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\u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nwo : IsWellOrder \u03b1 r\n\u22a2 Quotient.mk isEquivalent { \u03b1 := \u03b1, r := r, wo := wo } = type r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/UniformGroup.lean", "full_name": "UniformContinuous.pow_const", "start": [118, 1], "end": [125, 34], "traced_tactics": [{"tactic": "simp_rw [pow_zero]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : UniformContinuous f\n\u22a2 UniformContinuous fun x => f x ^ 0", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : UniformContinuous f\n\u22a2 UniformContinuous fun x => 1"}, {"tactic": "exact uniformContinuous_const", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : UniformContinuous f\n\u22a2 UniformContinuous fun x => 1", "state_after": "no goals"}, {"tactic": "simp_rw [pow_succ]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : UniformContinuous f\nn : \u2115\n\u22a2 UniformContinuous fun x => f x ^ (n + 1)", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : UniformSpace 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"Real.arctan_eq_arcsin", "start": [164, 1], "end": [165, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Monoid.lean", "full_name": "Set.image_const_add_Icc", "start": [126, 1], "end": [127, 46], "traced_tactics": [{"tactic": "simp only [add_comm a, image_add_const_Icc]", "state_before": "M : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 (fun x => a + x) '' Icc b c = Icc (a + b) (a + c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Stream/Init.lean", "full_name": "Stream'.cycle_eq", "start": [643, 1], "end": [656, 12], "traced_tactics": [{"tactic": "intro l'", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\n\u22a2 \u2200 (l' : List \u03b1) (a' : \u03b1),\n    corec Stream'.cycleF Stream'.cycleG (a', l', a, l) = a' :: l' ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\nl' : List \u03b1\n\u22a2 \u2200 (a' : \u03b1),\n    corec Stream'.cycleF Stream'.cycleG (a', l', a, l) = a' :: l' ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)"}, {"tactic": "induction' l' with a\u2081 l\u2081 ih", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\nl' : List \u03b1\n\u22a2 \u2200 (a' : \u03b1),\n    corec Stream'.cycleF Stream'.cycleG (a', l', a, l) = a' :: l' ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)", "state_after": "case nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\n\u22a2 \u2200 (a' : \u03b1),\n    corec Stream'.cycleF Stream'.cycleG (a', [], a, l) = [a'] ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)\n\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\na\u2081 : \u03b1\nl\u2081 : List \u03b1\nih :\n  \u2200 (a' : \u03b1),\n    corec Stream'.cycleF Stream'.cycleG (a', l\u2081, a, l) = a' :: l\u2081 ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)\n\u22a2 \u2200 (a' : \u03b1),\n    corec Stream'.cycleF Stream'.cycleG (a', a\u2081 :: l\u2081, a, l) =\n      a' :: a\u2081 :: l\u2081 ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)"}, {"tactic": "intros", "state_before": "case nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\n\u22a2 \u2200 (a' : \u03b1),\n    corec Stream'.cycleF Stream'.cycleG (a', [], a, l) = [a'] ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)", "state_after": "case nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\na'\u271d : \u03b1\n\u22a2 corec Stream'.cycleF Stream'.cycleG (a'\u271d, [], a, l) = [a'\u271d] ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)"}, {"tactic": "rw [corec_eq]", "state_before": "case nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\na'\u271d : \u03b1\n\u22a2 corec Stream'.cycleF Stream'.cycleG (a'\u271d, [], a, l) = [a'\u271d] ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)", "state_after": "case nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\na'\u271d : \u03b1\n\u22a2 Stream'.cycleF (a'\u271d, [], a, l) :: corec Stream'.cycleF Stream'.cycleG (Stream'.cycleG (a'\u271d, [], a, l)) =\n    [a'\u271d] ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)"}, {"tactic": "rfl", "state_before": "case nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\na'\u271d : \u03b1\n\u22a2 Stream'.cycleF (a'\u271d, [], a, l) :: corec Stream'.cycleF Stream'.cycleG (Stream'.cycleG (a'\u271d, [], a, l)) =\n    [a'\u271d] ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)", "state_after": "no goals"}, {"tactic": "intros", "state_before": "case cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\na\u2081 : \u03b1\nl\u2081 : List \u03b1\nih :\n  \u2200 (a' : \u03b1),\n    corec Stream'.cycleF Stream'.cycleG (a', l\u2081, a, l) = a' :: l\u2081 ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)\n\u22a2 \u2200 (a' : \u03b1),\n    corec Stream'.cycleF Stream'.cycleG (a', a\u2081 :: l\u2081, a, l) =\n      a' :: a\u2081 :: l\u2081 ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)", "state_after": "case cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\na\u2081 : \u03b1\nl\u2081 : List \u03b1\nih :\n  \u2200 (a' : \u03b1),\n    corec Stream'.cycleF Stream'.cycleG (a', l\u2081, a, l) = a' :: l\u2081 ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)\na'\u271d : \u03b1\n\u22a2 corec Stream'.cycleF Stream'.cycleG (a'\u271d, a\u2081 :: l\u2081, a, l) =\n    a'\u271d :: a\u2081 :: l\u2081 ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)"}, {"tactic": "rw [corec_eq, Stream'.cycle_g_cons, ih a\u2081]", "state_before": "case cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\na\u2081 : \u03b1\nl\u2081 : List \u03b1\nih :\n  \u2200 (a' : \u03b1),\n    corec Stream'.cycleF Stream'.cycleG (a', l\u2081, a, l) = a' :: l\u2081 ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)\na'\u271d : \u03b1\n\u22a2 corec Stream'.cycleF Stream'.cycleG (a'\u271d, a\u2081 :: l\u2081, a, l) =\n    a'\u271d :: a\u2081 :: l\u2081 ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)", "state_after": "case cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\na\u2081 : \u03b1\nl\u2081 : List \u03b1\nih :\n  \u2200 (a' : \u03b1),\n    corec Stream'.cycleF Stream'.cycleG (a', l\u2081, a, l) = a' :: l\u2081 ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)\na'\u271d : \u03b1\n\u22a2 Stream'.cycleF (a'\u271d, a\u2081 :: l\u2081, a, l) :: (a\u2081 :: l\u2081 ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)) =\n    a'\u271d :: a\u2081 :: l\u2081 ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)"}, {"tactic": "rfl", "state_before": "case cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\na : \u03b1\nl : List \u03b1\nx\u271d : a :: l \u2260 []\na\u2081 : \u03b1\nl\u2081 : List \u03b1\nih :\n  \u2200 (a' : \u03b1),\n    corec Stream'.cycleF Stream'.cycleG (a', l\u2081, a, l) = a' :: l\u2081 ++\u209b corec Stream'.cycleF Stream'.cycleG (a, l, a, l)\na'\u271d 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: M\nX : C\n\u22a2 (F.obj n).map (F.\u03b5.app X) = (F.map (\u03bb_ n).inv).app X \u226b (MonoidalFunctor.\u03bcIso F (\ud835\udfd9_ M) n).inv.app X", "state_after": "C : Type u\ninst\u271d\u00b2 : Category C\nM : Type u_2\ninst\u271d\u00b9 : Category M\ninst\u271d : MonoidalCategory M\nF : MonoidalFunctor M (C \u2964 C)\nn : M\nX : C\n\u22a2 (F.obj n).map (F.\u03b5.app X) =\n    \ud835\udfd9 ((F.obj n).obj ((\ud835\udfd9_ (C \u2964 C)).obj X)) \u226b (F.map (\u03bb_ n).inv).app X \u226b (MonoidalFunctor.\u03bcIso F (\ud835\udfd9_ M) n).inv.app X"}, {"tactic": "rw [\u2190 Category.assoc, \u2190 IsIso.comp_inv_eq, \u2190 IsIso.comp_inv_eq, Category.assoc]", "state_before": "C : Type u\ninst\u271d\u00b2 : Category C\nM : Type u_2\ninst\u271d\u00b9 : Category M\ninst\u271d : MonoidalCategory M\nF : MonoidalFunctor M (C \u2964 C)\nn : M\nX : C\n\u22a2 (F.obj n).map (F.