u03b9) in l, x x_2 \u2264 x_1\nr : \u211d\u22650\n\u22a2 \u2200\u1da0 (a : \u03b9) in l, r \u2264 \u2191Real.nnabs (x a)", "state_after": "no goals"}, {"tactic": "isBoundedDefault", "state_before": "\u03b1 : Type ?u.186400\n\u03b2 : Type ?u.186403\n\u03b3 : Type ?u.186406\na b c d : \u211d\u22650\u221e\nr\u271d p q : \u211d\u22650\nx\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Type u_1\nl : Filter \u03b9\nx : \u03b9 \u2192 \u211d\nhx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4\nh : \u2200 (x_1 : \u211d), \u2200\u1da0 (x_2 : \u03b9) in l, x x_2 \u2264 x_1\nr : \u211d\u22650\n\u22a2 IsCobounded (fun x x_1 => x \u2265 x_1) (map (fun n => \u2191(\u2191Real.nnabs (x n))) l)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Sum/Order.lean", "full_name": "Sum.LiftRel.trans", "start": [57, 1], "end": [60, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.val_neg'", "start": [77, 1], "end": [78, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/ArithmeticFunction.lean", "full_name": "Nat.ArithmeticFunction.pmul_zeta", "start": [526, 1], "end": [528, 38], "traced_tactics": [{"tactic": "ext x", "state_before": "R : Type u_1\ninst\u271d : NonAssocSemiring R\nf : ArithmeticFunction R\n\u22a2 pmul f \u2191\u03b6 = f", "state_after": "case h\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : ArithmeticFunction R\nx : \u2115\n\u22a2 \u2191(pmul f \u2191\u03b6) x = \u2191f x"}, {"tactic": "cases x <;> simp [Nat.succ_ne_zero]", "state_before": "case h\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : ArithmeticFunction R\nx : \u2115\n\u22a2 \u2191(pmul f \u2191\u03b6) x = \u2191f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "IsClosed.isClosed_le", "start": [226, 1], "end": [228, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean", "full_name": "CategoryTheory.Limits.BinaryFan.isLimit_iff_isIso_snd", "start": [428, 1], "end": [433, 75], "traced_tactics": [{"tactic": "refine' Iff.trans _ (BinaryFan.isLimit_iff_isIso_fst h (BinaryFan.mk c.snd c.fst))", "state_before": "C : Type u\ninst\u271d : Category C\nX\u271d Y\u271d X Y : C\nh : IsTerminal X\nc : BinaryFan X Y\n\u22a2 Nonempty (IsLimit c) \u2194 IsIso (snd c)", "state_after": "C : Type u\ninst\u271d : Category C\nX\u271d Y\u271d X Y : C\nh : IsTerminal X\nc : BinaryFan X Y\n\u22a2 Nonempty (IsLimit c) \u2194 Nonempty (IsLimit (mk (snd c) (fst c)))"}, {"tactic": "exact\n  \u27e8fun h => \u27e8BinaryFan.isLimitFlip h.some\u27e9, fun h =>\n    \u27e8(BinaryFan.isLimitFlip h.some).ofIsoLimit (isoBinaryFanMk c).symm\u27e9\u27e9", "state_before": "C : Type u\ninst\u271d : Category C\nX\u271d Y\u271d X Y : C\nh : IsTerminal X\nc : BinaryFan X Y\n\u22a2 Nonempty (IsLimit c) \u2194 Nonempty (IsLimit (mk (snd c) (fst c)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.singleton_pi", "start": [731, 1], "end": [733, 12], "traced_tactics": [{"tactic": "ext", "state_before": "\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_1\n\u03b2 : \u03b9 \u2192 Type ?u.134307\ns s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d i : \u03b9\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\n\u22a2 pi {i} t = eval i \u207b\u00b9' t i", "state_after": "case h\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_1\n\u03b2 : \u03b9 \u2192 Type ?u.134307\ns s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d i : \u03b9\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 x\u271d \u2208 pi {i} t \u2194 x\u271d \u2208 eval i \u207b\u00b9' t i"}, {"tactic": "simp [pi]", "state_before": "case h\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_1\n\u03b2 : \u03b9 \u2192 Type ?u.134307\ns s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d i : \u03b9\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nx\u271d : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 x\u271d \u2208 pi {i} t \u2194 x\u271d \u2208 eval i \u207b\u00b9' t i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Setoid/Partition.lean", "full_name": "IndexedPartition.some_index", "start": [401, 1], "end": [402, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GroupWithZero/Power.lean", "full_name": "zpow_bit1'", "start": [168, 1], "end": [169, 45], "traced_tactics": [{"tactic": "rw [zpow_bit1\u2080, (Commute.refl a).mul_zpow]", "state_before": "G\u2080 : Type u_1\ninst\u271d : GroupWithZero G\u2080\na : G\u2080\nn : \u2124\n\u22a2 a ^ bit1 n = (a * a) ^ n * a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Opposites.lean", "full_name": "AddOpposite.unop_div", "start": [428, 1], "end": [429, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Digits.lean", "full_name": "Nat.modEq_eleven_digits_sum", "start": [578, 1], "end": [581, 30], "traced_tactics": [{"tactic": "have t := zmodeq_ofDigits_digits 11 10 (-1 : \u2124) (by unfold Int.ModEq; norm_num) n", "state_before": "n\u271d n : \u2115\n\u22a2 \u2191n \u2261 List.alternatingSum (List.map (fun n => \u2191n) (digits 10 n)) [ZMOD 11]", "state_after": "n\u271d n : \u2115\nt : \u2191n \u2261 ofDigits (-1) (digits 10 n) [ZMOD \u219111]\n\u22a2 \u2191n \u2261 List.alternatingSum (List.map (fun n => \u2191n) (digits 10 n)) [ZMOD 11]"}, {"tactic": "rwa [ofDigits_neg_one] at t", "state_before": "n\u271d n : \u2115\nt : \u2191n \u2261 ofDigits (-1) (digits 10 n) [ZMOD \u219111]\n\u22a2 \u2191n \u2261 List.alternatingSum (List.map (fun n => \u2191n) (digits 10 n)) [ZMOD 11]", "state_after": "no goals"}, {"tactic": "unfold Int.ModEq", "state_before": "n\u271d n : \u2115\n\u22a2 \u219110 \u2261 -1 [ZMOD \u219111]", "state_after": "n\u271d n : \u2115\n\u22a2 \u219110 % \u219111 = -1 % \u219111"}, {"tactic": "norm_num", "state_before": "n\u271d n : \u2115\n\u22a2 \u219110 % \u219111 = -1 % \u219111", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/Basic.lean", "full_name": "uniformSpace_comap_id", "start": [1272, 1], "end": [1274, 47], "traced_tactics": [{"tactic": "ext : 2", "state_before": "\u03b1\u271d : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort ?u.137256\n\u03b1 : Type u_1\n\u22a2 UniformSpace.comap id = id", "state_after": "case h.a\n\u03b1\u271d : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort ?u.137256\n\u03b1 : Type u_1\nx\u271d : UniformSpace \u03b1\n\u22a2 \ud835\udce4 \u03b1 = \ud835\udce4 \u03b1"}, {"tactic": "rw [uniformity_comap, Prod.map_id, comap_id]", "state_before": "case h.a\n\u03b1\u271d : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort ?u.137256\n\u03b1 : Type u_1\nx\u271d : UniformSpace \u03b1\n\u22a2 \ud835\udce4 \u03b1 = \ud835\udce4 \u03b1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/StructuredArrow.lean", "full_name": "CategoryTheory.CostructuredArrow.epi_of_epi_left", "start": [383, 1], "end": [384, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/MeasurableSpace.lean", "full_name": "measurable_tProd_mk", "start": [947, 1], "end": [950, 45], "traced_tactics": [{"tactic": "induction' l with i l ih", "state_before": "\u03b1 : Type ?u.370249\n\u03b2 : Type ?u.370252\n\u03b3 : Type ?u.370255\n\u03b4 : Type u_1\n\u03b4' : Type ?u.370261\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_2\ninst\u271d : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\nl : List \u03b4\n\u22a2 Measurable (TProd.mk l)", "state_after": "case nil\n\u03b1 : Type ?u.370249\n\u03b2 : Type ?u.370252\n\u03b3 : Type ?u.370255\n\u03b4 : Type u_1\n\u03b4' : Type ?u.370261\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_2\ninst\u271d : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\n\u22a2 Measurable (TProd.mk [])\n\ncase cons\n\u03b1 : Type ?u.370249\n\u03b2 : Type ?u.370252\n\u03b3 : Type ?u.370255\n\u03b4 : Type u_1\n\u03b4' : Type ?u.370261\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_2\ninst\u271d : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\ni : \u03b4\nl : List \u03b4\nih : Measurable (TProd.mk l)\n\u22a2 Measurable (TProd.mk (i :: l))"}, {"tactic": "exact measurable_const", "state_before": "case nil\n\u03b1 : Type ?u.370249\n\u03b2 : Type ?u.370252\n\u03b3 : Type ?u.370255\n\u03b4 : Type u_1\n\u03b4' : Type ?u.370261\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_2\ninst\u271d : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\n\u22a2 Measurable (TProd.mk [])", "state_after": "no goals"}, {"tactic": "exact (measurable_pi_apply i).prod_mk ih", "state_before": "case cons\n\u03b1 : Type ?u.370249\n\u03b2 : Type ?u.370252\n\u03b3 : Type ?u.370255\n\u03b4 : Type u_1\n\u03b4' : Type ?u.370261\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_2\ninst\u271d : (x : \u03b4) \u2192 MeasurableSpace (\u03c0 x)\ni : \u03b4\nl : List \u03b4\nih : Measurable (TProd.mk l)\n\u22a2 Measurable (TProd.mk (i :: l))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Cardinal/Cofinality.lean", "full_name": "Cardinal.mk_bounded_subset", "start": [897, 1], "end": [925, 51], "traced_tactics": [{"tactic": "rcases eq_or_ne (#\u03b1) 0 with (ha | ha)", "state_before": "\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\n\u22a2 (#{ s // Bounded r s }) = (#\u03b1)", "state_after": "case inl\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha : (#\u03b1) = 0\n\u22a2 (#{ s // Bounded r s }) = (#\u03b1)\n\ncase inr\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha : (#\u03b1) \u2260 0\n\u22a2 (#{ s // Bounded r s }) = (#\u03b1)"}, {"tactic": "have h' : IsStrongLimit (#\u03b1) := \u27e8ha, h\u27e9", "state_before": "case inr\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha : (#\u03b1) \u2260 0\n\u22a2 (#{ s // Bounded r s }) = (#\u03b1)", "state_after": "case inr\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\n\u22a2 (#{ s // Bounded r s }) = (#\u03b1)"}, {"tactic": "have ha := h'.isLimit.aleph0_le", "state_before": "case inr\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\n\u22a2 (#{ s // Bounded r s }) = (#\u03b1)", "state_after": "case inr\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\n\u22a2 (#{ s // Bounded r s }) = (#\u03b1)"}, {"tactic": "apply le_antisymm", "state_before": "case inr\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\n\u22a2 (#{ s // Bounded r s }) = (#\u03b1)", "state_after": "case inr.a\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\n\u22a2 (#{ s // Bounded r s }) \u2264 (#\u03b1)\n\ncase inr.a\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\n\u22a2 (#\u03b1) \u2264 (#{ s // Bounded r s })"}, {"tactic": "rw [ha]", "state_before": "case inl\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha : (#\u03b1) = 0\n\u22a2 (#{ s // Bounded r s }) = (#\u03b1)", "state_after": "case inl\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha : (#\u03b1) = 0\n\u22a2 (#{ s // Bounded r s }) = 0"}, {"tactic": "haveI := mk_eq_zero_iff.1 ha", "state_before": "case inl\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha : (#\u03b1) = 0\n\u22a2 (#{ s // Bounded r s }) = 0", "state_after": "case inl\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha : (#\u03b1) = 0\nthis : IsEmpty \u03b1\n\u22a2 (#{ s // Bounded r s }) = 0"}, {"tactic": "rw [mk_eq_zero_iff]", "state_before": "case inl\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha : (#\u03b1) = 0\nthis : IsEmpty \u03b1\n\u22a2 (#{ s // Bounded r s }) = 0", "state_after": "case inl\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha : (#\u03b1) = 0\nthis : IsEmpty \u03b1\n\u22a2 IsEmpty { s // Bounded r s }"}, {"tactic": "constructor", "state_before": "case inl\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha : (#\u03b1) = 0\nthis : IsEmpty \u03b1\n\u22a2 IsEmpty { s // Bounded r s }", "state_after": "case inl.false\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha : (#\u03b1) = 0\nthis : IsEmpty \u03b1\n\u22a2 { s // Bounded r s } \u2192 False"}, {"tactic": "rintro \u27e8s, hs\u27e9", "state_before": "case inl.false\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha : (#\u03b1) = 0\nthis : IsEmpty \u03b1\n\u22a2 { s // Bounded r s } \u2192 False", "state_after": "case inl.false.mk\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha : (#\u03b1) = 0\nthis : IsEmpty \u03b1\ns : Set \u03b1\nhs : Bounded r s\n\u22a2 False"}, {"tactic": "exact (not_unbounded_iff s).2 hs (unbounded_of_isEmpty s)", "state_before": "case inl.false.mk\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha : (#\u03b1) = 0\nthis : IsEmpty \u03b1\ns : Set \u03b1\nhs : Bounded r s\n\u22a2 False", "state_after": "no goals"}, {"tactic": "have : { s : Set \u03b1 | Bounded r s } = \u22c3 i, \ud835\udcab{ j | r j i } := setOf_exists _", "state_before": "case inr.a\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\n\u22a2 (#{ s // Bounded r s }) \u2264 (#\u03b1)", "state_after": "case inr.a\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nthis : {s | Bounded r s} = \u22c3 (i : \u03b1), \ud835\udcab{j | r j i}\n\u22a2 (#{ s // Bounded r s }) \u2264 (#\u03b1)"}, {"tactic": "rw [\u2190 coe_setOf, this]", "state_before": "case inr.a\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nthis : {s | Bounded r s} = \u22c3 (i : \u03b1), \ud835\udcab{j | r j i}\n\u22a2 (#{ s // Bounded r s }) \u2264 (#\u03b1)", "state_after": "case inr.a\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nthis : {s | Bounded r s} = \u22c3 (i : \u03b1), \ud835\udcab{j | r j i}\n\u22a2 (#\u2191(\u22c3 (i : \u03b1), \ud835\udcab{j | r j i})) \u2264 (#\u03b1)"}, {"tactic": "refine mk_iUnion_le_sum_mk.trans ((sum_le_iSup (fun i => #(\ud835\udcab{ j | r j i }))).trans\n  ((mul_le_max_of_aleph0_le_left ha).trans ?_))", "state_before": "case inr.a\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nthis : {s | Bounded r s} = \u22c3 (i : \u03b1), \ud835\udcab{j | r j i}\n\u22a2 (#\u2191(\u22c3 (i : \u03b1), \ud835\udcab{j | r j i})) \u2264 (#\u03b1)", "state_after": "case inr.a\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nthis : {s | Bounded r s} = \u22c3 (i : \u03b1), \ud835\udcab{j | r j i}\n\u22a2 max (#\u03b1) (\u2a06 (i : \u03b1), #\u2191(\ud835\udcab{j | r j i})) \u2264 (#\u03b1)"}, {"tactic": "rw [max_eq_left]", "state_before": "case inr.a\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nthis : {s | Bounded r s} = \u22c3 (i : \u03b1), \ud835\udcab{j | r j i}\n\u22a2 max (#\u03b1) (\u2a06 (i : \u03b1), #\u2191(\ud835\udcab{j | r j i})) \u2264 (#\u03b1)", "state_after": "case inr.a\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nthis : {s | Bounded r s} = \u22c3 (i : \u03b1), \ud835\udcab{j | r j i}\n\u22a2 (\u2a06 (i : \u03b1), #\u2191(\ud835\udcab{j | r j i})) \u2264 (#\u03b1)"}, {"tactic": "apply ciSup_le' _", "state_before": "case inr.a\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nthis : {s | Bounded r s} = \u22c3 (i : \u03b1), \ud835\udcab{j | r j i}\n\u22a2 (\u2a06 (i : \u03b1), #\u2191(\ud835\udcab{j | r j i})) \u2264 (#\u03b1)", "state_after": "\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nthis : {s | Bounded r s} = \u22c3 (i : \u03b1), \ud835\udcab{j | r j i}\n\u22a2 \u2200 (i : \u03b1), (#\u2191(\ud835\udcab{j | r j i})) \u2264 (#\u03b1)"}, {"tactic": "intro i", "state_before": "\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nthis : {s | Bounded r s} = \u22c3 (i : \u03b1), \ud835\udcab{j | r j i}\n\u22a2 \u2200 (i : \u03b1), (#\u2191(\ud835\udcab{j | r j i})) \u2264 (#\u03b1)", "state_after": "\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nthis : {s | Bounded r s} = \u22c3 (i : \u03b1), \ud835\udcab{j | r j i}\ni : \u03b1\n\u22a2 (#\u2191(\ud835\udcab{j | r j i})) \u2264 (#\u03b1)"}, {"tactic": "rw [mk_powerset]", "state_before": "\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nthis : {s | Bounded r s} = \u22c3 (i : \u03b1), \ud835\udcab{j | r j i}\ni : \u03b1\n\u22a2 (#\u2191(\ud835\udcab{j | r j i})) \u2264 (#\u03b1)", "state_after": "\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nthis : {s | Bounded r s} = \u22c3 (i : \u03b1), \ud835\udcab{j | r j i}\ni : \u03b1\n\u22a2 2 ^ (#\u2191{j | r j i}) \u2264 (#\u03b1)"}, {"tactic": "apply (h'.two_power_lt _).le", "state_before": "\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nthis : {s | Bounded r s} = \u22c3 (i : \u03b1), \ud835\udcab{j | r j i}\ni : \u03b1\n\u22a2 2 ^ (#\u2191{j | r j i}) \u2264 (#\u03b1)", "state_after": "\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nthis : {s | Bounded r s} = \u22c3 (i : \u03b1), \ud835\udcab{j | r j i}\ni : \u03b1\n\u22a2 (#\u2191{j | r j i}) < (#\u03b1)"}, {"tactic": "rw [coe_setOf, card_typein, \u2190 lt_ord, hr]", "state_before": "\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nthis : {s | Bounded r s} = \u22c3 (i : \u03b1), \ud835\udcab{j | r j i}\ni : \u03b1\n\u22a2 (#\u2191{j | r j i}) < (#\u03b1)", "state_after": "\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nthis : {s | Bounded r s} = \u22c3 (i : \u03b1), \ud835\udcab{j | r j i}\ni : \u03b1\n\u22a2 typein (fun x => r x) i < type r"}, {"tactic": "apply typein_lt_type", "state_before": "\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nthis : {s | Bounded r s} = \u22c3 (i : \u03b1), \ud835\udcab{j | r j i}\ni : \u03b1\n\u22a2 typein (fun x => r x) i < type r", "state_after": "no goals"}, {"tactic": "refine' @mk_le_of_injective \u03b1 _ (fun x => Subtype.mk {x} _) _", "state_before": "case inr.a\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\n\u22a2 (#\u03b1) \u2264 (#{ s // Bounded r s })", "state_after": "case inr.a.refine'_1\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nx : \u03b1\n\u22a2 Bounded r {x}\n\ncase inr.a.refine'_2\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\n\u22a2 Injective fun x => { val := {x}, property := (_ : Bounded r {x}) }"}, {"tactic": "apply bounded_singleton", "state_before": "case inr.a.refine'_1\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nx : \u03b1\n\u22a2 Bounded r {x}", "state_after": "case inr.a.refine'_1.hr\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nx : \u03b1\n\u22a2 Ordinal.IsLimit (type r)"}, {"tactic": "rw [\u2190 hr]", "state_before": "case inr.a.refine'_1.hr\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nx : \u03b1\n\u22a2 Ordinal.IsLimit (type r)", "state_after": "case inr.a.refine'_1.hr\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nx : \u03b1\n\u22a2 Ordinal.IsLimit (ord (#\u03b1))"}, {"tactic": "apply ord_isLimit ha", "state_before": "case inr.a.refine'_1.hr\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\nx : \u03b1\n\u22a2 Ordinal.IsLimit (ord (#\u03b1))", "state_after": "no goals"}, {"tactic": "intro a b hab", "state_before": "case inr.a.refine'_2\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\n\u22a2 Injective fun x => { val := {x}, property := (_ : Bounded r {x}) }", "state_after": "case inr.a.refine'_2\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\na b : \u03b1\nhab :\n  (fun x => { val := {x}, property := (_ : Bounded r {x}) }) a =\n    (fun x => { val := {x}, property := (_ : Bounded r {x}) }) b\n\u22a2 a = b"}, {"tactic": "simpa [singleton_eq_singleton_iff] using hab", "state_before": "case inr.a.refine'_2\n\u03b1\u271d : Type ?u.115548\nr\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_1\nh : \u2200 (x : Cardinal), x < (#\u03b1) \u2192 2 ^ x < (#\u03b1)\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsWellOrder \u03b1 r\nhr : ord (#\u03b1) = type r\nha\u271d : (#\u03b1) \u2260 0\nh' : IsStrongLimit (#\u03b1)\nha : \u2135\u2080 \u2264 (#\u03b1)\na b : \u03b1\nhab :\n  (fun x => { val := {x}, property := (_ : Bounded r {x}) }) a =\n    (fun x => { val := {x}, property := (_ : Bounded r {x}) }) b\n\u22a2 a = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Eval.lean", "full_name": "Polynomial.map_comp", "start": [947, 1], "end": [954, 81], "traced_tactics": [{"tactic": "simp only [map_C, forall_const, C_comp, eq_self_iff_true]", "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q\u271d r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np q : R[X]\n\u22a2 \u2200 (a : R), map f (comp (\u2191C a) q) = comp (map f (\u2191C a)) (map f q)", "state_after": "no goals"}, {"tactic": "simp (config := { contextual := true }) only [Polynomial.map_add, add_comp, forall_const,\n  imp_true_iff, eq_self_iff_true]", "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q\u271d r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np q : R[X]\n\u22a2 \u2200 (p q_1 : R[X]),\n    map f (comp p q) = comp (map f p) (map f q) \u2192\n      map f (comp q_1 q) = comp (map f q_1) (map f q) \u2192 map f (comp (p + q_1) q) = comp (map f (p + q_1)) (map f q)", "state_after": "no goals"}, {"tactic": "simp (config := { contextual := true }) only [pow_succ', \u2190 mul_assoc, comp, forall_const,\n  eval\u2082_mul_X, imp_true_iff, eq_self_iff_true, map_X, Polynomial.map_mul]", "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q\u271d r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\np q : R[X]\n\u22a2 \u2200 (n : \u2115) (a : R),\n    map f (comp (\u2191C a * X ^ n) q) = comp (map f (\u2191C a * X ^ n)) (map f q) \u2192\n      map f (comp (\u2191C a * X ^ (n + 1)) q) = comp (map f (\u2191C a * X ^ (n + 1))) (map f q)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Matrix/Basic.lean", "full_name": "Matrix.mul_mul_left", "start": [1226, 1], "end": [1228, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Complex/Basic.lean", "full_name": "Complex.not_lt_zero_iff", "start": [1187, 1], "end": [1188, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/FiniteType.lean", "full_name": "RingHom.FiniteType.comp", "start": [236, 1], "end": [246, 10], "traced_tactics": [{"tactic": "let _ : Algebra A B := f.toAlgebra", "state_before": "A : Type u_3\nB : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : CommRing B\ninst\u271d : CommRing C\ng : B \u2192+* C\nf : A \u2192+* B\nhg : FiniteType g\nhf : FiniteType f\n\u22a2 FiniteType (RingHom.comp g f)", "state_after": "A : Type u_3\nB : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : CommRing B\ninst\u271d : CommRing C\ng : B \u2192+* C\nf : A \u2192+* B\nhg : FiniteType g\nhf : FiniteType f\nx\u271d : Algebra A B := toAlgebra f\n\u22a2 FiniteType (RingHom.comp g f)"}, {"tactic": "let _ : Algebra A C := (g.comp f).toAlgebra", "state_before": "A : Type u_3\nB : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : CommRing B\ninst\u271d : CommRing C\ng : B \u2192+* C\nf : A \u2192+* B\nhg : FiniteType g\nhf : FiniteType f\nx\u271d : Algebra A B := toAlgebra f\n\u22a2 FiniteType (RingHom.comp g f)", "state_after": "A : Type u_3\nB : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : CommRing B\ninst\u271d : CommRing C\ng : B \u2192+* C\nf : A \u2192+* B\nhg : FiniteType g\nhf : FiniteType f\nx\u271d\u00b9 : Algebra A B := toAlgebra f\nx\u271d : Algebra A C := toAlgebra (RingHom.comp g f)\n\u22a2 FiniteType (RingHom.comp g f)"}, {"tactic": "let _ : Algebra B C := g.toAlgebra", "state_before": "A : Type u_3\nB : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : CommRing B\ninst\u271d : CommRing C\ng : B \u2192+* C\nf : A \u2192+* B\nhg : FiniteType g\nhf : FiniteType f\nx\u271d\u00b9 : Algebra A B := toAlgebra f\nx\u271d : Algebra A C := toAlgebra (RingHom.comp g f)\n\u22a2 FiniteType (RingHom.comp g f)", "state_after": "A : Type u_3\nB : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : CommRing B\ninst\u271d : CommRing C\ng : B \u2192+* C\nf : A \u2192+* B\nhg : FiniteType g\nhf : FiniteType f\nx\u271d\u00b2 : Algebra A B := toAlgebra f\nx\u271d\u00b9 : Algebra A C := toAlgebra (RingHom.comp g f)\nx\u271d : Algebra B C := toAlgebra g\n\u22a2 FiniteType (RingHom.comp g f)"}, {"tactic": "exact @Algebra.FiniteType.trans A B C _ _ f.toAlgebra _ (g.comp f).toAlgebra g.toAlgebra\n  \u27e8by\n    intro a b c\n    simp [Algebra.smul_def, RingHom.map_mul, mul_assoc]\n    rfl\u27e9\n  hf hg", "state_before": "A : Type u_3\nB : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : CommRing B\ninst\u271d : CommRing C\ng : B \u2192+* C\nf : A \u2192+* B\nhg : FiniteType g\nhf : FiniteType f\nx\u271d\u00b2 : Algebra A B := toAlgebra f\nx\u271d\u00b9 : Algebra A C := toAlgebra (RingHom.comp g f)\nx\u271d : Algebra B C := toAlgebra g\n\u22a2 FiniteType (RingHom.comp g f)", "state_after": "no goals"}, {"tactic": "intro a b c", "state_before": "A : Type u_3\nB : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : CommRing B\ninst\u271d : CommRing C\ng : B \u2192+* C\nf : A \u2192+* B\nhg : FiniteType g\nhf : FiniteType f\nx\u271d\u00b2 : Algebra A B := toAlgebra f\nx\u271d\u00b9 : Algebra A C := toAlgebra (RingHom.comp g f)\nx\u271d : Algebra B C := toAlgebra g\n\u22a2 \u2200 (x : A) (y : B) (z : C), (x \u2022 y) \u2022 z = x \u2022 y \u2022 z", "state_after": "A : Type u_3\nB : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : CommRing B\ninst\u271d : CommRing C\ng : B \u2192+* C\nf : A \u2192+* B\nhg : FiniteType g\nhf : FiniteType f\nx\u271d\u00b2 : Algebra A B := toAlgebra f\nx\u271d\u00b9 : Algebra A C := toAlgebra (RingHom.comp g f)\nx\u271d : Algebra B C := toAlgebra g\na : A\nb : B\nc : C\n\u22a2 (a \u2022 b) \u2022 c = a \u2022 b \u2022 c"}, {"tactic": "simp [Algebra.smul_def, RingHom.map_mul, mul_assoc]", "state_before": "A : Type u_3\nB : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : CommRing B\ninst\u271d : CommRing C\ng : B \u2192+* C\nf : A \u2192+* B\nhg : FiniteType g\nhf : FiniteType f\nx\u271d\u00b2 : Algebra A B := toAlgebra f\nx\u271d\u00b9 : Algebra A C := toAlgebra (RingHom.comp g f)\nx\u271d : Algebra B C := toAlgebra g\na : A\nb : B\nc : C\n\u22a2 (a \u2022 b) \u2022 c = a \u2022 b \u2022 c", "state_after": "A : Type u_3\nB : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : CommRing B\ninst\u271d : CommRing C\ng : B \u2192+* C\nf : A \u2192+* B\nhg : FiniteType g\nhf : FiniteType f\nx\u271d\u00b2 : Algebra A B := toAlgebra f\nx\u271d\u00b9 : Algebra A C := toAlgebra (RingHom.comp g f)\nx\u271d : Algebra B C := toAlgebra g\na : A\nb : B\nc : C\n\u22a2 \u2191(algebraMap B C) (\u2191(algebraMap A B) a) * (\u2191(algebraMap B C) b * c) = \u2191(algebraMap A C) a * (\u2191(algebraMap B C) b * c)"}, {"tactic": "rfl", "state_before": "A : Type u_3\nB : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : CommRing B\ninst\u271d : CommRing C\ng : B \u2192+* C\nf : A \u2192+* B\nhg : FiniteType g\nhf : FiniteType f\nx\u271d\u00b2 : Algebra A B := toAlgebra f\nx\u271d\u00b9 : Algebra A C := toAlgebra (RingHom.comp g f)\nx\u271d : Algebra B C := toAlgebra g\na : A\nb : B\nc : C\n\u22a2 \u2191(algebraMap B C) (\u2191(algebraMap A B) a) * (\u2191(algebraMap B C) b * c) = \u2191(algebraMap A C) a * (\u2191(algebraMap B C) b * c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Heyting/Basic.lean", "full_name": "compl_compl_himp_distrib", "start": [936, 1], "end": [943, 22], "traced_tactics": [{"tactic": "refine' le_antisymm _ _", "state_before": "\u03b9 : Type ?u.165630\n\u03b1 : Type u_1\n\u03b2 : Type ?u.165636\ninst\u271d : HeytingAlgebra \u03b1\na\u271d b\u271d c a b : \u03b1\n\u22a2 (a \u21e8 b)\u1d9c\u1d9c = a\u1d9c\u1d9c \u21e8 b\u1d9c\u1d9c", "state_after": "case refine'_1\n\u03b9 : Type ?u.165630\n\u03b1 : Type u_1\n\u03b2 : Type ?u.165636\ninst\u271d : HeytingAlgebra \u03b1\na\u271d b\u271d c a b : \u03b1\n\u22a2 (a \u21e8 b)\u1d9c\u1d9c \u2264 a\u1d9c\u1d9c \u21e8 b\u1d9c\u1d9c\n\ncase refine'_2\n\u03b9 : Type ?u.165630\n\u03b1 : Type u_1\n\u03b2 : Type ?u.165636\ninst\u271d : HeytingAlgebra \u03b1\na\u271d b\u271d c a b : \u03b1\n\u22a2 a\u1d9c\u1d9c \u21e8 b\u1d9c\u1d9c \u2264 (a \u21e8 b)\u1d9c\u1d9c"}, {"tactic": "rw [le_himp_iff, \u2190 compl_compl_inf_distrib]", "state_before": "case refine'_1\n\u03b9 : Type ?u.165630\n\u03b1 : Type u_1\n\u03b2 : Type ?u.165636\ninst\u271d : HeytingAlgebra \u03b1\na\u271d b\u271d c a b : \u03b1\n\u22a2 (a \u21e8 b)\u1d9c\u1d9c \u2264 a\u1d9c\u1d9c \u21e8 b\u1d9c\u1d9c", "state_after": "case refine'_1\n\u03b9 : Type ?u.165630\n\u03b1 : Type u_1\n\u03b2 : Type ?u.165636\ninst\u271d : HeytingAlgebra \u03b1\na\u271d b\u271d c a b : \u03b1\n\u22a2 ((a \u21e8 b) \u2293 a)\u1d9c\u1d9c \u2264 b\u1d9c\u1d9c"}, {"tactic": "exact compl_anti (compl_anti himp_inf_le)", "state_before": "case refine'_1\n\u03b9 : Type ?u.165630\n\u03b1 : Type u_1\n\u03b2 : Type ?u.165636\ninst\u271d : HeytingAlgebra \u03b1\na\u271d b\u271d c a b : \u03b1\n\u22a2 ((a \u21e8 b) \u2293 a)\u1d9c\u1d9c \u2264 b\u1d9c\u1d9c", "state_after": "no goals"}, {"tactic": "refine' le_compl_comm.1 ((compl_anti compl_sup_le_himp).trans _)", "state_before": "case refine'_2\n\u03b9 : Type ?u.165630\n\u03b1 : Type u_1\n\u03b2 : Type ?u.165636\ninst\u271d : HeytingAlgebra \u03b1\na\u271d b\u271d c a b : \u03b1\n\u22a2 a\u1d9c\u1d9c \u21e8 b\u1d9c\u1d9c \u2264 (a \u21e8 b)\u1d9c\u1d9c", "state_after": "case refine'_2\n\u03b9 : Type ?u.165630\n\u03b1 : Type u_1\n\u03b2 : Type ?u.165636\ninst\u271d : HeytingAlgebra \u03b1\na\u271d b\u271d c a b : \u03b1\n\u22a2 (a\u1d9c \u2294 b)\u1d9c \u2264 (a\u1d9c\u1d9c \u21e8 b\u1d9c\u1d9c)\u1d9c"}, {"tactic": "rw [compl_sup_distrib, le_compl_iff_disjoint_right, disjoint_right_comm, \u2190\n  le_compl_iff_disjoint_right]", "state_before": "case refine'_2\n\u03b9 : Type ?u.165630\n\u03b1 : Type u_1\n\u03b2 : Type ?u.165636\ninst\u271d : HeytingAlgebra \u03b1\na\u271d b\u271d c a b : \u03b1\n\u22a2 (a\u1d9c \u2294 b)\u1d9c \u2264 (a\u1d9c\u1d9c \u21e8 b\u1d9c\u1d9c)\u1d9c", "state_after": "case refine'_2\n\u03b9 : Type ?u.165630\n\u03b1 : Type u_1\n\u03b2 : Type ?u.165636\ninst\u271d : HeytingAlgebra \u03b1\na\u271d b\u271d c a b : \u03b1\n\u22a2 a\u1d9c\u1d9c \u2293 (a\u1d9c\u1d9c \u21e8 b\u1d9c\u1d9c) \u2264 b\u1d9c\u1d9c"}, {"tactic": "exact inf_himp_le", "state_before": "case refine'_2\n\u03b9 : Type ?u.165630\n\u03b1 : Type u_1\n\u03b2 : Type ?u.165636\ninst\u271d : HeytingAlgebra \u03b1\na\u271d b\u271d c a b : \u03b1\n\u22a2 a\u1d9c\u1d9c \u2293 (a\u1d9c\u1d9c \u21e8 b\u1d9c\u1d9c) \u2264 b\u1d9c\u1d9c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/SuccPred/Basic.lean", "full_name": "Order.Iic_pred", "start": [735, 1], "end": [736, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/Independent.lean", "full_name": "AffineIndependent.injective", "start": [279, 11], "end": [285, 24], "traced_tactics": [{"tactic": "intro i j hij", "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nha : AffineIndependent k p\n\u22a2 Injective p", "state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ni j : \u03b9\nhij : p i = p j\n\u22a2 i = j"}, {"tactic": "rw [affineIndependent_iff_linearIndependent_vsub _ _ j] at ha", "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\nha : AffineIndependent k p\ni j : \u03b9\nhij : p i = p j\n\u22a2 i = j", "state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\ni j : \u03b9\nha : LinearIndependent k fun i => p \u2191i -\u1d65 p j\nhij : p i = p j\n\u22a2 i = j"}, {"tactic": "by_contra hij'", "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\ni j : \u03b9\nha : LinearIndependent k fun i => p \u2191i -\u1d65 p j\nhij : p i = p j\n\u22a2 i = j", "state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\ni j : \u03b9\nha : LinearIndependent k fun i => p \u2191i -\u1d65 p j\nhij : p i = p j\nhij' : \u00aci = j\n\u22a2 False"}, {"tactic": "refine' ha.ne_zero \u27e8i, hij'\u27e9 (vsub_eq_zero_iff_eq.mpr _)", "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\ni j : \u03b9\nha : LinearIndependent k fun i => p \u2191i -\u1d65 p j\nhij : p i = p j\nhij' : \u00aci = j\n\u22a2 False", "state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\ni j : \u03b9\nha : LinearIndependent k fun i => p \u2191i -\u1d65 p j\nhij : p i = p j\nhij' : \u00aci = j\n\u22a2 p \u2191{ val := i, property := hij' } = p j"}, {"tactic": "simp_all only [ne_eq]", "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d : Nontrivial k\np : \u03b9 \u2192 P\ni j : \u03b9\nha : LinearIndependent k fun i => p \u2191i -\u1d65 p j\nhij : p i = p j\nhij' : \u00aci = j\n\u22a2 p \u2191{ val := i, property := hij' } = p j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GeomSum.lean", "full_name": "Commute.geom_sum\u2082_succ_eq", "start": [292, 11], "end": [303, 62], "traced_tactics": [{"tactic": "simp_rw [mul_sum, sum_range_succ_comm, tsub_self, pow_zero, mul_one, add_right_inj, \u2190 mul_assoc,\n  (h.symm.pow_right _).eq, mul_assoc, \u2190 pow_succ]", "state_before": "\u03b1\u271d \u03b1 : Type u\ninst\u271d : Ring \u03b1\nx y : \u03b1\nh : Commute x y\nn : \u2115\n\u22a2 \u2211 i in range (n + 1), x ^ i * y ^ (n - i) = x ^ n + y * \u2211 i in range n, x ^ i * y ^ (n - 1 - i)", "state_after": "\u03b1\u271d \u03b1 : Type u\ninst\u271d : Ring \u03b1\nx y : \u03b1\nh : Commute x y\nn : \u2115\n\u22a2 \u2211 i in range n, x ^ i * y ^ (n - i) = \u2211 x_1 in range n, x ^ x_1 * y ^ (n - 1 - x_1 + 1)"}, {"tactic": "refine' sum_congr rfl fun i hi => _", "state_before": "\u03b1\u271d \u03b1 : Type u\ninst\u271d : Ring \u03b1\nx y : \u03b1\nh : Commute x y\nn : \u2115\n\u22a2 \u2211 i in range n, x ^ i * y ^ (n - i) = \u2211 x_1 in range n, x ^ x_1 * y ^ (n - 1 - x_1 + 1)", "state_after": "\u03b1\u271d \u03b1 : Type u\ninst\u271d : Ring \u03b1\nx y : \u03b1\nh : Commute x y\nn i : \u2115\nhi : i \u2208 range n\n\u22a2 x ^ i * y ^ (n - i) = x ^ i * y ^ (n - 1 - i + 1)"}, {"tactic": "suffices n - 1 - i + 1 = n - i by rw [this]", "state_before": "\u03b1\u271d \u03b1 : Type u\ninst\u271d : Ring \u03b1\nx y : \u03b1\nh : Commute x y\nn i : \u2115\nhi : i \u2208 range n\n\u22a2 x ^ i * y ^ (n - i) = x ^ i * y ^ (n - 1 - i + 1)", "state_after": "\u03b1\u271d \u03b1 : Type u\ninst\u271d : Ring \u03b1\nx y : \u03b1\nh : Commute x y\nn i : \u2115\nhi : i \u2208 range n\n\u22a2 n - 1 - i + 1 = n - i"}, {"tactic": "cases' n with n", "state_before": "\u03b1\u271d \u03b1 : Type u\ninst\u271d : Ring \u03b1\nx y : \u03b1\nh : Commute x y\nn i : \u2115\nhi : i \u2208 range n\n\u22a2 n - 1 - i + 1 = n - i", "state_after": "case zero\n\u03b1\u271d \u03b1 : Type u\ninst\u271d : Ring \u03b1\nx y : \u03b1\nh : Commute x y\ni : \u2115\nhi : i \u2208 range Nat.zero\n\u22a2 Nat.zero - 1 - i + 1 = Nat.zero - i\n\ncase succ\n\u03b1\u271d \u03b1 : Type u\ninst\u271d : Ring \u03b1\nx y : \u03b1\nh : Commute x y\ni n : \u2115\nhi : i \u2208 range (Nat.succ n)\n\u22a2 Nat.succ n - 1 - i + 1 = Nat.succ n - i"}, {"tactic": "rw [this]", "state_before": "\u03b1\u271d \u03b1 : Type u\ninst\u271d : Ring \u03b1\nx y : \u03b1\nh : Commute x y\nn i : \u2115\nhi : i \u2208 range n\nthis : n - 1 - i + 1 = n - i\n\u22a2 x ^ i * y ^ (n - i) = x ^ i * y ^ (n - 1 - i + 1)", "state_after": "no goals"}, {"tactic": "exact absurd (List.mem_range.mp hi) i.not_lt_zero", "state_before": "case zero\n\u03b1\u271d \u03b1 : Type u\ninst\u271d : Ring \u03b1\nx y : \u03b1\nh : Commute x y\ni : \u2115\nhi : i \u2208 range Nat.zero\n\u22a2 Nat.zero - 1 - i + 1 = Nat.zero - i", "state_after": "no goals"}, {"tactic": "rw [tsub_add_eq_add_tsub (Nat.le_pred_of_lt (List.mem_range.mp hi)),\n  tsub_add_cancel_of_le (Nat.succ_le_iff.mpr n.succ_pos)]", "state_before": "case succ\n\u03b1\u271d \u03b1 : Type u\ninst\u271d : Ring \u03b1\nx y : \u03b1\nh : Commute x y\ni n : \u2115\nhi : i \u2208 range (Nat.succ n)\n\u22a2 Nat.succ n - 1 - i + 1 = Nat.succ n - i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/TrivSqZeroExt.lean", "full_name": "TrivSqZeroExt.fst_sum", "start": [297, 1], "end": [299, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Logic.lean", "full_name": "exists_imp", "start": [367, 1], "end": [367, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Prime.lean", "full_name": "Nat.Prime.not_coprime_iff_dvd", "start": [553, 1], "end": [561, 74], "traced_tactics": [{"tactic": "apply Iff.intro", "state_before": "m n : \u2115\n\u22a2 \u00accoprime m n \u2194 \u2203 p, Prime p \u2227 p \u2223 m \u2227 p \u2223 n", "state_after": "case mp\nm n : \u2115\n\u22a2 \u00accoprime m n \u2192 \u2203 p, Prime p \u2227 p \u2223 m \u2227 p \u2223 n\n\ncase mpr\nm n : \u2115\n\u22a2 (\u2203 p, Prime p \u2227 p \u2223 m \u2227 p \u2223 n) \u2192 \u00accoprime m n"}, {"tactic": "intro h", "state_before": "case mp\nm n : \u2115\n\u22a2 \u00accoprime m n \u2192 \u2203 p, Prime p \u2227 p \u2223 m \u2227 p \u2223 n", "state_after": "case mp\nm n : \u2115\nh : \u00accoprime m n\n\u22a2 \u2203 p, Prime p \u2227 p \u2223 m \u2227 p \u2223 n"}, {"tactic": "exact\n  \u27e8minFac (gcd m n), minFac_prime h, (minFac_dvd (gcd m n)).trans (gcd_dvd_left m n),\n    (minFac_dvd (gcd m n)).trans (gcd_dvd_right m n)\u27e9", "state_before": "case mp\nm n : \u2115\nh : \u00accoprime m n\n\u22a2 \u2203 p, Prime p \u2227 p \u2223 m \u2227 p \u2223 n", "state_after": "no goals"}, {"tactic": "intro h", "state_before": "case mpr\nm n : \u2115\n\u22a2 (\u2203 p, Prime p \u2227 p \u2223 m \u2227 p \u2223 n) \u2192 \u00accoprime m n", "state_after": "case mpr\nm n : \u2115\nh : \u2203 p, Prime p \u2227 p \u2223 m \u2227 p \u2223 n\n\u22a2 \u00accoprime m n"}, {"tactic": "cases' h with p hp", "state_before": "case mpr\nm n : \u2115\nh : \u2203 p, Prime p \u2227 p \u2223 m \u2227 p \u2223 n\n\u22a2 \u00accoprime m n", "state_after": "case mpr.intro\nm n p : \u2115\nhp : Prime p \u2227 p \u2223 m \u2227 p \u2223 n\n\u22a2 \u00accoprime m n"}, {"tactic": "apply Nat.not_coprime_of_dvd_of_dvd (Prime.one_lt hp.1) hp.2.1 hp.2.2", "state_before": "case mpr.intro\nm n p : \u2115\nhp : Prime p \u2227 p \u2223 m \u2227 p \u2223 n\n\u22a2 \u00accoprime m n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/FreeGroup.lean", "full_name": "FreeGroup.Red.enum.complete", "start": [1387, 1], "end": [1388, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Rearrangement.lean", "full_name": "Antivary.sum_smul_le_sum_smul_comp_perm", "start": [293, 1], "end": [295, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Module/LocalizedModule.lean", "full_name": "LocalizedModule.smul_add'", "start": [353, 9], "end": [362, 10], "traced_tactics": [{"tactic": "induction' x using Localization.induction_on with data", "state_before": "R : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Submonoid R\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nx : Localization S\ny z : LocalizedModule S M\n\u22a2 x \u2022 (y + z) = x \u2022 y + x \u2022 z", "state_after": "case H\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Submonoid R\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ny z : LocalizedModule S M\ndata : R \u00d7 { x // x \u2208 S }\n\u22a2 Localization.mk data.fst data.snd \u2022 (y + z) =\n    Localization.mk data.fst data.snd \u2022 y + Localization.mk data.fst data.snd \u2022 z"}, {"tactic": "rcases data with \u27e8r, u\u27e9", "state_before": "case H\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Submonoid R\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ny z : LocalizedModule S M\ndata : R \u00d7 { x // x \u2208 S }\n\u22a2 Localization.mk data.fst data.snd \u2022 (y + z) =\n    Localization.mk data.fst data.snd \u2022 y + Localization.mk data.fst data.snd \u2022 z", "state_after": "case H.mk\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Submonoid R\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ny z : LocalizedModule S M\nr : R\nu : { x // x \u2208 S }\n\u22a2 Localization.mk (r, u).fst (r, u).snd \u2022 (y + z) =\n    Localization.mk (r, u).fst (r, u).snd \u2022 y + Localization.mk (r, u).fst (r, u).snd \u2022 z"}, {"tactic": "induction' y using LocalizedModule.induction_on with m s", "state_before": "case H.mk\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Submonoid R\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\ny z : LocalizedModule S M\nr : R\nu : { x // x \u2208 S }\n\u22a2 Localization.mk (r, u).fst (r, u).snd \u2022 (y + z) =\n    Localization.mk (r, u).fst (r, u).snd \u2022 y + Localization.mk (r, u).fst (r, u).snd \u2022 z", "state_after": "case H.mk.h\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Submonoid R\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nz : LocalizedModule S M\nr : R\nu : { x // x \u2208 S }\nm : M\ns : { x // x \u2208 S }\n\u22a2 Localization.mk (r, u).fst (r, u).snd \u2022 (mk m s + z) =\n    Localization.mk (r, u).fst (r, u).snd \u2022 mk m s + Localization.mk (r, u).fst (r, u).snd \u2022 z"}, {"tactic": "induction' z using LocalizedModule.induction_on with n t", "state_before": "case H.mk.h\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Submonoid R\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nz : LocalizedModule S M\nr : R\nu : { x // x \u2208 S }\nm : M\ns : { x // x \u2208 S }\n\u22a2 Localization.mk (r, u).fst (r, u).snd \u2022 (mk m s + z) =\n    Localization.mk (r, u).fst (r, u).snd \u2022 mk m s + Localization.mk (r, u).fst (r, u).snd \u2022 z", "state_after": "case H.mk.h.h\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Submonoid R\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nr : R\nu : { x // x \u2208 S }\nm : M\ns : { x // x \u2208 S }\nn : M\nt : { x // x \u2208 S }\n\u22a2 Localization.mk (r, u).fst (r, u).snd \u2022 (mk m s + mk n t) =\n    Localization.mk (r, u).fst (r, u).snd \u2022 mk m s + Localization.mk (r, u).fst (r, u).snd \u2022 mk n t"}, {"tactic": "rw [mk_smul_mk, mk_smul_mk, mk_add_mk, mk_smul_mk, mk_add_mk, mk_eq]", "state_before": "case H.mk.h.h\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Submonoid R\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nr : R\nu : { x // x \u2208 S }\nm : M\ns : { x // x \u2208 S }\nn : M\nt : { x // x \u2208 S }\n\u22a2 Localization.mk (r, u).fst (r, u).snd \u2022 (mk m s + mk n t) =\n    Localization.mk (r, u).fst (r, u).snd \u2022 mk m s + Localization.mk (r, u).fst (r, u).snd \u2022 mk n t", "state_after": "case H.mk.h.h\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Submonoid R\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nr : R\nu : { x // x \u2208 S }\nm : M\ns : { x // x \u2208 S }\nn : M\nt : { x // x \u2208 S }\n\u22a2 \u2203 u_1,\n    u_1 \u2022 ((r, u).snd * s * ((r, u).snd * t)) \u2022 (r, u).fst \u2022 (t \u2022 m + s \u2022 n) =\n      u_1 \u2022 ((r, u).snd * (s * t)) \u2022 (((r, u).snd * t) \u2022 (r, u).fst \u2022 m + ((r, u).snd * s) \u2022 (r, u).fst \u2022 n)"}, {"tactic": "use 1", "state_before": "case H.mk.h.h\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Submonoid R\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nr : R\nu : { x // x \u2208 S }\nm : M\ns : { x // x \u2208 S }\nn : M\nt : { x // x \u2208 S }\n\u22a2 \u2203 u_1,\n    u_1 \u2022 ((r, u).snd * s * ((r, u).snd * t)) \u2022 (r, u).fst \u2022 (t \u2022 m + s \u2022 n) =\n      u_1 \u2022 ((r, u).snd * (s * t)) \u2022 (((r, u).snd * t) \u2022 (r, u).fst \u2022 m + ((r, u).snd * s) \u2022 (r, u).fst \u2022 n)", "state_after": "case H.mk.h.h\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Submonoid R\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nr : R\nu : { x // x \u2208 S }\nm : M\ns : { x // x \u2208 S }\nn : M\nt : { x // x \u2208 S }\n\u22a2 1 \u2022 ((r, u).snd * s * ((r, u).snd * t)) \u2022 (r, u).fst \u2022 (t \u2022 m + s \u2022 n) =\n    1 \u2022 ((r, u).snd * (s * t)) \u2022 (((r, u).snd * t) \u2022 (r, u).fst \u2022 m + ((r, u).snd * s) \u2022 (r, u).fst \u2022 n)"}, {"tactic": "simp only [one_smul, smul_add, \u2190 mul_smul, Submonoid.smul_def, Submonoid.coe_mul]", "state_before": "case H.mk.h.h\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Submonoid R\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nr : R\nu : { x // x \u2208 S }\nm : M\ns : { x // x \u2208 S }\nn : M\nt : { x // x \u2208 S }\n\u22a2 1 \u2022 ((r, u).snd * s * ((r, u).snd * t)) \u2022 (r, u).fst \u2022 (t \u2022 m + s \u2022 n) =\n    1 \u2022 ((r, u).snd * (s * t)) \u2022 (((r, u).snd * t) \u2022 (r, u).fst \u2022 m + ((r, u).snd * s) \u2022 (r, u).fst \u2022 n)", "state_after": "case H.mk.h.h\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Submonoid R\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nr : R\nu : { x // x \u2208 S }\nm : M\ns : { x // x \u2208 S }\nn : M\nt : { x // x \u2208 S }\n\u22a2 (\u2191u * \u2191s * (\u2191u * \u2191t) * (r * \u2191t)) \u2022 m + (\u2191u * \u2191s * (\u2191u * \u2191t) * (r * \u2191s)) \u2022 n =\n    (\u2191u * (\u2191s * \u2191t) * (\u2191u * \u2191t * r)) \u2022 m + (\u2191u * (\u2191s * \u2191t) * (\u2191u * \u2191s * r)) \u2022 n"}, {"tactic": "ring_nf", "state_before": "case H.mk.h.h\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Submonoid R\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nr : R\nu : { x // x \u2208 S }\nm : M\ns : { x // x \u2208 S }\nn : M\nt : { x // x \u2208 S }\n\u22a2 (\u2191u * \u2191s * (\u2191u * \u2191t) * (r * \u2191t)) \u2022 m + (\u2191u * \u2191s * (\u2191u * \u2191t) * (r * \u2191s)) \u2022 n =\n    (\u2191u * (\u2191s * \u2191t) * (\u2191u * \u2191t * r)) \u2022 m + (\u2191u * (\u2191s * \u2191t) * (\u2191u * \u2191s * r)) \u2022 n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Order/IntermediateValue.lean", "full_name": "IsPreconnected.Ioi_csInf_subset", "start": [263, 1], "end": [268, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Order/Floor.lean", "full_name": "tendsto_fract_right'", "start": [180, 1], "end": [182, 98], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.iUnion_smul_right_image", "start": [237, 1], "end": [238, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/Equicontinuity.lean", "full_name": "Equicontinuous.continuous", "start": [166, 1], "end": [168, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "full_name": "PowerSeries.smul_eq_C_mul", "start": [1514, 1], "end": [1516, 7], "traced_tactics": [{"tactic": "ext", "state_before": "R : Type u_1\ninst\u271d : Semiring R\nf : PowerSeries R\na : R\n\u22a2 a \u2022 f = \u2191(C R) a * f", "state_after": "case h\nR : Type u_1\ninst\u271d : Semiring R\nf : PowerSeries R\na : R\nn\u271d : \u2115\n\u22a2 \u2191(coeff R n\u271d) (a \u2022 f) = \u2191(coeff R n\u271d) (\u2191(C R) a * f)"}, {"tactic": "simp", "state_before": "case h\nR : Type u_1\ninst\u271d : Semiring R\nf : PowerSeries R\na : R\nn\u271d : \u2115\n\u22a2 \u2191(coeff R n\u271d) (a \u2022 f) = \u2191(coeff R n\u271d) (\u2191(C R) a * f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Infsep.lean", "full_name": "Set.einfsep_of_fintype", "start": [176, 1], "end": [180, 46], "traced_tactics": [{"tactic": "refine' eq_of_forall_le_iff fun _ => _", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.30517\ninst\u271d\u00b2 : EDist \u03b1\nx y : \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u2191s\n\u22a2 einfsep s = Finset.inf (toFinset (offDiag s)) (uncurry edist)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.30517\ninst\u271d\u00b2 : EDist \u03b1\nx y : \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u2191s\nx\u271d : \u211d\u22650\u221e\n\u22a2 x\u271d \u2264 einfsep s \u2194 x\u271d \u2264 Finset.inf (toFinset (offDiag s)) (uncurry edist)"}, {"tactic": "simp_rw [le_einfsep_iff, imp_forall_iff, Finset.le_inf_iff, mem_toFinset, mem_offDiag,\n  Prod.forall, uncurry_apply_pair, and_imp]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.30517\ninst\u271d\u00b2 : EDist \u03b1\nx y : \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u2191s\nx\u271d : \u211d\u22650\u221e\n\u22a2 x\u271d \u2264 einfsep s \u2194 x\u271d \u2264 Finset.inf (toFinset (offDiag s)) (uncurry edist)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/PrimeIdeal.lean", "full_name": "Order.Ideal.PrimePair.compl_I_eq_F", "start": [60, 1], "end": [61, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/GroupAction/Group.lean", "full_name": "MulAction.surjective", "start": [159, 11], "end": [160, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.erase_insert", "start": [1913, 1], "end": [1914, 52], "traced_tactics": [{"tactic": "rw [erase_insert_eq_erase, erase_eq_of_not_mem h]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.209994\n\u03b3 : Type ?u.209997\ninst\u271d : DecidableEq \u03b1\ns\u271d t u v : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\nh : \u00aca \u2208 s\n\u22a2 erase (insert a s) a = s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Matrix/Kronecker.lean", "full_name": "Matrix.kronecker_assoc'", "start": [372, 1], "end": [375, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Range/Lemmas.lean", "full_name": "Std.Range.forIn_eq_forIn_range'", "start": [88, 1], "end": [100, 68], "traced_tactics": [{"tactic": "refine Eq.trans ?_ <| (forIn'_eq_forIn_range' r init (fun x _ => f x)).trans ?_", "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 forIn r init f = forIn (List.range' r.start (numElems r) r.step) init f", "state_after": "case refine_1\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 forIn r init f = forIn' r init fun x x_1 => f x\n\ncase refine_2\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 (forIn\n      (List.pmap Subtype.mk (List.range' r.start (numElems r) r.step)\n        (_ : \u2200 (x : Nat), x \u2208 List.range' r.start (numElems r) r.step \u2192 x \u2208 r))\n      init fun x =>\n      match (motive := { x // x \u2208 r } \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) x with\n      | { val := a, property := h } => f a) =\n    forIn (List.range' r.start (numElems r) r.step) init f"}, {"tactic": "simp [forIn, forIn', Range.forIn, Range.forIn']", "state_before": "case refine_1\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 forIn r init f = forIn' r init fun x x_1 => f x", "state_after": "case refine_1\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 forIn.loop f r.stop r.start r.stop r.step init =\n    forIn'.loop r.start r.stop r.step (fun x x_1 => f x) r.stop r.start (_ : r.start \u2264 r.start) init"}, {"tactic": "suffices \u2200 fuel i hl b, forIn'.loop r.start r.stop r.step (fun x _ => f x) fuel i hl b =\n    forIn.loop f fuel i r.stop r.step b from (this _ ..).symm", "state_before": "case refine_1\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 forIn.loop f r.stop r.start r.stop r.step init =\n    forIn'.loop r.start r.stop r.step (fun x x_1 => f x) r.stop r.start (_ : r.start \u2264 r.start) init", "state_after": "case refine_1\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 \u2200 (fuel i : Nat) (hl : r.start \u2264 i) (b : \u03b2),\n    forIn'.loop r.start r.stop r.step (fun x x_1 => f x) fuel i hl b = forIn.loop f fuel i r.stop r.step b"}, {"tactic": "intro fuel", "state_before": "case refine_1\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 \u2200 (fuel i : Nat) (hl : r.start \u2264 i) (b : \u03b2),\n    forIn'.loop r.start r.stop r.step (fun x x_1 => f x) fuel i hl b = forIn.loop f fuel i r.stop r.step b", "state_after": "case refine_1\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nfuel : Nat\n\u22a2 \u2200 (i : Nat) (hl : r.start \u2264 i) (b : \u03b2),\n    forIn'.loop r.start r.stop r.step (fun x x_1 => f x) fuel i hl b = forIn.loop f fuel i r.stop r.step b"}, {"tactic": "induction fuel <;> intro i hl b <;>\nunfold forIn.loop forIn'.loop <;> simp [*] <;> split <;> simp", "state_before": "case refine_1\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nfuel : Nat\n\u22a2 \u2200 (i : Nat) (hl : r.start \u2264 i) (b : \u03b2),\n    forIn'.loop r.start r.stop r.step (fun x x_1 => f x) fuel i hl b = forIn.loop f fuel i r.stop r.step b", "state_after": "case refine_1.succ.inl\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nn\u271d : Nat\nn_ih\u271d :\n  \u2200 (i : Nat) (hl : r.start \u2264 i) (b : \u03b2),\n    forIn'.loop r.start r.stop r.step (fun x x_1 => f x) n\u271d i hl b = forIn.loop f n\u271d i r.stop r.step b\ni : Nat\nhl : r.start \u2264 i\nb : \u03b2\nh\u271d : i < r.stop\n\u22a2 (do\n      let __do_lift \u2190 f i b\n      match __do_lift with\n        | ForInStep.done b => pure b\n        | ForInStep.yield b => forIn.loop f n\u271d (i + r.step) r.stop r.step b) =\n    if i \u2265 r.stop then pure b\n    else do\n      let __do_lift \u2190 f i b\n      match __do_lift with\n        | ForInStep.done b => pure b\n        | ForInStep.yield b => forIn.loop f n\u271d (i + r.step) r.stop r.step b\n\ncase refine_1.succ.inr\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nn\u271d : Nat\nn_ih\u271d :\n  \u2200 (i : Nat) (hl : r.start \u2264 i) (b : \u03b2),\n    forIn'.loop r.start r.stop r.step (fun x x_1 => f x) n\u271d i hl b = forIn.loop f n\u271d i r.stop r.step b\ni : Nat\nhl : r.start \u2264 i\nb : \u03b2\nh\u271d : \u00aci < r.stop\n\u22a2 pure b =\n    if i \u2265 r.stop then pure b\n    else do\n      let __do_lift \u2190 f i b\n      match __do_lift with\n        | ForInStep.done b => pure b\n        | ForInStep.yield b => forIn.loop f n\u271d (i + r.step) r.stop r.step b"}, {"tactic": "simp [if_neg (Nat.not_le.2 \u2039_\u203a)]", "state_before": "case refine_1.succ.inl\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nn\u271d : Nat\nn_ih\u271d :\n  \u2200 (i : Nat) (hl : r.start \u2264 i) (b : \u03b2),\n    forIn'.loop r.start r.stop r.step (fun x x_1 => f x) n\u271d i hl b = forIn.loop f n\u271d i r.stop r.step b\ni : Nat\nhl : r.start \u2264 i\nb : \u03b2\nh\u271d : i < r.stop\n\u22a2 (do\n      let __do_lift \u2190 f i b\n      match __do_lift with\n        | ForInStep.done b => pure b\n        | ForInStep.yield b => forIn.loop f n\u271d (i + r.step) r.stop r.step b) =\n    if i \u2265 r.stop then pure b\n    else do\n      let __do_lift \u2190 f i b\n      match __do_lift with\n        | ForInStep.done b => pure b\n        | ForInStep.yield b => forIn.loop f n\u271d (i + r.step) r.stop r.step b", "state_after": "no goals"}, {"tactic": "simp [if_pos (Nat.not_lt.1 \u2039_\u203a)]", "state_before": "case refine_1.succ.inr\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nn\u271d : Nat\nn_ih\u271d :\n  \u2200 (i : Nat) (hl : r.start \u2264 i) (b : \u03b2),\n    forIn'.loop r.start r.stop r.step (fun x x_1 => f x) n\u271d i hl b = forIn.loop f n\u271d i r.stop r.step b\ni : Nat\nhl : r.start \u2264 i\nb : \u03b2\nh\u271d : \u00aci < r.stop\n\u22a2 pure b =\n    if i \u2265 r.stop then pure b\n    else do\n      let __do_lift \u2190 f i b\n      match __do_lift with\n        | ForInStep.done b => pure b\n        | ForInStep.yield b => forIn.loop f n\u271d (i + r.step) r.stop r.step b", "state_after": "no goals"}, {"tactic": "intro L", "state_before": "case refine_2\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 \u2200 (L : List Nat) (H : \u2200 (a : Nat), a \u2208 L \u2192 a \u2208 r),\n    (forIn (List.pmap Subtype.mk L H) init fun x => f x.val) = forIn L init f", "state_after": "case refine_2\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat\n\u22a2 \u2200 (H : \u2200 (a : Nat), a \u2208 L \u2192 a \u2208 r), (forIn (List.pmap Subtype.mk L H) init fun x => f x.val) = forIn L init f"}, {"tactic": "induction L generalizing init <;> intro H <;> simp [*]", "state_before": "case refine_2\nm : Type u_1 \u2192 Type u_2\n\u03b2 : Type u_1\ninst\u271d : Monad m\nr : Range\ninit : \u03b2\nf : Nat \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nL : List Nat\n\u22a2 \u2200 (H : \u2200 (a : Nat), a \u2208 L \u2192 a \u2208 r), (forIn (List.pmap Subtype.mk L H) init fun x => f x.val) = forIn L init f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/LocalRing.lean", "full_name": "LocalRing.of_nonunits_add", "start": [63, 1], "end": [65, 90], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/RootsOfUnity/Basic.lean", "full_name": "IsPrimitiveRoot.card_rootsOfUnity'", "start": [801, 1], "end": [809, 25], "traced_tactics": [{"tactic": "let e := h.zmodEquivZpowers", "state_before": "M : Type ?u.3542232\nN : Type ?u.3542235\nG : Type ?u.3542238\nR : Type u_1\nS : Type ?u.3542244\nF : Type ?u.3542247\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6 : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6 k\ninst\u271d : IsDomain R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 Fintype.card { x // x \u2208 rootsOfUnity n R } = \u2191n", "state_after": "M : Type ?u.3542232\nN : Type ?u.3542235\nG : Type ?u.3542238\nR : Type u_1\nS : Type ?u.3542244\nF : Type ?u.3542247\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6 : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6 k\ninst\u271d : IsDomain R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\ne : ZMod \u2191n \u2243+ Additive { x // x \u2208 Subgroup.zpowers \u03b6 } := zmodEquivZpowers h\n\u22a2 Fintype.card { x // x \u2208 rootsOfUnity n R } = \u2191n"}, {"tactic": "haveI F : Fintype (Subgroup.zpowers \u03b6) := Fintype.ofEquiv _ e.toEquiv", "state_before": "M : Type ?u.3542232\nN : Type ?u.3542235\nG : Type ?u.3542238\nR : Type u_1\nS : Type ?u.3542244\nF : Type ?u.3542247\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6 : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6 k\ninst\u271d : IsDomain R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\ne : ZMod \u2191n \u2243+ Additive { x // x \u2208 Subgroup.zpowers \u03b6 } := zmodEquivZpowers h\n\u22a2 Fintype.card { x // x \u2208 rootsOfUnity n R } = \u2191n", "state_after": "M : Type ?u.3542232\nN : Type ?u.3542235\nG : Type ?u.3542238\nR : Type u_1\nS : Type ?u.3542244\nF\u271d : Type ?u.3542247\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6 : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6 k\ninst\u271d : IsDomain R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\ne : ZMod \u2191n \u2243+ Additive { x // x \u2208 Subgroup.zpowers \u03b6 } := zmodEquivZpowers h\nF : Fintype { x // x \u2208 Subgroup.zpowers \u03b6 }\n\u22a2 Fintype.card { x // x \u2208 rootsOfUnity n R } = \u2191n"}, {"tactic": "calc\n  Fintype.card (rootsOfUnity n R) = Fintype.card (Subgroup.zpowers \u03b6) :=\n    Fintype.card_congr <| by rw [h.zpowers_eq]\n  _ = Fintype.card (ZMod n) := (Fintype.card_congr e.toEquiv.symm)\n  _ = n := ZMod.card n", "state_before": "M : Type ?u.3542232\nN : Type ?u.3542235\nG : Type ?u.3542238\nR : Type u_1\nS : Type ?u.3542244\nF\u271d : Type ?u.3542247\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6 : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6 k\ninst\u271d : IsDomain R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\ne : ZMod \u2191n \u2243+ Additive { x // x \u2208 Subgroup.zpowers \u03b6 } := zmodEquivZpowers h\nF : Fintype { x // x \u2208 Subgroup.zpowers \u03b6 }\n\u22a2 Fintype.card { x // x \u2208 rootsOfUnity n R } = \u2191n", "state_after": "no goals"}, {"tactic": "rw [h.zpowers_eq]", "state_before": "M : Type ?u.3542232\nN : Type ?u.3542235\nG : Type ?u.3542238\nR : Type u_1\nS : Type ?u.3542244\nF\u271d : Type ?u.3542247\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6 : R\u02e3\nh\u271d : IsPrimitiveRoot \u03b6 k\ninst\u271d : IsDomain R\nn : \u2115+\nh : IsPrimitiveRoot \u03b6 \u2191n\ne : ZMod \u2191n \u2243+ Additive { x // x \u2208 Subgroup.zpowers \u03b6 } := zmodEquivZpowers h\nF : Fintype { x // x \u2208 Subgroup.zpowers \u03b6 }\n\u22a2 { x // x \u2208 rootsOfUnity n R } \u2243 { x // x \u2208 Subgroup.zpowers \u03b6 }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "full_name": "mul_nonpos_iff", "start": [1092, 1], "end": [1093, 80], "traced_tactics": [{"tactic": "rw [\u2190 neg_nonneg, neg_mul_eq_mul_neg, mul_nonneg_iff, neg_nonneg, neg_nonpos]", "state_before": "\u03b1 : Type u\n\u03b2 : Type ?u.181819\ninst\u271d : LinearOrderedRing \u03b1\na b c : \u03b1\n\u22a2 a * b \u2264 0 \u2194 0 \u2264 a \u2227 b \u2264 0 \u2228 a \u2264 0 \u2227 0 \u2264 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Homotopy/Basic.lean", "full_name": "ContinuousMap.HomotopyWith.extendProp", "start": [476, 1], "end": [476, 94], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "full_name": "SimpleGraph.ConnectedComponent.supp_inj", "start": [2148, 1], "end": [2149, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Init/Data/Int/Order.lean", "full_name": "Int.eq_zero_or_eq_zero_of_mul_eq_zero", "start": [76, 11], "end": [100, 38], "traced_tactics": [{"tactic": "have : 0 < a * b := Int.mul_pos hlt\u2081 hlt\u2082", "state_before": "a b : \u2124\nh : a * b = 0\nhlt\u2081 : 0 < a\nhlt\u2082 : 0 < b\n\u22a2 a = 0 \u2228 b = 0", "state_after": "a b : \u2124\nh : a * b = 0\nhlt\u2081 : 0 < a\nhlt\u2082 : 0 < b\nthis : 0 < a * b\n\u22a2 a = 0 \u2228 b = 0"}, {"tactic": "rw [h] at this", "state_before": "a b : \u2124\nh : a * b = 0\nhlt\u2081 : 0 < a\nhlt\u2082 : 0 < b\nthis : 0 < a * b\n\u22a2 a = 0 \u2228 b = 0", "state_after": "a b : \u2124\nh : a * b = 0\nhlt\u2081 : 0 < a\nhlt\u2082 : 0 < b\nthis : 0 < 0\n\u22a2 a = 0 \u2228 b = 0"}, {"tactic": "exact absurd this (lt_irrefl _)", "state_before": "a b : \u2124\nh : a * b = 0\nhlt\u2081 : 0 < a\nhlt\u2082 : 0 < b\nthis : 0 < 0\n\u22a2 a = 0 \u2228 b = 0", "state_after": "no goals"}, {"tactic": "have : 0 > a * b := Int.mul_neg_of_pos_of_neg hlt\u2081 hgt\u2082", "state_before": "a b : \u2124\nh : a * b = 0\nhlt\u2081 : 0 < a\nhgt\u2082 : b < 0\n\u22a2 a = 0 \u2228 b = 0", "state_after": "a b : \u2124\nh : a * b = 0\nhlt\u2081 : 0 < a\nhgt\u2082 : b < 0\nthis : 0 > a * b\n\u22a2 a = 0 \u2228 b = 0"}, {"tactic": "rw [h] at this", "state_before": "a b : \u2124\nh : a * b = 0\nhlt\u2081 : 0 < a\nhgt\u2082 : b < 0\nthis : 0 > a * b\n\u22a2 a = 0 \u2228 b = 0", "state_after": "a b : \u2124\nh : a * b = 0\nhlt\u2081 : 0 < a\nhgt\u2082 : b < 0\nthis : 0 > 0\n\u22a2 a = 0 \u2228 b = 0"}, {"tactic": "exact absurd this (lt_irrefl _)", "state_before": "a b : \u2124\nh : a * b = 0\nhlt\u2081 : 0 < a\nhgt\u2082 : b < 0\nthis : 0 > 0\n\u22a2 a = 0 \u2228 b = 0", "state_after": "no goals"}, {"tactic": "have : 0 > a * b := Int.mul_neg_of_neg_of_pos hgt\u2081 hlt\u2082", "state_before": "a b : \u2124\nh : a * b = 0\nhgt\u2081 : a < 0\nhlt\u2082 : 0 < b\n\u22a2 a = 0 \u2228 b = 0", "state_after": "a b : \u2124\nh : a * b = 0\nhgt\u2081 : a < 0\nhlt\u2082 : 0 < b\nthis : 0 > a * b\n\u22a2 a = 0 \u2228 b = 0"}, {"tactic": "rw [h] at this", "state_before": "a b : \u2124\nh : a * b = 0\nhgt\u2081 : a < 0\nhlt\u2082 : 0 < b\nthis : 0 > a * b\n\u22a2 a = 0 \u2228 b = 0", "state_after": "a b : \u2124\nh : a * b = 0\nhgt\u2081 : a < 0\nhlt\u2082 : 0 < b\nthis : 0 > 0\n\u22a2 a = 0 \u2228 b = 0"}, {"tactic": "exact absurd this (lt_irrefl _)", "state_before": "a b : \u2124\nh : a * b = 0\nhgt\u2081 : a < 0\nhlt\u2082 : 0 < b\nthis : 0 > 0\n\u22a2 a = 0 \u2228 b = 0", "state_after": "no goals"}, {"tactic": "have : 0 < a * b := Int.mul_pos_of_neg_of_neg hgt\u2081 hgt\u2082", "state_before": "a b : \u2124\nh : a * b = 0\nhgt\u2081 : a < 0\nhgt\u2082 : b < 0\n\u22a2 a = 0 \u2228 b = 0", "state_after": "a b : \u2124\nh : a * b = 0\nhgt\u2081 : a < 0\nhgt\u2082 : b < 0\nthis : 0 < a * b\n\u22a2 a = 0 \u2228 b = 0"}, {"tactic": "rw [h] at this", "state_before": "a b : \u2124\nh : a * b = 0\nhgt\u2081 : a < 0\nhgt\u2082 : b < 0\nthis : 0 < a * b\n\u22a2 a = 0 \u2228 b = 0", "state_after": "a b : \u2124\nh : a * b = 0\nhgt\u2081 : a < 0\nhgt\u2082 : b < 0\nthis : 0 < 0\n\u22a2 a = 0 \u2228 b = 0"}, {"tactic": "exact absurd this (lt_irrefl _)", "state_before": "a b : \u2124\nh : a * b = 0\nhgt\u2081 : a < 0\nhgt\u2082 : b < 0\nthis : 0 < 0\n\u22a2 a = 0 \u2228 b = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Sets/Compacts.lean", "full_name": "TopologicalSpace.CompactOpens.coe_map", "start": [582, 1], "end": [584, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Rat/Lemmas.lean", "full_name": "Rat.inv_def''", "start": [178, 1], "end": [180, 36], "traced_tactics": [{"tactic": "conv_lhs => rw [\u2190 @num_den q]", "state_before": "q : \u211a\n\u22a2 q\u207b\u00b9 = \u2191q.den / \u2191q.num", "state_after": "q : \u211a\n\u22a2 (q.num /. \u2191q.den)\u207b\u00b9 = \u2191q.den / \u2191q.num"}, {"tactic": "rw [inv_def', divInt_eq_div]", "state_before": "q : \u211a\n\u22a2 (q.num /. \u2191q.den)\u207b\u00b9 = \u2191q.den / \u2191q.num", "state_after": "q : \u211a\n\u22a2 \u2191\u2191q.den / \u2191q.num = \u2191q.den / \u2191q.num"}, {"tactic": "rfl", "state_before": "q : \u211a\n\u22a2 \u2191\u2191q.den / \u2191q.num = \u2191q.den / \u2191q.num", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_const", "start": [166, 1], "end": [168, 6], "traced_tactics": [{"tactic": "rw [\u2190 SimpleFunc.const_lintegral, \u2190 SimpleFunc.lintegral_eq_lintegral, SimpleFunc.coe_const]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.74036\n\u03b3 : Type ?u.74039\n\u03b4 : Type ?u.74042\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : \u211d\u22650\u221e\n\u22a2 (\u222b\u207b (x : \u03b1), c \u2202\u03bc) = c * \u2191\u2191\u03bc univ", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.74036\n\u03b3 : Type ?u.74039\n\u03b4 : Type ?u.74042\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : \u211d\u22650\u221e\n\u22a2 (\u222b\u207b (x : \u03b1), c \u2202\u03bc) = \u222b\u207b (a : \u03b1), Function.const \u03b1 c a \u2202\u03bc"}, {"tactic": "rfl", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.74036\n\u03b3 : Type ?u.74039\n\u03b4 : Type ?u.74042\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nc : \u211d\u22650\u221e\n\u22a2 (\u222b\u207b (x : \u03b1), c \u2202\u03bc) = \u222b\u207b (a : \u03b1), Function.const \u03b1 c a \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "full_name": "PowerSeries.ext", "start": [1325, 1], "end": [1329, 8], "traced_tactics": [{"tactic": "rw [\u2190 coeff_def]", "state_before": "R : Type u_1\ninst\u271d : Semiring R\n\u03c6 \u03c8 : PowerSeries R\nh : \u2200 (n : \u2115), \u2191(coeff R n) \u03c6 = \u2191(coeff R n) \u03c8\nn : Unit \u2192\u2080 \u2115\n\u22a2 \u2191(MvPowerSeries.coeff R n) \u03c6 = \u2191(MvPowerSeries.coeff R n) \u03c8", "state_after": "R : Type u_1\ninst\u271d : Semiring R\n\u03c6 \u03c8 : PowerSeries R\nh : \u2200 (n : \u2115), \u2191(coeff R n) \u03c6 = \u2191(coeff R n) \u03c8\nn : Unit \u2192\u2080 \u2115\n\u22a2 \u2191(coeff R ?m.2202640) \u03c6 = \u2191(coeff R ?m.2202640) \u03c8\n\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6 \u03c8 : PowerSeries R\nh : \u2200 (n : \u2115), \u2191(coeff R n) \u03c6 = \u2191(coeff R n) \u03c8\nn : Unit \u2192\u2080 \u2115\n\u22a2 \u2191n () = ?m.2202640\n\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6 \u03c8 : PowerSeries R\nh : \u2200 (n : \u2115), \u2191(coeff R n) \u03c6 = \u2191(coeff R n) \u03c8\nn : Unit \u2192\u2080 \u2115\n\u22a2 \u2115"}, {"tactic": "rfl", "state_before": "R : Type u_1\ninst\u271d : Semiring R\n\u03c6 \u03c8 : PowerSeries R\nh : \u2200 (n : \u2115), \u2191(coeff R n) \u03c6 = \u2191(coeff R n) \u03c8\nn : Unit \u2192\u2080 \u2115\n\u22a2 \u2191n () = ?m.2202640\n\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6 \u03c8 : PowerSeries R\nh : \u2200 (n : \u2115), \u2191(coeff R n) \u03c6 = \u2191(coeff R n) \u03c8\nn : Unit \u2192\u2080 \u2115\n\u22a2 \u2115", "state_after": "no goals"}, {"tactic": "apply h", "state_before": "R : Type u_1\ninst\u271d : Semiring R\n\u03c6 \u03c8 : PowerSeries R\nh : \u2200 (n : \u2115), \u2191(coeff R n) \u03c6 = \u2191(coeff R n) \u03c8\nn : Unit \u2192\u2080 \u2115\n\u22a2 \u2191(coeff R ?m.2202640) \u03c6 = \u2191(coeff R ?m.2202640) \u03c8", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "full_name": "InnerProductSpace.Core.inner_add_right", "start": [225, 1], "end": [226, 87], "traced_tactics": [{"tactic": "rw [\u2190 inner_conj_symm, inner_add_left, RingHom.map_add]", "state_before": "\ud835\udd5c : Type u_1\nE : Type ?u.362411\nF : Type u_2\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y z : F\n\u22a2 inner x (y + z) = inner x y + inner x z", "state_after": "\ud835\udd5c : Type u_1\nE : Type ?u.362411\nF : Type u_2\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y z : F\n\u22a2 \u2191(starRingEnd \ud835\udd5c) (inner y x) + \u2191(starRingEnd \ud835\udd5c) (inner z x) = inner x y + inner x z"}, {"tactic": "simp only [inner_conj_symm]", "state_before": "\ud835\udd5c : Type u_1\nE : Type ?u.362411\nF : Type u_2\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y z : F\n\u22a2 \u2191(starRingEnd \ud835\udd5c) (inner y x) + \u2191(starRingEnd \ud835\udd5c) (inner z x) = inner x y + inner x z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Count.lean", "full_name": "List.count_erase_of_ne", "start": [357, 1], "end": [358, 44], "traced_tactics": [{"tactic": "rw [count_erase, if_neg ab, tsub_zero]", "state_before": "\u03b1 : Type u_1\nl\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\nab : a \u2260 b\nl : List \u03b1\n\u22a2 count a (List.erase l b) = count a l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "full_name": "pos_of_mul_pos_right", "start": [395, 1], "end": [396, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Prod.lean", "full_name": "MonoidHom.prod_comp_prodMap", "start": [614, 1], "end": [616, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Basic.lean", "full_name": "ball_true_iff", "start": [1096, 1], "end": [1097, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/PNat/Defs.lean", "full_name": "PNat.eq", "start": [150, 1], "end": [151, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "sSup_union", "start": [467, 1], "end": [468, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Heyting/Basic.lean", "full_name": "hnot_inf_distrib", "start": [992, 1], "end": [993, 44], "traced_tactics": [{"tactic": "simp_rw [\u2190 top_sdiff', sdiff_inf_distrib]", "state_before": "\u03b9 : Type ?u.170618\n\u03b1 : Type u_1\n\u03b2 : Type ?u.170624\ninst\u271d : CoheytingAlgebra \u03b1\na\u271d b\u271d c a b : \u03b1\n\u22a2 \uffe2(a \u2293 b) = \uffe2a \u2294 \uffe2b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/Star/Multiplier.lean", "full_name": "DoubleCentralizer.coe_fst", "start": [500, 1], "end": [501, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Seq/Computation.lean", "full_name": "Computation.get_promises", "start": [488, 1], "end": [488, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.subtype_val_iff", "start": [1234, 1], "end": [1241, 12], "traced_tactics": [{"tactic": "letI := Primcodable.subtype hp", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.354442\n\u03c3 : Type ?u.354445\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : PrimrecPred p\nf : \u03b1 \u2192 Subtype p\n\u22a2 (Primrec fun a => \u2191(f a)) \u2194 Primrec f", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.354442\n\u03c3 : Type ?u.354445\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : PrimrecPred p\nf : \u03b1 \u2192 Subtype p\nthis : Primcodable (Subtype p) := Primcodable.subtype hp\n\u22a2 (Primrec fun a => \u2191(f a)) \u2194 Primrec f"}, {"tactic": "refine' \u27e8fun h => _, fun hf => subtype_val.comp hf\u27e9", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.354442\n\u03c3 : Type ?u.354445\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : PrimrecPred p\nf : \u03b1 \u2192 Subtype p\nthis : Primcodable (Subtype p) := Primcodable.subtype hp\n\u22a2 (Primrec fun a => \u2191(f a)) \u2194 Primrec f", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.354442\n\u03c3 : Type ?u.354445\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : PrimrecPred p\nf : \u03b1 \u2192 Subtype p\nthis : Primcodable (Subtype p) := Primcodable.subtype hp\nh : Primrec fun a => \u2191(f a)\n\u22a2 Primrec f"}, {"tactic": "refine' Nat.Primrec.of_eq h fun n => _", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.354442\n\u03c3 : Type ?u.354445\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : PrimrecPred p\nf : \u03b1 \u2192 Subtype p\nthis : Primcodable (Subtype p) := Primcodable.subtype hp\nh : Primrec fun a => \u2191(f a)\n\u22a2 Primrec f", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.354442\n\u03c3 : Type ?u.354445\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : PrimrecPred p\nf : \u03b1 \u2192 Subtype p\nthis : Primcodable (Subtype p) := Primcodable.subtype hp\nh : Primrec fun a => \u2191(f a)\nn : \u2115\n\u22a2 encode (Option.map (fun a => \u2191(f a)) (decode n)) = encode (Option.map f (decode n))"}, {"tactic": "cases' @decode \u03b1 _ n with a", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.354442\n\u03c3 : Type ?u.354445\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : PrimrecPred p\nf : \u03b1 \u2192 Subtype p\nthis : Primcodable (Subtype p) := Primcodable.subtype hp\nh : Primrec fun a => \u2191(f a)\nn : \u2115\n\u22a2 encode (Option.map (fun a => \u2191(f a)) (decode n)) = encode (Option.map f (decode n))", "state_after": "case none\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.354442\n\u03c3 : Type ?u.354445\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : PrimrecPred p\nf : \u03b1 \u2192 Subtype p\nthis : Primcodable (Subtype p) := Primcodable.subtype hp\nh : Primrec fun a => \u2191(f a)\nn : \u2115\n\u22a2 encode (Option.map (fun a => \u2191(f a)) none) = encode (Option.map f none)\n\ncase some\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.354442\n\u03c3 : Type ?u.354445\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : PrimrecPred p\nf : \u03b1 \u2192 Subtype p\nthis : Primcodable (Subtype p) := Primcodable.subtype hp\nh : Primrec fun a => \u2191(f a)\nn : \u2115\na : \u03b1\n\u22a2 encode (Option.map (fun a => \u2191(f a)) (some a)) = encode (Option.map f (some a))"}, {"tactic": "simp", "state_before": "case some\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.354442\n\u03c3 : Type ?u.354445\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : PrimrecPred p\nf : \u03b1 \u2192 Subtype p\nthis : Primcodable (Subtype p) := Primcodable.subtype hp\nh : Primrec fun a => \u2191(f a)\nn : \u2115\na : \u03b1\n\u22a2 encode (Option.map (fun a => \u2191(f a)) (some a)) = encode (Option.map f (some a))", "state_after": "case some\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.354442\n\u03c3 : Type ?u.354445\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : PrimrecPred p\nf : \u03b1 \u2192 Subtype p\nthis : Primcodable (Subtype p) := Primcodable.subtype hp\nh : Primrec fun a => \u2191(f a)\nn : \u2115\na : \u03b1\n\u22a2 Nat.succ (encode \u2191(f a)) = encode (some (f a))"}, {"tactic": "rfl", "state_before": "case some\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.354442\n\u03c3 : Type ?u.354445\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : PrimrecPred p\nf : \u03b1 \u2192 Subtype p\nthis : Primcodable (Subtype p) := Primcodable.subtype hp\nh : Primrec fun a => \u2191(f a)\nn : \u2115\na : \u03b1\n\u22a2 Nat.succ (encode \u2191(f a)) = encode (some (f a))", "state_after": "no goals"}, {"tactic": "rfl", "state_before": "case none\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.354442\n\u03c3 : Type ?u.354445\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03c3\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : PrimrecPred p\nf : \u03b1 \u2192 Subtype p\nthis : Primcodable (Subtype p) := Primcodable.subtype hp\nh : Primrec fun a => \u2191(f a)\nn : \u2115\n\u22a2 encode (Option.map (fun a => \u2191(f a)) none) = encode (Option.map f none)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Control/ForInStep/Lemmas.lean", "full_name": "ForInStep.bindList_append", "start": [44, 9], "end": [47, 43], "traced_tactics": [{"tactic": "induction l\u2081 generalizing s <;> simp [*]", "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nf : \u03b1 \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\ns : ForInStep \u03b2\nl\u2081 l\u2082 : List \u03b1\n\u22a2 bindList f (l\u2081 ++ l\u2082) s = do\n    let x \u2190 bindList f l\u2081 s\n    bindList f l\u2082 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/ConcreteCategory.lean", "full_name": "CategoryTheory.Limits.Concrete.from_union_surjective_of_isColimit", "start": [175, 1], "end": [199, 22], "traced_tactics": [{"tactic": "intro ff", "state_before": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\n\u22a2 let ff := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd;\n  Function.Surjective ff", "state_after": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\n\u22a2 Function.Surjective ff"}, {"tactic": "let E := (forget C).mapCocone D", "state_before": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\n\u22a2 Function.Surjective ff", "state_after": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\n\u22a2 Function.Surjective ff"}, {"tactic": "let hE : IsColimit E := isColimitOfPreserves _ hD", "state_before": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\n\u22a2 Function.Surjective ff", "state_after": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\n\u22a2 Function.Surjective ff"}, {"tactic": "let G := Types.colimitCocone.{v, v} (F \u22d9 forget C)", "state_before": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\n\u22a2 Function.Surjective ff", "state_after": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\n\u22a2 Function.Surjective ff"}, {"tactic": "let hG := Types.colimitCoconeIsColimit.{v, v} (F \u22d9 forget C)", "state_before": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\n\u22a2 Function.Surjective ff", "state_after": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\n\u22a2 Function.Surjective ff"}, {"tactic": "let T : E \u2245 G := hE.uniqueUpToIso hG", "state_before": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\n\u22a2 Function.Surjective ff", "state_after": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\n\u22a2 Function.Surjective ff"}, {"tactic": "let TX : E.pt \u2245 G.pt := (Cocones.forget _).mapIso T", "state_before": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\n\u22a2 Function.Surjective ff", "state_after": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\n\u22a2 Function.Surjective ff"}, {"tactic": "suffices Function.Surjective (TX.hom \u2218 ff) by\n  intro a\n  obtain \u27e8b, hb\u27e9 := this (TX.hom a)\n  refine' \u27e8b, _\u27e9\n  replace hb := congr_arg TX.inv hb\n  change (TX.hom \u226b TX.inv) (ff b) = (TX.hom \u226b TX.inv) _ at hb\n  simpa only [TX.hom_inv_id] using hb", "state_before": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\n\u22a2 Function.Surjective ff", "state_after": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\n\u22a2 Function.Surjective (TX.hom \u2218 ff)"}, {"tactic": "have : TX.hom \u2218 ff = fun a => G.\u03b9.app a.1 a.2 := by\n  ext a\n  change (E.\u03b9.app a.1 \u226b hE.desc G) a.2 = _\n  rw [hE.fac]", "state_before": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\n\u22a2 Function.Surjective (TX.hom \u2218 ff)", "state_after": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\nthis : TX.hom \u2218 ff = fun a => G.\u03b9.app a.fst a.snd\n\u22a2 Function.Surjective (TX.hom \u2218 ff)"}, {"tactic": "rw [this]", "state_before": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\nthis : TX.hom \u2218 ff = fun a => G.\u03b9.app a.fst a.snd\n\u22a2 Function.Surjective (TX.hom \u2218 ff)", "state_after": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\nthis : TX.hom \u2218 ff = fun a => G.\u03b9.app a.fst a.snd\n\u22a2 Function.Surjective fun a => G.\u03b9.app a.fst a.snd"}, {"tactic": "rintro \u27e8\u27e8j, a\u27e9\u27e9", "state_before": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\nthis : TX.hom \u2218 ff = fun a => G.\u03b9.app a.fst a.snd\n\u22a2 Function.Surjective fun a => G.\u03b9.app a.fst a.snd", "state_after": "case mk.mk\nC : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\nthis : TX.hom \u2218 ff = fun a => G.\u03b9.app a.fst a.snd\nb\u271d : G.pt\nj : J\na : (F \u22d9 forget C).obj j\n\u22a2 \u2203 a_1, (fun a => G.\u03b9.app a.fst a.snd) a_1 = Quot.mk (Types.Quot.Rel (F \u22d9 forget C)) { fst := j, snd := a }"}, {"tactic": "exact \u27e8\u27e8j, a\u27e9, rfl\u27e9", "state_before": "case mk.mk\nC : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\nthis : TX.hom \u2218 ff = fun a => G.\u03b9.app a.fst a.snd\nb\u271d : G.pt\nj : J\na : (F \u22d9 forget C).obj j\n\u22a2 \u2203 a_1, (fun a => G.\u03b9.app a.fst a.snd) a_1 = Quot.mk (Types.Quot.Rel (F \u22d9 forget C)) { fst := j, snd := a }", "state_after": "no goals"}, {"tactic": "intro a", "state_before": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\nthis : Function.Surjective (TX.hom \u2218 ff)\n\u22a2 Function.Surjective ff", "state_after": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\nthis : Function.Surjective (TX.hom \u2218 ff)\na : (forget C).obj D.pt\n\u22a2 \u2203 a_1, ff a_1 = a"}, {"tactic": "obtain \u27e8b, hb\u27e9 := this (TX.hom a)", "state_before": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\nthis : Function.Surjective (TX.hom \u2218 ff)\na : (forget C).obj D.pt\n\u22a2 \u2203 a_1, ff a_1 = a", "state_after": "case intro\nC : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\nthis : Function.Surjective (TX.hom \u2218 ff)\na : (forget C).obj D.pt\nb : (j : J) \u00d7 (forget C).obj (F.obj j)\nhb : (TX.hom \u2218 ff) b = TX.hom a\n\u22a2 \u2203 a_1, ff a_1 = a"}, {"tactic": "refine' \u27e8b, _\u27e9", "state_before": "case intro\nC : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\nthis : Function.Surjective (TX.hom \u2218 ff)\na : (forget C).obj D.pt\nb : (j : J) \u00d7 (forget C).obj (F.obj j)\nhb : (TX.hom \u2218 ff) b = TX.hom a\n\u22a2 \u2203 a_1, ff a_1 = a", "state_after": "case intro\nC : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\nthis : Function.Surjective (TX.hom \u2218 ff)\na : (forget C).obj D.pt\nb : (j : J) \u00d7 (forget C).obj (F.obj j)\nhb : (TX.hom \u2218 ff) b = TX.hom a\n\u22a2 ff b = a"}, {"tactic": "replace hb := congr_arg TX.inv hb", "state_before": "case intro\nC : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\nthis : Function.Surjective (TX.hom \u2218 ff)\na : (forget C).obj D.pt\nb : (j : J) \u00d7 (forget C).obj (F.obj j)\nhb : (TX.hom \u2218 ff) b = TX.hom a\n\u22a2 ff b = a", "state_after": "case intro\nC : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\nthis : Function.Surjective (TX.hom \u2218 ff)\na : (forget C).obj D.pt\nb : (j : J) \u00d7 (forget C).obj (F.obj j)\nhb : TX.inv ((TX.hom \u2218 ff) b) = TX.inv (TX.hom a)\n\u22a2 ff b = a"}, {"tactic": "change (TX.hom \u226b TX.inv) (ff b) = (TX.hom \u226b TX.inv) _ at hb", "state_before": "case intro\nC : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\nthis : Function.Surjective (TX.hom \u2218 ff)\na : (forget C).obj D.pt\nb : (j : J) \u00d7 (forget C).obj (F.obj j)\nhb : TX.inv ((TX.hom \u2218 ff) b) = TX.inv (TX.hom a)\n\u22a2 ff b = a", "state_after": "case intro\nC : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\nthis : Function.Surjective (TX.hom \u2218 ff)\na : (forget C).obj D.pt\nb : (j : J) \u00d7 (forget C).obj (F.obj j)\nhb : (TX.hom \u226b TX.inv) (ff b) = (TX.hom \u226b TX.inv) a\n\u22a2 ff b = a"}, {"tactic": "simpa only [TX.hom_inv_id] using hb", "state_before": "case intro\nC : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\nthis : Function.Surjective (TX.hom \u2218 ff)\na : (forget C).obj D.pt\nb : (j : J) \u00d7 (forget C).obj (F.obj j)\nhb : (TX.hom \u226b TX.inv) (ff b) = (TX.hom \u226b TX.inv) a\n\u22a2 ff b = a", "state_after": "no goals"}, {"tactic": "ext a", "state_before": "C : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\n\u22a2 TX.hom \u2218 ff = fun a => G.\u03b9.app a.fst a.snd", "state_after": "case h\nC : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\na : (j : J) \u00d7 (forget C).obj (F.obj j)\n\u22a2 (TX.hom \u2218 ff) a = G.\u03b9.app a.fst a.snd"}, {"tactic": "change (E.\u03b9.app a.1 \u226b hE.desc G) a.2 = _", "state_before": "case h\nC : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\na : (j : J) \u00d7 (forget C).obj (F.obj j)\n\u22a2 (TX.hom \u2218 ff) a = G.\u03b9.app a.fst a.snd", "state_after": "case h\nC : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\na : (j : J) \u00d7 (forget C).obj (F.obj j)\n\u22a2 (E.\u03b9.app a.fst \u226b IsColimit.desc hE G) a.snd = G.\u03b9.app a.fst a.snd"}, {"tactic": "rw [hE.fac]", "state_before": "case h\nC : Type u\ninst\u271d\u00b3 : Category C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\nF : J \u2964 C\ninst\u271d : PreservesColimit F (forget C)\nD : Cocone F\nhD : IsColimit D\nff : (j : J) \u00d7 (forget C).obj (F.obj j) \u2192 (forget C).obj D.pt := fun a => (forget C).map (D.\u03b9.app a.fst) a.snd\nE : Cocone (F \u22d9 forget C) := (forget C).mapCocone D\nhE : IsColimit E := isColimitOfPreserves (forget C) hD\nG : Cocone (F \u22d9 forget C) := Types.colimitCocone (F \u22d9 forget C)\nhG : IsColimit (Types.colimitCocone (F \u22d9 forget C)) := Types.colimitCoconeIsColimit (F \u22d9 forget C)\nT : E \u2245 G := IsColimit.uniqueUpToIso hE hG\nTX : E.pt \u2245 G.pt := (Cocones.forget (F \u22d9 forget C)).mapIso T\na : (j : J) \u00d7 (forget C).obj (F.obj j)\n\u22a2 (E.\u03b9.app a.fst \u226b IsColimit.desc hE G) a.snd = G.\u03b9.app a.fst a.snd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/ContinuousOn.lean", "full_name": "nhdsWithin_eq_nhdsWithin", "start": [215, 1], "end": [217, 68], "traced_tactics": [{"tactic": "rw [nhdsWithin_restrict t h\u2080 h\u2081, nhdsWithin_restrict u h\u2080 h\u2081, h\u2082]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.12224\n\u03b3 : Type ?u.12227\n\u03b4 : Type ?u.12230\ninst\u271d : TopologicalSpace \u03b1\na : \u03b1\ns t u : Set \u03b1\nh\u2080 : a \u2208 s\nh\u2081 : IsOpen s\nh\u2082 : t \u2229 s = u \u2229 s\n\u22a2 \ud835\udcdd[t] a = \ud835\udcdd[u] a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Semiconj.lean", "full_name": "SemiconjBy.inv_inv_symm", "start": [193, 1], "end": [194, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/LiminfLimsup.lean", "full_name": "Filter.bliminf_true", "start": [405, 1], "end": [406, 31], "traced_tactics": [{"tactic": "simp [bliminf_eq, liminf_eq]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.57903\n\u03b9 : Type ?u.57906\ninst\u271d : ConditionallyCompleteLattice \u03b1\nf : Filter \u03b2\nu : \u03b2 \u2192 \u03b1\n\u22a2 (bliminf u f fun x => True) = liminf u f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Core.lean", "full_name": "Ne.symm", "start": [575, 1], "end": [576, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.map\u2097_mk_apply_of_aemeasurable", "start": [1160, 1], "end": [1161, 52], "traced_tactics": [{"tactic": "simp [map, hf]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.175590\n\u03b4 : Type ?u.175593\n\u03b9 : Type ?u.175596\nR : Type ?u.175599\nR' : Type ?u.175602\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\n\u22a2 \u2191(map\u2097 (AEMeasurable.mk f hf)) \u03bc = map f \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean", "full_name": "Real.differentiableOn_log", "start": [61, 1], "end": [62, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/UniformEmbedding.lean", "full_name": "IsClosed.completeSpace_coe", "start": [328, 1], "end": [330, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.append_right_injective", "start": [406, 1], "end": [407, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/ODE/PicardLindelof.lean", "full_name": "PicardLindelof.FunSpace.mem_closedBall", "start": [212, 11], "end": [217, 45], "traced_tactics": [{"tactic": "rw [f.map_t\u2080']", "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nv : PicardLindelof E\nf : FunSpace v\nt : \u2191(Icc v.tMin v.tMax)\n\u22a2 dist (toFun f t) v.x\u2080 = dist (toFun f t) (toFun f v.t\u2080)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/MonoidLocalization.lean", "full_name": "Submonoid.LocalizationMap.ext_iff", "start": [564, 1], "end": [565, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Category/Bipointed.lean", "full_name": "pointedToBipointedSnd_comp_swap", "start": [195, 1], "end": [197, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.bddAbove_of_small", "start": [969, 1], "end": [970, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "full_name": "Subgroup.le_comap_map", "start": [3001, 1], "end": [3002, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/MvPolynomial/Variables.lean", "full_name": "MvPolynomial.totalDegree_add_eq_right_of_totalDegree_lt", "start": [678, 1], "end": [680, 61], "traced_tactics": [{"tactic": "rw [add_comm, totalDegree_add_eq_left_of_totalDegree_lt h]", "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type ?u.448839\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np\u271d q\u271d p q : MvPolynomial \u03c3 R\nh : totalDegree q < totalDegree p\n\u22a2 totalDegree (q + p) = totalDegree p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Dual.lean", "full_name": "LinearMap.dualMap_comp_dualMap", "start": [221, 1], "end": [223, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/ContinuedFractions/Translations.lean", "full_name": "GeneralizedContinuedFraction.exists_conts_b_of_denom", "start": [117, 1], "end": [118, 63], "traced_tactics": [{"tactic": "simpa", "state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\nB : K\nnth_denom_eq : denominators g n = B\n\u22a2 \u2203 conts, continuants g n = conts \u2227 conts.b = B", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.IsLimit.succ_lt", "start": [244, 1], "end": [245, 8], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Associated.lean", "full_name": "Associated.dvd_iff_dvd_right", "start": [575, 1], "end": [577, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Subobject/Basic.lean", "full_name": "CategoryTheory.Subobject.map_id", "start": [584, 1], "end": [586, 46], "traced_tactics": [{"tactic": "induction' x using Quotient.inductionOn' with f", "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category D\nx : Subobject X\n\u22a2 (map (\ud835\udfd9 X)).obj x = x", "state_after": "case h\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category D\nf : MonoOver X\n\u22a2 (map (\ud835\udfd9 X)).obj (Quotient.mk'' f) = Quotient.mk'' f"}, {"tactic": "exact Quotient.sound \u27e8MonoOver.mapId.app f\u27e9", "state_before": "case h\nC : Type u\u2081\ninst\u271d\u00b9 : Category C\nX Y Z : C\nD : Type u\u2082\ninst\u271d : Category D\nf : MonoOver X\n\u22a2 (map (\ud835\udfd9 X)).obj (Quotient.mk'' f) = Quotient.mk'' f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/BoxIntegral/Basic.lean", "full_name": "BoxIntegral.integrable_zero", "start": [334, 1], "end": [335, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Quot.lift_mk", "start": [99, 1], "end": [101, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Heyting/Basic.lean", "full_name": "sup_sdiff", "start": [653, 1], "end": [654, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "full_name": "Set.preimage_const_sub_uIcc", "start": [450, 1], "end": [452, 91], "traced_tactics": [{"tactic": "simp_rw [\u2190 Icc_min_max, preimage_const_sub_Icc]", "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 (fun x => a - x) \u207b\u00b9' [[b, c]] = [[a - b, a - c]]", "state_after": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 Icc (a - max b c) (a - min b c) = Icc (min (a - b) (a - c)) (max (a - b) (a - c))"}, {"tactic": "simp only [sub_eq_add_neg, min_add_add_left, max_add_add_left, min_neg_neg, max_neg_neg]", "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 Icc (a - max b c) (a - min b c) = Icc (min (a - b) (a - c)) (max (a - b) (a - c))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fintype/Pi.lean", "full_name": "Fintype.piFinset_singleton", "start": [66, 1], "end": [67, 87], "traced_tactics": [{"tactic": "simp only [Function.funext_iff, Fintype.mem_piFinset, mem_singleton]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03b4 : \u03b1 \u2192 Type u_2\nf x\u271d : (a : \u03b1) \u2192 \u03b4 a\n\u22a2 (x\u271d \u2208 piFinset fun i => {f i}) \u2194 x\u271d \u2208 {f}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Ring/Prod.lean", "full_name": "NonUnitalRingHom.snd_comp_prod", "start": [149, 1], "end": [150, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Ring/Equiv.lean", "full_name": "RingEquiv.coe_ringHom_refl", "start": [527, 1], "end": [528, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fintype/Card.lean", "full_name": "Fintype.card_compl_set", "start": [295, 1], "end": [297, 99], "traced_tactics": [{"tactic": "classical rw [\u2190 Set.toFinset_card, \u2190 Set.toFinset_card, \u2190 Finset.card_compl, Set.toFinset_compl]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.9460\n\u03b3 : Type ?u.9463\ninst\u271d\u00b2 : Fintype \u03b1\ns : Set \u03b1\ninst\u271d\u00b9 : Fintype \u2191s\ninst\u271d : Fintype \u2191(s\u1d9c)\n\u22a2 card \u2191(s\u1d9c) = card \u03b1 - card \u2191s", "state_after": "no goals"}, {"tactic": "rw [\u2190 Set.toFinset_card, \u2190 Set.toFinset_card, \u2190 Finset.card_compl, Set.toFinset_compl]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.9460\n\u03b3 : Type ?u.9463\ninst\u271d\u00b2 : Fintype \u03b1\ns : Set \u03b1\ninst\u271d\u00b9 : Fintype \u2191s\ninst\u271d : Fintype \u2191(s\u1d9c)\n\u22a2 card \u2191(s\u1d9c) = card \u03b1 - card \u2191s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/MeasurableSpace.lean", "full_name": "MeasurableSpace.comap_generateFrom", "start": [189, 1], "end": [194, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "full_name": "Subgroup.comap_equiv_eq_map_symm", "start": [1478, 1], "end": [1480, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "generateFrom_Ico_mem_le_borel", "start": [568, 1], "end": [575, 26], "traced_tactics": [{"tactic": "apply generateFrom_le", "state_before": "\u03b1\u271d : Type ?u.606888\n\u03b2 : Type ?u.606891\n\u03b3 : Type ?u.606894\n\u03b3\u2082 : Type ?u.606897\n\u03b4 : Type ?u.606900\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b9\u2079 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2075 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b3\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\u2082\ninst\u271d\u2079 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2078 : BorelSpace \u03b3\u2082\ninst\u271d\u2077 : MeasurableSpace \u03b4\n\u03b1' : Type ?u.606993\ninst\u271d\u2076 : TopologicalSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : LinearOrder \u03b1\u271d\ninst\u271d\u00b3 : OrderClosedTopology \u03b1\u271d\na b x : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderClosedTopology \u03b1\ns t : Set \u03b1\n\u22a2 MeasurableSpace.generateFrom {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 t \u2227 l < u \u2227 Ico l u = S} \u2264 borel \u03b1", "state_after": "case h\n\u03b1\u271d : Type ?u.606888\n\u03b2 : Type ?u.606891\n\u03b3 : Type ?u.606894\n\u03b3\u2082 : Type ?u.606897\n\u03b4 : Type ?u.606900\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b9\u2079 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2075 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b3\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\u2082\ninst\u271d\u2079 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2078 : BorelSpace \u03b3\u2082\ninst\u271d\u2077 : MeasurableSpace \u03b4\n\u03b1' : Type ?u.606993\ninst\u271d\u2076 : TopologicalSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : LinearOrder \u03b1\u271d\ninst\u271d\u00b3 : OrderClosedTopology \u03b1\u271d\na b x : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderClosedTopology \u03b1\ns t : Set \u03b1\n\u22a2 \u2200 (t_1 : Set \u03b1), t_1 \u2208 {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 t \u2227 l < u \u2227 Ico l u = S} \u2192 MeasurableSet t_1"}, {"tactic": "borelize \u03b1", "state_before": "case h\n\u03b1\u271d : Type ?u.606888\n\u03b2 : Type ?u.606891\n\u03b3 : Type ?u.606894\n\u03b3\u2082 : Type ?u.606897\n\u03b4 : Type ?u.606900\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b9\u2079 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2075 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b3\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\u2082\ninst\u271d\u2079 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2078 : BorelSpace \u03b3\u2082\ninst\u271d\u2077 : MeasurableSpace \u03b4\n\u03b1' : Type ?u.606993\ninst\u271d\u2076 : TopologicalSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : LinearOrder \u03b1\u271d\ninst\u271d\u00b3 : OrderClosedTopology \u03b1\u271d\na b x : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderClosedTopology \u03b1\ns t : Set \u03b1\n\u22a2 \u2200 (t_1 : Set \u03b1), t_1 \u2208 {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 t \u2227 l < u \u2227 Ico l u = S} \u2192 MeasurableSet t_1", "state_after": "case h\n\u03b1\u271d : Type ?u.606888\n\u03b2 : Type ?u.606891\n\u03b3 : Type ?u.606894\n\u03b3\u2082 : Type ?u.606897\n\u03b4 : Type ?u.606900\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b9\u2079 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2075 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b3\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\u2082\ninst\u271d\u2079 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2078 : BorelSpace \u03b3\u2082\ninst\u271d\u2077 : MeasurableSpace \u03b4\n\u03b1' : Type ?u.606993\ninst\u271d\u2076 : TopologicalSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : LinearOrder \u03b1\u271d\ninst\u271d\u00b3 : OrderClosedTopology \u03b1\u271d\na b x : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderClosedTopology \u03b1\ns t : Set \u03b1\nthis\u271d\u00b9 : MeasurableSpace \u03b1 := borel \u03b1\nthis\u271d : BorelSpace \u03b1\n\u22a2 \u2200 (t_1 : Set \u03b1), t_1 \u2208 {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 t \u2227 l < u \u2227 Ico l u = S} \u2192 MeasurableSet t_1"}, {"tactic": "rintro _ \u27e8a, -, b, -, -, rfl\u27e9", "state_before": "case h\n\u03b1\u271d : Type ?u.606888\n\u03b2 : Type ?u.606891\n\u03b3 : Type ?u.606894\n\u03b3\u2082 : Type ?u.606897\n\u03b4 : Type ?u.606900\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b9\u2079 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2075 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b3\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\u2082\ninst\u271d\u2079 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2078 : BorelSpace \u03b3\u2082\ninst\u271d\u2077 : MeasurableSpace \u03b4\n\u03b1' : Type ?u.606993\ninst\u271d\u2076 : TopologicalSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : LinearOrder \u03b1\u271d\ninst\u271d\u00b3 : OrderClosedTopology \u03b1\u271d\na b x : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderClosedTopology \u03b1\ns t : Set \u03b1\nthis\u271d\u00b9 : MeasurableSpace \u03b1 := borel \u03b1\nthis\u271d : BorelSpace \u03b1\n\u22a2 \u2200 (t_1 : Set \u03b1), t_1 \u2208 {S | \u2203 l, l \u2208 s \u2227 \u2203 u, u \u2208 t \u2227 l < u \u2227 Ico l u = S} \u2192 MeasurableSet t_1", "state_after": "case h.intro.intro.intro.intro.intro\n\u03b1\u271d : Type ?u.606888\n\u03b2 : Type ?u.606891\n\u03b3 : Type ?u.606894\n\u03b3\u2082 : Type ?u.606897\n\u03b4 : Type ?u.606900\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b9\u2079 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2075 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b3\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\u2082\ninst\u271d\u2079 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2078 : BorelSpace \u03b3\u2082\ninst\u271d\u2077 : MeasurableSpace \u03b4\n\u03b1' : Type ?u.606993\ninst\u271d\u2076 : TopologicalSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : LinearOrder \u03b1\u271d\ninst\u271d\u00b3 : OrderClosedTopology \u03b1\u271d\na\u271d b\u271d x : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderClosedTopology \u03b1\ns t : Set \u03b1\nthis\u271d\u00b9 : MeasurableSpace \u03b1 := borel \u03b1\nthis\u271d : BorelSpace \u03b1\na b : \u03b1\n\u22a2 MeasurableSet (Ico a b)"}, {"tactic": "exact measurableSet_Ico", "state_before": "case h.intro.intro.intro.intro.intro\n\u03b1\u271d : Type ?u.606888\n\u03b2 : Type ?u.606891\n\u03b3 : Type ?u.606894\n\u03b3\u2082 : Type ?u.606897\n\u03b4 : Type ?u.606900\n\u03b9 : Sort y\ns\u271d t\u271d u : Set \u03b1\u271d\ninst\u271d\u00b9\u2079 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9\u2078 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2077 : OpensMeasurableSpace \u03b1\u271d\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u2075 : MeasurableSpace \u03b2\ninst\u271d\u00b9\u2074 : OpensMeasurableSpace \u03b2\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b2 : MeasurableSpace \u03b3\ninst\u271d\u00b9\u00b9 : BorelSpace \u03b3\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\u2082\ninst\u271d\u2079 : MeasurableSpace \u03b3\u2082\ninst\u271d\u2078 : BorelSpace \u03b3\u2082\ninst\u271d\u2077 : MeasurableSpace \u03b4\n\u03b1' : Type ?u.606993\ninst\u271d\u2076 : TopologicalSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : LinearOrder \u03b1\u271d\ninst\u271d\u00b3 : OrderClosedTopology \u03b1\u271d\na\u271d b\u271d x : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : OrderClosedTopology \u03b1\ns t : Set \u03b1\nthis\u271d\u00b9 : MeasurableSpace \u03b1 := borel \u03b1\nthis\u271d : BorelSpace \u03b1\na b : \u03b1\n\u22a2 MeasurableSet (Ico a b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Disjoint.union_left", "start": [2952, 1], "end": [2953, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Constructions/Pi.lean", "full_name": "MeasureTheory.Measure.pi_eq_generateFrom", "start": [364, 1], "end": [377, 59], "traced_tactics": [{"tactic": "apply_assumption", "state_before": "\u03b9 : Type ?u.2834565\n\u03b9' : Type ?u.2834568\n\u03b1 : \u03b9 \u2192 Type ?u.2834573\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\ni : \u03b9\n\u22a2 MeasurableSpace (\u03b1 i)", "state_after": "no goals"}, {"tactic": "have h4C : \u2200 (i) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s := by\n  intro i s hs; rw [\u2190 hC]; exact measurableSet_generateFrom hs", "state_before": "\u03b9 : Type u_2\n\u03b9' : Type ?u.2834568\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\n\u22a2 Measure.pi \u03bc = \u03bc\u03bd", "state_after": "\u03b9 : Type u_2\n\u03b9' : Type ?u.2834568\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\nh4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s\n\u22a2 Measure.pi \u03bc = \u03bc\u03bd"}, {"tactic": "refine'\n  (FiniteSpanningSetsIn.pi h3C).ext\n    (generateFrom_eq_pi hC fun i => (h3C i).isCountablySpanning).symm (IsPiSystem.pi h2C) _", "state_before": "\u03b9 : Type u_2\n\u03b9' : Type ?u.2834568\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\nh4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s\n\u22a2 Measure.pi \u03bc = \u03bc\u03bd", "state_after": "\u03b9 : Type u_2\n\u03b9' : Type ?u.2834568\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\nh4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s\n\u22a2 \u2200 (s : Set ((i : \u03b9) \u2192 \u03b1 i)), (s \u2208 Set.pi univ '' Set.pi univ fun i => C i) \u2192 \u2191\u2191(Measure.pi \u03bc) s = \u2191\u2191\u03bc\u03bd s"}, {"tactic": "rintro _ \u27e8s, hs, rfl\u27e9", "state_before": "\u03b9 : Type u_2\n\u03b9' : Type ?u.2834568\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\nh4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s\n\u22a2 \u2200 (s : Set ((i : \u03b9) \u2192 \u03b1 i)), (s \u2208 Set.pi univ '' Set.pi univ fun i => C i) \u2192 \u2191\u2191(Measure.pi \u03bc) s = \u2191\u2191\u03bc\u03bd s", "state_after": "case intro.intro\n\u03b9 : Type u_2\n\u03b9' : Type ?u.2834568\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\nh4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs : s \u2208 Set.pi univ fun i => C i\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) = \u2191\u2191\u03bc\u03bd (Set.pi univ s)"}, {"tactic": "rw [mem_univ_pi] at hs", "state_before": "case intro.intro\n\u03b9 : Type u_2\n\u03b9' : Type ?u.2834568\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\nh4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs : s \u2208 Set.pi univ fun i => C i\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) = \u2191\u2191\u03bc\u03bd (Set.pi univ s)", "state_after": "case intro.intro\n\u03b9 : Type u_2\n\u03b9' : Type ?u.2834568\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\nh4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs : \u2200 (i : \u03b9), s i \u2208 C i\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) = \u2191\u2191\u03bc\u03bd (Set.pi univ s)"}, {"tactic": "haveI := fun i => (h3C i).sigmaFinite", "state_before": "case intro.intro\n\u03b9 : Type u_2\n\u03b9' : Type ?u.2834568\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\nh4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs : \u2200 (i : \u03b9), s i \u2208 C i\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) = \u2191\u2191\u03bc\u03bd (Set.pi univ s)", "state_after": "case intro.intro\n\u03b9 : Type u_2\n\u03b9' : Type ?u.2834568\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\nh4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs : \u2200 (i : \u03b9), s i \u2208 C i\nthis : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) = \u2191\u2191\u03bc\u03bd (Set.pi univ s)"}, {"tactic": "simp_rw [h\u2081 s hs, pi_pi_aux \u03bc s fun i => h4C i _ (hs i)]", "state_before": "case intro.intro\n\u03b9 : Type u_2\n\u03b9' : Type ?u.2834568\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\nh4C : \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nhs : \u2200 (i : \u03b9), s i \u2208 C i\nthis : \u2200 (i : \u03b9), SigmaFinite (\u03bc i)\n\u22a2 \u2191\u2191(Measure.pi \u03bc) (Set.pi univ s) = \u2191\u2191\u03bc\u03bd (Set.pi univ s)", "state_after": "no goals"}, {"tactic": "intro i s hs", "state_before": "\u03b9 : Type u_2\n\u03b9' : Type ?u.2834568\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\n\u22a2 \u2200 (i : \u03b9) (s : Set (\u03b1 i)), s \u2208 C i \u2192 MeasurableSet s", "state_after": "\u03b9 : Type u_2\n\u03b9' : Type ?u.2834568\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\ni : \u03b9\ns : Set (\u03b1 i)\nhs : s \u2208 C i\n\u22a2 MeasurableSet s"}, {"tactic": "rw [\u2190 hC]", "state_before": "\u03b9 : Type u_2\n\u03b9' : Type ?u.2834568\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\ni : \u03b9\ns : Set (\u03b1 i)\nhs : s \u2208 C i\n\u22a2 MeasurableSet s", "state_after": "\u03b9 : Type u_2\n\u03b9' : Type ?u.2834568\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\ni : \u03b9\ns : Set (\u03b1 i)\nhs : s \u2208 C i\n\u22a2 MeasurableSet s"}, {"tactic": "exact measurableSet_generateFrom hs", "state_before": "\u03b9 : Type u_2\n\u03b9' : Type ?u.2834568\n\u03b1 : \u03b9 \u2192 Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\nm : (i : \u03b9) \u2192 OuterMeasure (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 MeasurableSpace (\u03b1 i)\n\u03bc : (i : \u03b9) \u2192 Measure (\u03b1 i)\nC : (i : \u03b9) \u2192 Set (Set (\u03b1 i))\nhC : \u2200 (i : \u03b9), generateFrom (C i) = inst\u271d i\nh2C : \u2200 (i : \u03b9), IsPiSystem (C i)\nh3C : (i : \u03b9) \u2192 FiniteSpanningSetsIn (\u03bc i) (C i)\n\u03bc\u03bd : Measure ((i : \u03b9) \u2192 \u03b1 i)\nh\u2081 : \u2200 (s : (i : \u03b9) \u2192 Set (\u03b1 i)), (\u2200 (i : \u03b9), s i \u2208 C i) \u2192 \u2191\u2191\u03bc\u03bd (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191(\u03bc i) (s i)\ni : \u03b9\ns : Set (\u03b1 i)\nhs : s \u2208 C i\n\u22a2 MeasurableSet s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.eqOn_of_leftInvOn_of_rightInvOn", "start": [1151, 1], "end": [1155, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Factorial/Basic.lean", "full_name": "Nat.factorial_mul_pow_sub_le_factorial", "start": [209, 1], "end": [216, 13], "traced_tactics": [{"tactic": "suffices n ! * (n + 1) ^ (m - n) \u2264 m ! from by\n  apply LE.le.trans _ this\n  apply mul_le_mul_left\n  apply pow_le_pow_of_le_left (le_succ n)", "state_before": "m\u271d n\u271d n m : \u2115\nhnm : n \u2264 m\n\u22a2 n ! * n ^ (m - n) \u2264 m !", "state_after": "m\u271d n\u271d n m : \u2115\nhnm : n \u2264 m\n\u22a2 n ! * (n + 1) ^ (m - n) \u2264 m !"}, {"tactic": "have := @Nat.factorial_mul_pow_le_factorial n (m - n)", "state_before": "m\u271d n\u271d n m : \u2115\nhnm : n \u2264 m\n\u22a2 n ! * (n + 1) ^ (m - n) \u2264 m !", "state_after": "m\u271d n\u271d n m : \u2115\nhnm : n \u2264 m\nthis : n ! * succ n ^ (m - n) \u2264 (n + (m - n))!\n\u22a2 n ! * (n + 1) ^ (m - n) \u2264 m !"}, {"tactic": "simp [hnm] at this", "state_before": "m\u271d n\u271d n m : \u2115\nhnm : n \u2264 m\nthis : n ! * succ n ^ (m - n) \u2264 (n + (m - n))!\n\u22a2 n ! * (n + 1) ^ (m - n) \u2264 m !", "state_after": "m\u271d n\u271d n m : \u2115\nhnm : n \u2264 m\nthis : n ! * succ n ^ (m - n) \u2264 m !\n\u22a2 n ! * (n + 1) ^ (m - n) \u2264 m !"}, {"tactic": "exact this", "state_before": "m\u271d n\u271d n m : \u2115\nhnm : n \u2264 m\nthis : n ! * succ n ^ (m - n) \u2264 m !\n\u22a2 n ! * (n + 1) ^ (m - n) \u2264 m !", "state_after": "no goals"}, {"tactic": "apply LE.le.trans _ this", "state_before": "m\u271d n\u271d n m : \u2115\nhnm : n \u2264 m\nthis : n ! * (n + 1) ^ (m - n) \u2264 m !\n\u22a2 n ! * n ^ (m - n) \u2264 m !", "state_after": "m\u271d n\u271d n m : \u2115\nhnm : n \u2264 m\nthis : n ! * (n + 1) ^ (m - n) \u2264 m !\n\u22a2 n ! * n ^ (m - n) \u2264 n ! * (n + 1) ^ (m - n)"}, {"tactic": "apply mul_le_mul_left", "state_before": "m\u271d n\u271d n m : \u2115\nhnm : n \u2264 m\nthis : n ! * (n + 1) ^ (m - n) \u2264 m !\n\u22a2 n ! * n ^ (m - n) \u2264 n ! * (n + 1) ^ (m - n)", "state_after": "case h\nm\u271d n\u271d n m : \u2115\nhnm : n \u2264 m\nthis : n ! * (n + 1) ^ (m - n) \u2264 m !\n\u22a2 n ^ (m - n) \u2264 (n + 1) ^ (m - n)"}, {"tactic": "apply pow_le_pow_of_le_left (le_succ n)", "state_before": "case h\nm\u271d n\u271d n m : \u2115\nhnm : n \u2264 m\nthis : n ! * (n + 1) ^ (m - n) \u2264 m !\n\u22a2 n ^ (m - n) \u2264 (n + 1) ^ (m - n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.power_add", "start": [517, 1], "end": [518, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/LocallyConstant/Basic.lean", "full_name": "LocallyConstant.coe_continuousMap", "start": [317, 9], "end": [317, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/ContDiffDef.lean", "full_name": "Filter.EventuallyEq.iteratedFDerivWithin", "start": [889, 11], "end": [891, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/PartitionOfUnity.lean", "full_name": "BumpCovering.support_toPouFun_subset", "start": [385, 1], "end": [386, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/PEquiv.lean", "full_name": "PEquiv.ofSet_symm", "start": [248, 1], "end": [249, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/SymmDiff.lean", "full_name": "Bool.symmDiff_eq_xor", "start": [100, 1], "end": [100, 74], "traced_tactics": [{"tactic": "decide", "state_before": "\u03b9 : Type ?u.16365\n\u03b1 : Type ?u.16368\n\u03b2 : Type ?u.16371\n\u03c0 : \u03b9 \u2192 Type ?u.16376\n\u22a2 \u2200 (p q : Bool), p \u2206 q = xor p q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Order/MonotoneConvergence.lean", "full_name": "tendsto_atBot_ciSup", "start": [135, 1], "end": [136, 99], "traced_tactics": [{"tactic": "convert tendsto_atTop_ciSup h_anti.dual hbdd.dual using 1", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.18385\n\u03b9 : Type u_1\ninst\u271d\u00b3 : Preorder \u03b9\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : ConditionallyCompleteLattice \u03b1\ninst\u271d : SupConvergenceClass \u03b1\nf : \u03b9 \u2192 \u03b1\na : \u03b1\nh_anti : Antitone f\nhbdd : BddAbove (range f)\n\u22a2 Tendsto f atBot (\ud835\udcdd (\u2a06 (i : \u03b9), f i))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/Basic.lean", "full_name": "to_nhds_mono", "start": [1325, 1], "end": [1328, 92], "traced_tactics": [{"tactic": "rw [@nhds_eq_uniformity \u03b1 u\u2081 a, @nhds_eq_uniformity \u03b1 u\u2082 a]", "state_before": "\u03b1 : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort ?u.141492\nu\u2081 u\u2082 : UniformSpace \u03b1\nh : u\u2081 \u2264 u\u2082\na : \u03b1\n\u22a2 \ud835\udcdd a \u2264 \ud835\udcdd a", "state_after": "\u03b1 : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort ?u.141492\nu\u2081 u\u2082 : UniformSpace \u03b1\nh : u\u2081 \u2264 u\u2082\na : \u03b1\n\u22a2 Filter.lift' (\ud835\udce4 \u03b1) (UniformSpace.ball a) \u2264 Filter.lift' (\ud835\udce4 \u03b1) (UniformSpace.ball a)"}, {"tactic": "exact lift'_mono h le_rfl", "state_before": "\u03b1 : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort ?u.141492\nu\u2081 u\u2082 : UniformSpace \u03b1\nh : u\u2081 \u2264 u\u2082\na : \u03b1\n\u22a2 Filter.lift' (\ud835\udce4 \u03b1) (UniformSpace.ball a) \u2264 Filter.lift' (\ud835\udce4 \u03b1) (UniformSpace.ball a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Alternating.lean", "full_name": "AlternatingMap.map_swap", "start": [703, 1], "end": [704, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean", "full_name": "Cardinal.continuum_add_nat", "start": [146, 1], "end": [147, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Congruence.lean", "full_name": "RingCon.eq", "start": [175, 11], "end": [176, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/MorphismProperty.lean", "full_name": "CategoryTheory.MorphismProperty.StableUnderComposition.unop", "start": [117, 1], "end": [118, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.exists_mem_of_nonempty", "start": [706, 1], "end": [707, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Interval.lean", "full_name": "Interval.inv_bot", "start": [520, 1], "end": [521, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Constructions.lean", "full_name": "DiscreteTopology.of_subset", "start": [1129, 1], "end": [1131, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/SuccPred/Basic.lean", "full_name": "Order.succ_le_iff", "start": [341, 1], "end": [342, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean", "full_name": "Complex.hasStrictDerivAt_const_cpow", "start": [55, 1], "end": [63, 55], "traced_tactics": [{"tactic": "rcases em (x = 0) with (rfl | hx)", "state_before": "x y : \u2102\nh : x \u2260 0 \u2228 y \u2260 0\n\u22a2 HasStrictDerivAt (fun y => x ^ y) (x ^ y * log x) y", "state_after": "case inl\ny : \u2102\nh : 0 \u2260 0 \u2228 y \u2260 0\n\u22a2 HasStrictDerivAt (fun y => 0 ^ y) (0 ^ y * log 0) y\n\ncase inr\nx y : \u2102\nh : x \u2260 0 \u2228 y \u2260 0\nhx : \u00acx = 0\n\u22a2 HasStrictDerivAt (fun y => x ^ y) (x ^ y * log x) y"}, {"tactic": "replace h := h.neg_resolve_left rfl", "state_before": "case inl\ny : \u2102\nh : 0 \u2260 0 \u2228 y \u2260 0\n\u22a2 HasStrictDerivAt (fun y => 0 ^ y) (0 ^ y * log 0) y", "state_after": "case inl\ny : \u2102\nh : y \u2260 0\n\u22a2 HasStrictDerivAt (fun y => 0 ^ y) (0 ^ y * log 0) y"}, {"tactic": "rw [log_zero, MulZeroClass.mul_zero]", "state_before": "case inl\ny : \u2102\nh : y \u2260 0\n\u22a2 HasStrictDerivAt (fun y => 0 ^ y) (0 ^ y * log 0) y", "state_after": "case inl\ny : \u2102\nh : y \u2260 0\n\u22a2 HasStrictDerivAt (fun y => 0 ^ y) 0 y"}, {"tactic": "refine' (hasStrictDerivAt_const _ 0).congr_of_eventuallyEq _", "state_before": "case inl\ny : \u2102\nh : y \u2260 0\n\u22a2 HasStrictDerivAt (fun y => 0 ^ y) 0 y", "state_after": "case inl\ny : \u2102\nh : y \u2260 0\n\u22a2 (fun x => 0) =\u1da0[\ud835\udcdd y] fun y => 0 ^ y"}, {"tactic": "exact (isOpen_ne.eventually_mem h).mono fun y hy => (zero_cpow hy).symm", "state_before": "case inl\ny : \u2102\nh : y \u2260 0\n\u22a2 (fun x => 0) =\u1da0[\ud835\udcdd y] fun y => 0 ^ y", "state_after": "no goals"}, {"tactic": "simpa only [cpow_def_of_ne_zero hx, mul_one] using\n  ((hasStrictDerivAt_id y).const_mul (log x)).cexp", "state_before": "case inr\nx y : \u2102\nh : x \u2260 0 \u2228 y \u2260 0\nhx : \u00acx = 0\n\u22a2 HasStrictDerivAt (fun y => x ^ y) (x ^ y * log x) y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.exists_seq_forall_of_frequently", "start": [1883, 1], "end": [1889, 40], "traced_tactics": [{"tactic": "rw [frequently_iff_seq_frequently] at h", "state_before": "\u03b9\u271d : Type ?u.371978\n\u03b9' : Type ?u.371981\n\u03b1 : Type ?u.371984\n\u03b2 : Type ?u.371987\n\u03b3 : Type ?u.371990\n\u03b9 : Type u_1\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nh : \u2203\u1da0 (n : \u03b9) in l, p n\n\u22a2 \u2203 ns, Tendsto ns atTop l \u2227 \u2200 (n : \u2115), p (ns n)", "state_after": "\u03b9\u271d : Type ?u.371978\n\u03b9' : Type ?u.371981\n\u03b1 : Type ?u.371984\n\u03b2 : Type ?u.371987\n\u03b3 : Type ?u.371990\n\u03b9 : Type u_1\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nh : \u2203 x, Tendsto x atTop l \u2227 \u2203\u1da0 (n : \u2115) in atTop, p (x n)\n\u22a2 \u2203 ns, Tendsto ns atTop l \u2227 \u2200 (n : \u2115), p (ns n)"}, {"tactic": "obtain \u27e8x, hx_tendsto, hx_freq\u27e9 := h", "state_before": "\u03b9\u271d : Type ?u.371978\n\u03b9' : Type ?u.371981\n\u03b1 : Type ?u.371984\n\u03b2 : Type ?u.371987\n\u03b3 : Type ?u.371990\n\u03b9 : Type u_1\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nh : \u2203 x, Tendsto x atTop l \u2227 \u2203\u1da0 (n : \u2115) in atTop, p (x n)\n\u22a2 \u2203 ns, Tendsto ns atTop l \u2227 \u2200 (n : \u2115), p (ns n)", "state_after": "case intro.intro\n\u03b9\u271d : Type ?u.371978\n\u03b9' : Type ?u.371981\n\u03b1 : Type ?u.371984\n\u03b2 : Type ?u.371987\n\u03b3 : Type ?u.371990\n\u03b9 : Type u_1\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nx : \u2115 \u2192 \u03b9\nhx_tendsto : Tendsto x atTop l\nhx_freq : \u2203\u1da0 (n : \u2115) in atTop, p (x n)\n\u22a2 \u2203 ns, Tendsto ns atTop l \u2227 \u2200 (n : \u2115), p (ns n)"}, {"tactic": "obtain \u27e8n_to_n, h_tendsto, h_freq\u27e9 := subseq_forall_of_frequently hx_tendsto hx_freq", "state_before": "case intro.intro\n\u03b9\u271d : Type ?u.371978\n\u03b9' : Type ?u.371981\n\u03b1 : Type ?u.371984\n\u03b2 : Type ?u.371987\n\u03b3 : Type ?u.371990\n\u03b9 : Type u_1\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nx : \u2115 \u2192 \u03b9\nhx_tendsto : Tendsto x atTop l\nhx_freq : \u2203\u1da0 (n : \u2115) in atTop, p (x n)\n\u22a2 \u2203 ns, Tendsto ns atTop l \u2227 \u2200 (n : \u2115), p (ns n)", "state_after": "case intro.intro.intro.intro\n\u03b9\u271d : Type ?u.371978\n\u03b9' : Type ?u.371981\n\u03b1 : Type ?u.371984\n\u03b2 : Type ?u.371987\n\u03b3 : Type ?u.371990\n\u03b9 : Type u_1\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nx : \u2115 \u2192 \u03b9\nhx_tendsto : Tendsto x atTop l\nhx_freq : \u2203\u1da0 (n : \u2115) in atTop, p (x n)\nn_to_n : \u2115 \u2192 \u2115\nh_tendsto : Tendsto (fun n => x (n_to_n n)) atTop l\nh_freq : \u2200 (n : \u2115), p (x (n_to_n n))\n\u22a2 \u2203 ns, Tendsto ns atTop l \u2227 \u2200 (n : \u2115), p (ns n)"}, {"tactic": "exact \u27e8x \u2218 n_to_n, h_tendsto, h_freq\u27e9", "state_before": "case intro.intro.intro.intro\n\u03b9\u271d : Type ?u.371978\n\u03b9' : Type ?u.371981\n\u03b1 : Type ?u.371984\n\u03b2 : Type ?u.371987\n\u03b3 : Type ?u.371990\n\u03b9 : Type u_1\nl : Filter \u03b9\np : \u03b9 \u2192 Prop\nhl : IsCountablyGenerated l\nx : \u2115 \u2192 \u03b9\nhx_tendsto : Tendsto x atTop l\nhx_freq : \u2203\u1da0 (n : \u2115) in atTop, p (x n)\nn_to_n : \u2115 \u2192 \u2115\nh_tendsto : Tendsto (fun n => x (n_to_n n)) atTop l\nh_freq : \u2200 (n : \u2115), p (x (n_to_n n))\n\u22a2 \u2203 ns, Tendsto ns atTop l \u2227 \u2200 (n : \u2115), p (ns n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fintype/Basic.lean", "full_name": "Finset.univ_eq_empty", "start": [117, 1], "end": [118, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.replace_toFinmap", "start": [373, 1], "end": [375, 17], "traced_tactics": [{"tactic": "simp [replace]", "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nb : \u03b2 a\ns : AList \u03b2\n\u22a2 replace a b \u27e6s\u27e7 = \u27e6AList.replace a b s\u27e7", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.inter_subset_ite", "start": [2303, 1], "end": [2304, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Sum/Basic.lean", "full_name": "Sum.isLeft_swap", "start": [369, 1], "end": [369, 84], "traced_tactics": [{"tactic": "cases x <;> rfl", "state_before": "\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type ?u.22858\n\u03b4 : Type ?u.22861\nx : \u03b1 \u2295 \u03b2\n\u22a2 isLeft (swap x) = isRight x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/LiminfLimsup.lean", "full_name": "Filter.blimsup_eq_iInf_biSup_of_nat", "start": [744, 1], "end": [749, 68], "traced_tactics": [{"tactic": "simp only [blimsup_eq_limsup_subtype, Function.comp,\n  (atTop_basis.comap ((\u2191) : { x | p x } \u2192 \u2115)).limsup_eq_iInf_iSup, iSup_subtype, iSup_and]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.113892\n\u03b3 : Type ?u.113895\n\u03b9 : Type ?u.113898\ninst\u271d : CompleteLattice \u03b1\np : \u2115 \u2192 Prop\nu : \u2115 \u2192 \u03b1\n\u22a2 blimsup u atTop p = \u2a05 (i : \u2115), \u2a06 (j : \u2115) (_ : p j \u2227 i \u2264 j), u j", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.113892\n\u03b3 : Type ?u.113895\n\u03b9 : Type ?u.113898\ninst\u271d : CompleteLattice \u03b1\np : \u2115 \u2192 Prop\nu : \u2115 \u2192 \u03b1\n\u22a2 (\u2a05 (i : \u2115) (_ : True),\n      \u2a06 (i_1 : \u2115) (x : i_1 \u2208 {x | p x}) (_ : { val := i_1, property := (_ : i_1 \u2208 {x | p x}) } \u2208 Subtype.val \u207b\u00b9' Ici i),\n        u i_1) =\n    \u2a05 (i : \u2115), \u2a06 (j : \u2115) (_ : p j) (_ : i \u2264 j), u j"}, {"tactic": "simp only [mem_setOf_eq, mem_preimage, mem_Ici, not_le, iInf_pos]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.113892\n\u03b3 : Type ?u.113895\n\u03b9 : Type ?u.113898\ninst\u271d : CompleteLattice \u03b1\np : \u2115 \u2192 Prop\nu : \u2115 \u2192 \u03b1\n\u22a2 (\u2a05 (i : \u2115) (_ : True),\n      \u2a06 (i_1 : \u2115) (x : i_1 \u2208 {x | p x}) (_ : { val := i_1, property := (_ : i_1 \u2208 {x | p x}) } \u2208 Subtype.val \u207b\u00b9' Ici i),\n        u i_1) =\n    \u2a05 (i : \u2115), \u2a06 (j : \u2115) (_ : p j) (_ : i \u2264 j), u j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Integrals.lean", "full_name": "intervalIntegral.inv_mul_integral_comp_sub_div", "start": [307, 1], "end": [309, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "full_name": "CategoryTheory.MonoidalCategory.triangle_assoc_comp_left_inv", "start": [311, 1], "end": [315, 74], "traced_tactics": [{"tactic": "apply (cancel_mono ((\u03c1_ X).hom \u2297 \ud835\udfd9 Y)).1", "state_before": "C\u271d : Type u\n\ud835\udc9e : Category C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y\u271d Z X Y : C\n\u22a2 (\ud835\udfd9 X \u2297 (\u03bb_ Y).inv) \u226b (\u03b1_ X (\ud835\udfd9_ C) Y).inv = (\u03c1_ X).inv \u2297 \ud835\udfd9 Y", "state_after": "C\u271d : Type u\n\ud835\udc9e : Category C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y\u271d Z X Y : C\n\u22a2 ((\ud835\udfd9 X \u2297 (\u03bb_ Y).inv) \u226b (\u03b1_ X (\ud835\udfd9_ C) Y).inv) \u226b ((\u03c1_ X).hom \u2297 \ud835\udfd9 Y) = ((\u03c1_ X).inv \u2297 \ud835\udfd9 Y) \u226b ((\u03c1_ X).hom \u2297 \ud835\udfd9 Y)"}, {"tactic": "simp only [triangle_assoc_comp_right, assoc]", "state_before": "C\u271d : Type u\n\ud835\udc9e : Category C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y\u271d Z X Y : C\n\u22a2 ((\ud835\udfd9 X \u2297 (\u03bb_ Y).inv) \u226b (\u03b1_ X (\ud835\udfd9_ C) Y).inv) \u226b ((\u03c1_ X).hom \u2297 \ud835\udfd9 Y) = ((\u03c1_ X).inv \u2297 \ud835\udfd9 Y) \u226b ((\u03c1_ X).hom \u2297 \ud835\udfd9 Y)", "state_after": "C\u271d : Type u\n\ud835\udc9e : Category C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y\u271d Z X Y : C\n\u22a2 (\ud835\udfd9 X \u2297 (\u03bb_ Y).inv) \u226b (\ud835\udfd9 X \u2297 (\u03bb_ Y).hom) = ((\u03c1_ X).inv \u2297 \ud835\udfd9 Y) \u226b ((\u03c1_ X).hom \u2297 \ud835\udfd9 Y)"}, {"tactic": "rw [\u2190 id_tensor_comp, Iso.inv_hom_id, \u2190 comp_tensor_id, Iso.inv_hom_id]", "state_before": "C\u271d : Type u\n\ud835\udc9e : Category C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category C\ninst\u271d : MonoidalCategory C\nU V W X\u271d Y\u271d Z X Y : C\n\u22a2 (\ud835\udfd9 X \u2297 (\u03bb_ Y).inv) \u226b (\ud835\udfd9 X \u2297 (\u03bb_ Y).hom) = ((\u03c1_ X).inv \u2297 \ud835\udfd9 Y) \u226b ((\u03c1_ X).hom \u2297 \ud835\udfd9 Y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/TensorProduct.lean", "full_name": "TensorProduct.leftComm_symm_tmul", "start": [901, 1], "end": [903, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean", "full_name": "Real.sin_arcsin", "start": [69, 1], "end": [70, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Trace.lean", "full_name": "LinearMap.trace_one", "start": [189, 1], "end": [192, 7], "traced_tactics": [{"tactic": "have b := Module.Free.chooseBasis R M", "state_before": "R : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Type u_2\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\nN : Type ?u.389113\ninst\u271d\u2076 : AddCommGroup N\ninst\u271d\u2075 : Module R N\n\u03b9 : Type ?u.389655\ninst\u271d\u2074 : Module.Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Module.Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\n\u22a2 \u2191(trace R M) 1 = \u2191(finrank R M)", "state_after": "R : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Type u_2\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\nN : Type ?u.389113\ninst\u271d\u2076 : AddCommGroup N\ninst\u271d\u2075 : Module R N\n\u03b9 : Type ?u.389655\ninst\u271d\u2074 : Module.Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Module.Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\nb : Basis (Module.Free.ChooseBasisIndex R M) R M\n\u22a2 \u2191(trace R M) 1 = \u2191(finrank R M)"}, {"tactic": "rw [trace_eq_matrix_trace R b, toMatrix_one, finrank_eq_card_chooseBasisIndex]", "state_before": "R : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Type u_2\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\nN : Type ?u.389113\ninst\u271d\u2076 : AddCommGroup N\ninst\u271d\u2075 : Module R N\n\u03b9 : Type ?u.389655\ninst\u271d\u2074 : Module.Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Module.Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\nb : Basis (Module.Free.ChooseBasisIndex R M) R M\n\u22a2 \u2191(trace R M) 1 = \u2191(finrank R M)", "state_after": "R : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Type u_2\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\nN : Type ?u.389113\ninst\u271d\u2076 : AddCommGroup N\ninst\u271d\u2075 : Module R N\n\u03b9 : Type ?u.389655\ninst\u271d\u2074 : Module.Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Module.Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\nb : Basis (Module.Free.ChooseBasisIndex R M) R M\n\u22a2 Matrix.trace 1 = \u2191(Fintype.card (Module.Free.ChooseBasisIndex R M))"}, {"tactic": "simp", "state_before": "R : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Type u_2\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : Module R M\nN : Type ?u.389113\ninst\u271d\u2076 : AddCommGroup N\ninst\u271d\u2075 : Module R N\n\u03b9 : Type ?u.389655\ninst\u271d\u2074 : Module.Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Module.Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\nb : Basis (Module.Free.ChooseBasisIndex R M) R M\n\u22a2 Matrix.trace 1 = \u2191(Fintype.card (Module.Free.ChooseBasisIndex R M))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Bits.lean", "full_name": "Nat.bit0_eq_bit0", "start": [73, 1], "end": [74, 43], "traced_tactics": [{"tactic": "subst h", "state_before": "n\u271d m n : \u2115\nh : m = n\n\u22a2 bit0 m = bit0 n", "state_after": "n m : \u2115\n\u22a2 bit0 m = bit0 m"}, {"tactic": "rfl", "state_before": "n m : \u2115\n\u22a2 bit0 m = bit0 m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.Icc_subset_Ico_iff", "start": [257, 1], "end": [258, 65], "traced_tactics": [{"tactic": "rw [\u2190 coe_subset, coe_Icc, coe_Ico, Set.Icc_subset_Ico_iff h\u2081]", "state_before": "\u03b9 : Type ?u.26269\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nh\u2081 : a\u2081 \u2264 b\u2081\n\u22a2 Icc a\u2081 b\u2081 \u2286 Ico a\u2082 b\u2082 \u2194 a\u2082 \u2264 a\u2081 \u2227 b\u2081 < b\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.comap_neBot_iff", "start": [2337, 1], "end": [2339, 94], "traced_tactics": [{"tactic": "simp only [\u2190 forall_mem_nonempty_iff_neBot, mem_comap, forall_exists_index, and_imp]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.262344\n\u03b9 : Sort x\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nm\u271d : \u03b1 \u2192 \u03b2\nm' : \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nf : Filter \u03b2\nm : \u03b1 \u2192 \u03b2\n\u22a2 NeBot (comap m f) \u2194 \u2200 (t : Set \u03b2), t \u2208 f \u2192 \u2203 a, m a \u2208 t", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.262344\n\u03b9 : Sort x\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nm\u271d : \u03b1 \u2192 \u03b2\nm' : \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nf : Filter \u03b2\nm : \u03b1 \u2192 \u03b2\n\u22a2 (\u2200 (s : Set \u03b1) (x : Set \u03b2), x \u2208 f \u2192 m \u207b\u00b9' x \u2286 s \u2192 Set.Nonempty s) \u2194 \u2200 (t : Set \u03b2), t \u2208 f \u2192 \u2203 a, m a \u2208 t"}, {"tactic": "exact \u27e8fun h t t_in => h (m \u207b\u00b9' t) t t_in Subset.rfl, fun h s t ht hst => (h t ht).imp hst\u27e9", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.262344\n\u03b9 : Sort x\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nm\u271d : \u03b1 \u2192 \u03b2\nm' : \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nf : Filter \u03b2\nm : \u03b1 \u2192 \u03b2\n\u22a2 (\u2200 (s : Set \u03b1) (x : Set \u03b2), x \u2208 f \u2192 m \u207b\u00b9' x \u2286 s \u2192 Set.Nonempty s) \u2194 \u2200 (t : Set \u03b2), t \u2208 f \u2192 \u2203 a, m a \u2208 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "full_name": "map_mk_unit_aux", "start": [1009, 9], "end": [1012, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.getLast?_append_cons", "start": [834, 1], "end": [839, 51], "traced_tactics": [{"tactic": "rw [cons_append, cons_append, getLast?_cons_cons,\n\u2190 cons_append, getLast?_append_cons (c :: l\u2081)]", "state_before": "\u03b9 : Type ?u.40235\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081\u271d l\u2082\u271d : List \u03b1\nb c : \u03b1\nl\u2081 : List \u03b1\na : \u03b1\nl\u2082 : List \u03b1\n\u22a2 getLast? (b :: c :: l\u2081 ++ a :: l\u2082) = getLast? (a :: l\u2082)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "full_name": "ModelWithCorners.preimage_image", "start": [287, 1], "end": [288, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/DiscreteValuationRing/Basic.lean", "full_name": "DiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.toUniqueFactorizationMonoid", "start": [202, 1], "end": [228, 46], "traced_tactics": [{"tactic": "use Multiset.replicate (Classical.choose (spec.2 hx)) p", "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\n\u22a2 \u2203 f, (\u2200 (b : R), b \u2208 f \u2192 Prime b) \u2227 Associated (Multiset.prod f) x", "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\n\u22a2 (\u2200 (b : R), b \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p \u2192 Prime b) \u2227\n    Associated (Multiset.prod (Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p)) x"}, {"tactic": "constructor", "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\n\u22a2 (\u2200 (b : R), b \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p \u2192 Prime b) \u2227\n    Associated (Multiset.prod (Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p)) x", "state_after": "case left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\n\u22a2 \u2200 (b : R), b \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p \u2192 Prime b\n\ncase right\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\n\u22a2 Associated (Multiset.prod (Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p)) x"}, {"tactic": "intro q hq", "state_before": "case left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\n\u22a2 \u2200 (b : R), b \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p \u2192 Prime b", "state_after": "case left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\n\u22a2 Prime q"}, {"tactic": "have hpq := Multiset.eq_of_mem_replicate hq", "state_before": "case left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\n\u22a2 Prime q", "state_after": "case left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\n\u22a2 Prime q"}, {"tactic": "rw [hpq]", "state_before": "case left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\n\u22a2 Prime q", "state_after": "case left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\n\u22a2 Prime p"}, {"tactic": "refine' \u27e8spec.1.ne_zero, spec.1.not_unit, _\u27e9", "state_before": "case left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\n\u22a2 Prime p", "state_after": "case left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\n\u22a2 \u2200 (a b : R), p \u2223 a * b \u2192 p \u2223 a \u2228 p \u2223 b"}, {"tactic": "intro a b h", "state_before": "case left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\n\u22a2 \u2200 (a b : R), p \u2223 a * b \u2192 p \u2223 a \u2228 p \u2223 b", "state_after": "case left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\na b : R\nh : p \u2223 a * b\n\u22a2 p \u2223 a \u2228 p \u2223 b"}, {"tactic": "by_cases ha : a = 0", "state_before": "case left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\na b : R\nh : p \u2223 a * b\n\u22a2 p \u2223 a \u2228 p \u2223 b", "state_after": "case pos\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\na b : R\nh : p \u2223 a * b\nha : a = 0\n\u22a2 p \u2223 a \u2228 p \u2223 b\n\ncase neg\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\na b : R\nh : p \u2223 a * b\nha : \u00aca = 0\n\u22a2 p \u2223 a \u2228 p \u2223 b"}, {"tactic": "obtain \u27e8m, u, rfl\u27e9 := spec.2 ha", "state_before": "case neg\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\na b : R\nh : p \u2223 a * b\nha : \u00aca = 0\n\u22a2 p \u2223 a \u2228 p \u2223 b", "state_after": "case neg.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 choose hR ^ m * \u2191u * b\nha : \u00acchoose hR ^ m * \u2191u = 0\n\u22a2 p \u2223 choose hR ^ m * \u2191u \u2228 p \u2223 b"}, {"tactic": "rw [mul_assoc, mul_left_comm, IsUnit.dvd_mul_left _ _ _ (Units.isUnit _)] at h", "state_before": "case neg.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 choose hR ^ m * \u2191u * b\nha : \u00acchoose hR ^ m * \u2191u = 0\n\u22a2 p \u2223 choose hR ^ m * \u2191u \u2228 p \u2223 b", "state_after": "case neg.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 choose hR ^ m * b\nha : \u00acchoose hR ^ m * \u2191u = 0\n\u22a2 p \u2223 choose hR ^ m * \u2191u \u2228 p \u2223 b"}, {"tactic": "rw [IsUnit.dvd_mul_right (Units.isUnit _)]", "state_before": "case neg.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 choose hR ^ m * b\nha : \u00acchoose hR ^ m * \u2191u = 0\n\u22a2 p \u2223 choose hR ^ m * \u2191u \u2228 p \u2223 b", "state_after": "case neg.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 choose hR ^ m * b\nha : \u00acchoose hR ^ m * \u2191u = 0\n\u22a2 p \u2223 choose hR ^ m \u2228 p \u2223 b"}, {"tactic": "by_cases hm : m = 0", "state_before": "case neg.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 choose hR ^ m * b\nha : \u00acchoose hR ^ m * \u2191u = 0\n\u22a2 p \u2223 choose hR ^ m \u2228 p \u2223 b", "state_after": "case pos\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 choose hR ^ m * b\nha : \u00acchoose hR ^ m * \u2191u = 0\nhm : m = 0\n\u22a2 p \u2223 choose hR ^ m \u2228 p \u2223 b\n\ncase neg\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 choose hR ^ m * b\nha : \u00acchoose hR ^ m * \u2191u = 0\nhm : \u00acm = 0\n\u22a2 p \u2223 choose hR ^ m \u2228 p \u2223 b"}, {"tactic": "left", "state_before": "case neg\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 choose hR ^ m * b\nha : \u00acchoose hR ^ m * \u2191u = 0\nhm : \u00acm = 0\n\u22a2 p \u2223 choose hR ^ m \u2228 p \u2223 b", "state_after": "case neg.h\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 choose hR ^ m * b\nha : \u00acchoose hR ^ m * \u2191u = 0\nhm : \u00acm = 0\n\u22a2 p \u2223 choose hR ^ m"}, {"tactic": "obtain \u27e8m, rfl\u27e9 := Nat.exists_eq_succ_of_ne_zero hm", "state_before": "case neg.h\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 choose hR ^ m * b\nha : \u00acchoose hR ^ m * \u2191u = 0\nhm : \u00acm = 0\n\u22a2 p \u2223 choose hR ^ m", "state_after": "case neg.h.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\nb : R\nu : R\u02e3\nm : \u2115\nh : p \u2223 choose hR ^ Nat.succ m * b\nha : \u00acchoose hR ^ Nat.succ m * \u2191u = 0\nhm : \u00acNat.succ m = 0\n\u22a2 p \u2223 choose hR ^ Nat.succ m"}, {"tactic": "rw [pow_succ]", "state_before": "case neg.h.