\u03b5.app X) =\n    \ud835\udfd9 ((F.obj n).obj ((\ud835\udfd9_ (C \u2964 C)).obj X)) \u226b (F.map (\u03bb_ n).inv).app X 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: Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\n\u22a2 u n =\n    if h : n < E.order then init { val := n, isLt := h }\n    else\n      \u2211 k : Fin E.order,\n        let_fun x := (_ : n - E.order + \u2191k < n);\n        coeffs E k * mkSol E init (n - E.order + \u2191k)"}, {"tactic": "split_ifs with h'", "state_before": "\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\n\u22a2 u n =\n    if h : n < E.order then init { val := n, isLt := h }\n    else\n      \u2211 k : Fin E.order,\n        let_fun x := (_ : n - E.order + \u2191k < n);\n        coeffs E k * mkSol E init (n - E.order + \u2191k)", "state_after": "case inl\n\u03b1 : Type 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n =\n    \u2211 k : Fin E.order,\n      let_fun x := (_ : n - E.order + \u2191k < n);\n      coeffs E k * mkSol E init (n - E.order + \u2191k)", "state_after": "case inr\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\n\u22a2 u n = \u2211 x : Fin E.order, coeffs E x * mkSol E init (n - E.order + \u2191x)"}, {"tactic": "rw [\u2190 tsub_add_cancel_of_le (le_of_not_lt h'), h (n - E.order)]", "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\n\u22a2 u n = \u2211 x : Fin E.order, coeffs E x * mkSol E init (n - E.order + \u2191x)", "state_after": "case inr\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\n\u22a2 \u2211 i : Fin E.order, coeffs E i * u (n - E.order + \u2191i) =\n    \u2211 x : Fin E.order, coeffs E x * mkSol E init (n - E.order + E.order - E.order + \u2191x)"}, {"tactic": "congr with k", "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\n\u22a2 \u2211 i : Fin E.order, coeffs E i * u (n - E.order + \u2191i) =\n    \u2211 x : Fin E.order, coeffs E x * mkSol E init (n - E.order + E.order - E.order + \u2191x)", "state_after": "case inr.e_f.h\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\nk : Fin E.order\n\u22a2 coeffs E k * u (n - E.order + \u2191k) = coeffs E k * mkSol E init (n - E.order + E.order - E.order + \u2191k)"}, {"tactic": "rw [eq_mk_of_is_sol_of_eq_init h heq (n - E.order + k)]", "state_before": "case inr.e_f.h\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\nk : Fin E.order\nthis : n - E.order + \u2191k < n\n\u22a2 coeffs E k * u (n - E.order + \u2191k) = coeffs E k * mkSol E init (n - E.order + E.order - E.order + \u2191k)", "state_after": "case inr.e_f.h\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\nk : Fin E.order\nthis : n - E.order + \u2191k < n\n\u22a2 coeffs E k * mkSol E (fun n => init n) (n - E.order + \u2191k) =\n    coeffs E k * mkSol E init (n - E.order + E.order - E.order + \u2191k)"}, {"tactic": "simp", "state_before": "case inr.e_f.h\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\nk : Fin E.order\nthis : n - E.order + \u2191k < n\n\u22a2 coeffs E k * mkSol E (fun n => init n) (n - E.order + \u2191k) =\n    coeffs E k * mkSol E init (n - E.order + E.order - E.order + \u2191k)", "state_after": "no goals"}, {"tactic": "exact_mod_cast heq \u27e8n, h'\u27e9", "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : n < E.order\n\u22a2 u n = init { val := n, isLt := h' }", "state_after": "no goals"}, {"tactic": "rw [add_comm, \u2190 add_tsub_assoc_of_le (not_lt.mp h'), tsub_lt_iff_left]", "state_before": "\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\nk : Fin E.order\n\u22a2 n - E.order + \u2191k < n", "state_after": "\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\nk : Fin E.order\n\u22a2 \u2191k + n < E.order + n\n\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\nk : Fin E.order\n\u22a2 E.order \u2264 \u2191k + n"}, {"tactic": "exact add_lt_add_right k.is_lt n", "state_before": "\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\nk : Fin E.order\n\u22a2 \u2191k + n < E.order + n", "state_after": "no goals"}, {"tactic": "convert add_le_add (zero_le (k : \u2115)) (not_lt.mp h')", "state_before": "\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\nk : Fin E.order\n\u22a2 E.order \u2264 \u2191k + n", "state_after": "case h.e'_3\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\nk : Fin E.order\n\u22a2 E.order = 0 + E.order"}, {"tactic": "simp only [zero_add]", "state_before": "case h.e'_3\n\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\nu : \u2115 \u2192 \u03b1\ninit : Fin E.order \u2192 \u03b1\nh : IsSolution E u\nheq : \u2200 (n : Fin E.order), u \u2191n = init n\nn : \u2115\nh' : \u00acn < E.order\nk : Fin E.order\n\u22a2 E.order = 0 + E.order", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Module/LinearMap.lean", "full_name": "LinearMap.coe_restrictScalars", "start": [460, 1], "end": [461, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "full_name": "SimpleGraph.Walk.concat_nil", "start": [284, 1], "end": [284, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.preimage_eq_iff_eq_image", "start": [1562, 1], "end": [1563, 53], "traced_tactics": [{"tactic": "rw [\u2190 image_eq_image hf.1, hf.2.image_preimage]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf\u271d f : \u03b1 \u2192 \u03b2\nhf : Bijective f\ns : Set \u03b2\nt : Set \u03b1\n\u22a2 f \u207b\u00b9' s = t \u2194 s = f '' t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Prelude.lean", "full_name": "eq_true_of_ne_false", "start": [647, 1], "end": [649, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Fin.lean", "full_name": "Fin.prod_Ioi_succ", "start": [182, 1], "end": [184, 82], "traced_tactics": [{"tactic": "rw [Ioi_succ, Finset.prod_map, RelEmbedding.coe_toEmbedding, val_succEmbedding]", "state_before": "\u03b1 : Type ?u.64433\n\u03b2 : Type ?u.64436\nM : Type u_1\ninst\u271d : CommMonoid M\nn : \u2115\ni : Fin n\nv : Fin (Nat.succ n) \u2192 M\n\u22a2 \u220f j in Ioi (succ i), v j = \u220f j in Ioi i, v (succ j)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean", "full_name": "UniformCauchySeqOnFilter.one_smulRight", "start": [462, 1], "end": [481, 9], "traced_tactics": [{"tactic": "rw [SeminormedAddGroup.uniformCauchySeqOnFilter_iff_tendstoUniformlyOnFilter_zero,\n  Metric.tendstoUniformlyOnFilter_iff] at hf' \u22a2", "state_before": "\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' : UniformCauchySeqOnFilter f' l l'\n\u22a2 UniformCauchySeqOnFilter (fun n z => ContinuousLinearMap.smulRight 1 (f' n z)) l l'", "state_after": "\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < \u03b5\n\u22a2 \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l',\n        dist (OfNat.ofNat 0 n.snd)\n            (ContinuousLinearMap.smulRight 1 (f' n.fst.fst n.snd) -\n              ContinuousLinearMap.smulRight 1 (f' n.fst.snd n.snd)) <\n          \u03b5"}, {"tactic": "intro \u03b5 h\u03b5", "state_before": "\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < \u03b5\n\u22a2 \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l',\n        dist (OfNat.ofNat 0 n.snd)\n            (ContinuousLinearMap.smulRight 1 (f' n.fst.fst n.snd) -\n              ContinuousLinearMap.smulRight 1 (f' n.fst.snd n.snd)) <\n          \u03b5", "state_after": "\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l',\n    dist (OfNat.ofNat 0 n.snd)\n        (ContinuousLinearMap.smulRight 1 (f' n.fst.fst n.snd) - ContinuousLinearMap.smulRight 1 (f' n.fst.snd n.snd)) <\n      \u03b5"}, {"tactic": "obtain \u27e8q, hq, hq'\u27e9 := exists_between h\u03b5.lt", "state_before": "\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l',\n    dist (OfNat.ofNat 0 n.snd)\n        (ContinuousLinearMap.smulRight 1 (f' n.fst.fst n.snd) - ContinuousLinearMap.smulRight 1 (f' n.fst.snd n.snd)) <\n      \u03b5", "state_after": "case intro.intro\n\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nq : \u211d\nhq : 0 < q\nhq' : q < \u03b5\n\u22a2 \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l',\n    dist (OfNat.ofNat 0 n.snd)\n        (ContinuousLinearMap.smulRight 1 (f' n.fst.fst n.snd) - ContinuousLinearMap.smulRight 1 (f' n.fst.snd n.snd)) <\n      \u03b5"}, {"tactic": "apply (hf' q hq).mono", "state_before": "case intro.intro\n\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nq : \u211d\nhq : 0 < q\nhq' : q < \u03b5\n\u22a2 \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l',\n    dist (OfNat.ofNat 0 n.snd)\n        (ContinuousLinearMap.smulRight 1 (f' n.fst.fst n.snd) - ContinuousLinearMap.smulRight 1 (f' n.fst.snd n.snd)) <\n      \u03b5", "state_after": "case intro.intro\n\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nq : \u211d\nhq : 0 < q\nhq' : q < \u03b5\n\u22a2 \u2200 (x : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c),\n    dist (OfNat.ofNat 0 x.snd) (f' x.fst.fst x.snd - f' x.fst.snd x.snd) < q \u2192\n      dist (OfNat.ofNat 0 x.snd)\n          (ContinuousLinearMap.smulRight 1 (f' x.fst.fst x.snd) -\n            ContinuousLinearMap.smulRight 1 (f' x.fst.snd x.snd)) <\n        \u03b5"}, {"tactic": "intro n hn", "state_before": "case intro.intro\n\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nq : \u211d\nhq : 0 < q\nhq' : q < \u03b5\n\u22a2 \u2200 (x : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c),\n    dist (OfNat.ofNat 0 x.snd) (f' x.fst.fst x.snd - f' x.fst.snd x.snd) < q \u2192\n      dist (OfNat.ofNat 0 x.snd)\n          (ContinuousLinearMap.smulRight 1 (f' x.fst.fst x.snd) -\n            ContinuousLinearMap.smulRight 1 (f' x.fst.snd x.snd)) <\n        \u03b5", "state_after": "case intro.intro\n\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nq : \u211d\nhq : 0 < q\nhq' : q < \u03b5\nn : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c\nhn : dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < q\n\u22a2 dist (OfNat.ofNat 0 n.snd)\n      (ContinuousLinearMap.smulRight 1 (f' n.fst.fst n.snd) - ContinuousLinearMap.smulRight 1 (f' n.fst.snd n.snd)) <\n    \u03b5"}, {"tactic": "refine' lt_of_le_of_lt _ hq'", "state_before": "case intro.intro\n\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nq : \u211d\nhq : 0 < q\nhq' : q < \u03b5\nn : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c\nhn : dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < q\n\u22a2 dist (OfNat.ofNat 0 n.snd)\n      (ContinuousLinearMap.smulRight 1 (f' n.fst.fst n.snd) - ContinuousLinearMap.smulRight 1 (f' n.fst.snd n.snd)) <\n    \u03b5", "state_after": "case intro.intro\n\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nq : \u211d\nhq : 0 < q\nhq' : q < \u03b5\nn : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c\nhn : dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < q\n\u22a2 dist (OfNat.ofNat 0 n.snd)\n      (ContinuousLinearMap.smulRight 1 (f' n.fst.fst n.snd) - ContinuousLinearMap.smulRight 1 (f' n.fst.snd n.snd)) \u2264\n    q"}, {"tactic": "simp only [dist_eq_norm, Pi.zero_apply, zero_sub, norm_neg] at hn \u22a2", "state_before": "case intro.intro\n\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nq : \u211d\nhq : 0 < q\nhq' : q < \u03b5\nn : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c\nhn : dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < q\n\u22a2 dist (OfNat.ofNat 0 n.snd)\n      (ContinuousLinearMap.smulRight 1 (f' n.fst.fst n.snd) - ContinuousLinearMap.smulRight 1 (f' n.fst.snd n.snd)) \u2264\n    q", "state_after": "case intro.intro\n\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nq : \u211d\nhq : 0 < q\nhq' : q < \u03b5\nn : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c\nhn : \u2016f' n.fst.fst n.snd - f' n.fst.snd n.snd\u2016 < q\n\u22a2 \u2016ContinuousLinearMap.smulRight 1 (f' n.fst.fst n.snd) - ContinuousLinearMap.smulRight 1 (f' n.fst.snd n.snd)\u2016 \u2264 q"}, {"tactic": "refine' ContinuousLinearMap.op_norm_le_bound _ hq.le _", "state_before": "case intro.intro\n\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nq : \u211d\nhq : 0 < q\nhq' : q < \u03b5\nn : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c\nhn : \u2016f' n.fst.fst n.snd - f' n.fst.snd n.snd\u2016 < q\n\u22a2 \u2016ContinuousLinearMap.smulRight 1 (f' n.fst.fst n.snd) - ContinuousLinearMap.smulRight 1 (f' n.fst.snd n.snd)\u2016 \u2264 q", "state_after": "case intro.intro\n\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nq : \u211d\nhq : 0 < q\nhq' : q < \u03b5\nn : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c\nhn : \u2016f' n.fst.fst n.snd - f' n.fst.snd n.snd\u2016 < q\n\u22a2 \u2200 (x : \ud835\udd5c),\n    \u2016\u2191(ContinuousLinearMap.smulRight 1 (f' n.fst.fst n.snd) - ContinuousLinearMap.smulRight 1 (f' n.fst.snd n.snd)) x\u2016 \u2264\n      q * \u2016x\u2016"}, {"tactic": "intro z", "state_before": "case intro.intro\n\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nq : \u211d\nhq : 0 < q\nhq' : q < \u03b5\nn : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c\nhn : \u2016f' n.fst.fst n.snd - f' n.fst.snd n.snd\u2016 < q\n\u22a2 \u2200 (x : \ud835\udd5c),\n    \u2016\u2191(ContinuousLinearMap.smulRight 1 (f' n.fst.fst n.snd) - ContinuousLinearMap.smulRight 1 (f' n.fst.snd n.snd)) x\u2016 \u2264\n      q * \u2016x\u2016", "state_after": "case intro.intro\n\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nq : \u211d\nhq : 0 < q\nhq' : q < \u03b5\nn : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c\nhn : \u2016f' n.fst.fst n.snd - f' n.fst.snd n.snd\u2016 < q\nz : \ud835\udd5c\n\u22a2 \u2016\u2191(ContinuousLinearMap.smulRight 1 (f' n.fst.fst n.snd) - ContinuousLinearMap.smulRight 1 (f' n.fst.snd n.snd)) z\u2016 \u2264\n    q * \u2016z\u2016"}, {"tactic": "simp only [ContinuousLinearMap.coe_sub', Pi.sub_apply, ContinuousLinearMap.smulRight_apply,\n  ContinuousLinearMap.one_apply]", "state_before": "case intro.intro\n\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nq : \u211d\nhq : 0 < q\nhq' : q < \u03b5\nn : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c\nhn : \u2016f' n.fst.fst n.snd - f' n.fst.snd n.snd\u2016 < q\nz : \ud835\udd5c\n\u22a2 \u2016\u2191(ContinuousLinearMap.smulRight 1 (f' n.fst.fst n.snd) - ContinuousLinearMap.smulRight 1 (f' n.fst.snd n.snd)) z\u2016 \u2264\n    q * \u2016z\u2016", "state_after": "case intro.intro\n\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nq : \u211d\nhq : 0 < q\nhq' : q < \u03b5\nn : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c\nhn : \u2016f' n.fst.fst n.snd - f' n.fst.snd n.snd\u2016 < q\nz : \ud835\udd5c\n\u22a2 \u2016z \u2022 f' n.fst.fst n.snd - z \u2022 f' n.fst.snd n.snd\u2016 \u2264 q * \u2016z\u2016"}, {"tactic": "rw [\u2190 smul_sub, norm_smul, mul_comm]", "state_before": "case intro.intro\n\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nq : \u211d\nhq : 0 < q\nhq' : q < \u03b5\nn : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c\nhn : \u2016f' n.fst.fst n.snd - f' n.fst.snd n.snd\u2016 < q\nz : \ud835\udd5c\n\u22a2 \u2016z \u2022 f' n.fst.fst n.snd - z \u2022 f' n.fst.snd n.snd\u2016 \u2264 q * \u2016z\u2016", "state_after": "case intro.intro\n\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nq : \u211d\nhq : 0 < q\nhq' : q < \u03b5\nn : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c\nhn : \u2016f' n.fst.fst n.snd - f' n.fst.snd n.snd\u2016 < q\nz : \ud835\udd5c\n\u22a2 \u2016f' n.fst.fst n.snd - f' n.fst.snd n.snd\u2016 * \u2016z\u2016 \u2264 q * \u2016z\u2016"}, {"tactic": "gcongr", "state_before": "case intro.intro\n\u03b9 : Type u_3\nl : Filter \u03b9\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nG : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng : \ud835\udd5c \u2192 G\nf' : \u03b9 \u2192 \ud835\udd5c \u2192 G\ng' : \ud835\udd5c \u2192 G\nx : \ud835\udd5c\nl' : Filter \ud835\udd5c\nhf' :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nq : \u211d\nhq : 0 < q\nhq' : q < \u03b5\nn : (\u03b9 \u00d7 \u03b9) \u00d7 \ud835\udd5c\nhn : \u2016f' n.fst.fst n.snd - f' n.fst.snd n.snd\u2016 < q\nz : \ud835\udd5c\n\u22a2 \u2016f' n.fst.fst n.snd - f' n.fst.snd n.snd\u2016 * \u2016z\u2016 \u2264 q * \u2016z\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "full_name": "abs_norm_sub_norm_le'", "start": [559, 1], "end": [560, 55], "traced_tactics": [{"tactic": "simpa [dist_eq_norm_div] using abs_dist_sub_le a b 1", "state_before": "\ud835\udcd5 : Type ?u.82498\n\ud835\udd5c : Type ?u.82501\n\u03b1 : Type ?u.82504\n\u03b9 : Type ?u.82507\n\u03ba : Type ?u.82510\nE : Type u_1\nF : Type ?u.82516\nG : Type ?u.82519\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 : E\n\u22a2 abs (\u2016a\u2016 - \u2016b\u2016) \u2264 \u2016a / b\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Alternating.lean", "full_name": "AlternatingMap.coe_compLinearMap", "start": [545, 1], "end": [547, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "full_name": "Polynomial.degree_C", "start": [249, 1], "end": [251, 22], "traced_tactics": [{"tactic": "rw [degree, \u2190 monomial_zero_left, support_monomial 0 ha, max_eq_sup_coe, sup_singleton,\n  WithBot.coe_zero]", "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nha : a \u2260 0\n\u22a2 degree (\u2191C a) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "full_name": "Ordinal.le_enum_succ", "start": [1177, 1], "end": [1184, 23], "traced_tactics": [{"tactic": "rw [type_lt]", "state_before": "\u03b1 : Type ?u.196040\n\u03b2 : Type ?u.196043\n\u03b3 : Type ?u.196046\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal\na : (Quotient.out (succ o)).