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\nb : R\nu : R\u02e3\nm : \u2115\nh : p \u2223 choose hR ^ Nat.succ m * b\nha : \u00acchoose hR ^ Nat.succ m * \u2191u = 0\nhm : \u00acNat.succ m = 0\n\u22a2 p \u2223 choose hR ^ Nat.succ m", "state_after": "case neg.h.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\nb : R\nu : R\u02e3\nm : \u2115\nh : p \u2223 choose hR ^ Nat.succ m * b\nha : \u00acchoose hR ^ Nat.succ m * \u2191u = 0\nhm : \u00acNat.succ m = 0\n\u22a2 p \u2223 choose hR * choose hR ^ m"}, {"tactic": "apply dvd_mul_of_dvd_left dvd_rfl _", "state_before": "case neg.h.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\nb : R\nu : R\u02e3\nm : \u2115\nh : p \u2223 choose hR ^ Nat.succ m * b\nha : \u00acchoose hR ^ Nat.succ m * \u2191u = 0\nhm : \u00acNat.succ m = 0\n\u22a2 p \u2223 choose hR * choose hR ^ m", "state_after": "no goals"}, {"tactic": "rw [ha]", "state_before": "case pos\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\na b : R\nh : p \u2223 a * b\nha : a = 0\n\u22a2 p \u2223 a \u2228 p \u2223 b", "state_after": "case pos\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\na b : R\nh : p \u2223 a * b\nha : a = 0\n\u22a2 p \u2223 0 \u2228 p \u2223 b"}, {"tactic": "simp only [true_or_iff, dvd_zero]", "state_before": "case pos\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\na b : R\nh : p \u2223 a * b\nha : a = 0\n\u22a2 p \u2223 0 \u2228 p \u2223 b", "state_after": "no goals"}, {"tactic": "simp only [hm, one_mul, pow_zero] at h\u22a2", "state_before": "case pos\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nh : p \u2223 choose hR ^ m * b\nha : \u00acchoose hR ^ m * \u2191u = 0\nhm : m = 0\n\u22a2 p \u2223 choose hR ^ m \u2228 p \u2223 b", "state_after": "case pos\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nha : \u00acchoose hR ^ m * \u2191u = 0\nhm : m = 0\nh : choose hR \u2223 b\n\u22a2 choose hR \u2223 1 \u2228 choose hR \u2223 b"}, {"tactic": "right", "state_before": "case pos\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nha : \u00acchoose hR ^ m * \u2191u = 0\nhm : m = 0\nh : choose hR \u2223 b\n\u22a2 choose hR \u2223 1 \u2228 choose hR \u2223 b", "state_after": "case pos.h\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nha : \u00acchoose hR ^ m * \u2191u = 0\nhm : m = 0\nh : choose hR \u2223 b\n\u22a2 choose hR \u2223 b"}, {"tactic": "exact h", "state_before": "case pos.h\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\nq : R\nhq : q \u2208 Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p\nhpq : q = p\nb : R\nm : \u2115\nu : R\u02e3\nha : \u00acchoose hR ^ m * \u2191u = 0\nhm : m = 0\nh : choose hR \u2223 b\n\u22a2 choose hR \u2223 b", "state_after": "no goals"}, {"tactic": "rw [Multiset.prod_replicate]", "state_before": "case right\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\n\u22a2 Associated (Multiset.prod (Multiset.replicate (choose (_ : \u2203 n, Associated (choose hR ^ n) x)) p)) x", "state_after": "case right\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\n\u22a2 Associated (p ^ choose (_ : \u2203 n, Associated (choose hR ^ n) x)) x"}, {"tactic": "exact Classical.choose_spec (spec.2 hx)", "state_before": "case right\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nhR : HasUnitMulPowIrreducibleFactorization R\ninst\u271d : IsDomain R\np : R := choose hR\nspec : Irreducible (choose hR) \u2227 \u2200 {x : R}, x \u2260 0 \u2192 \u2203 n, Associated (choose hR ^ n) x := choose_spec hR\nx : R\nhx : x \u2260 0\n\u22a2 Associated (p ^ choose (_ : \u2203 n, Associated (choose hR ^ n) x)) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Types.lean", "full_name": "CategoryTheory.mono_iff_injective", "start": [243, 1], "end": [249, 49], "traced_tactics": [{"tactic": "constructor", "state_before": "X Y : Type u\nf : X \u27f6 Y\n\u22a2 Mono f \u2194 Function.Injective f", "state_after": "case mp\nX Y : Type u\nf : X \u27f6 Y\n\u22a2 Mono f \u2192 Function.Injective f\n\ncase mpr\nX Y : Type u\nf : X \u27f6 Y\n\u22a2 Function.Injective f \u2192 Mono f"}, {"tactic": "intro H x x' h", "state_before": "case mp\nX Y : Type u\nf : X \u27f6 Y\n\u22a2 Mono f \u2192 Function.Injective f", "state_after": "case mp\nX Y : Type u\nf : X \u27f6 Y\nH : Mono f\nx x' : X\nh : f x = f x'\n\u22a2 x = x'"}, {"tactic": "skip", "state_before": "case mp\nX Y : Type u\nf : X \u27f6 Y\nH : Mono f\nx x' : X\nh : f x = f x'\n\u22a2 x = x'", "state_after": "case mp\nX Y : Type u\nf : X \u27f6 Y\nH : Mono f\nx x' : X\nh : f x = f x'\n\u22a2 x = x'"}, {"tactic": "rw [\u2190 homOfElement_eq_iff] at h\u22a2", "state_before": "case mp\nX Y : Type u\nf : X \u27f6 Y\nH : Mono f\nx x' : X\nh : f x = f x'\n\u22a2 x = x'", "state_after": "case mp\nX Y : Type u\nf : X \u27f6 Y\nH : Mono f\nx x' : X\nh : homOfElement (f x) = homOfElement (f x')\n\u22a2 homOfElement x = homOfElement x'"}, {"tactic": "exact (cancel_mono f).mp h", "state_before": "case mp\nX Y : Type u\nf : X \u27f6 Y\nH : Mono f\nx x' : X\nh : homOfElement (f x) = homOfElement (f x')\n\u22a2 homOfElement x = homOfElement x'", "state_after": "no goals"}, {"tactic": "exact fun H => \u27e8fun g g' h => H.comp_left h\u27e9", "state_before": "case mpr\nX Y : Type u\nf : X \u27f6 Y\n\u22a2 Function.Injective f \u2192 Mono f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Submonoid/Operations.lean", "full_name": "Submonoid.apply_coe_mem_map", "start": [250, 1], "end": [251, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Quaternion.lean", "full_name": "QuaternionAlgebra.coe_inj", "start": [163, 1], "end": [164, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/PythagoreanTriples.lean", "full_name": "PythagoreanTriple.ne_zero_of_coprime", "start": [228, 1], "end": [238, 70], "traced_tactics": [{"tactic": "suffices 0 < z * z by\n  rintro rfl\n  norm_num at this", "state_before": "x y z : \u2124\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\n\u22a2 z \u2260 0", "state_after": "x y z : \u2124\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\n\u22a2 0 < z * z"}, {"tactic": "rw [\u2190 h.eq, \u2190 sq, \u2190 sq]", "state_before": "x y z : \u2124\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\n\u22a2 0 < z * z", "state_after": "x y z : \u2124\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\n\u22a2 0 < x ^ 2 + y ^ 2"}, {"tactic": "have hc' : Int.gcd x y \u2260 0 := by\n  rw [hc]\n  exact one_ne_zero", "state_before": "x y z : \u2124\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\n\u22a2 0 < x ^ 2 + y ^ 2", "state_after": "x y z : \u2124\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\nhc' : Int.gcd x y \u2260 0\n\u22a2 0 < x ^ 2 + y ^ 2"}, {"tactic": "cases' Int.ne_zero_of_gcd hc' with hxz hyz", "state_before": "x y z : \u2124\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\nhc' : Int.gcd x y \u2260 0\n\u22a2 0 < x ^ 2 + y ^ 2", "state_after": "case inl\nx y z : \u2124\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\nhc' : Int.gcd x y \u2260 0\nhxz : x \u2260 0\n\u22a2 0 < x ^ 2 + y ^ 2\n\ncase inr\nx y z : \u2124\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\nhc' : Int.gcd x y \u2260 0\nhyz : y \u2260 0\n\u22a2 0 < x ^ 2 + y ^ 2"}, {"tactic": "rintro rfl", "state_before": "x y z : \u2124\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\nthis : 0 < z * z\n\u22a2 z \u2260 0", "state_after": "x y : \u2124\nhc : Int.gcd x y = 1\nh : PythagoreanTriple x y 0\nthis : 0 < 0 * 0\n\u22a2 False"}, {"tactic": "norm_num at this", "state_before": "x y : \u2124\nhc : Int.gcd x y = 1\nh : PythagoreanTriple x y 0\nthis : 0 < 0 * 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rw [hc]", "state_before": "x y z : \u2124\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\n\u22a2 Int.gcd x y \u2260 0", "state_after": "x y z : \u2124\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\n\u22a2 1 \u2260 0"}, {"tactic": "exact one_ne_zero", "state_before": "x y z : \u2124\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\n\u22a2 1 \u2260 0", "state_after": "no goals"}, {"tactic": "apply lt_add_of_pos_of_le (sq_pos_of_ne_zero x hxz) (sq_nonneg y)", "state_before": "case inl\nx y z : \u2124\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\nhc' : Int.gcd x y \u2260 0\nhxz : x \u2260 0\n\u22a2 0 < x ^ 2 + y ^ 2", "state_after": "no goals"}, {"tactic": "apply lt_add_of_le_of_pos (sq_nonneg x) (sq_pos_of_ne_zero y hyz)", "state_before": "case inr\nx y z : \u2124\nh : PythagoreanTriple x y z\nhc : Int.gcd x y = 1\nhc' : Int.gcd x y \u2260 0\nhyz : y \u2260 0\n\u22a2 0 < x ^ 2 + y ^ 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.Nonempty.vsub", "start": [636, 1], "end": [637, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.singletonMonoidHom_apply", "start": [820, 1], "end": [821, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean", "full_name": "MeasureTheory.SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge", "start": [97, 1], "end": [157, 9], "traced_tactics": [{"tactic": "induction' f using MeasureTheory.SimpleFunc.induction with c s hs f\u2081 f\u2082 _ h\u2081 h\u2082 generalizing \u03b5", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\nf : \u03b1 \u2192\u209b \u211d\u22650\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 g, (\u2200 (x : \u03b1), \u2191f x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) + \u03b5", "state_after": "case h_ind\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5\n\ncase h_add\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5\nh\u2082 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f\u2081 + f\u2082) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc) + \u03b5"}, {"tactic": "let f := SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)", "state_before": "case h_ind\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5", "state_after": "case h_ind\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5"}, {"tactic": "by_cases h : (\u222b\u207b x, f x \u2202\u03bc) = \u22a4", "state_before": "case h_ind\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : (\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5"}, {"tactic": "by_cases hc : c = 0", "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : c = 0\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5"}, {"tactic": "have : \u03bc s < \u03bc s + \u03b5 / c := by\n  have : (0 : \u211d\u22650\u221e) < \u03b5 / c := ENNReal.div_pos_iff.2 \u27e8\u03b50, ENNReal.coe_ne_top\u27e9\n  simpa using ENNReal.add_lt_add_left ?aux this", "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5", "state_after": "case aux\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : 0 < \u03b5 / \u2191c\n\u22a2 \u2191\u2191\u03bc s \u2260 \u22a4\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5"}, {"tactic": "case aux =>\n  classical\n  simpa [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const,\n    Function.const_apply, lintegral_const, ENNReal.coe_indicator, Set.univ_inter,\n    ENNReal.coe_ne_top, MeasurableSet.univ, ENNReal.mul_eq_top, SimpleFunc.const_zero,\n    or_false_iff, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,\n    SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, false_and_iff,\n    restrict_apply] using h", "state_before": "case aux\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : 0 < \u03b5 / \u2191c\n\u22a2 \u2191\u2191\u03bc s \u2260 \u22a4\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5"}, {"tactic": "obtain \u27e8u, su, u_open, \u03bcu\u27e9 : \u2203 (u : _), u \u2287 s \u2227 IsOpen u \u2227 \u03bc u < \u03bc s + \u03b5 / c :=\n  s.exists_isOpen_lt_of_lt _ this", "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5", "state_after": "case neg.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5"}, {"tactic": "refine'\n  \u27e8Set.indicator u fun _ => c, fun x => _, u_open.lowerSemicontinuous_indicator (zero_le _), _\u27e9", "state_before": "case neg.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5", "state_after": "case neg.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nx : \u03b1\n\u22a2 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 Set.indicator u (fun x => c) x\n\ncase neg.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 (\u222b\u207b (x : \u03b1), \u2191(Set.indicator u (fun x => c) x) \u2202\u03bc) \u2264\n    (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5"}, {"tactic": "refine'\n  \u27e8fun _ => c, fun x => _, lowerSemicontinuous_const, by\n    simp only [_root_.top_add, le_top, h]\u27e9", "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : (\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : (\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nx : \u03b1\n\u22a2 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 (fun x => c) x"}, {"tactic": "simp only [SimpleFunc.coe_const, SimpleFunc.const_zero, SimpleFunc.coe_zero,\n  Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise]", "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : (\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nx : \u03b1\n\u22a2 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 (fun x => c) x", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : (\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nx : \u03b1\n\u22a2 Set.piecewise s (Function.const \u03b1 c) 0 x \u2264 c"}, {"tactic": "exact Set.indicator_le_self _ _ _", "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : (\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nx : \u03b1\n\u22a2 Set.piecewise s (Function.const \u03b1 c) 0 x \u2264 c", "state_after": "no goals"}, {"tactic": "simp only [_root_.top_add, le_top, h]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : (\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\n\u22a2 (\u222b\u207b (x : \u03b1), \u2191((fun x => c) x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5", "state_after": "no goals"}, {"tactic": "refine' \u27e8fun _ => 0, _, lowerSemicontinuous_const, _\u27e9", "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : c = 0\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5", "state_after": "case pos.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : c = 0\n\u22a2 \u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 (fun x => 0) x\n\ncase pos.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : c = 0\n\u22a2 (\u222b\u207b (x : \u03b1), \u2191((fun x => 0) x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5"}, {"tactic": "classical\nsimp only [hc, Set.indicator_zero', Pi.zero_apply, SimpleFunc.const_zero, imp_true_iff,\n  eq_self_iff_true, SimpleFunc.coe_zero, Set.piecewise_eq_indicator,\n  SimpleFunc.coe_piecewise, le_zero_iff]", "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : c = 0\n\u22a2 \u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 (fun x => 0) x", "state_after": "no goals"}, {"tactic": "simp only [hc, Set.indicator_zero', Pi.zero_apply, SimpleFunc.const_zero, imp_true_iff,\n  eq_self_iff_true, SimpleFunc.coe_zero, Set.piecewise_eq_indicator,\n  SimpleFunc.coe_piecewise, le_zero_iff]", "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : c = 0\n\u22a2 \u2200 (x : \u03b1), \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 (fun x => 0) x", "state_after": "no goals"}, {"tactic": "simp only [lintegral_const, MulZeroClass.zero_mul, zero_le, ENNReal.coe_zero]", "state_before": "case pos.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : c = 0\n\u22a2 (\u222b\u207b (x : \u03b1), \u2191((fun x => 0) x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5", "state_after": "no goals"}, {"tactic": "have : (0 : \u211d\u22650\u221e) < \u03b5 / c := ENNReal.div_pos_iff.2 \u27e8\u03b50, ENNReal.coe_ne_top\u27e9", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\n\u22a2 \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : 0 < \u03b5 / \u2191c\n\u22a2 \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c"}, {"tactic": "simpa using ENNReal.add_lt_add_left ?aux this", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : 0 < \u03b5 / \u2191c\n\u22a2 \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c", "state_after": "no goals"}, {"tactic": "classical\nsimpa [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const,\n  Function.const_apply, lintegral_const, ENNReal.coe_indicator, Set.univ_inter,\n  ENNReal.coe_ne_top, MeasurableSet.univ, ENNReal.mul_eq_top, SimpleFunc.const_zero,\n  or_false_iff, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,\n  SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, false_and_iff,\n  restrict_apply] using h", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : 0 < \u03b5 / \u2191c\n\u22a2 \u2191\u2191\u03bc s \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "simpa [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const,\n  Function.const_apply, lintegral_const, ENNReal.coe_indicator, Set.univ_inter,\n  ENNReal.coe_ne_top, MeasurableSet.univ, ENNReal.mul_eq_top, SimpleFunc.const_zero,\n  or_false_iff, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,\n  SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, false_and_iff,\n  restrict_apply] using h", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : 0 < \u03b5 / \u2191c\n\u22a2 \u2191\u2191\u03bc s \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "simp only [SimpleFunc.coe_const, SimpleFunc.const_zero, SimpleFunc.coe_zero,\n  Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise]", "state_before": "case neg.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nx : \u03b1\n\u22a2 \u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x \u2264 Set.indicator u (fun x => c) x", "state_after": "case neg.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nx : \u03b1\n\u22a2 Set.piecewise s (Function.const \u03b1 c) 0 x \u2264 Set.indicator u (fun x => c) x"}, {"tactic": "exact Set.indicator_le_indicator_of_subset su (fun x => zero_le _) _", "state_before": "case neg.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nx : \u03b1\n\u22a2 Set.piecewise s (Function.const \u03b1 c) 0 x \u2264 Set.indicator u (fun x => c) x", "state_after": "no goals"}, {"tactic": "suffices (c : \u211d\u22650\u221e) * \u03bc u \u2264 c * \u03bc s + \u03b5 by\n  classical\n  simpa only [ENNReal.coe_indicator, u_open.measurableSet, lintegral_indicator,\n    lintegral_const, MeasurableSet.univ, Measure.restrict_apply, Set.univ_inter, const_zero,\n    coe_piecewise, coe_const, coe_zero, Set.piecewise_eq_indicator, Function.const_apply, hs]", "state_before": "case neg.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 (\u222b\u207b (x : \u03b1), \u2191(Set.indicator u (fun x => c) x) \u2202\u03bc) \u2264\n    (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5", "state_after": "case neg.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 \u2191c * \u2191\u2191\u03bc u \u2264 \u2191c * \u2191\u2191\u03bc s + \u03b5"}, {"tactic": "calc\n  (c : \u211d\u22650\u221e) * \u03bc u \u2264 c * (\u03bc s + \u03b5 / c) := mul_le_mul_left' \u03bcu.le _\n  _ = c * \u03bc s + \u03b5 := by\n    simp_rw [mul_add]\n    rw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top]\n    simpa using hc", "state_before": "case neg.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 \u2191c * \u2191\u2191\u03bc u \u2264 \u2191c * \u2191\u2191\u03bc s + \u03b5", "state_after": "no goals"}, {"tactic": "classical\nsimpa only [ENNReal.coe_indicator, u_open.measurableSet, lintegral_indicator,\n  lintegral_const, MeasurableSet.univ, Measure.restrict_apply, Set.univ_inter, const_zero,\n  coe_piecewise, coe_const, coe_zero, Set.piecewise_eq_indicator, Function.const_apply, hs]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis\u271d : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nthis : \u2191c * \u2191\u2191\u03bc u \u2264 \u2191c * \u2191\u2191\u03bc s + \u03b5\n\u22a2 (\u222b\u207b (x : \u03b1), \u2191(Set.indicator u (fun x => c) x) \u2202\u03bc) \u2264\n    (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5", "state_after": "no goals"}, {"tactic": "simpa only [ENNReal.coe_indicator, u_open.measurableSet, lintegral_indicator,\n  lintegral_const, MeasurableSet.univ, Measure.restrict_apply, Set.univ_inter, const_zero,\n  coe_piecewise, coe_const, coe_zero, Set.piecewise_eq_indicator, Function.const_apply, hs]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis\u271d : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nthis : \u2191c * \u2191\u2191\u03bc u \u2264 \u2191c * \u2191\u2191\u03bc s + \u03b5\n\u22a2 (\u222b\u207b (x : \u03b1), \u2191(Set.indicator u (fun x => c) x) \u2202\u03bc) \u2264\n    (\u222b\u207b (x : \u03b1), \u2191(\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0)) x) \u2202\u03bc) + \u03b5", "state_after": "no goals"}, {"tactic": "simp_rw [mul_add]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 \u2191c * (\u2191\u2191\u03bc s + \u03b5 / \u2191c) = \u2191c * \u2191\u2191\u03bc s + \u03b5", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 \u2191c * \u2191\u2191\u03bc s + \u2191c * (\u03b5 / \u2191c) = \u2191c * \u2191\u2191\u03bc s + \u03b5"}, {"tactic": "rw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 \u2191c * \u2191\u2191\u03bc s + \u2191c * (\u03b5 / \u2191c) = \u2191c * \u2191\u2191\u03bc s + \u03b5", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 \u2191c \u2260 0"}, {"tactic": "simpa using hc", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nc : \u211d\u22650\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\nf : \u03b1 \u2192\u209b \u211d\u22650 := piecewise s hs (const \u03b1 c) (const \u03b1 0)\nh : \u00ac(\u222b\u207b (x : \u03b1), \u2191(\u2191f x) \u2202\u03bc) = \u22a4\nhc : \u00acc = 0\nthis : \u2191\u2191\u03bc s < \u2191\u2191\u03bc s + \u03b5 / \u2191c\nu : Set \u03b1\nsu : u \u2287 s\nu_open : IsOpen u\n\u03bcu : \u2191\u2191\u03bc u < \u2191\u2191\u03bc s + \u03b5 / \u2191c\n\u22a2 \u2191c \u2260 0", "state_after": "no goals"}, {"tactic": "rcases h\u2081 (ENNReal.half_pos \u03b50).ne' with \u27e8g\u2081, f\u2081_le_g\u2081, g\u2081cont, g\u2081int\u27e9", "state_before": "case h_add\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5\nh\u2082 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f\u2081 + f\u2082) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc) + \u03b5", "state_after": "case h_add.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5\nh\u2082 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : (\u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5 / 2\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f\u2081 + f\u2082) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc) + \u03b5"}, {"tactic": "rcases h\u2082 (ENNReal.half_pos \u03b50).ne' with \u27e8g\u2082, f\u2082_le_g\u2082, g\u2082cont, g\u2082int\u27e9", "state_before": "case h_add.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5\nh\u2082 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : (\u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5 / 2\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f\u2081 + f\u2082) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc) + \u03b5", "state_after": "case h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5\nh\u2082 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : (\u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5 / 2\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f\u2081 + f\u2082) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc) + \u03b5"}, {"tactic": "refine'\n  \u27e8fun x => g\u2081 x + g\u2082 x, fun x => add_le_add (f\u2081_le_g\u2081 x) (f\u2082_le_g\u2082 x), g\u2081cont.add g\u2082cont, _\u27e9", "state_before": "case h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5\nh\u2082 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : (\u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5 / 2\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f\u2081 + f\u2082) x \u2264 g x) \u2227\n      LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc) + \u03b5", "state_after": "case h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5\nh\u2082 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : (\u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5 / 2\n\u22a2 (\u222b\u207b (x : \u03b1), \u2191((fun x => g\u2081 x + g\u2082 x) x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc) + \u03b5"}, {"tactic": "simp only [SimpleFunc.coe_add, ENNReal.coe_add, Pi.add_apply]", "state_before": "case h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5\nh\u2082 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : (\u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5 / 2\n\u22a2 (\u222b\u207b (x : \u03b1), \u2191((fun x => g\u2081 x + g\u2082 x) x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191(f\u2081 + f\u2082) x) \u2202\u03bc) + \u03b5", "state_after": "case h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5\nh\u2082 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : (\u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5 / 2\n\u22a2 (\u222b\u207b (x : \u03b1), \u2191(g\u2081 x) + \u2191(g\u2082 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) + \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5"}, {"tactic": "rw [lintegral_add_left f\u2081.measurable.coe_nnreal_ennreal,\n  lintegral_add_left g\u2081cont.measurable.coe_nnreal_ennreal]", "state_before": "case h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5\nh\u2082 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : (\u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5 / 2\n\u22a2 (\u222b\u207b (x : \u03b1), \u2191(g\u2081 x) + \u2191(g\u2082 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) + \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5", "state_after": "case h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5\nh\u2082 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : (\u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5 / 2\n\u22a2 ((\u222b\u207b (a : \u03b1), \u2191(g\u2081 a) \u2202\u03bc) + \u222b\u207b (a : \u03b1), \u2191(g\u2082 a) \u2202\u03bc) \u2264 ((\u222b\u207b (a : \u03b1), \u2191(\u2191f\u2081 a) \u2202\u03bc) + \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2082 a) \u2202\u03bc) + \u03b5"}, {"tactic": "convert add_le_add g\u2081int g\u2082int using 1", "state_before": "case h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5\nh\u2082 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : (\u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5 / 2\n\u22a2 ((\u222b\u207b (a : \u03b1), \u2191(g\u2081 a) \u2202\u03bc) + \u222b\u207b (a : \u03b1), \u2191(g\u2082 a) \u2202\u03bc) \u2264 ((\u222b\u207b (a : \u03b1), \u2191(\u2191f\u2081 a) \u2202\u03bc) + \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2082 a) \u2202\u03bc) + \u03b5", "state_after": "case h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5\nh\u2082 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : (\u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5 / 2\n\u22a2 ((\u222b\u207b (a : \u03b1), \u2191(\u2191f\u2081 a) \u2202\u03bc) + \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2082 a) \u2202\u03bc) + \u03b5 =\n    (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5 / 2 + ((\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5 / 2)"}, {"tactic": "simp only", "state_before": "case h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5\nh\u2082 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : (\u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5 / 2\n\u22a2 ((\u222b\u207b (a : \u03b1), \u2191(\u2191f\u2081 a) \u2202\u03bc) + \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2082 a) \u2202\u03bc) + \u03b5 =\n    (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5 / 2 + ((\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5 / 2)", "state_after": "case h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5\nh\u2082 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : (\u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5 / 2\n\u22a2 ((\u222b\u207b (a : \u03b1), \u2191(\u2191f\u2081 a) \u2202\u03bc) + \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2082 a) \u2202\u03bc) + \u03b5 =\n    (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5 / 2 + ((\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5 / 2)"}, {"tactic": "conv_lhs => rw [\u2190 ENNReal.add_halves \u03b5]", "state_before": "case h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5\nh\u2082 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : (\u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5 / 2\n\u22a2 ((\u222b\u207b (a : \u03b1), \u2191(\u2191f\u2081 a) \u2202\u03bc) + \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2082 a) \u2202\u03bc) + \u03b5 =\n    (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5 / 2 + ((\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5 / 2)", "state_after": "case h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5\nh\u2082 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : (\u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5 / 2\n\u22a2 ((\u222b\u207b (a : \u03b1), \u2191(\u2191f\u2081 a) \u2202\u03bc) + \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2082 a) \u2202\u03bc) + (\u03b5 / 2 + \u03b5 / 2) =\n    (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5 / 2 + ((\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5 / 2)"}, {"tactic": "abel", "state_before": "case h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : WeaklyRegular \u03bc\n\u03b5\u271d : \u211d\u22650\u221e\n\u03b50\u271d : \u03b5\u271d \u2260 0\nf\u2081 f\u2082 : \u03b1 \u2192\u209b \u211d\u22650\na\u271d : Disjoint (Function.support \u2191f\u2081) (Function.support \u2191f\u2082)\nh\u2081 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5\nh\u2082 :\n  \u2200 {\u03b5 : \u211d\u22650\u221e},\n    \u03b5 \u2260 0 \u2192\n      \u2203 g, (\u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g x) \u2227 LowerSemicontinuous g \u2227 (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : \u03b5 \u2260 0\ng\u2081 : \u03b1 \u2192 \u211d\u22650\nf\u2081_le_g\u2081 : \u2200 (x : \u03b1), \u2191f\u2081 x \u2264 g\u2081 x\ng\u2081cont : LowerSemicontinuous g\u2081\ng\u2081int : (\u222b\u207b (x : \u03b1), \u2191(g\u2081 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5 / 2\ng\u2082 : \u03b1 \u2192 \u211d\u22650\nf\u2082_le_g\u2082 : \u2200 (x : \u03b1), \u2191f\u2082 x \u2264 g\u2082 x\ng\u2082cont : LowerSemicontinuous g\u2082\ng\u2082int : (\u222b\u207b (x : \u03b1), \u2191(g\u2082 x) \u2202\u03bc) \u2264 (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5 / 2\n\u22a2 ((\u222b\u207b (a : \u03b1), \u2191(\u2191f\u2081 a) \u2202\u03bc) + \u222b\u207b (a : \u03b1), \u2191(\u2191f\u2082 a) \u2202\u03bc) + (\u03b5 / 2 + \u03b5 / 2) =\n    (\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2081 x) \u2202\u03bc) + \u03b5 / 2 + ((\u222b\u207b (x : \u03b1), \u2191(\u2191f\u2082 x) \u2202\u03bc) + \u03b5 / 2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Polynomial/Basic.lean", "full_name": "Polynomial.coeff_prod_mem_ideal_pow_tsub", "start": [618, 1], "end": [633, 76], "traced_tactics": [{"tactic": "induction' s using Finset.induction with a s ha hs generalizing k", "state_before": "R : Type u\nS : Type ?u.185353\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nI\u271d : Ideal R[X]\n\u03b9 : Type u_1\ns : Finset \u03b9\nf : \u03b9 \u2192 R[X]\nI : Ideal R\nn : \u03b9 \u2192 \u2115\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk : \u2115\n\u22a2 coeff (Finset.prod s f) k \u2208 I ^ (Finset.sum s n - k)", "state_after": "case empty\nR : Type u\nS : Type ?u.185353\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nI\u271d : Ideal R[X]\n\u03b9 : Type u_1\ns : Finset \u03b9\nf : \u03b9 \u2192 R[X]\nI : Ideal R\nn : \u03b9 \u2192 \u2115\nh\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk\u271d : \u2115\nh : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk : \u2115\n\u22a2 coeff (Finset.prod \u2205 f) k \u2208 I ^ (Finset.sum \u2205 n - k)\n\ncase insert\nR : Type u\nS : Type ?u.185353\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nI\u271d : Ideal R[X]\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R[X]\nI : Ideal R\nn : \u03b9 \u2192 \u2115\nh\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk\u271d : \u2115\na : \u03b9\ns : Finset \u03b9\nha : \u00aca \u2208 s\nhs :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)) \u2192\n    \u2200 (k : \u2115), coeff (Finset.prod s f) k \u2208 I ^ (Finset.sum s n - k)\nh : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk : \u2115\n\u22a2 coeff (Finset.prod (insert a s) f) k \u2208 I ^ (Finset.sum (insert a s) n - k)"}, {"tactic": "rw [sum_empty, prod_empty, coeff_one, zero_tsub, pow_zero, Ideal.one_eq_top]", "state_before": "case empty\nR : Type u\nS : Type ?u.185353\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nI\u271d : Ideal R[X]\n\u03b9 : Type u_1\ns : Finset \u03b9\nf : \u03b9 \u2192 R[X]\nI : Ideal R\nn : \u03b9 \u2192 \u2115\nh\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk\u271d : \u2115\nh : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk : \u2115\n\u22a2 coeff (Finset.prod \u2205 f) k \u2208 I ^ (Finset.sum \u2205 n - k)", "state_after": "case empty\nR : Type u\nS : Type ?u.185353\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nI\u271d : Ideal R[X]\n\u03b9 : Type u_1\ns : Finset \u03b9\nf : \u03b9 \u2192 R[X]\nI : Ideal R\nn : \u03b9 \u2192 \u2115\nh\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk\u271d : \u2115\nh : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk : \u2115\n\u22a2 (if 0 = k then 1 else 0) \u2208 \u22a4"}, {"tactic": "exact Submodule.mem_top", "state_before": "case empty\nR : Type u\nS : Type ?u.185353\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nI\u271d : Ideal R[X]\n\u03b9 : Type u_1\ns : Finset \u03b9\nf : \u03b9 \u2192 R[X]\nI : Ideal R\nn : \u03b9 \u2192 \u2115\nh\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk\u271d : \u2115\nh : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk : \u2115\n\u22a2 (if 0 = k then 1 else 0) \u2208 \u22a4", "state_after": "no goals"}, {"tactic": "rw [sum_insert ha, prod_insert ha, coeff_mul]", "state_before": "case insert\nR : Type u\nS : Type ?u.185353\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nI\u271d : Ideal R[X]\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R[X]\nI : Ideal R\nn : \u03b9 \u2192 \u2115\nh\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk\u271d : \u2115\na : \u03b9\ns : Finset \u03b9\nha : \u00aca \u2208 s\nhs :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)) \u2192\n    \u2200 (k : \u2115), coeff (Finset.prod s f) k \u2208 I ^ (Finset.sum s n - k)\nh : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk : \u2115\n\u22a2 coeff (Finset.prod (insert a s) f) k \u2208 I ^ (Finset.sum (insert a s) n - k)", "state_after": "case insert\nR : Type u\nS : Type ?u.185353\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nI\u271d : Ideal R[X]\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R[X]\nI : Ideal R\nn : \u03b9 \u2192 \u2115\nh\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk\u271d : \u2115\na : \u03b9\ns : Finset \u03b9\nha : \u00aca \u2208 s\nhs :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)) \u2192\n    \u2200 (k : \u2115), coeff (Finset.prod s f) k \u2208 I ^ (Finset.sum s n - k)\nh : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk : \u2115\n\u22a2 \u2211 x in Nat.antidiagonal k, coeff (f a) x.fst * coeff (\u220f x in s, f x) x.snd \u2208 I ^ (n a + \u2211 x in s, n x - k)"}, {"tactic": "apply sum_mem", "state_before": "case insert\nR : Type u\nS : Type ?u.185353\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nI\u271d : Ideal R[X]\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R[X]\nI : Ideal R\nn : \u03b9 \u2192 \u2115\nh\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk\u271d : \u2115\na : \u03b9\ns : Finset \u03b9\nha : \u00aca \u2208 s\nhs :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)) \u2192\n    \u2200 (k : \u2115), coeff (Finset.prod s f) k \u2208 I ^ (Finset.sum s n - k)\nh : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk : \u2115\n\u22a2 \u2211 x in Nat.antidiagonal k, coeff (f a) x.fst * coeff (\u220f x in s, f x) x.snd \u2208 I ^ (n a + \u2211 x in s, n x - k)", "state_after": "case insert.a\nR : Type u\nS : Type ?u.185353\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nI\u271d : Ideal R[X]\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R[X]\nI : Ideal R\nn : \u03b9 \u2192 \u2115\nh\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk\u271d : \u2115\na : \u03b9\ns : Finset \u03b9\nha : \u00aca \u2208 s\nhs :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)) \u2192\n    \u2200 (k : \u2115), coeff (Finset.prod s f) k \u2208 I ^ (Finset.sum s n - k)\nh : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk : \u2115\n\u22a2 \u2200 (c : \u2115 \u00d7 \u2115),\n    c \u2208 Nat.antidiagonal k \u2192 coeff (f a) c.fst * coeff (\u220f x in s, f x) c.snd \u2208 I ^ (n a + \u2211 x in s, n x - k)"}, {"tactic": "rintro \u27e8i, j\u27e9 e", "state_before": "case insert.a\nR : Type u\nS : Type ?u.185353\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nI\u271d : Ideal R[X]\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R[X]\nI : Ideal R\nn : \u03b9 \u2192 \u2115\nh\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk\u271d : \u2115\na : \u03b9\ns : Finset \u03b9\nha : \u00aca \u2208 s\nhs :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)) \u2192\n    \u2200 (k : \u2115), coeff (Finset.prod s f) k \u2208 I ^ (Finset.sum s n - k)\nh : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk : \u2115\n\u22a2 \u2200 (c : \u2115 \u00d7 \u2115),\n    c \u2208 Nat.antidiagonal k \u2192 coeff (f a) c.fst * coeff (\u220f x in s, f x) c.snd \u2208 I ^ (n a + \u2211 x in s, n x - k)", "state_after": "case insert.a.mk\nR : Type u\nS : Type ?u.185353\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nI\u271d : Ideal R[X]\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R[X]\nI : Ideal R\nn : \u03b9 \u2192 \u2115\nh\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk\u271d : \u2115\na : \u03b9\ns : Finset \u03b9\nha : \u00aca \u2208 s\nhs :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)) \u2192\n    \u2200 (k : \u2115), coeff (Finset.prod s f) k \u2208 I ^ (Finset.sum s n - k)\nh : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk i j : \u2115\ne : (i, j) \u2208 Nat.antidiagonal k\n\u22a2 coeff (f a) (i, j).fst * coeff (\u220f x in s, f x) (i, j).snd \u2208 I ^ (n a + \u2211 x in s, n x - k)"}, {"tactic": "obtain rfl : i + j = k := Nat.mem_antidiagonal.mp e", "state_before": "case insert.a.mk\nR : Type u\nS : Type ?u.185353\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nI\u271d : Ideal R[X]\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R[X]\nI : Ideal R\nn : \u03b9 \u2192 \u2115\nh\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk\u271d : \u2115\na : \u03b9\ns : Finset \u03b9\nha : \u00aca \u2208 s\nhs :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)) \u2192\n    \u2200 (k : \u2115), coeff (Finset.prod s f) k \u2208 I ^ (Finset.sum s n - k)\nh : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk i j : \u2115\ne : (i, j) \u2208 Nat.antidiagonal k\n\u22a2 coeff (f a) (i, j).fst * coeff (\u220f x in s, f x) (i, j).snd \u2208 I ^ (n a + \u2211 x in s, n x - k)", "state_after": "case insert.a.mk\nR : Type u\nS : Type ?u.185353\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nI\u271d : Ideal R[X]\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R[X]\nI : Ideal R\nn : \u03b9 \u2192 \u2115\nh\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk : \u2115\na : \u03b9\ns : Finset \u03b9\nha : \u00aca \u2208 s\nhs :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)) \u2192\n    \u2200 (k : \u2115), coeff (Finset.prod s f) k \u2208 I ^ (Finset.sum s n - k)\nh : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\ni j : \u2115\ne : (i, j) \u2208 Nat.antidiagonal (i + j)\n\u22a2 coeff (f a) (i, j).fst * coeff (\u220f x in s, f x) (i, j).snd \u2208 I ^ (n a + \u2211 x in s, n x - (i + j))"}, {"tactic": "apply Ideal.pow_le_pow add_tsub_add_le_tsub_add_tsub", "state_before": "case insert.a.mk\nR : Type u\nS : Type ?u.185353\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nI\u271d : Ideal R[X]\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R[X]\nI : Ideal R\nn : \u03b9 \u2192 \u2115\nh\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk : \u2115\na : \u03b9\ns : Finset \u03b9\nha : \u00aca \u2208 s\nhs :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)) \u2192\n    \u2200 (k : \u2115), coeff (Finset.prod s f) k \u2208 I ^ (Finset.sum s n - k)\nh : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\ni j : \u2115\ne : (i, j) \u2208 Nat.antidiagonal (i + j)\n\u22a2 coeff (f a) (i, j).fst * coeff (\u220f x in s, f x) (i, j).snd \u2208 I ^ (n a + \u2211 x in s, n x - (i + j))", "state_after": "case insert.a.mk.a\nR : Type u\nS : Type ?u.185353\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nI\u271d : Ideal R[X]\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R[X]\nI : Ideal R\nn : \u03b9 \u2192 \u2115\nh\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk : \u2115\na : \u03b9\ns : Finset \u03b9\nha : \u00aca \u2208 s\nhs :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)) \u2192\n    \u2200 (k : \u2115), coeff (Finset.prod s f) k \u2208 I ^ (Finset.sum s n - k)\nh : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\ni j : \u2115\ne : (i, j) \u2208 Nat.antidiagonal (i + j)\n\u22a2 coeff (f a) (i, j).fst * coeff (\u220f x in s, f x) (i, j).snd \u2208 I ^ (n a - i + (\u2211 x in s, n x - j))"}, {"tactic": "rw [pow_add]", "state_before": "case insert.a.mk.a\nR : Type u\nS : Type ?u.185353\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nI\u271d : Ideal R[X]\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R[X]\nI : Ideal R\nn : \u03b9 \u2192 \u2115\nh\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk : \u2115\na : \u03b9\ns : Finset \u03b9\nha : \u00aca \u2208 s\nhs :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)) \u2192\n    \u2200 (k : \u2115), coeff (Finset.prod s f) k \u2208 I ^ (Finset.sum s n - k)\nh : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\ni j : \u2115\ne : (i, j) \u2208 Nat.antidiagonal (i + j)\n\u22a2 coeff (f a) (i, j).fst * coeff (\u220f x in s, f x) (i, j).snd \u2208 I ^ (n a - i + (\u2211 x in s, n x - j))", "state_after": "case insert.a.mk.a\nR : Type u\nS : Type ?u.185353\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nI\u271d : Ideal R[X]\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R[X]\nI : Ideal R\nn : \u03b9 \u2192 \u2115\nh\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk : \u2115\na : \u03b9\ns : Finset \u03b9\nha : \u00aca \u2208 s\nhs :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)) \u2192\n    \u2200 (k : \u2115), coeff (Finset.prod s f) k \u2208 I ^ (Finset.sum s n - k)\nh : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\ni j : \u2115\ne : (i, j) \u2208 Nat.antidiagonal (i + j)\n\u22a2 coeff (f a) (i, j).fst * coeff (\u220f x in s, f x) (i, j).snd \u2208 I ^ (n a - i) * I ^ (\u2211 x in s, n x - j)"}, {"tactic": "exact\n  Ideal.mul_mem_mul (h _ (Finset.mem_insert.mpr <| Or.inl rfl) _)\n    (hs (fun i hi k => h _ (Finset.mem_insert.mpr <| Or.inr hi) _) j)", "state_before": "case insert.a.mk.a\nR : Type u\nS : Type ?u.185353\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Semiring S\nI\u271d : Ideal R[X]\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R[X]\nI : Ideal R\nn : \u03b9 \u2192 \u2115\nh\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\nk : \u2115\na : \u03b9\ns : Finset \u03b9\nha : \u00aca \u2208 s\nhs :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)) \u2192\n    \u2200 (k : \u2115), coeff (Finset.prod s f) k \u2208 I ^ (Finset.sum s n - k)\nh : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 \u2200 (k : \u2115), coeff (f i) k \u2208 I ^ (n i - k)\ni j : \u2115\ne : (i, j) \u2208 Nat.antidiagonal (i + j)\n\u22a2 coeff (f a) (i, j).fst * coeff (\u220f x in s, f x) (i, j).snd \u2208 I ^ (n a - i) * I ^ (\u2211 x in s, n x - j)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean", "full_name": "Matrix.pow_eq_aeval_mod_charpoly", "start": [237, 1], "end": [238, 101], "traced_tactics": [{"tactic": "rw [\u2190 aeval_eq_aeval_mod_charpoly, map_pow, aeval_X]", "state_before": "R : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nM : Matrix n n R\nk : \u2115\n\u22a2 M ^ k = \u2191(aeval M) (X ^ k %\u2098 charpoly M)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Between.lean", "full_name": "Collinear.wbtw_or_wbtw_or_wbtw", "start": [863, 1], "end": [885, 72], "traced_tactics": [{"tactic": "rw [collinear_iff_of_mem (Set.mem_insert _ _)] at h", "state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : Collinear R {x, y, z}\n\u22a2 Wbtw R x y z \u2228 Wbtw R y z x \u2228 Wbtw R z x y", "state_after": "R : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : \u2203 v, \u2200 (p : P), p \u2208 {x, y, z} \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x y z \u2228 Wbtw R y z x \u2228 Wbtw R z x y"}, {"tactic": "rcases h with \u27e8v, h\u27e9", "state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : \u2203 v, \u2200 (p : P), p \u2208 {x, y, z} \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x y z \u2228 Wbtw R y z x \u2228 Wbtw R z x y", "state_after": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nv : V\nh : \u2200 (p : P), p \u2208 {x, y, z} \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x y z \u2228 Wbtw R y z x \u2228 Wbtw R z x y"}, {"tactic": "simp_rw [Set.mem_insert_iff, Set.mem_singleton_iff] at h", "state_before": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nv : V\nh : \u2200 (p : P), p \u2208 {x, y, z} \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x y z \u2228 Wbtw R y z x \u2228 Wbtw R z x y", "state_after": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nv : V\nh : \u2200 (p : P), p = x \u2228 p = y \u2228 p = z \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x y z \u2228 Wbtw R y z x \u2228 Wbtw R z x y"}, {"tactic": "have hy := h y (Or.inr (Or.inl rfl))", "state_before": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nv : V\nh : \u2200 (p : P), p = x \u2228 p = y \u2228 p = z \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x y z \u2228 Wbtw R y z x \u2228 Wbtw R z x y", "state_after": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nv : V\nh : \u2200 (p : P), p = x \u2228 p = y \u2228 p = z \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy : \u2203 r, y = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x y z \u2228 Wbtw R y z x \u2228 Wbtw R z x y"}, {"tactic": "have hz := h z (Or.inr (Or.inr rfl))", "state_before": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nv : V\nh : \u2200 (p : P), p = x \u2228 p = y \u2228 p = z \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy : \u2203 r, y = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x y z \u2228 Wbtw R y z x \u2228 Wbtw R z x y", "state_after": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nv : V\nh : \u2200 (p : P), p = x \u2228 p = y \u2228 p = z \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy : \u2203 r, y = r \u2022 v +\u1d65 x\nhz : \u2203 r, z = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x y z \u2228 Wbtw R y z x \u2228 Wbtw R z x y"}, {"tactic": "rcases hy with \u27e8ty, rfl\u27e9", "state_before": "case intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nv : V\nh : \u2200 (p : P), p = x \u2228 p = y \u2228 p = z \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy : \u2203 r, y = r \u2022 v +\u1d65 x\nhz : \u2203 r, z = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x y z \u2228 Wbtw R y z x \u2228 Wbtw R z x y", "state_after": "case intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nv : V\nhz : \u2203 r, z = r \u2022 v +\u1d65 x\nty : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = z \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) z \u2228 Wbtw R (ty \u2022 v +\u1d65 x) z x \u2228 Wbtw R z x (ty \u2022 v +\u1d65 x)"}, {"tactic": "rcases hz with \u27e8tz, rfl\u27e9", "state_before": "case intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx z : P\nv : V\nhz : \u2203 r, z = r \u2022 v +\u1d65 x\nty : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = z \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) z \u2228 Wbtw R (ty \u2022 v +\u1d65 x) z x \u2228 Wbtw R z x (ty \u2022 v +\u1d65 x)", "state_after": "case intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)"}, {"tactic": "rcases lt_trichotomy ty 0 with (hy0 | rfl | hy0)", "state_before": "case intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)", "state_after": "case intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : ty < 0\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)\n\ncase intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\ntz : R\nh : \u2200 (p : P), p = x \u2228 p = 0 \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x (0 \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (0 \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (0 \u2022 v +\u1d65 x)\n\ncase intro.intro.intro.inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : 0 < ty\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)"}, {"tactic": "rcases lt_trichotomy tz 0 with (hz0 | rfl | hz0)", "state_before": "case intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : ty < 0\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)", "state_after": "case intro.intro.intro.inl.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : ty < 0\nhz0 : tz < 0\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)\n\ncase intro.intro.intro.inl.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty : R\nhy0 : ty < 0\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = 0 \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (0 \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (0 \u2022 v +\u1d65 x) x \u2228 Wbtw R (0 \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)\n\ncase intro.intro.intro.inl.inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : ty < 0\nhz0 : 0 < tz\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)"}, {"tactic": "rw [wbtw_comm (z := x)]", "state_before": "case intro.intro.intro.inl.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : ty < 0\nhz0 : tz < 0\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)", "state_after": "case intro.intro.intro.inl.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : ty < 0\nhz0 : tz < 0\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R x (tz \u2022 v +\u1d65 x) (ty \u2022 v +\u1d65 x) \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)"}, {"tactic": "rw [\u2190 or_assoc]", "state_before": "case intro.intro.intro.inl.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : ty < 0\nhz0 : tz < 0\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R x (tz \u2022 v +\u1d65 x) (ty \u2022 v +\u1d65 x) \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)", "state_after": "case intro.intro.intro.inl.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : ty < 0\nhz0 : tz < 0\n\u22a2 (Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R x (tz \u2022 v +\u1d65 x) (ty \u2022 v +\u1d65 x)) \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)"}, {"tactic": "exact Or.inl (wbtw_or_wbtw_smul_vadd_of_nonpos _ _ hy0.le hz0.le)", "state_before": "case intro.intro.intro.inl.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : ty < 0\nhz0 : tz < 0\n\u22a2 (Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R x (tz \u2022 v +\u1d65 x) (ty \u2022 v +\u1d65 x)) \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)", "state_after": "no goals"}, {"tactic": "simp", "state_before": "case intro.intro.intro.inl.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty : R\nhy0 : ty < 0\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = 0 \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (0 \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (0 \u2022 v +\u1d65 x) x \u2228 Wbtw R (0 \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)", "state_after": "no goals"}, {"tactic": "exact Or.inr (Or.inr (wbtw_smul_vadd_smul_vadd_of_nonneg_of_nonpos _ _ hz0.le hy0.le))", "state_before": "case intro.intro.intro.inl.inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : ty < 0\nhz0 : 0 < tz\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)", "state_after": "no goals"}, {"tactic": "simp", "state_before": "case intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\ntz : R\nh : \u2200 (p : P), p = x \u2228 p = 0 \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x (0 \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (0 \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (0 \u2022 v +\u1d65 x)", "state_after": "no goals"}, {"tactic": "rcases lt_trichotomy tz 0 with (hz0 | rfl | hz0)", "state_before": "case intro.intro.intro.inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : 0 < ty\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)", "state_after": "case intro.intro.intro.inr.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : 0 < ty\nhz0 : tz < 0\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)\n\ncase intro.intro.intro.inr.inr.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty : R\nhy0 : 0 < ty\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = 0 \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (0 \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (0 \u2022 v +\u1d65 x) x \u2228 Wbtw R (0 \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)\n\ncase intro.intro.intro.inr.inr.inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : 0 < ty\nhz0 : 0 < tz\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)"}, {"tactic": "refine' Or.inr (Or.inr (wbtw_smul_vadd_smul_vadd_of_nonpos_of_nonneg _ _ hz0.le hy0.le))", "state_before": "case intro.intro.intro.inr.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : 0 < ty\nhz0 : tz < 0\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)", "state_after": "no goals"}, {"tactic": "simp", "state_before": "case intro.intro.intro.inr.inr.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty : R\nhy0 : 0 < ty\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = 0 \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (0 \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (0 \u2022 v +\u1d65 x) x \u2228 Wbtw R (0 \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)", "state_after": "no goals"}, {"tactic": "rw [wbtw_comm (z := x)]", "state_before": "case intro.intro.intro.inr.inr.inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : 0 < ty\nhz0 : 0 < tz\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) x \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)", "state_after": "case intro.intro.intro.inr.inr.inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : 0 < ty\nhz0 : 0 < tz\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R x (tz \u2022 v +\u1d65 x) (ty \u2022 v +\u1d65 x) \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)"}, {"tactic": "rw [\u2190 or_assoc]", "state_before": "case intro.intro.intro.inr.inr.inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : 0 < ty\nhz0 : 0 < tz\n\u22a2 Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R x (tz \u2022 v +\u1d65 x) (ty \u2022 v +\u1d65 x) \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)", "state_after": "case intro.intro.intro.inr.inr.inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : 0 < ty\nhz0 : 0 < tz\n\u22a2 (Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R x (tz \u2022 v +\u1d65 x) (ty \u2022 v +\u1d65 x)) \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)"}, {"tactic": "exact Or.inl (wbtw_or_wbtw_smul_vadd_of_nonneg _ _ hy0.le hz0.le)", "state_before": "case intro.intro.intro.inr.inr.inr.inr\nR : Type u_1\nV : Type u_2\nV' : Type ?u.594033\nP : Type u_3\nP' : Type ?u.594039\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : \u2200 (p : P), p = x \u2228 p = ty \u2022 v +\u1d65 x \u2228 p = tz \u2022 v +\u1d65 x \u2192 \u2203 r, p = r \u2022 v +\u1d65 x\nhy0 : 0 < ty\nhz0 : 0 < tz\n\u22a2 (Wbtw R x (ty \u2022 v +\u1d65 x) (tz \u2022 v +\u1d65 x) \u2228 Wbtw R x (tz \u2022 v +\u1d65 x) (ty \u2022 v +\u1d65 x)) \u2228 Wbtw R (tz \u2022 v +\u1d65 x) x (ty \u2022 v +\u1d65 x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.Finite.toFinset_offDiag", "start": [1074, 1], "end": [1076, 24], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns : Set \u03b1\ninst\u271d : DecidableEq \u03b1\nhs : Set.Finite s\n\u22a2 \u2200 (a : \u03b1 \u00d7 \u03b1), a \u2208 Finite.toFinset (_ : Set.Finite (offDiag s)) \u2194 a \u2208 Finset.offDiag (Finite.toFinset hs)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Span.lean", "full_name": "Submodule.span_insert_eq_span", "start": [494, 1], "end": [495, 92], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/Haar/Quotient.lean", "full_name": "MeasureTheory.IsFundamentalDomain.isMulLeftInvariant_map", "start": [103, 1], "end": [115, 15], "traced_tactics": [{"tactic": "apply Measure.ext", "state_before": "G : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\n\u22a2 map (fun x_1 => x * x_1) (map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5)) =\n    map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5)", "state_after": "case h\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\n\u22a2 \u2200 (s : Set (G \u29f8 \u0393)),\n    MeasurableSet s \u2192\n      \u2191\u2191(map (fun x_1 => x * x_1) (map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5))) s =\n        \u2191\u2191(map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5)) s"}, {"tactic": "intro A hA", "state_before": "case h\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\n\u22a2 \u2200 (s : Set (G \u29f8 \u0393)),\n    MeasurableSet s \u2192\n      \u2191\u2191(map (fun x_1 => x * x_1) (map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5))) s =\n        \u2191\u2191(map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5)) s", "state_after": "case h\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u22a2 \u2191\u2191(map (fun x_1 => x * x_1) (map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5))) A =\n    \u2191\u2191(map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5)) A"}, {"tactic": "obtain \u27e8x\u2081, h\u27e9 := @Quotient.exists_rep _ (QuotientGroup.leftRel \u0393) x", "state_before": "case h\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u22a2 \u2191\u2191(map (fun x_1 => x * x_1) (map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5))) A =\n    \u2191\u2191(map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5)) A", "state_after": "case h.intro\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\nx\u2081 : G\nh : Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 = x\n\u22a2 \u2191\u2191(map (fun x_1 => x * x_1) (map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5))) A =\n    \u2191\u2191(map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5)) A"}, {"tactic": "haveI := h\ud835\udcd5.smulInvariantMeasure_map", "state_before": "case h.intro\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\nx\u2081 : G\nh : Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 = x\n\u22a2 \u2191\u2191(map (fun x_1 => x * x_1) (map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5))) A =\n    \u2191\u2191(map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5)) A", "state_after": "case h.intro\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\nx\u2081 : G\nh : Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 = x\nthis : SMulInvariantMeasure G (G \u29f8 \u0393) (map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5))\n\u22a2 \u2191\u2191(map (fun x_1 => x * x_1) (map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5))) A =\n    \u2191\u2191(map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5)) A"}, {"tactic": "convert measure_preimage_smul x\u2081 ((Measure.map QuotientGroup.mk) (\u03bc.restrict \ud835\udcd5)) A using 1", "state_before": "case h.intro\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\nx\u2081 : G\nh : Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 = x\nthis : SMulInvariantMeasure G (G \u29f8 \u0393) (map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5))\n\u22a2 \u2191\u2191(map (fun x_1 => x * x_1) (map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5))) A =\n    \u2191\u2191(map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5)) A", "state_after": "case h.e'_2\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\nx\u2081 : G\nh : Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 = x\nthis : SMulInvariantMeasure G (G \u29f8 \u0393) (map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5))\n\u22a2 \u2191\u2191(map (fun x_1 => x * x_1) (map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5))) A =\n    \u2191\u2191(map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5)) ((fun x => x\u2081 \u2022 x) \u207b\u00b9' A)"}, {"tactic": "rw [\u2190 h, Measure.map_apply]", "state_before": "case h.e'_2\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\nx\u2081 : G\nh : Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 = x\nthis : SMulInvariantMeasure G (G \u29f8 \u0393) (map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5))\n\u22a2 \u2191\u2191(map (fun x_1 => x * x_1) (map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5))) A =\n    \u2191\u2191(map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5)) ((fun x => x\u2081 \u2022 x) \u207b\u00b9' A)", "state_after": "case h.e'_2\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\nx\u2081 : G\nh : Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 = x\nthis : SMulInvariantMeasure G (G \u29f8 \u0393) (map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5))\n\u22a2 \u2191\u2191(map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5))\n      ((fun x => Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 * x) \u207b\u00b9' A) =\n    \u2191\u2191(map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5)) ((fun x => x\u2081 \u2022 x) \u207b\u00b9' A)\n\ncase h.e'_2.hf\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\nx\u2081 : G\nh : Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 = x\nthis : SMulInvariantMeasure G (G \u29f8 \u0393) (map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5))\n\u22a2 Measurable fun x => Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 * x\n\ncase h.e'_2.hs\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\nx\u2081 : G\nh : Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 = x\nthis : SMulInvariantMeasure G (G \u29f8 \u0393) (map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5))\n\u22a2 MeasurableSet A"}, {"tactic": "rfl", "state_before": "case h.e'_2\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\nx\u2081 : G\nh : Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 = x\nthis : SMulInvariantMeasure G (G \u29f8 \u0393) (map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5))\n\u22a2 \u2191\u2191(map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5))\n      ((fun x => Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 * x) \u207b\u00b9' A) =\n    \u2191\u2191(map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5)) ((fun x => x\u2081 \u2022 x) \u207b\u00b9' A)", "state_after": "no goals"}, {"tactic": "exact measurable_const_mul _", "state_before": "case h.e'_2.hf\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\nx\u2081 : G\nh : Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 = x\nthis : SMulInvariantMeasure G (G \u29f8 \u0393) (map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5))\n\u22a2 Measurable fun x => Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 * x", "state_after": "no goals"}, {"tactic": "exact hA", "state_before": "case h.e'_2.hs\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\nx\u2081 : G\nh : Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 = x\nthis : SMulInvariantMeasure G (G \u29f8 \u0393) (map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5))\n\u22a2 MeasurableSet A", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean", "full_name": "CategoryTheory.IsPullback.paste_horiz_iff", "start": [547, 1], "end": [551, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Artinian.lean", "full_name": "Function.Surjective.isArtinianRing", "start": [371, 1], "end": [374, 62], "traced_tactics": [{"tactic": "rw [isArtinianRing_iff, isArtinian_iff_wellFounded] at H\u22a2", "state_before": "R : Type u_1\ninst\u271d\u00b2 : Ring R\nS : Type u_2\ninst\u271d\u00b9 : Ring S\nF : Type u_3\ninst\u271d : RingHomClass F R S\nf : F\nhf : Surjective \u2191f\nH : IsArtinianRing R\n\u22a2 IsArtinianRing S", "state_after": "R : Type u_1\ninst\u271d\u00b2 : Ring R\nS : Type u_2\ninst\u271d\u00b9 : Ring S\nF : Type u_3\ninst\u271d : RingHomClass F R S\nf : F\nhf : Surjective \u2191f\nH : WellFounded fun x x_1 => x < x_1\n\u22a2 WellFounded fun x x_1 => x < x_1"}, {"tactic": "exact (Ideal.orderEmbeddingOfSurjective f hf).wellFounded H", "state_before": "R : Type u_1\ninst\u271d\u00b2 : Ring R\nS : Type u_2\ninst\u271d\u00b9 : Ring S\nF : Type u_3\ninst\u271d : RingHomClass F R S\nf : F\nhf : Surjective \u2191f\nH : WellFounded fun x x_1 => x < x_1\n\u22a2 WellFounded fun x x_1 => x < x_1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/lpSpace.lean", "full_name": "lp.norm_neg", "start": [486, 1], "end": [496, 84], "traced_tactics": [{"tactic": "rcases p.trichotomy with (rfl | rfl | hp)", "state_before": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\n\u22a2 \u2016-f\u2016 = \u2016f\u2016", "state_after": "case inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E 0 }\n\u22a2 \u2016-f\u2016 = \u2016f\u2016\n\ncase inr.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\n\u22a2 \u2016-f\u2016 = \u2016f\u2016\n\ncase inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\n\u22a2 \u2016-f\u2016 = \u2016f\u2016"}, {"tactic": "simp only [norm_eq_card_dsupport, coeFn_neg, Pi.neg_apply, ne_eq, neg_eq_zero]", "state_before": "case inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E 0 }\n\u22a2 \u2016-f\u2016 = \u2016f\u2016", "state_after": "no goals"}, {"tactic": "cases isEmpty_or_nonempty \u03b1", "state_before": "case inr.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\n\u22a2 \u2016-f\u2016 = \u2016f\u2016", "state_after": "case inr.inl.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\nh\u271d : IsEmpty \u03b1\n\u22a2 \u2016-f\u2016 = \u2016f\u2016\n\ncase inr.inl.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\nh\u271d : Nonempty \u03b1\n\u22a2 \u2016-f\u2016 = \u2016f\u2016"}, {"tactic": "apply (lp.isLUB_norm (-f)).unique", "state_before": "case inr.inl.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\nh\u271d : Nonempty \u03b1\n\u22a2 \u2016-f\u2016 = \u2016f\u2016", "state_after": "case inr.inl.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\nh\u271d : Nonempty \u03b1\n\u22a2 IsLUB (Set.range fun i => \u2016\u2191(-f) i\u2016) \u2016f\u2016"}, {"tactic": "simpa only [coeFn_neg, Pi.neg_apply, norm_neg] using lp.isLUB_norm f", "state_before": "case inr.inl.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\nh\u271d : Nonempty \u03b1\n\u22a2 IsLUB (Set.range fun i => \u2016\u2191(-f) i\u2016) \u2016f\u2016", "state_after": "no goals"}, {"tactic": "simp only [lp.eq_zero' f, neg_zero, norm_zero]", "state_before": "case inr.inl.inl\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\nq : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E \u22a4 }\nh\u271d : IsEmpty \u03b1\n\u22a2 \u2016-f\u2016 = \u2016f\u2016", "state_after": "no goals"}, {"tactic": "suffices \u2016-f\u2016 ^ p.toReal = \u2016f\u2016 ^ p.toReal by\n  exact Real.rpow_left_injOn hp.ne' (norm_nonneg' _) (norm_nonneg' _) this", "state_before": "case inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\n\u22a2 \u2016-f\u2016 = \u2016f\u2016", "state_after": "case inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\n\u22a2 \u2016-f\u2016 ^ ENNReal.toReal p = \u2016f\u2016 ^ ENNReal.toReal p"}, {"tactic": "apply (lp.hasSum_norm hp (-f)).unique", "state_before": "case inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\n\u22a2 \u2016-f\u2016 ^ ENNReal.toReal p = \u2016f\u2016 ^ ENNReal.toReal p", "state_after": "case inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\n\u22a2 HasSum (fun i => \u2016\u2191(-f) i\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p)"}, {"tactic": "simpa only [coeFn_neg, Pi.neg_apply, _root_.norm_neg] using lp.hasSum_norm hp f", "state_before": "case inr.inr\n\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\n\u22a2 HasSum (fun i => \u2016\u2191(-f) i\u2016 ^ ENNReal.toReal p) (\u2016f\u2016 ^ ENNReal.toReal p)", "state_after": "no goals"}, {"tactic": "exact Real.rpow_left_injOn hp.ne' (norm_nonneg' _) (norm_nonneg' _) this", "state_before": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\nf : { x // x \u2208 lp E p }\nhp : 0 < ENNReal.toReal p\nthis : \u2016-f\u2016 ^ ENNReal.toReal p = \u2016f\u2016 ^ ENNReal.toReal p\n\u22a2 \u2016-f\u2016 = \u2016f\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.disjoint_univ_pi", "start": [709, 1], "end": [710, 84], "traced_tactics": [{"tactic": "simp only [disjoint_iff_inter_eq_empty, \u2190 pi_inter_distrib, univ_pi_eq_empty_iff]", "state_before": "\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_1\n\u03b2 : \u03b9 \u2192 Type ?u.127302\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\n\u22a2 Disjoint (pi univ t\u2081) (pi univ t\u2082) \u2194 \u2203 i, Disjoint (t\u2081 i) (t\u2082 i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicTopology/DoldKan/Compatibility.lean", "full_name": "AlgebraicTopology.DoldKan.Compatibility.equivalence_functor", "start": [186, 1], "end": [187, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean", "full_name": "Real.differentiableAt_sinh", "start": [646, 1], "end": [647, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Analytic/Basic.lean", "full_name": "FormalMultilinearSeries.continuousOn", "start": [908, 11], "end": [912, 51], "traced_tactics": [{"tactic": "cases' (zero_le p.radius).eq_or_lt with h h", "state_before": "\ud835\udd5c : Type u_3\nE : Type u_2\nF : Type u_1\nG : Type ?u.1074397\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx : E\nr r' : \u211d\u22650\u221e\ninst\u271d : CompleteSpace F\n\u22a2 ContinuousOn (FormalMultilinearSeries.sum p) (EMetric.ball 0 (radius p))", "state_after": "case inl\n\ud835\udd5c : Type u_3\nE : Type u_2\nF : Type u_1\nG : Type ?u.1074397\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx : E\nr r' : \u211d\u22650\u221e\ninst\u271d : CompleteSpace F\nh : 0 = radius p\n\u22a2 ContinuousOn (FormalMultilinearSeries.sum p) (EMetric.ball 0 (radius p))\n\ncase inr\n\ud835\udd5c : Type u_3\nE : Type u_2\nF : Type u_1\nG : Type ?u.1074397\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx : E\nr r' : \u211d\u22650\u221e\ninst\u271d : CompleteSpace F\nh : 0 < radius p\n\u22a2 ContinuousOn (FormalMultilinearSeries.sum p) (EMetric.ball 0 (radius p))"}, {"tactic": "simp [\u2190 h, continuousOn_empty]", "state_before": "case inl\n\ud835\udd5c : Type u_3\nE : Type u_2\nF : Type u_1\nG : Type ?u.1074397\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx : E\nr r' : \u211d\u22650\u221e\ninst\u271d : CompleteSpace F\nh : 0 = radius p\n\u22a2 ContinuousOn (FormalMultilinearSeries.sum p) (EMetric.ball 0 (radius p))", "state_after": "no goals"}, {"tactic": "exact (p.hasFPowerSeriesOnBall h).continuousOn", "state_before": "case inr\n\ud835\udd5c : Type u_3\nE : Type u_2\nF : Type u_1\nG : Type ?u.1074397\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx : E\nr r' : \u211d\u22650\u221e\ninst\u271d : CompleteSpace F\nh : 0 < radius p\n\u22a2 ContinuousOn (FormalMultilinearSeries.sum p) (EMetric.ball 0 (radius p))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/MeasurableSpace.lean", "full_name": "measurableSet_pi", "start": [897, 1], "end": [901, 75], "traced_tactics": [{"tactic": "cases' (pi s t).eq_empty_or_nonempty with h h", "state_before": "\u03b1 : Type ?u.235303\n\u03b2 : Type ?u.235306\n\u03b3 : Type ?u.235309\n\u03b4 : Type u_1\n\u03b4' : Type ?u.235315\n\u03b9 : Sort u\u03b9\ns\u271d t\u271d u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\ns : Set \u03b4\nt : (i : \u03b4) \u2192 Set (\u03c0 i)\nhs : Set.Countable s\n\u22a2 MeasurableSet (Set.pi s t) \u2194 (\u2200 (i : \u03b4), i \u2208 s \u2192 MeasurableSet (t i)) \u2228 Set.pi s t = \u2205", "state_after": "case inl\n\u03b1 : Type ?u.235303\n\u03b2 : Type ?u.235306\n\u03b3 : Type ?u.235309\n\u03b4 : Type u_1\n\u03b4' : Type ?u.235315\n\u03b9 : Sort u\u03b9\ns\u271d t\u271d u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\ns : Set \u03b4\nt : (i : \u03b4) \u2192 Set (\u03c0 i)\nhs : Set.Countable s\nh : Set.pi s t = \u2205\n\u22a2 MeasurableSet (Set.pi s t) \u2194 (\u2200 (i : \u03b4), i \u2208 s \u2192 MeasurableSet (t i)) \u2228 Set.pi s t = \u2205\n\ncase inr\n\u03b1 : Type ?u.235303\n\u03b2 : Type ?u.235306\n\u03b3 : Type ?u.235309\n\u03b4 : Type u_1\n\u03b4' : Type ?u.235315\n\u03b9 : Sort u\u03b9\ns\u271d t\u271d u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\ns : Set \u03b4\nt : (i : \u03b4) \u2192 Set (\u03c0 i)\nhs : Set.Countable s\nh : Set.Nonempty (Set.pi s t)\n\u22a2 MeasurableSet (Set.pi s t) \u2194 (\u2200 (i : \u03b4), i \u2208 s \u2192 MeasurableSet (t i)) \u2228 Set.pi s t = \u2205"}, {"tactic": "simp [h]", "state_before": "case inl\n\u03b1 : Type ?u.235303\n\u03b2 : Type ?u.235306\n\u03b3 : Type ?u.235309\n\u03b4 : Type u_1\n\u03b4' : Type ?u.235315\n\u03b9 : Sort u\u03b9\ns\u271d t\u271d u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\ns : Set \u03b4\nt : (i : \u03b4) \u2192 Set (\u03c0 i)\nhs : Set.Countable s\nh : Set.pi s t = \u2205\n\u22a2 MeasurableSet (Set.pi s t) \u2194 (\u2200 (i : \u03b4), i \u2208 s \u2192 MeasurableSet (t i)) \u2228 Set.pi s t = \u2205", "state_after": "no goals"}, {"tactic": "simp [measurableSet_pi_of_nonempty hs, h, \u2190 not_nonempty_iff_eq_empty]", "state_before": "case inr\n\u03b1 : Type ?u.235303\n\u03b2 : Type ?u.235306\n\u03b3 : Type ?u.235309\n\u03b4 : Type u_1\n\u03b4' : Type ?u.235315\n\u03b9 : Sort u\u03b9\ns\u271d t\u271d u : Set \u03b1\n\u03c0 : \u03b4 \u2192 Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : (a : \u03b4) \u2192 MeasurableSpace (\u03c0 a)\ninst\u271d : MeasurableSpace \u03b3\ns : Set \u03b4\nt : (i : \u03b4) \u2192 Set (\u03c0 i)\nhs : Set.Countable s\nh : Set.Nonempty (Set.pi s t)\n\u22a2 MeasurableSet (Set.pi s t) \u2194 (\u2200 (i : \u03b4), i \u2208 s \u2192 MeasurableSet (t i)) \u2228 Set.pi s t = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "full_name": "Real.Angle.toReal_coe_eq_self_add_two_pi_iff", "start": [693, 1], "end": [695, 85], "traced_tactics": [{"tactic": "convert @toReal_coe_eq_self_sub_two_mul_int_mul_pi_iff \u03b8 (-1) using 2 <;> norm_num", "state_before": "\u03b8 : \u211d\n\u22a2 toReal \u2191\u03b8 = \u03b8 + 2 * \u03c0 \u2194 \u03b8 \u2208 Set.Ioc (-3 * \u03c0) (-\u03c0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean", "full_name": "ContinuousAffineMap.coe_mk_const_linear_eq_linear", "start": [79, 1], "end": [81, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Laurent.lean", "full_name": "LaurentPolynomial.single_zero_one_eq_one", "start": [130, 1], "end": [131, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.pred_mk", "start": [1534, 1], "end": [1538, 6], "traced_tactics": [{"tactic": "rwa [tsub_lt_iff_right (Nat.succ_le_of_lt <| Nat.pos_of_ne_zero\n  (by simpa using Fin.vne_of_ne w))]", "state_before": "n\u271d m n i : \u2115\nh : i < n + 1\nw : { val := i, isLt := h } \u2260 0\n\u22a2 i - 1 < n", "state_after": "no goals"}, {"tactic": "simpa using Fin.vne_of_ne w", "state_before": "n\u271d m n i : \u2115\nh : i < n + 1\nw : { val := i, isLt := h } \u2260 0\n\u22a2 i \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Order.lean", "full_name": "map_nhds_induced_of_surjective", "start": [850, 1], "end": [852, 48], "traced_tactics": [{"tactic": "rw [nhds_induced, map_comap_of_surjective hf]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type ?u.44989\nf\u271d : \u03b1 \u2192 \u03b2\n\u03b9 : Sort ?u.44996\nT : TopologicalSpace \u03b1\nf : \u03b2 \u2192 \u03b1\nhf : Surjective f\na : \u03b2\n\u22a2 map f (\ud835\udcdd a) = \ud835\udcdd (f a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Rdrop.lean", "full_name": "List.rtake_eq_reverse_take_reverse", "start": [83, 1], "end": [89, 44], "traced_tactics": [{"tactic": "rw [rtake]", "state_before": "\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nn : \u2115\n\u22a2 rtake l n = reverse (take n (reverse l))", "state_after": "\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nn : \u2115\n\u22a2 drop (length l - n) l = reverse (take n (reverse l))"}, {"tactic": "induction' l using List.reverseRecOn with xs x IH generalizing n", "state_before": "\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nn : \u2115\n\u22a2 drop (length l - n) l = reverse (take n (reverse l))", "state_after": "case H0\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nn\u271d n : \u2115\n\u22a2 drop (length [] - n) [] = reverse (take n (reverse []))\n\ncase H1\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nn\u271d : \u2115\nxs : List \u03b1\nx : \u03b1\nIH : \u2200 (n : \u2115), drop (length xs - n) xs = reverse (take n (reverse xs))\nn : \u2115\n\u22a2 drop (length (xs ++ [x]) - n) (xs ++ [x]) = reverse (take n (reverse (xs ++ [x])))"}, {"tactic": "simp", "state_before": "case H0\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nn\u271d n : \u2115\n\u22a2 drop (length [] - n) [] = reverse (take n (reverse []))", "state_after": "no goals"}, {"tactic": "cases n", "state_before": "case H1\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nn\u271d : \u2115\nxs : List \u03b1\nx : \u03b1\nIH : \u2200 (n : \u2115), drop (length xs - n) xs = reverse (take n (reverse xs))\nn : \u2115\n\u22a2 drop (length (xs ++ [x]) - n) (xs ++ [x]) = reverse (take n (reverse (xs ++ [x])))", "state_after": "case H1.zero\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nn : \u2115\nxs : List \u03b1\nx : \u03b1\nIH : \u2200 (n : \u2115), drop (length xs - n) xs = reverse (take n (reverse xs))\n\u22a2 drop (length (xs ++ [x]) - Nat.zero) (xs ++ [x]) = reverse (take Nat.zero (reverse (xs ++ [x])))\n\ncase H1.succ\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nn : \u2115\nxs : List \u03b1\nx : \u03b1\nIH : \u2200 (n : \u2115), drop (length xs - n) xs = reverse (take n (reverse xs))\nn\u271d : \u2115\n\u22a2 drop (length (xs ++ [x]) - Nat.succ n\u271d) (xs ++ [x]) = reverse (take (Nat.succ n\u271d) (reverse (xs ++ [x])))"}, {"tactic": "exact drop_length _", "state_before": "case H1.zero\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nn : \u2115\nxs : List \u03b1\nx : \u03b1\nIH : \u2200 (n : \u2115), drop (length xs - n) xs = reverse (take n (reverse xs))\n\u22a2 drop (length (xs ++ [x]) - Nat.zero) (xs ++ [x]) = reverse (take Nat.zero (reverse (xs ++ [x])))", "state_after": "no goals"}, {"tactic": "simp [drop_append_eq_append_drop, IH]", "state_before": "case H1.succ\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nn : \u2115\nxs : List \u03b1\nx : \u03b1\nIH : \u2200 (n : \u2115), drop (length xs - n) xs = reverse (take n (reverse xs))\nn\u271d : \u2115\n\u22a2 drop (length (xs ++ [x]) - Nat.succ n\u271d) (xs ++ [x]) = reverse (take (Nat.succ n\u271d) (reverse (xs ++ [x])))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Module/BigOperators.lean", "full_name": "List.sum_smul", "start": [26, 1], "end": [27, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Equiv.lean", "full_name": "AlgEquiv.toLinearEquiv_trans", "start": [558, 1], "end": [560, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "Real.exp_approx_end'", "start": [1788, 1], "end": [1792, 80], "traced_tactics": [{"tactic": "subst er", "state_before": "n : \u2115\nx a b : \u211d\nm : \u2115\ne\u2081 : n + 1 = m\nrm : \u211d\ner : \u2191m = rm\nh : abs' x \u2264 1\ne : abs' (1 - a) \u2264 b - abs' x / rm * ((rm + 1) / rm)\n\u22a2 abs' (exp x - expNear n x a) \u2264 abs' x ^ n / \u2191(Nat.factorial n) * b", "state_after": "n : \u2115\nx a b : \u211d\nm : \u2115\ne\u2081 : n + 1 = m\nh : abs' x \u2264 1\ne : abs' (1 - a) \u2264 b - abs' x / \u2191m * ((\u2191m + 1) / \u2191m)\n\u22a2 abs' (exp x - expNear n x a) \u2264 abs' x ^ n / \u2191(Nat.factorial n) * b"}, {"tactic": "exact exp_approx_succ _ e\u2081 _ _ (by simpa using e) (exp_approx_end _ _ _ e\u2081 h)", "state_before": "n : \u2115\nx a b : \u211d\nm : \u2115\ne\u2081 : n + 1 = m\nh : abs' x \u2264 1\ne : abs' (1 - a) \u2264 b - abs' x / \u2191m * ((\u2191m + 1) / \u2191m)\n\u22a2 abs' (exp x - expNear n x a) \u2264 abs' x ^ n / \u2191(Nat.factorial n) * b", "state_after": "no goals"}, {"tactic": "simpa using e", "state_before": "n : \u2115\nx a b : \u211d\nm : \u2115\ne\u2081 : n + 1 = m\nh : abs' x \u2264 1\ne : abs' (1 - a) \u2264 b - abs' x / \u2191m * ((\u2191m + 1) / \u2191m)\n\u22a2 abs' (1 + x / \u2191m * 0 - a) \u2264 b - abs' x / \u2191m * ((\u2191m + 1) / \u2191m)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/LocallyConvex/Basic.lean", "full_name": "balanced_iInter", "start": [199, 1], "end": [200, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicGeometry/SheafedSpace.lean", "full_name": "AlgebraicGeometry.SheafedSpace.\u0393_obj_op", "start": [209, 1], "end": [210, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.succ_ofInt'", "start": [1257, 1], "end": [1270, 60], "traced_tactics": [{"tactic": "change ZNum.ofInt' (n + 1 : \u2115) = ZNum.ofInt' (n : \u2115) + 1", "state_before": "\u03b1 : Type ?u.859157\nn : \u2115\n\u22a2 ZNum.ofInt' (\u2191n + 1) = ZNum.ofInt' \u2191n + 1", "state_after": "\u03b1 : Type ?u.859157\nn : \u2115\n\u22a2 ZNum.ofInt' \u2191(n + 1) = ZNum.ofInt' \u2191n + 1"}, {"tactic": "dsimp only [ZNum.ofInt', ZNum.ofInt']", "state_before": "\u03b1 : Type ?u.859157\nn : \u2115\n\u22a2 ZNum.ofInt' \u2191(n + 1) = ZNum.ofInt' \u2191n + 1", "state_after": "\u03b1 : Type ?u.859157\nn : \u2115\n\u22a2 toZNum (ofNat' (n + 1)) = toZNum (ofNat' n) + 1"}, {"tactic": "rw [Num.ofNat'_succ, Num.add_one, toZNum_succ, ZNum.add_one]", "state_before": "\u03b1 : Type ?u.859157\nn : \u2115\n\u22a2 toZNum (ofNat' (n + 1)) = toZNum (ofNat' n) + 1", "state_after": "no goals"}, {"tactic": "change ZNum.ofInt' 0 = ZNum.ofInt' (-[0+1]) + 1", "state_before": "\u03b1 : Type ?u.859157\n\u22a2 ZNum.ofInt' (-[0+1] + 1) = ZNum.ofInt' -[0+1] + 1", "state_after": "\u03b1 : Type ?u.859157\n\u22a2 ZNum.ofInt' 0 = ZNum.ofInt' -[0+1] + 1"}, {"tactic": "dsimp only [ZNum.ofInt', ZNum.ofInt']", "state_before": "\u03b1 : Type ?u.859157\n\u22a2 ZNum.ofInt' 0 = ZNum.ofInt' -[0+1] + 1", "state_after": "\u03b1 : Type ?u.859157\n\u22a2 toZNum (ofNat' 0) = toZNumNeg (ofNat' (0 + 1)) + 1"}, {"tactic": "rw [ofNat'_succ, ofNat'_zero]", "state_before": "\u03b1 : Type ?u.859157\n\u22a2 toZNum (ofNat' 0) = toZNumNeg (ofNat' (0 + 1)) + 1", "state_after": "\u03b1 : Type ?u.859157\n\u22a2 toZNum 0 = toZNumNeg (0 + 1) + 1"}, {"tactic": "rfl", "state_before": "\u03b1 : Type ?u.859157\n\u22a2 toZNum 0 = toZNumNeg (0 + 1) + 1", "state_after": "no goals"}, {"tactic": "change ZNum.ofInt' -[n+1] = ZNum.ofInt' -[(n + 1)+1] + 1", "state_before": "\u03b1 : Type ?u.859157\nn : \u2115\n\u22a2 ZNum.ofInt' (-[n + 1+1] + 1) = ZNum.ofInt' -[n + 1+1] + 1", "state_after": "\u03b1 : Type ?u.859157\nn : \u2115\n\u22a2 ZNum.ofInt' -[n+1] = ZNum.ofInt' -[n + 1+1] + 1"}, {"tactic": "dsimp only [ZNum.ofInt', ZNum.ofInt']", "state_before": "\u03b1 : Type ?u.859157\nn : \u2115\n\u22a2 ZNum.ofInt' -[n+1] = ZNum.ofInt' -[n + 1+1] + 1", "state_after": "\u03b1 : Type ?u.859157\nn : \u2115\n\u22a2 toZNumNeg (ofNat' (n + 1)) = toZNumNeg (ofNat' (n + 1 + 1)) + 1"}, {"tactic": "rw [@Num.ofNat'_succ (n + 1), Num.add_one, toZNumNeg_succ,\n  @ofNat'_succ n, Num.add_one, ZNum.add_one, pred_succ]", "state_before": "\u03b1 : Type ?u.859157\nn : \u2115\n\u22a2 toZNumNeg (ofNat' (n + 1)) = toZNumNeg (ofNat' (n + 1 + 1)) + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SetFamily/HarrisKleitman.lean", "full_name": "IsLowerSet.card_inter_le_finset", "start": [117, 1], "end": [121, 35], "traced_tactics": [{"tactic": "rw [inter_comm, mul_comm \ud835\udc9c.card]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\ns : Finset \u03b1\na : \u03b1\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsUpperSet \u2191\u212c\n\u22a2 2 ^ Fintype.card \u03b1 * card (\ud835\udc9c \u2229 \u212c) \u2264 card \ud835\udc9c * card \u212c", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\ns : Finset \u03b1\na : \u03b1\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsUpperSet \u2191\u212c\n\u22a2 2 ^ Fintype.card \u03b1 * card (\u212c \u2229 \ud835\udc9c) \u2264 card \u212c * card \ud835\udc9c"}, {"tactic": "exact h\u212c.card_inter_le_finset h\ud835\udc9c", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\ns : Finset \u03b1\na : \u03b1\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsLowerSet \u2191\ud835\udc9c\nh\u212c : IsUpperSet \u2191\u212c\n\u22a2 2 ^ Fintype.card \u03b1 * card (\u212c \u2229 \ud835\udc9c) \u2264 card \u212c * card \ud835\udc9c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean", "full_name": "CategoryTheory.Limits.HasBinaryBiproduct.mk", "start": [1182, 1], "end": [1183, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Matrix/Notation.lean", "full_name": "Matrix.empty_vecMulVec", "start": [331, 1], "end": [332, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/ModelTheory/FinitelyGenerated.lean", "full_name": "FirstOrder.Language.Substructure.FG.sup", "start": [78, 1], "end": [81, 77], "traced_tactics": [{"tactic": "rw [closure_union, ht\u2081.2, ht\u2082.2]", "state_before": "L : Language\nM : Type u_3\ninst\u271d : Structure L M\nN\u2081 N\u2082 : Substructure L M\nhN\u2081 : FG N\u2081\nhN\u2082 : FG N\u2082\nt\u2081 : Set M\nht\u2081 : Set.Finite t\u2081 \u2227 LowerAdjoint.toFun (closure L) t\u2081 = N\u2081\nt\u2082 : Set M\nht\u2082 : Set.Finite t\u2082 \u2227 LowerAdjoint.toFun (closure L) t\u2082 = N\u2082\n\u22a2 LowerAdjoint.toFun (closure L) (t\u2081 \u222a t\u2082) = N\u2081 \u2294 N\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/ThickenedIndicator.lean", "full_name": "thickenedIndicatorAux_lt_top", "start": [77, 1], "end": [79, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Control/Applicative.lean", "full_name": "Applicative.ext", "start": [44, 1], "end": [67, 85], "traced_tactics": [{"tactic": "obtain rfl : @p1 = @p2 := by\n  funext \u03b1 x\n  apply H1", "state_before": "F\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\np2 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns2 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nL1 : LawfulApplicative F\nL2 : LawfulApplicative F\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\n\u22a2 mk = mk", "state_after": "F\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\ns2 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nL1 : LawfulApplicative F\nL2 : LawfulApplicative F\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\n\u22a2 mk = mk"}, {"tactic": "obtain rfl : @s1 = @s2 := by\n  funext \u03b1 \u03b2 f x\n  exact H2 f (x Unit.unit)", "state_before": "F\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\ns2 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nL1 : LawfulApplicative F\nL2 : LawfulApplicative F\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\n\u22a2 mk = mk", "state_after": "F\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nL1 : LawfulApplicative F\nL2 : LawfulApplicative F\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\n\u22a2 mk = mk"}, {"tactic": "obtain \u27e8seqLeft_eq1, seqRight_eq1, pure_seq1, -\u27e9 := L1", "state_before": "F\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nL1 : LawfulApplicative F\nL2 : LawfulApplicative F\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\n\u22a2 mk = mk", "state_after": "case mk\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nL2 : LawfulApplicative F\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u22a2 mk = mk"}, {"tactic": "obtain \u27e8seqLeft_eq2, seqRight_eq2, pure_seq2, -\u27e9 := L2", "state_before": "case mk\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nL2 : LawfulApplicative F\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u22a2 mk = mk", "state_after": "case mk.mk\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d\u00b9 : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d\u00b9 : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d\u00b9 :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq2 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq2 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq2 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u22a2 mk = mk"}, {"tactic": "obtain rfl : F1 = F2 := by\n  apply Functor.ext\n  intros\n  exact (pure_seq1 _ _).symm.trans (pure_seq2 _ _)", "state_before": "case mk.mk\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d\u00b9 : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d\u00b9 : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d\u00b9 :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq2 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq2 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq2 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u22a2 mk = mk", "state_after": "case mk.mk\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\ntoLawfulFunctor\u271d\u00b9 : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d\u00b9 : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d\u00b9 :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq2 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq2 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq2 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u22a2 mk = mk"}, {"tactic": "congr <;> funext \u03b1 \u03b2 x y", "state_before": "case mk.mk\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\ntoLawfulFunctor\u271d\u00b9 : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d\u00b9 : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d\u00b9 :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq2 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq2 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq2 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u22a2 mk = mk", "state_after": "case mk.mk.e_toSeqLeft.e_seqLeft.h.h.h.h\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1\u271d \u03b2\u271d \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\ntoLawfulFunctor\u271d\u00b9 : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d\u00b9 : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d\u00b9 :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq2 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq2 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq2 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u03b1 \u03b2 : Type u\nx : F \u03b1\ny : Unit \u2192 F \u03b2\n\u22a2 sl1 x y = sl2 x y\n\ncase mk.mk.e_toSeqRight.e_seqRight.h.h.h.h\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1\u271d \u03b2\u271d \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\ntoLawfulFunctor\u271d\u00b9 : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d\u00b9 : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d\u00b9 :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq2 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq2 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq2 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u03b1 \u03b2 : Type u\nx : F \u03b1\ny : Unit \u2192 F \u03b2\n\u22a2 sr1 x y = sr2 x y"}, {"tactic": "funext \u03b1 x", "state_before": "F\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\np2 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns2 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nL1 : LawfulApplicative F\nL2 : LawfulApplicative F\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\n\u22a2 p1 = p2", "state_after": "case h.h\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1\u271d \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\np2 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns2 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nL1 : LawfulApplicative F\nL2 : LawfulApplicative F\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\n\u03b1 : Type u\nx : \u03b1\n\u22a2 p1 x = p2 x"}, {"tactic": "apply H1", "state_before": "case h.h\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1\u271d \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\np2 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns2 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nL1 : LawfulApplicative F\nL2 : LawfulApplicative F\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\n\u03b1 : Type u\nx : \u03b1\n\u22a2 p1 x = p2 x", "state_after": "no goals"}, {"tactic": "funext \u03b1 \u03b2 f x", "state_before": "F\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\ns2 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nL1 : LawfulApplicative F\nL2 : LawfulApplicative F\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\n\u22a2 s1 = s2", "state_after": "case h.h.h.h\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1\u271d \u03b2\u271d \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\ns2 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nL1 : LawfulApplicative F\nL2 : LawfulApplicative F\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\n\u03b1 \u03b2 : Type u\nf : F (\u03b1 \u2192 \u03b2)\nx : Unit \u2192 F \u03b1\n\u22a2 s1 f x = s2 f x"}, {"tactic": "exact H2 f (x Unit.unit)", "state_before": "case h.h.h.h\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1\u271d \u03b2\u271d \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\ns2 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nL1 : LawfulApplicative F\nL2 : LawfulApplicative F\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\n\u03b1 \u03b2 : Type u\nf : F (\u03b1 \u2192 \u03b2)\nx : Unit \u2192 F \u03b1\n\u22a2 s1 f x = s2 f x", "state_after": "no goals"}, {"tactic": "apply Functor.ext", "state_before": "F\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d\u00b9 : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d\u00b9 : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d\u00b9 :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq2 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq2 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq2 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u22a2 F1 = F2", "state_after": "case a\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d\u00b9 : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d\u00b9 : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d\u00b9 :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq2 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq2 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq2 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u22a2 \u2200 (\u03b1 \u03b2 : Type u) (f : \u03b1 \u2192 \u03b2) (x : F \u03b1), f <$> x = f <$> x"}, {"tactic": "intros", "state_before": "case a\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d\u00b9 : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d\u00b9 : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d\u00b9 :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq2 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq2 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq2 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u22a2 \u2200 (\u03b1 \u03b2 : Type u) (f : \u03b1 \u2192 \u03b2) (x : F \u03b1), f <$> x = f <$> x", "state_after": "case a\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d\u00b9 : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d\u00b9 : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d\u00b9 :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq2 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq2 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq2 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u03b1\u271d \u03b2\u271d : Type u\nf\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : F \u03b1\u271d\n\u22a2 f\u271d <$> x\u271d = f\u271d <$> x\u271d"}, {"tactic": "exact (pure_seq1 _ _).symm.trans (pure_seq2 _ _)", "state_before": "case a\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1 \u03b2 \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nF2 : Functor F\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d\u00b9 : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d\u00b9 : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d\u00b9 :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq2 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq2 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq2 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u03b1\u271d \u03b2\u271d : Type u\nf\u271d : \u03b1\u271d \u2192 \u03b2\u271d\nx\u271d : F \u03b1\u271d\n\u22a2 f\u271d <$> x\u271d = f\u271d <$> x\u271d", "state_after": "no goals"}, {"tactic": "exact (seqLeft_eq1 _ (y Unit.unit)).trans (seqLeft_eq2 _ _).symm", "state_before": "case mk.mk.e_toSeqLeft.e_seqLeft.h.h.h.h\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1\u271d \u03b2\u271d \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\ntoLawfulFunctor\u271d\u00b9 : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d\u00b9 : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d\u00b9 :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq2 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq2 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq2 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u03b1 \u03b2 : Type u\nx : F \u03b1\ny : Unit \u2192 F \u03b2\n\u22a2 sl1 x y = sl2 x y", "state_after": "no goals"}, {"tactic": "exact (seqRight_eq1 _ (y Unit.unit)).trans (seqRight_eq2 _ (y Unit.unit)).symm", "state_before": "case mk.mk.e_toSeqRight.e_seqRight.h.h.h.h\nF\u271d : Type u \u2192 Type v\ninst\u271d\u00b9 : Applicative F\u271d\ninst\u271d : LawfulApplicative F\u271d\n\u03b1\u271d \u03b2\u271d \u03b3 \u03c3 : Type u\nF : Type u \u2192 Type u_1\nF1 : Functor F\np1 : {\u03b1 : Type u} \u2192 \u03b1 \u2192 F \u03b1\ns1 : {\u03b1 \u03b2 : Type u} \u2192 F (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 F \u03b1) \u2192 F \u03b2\nsl1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr1 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\nsl2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b1\nsr2 : {\u03b1 \u03b2 : Type u} \u2192 F \u03b1 \u2192 (Unit \u2192 F \u03b2) \u2192 F \u03b2\ntoLawfulFunctor\u271d\u00b9 : LawfulFunctor F\nseqLeft_eq1 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq1 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq1 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d\u00b9 : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d\u00b9 :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\nH1 : \u2200 {\u03b1 : Type u} (x : \u03b1), pure x = pure x\nH2 : \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1), (Seq.seq f fun x_1 => x) = Seq.seq f fun x_1 => x\ntoLawfulFunctor\u271d : LawfulFunctor F\nseqLeft_eq2 : \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqLeft.seqLeft x fun x => y) = Seq.seq (const \u03b2 <$> x) fun x => y\nseqRight_eq2 :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F \u03b1) (y : F \u03b2), (SeqRight.seqRight x fun x => y) = Seq.seq (const \u03b1 id <$> x) fun x => y\npure_seq2 : \u2200 {\u03b1 \u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1), (Seq.seq (pure g) fun x_1 => x) = g <$> x\nseq_pure\u271d : \u2200 {\u03b1 \u03b2 : Type u} (g : F (\u03b1 \u2192 \u03b2)) (x : \u03b1), (Seq.seq g fun x_1 => pure x) = (fun h => h x) <$> g\nseq_assoc\u271d :\n  \u2200 {\u03b1 \u03b2 \u03b3 : Type u} (x : F \u03b1) (g : F (\u03b1 \u2192 \u03b2)) (h : F (\u03b2 \u2192 \u03b3)),\n    (Seq.seq h fun x_1 => Seq.seq g fun x_2 => x) = Seq.seq (Seq.seq (comp <$> h) fun x => g) fun x_1 => x\n\u03b1 \u03b2 : Type u\nx : F \u03b1\ny : Unit \u2192 F \u03b2\n\u22a2 sr1 x y = sr2 x y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/Basic.lean", "full_name": "Real.mk_le", "start": [359, 1], "end": [359, 90], "traced_tactics": [{"tactic": "simp [le_def', mk_eq]", "state_before": "x y : \u211d\nf g : CauSeq \u211a abs\n\u22a2 mk f \u2264 mk g \u2194 f \u2264 g", "state_after": "x y : \u211d\nf g : CauSeq \u211a abs\n\u22a2 f < g \u2228 f \u2248 g \u2194 f \u2264 g"}, {"tactic": "rfl", "state_before": "x y : \u211d\nf g : CauSeq \u211a abs\n\u22a2 f < g \u2228 f \u2248 g \u2194 f \u2264 g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Localization/Submodule.lean", "full_name": "IsFractionRing.coeSubmodule_injective", "start": [207, 1], "end": [208, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/Partrec.lean", "full_name": "Computable.list_concat", "start": [361, 1], "end": [362, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "full_name": "FiniteDimensional.of_finite_basis", "start": [166, 1], "end": [169, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean", "full_name": "Pmf.bind_apply", "start": [120, 1], "end": [120, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Normed/Group/Hom.lean", "full_name": "NormedAddGroupHom.coe_zero", "start": [402, 1], "end": [403, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Basic.lean", "full_name": "Finset.prod_pow_boole", "start": [1723, 1], "end": [1724, 71], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b9 : Type ?u.805362\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b1\n\u22a2 (\u220f x in s, f x ^ if a = x then 1 else 0) = if a \u2208 s then f a else 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "full_name": "HasFDerivWithinAt.differentiableWithinAt", "start": [376, 1], "end": [378, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean", "full_name": "AlgebraicTopology.DoldKan.