\u03b1\n\u22a2 o < type fun x x_1 => x < x_1", "state_after": "\u03b1 : Type ?u.196040\n\u03b2 : Type ?u.196043\n\u03b3 : Type ?u.196046\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal\na : (Quotient.out (succ o)).\u03b1\n\u22a2 o < succ o"}, {"tactic": "exact lt_succ o", "state_before": "\u03b1 : Type ?u.196040\n\u03b2 : Type ?u.196043\n\u03b3 : Type ?u.196046\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal\na : (Quotient.out (succ o)).\u03b1\n\u22a2 o < succ o", "state_after": "no goals"}, {"tactic": "apply typein_lt_self", "state_before": "\u03b1 : Type ?u.196040\n\u03b2 : Type ?u.196043\n\u03b3 : Type ?u.196046\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal\na : (Quotient.out (succ o)).\u03b1\n\u22a2 typein (fun x x_1 => x < x_1) a < succ o", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/Deriv/Inv.lean", "full_name": "hasFDerivAt_inv", "start": [113, 1], "end": [115, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Int/Lemmas.lean", "full_name": "Int.natAbs_lt_iff_sq_lt", "start": [54, 1], "end": [56, 34], "traced_tactics": [{"tactic": "rw [sq, sq]", "state_before": "a\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\n\u22a2 natAbs a < natAbs b \u2194 a ^ 2 < b ^ 2", "state_after": "a\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\n\u22a2 natAbs a < natAbs b \u2194 a * a < b * b"}, {"tactic": "exact natAbs_lt_iff_mul_self_lt", "state_before": "a\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\n\u22a2 natAbs a < natAbs b \u2194 a * a < b * b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean", "full_name": "isAdic_iff", "start": [162, 1], "end": [188, 48], "traced_tactics": [{"tactic": "constructor", "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\n\u22a2 IsAdic J \u2194 (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s", "state_after": "case mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\n\u22a2 IsAdic J \u2192 (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\n\ncase mpr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\n\u22a2 ((\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s) \u2192 IsAdic J"}, {"tactic": "intro H", "state_before": "case mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\n\u22a2 IsAdic J \u2192 (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s", "state_after": "case mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : IsAdic J\n\u22a2 (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s"}, {"tactic": "change _ = _ at H", "state_before": "case mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : IsAdic J\n\u22a2 (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s", "state_after": "case mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : top = Ideal.adicTopology J\n\u22a2 (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s"}, {"tactic": "rw [H]", "state_before": "case mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : top = Ideal.adicTopology J\n\u22a2 (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s", "state_after": "case mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : top = Ideal.adicTopology J\n\u22a2 (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s"}, {"tactic": "letI := J.adicTopology", "state_before": "case mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : top = Ideal.adicTopology J\n\u22a2 (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s", "state_after": "case mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : top = Ideal.adicTopology J\nthis : TopologicalSpace R := Ideal.adicTopology J\n\u22a2 (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s"}, {"tactic": "constructor", "state_before": "case mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : top = Ideal.adicTopology J\nthis : TopologicalSpace R := Ideal.adicTopology J\n\u22a2 (\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s", "state_after": "case mp.left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : top = Ideal.adicTopology J\nthis : TopologicalSpace R := Ideal.adicTopology J\n\u22a2 \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\n\ncase mp.right\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : top = Ideal.adicTopology J\nthis : TopologicalSpace R := Ideal.adicTopology J\n\u22a2 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s"}, {"tactic": "intro n", "state_before": "case mp.left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : top = Ideal.adicTopology J\nthis : TopologicalSpace R := Ideal.adicTopology J\n\u22a2 \u2200 (n : \u2115), IsOpen \u2191(J ^ n)", "state_after": "case mp.left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : top = Ideal.adicTopology J\nthis : TopologicalSpace R := Ideal.adicTopology J\nn : \u2115\n\u22a2 IsOpen \u2191(J ^ n)"}, {"tactic": "exact (J.openAddSubgroup n).isOpen'", "state_before": "case mp.left\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : top = Ideal.adicTopology J\nthis : TopologicalSpace R := Ideal.adicTopology J\nn : \u2115\n\u22a2 IsOpen \u2191(J ^ n)", "state_after": "no goals"}, {"tactic": "intro s hs", "state_before": "case mp.right\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : top = Ideal.adicTopology J\nthis : TopologicalSpace R := Ideal.adicTopology J\n\u22a2 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s", "state_after": "case mp.right\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : top = Ideal.adicTopology J\nthis : TopologicalSpace R := Ideal.adicTopology J\ns : Set R\nhs : s \u2208 \ud835\udcdd 0\n\u22a2 \u2203 n, \u2191(J ^ n) \u2286 s"}, {"tactic": "simpa using J.hasBasis_nhds_zero_adic.mem_iff.mp hs", "state_before": "case mp.right\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH : top = Ideal.adicTopology J\nthis : TopologicalSpace R := Ideal.adicTopology J\ns : Set R\nhs : s \u2208 \ud835\udcdd 0\n\u22a2 \u2203 n, \u2191(J ^ n) \u2286 s", "state_after": "no goals"}, {"tactic": "rintro \u27e8H\u2081, H\u2082\u27e9", "state_before": "case mpr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\n\u22a2 ((\u2200 (n : \u2115), IsOpen \u2191(J ^ n)) \u2227 \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s) \u2192 IsAdic J", "state_after": "case mpr.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\n\u22a2 IsAdic J"}, {"tactic": "apply TopologicalAddGroup.ext", "state_before": "case mpr.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\n\u22a2 IsAdic J", "state_after": "case mpr.intro.tg\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\n\u22a2 TopologicalAddGroup R\n\ncase mpr.intro.tg'\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\n\u22a2 TopologicalAddGroup R\n\ncase mpr.intro.h\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\n\u22a2 \ud835\udcdd 0 = \ud835\udcdd 0"}, {"tactic": "apply @TopologicalRing.to_topologicalAddGroup", "state_before": "case mpr.intro.tg\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\n\u22a2 TopologicalAddGroup R", "state_after": "no goals"}, {"tactic": "apply (RingSubgroupsBasis.toRingFilterBasis _).toAddGroupFilterBasis.isTopologicalAddGroup", "state_before": "case mpr.intro.tg'\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\n\u22a2 TopologicalAddGroup R", "state_after": "no goals"}, {"tactic": "ext s", "state_before": "case mpr.intro.h\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\n\u22a2 \ud835\udcdd 0 = \ud835\udcdd 0", "state_after": "case mpr.intro.h.a\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\n\u22a2 s \u2208 \ud835\udcdd 0 \u2194 s \u2208 \ud835\udcdd 0"}, {"tactic": "letI := Ideal.adic_basis J", "state_before": "case mpr.intro.h.a\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\n\u22a2 s \u2208 \ud835\udcdd 0 \u2194 s \u2208 \ud835\udcdd 0", "state_after": "case mpr.intro.h.a\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\n\u22a2 s \u2208 \ud835\udcdd 0 \u2194 s \u2208 \ud835\udcdd 0"}, {"tactic": "rw [J.hasBasis_nhds_zero_adic.mem_iff]", "state_before": "case mpr.intro.h.a\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\n\u22a2 s \u2208 \ud835\udcdd 0 \u2194 s \u2208 \ud835\udcdd 0", "state_after": "case mpr.intro.h.a\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\n\u22a2 s \u2208 \ud835\udcdd 0 \u2194 \u2203 i, True \u2227 \u2191(J ^ i) \u2286 s"}, {"tactic": "constructor <;> intro H", "state_before": "case mpr.intro.h.a\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\n\u22a2 s \u2208 \ud835\udcdd 0 \u2194 \u2203 i, True \u2227 \u2191(J ^ i) \u2286 s", "state_after": "case mpr.intro.h.a.mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\nH : s \u2208 \ud835\udcdd 0\n\u22a2 \u2203 i, True \u2227 \u2191(J ^ i) \u2286 s\n\ncase mpr.intro.h.a.mpr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\nH : \u2203 i, True \u2227 \u2191(J ^ i) \u2286 s\n\u22a2 s \u2208 \ud835\udcdd 0"}, {"tactic": "rcases H\u2082 s H with \u27e8n, h\u27e9", "state_before": "case mpr.intro.h.a.mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\nH : s \u2208 \ud835\udcdd 0\n\u22a2 \u2203 i, True \u2227 \u2191(J ^ i) \u2286 s", "state_after": "case mpr.intro.h.a.mp.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\nH : s \u2208 \ud835\udcdd 0\nn : \u2115\nh : \u2191(J ^ n) \u2286 s\n\u22a2 \u2203 i, True \u2227 \u2191(J ^ i) \u2286 s"}, {"tactic": "exact \u27e8n, trivial, h\u27e9", "state_before": "case mpr.intro.h.a.mp.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\nH : s \u2208 \ud835\udcdd 0\nn : \u2115\nh : \u2191(J ^ n) \u2286 s\n\u22a2 \u2203 i, True \u2227 \u2191(J ^ i) \u2286 s", "state_after": "no goals"}, {"tactic": "rcases H with \u27e8n, -, hn\u27e9", "state_before": "case mpr.intro.h.a.mpr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\nH : \u2203 i, True \u2227 \u2191(J ^ i) \u2286 s\n\u22a2 s \u2208 \ud835\udcdd 0", "state_after": "case mpr.intro.h.a.mpr.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\nn : \u2115\nhn : \u2191(J ^ n) \u2286 s\n\u22a2 s \u2208 \ud835\udcdd 0"}, {"tactic": "rw [mem_nhds_iff]", "state_before": "case mpr.intro.h.a.mpr.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\nn : \u2115\nhn : \u2191(J ^ n) \u2286 s\n\u22a2 s \u2208 \ud835\udcdd 0", "state_after": "case mpr.intro.h.a.mpr.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\nn : \u2115\nhn : \u2191(J ^ n) \u2286 s\n\u22a2 \u2203 t, t \u2286 s \u2227 IsOpen t \u2227 0 \u2208 t"}, {"tactic": "refine' \u27e8_, hn, H\u2081 n, (J ^ n).zero_mem\u27e9", "state_before": "case mpr.intro.h.a.mpr.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ntop : TopologicalSpace R\ninst\u271d : TopologicalRing R\nJ : Ideal R\nH\u2081 : \u2200 (n : \u2115), IsOpen \u2191(J ^ n)\nH\u2082 : \u2200 (s : Set R), s \u2208 \ud835\udcdd 0 \u2192 \u2203 n, \u2191(J ^ n) \u2286 s\ns : Set R\nthis : SubmodulesRingBasis fun n => J ^ n \u2022 \u22a4 := Ideal.adic_basis J\nn : \u2115\nhn : \u2191(J ^ n) \u2286 s\n\u22a2 \u2203 t, t \u2286 s \u2227 IsOpen t \u2227 0 \u2208 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Sets/Closeds.lean", "full_name": "TopologicalSpace.Closeds.coe_finset_inf", "start": [153, 1], "end": [155, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fintype/Card.lean", "full_name": "Fintype.card_coe", "start": [727, 1], "end": [728, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Group.lean", "full_name": "Set.add_mem_Ico_iff_left", "start": [69, 1], "end": [70, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Hom/Centroid.lean", "full_name": "CentroidHom.toAddMonoidHom_eq_coe", "start": [127, 1], "end": [128, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/TrivSqZeroExt.lean", "full_name": "TrivSqZeroExt.algebraMap_eq_inl'", "start": [770, 1], "end": [771, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/Reduce.lean", "full_name": "OneOneReducible.trans", "start": [99, 1], "end": [103, 76], "traced_tactics": [{"tactic": "erw [\u2190 h\u2082, \u2190 h\u2081]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03b3\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nr : \u03b3 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nc\u2081 : Computable f\ni\u2081 : Injective f\nh\u2081 : \u2200 (a : \u03b1), p a \u2194 q (f a)\ng : \u03b2 \u2192 \u03b3\nc\u2082 : Computable g\ni\u2082 : Injective g\nh\u2082 : \u2200 (a : \u03b2), q a \u2194 r (g a)\na : \u03b1\nh : p a\n\u22a2 r ((g \u2218 f) a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03b3\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nr : \u03b3 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nc\u2081 : Computable f\ni\u2081 : Injective f\nh\u2081 : \u2200 (a : \u03b1), p a \u2194 q (f a)\ng : \u03b2 \u2192 \u03b3\nc\u2082 : Computable g\ni\u2082 : Injective g\nh\u2082 : \u2200 (a : \u03b2), q a \u2194 r (g a)\na : \u03b1\nh : p a\n\u22a2 p a"}, {"tactic": "assumption", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03b3\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nr : \u03b3 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nc\u2081 : Computable f\ni\u2081 : Injective f\nh\u2081 : \u2200 (a : \u03b1), p a \u2194 q (f a)\ng : \u03b2 \u2192 \u03b3\nc\u2082 : Computable g\ni\u2082 : Injective g\nh\u2082 : \u2200 (a : \u03b2), q a \u2194 r (g a)\na : \u03b1\nh : p a\n\u22a2 p a", "state_after": "no goals"}, {"tactic": "rwa [h\u2081, h\u2082]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Primcodable \u03b1\ninst\u271d\u00b9 : Primcodable \u03b2\ninst\u271d : Primcodable \u03b3\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nr : \u03b3 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nc\u2081 : Computable f\ni\u2081 : Injective f\nh\u2081 : \u2200 (a : \u03b1), p a \u2194 q (f a)\ng : \u03b2 \u2192 \u03b3\nc\u2082 : Computable g\ni\u2082 : Injective g\nh\u2082 : \u2200 (a : \u03b2), q a \u2194 r (g a)\na : \u03b1\nh : r ((g \u2218 f) a)\n\u22a2 p a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.inter_singleton_of_not_mem", "start": [1691, 1], "end": [1692, 48], "traced_tactics": [{"tactic": "rw [inter_comm, singleton_inter_of_not_mem h]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.188434\n\u03b3 : Type ?u.188437\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\nh : \u00aca \u2208 s\n\u22a2 s \u2229 {a} = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/GroupAction/SubMulAction.lean", "full_name": "SubMulAction.neg_mem_iff", "start": [364, 1], "end": [367, 34], "traced_tactics": [{"tactic": "rw [\u2190 neg_neg x]", "state_before": "S : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\np p' : SubMulAction R M\nr : R\nx y : M\nh : -x \u2208 p\n\u22a2 x \u2208 p", "state_after": "S : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\np p' : SubMulAction R M\nr : R\nx y : M\nh : -x \u2208 p\n\u22a2 - -x \u2208 p"}, {"tactic": "exact neg_mem _ h", "state_before": "S : Type u'\nT : Type u''\nR : Type u\nM : Type v\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\np p' : SubMulAction R M\nr : R\nx y : M\nh : -x \u2208 p\n\u22a2 - -x \u2208 p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/ContDiff.lean", "full_name": "contDiffOn_succ_of_fderiv_apply", "start": [1910, 1], "end": [1913, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/QuotientGroup.lean", "full_name": "QuotientGroup.homQuotientZPowOfHom_id", "start": [509, 1], "end": [510, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Basis.lean", "full_name": "Basis.repr_range", "start": [192, 1], "end": [193, 49], "traced_tactics": [{"tactic": "rw [LinearEquiv.range, Finsupp.supported_univ]", "state_before": "\u03b9 : Type u_1\n\u03b9' : Type ?u.55731\nR : Type u_2\nR\u2082 : Type ?u.55737\nK : Type ?u.55740\nM : Type u_3\nM' : Type ?u.55746\nM'' : Type ?u.55749\nV : Type u\nV' : Type ?u.55754\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nb b\u2081 : Basis \u03b9 R M\ni : \u03b9\nc : R\nx : M\n\u22a2 LinearMap.range \u2191b.repr = Finsupp.supported R R univ", "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_le_natTrailingDegree", "start": [192, 