\u0393\u2080.Obj.Termwise.mapMono_eq_zero", "start": [125, 1], "end": [129, 6], "traced_tactics": [{"tactic": "unfold mapMono", "state_before": "C : Type u_2\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\nK K' : ChainComplex C \u2115\nf : K \u27f6 K'\n\u0394 \u0394' \u0394'' : SimplexCategory\ni : \u0394' \u27f6 \u0394\ninst\u271d : Mono i\nh\u2081 : \u0394 \u2260 \u0394'\nh\u2082 : \u00acIs\u03b4\u2080 i\n\u22a2 mapMono K i = 0", "state_after": "C : Type u_2\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\nK K' : ChainComplex C \u2115\nf : K \u27f6 K'\n\u0394 \u0394' \u0394'' : SimplexCategory\ni : \u0394' \u27f6 \u0394\ninst\u271d : Mono i\nh\u2081 : \u0394 \u2260 \u0394'\nh\u2082 : \u00acIs\u03b4\u2080 i\n\u22a2 (if h : \u0394 = \u0394' then eqToHom (_ : HomologicalComplex.X K (len \u0394) = HomologicalComplex.X K (len \u0394'))\n    else if h : Is\u03b4\u2080 i then HomologicalComplex.d K (len \u0394) (len \u0394') else 0) =\n    0"}, {"tactic": "rw [Ne.def] at h\u2081", "state_before": "C : Type u_2\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\nK K' : ChainComplex C \u2115\nf : K \u27f6 K'\n\u0394 \u0394' \u0394'' : SimplexCategory\ni : \u0394' \u27f6 \u0394\ninst\u271d : Mono i\nh\u2081 : \u0394 \u2260 \u0394'\nh\u2082 : \u00acIs\u03b4\u2080 i\n\u22a2 (if h : \u0394 = \u0394' then eqToHom (_ : HomologicalComplex.X K (len \u0394) = HomologicalComplex.X K (len \u0394'))\n    else if h : Is\u03b4\u2080 i then HomologicalComplex.d K (len \u0394) (len \u0394') else 0) =\n    0", "state_after": "C : Type u_2\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\nK K' : ChainComplex C \u2115\nf : K \u27f6 K'\n\u0394 \u0394' \u0394'' : SimplexCategory\ni : \u0394' \u27f6 \u0394\ninst\u271d : Mono i\nh\u2081 : \u00ac\u0394 = \u0394'\nh\u2082 : \u00acIs\u03b4\u2080 i\n\u22a2 (if h : \u0394 = \u0394' then eqToHom (_ : HomologicalComplex.X K (len \u0394) = HomologicalComplex.X K (len \u0394'))\n    else if h : Is\u03b4\u2080 i then HomologicalComplex.d K (len \u0394) (len \u0394') else 0) =\n    0"}, {"tactic": "split_ifs", "state_before": "C : Type u_2\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\nK K' : ChainComplex C \u2115\nf : K \u27f6 K'\n\u0394 \u0394' \u0394'' : SimplexCategory\ni : \u0394' \u27f6 \u0394\ninst\u271d : Mono i\nh\u2081 : \u00ac\u0394 = \u0394'\nh\u2082 : \u00acIs\u03b4\u2080 i\n\u22a2 (if h : \u0394 = \u0394' then eqToHom (_ : HomologicalComplex.X K (len \u0394) = HomologicalComplex.X K (len \u0394'))\n    else if h : Is\u03b4\u2080 i then HomologicalComplex.d K (len \u0394) (len \u0394') else 0) =\n    0", "state_after": "C : Type u_2\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\nK K' : ChainComplex C \u2115\nf : K \u27f6 K'\n\u0394 \u0394' \u0394'' : SimplexCategory\ni : \u0394' \u27f6 \u0394\ninst\u271d : Mono i\nh\u2081 : \u00ac\u0394 = \u0394'\nh\u2082 : \u00acIs\u03b4\u2080 i\n\u22a2 0 = 0"}, {"tactic": "rfl", "state_before": "C : Type u_2\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : Preadditive C\nK K' : ChainComplex C \u2115\nf : K \u27f6 K'\n\u0394 \u0394' \u0394'' : SimplexCategory\ni : \u0394' \u27f6 \u0394\ninst\u271d : Mono i\nh\u2081 : \u00ac\u0394 = \u0394'\nh\u2082 : \u00acIs\u03b4\u2080 i\n\u22a2 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "full_name": "LinearIsometry.map_neg", "start": [213, 11], "end": [214, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/TensorPower.lean", "full_name": "TensorPower.cast_symm", "start": [123, 1], "end": [124, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "full_name": "MeasureTheory.AEStronglyMeasurable.stronglyMeasurableAtFilter", "start": [61, 11], "end": [63, 51], "traced_tactics": [{"tactic": "rwa [Measure.restrict_univ]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type ?u.1262952\nF : Type ?u.1262955\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nl l' : Filter \u03b1\nf g : \u03b1 \u2192 \u03b2\n\u03bc \u03bd : Measure \u03b1\nh : AEStronglyMeasurable f \u03bc\n\u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc 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\u2115\nn m : \u03c3\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\n\u03b9 : Type u_1\nf : \u03b9 \u2192 MvPolynomial \u03c3 R\na : \u03b9\ns : Finset \u03b9\nhas : \u00aca \u2208 s\nhind : totalDegree (Finset.sum s f) \u2264 Finset.sup s fun i => totalDegree (f i)\n\u22a2 totalDegree (f a + \u2211 x in s, f x) \u2264 max (totalDegree (f a)) (Finset.sup s fun i => totalDegree (f i))"}, {"tactic": "exact (MvPolynomial.totalDegree_add _ _).trans (max_le_max le_rfl hind)", "state_before": "case cons\nR : Type u\nS : Type v\n\u03c3 : Type u_2\n\u03c4 : Type ?u.472729\nr : R\ne : \u2115\nn m : \u03c3\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\np q : MvPolynomial \u03c3 R\n\u03b9 : Type u_1\nf : \u03b9 \u2192 MvPolynomial \u03c3 R\na : \u03b9\ns : Finset \u03b9\nhas : \u00aca \u2208 s\nhind : totalDegree (Finset.sum s f) \u2264 Finset.sup s fun i => totalDegree (f i)\n\u22a2 totalDegree (f a + \u2211 x in s, f x) \u2264 max (totalDegree (f a)) (Finset.sup s fun i => totalDegree (f i))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Hom/Set.lean", "full_name": "OrderIso.image_symm_image", "start": [40, 1], "end": [41, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean", "full_name": "MvPolynomial.weightedHomogeneousSubmodule_eq_finsupp_supported", "start": [167, 1], "end": [172, 6], "traced_tactics": [{"tactic": "ext x", "state_before": "R : Type u_1\nM : Type u_3\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_2\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm : M\n\u22a2 weightedHomogeneousSubmodule R w m = supported R R {d | \u2191(weightedDegree' w) d = m}", "state_after": "case h\nR : Type u_1\nM : Type u_3\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_2\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm : M\nx : MvPolynomial \u03c3 R\n\u22a2 x \u2208 weightedHomogeneousSubmodule R w m \u2194 x \u2208 supported R R {d | \u2191(weightedDegree' w) d = m}"}, {"tactic": "rw [mem_supported, Set.subset_def]", "state_before": "case h\nR : Type u_1\nM : Type u_3\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_2\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm : M\nx : MvPolynomial \u03c3 R\n\u22a2 x \u2208 weightedHomogeneousSubmodule R w m \u2194 x \u2208 supported R R {d | \u2191(weightedDegree' w) d = m}", "state_after": "case h\nR : Type u_1\nM : Type u_3\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_2\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm : M\nx : MvPolynomial \u03c3 R\n\u22a2 x \u2208 weightedHomogeneousSubmodule R w m \u2194 \u2200 (x_1 : \u03c3 \u2192\u2080 \u2115), x_1 \u2208 \u2191x.support \u2192 x_1 \u2208 {d | \u2191(weightedDegree' w) d = m}"}, {"tactic": "simp only [Finsupp.mem_support_iff, mem_coe]", "state_before": "case h\nR : Type u_1\nM : Type u_3\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_2\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm : M\nx : MvPolynomial \u03c3 R\n\u22a2 x \u2208 weightedHomogeneousSubmodule R w m \u2194 \u2200 (x_1 : \u03c3 \u2192\u2080 \u2115), x_1 \u2208 \u2191x.support \u2192 x_1 \u2208 {d | \u2191(weightedDegree' w) d = m}", "state_after": "case h\nR : Type u_1\nM : Type u_3\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_2\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm : M\nx : MvPolynomial \u03c3 R\n\u22a2 x \u2208 weightedHomogeneousSubmodule R w m \u2194 \u2200 (x_1 : \u03c3 \u2192\u2080 \u2115), \u2191x x_1 \u2260 0 \u2192 x_1 \u2208 {d | \u2191(weightedDegree' w) d = m}"}, {"tactic": "rfl", "state_before": "case h\nR : Type u_1\nM : Type u_3\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_2\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nm : M\nx : MvPolynomial \u03c3 R\n\u22a2 x \u2208 weightedHomogeneousSubmodule R w m \u2194 \u2200 (x_1 : \u03c3 \u2192\u2080 \u2115), \u2191x x_1 \u2260 0 \u2192 x_1 \u2208 {d | \u2191(weightedDegree' w) d = m}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Group.lean", "full_name": "Set.pairwise_disjoint_Ioc_mul_zpow", "start": [172, 1], "end": [183, 26], "traced_tactics": [{"tactic": "simp_rw [Function.onFun, Set.disjoint_iff]", "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\n\u22a2 Pairwise (Disjoint on fun n => Ioc (a * b ^ n) (a * b ^ (n + 1)))", "state_after": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\n\u22a2 Pairwise fun x y => Ioc (a * b ^ x) (a * b ^ (x + 1)) \u2229 Ioc (a * b ^ y) (a * b ^ (y + 1)) \u2286 \u2205"}, {"tactic": "intro m n hmn x hx", "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\n\u22a2 Pairwise fun x y => Ioc (a * b ^ x) (a * b ^ (x + 1)) \u2229 Ioc (a * b ^ y) (a * b ^ (y + 1)) \u2286 \u2205", "state_after": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ioc (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ioc (a * b ^ n) (a * b ^ (n + 1))\n\u22a2 x \u2208 \u2205"}, {"tactic": "apply hmn", "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ioc (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ioc (a * b ^ n) (a * b ^ (n + 1))\n\u22a2 x \u2208 \u2205", "state_after": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ioc (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ioc (a * b ^ n) (a * b ^ (n + 1))\n\u22a2 m = n"}, {"tactic": "have hb : 1 < b := by\n  have : a * b ^ m < a * b ^ (m + 1) := hx.1.1.trans_le hx.1.2\n  rwa [mul_lt_mul_iff_left, \u2190 mul_one (b ^ m), zpow_add_one, mul_lt_mul_iff_left] at this", "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ioc (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ioc (a * b ^ n) (a * b ^ (n + 1))\n\u22a2 m = n", "state_after": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ioc (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ioc (a * b ^ n) (a * b ^ (n + 1))\nhb : 1 < b\n\u22a2 m = n"}, {"tactic": "have i1 := hx.1.1.trans_le hx.2.2", "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ioc (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ioc (a * b ^ n) (a * b ^ (n + 1))\nhb : 1 < b\n\u22a2 m = n", "state_after": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ioc (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ioc (a * b ^ n) (a * b ^ (n + 1))\nhb : 1 < b\ni1 : a * b ^ m < a * b ^ (n + 1)\n\u22a2 m = n"}, {"tactic": "have i2 := hx.2.1.trans_le hx.1.2", "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ioc (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ioc (a * b ^ n) (a * b ^ (n + 1))\nhb : 1 < b\ni1 : a * b ^ m < a * b ^ (n + 1)\n\u22a2 m = n", "state_after": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ioc (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ioc (a * b ^ n) (a * b ^ (n + 1))\nhb : 1 < b\ni1 : a * b ^ m < a * b ^ (n + 1)\ni2 : a * b ^ n < a * b ^ (m + 1)\n\u22a2 m = n"}, {"tactic": "rw [mul_lt_mul_iff_left, zpow_lt_zpow_iff hb, Int.lt_add_one_iff] at i1 i2", "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ioc (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ioc (a * b ^ n) (a * b ^ (n + 1))\nhb : 1 < b\ni1 : a * b ^ m < a * b ^ (n + 1)\ni2 : a * b ^ n < a * b ^ (m + 1)\n\u22a2 m = n", "state_after": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ioc (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ioc (a * b ^ n) (a * b ^ (n + 1))\nhb : 1 < b\ni1 : m \u2264 n\ni2 : n \u2264 m\n\u22a2 m = n"}, {"tactic": "exact le_antisymm i1 i2", "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ioc (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ioc (a * b ^ n) (a * b ^ (n + 1))\nhb : 1 < b\ni1 : m \u2264 n\ni2 : n \u2264 m\n\u22a2 m = n", "state_after": "no goals"}, {"tactic": "have : a * b ^ m < a * b ^ (m + 1) := hx.1.1.trans_le hx.1.2", "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ioc (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ioc (a * b ^ n) (a * b ^ (n + 1))\n\u22a2 1 < b", "state_after": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ioc (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ioc (a * b ^ n) (a * b ^ (n + 1))\nthis : a * b ^ m < a * b ^ (m + 1)\n\u22a2 1 < b"}, {"tactic": "rwa [mul_lt_mul_iff_left, \u2190 mul_one (b ^ m), zpow_add_one, mul_lt_mul_iff_left] at this", "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ioc (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ioc (a * b ^ n) (a * b ^ (n + 1))\nthis : a * b ^ m < a * b ^ (m + 1)\n\u22a2 1 < b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecificLimits/Normed.lean", "full_name": "mul_neg_geom_series", "start": [492, 1], "end": [498, 42], "traced_tactics": [{"tactic": "have := (NormedRing.summable_geometric_of_norm_lt_1 x h).hasSum.mul_left (1 - x)", "state_before": "\u03b1 : Type ?u.1257326\n\u03b2 : Type ?u.1257329\n\u03b9 : Type ?u.1257332\nR : Type u_1\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\nh : \u2016x\u2016 < 1\n\u22a2 ((1 - x) * \u2211' (i : \u2115), x ^ i) = 1", "state_after": "\u03b1 : Type ?u.1257326\n\u03b2 : Type ?u.1257329\n\u03b9 : Type ?u.1257332\nR : Type u_1\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\nh : \u2016x\u2016 < 1\nthis : HasSum (fun i => (1 - x) * x ^ i) ((1 - x) * \u2211' (b : \u2115), x ^ b)\n\u22a2 ((1 - x) * \u2211' (i : \u2115), x ^ i) = 1"}, {"tactic": "refine' tendsto_nhds_unique this.tendsto_sum_nat _", "state_before": "\u03b1 : Type ?u.1257326\n\u03b2 : Type ?u.1257329\n\u03b9 : Type ?u.1257332\nR : Type u_1\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\nh : \u2016x\u2016 < 1\nthis : HasSum (fun i => (1 - x) * x ^ i) ((1 - x) * \u2211' (b : \u2115), x ^ b)\n\u22a2 ((1 - x) * \u2211' (i : \u2115), x ^ i) = 1", "state_after": "\u03b1 : Type ?u.1257326\n\u03b2 : Type ?u.1257329\n\u03b9 : Type ?u.1257332\nR : Type u_1\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\nh : \u2016x\u2016 < 1\nthis : HasSum (fun i => (1 - x) * x ^ i) ((1 - x) * \u2211' (b : \u2115), x ^ b)\n\u22a2 Tendsto (fun n => \u2211 i in Finset.range n, (1 - x) * x ^ i) atTop (\ud835\udcdd 1)"}, {"tactic": "have : Tendsto (fun n : \u2115 \u21a6 1 - x ^ n) atTop (nhds 1) := by\n  simpa using tendsto_const_nhds.sub (tendsto_pow_atTop_nhds_0_of_norm_lt_1 h)", "state_before": "\u03b1 : Type ?u.1257326\n\u03b2 : Type ?u.1257329\n\u03b9 : Type ?u.1257332\nR : Type u_1\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\nh : \u2016x\u2016 < 1\nthis : HasSum (fun i => (1 - x) * x ^ i) ((1 - x) * \u2211' (b : \u2115), x ^ b)\n\u22a2 Tendsto (fun n => \u2211 i in Finset.range n, (1 - x) * x ^ i) atTop (\ud835\udcdd 1)", "state_after": "\u03b1 : Type ?u.1257326\n\u03b2 : Type ?u.1257329\n\u03b9 : Type ?u.1257332\nR : Type u_1\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\nh : \u2016x\u2016 < 1\nthis\u271d : HasSum (fun i => (1 - x) * x ^ i) ((1 - x) * \u2211' (b : \u2115), x ^ b)\nthis : Tendsto (fun n => 1 - x ^ n) atTop (\ud835\udcdd 1)\n\u22a2 Tendsto (fun n => \u2211 i in Finset.range n, (1 - x) * x ^ i) atTop (\ud835\udcdd 1)"}, {"tactic": "convert\u2190 this", "state_before": "\u03b1 : Type ?u.1257326\n\u03b2 : Type ?u.1257329\n\u03b9 : Type ?u.1257332\nR : Type u_1\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\nh : \u2016x\u2016 < 1\nthis\u271d : HasSum (fun i => (1 - x) * x ^ i) ((1 - x) * \u2211' (b : \u2115), x ^ b)\nthis : Tendsto (fun n => 1 - x ^ n) atTop (\ud835\udcdd 1)\n\u22a2 Tendsto (fun n => \u2211 i in Finset.range n, (1 - x) * x ^ i) atTop (\ud835\udcdd 1)", "state_after": "case h.e'_3.h\n\u03b1 : Type ?u.1257326\n\u03b2 : Type ?u.1257329\n\u03b9 : Type ?u.1257332\nR : Type u_1\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\nh : \u2016x\u2016 < 1\nthis\u271d : HasSum (fun i => (1 - x) * x ^ i) ((1 - x) * \u2211' (b : \u2115), x ^ b)\nthis : Tendsto (fun n => 1 - x ^ n) atTop (\ud835\udcdd 1)\nx\u271d : \u2115\n\u22a2 1 - x ^ x\u271d = \u2211 i in Finset.range x\u271d, (1 - x) * x ^ i"}, {"tactic": "rw [\u2190 mul_neg_geom_sum, Finset.mul_sum]", "state_before": "case h.e'_3.h\n\u03b1 : Type ?u.1257326\n\u03b2 : Type ?u.1257329\n\u03b9 : Type ?u.1257332\nR : Type u_1\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\nh : \u2016x\u2016 < 1\nthis\u271d : HasSum (fun i => (1 - x) * x ^ i) ((1 - x) * \u2211' (b : \u2115), x ^ b)\nthis : Tendsto (fun n => 1 - x ^ n) atTop (\ud835\udcdd 1)\nx\u271d : \u2115\n\u22a2 1 - x ^ x\u271d = \u2211 i in Finset.range x\u271d, (1 - x) * x ^ i", "state_after": "no goals"}, {"tactic": "simpa using tendsto_const_nhds.sub (tendsto_pow_atTop_nhds_0_of_norm_lt_1 h)", "state_before": "\u03b1 : Type ?u.1257326\n\u03b2 : Type ?u.1257329\n\u03b9 : Type ?u.1257332\nR : Type u_1\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\nh : \u2016x\u2016 < 1\nthis : HasSum (fun i => (1 - x) * x ^ i) ((1 - x) * \u2211' (b : \u2115), x ^ b)\n\u22a2 Tendsto (fun n => 1 - x ^ n) atTop (\ud835\udcdd 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Prod.lean", "full_name": "MonoidHom.inl_apply", "start": [505, 1], "end": [506, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Finprod.lean", "full_name": "finprod_mem_union''", "start": [819, 1], "end": [823, 92], "traced_tactics": [{"tactic": "rw [\u2190 finprod_mem_inter_mulSupport f s, \u2190 finprod_mem_inter_mulSupport f t, \u2190\n  finprod_mem_union hst hs ht, \u2190 union_inter_distrib_right, finprod_mem_inter_mulSupport]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.337656\n\u03b9 : Type ?u.337659\nG : Type ?u.337662\nM : Type u_2\nN : Type ?u.337668\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf g : \u03b1 \u2192 M\na b : \u03b1\ns t : Set \u03b1\nhst : Disjoint (s \u2229 mulSupport f) (t \u2229 mulSupport f)\nhs : Set.Finite (s \u2229 mulSupport f)\nht : Set.Finite (t \u2229 mulSupport f)\n\u22a2 (\u220f\u1da0 (i : \u03b1) (_ : i \u2208 s \u222a t), f i) = (\u220f\u1da0 (i : \u03b1) (_ : i \u2208 s), f i) * \u220f\u1da0 (i : \u03b1) (_ : i \u2208 t), f i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Matrix/Basic.lean", "full_name": "Matrix.updateRow_ne", "start": [2761, 1], "end": [2763, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Eval.lean", "full_name": "Polynomial.natDegree_map_le", "start": [843, 1], "end": [844, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/PerfectClosure.lean", "full_name": "PerfectClosure.of_apply", "start": [475, 1], "end": [476, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.keys_val", "start": [196, 1], "end": [197, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "Complex.exp_add", "start": [498, 1], "end": [517, 61], "traced_tactics": [{"tactic": "have hj : \u2200 j : \u2115, (\u2211 m in range j, (x + y) ^ m / m.factorial) =\n      \u2211 i in range j, \u2211 k in range (i + 1), x ^ k / k.factorial *\n        (y ^ (i - k) / (i - k).factorial) := by\n  intro j\n  refine' Finset.sum_congr rfl fun m _ => _\n  rw [add_pow, div_eq_mul_inv, sum_mul]\n  refine' Finset.sum_congr rfl fun I hi => _\n  have h\u2081 : (m.choose I : \u2102) \u2260 0 :=\n    Nat.cast_ne_zero.2 (pos_iff_ne_zero.1 (Nat.choose_pos (Nat.le_of_lt_succ (mem_range.1 hi))))\n  have h\u2082 := Nat.choose_mul_factorial_mul_factorial (Nat.le_of_lt_succ <| Finset.mem_range.1 hi)\n  rw [\u2190 h\u2082, Nat.cast_mul, Nat.cast_mul, mul_inv, mul_inv]\n  simp only [mul_left_comm (m.choose I : \u2102), mul_assoc, mul_left_comm (m.choose I : \u2102)\u207b\u00b9,\n    mul_comm (m.choose I : \u2102)]\n  rw [inv_mul_cancel h\u2081]\n  simp [div_eq_mul_inv, mul_comm, mul_assoc, mul_left_comm]", "state_before": "x y : \u2102\n\u22a2 exp (x + y) = exp x * exp y", "state_after": "x y : \u2102\nhj :\n  \u2200 (j : \u2115),\n    \u2211 m in range j, (x + y) ^ m / \u2191(Nat.factorial m) =\n      \u2211 i in range j, \u2211 k in range (i + 1), x ^ k / \u2191(Nat.factorial k) * (y ^ (i - k) / \u2191(Nat.factorial (i - k)))\n\u22a2 exp (x + y) = exp x * exp y"}, {"tactic": "simp_rw [exp, exp', lim_mul_lim]", "state_before": "x y : \u2102\nhj :\n  \u2200 (j : \u2115),\n    \u2211 m in range j, (x + y) ^ m / \u2191(Nat.factorial m) =\n      \u2211 i in range j, \u2211 k in range (i + 1), x ^ k / \u2191(Nat.factorial k) * (y ^ (i - k) / \u2191(Nat.factorial (i - k)))\n\u22a2 exp (x + y) = exp x * exp y", "state_after": "x y : \u2102\nhj :\n  \u2200 (j : \u2115),\n    \u2211 m in range j, (x + y) ^ m / \u2191(Nat.factorial m) =\n      \u2211 i in range j, \u2211 k in range (i + 1), x ^ k / \u2191(Nat.factorial k) * (y ^ (i - k) / \u2191(Nat.factorial (i - k)))\n\u22a2 CauSeq.lim\n      { val := fun n => \u2211 m in range n, (x + y) ^ m / \u2191(Nat.factorial m),\n        property := (_ : IsCauSeq \u2191abs fun n => \u2211 m in range n, (x + y) ^ m / \u2191(Nat.factorial m)) } =\n    CauSeq.lim\n      ({ val := fun n => \u2211 m in range n, x ^ m / \u2191(Nat.factorial m),\n          property := (_ : IsCauSeq \u2191abs fun n => \u2211 m in range n, x ^ m / \u2191(Nat.factorial m)) } *\n        { val := fun n => \u2211 m in range n, y ^ m / \u2191(Nat.factorial m),\n          property := (_ : IsCauSeq \u2191abs fun n => \u2211 m in range n, y ^ m / \u2191(Nat.factorial m)) })"}, {"tactic": "apply (lim_eq_lim_of_equiv _).symm", "state_before": "x y : \u2102\nhj :\n  \u2200 (j : \u2115),\n    \u2211 m in range j, (x + y) ^ m / \u2191(Nat.factorial m) =\n      \u2211 i in range j, \u2211 k in range (i + 1), x ^ k / \u2191(Nat.factorial k) * (y ^ (i - k) / \u2191(Nat.factorial (i - k)))\n\u22a2 CauSeq.lim\n      { val := fun n => \u2211 m in range n, (x + y) ^ m / \u2191(Nat.factorial m),\n        property := (_ : IsCauSeq \u2191abs fun n => \u2211 m in range n, (x + y) ^ m / \u2191(Nat.factorial m)) } =\n    CauSeq.lim\n      ({ val := fun n => \u2211 m in range n, x ^ m / \u2191(Nat.factorial m),\n          property := (_ : IsCauSeq \u2191abs fun n => \u2211 m in range n, x ^ m / \u2191(Nat.factorial m)) } *\n        { val := fun n => \u2211 m in range n, y ^ m / \u2191(Nat.factorial m),\n          property := (_ : IsCauSeq \u2191abs fun n => \u2211 m in range n, y ^ m / \u2191(Nat.factorial m)) })", "state_after": "x y : \u2102\nhj :\n  \u2200 (j : \u2115),\n    \u2211 m in range j, (x + y) ^ m / \u2191(Nat.factorial m) =\n      \u2211 i in range j, \u2211 k in range (i + 1), x ^ k / \u2191(Nat.factorial k) * (y ^ (i - k) / \u2191(Nat.factorial (i - k)))\n\u22a2 { val := fun n => \u2211 m in range n, x ^ m / \u2191(Nat.factorial m),\n        property := (_ : IsCauSeq \u2191abs fun n => \u2211 m in range n, x ^ m / \u2191(Nat.factorial m)) } *\n      { val := fun n => \u2211 m in range n, y ^ m / \u2191(Nat.factorial m),\n        property := (_ : IsCauSeq \u2191abs fun n => \u2211 m in range n, y ^ m / \u2191(Nat.factorial m)) } \u2248\n    { val := fun n => \u2211 m in range n, (x + y) ^ m / \u2191(Nat.factorial m),\n      property := (_ : IsCauSeq \u2191abs fun n => \u2211 m in range n, (x + y) ^ m / \u2191(Nat.factorial m)) }"}, {"tactic": "simp only [hj]", "state_before": "x y : \u2102\nhj :\n  \u2200 (j : \u2115),\n    \u2211 m in range j, (x + y) ^ m / \u2191(Nat.factorial m) =\n      \u2211 i in range j, \u2211 k in range (i + 1), x ^ k / \u2191(Nat.factorial k) * (y ^ (i - k) / \u2191(Nat.factorial (i - k)))\n\u22a2 { val := fun n => \u2211 m in range n, x ^ m / \u2191(Nat.factorial m),\n        property := (_ : IsCauSeq \u2191abs fun n => \u2211 m in range n, x ^ m / \u2191(Nat.factorial m)) } *\n      { val := fun n => \u2211 m in range n, y ^ m / \u2191(Nat.factorial m),\n        property := (_ : IsCauSeq \u2191abs fun n => \u2211 m in range n, y ^ m / \u2191(Nat.factorial m)) } \u2248\n    { val := fun n => \u2211 m in range n, (x + y) ^ m / \u2191(Nat.factorial m),\n      property := (_ : IsCauSeq \u2191abs fun n => \u2211 m in range n, (x + y) ^ m / \u2191(Nat.factorial m)) }", "state_after": "x y : \u2102\nhj :\n  \u2200 (j : \u2115),\n    \u2211 m in range j, (x + y) ^ m / \u2191(Nat.factorial m) =\n      \u2211 i in range j, \u2211 k in range (i + 1), x ^ k / \u2191(Nat.factorial k) * (y ^ (i - k) / \u2191(Nat.factorial (i - k)))\n\u22a2 { val := fun n => \u2211 m in range n, x ^ m / \u2191(Nat.factorial m),\n        property := (_ : IsCauSeq \u2191abs fun n => \u2211 m in range n, x ^ m / \u2191(Nat.factorial m)) } *\n      { val := fun n => \u2211 m in range n, y ^ m / \u2191(Nat.factorial m),\n        property := (_ : IsCauSeq \u2191abs fun n => \u2211 m in range n, y ^ m / \u2191(Nat.factorial m)) } \u2248\n    {\n      val := fun n =>\n        \u2211 i in range n, \u2211 k in range (i + 1), x ^ k / \u2191(Nat.factorial k) * (y ^ (i - k) / \u2191(Nat.factorial (i - k))),\n      property :=\n        (_ :\n          (fun f => IsCauSeq (\u2191abs) f) fun n =>\n            \u2211 i in range n,\n              \u2211 k in range (i + 1), x ^ k / \u2191(Nat.factorial k) * (y ^ (i - k) / \u2191(Nat.factorial (i - k)))) }"}, {"tactic": "exact cauchy_product (isCauSeq_abs_exp x) (isCauSeq_exp y)", "state_before": "x y : \u2102\nhj :\n  \u2200 (j : \u2115),\n    \u2211 m in range j, (x + y) ^ m / \u2191(Nat.factorial m) =\n      \u2211 i in range j, \u2211 k in range (i + 1), x ^ k / \u2191(Nat.factorial k) * (y ^ (i - k) / \u2191(Nat.factorial (i - k)))\n\u22a2 { val := fun n => \u2211 m in range n, x ^ m / \u2191(Nat.factorial m),\n        property := (_ : IsCauSeq \u2191abs fun n => \u2211 m in range n, x ^ m / \u2191(Nat.factorial m)) } *\n      { val := fun n => \u2211 m in range n, y ^ m / \u2191(Nat.factorial m),\n        property := (_ : IsCauSeq \u2191abs fun n => \u2211 m in range n, y ^ m / \u2191(Nat.factorial m)) } \u2248\n    {\n      val := fun n =>\n        \u2211 i in range n, \u2211 k in range (i + 1), x ^ k / \u2191(Nat.factorial k) * (y ^ (i - k) / \u2191(Nat.factorial (i - k))),\n      property :=\n        (_ :\n          (fun f => IsCauSeq (\u2191abs) f) fun n =>\n            \u2211 i in range n,\n              \u2211 k in range (i + 1), x ^ k / \u2191(Nat.factorial k) * (y ^ (i - k) / \u2191(Nat.factorial (i - k)))) }", "state_after": "no goals"}, {"tactic": "intro j", "state_before": "x y : \u2102\n\u22a2 \u2200 (j : \u2115),\n    \u2211 m in range j, (x + y) ^ m / \u2191(Nat.factorial m) =\n      \u2211 i in range j, \u2211 k in range (i + 1), x ^ k / \u2191(Nat.factorial k) * (y ^ (i - k) / \u2191(Nat.factorial (i - k)))", "state_after": "x y : \u2102\nj : \u2115\n\u22a2 \u2211 m in range j, (x + y) ^ m / \u2191(Nat.factorial m) =\n    \u2211 i in range j, \u2211 k in range (i + 1), x ^ k / \u2191(Nat.factorial k) * (y ^ (i - k) / \u2191(Nat.factorial (i - k)))"}, {"tactic": "refine' Finset.sum_congr rfl fun m _ => _", "state_before": "x y : \u2102\nj : \u2115\n\u22a2 \u2211 m in range j, (x + y) ^ m / \u2191(Nat.factorial m) =\n    \u2211 i in range j, \u2211 k in range (i + 1), x ^ k / \u2191(Nat.factorial k) * (y ^ (i - k) / \u2191(Nat.factorial (i - k)))", "state_after": "x y : \u2102\nj m : \u2115\nx\u271d : m \u2208 range j\n\u22a2 (x + y) ^ m / \u2191(Nat.factorial m) =\n    \u2211 k in range (m + 1), x ^ k / \u2191(Nat.factorial k) * (y ^ (m - k) / \u2191(Nat.factorial (m - k)))"}, {"tactic": "rw [add_pow, div_eq_mul_inv, sum_mul]", "state_before": "x y : \u2102\nj m : \u2115\nx\u271d : m \u2208 range j\n\u22a2 (x + y) ^ m / \u2191(Nat.factorial m) =\n    \u2211 k in range (m + 1), x ^ k / \u2191(Nat.factorial k) * (y ^ (m - k) / \u2191(Nat.factorial (m - k)))", "state_after": "x y : \u2102\nj m : \u2115\nx\u271d : m \u2208 range j\n\u22a2 \u2211 x_1 in range (m + 1), x ^ x_1 * y ^ (m - x_1) * \u2191(Nat.choose m x_1) * (\u2191(Nat.factorial m))\u207b\u00b9 =\n    \u2211 k in range (m + 1), x ^ k / \u2191(Nat.factorial k) * (y ^ (m - k) / \u2191(Nat.factorial (m - k)))"}, {"tactic": "refine' Finset.sum_congr rfl fun I hi => _", "state_before": "x y : \u2102\nj m : \u2115\nx\u271d : m \u2208 range j\n\u22a2 \u2211 x_1 in range (m + 1), x ^ x_1 * y ^ (m - x_1) * \u2191(Nat.choose m x_1) * (\u2191(Nat.factorial m))\u207b\u00b9 =\n    \u2211 k in range (m + 1), x ^ k / \u2191(Nat.factorial k) * (y ^ (m - k) / \u2191(Nat.factorial (m - k)))", "state_after": "x y : \u2102\nj m : \u2115\nx\u271d : m \u2208 range j\nI : \u2115\nhi : I \u2208 range (m + 1)\n\u22a2 x ^ I * y ^ (m - I) * \u2191(Nat.choose m I) * (\u2191(Nat.factorial m))\u207b\u00b9 =\n    x ^ I / \u2191(Nat.factorial I) * (y ^ (m - I) / \u2191(Nat.factorial (m - I)))"}, {"tactic": "have h\u2081 : (m.choose I : \u2102) \u2260 0 :=\n  Nat.cast_ne_zero.2 (pos_iff_ne_zero.1 (Nat.choose_pos (Nat.le_of_lt_succ (mem_range.1 hi))))", "state_before": "x y : \u2102\nj m : \u2115\nx\u271d : m \u2208 range j\nI : \u2115\nhi : I \u2208 range (m + 1)\n\u22a2 x ^ I * y ^ (m - I) * \u2191(Nat.choose m I) * (\u2191(Nat.factorial m))\u207b\u00b9 =\n    x ^ I / \u2191(Nat.factorial I) * (y ^ (m - I) / \u2191(Nat.factorial (m - I)))", "state_after": "x y : \u2102\nj m : \u2115\nx\u271d : m \u2208 range j\nI : \u2115\nhi : I \u2208 range (m + 1)\nh\u2081 : \u2191(Nat.choose m I) \u2260 0\n\u22a2 x ^ I * y ^ (m - I) * \u2191(Nat.choose m I) * (\u2191(Nat.factorial m))\u207b\u00b9 =\n    x ^ I / \u2191(Nat.factorial I) * (y ^ (m - I) / \u2191(Nat.factorial (m - I)))"}, {"tactic": "have h\u2082 := Nat.choose_mul_factorial_mul_factorial (Nat.le_of_lt_succ <| Finset.mem_range.1 hi)", "state_before": "x y : \u2102\nj m : \u2115\nx\u271d : m \u2208 range j\nI : \u2115\nhi : I \u2208 range (m + 1)\nh\u2081 : \u2191(Nat.choose m I) \u2260 0\n\u22a2 x ^ I * y ^ (m - I) * \u2191(Nat.choose m I) * (\u2191(Nat.factorial m))\u207b\u00b9 =\n    x ^ I / \u2191(Nat.factorial I) * (y ^ (m - I) / \u2191(Nat.factorial (m - I)))", "state_after": "x y : \u2102\nj m : \u2115\nx\u271d : m \u2208 range j\nI : \u2115\nhi : I \u2208 range (m + 1)\nh\u2081 : \u2191(Nat.choose m I) \u2260 0\nh\u2082 : Nat.choose m I * Nat.factorial I * Nat.factorial (m - I) = Nat.factorial m\n\u22a2 x ^ I * y ^ (m - I) * \u2191(Nat.choose m I) * (\u2191(Nat.factorial m))\u207b\u00b9 =\n    x ^ I / \u2191(Nat.factorial I) * (y ^ (m - I) / \u2191(Nat.factorial (m - I)))"}, {"tactic": "rw [\u2190 h\u2082, Nat.cast_mul, Nat.cast_mul, mul_inv, mul_inv]", "state_before": "x y : \u2102\nj m : \u2115\nx\u271d : m \u2208 range j\nI : \u2115\nhi : I \u2208 range (m + 1)\nh\u2081 : \u2191(Nat.choose m I) \u2260 0\nh\u2082 : Nat.choose m I * Nat.factorial I * Nat.factorial (m - I) = Nat.factorial m\n\u22a2 x ^ I * y ^ (m - I) * \u2191(Nat.choose m I) * (\u2191(Nat.factorial m))\u207b\u00b9 =\n    x ^ I / \u2191(Nat.factorial I) * (y ^ (m - I) / \u2191(Nat.factorial (m - I)))", "state_after": "x y : \u2102\nj m : \u2115\nx\u271d : m \u2208 range j\nI : \u2115\nhi : I \u2208 range (m + 1)\nh\u2081 : \u2191(Nat.choose m I) \u2260 0\nh\u2082 : Nat.choose m I * Nat.factorial I * Nat.factorial (m - I) = Nat.factorial m\n\u22a2 x ^ I * y ^ (m - I) * \u2191(Nat.choose m I) *\n      ((\u2191(Nat.choose m I))\u207b\u00b9 * (\u2191(Nat.factorial I))\u207b\u00b9 * (\u2191(Nat.factorial (m - I)))\u207b\u00b9) =\n    x ^ I / \u2191(Nat.factorial I) * (y ^ (m - I) / \u2191(Nat.factorial (m - I)))"}, {"tactic": "simp only [mul_left_comm (m.choose I : \u2102), mul_assoc, mul_left_comm (m.choose I : \u2102)\u207b\u00b9,\n  mul_comm (m.choose I : \u2102)]", "state_before": "x y : \u2102\nj m : \u2115\nx\u271d : m \u2208 range j\nI : \u2115\nhi : I \u2208 range (m + 1)\nh\u2081 : \u2191(Nat.choose m I) \u2260 0\nh\u2082 : Nat.choose m I * Nat.factorial I * Nat.factorial (m - I) = Nat.factorial m\n\u22a2 x ^ I * y ^ (m - I) * \u2191(Nat.choose m I) *\n      ((\u2191(Nat.choose m I))\u207b\u00b9 * (\u2191(Nat.factorial I))\u207b\u00b9 * (\u2191(Nat.factorial (m - I)))\u207b\u00b9) =\n    x ^ I / \u2191(Nat.factorial I) * (y ^ (m - I) / \u2191(Nat.factorial (m - I)))", "state_after": "x y : \u2102\nj m : \u2115\nx\u271d : m \u2208 range j\nI : \u2115\nhi : I \u2208 range (m + 1)\nh\u2081 : \u2191(Nat.choose m I) \u2260 0\nh\u2082 : Nat.choose m I * Nat.factorial I * Nat.factorial (m - I) = Nat.factorial m\n\u22a2 x ^ I *\n      (y ^ (m - I) *\n        ((\u2191(Nat.factorial I))\u207b\u00b9 * ((\u2191(Nat.factorial (m - I)))\u207b\u00b9 * ((\u2191(Nat.choose m I))\u207b\u00b9 * \u2191(Nat.choose m I))))) =\n    x ^ I / \u2191(Nat.factorial I) * (y ^ (m - I) / \u2191(Nat.factorial (m - I)))"}, {"tactic": "rw [inv_mul_cancel h\u2081]", "state_before": "x y : \u2102\nj m : \u2115\nx\u271d : m \u2208 range j\nI : \u2115\nhi : I \u2208 range (m + 1)\nh\u2081 : \u2191(Nat.choose m I) \u2260 0\nh\u2082 : Nat.choose m I * Nat.factorial I * Nat.factorial (m - I) = Nat.factorial m\n\u22a2 x ^ I *\n      (y ^ (m - I) *\n        ((\u2191(Nat.factorial I))\u207b\u00b9 * ((\u2191(Nat.factorial (m - I)))\u207b\u00b9 * ((\u2191(Nat.choose m I))\u207b\u00b9 * \u2191(Nat.choose m I))))) =\n    x ^ I / \u2191(Nat.factorial I) * (y ^ (m - I) / \u2191(Nat.factorial (m - I)))", "state_after": "x y : \u2102\nj m : \u2115\nx\u271d : m \u2208 range j\nI : \u2115\nhi : I \u2208 range (m + 1)\nh\u2081 : \u2191(Nat.choose m I) \u2260 0\nh\u2082 : Nat.choose m I * Nat.factorial I * Nat.factorial (m - I) = Nat.factorial m\n\u22a2 x ^ I * (y ^ (m - I) * ((\u2191(Nat.factorial I))\u207b\u00b9 * ((\u2191(Nat.factorial (m - I)))\u207b\u00b9 * 1))) =\n    x ^ I / \u2191(Nat.factorial I) * (y ^ (m - I) / \u2191(Nat.factorial (m - I)))"}, {"tactic": "simp [div_eq_mul_inv, mul_comm, mul_assoc, mul_left_comm]", "state_before": "x y : \u2102\nj m : \u2115\nx\u271d : m \u2208 range j\nI : \u2115\nhi : I \u2208 range (m + 1)\nh\u2081 : \u2191(Nat.choose m I) \u2260 0\nh\u2082 : Nat.choose m I * Nat.factorial I * Nat.factorial (m - I) = Nat.factorial m\n\u22a2 x ^ I * (y ^ (m - I) * ((\u2191(Nat.factorial I))\u207b\u00b9 * ((\u2191(Nat.factorial (m - I)))\u207b\u00b9 * 1))) =\n    x ^ I / \u2191(Nat.factorial I) * (y ^ (m - I) / \u2191(Nat.factorial (m - I)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Field/Power.lean", "full_name": "zpow_neg_two_nonneg", "start": [137, 1], "end": [138, 26], "traced_tactics": []}]