1], "end": [198, 33], "traced_tactics": [{"tactic": "by_cases hp : p = 0", "state_before": "R : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhq : q \u2260 0\nhpq : trailingDegree p \u2264 trailingDegree q\n\u22a2 natTrailingDegree p \u2264 natTrailingDegree q", "state_after": "case pos\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhq : q \u2260 0\nhpq : trailingDegree p \u2264 trailingDegree q\nhp : p = 0\n\u22a2 natTrailingDegree p \u2264 natTrailingDegree q\n\ncase neg\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhq : q \u2260 0\nhpq : trailingDegree p \u2264 trailingDegree q\nhp : \u00acp = 0\n\u22a2 natTrailingDegree p \u2264 natTrailingDegree q"}, {"tactic": "rw [trailingDegree_eq_natTrailingDegree hp, trailingDegree_eq_natTrailingDegree hq] at hpq", "state_before": "case neg\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhq : q \u2260 0\nhpq : trailingDegree p \u2264 trailingDegree q\nhp : \u00acp = 0\n\u22a2 natTrailingDegree p \u2264 natTrailingDegree q", "state_after": "case neg\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhq : q \u2260 0\nhpq : \u2191(natTrailingDegree p) \u2264 \u2191(natTrailingDegree q)\nhp : \u00acp = 0\n\u22a2 natTrailingDegree p \u2264 natTrailingDegree q"}, {"tactic": "exact WithTop.coe_le_coe.1 hpq", "state_before": "case neg\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhq : q \u2260 0\nhpq : \u2191(natTrailingDegree p) \u2264 \u2191(natTrailingDegree q)\nhp : \u00acp = 0\n\u22a2 natTrailingDegree p \u2264 natTrailingDegree q", "state_after": "no goals"}, {"tactic": "rw [hp, natTrailingDegree_zero]", "state_before": "case pos\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhq : q \u2260 0\nhpq : trailingDegree p \u2264 trailingDegree q\nhp : p = 0\n\u22a2 natTrailingDegree p \u2264 natTrailingDegree q", "state_after": "case pos\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhq : q \u2260 0\nhpq : trailingDegree p \u2264 trailingDegree q\nhp : p = 0\n\u22a2 0 \u2264 natTrailingDegree q"}, {"tactic": "exact zero_le _", "state_before": "case pos\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nhq : q \u2260 0\nhpq : trailingDegree p \u2264 trailingDegree q\nhp : p = 0\n\u22a2 0 \u2264 natTrailingDegree q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.tendsto_setToFun_of_dominated_convergence", "start": [1720, 1], "end": [1759, 51], "traced_tactics": [{"tactic": "have f_measurable : AEStronglyMeasurable f \u03bc :=\n  aestronglyMeasurable_of_tendsto_ae _ fs_measurable h_lim", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\n\u22a2 Tendsto (fun n => setToFun \u03bc T hT (fs n)) atTop (\ud835\udcdd (setToFun \u03bc T hT f))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\n\u22a2 Tendsto (fun n => setToFun \u03bc T hT (fs n)) atTop (\ud835\udcdd (setToFun \u03bc T hT f))"}, {"tactic": "have fs_int : \u2200 n, Integrable (fs n) \u03bc := fun n =>\n  bound_integrable.mono' (fs_measurable n) (h_bound _)", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\n\u22a2 Tendsto (fun n => setToFun \u03bc T hT (fs n)) atTop (\ud835\udcdd (setToFun \u03bc T hT f))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\n\u22a2 Tendsto (fun n => setToFun \u03bc T hT (fs n)) atTop (\ud835\udcdd (setToFun \u03bc T hT f))"}, {"tactic": "have f_int : Integrable f \u03bc :=\n  \u27e8f_measurable,\n    hasFiniteIntegral_of_dominated_convergence bound_integrable.hasFiniteIntegral h_bound\n      h_lim\u27e9", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\n\u22a2 Tendsto (fun n => setToFun \u03bc T hT (fs n)) atTop (\ud835\udcdd (setToFun \u03bc T hT f))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\n\u22a2 Tendsto (fun n => setToFun \u03bc T hT (fs n)) atTop (\ud835\udcdd (setToFun \u03bc T hT f))"}, {"tactic": "refine' L1.tendsto_setToL1 hT _ _ _", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\n\u22a2 Tendsto (fun n => \u2191(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))) atTop\n    (\ud835\udcdd (\u2191(L1.setToL1 hT) (Integrable.toL1 f f_int)))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\n\u22a2 Tendsto (fun n => Integrable.toL1 (fs n) (_ : Integrable (fs n))) atTop (\ud835\udcdd (Integrable.toL1 f f_int))"}, {"tactic": "rw [tendsto_iff_norm_tendsto_zero]", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\n\u22a2 Tendsto (fun n => Integrable.toL1 (fs n) (_ : Integrable (fs n))) atTop (\ud835\udcdd (Integrable.toL1 f f_int))", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\n\u22a2 Tendsto (fun e => \u2016Integrable.toL1 (fs e) (_ : Integrable (fs e)) - Integrable.toL1 f f_int\u2016) atTop (\ud835\udcdd 0)"}, {"tactic": "have lintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal <| \u222b\u207b a, ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc) atTop (\ud835\udcdd 0) :=\n  (tendsto_toReal zero_ne_top).comp\n    (tendsto_lintegral_norm_of_dominated_convergence fs_measurable\n      bound_integrable.hasFiniteIntegral h_bound h_lim)", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\n\u22a2 Tendsto (fun e => \u2016Integrable.toL1 (fs e) (_ : Integrable (fs e)) - Integrable.toL1 f f_int\u2016) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun e => \u2016Integrable.toL1 (fs e) (_ : Integrable (fs e)) - Integrable.toL1 f f_int\u2016) atTop (\ud835\udcdd 0)"}, {"tactic": "convert lintegral_norm_tendsto_zero with n", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun e => \u2016Integrable.toL1 (fs e) (_ : Integrable (fs e)) - Integrable.toL1 f f_int\u2016) atTop (\ud835\udcdd 0)", "state_after": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 \u2016Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int\u2016 =\n    ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)"}, {"tactic": "rw [L1.norm_def]", "state_before": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 \u2016Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int\u2016 =\n    ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)", "state_after": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int) a\u2016\u208a \u2202\u03bc) =\n    ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)"}, {"tactic": "congr 1", "state_before": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int) a\u2016\u208a \u2202\u03bc) =\n    ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)", "state_after": "case h.e'_3.h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int) a\u2016\u208a \u2202\u03bc) =\n    \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc"}, {"tactic": "refine' lintegral_congr_ae _", "state_before": "case h.e'_3.h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int) a\u2016\u208a \u2202\u03bc) =\n    \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc", "state_after": "case h.e'_3.h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 (fun a => \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int) a\u2016\u208a) =\u1d50[\u03bc] fun a =>\n    ENNReal.ofReal \u2016fs n a - f a\u2016"}, {"tactic": "rw [\u2190 Integrable.toL1_sub]", "state_before": "case h.e'_3.h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 (fun a => \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n) (_ : Integrable (fs n)) - Integrable.toL1 f f_int) a\u2016\u208a) =\u1d50[\u03bc] fun a =>\n    ENNReal.ofReal \u2016fs n a - f a\u2016", "state_after": "case h.e'_3.h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 (fun a => \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) a\u2016\u208a) =\u1d50[\u03bc] fun a =>\n    ENNReal.ofReal \u2016fs n a - f a\u2016"}, {"tactic": "refine' ((fs_int n).sub f_int).coeFn_toL1.mono fun x hx => _", "state_before": "case h.e'_3.h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 (fun a => \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) a\u2016\u208a) =\u1d50[\u03bc] fun a =>\n    ENNReal.ofReal \u2016fs n a - f a\u2016", "state_after": "case h.e'_3.h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\nx : \u03b1\nhx : \u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) x = (fs n - f) x\n\u22a2 (fun a => \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) a\u2016\u208a) x =\n    (fun a => ENNReal.ofReal \u2016fs n a - f a\u2016) x"}, {"tactic": "dsimp only", "state_before": "case h.e'_3.h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\nx : \u03b1\nhx : \u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) x = (fs n - f) x\n\u22a2 (fun a => \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) a\u2016\u208a) x =\n    (fun a => ENNReal.ofReal \u2016fs n a - f a\u2016) x", "state_after": "case h.e'_3.h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\nx : \u03b1\nhx : \u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) x = (fs n - f) x\n\u22a2 \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) x\u2016\u208a = ENNReal.ofReal \u2016fs n x - f x\u2016"}, {"tactic": "rw [hx, ofReal_norm_eq_coe_nnnorm, Pi.sub_apply]", "state_before": "case h.e'_3.h.e_a\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nlintegral_norm_tendsto_zero :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016fs n a - f a\u2016 \u2202\u03bc)) atTop (\ud835\udcdd 0)\nn : \u2115\nx : \u03b1\nhx : \u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) x = (fs n - f) x\n\u22a2 \u2191\u2016\u2191\u2191(Integrable.toL1 (fs n - f) (_ : Integrable (fs n - f))) x\u2016\u208a = ENNReal.ofReal \u2016fs n x - f x\u2016", "state_after": "no goals"}, {"tactic": "convert this with n", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nthis :\n  Tendsto (fun n => \u2191(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))) atTop\n    (\ud835\udcdd (\u2191(L1.setToL1 hT) (Integrable.toL1 f f_int)))\n\u22a2 Tendsto (fun n => setToFun \u03bc T hT (fs n)) atTop (\ud835\udcdd (setToFun \u03bc T hT f))", "state_after": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nthis :\n  Tendsto (fun n => \u2191(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))) atTop\n    (\ud835\udcdd (\u2191(L1.setToL1 hT) (Integrable.toL1 f f_int)))\nn : \u2115\n\u22a2 setToFun \u03bc T hT (fs n) = \u2191(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))\n\ncase h.e'_5.h.e'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nthis :\n  Tendsto (fun n => \u2191(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))) atTop\n    (\ud835\udcdd (\u2191(L1.setToL1 hT) (Integrable.toL1 f f_int)))\n\u22a2 setToFun \u03bc T hT f = \u2191(L1.setToL1 hT) (Integrable.toL1 f f_int)"}, {"tactic": "exact setToFun_eq hT (fs_int n)", "state_before": "case h.e'_3.h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nthis :\n  Tendsto (fun n => \u2191(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))) atTop\n    (\ud835\udcdd (\u2191(L1.setToL1 hT) (Integrable.toL1 f f_int)))\nn : \u2115\n\u22a2 setToFun \u03bc T hT (fs n) = \u2191(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))", "state_after": "no goals"}, {"tactic": "exact setToFun_eq hT f_int", "state_before": "case h.e'_5.h.e'_3\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1702608\nG : Type ?u.1702611\n\ud835\udd5c : Type ?u.1702614\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nfs : \u2115 \u2192 \u03b1 \u2192 E\nf : \u03b1 \u2192 E\nbound : \u03b1 \u2192 \u211d\nfs_measurable : \u2200 (n : \u2115), AEStronglyMeasurable (fs n) \u03bc\nbound_integrable : Integrable bound\nh_bound : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016fs n a\u2016 \u2264 bound a\nh_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => fs n a) atTop (\ud835\udcdd (f a))\nf_measurable : AEStronglyMeasurable f \u03bc\nfs_int : \u2200 (n : \u2115), Integrable (fs n)\nf_int : Integrable f\nthis :\n  Tendsto (fun n => \u2191(L1.setToL1 hT) (Integrable.toL1 (fs n) (_ : Integrable (fs n)))) atTop\n    (\ud835\udcdd (\u2191(L1.setToL1 hT) (Integrable.toL1 f f_int)))\n\u22a2 setToFun \u03bc T hT f = \u2191(L1.setToL1 hT) (Integrable.toL1 f f_int)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Coprime/Basic.lean", "full_name": "IsCoprime.of_isCoprime_of_dvd_left", "start": [157, 1], "end": [159, 37], "traced_tactics": [{"tactic": "obtain \u27e8d, rfl\u27e9 := hdvd", "state_before": "R : Type u\ninst\u271d : CommSemiring R\nx y z : R\nh : IsCoprime y z\nhdvd : x \u2223 y\n\u22a2 IsCoprime x z", "state_after": "case intro\nR : Type u\ninst\u271d : CommSemiring R\nx z d : R\nh : IsCoprime (x * d) z\n\u22a2 IsCoprime x z"}, {"tactic": "exact IsCoprime.of_mul_left_left h", "state_before": "case intro\nR : Type u\ninst\u271d : CommSemiring R\nx z d : R\nh : IsCoprime (x * d) z\n\u22a2 IsCoprime x z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Degree/CardPowDegree.lean", "full_name": "Polynomial.cardPowDegree_nonzero", "start": [90, 1], "end": [92, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Interval.lean", "full_name": "Interval.length_nonneg", "start": [689, 1], "end": [691, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/OrderOfElement.lean", "full_name": "Commute.orderOf_mul_eq_right_of_forall_prime_mul_dvd", "start": [430, 1], "end": [441, 83], "traced_tactics": [{"tactic": "have hoy := orderOf_pos' hy", "state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn m : \u2115\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nh : _root_.Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\n\u22a2 orderOf (x * y) = orderOf y", "state_after": "G : Type u\nA : Type v\nx y : G\na b : A\nn m : \u2115\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nh : _root_.Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\n\u22a2 orderOf (x * y) = orderOf y"}, {"tactic": "have hxy := dvd_of_forall_prime_mul_dvd hdvd", "state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn m : \u2115\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nh : _root_.Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\n\u22a2 orderOf (x * y) = orderOf y", "state_after": "G : Type u\nA : Type v\nx y : G\na b : A\nn m : \u2115\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nh : _root_.Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\n\u22a2 orderOf (x * y) = orderOf y"}, {"tactic": "apply orderOf_eq_of_pow_and_pow_div_prime hoy <;> simp only [Ne, \u2190 orderOf_dvd_iff_pow_eq_one]", "state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn m : \u2115\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nh : _root_.Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\n\u22a2 orderOf (x * y) = orderOf y", "state_after": "case hx\nG : Type u\nA : Type v\nx y : G\na b : A\nn m : \u2115\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nh : _root_.Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\n\u22a2 orderOf (x * y) \u2223 orderOf y\n\ncase hd\nG : Type u\nA : Type v\nx y : G\na b : A\nn m : \u2115\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nh : _root_.Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\n\u22a2 \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf y \u2192 \u00acorderOf (x * y) \u2223 orderOf y / p"}, {"tactic": "refine' fun p hp hpy hd => hp.ne_one _", "state_before": "case hd\nG : Type u\nA : Type v\nx y : G\na b : A\nn m : \u2115\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nh : _root_.Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\n\u22a2 \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf y \u2192 \u00acorderOf (x * y) \u2223 orderOf y / p", "state_after": "case hd\nG : Type u\nA : Type v\nx y : G\na b : A\nn m : \u2115\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nh : _root_.Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\np : \u2115\nhp : Nat.Prime p\nhpy : p \u2223 orderOf y\nhd : orderOf (x * y) \u2223 orderOf y / p\n\u22a2 p = 1"}, {"tactic": "rw [\u2190 Nat.dvd_one, \u2190 mul_dvd_mul_iff_right hoy.ne', one_mul, \u2190 dvd_div_iff hpy]", "state_before": "case hd\nG : Type u\nA : Type v\nx y : G\na b : A\nn m : \u2115\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nh : _root_.Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\np : \u2115\nhp : Nat.Prime p\nhpy : p \u2223 orderOf y\nhd : orderOf (x * y) \u2223 orderOf y / p\n\u22a2 p = 1", "state_after": "case hd\nG : Type u\nA : Type v\nx y : G\na b : A\nn m : \u2115\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nh : _root_.Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\np : \u2115\nhp : Nat.Prime p\nhpy : p \u2223 orderOf y\nhd : orderOf (x * y) \u2223 orderOf y / p\n\u22a2 orderOf y \u2223 orderOf y / p"}, {"tactic": "refine' (orderOf_dvd_lcm_mul h).trans (lcm_dvd ((dvd_div_iff hpy).2 _) hd)", "state_before": "case hd\nG : Type u\nA : Type v\nx y : G\na b : A\nn m : \u2115\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nh : _root_.Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\np : \u2115\nhp : Nat.Prime p\nhpy : p \u2223 orderOf y\nhd : orderOf (x * y) \u2223 orderOf y / p\n\u22a2 orderOf y \u2223 orderOf y / p", "state_after": "case hd\nG : Type u\nA : Type v\nx y : G\na b : A\nn m : \u2115\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nh : _root_.Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\np : \u2115\nhp : Nat.Prime p\nhpy : p \u2223 orderOf y\nhd : orderOf (x * y) \u2223 orderOf y / p\n\u22a2 p * orderOf x \u2223 orderOf y"}, {"tactic": "by_cases h : p \u2223 orderOf x", "state_before": "case hd\nG : Type u\nA : Type v\nx y : G\na b : A\nn m : \u2115\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nh : _root_.Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\np : \u2115\nhp : Nat.Prime p\nhpy : p \u2223 orderOf y\nhd : orderOf (x * y) \u2223 orderOf y / p\n\u22a2 p * orderOf x \u2223 orderOf y", "state_after": "case pos\nG : Type u\nA : Type v\nx y : G\na b : A\nn m : \u2115\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nh\u271d : _root_.Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\np : \u2115\nhp : Nat.Prime p\nhpy : p \u2223 orderOf y\nhd : orderOf (x * y) \u2223 orderOf y / p\nh : p \u2223 orderOf x\n\u22a2 p * orderOf x \u2223 orderOf y\n\ncase neg\nG : Type u\nA : Type v\nx y : G\na b : A\nn m : \u2115\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nh\u271d : _root_.Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\np : \u2115\nhp : Nat.Prime p\nhpy : p \u2223 orderOf y\nhd : orderOf (x * y) \u2223 orderOf y / p\nh : \u00acp \u2223 orderOf x\n\u22a2 p * orderOf x \u2223 orderOf y"}, {"tactic": "exacts [hdvd p hp h, (hp.coprime_iff_not_dvd.2 h).mul_dvd_of_dvd_of_dvd hpy hxy]", "state_before": "case pos\nG : Type u\nA : Type v\nx y : G\na b : A\nn m : \u2115\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nh\u271d : _root_.Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\np : \u2115\nhp : Nat.Prime p\nhpy : p \u2223 orderOf y\nhd : orderOf (x * y) \u2223 orderOf y / p\nh : p \u2223 orderOf x\n\u22a2 p * orderOf x \u2223 orderOf y\n\ncase neg\nG : Type u\nA : Type v\nx y : G\na b : A\nn m : \u2115\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nh\u271d : _root_.Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\np : \u2115\nhp : Nat.Prime p\nhpy : p \u2223 orderOf y\nhd : orderOf (x * y) \u2223 orderOf y / p\nh : \u00acp \u2223 orderOf x\n\u22a2 p * orderOf x \u2223 orderOf y", "state_after": "no goals"}, {"tactic": "exact h.orderOf_mul_dvd_lcm.trans (lcm_dvd hxy dvd_rfl)", "state_before": "case hx\nG : Type u\nA : Type v\nx y : G\na b : A\nn m : \u2115\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nh : _root_.Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\n\u22a2 orderOf (x * y) \u2223 orderOf y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Lie/Solvable.lean", "full_name": "LieIdeal.derivedSeries_add_eq_bot", "start": [174, 1], "end": [181, 34], "traced_tactics": [{"tactic": "rw [LieIdeal.derivedSeries_eq_bot_iff] at hI hJ \u22a2", "state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\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'\nI\u271d J\u271d : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nk l : \u2115\nI J : LieIdeal R L\nhI : derivedSeries R { x // x \u2208 \u2191I } k = \u22a5\nhJ : derivedSeries R { x // x \u2208 \u2191J } l = \u22a5\n\u22a2 derivedSeries R { x // x \u2208 \u2191(I + J) } (k + l) = \u22a5", "state_after": "R : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\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'\nI\u271d J\u271d : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nk l : \u2115\nI J : LieIdeal R L\nhI : derivedSeriesOfIdeal R L k I = \u22a5\nhJ : derivedSeriesOfIdeal R L l J = \u22a5\n\u22a2 derivedSeriesOfIdeal R L (k + l) (I + J) = \u22a5"}, {"tactic": "rw [\u2190 le_bot_iff]", "state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\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'\nI\u271d J\u271d : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nk l : \u2115\nI J : LieIdeal R L\nhI : derivedSeriesOfIdeal R L k I = \u22a5\nhJ : derivedSeriesOfIdeal R L l J = \u22a5\n\u22a2 derivedSeriesOfIdeal R L (k + l) (I + J) = \u22a5", "state_after": "R : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\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'\nI\u271d J\u271d : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nk l : \u2115\nI J : LieIdeal R L\nhI : derivedSeriesOfIdeal R L k I = \u22a5\nhJ : derivedSeriesOfIdeal R L l J = \u22a5\n\u22a2 derivedSeriesOfIdeal R L (k + l) (I + J) \u2264 \u22a5"}, {"tactic": "let D := derivedSeriesOfIdeal R L", "state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\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'\nI\u271d J\u271d : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nk l : \u2115\nI J : LieIdeal R L\nhI : derivedSeriesOfIdeal R L k I = \u22a5\nhJ : derivedSeriesOfIdeal R L l J = \u22a5\n\u22a2 derivedSeriesOfIdeal R L (k + l) (I + J) \u2264 \u22a5", "state_after": "R : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\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'\nI\u271d J\u271d : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nk l : \u2115\nI J : LieIdeal R L\nhI : derivedSeriesOfIdeal R L k I = \u22a5\nhJ : derivedSeriesOfIdeal R L l J = \u22a5\nD : \u2115 \u2192 LieIdeal R L \u2192 LieIdeal R L := derivedSeriesOfIdeal R L\n\u22a2 derivedSeriesOfIdeal R L (k + l) (I + J) \u2264 \u22a5"}, {"tactic": "change D k I = \u22a5 at hI", "state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\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'\nI\u271d J\u271d : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nk l : \u2115\nI J : LieIdeal R L\nhI : derivedSeriesOfIdeal R L k I = \u22a5\nhJ : derivedSeriesOfIdeal R L l J = \u22a5\nD : \u2115 \u2192 LieIdeal R L \u2192 LieIdeal R L := derivedSeriesOfIdeal R L\n\u22a2 derivedSeriesOfIdeal R L (k + l) (I + J) \u2264 \u22a5", "state_after": "R : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\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'\nI\u271d J\u271d : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nk l : \u2115\nI J : LieIdeal R L\nhJ : derivedSeriesOfIdeal R L l J = \u22a5\nD : \u2115 \u2192 LieIdeal R L \u2192 LieIdeal R L := derivedSeriesOfIdeal R L\nhI : D k I = \u22a5\n\u22a2 derivedSeriesOfIdeal R L (k + l) (I + J) \u2264 \u22a5"}, {"tactic": "change D l J = \u22a5 at hJ", "state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\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'\nI\u271d J\u271d : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nk l : \u2115\nI J : LieIdeal R L\nhJ : derivedSeriesOfIdeal R L l J = \u22a5\nD : \u2115 \u2192 LieIdeal R L \u2192 LieIdeal R L := derivedSeriesOfIdeal R L\nhI : D k I = \u22a5\n\u22a2 derivedSeriesOfIdeal R L (k + l) (I + J) \u2264 \u22a5", "state_after": "R : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\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'\nI\u271d J\u271d : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nk l : \u2115\nI J : LieIdeal R L\nD : \u2115 \u2192 LieIdeal R L \u2192 LieIdeal R L := derivedSeriesOfIdeal R L\nhI : D k I = \u22a5\nhJ : D l J = \u22a5\n\u22a2 derivedSeriesOfIdeal R L (k + l) (I + J) \u2264 \u22a5"}, {"tactic": "calc\n  D (k + l) (I + J) \u2264 D k I + D l J := derivedSeriesOfIdeal_add_le_add I J k l\n  _ \u2264 \u22a5 := by rw [hI, hJ]; simp", "state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\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'\nI\u271d J\u271d : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nk l : \u2115\nI J : LieIdeal R L\nD : \u2115 \u2192 LieIdeal R L \u2192 LieIdeal R L := derivedSeriesOfIdeal R L\nhI : D k I = \u22a5\nhJ : D l J = \u22a5\n\u22a2 derivedSeriesOfIdeal R L (k + l) (I + J) \u2264 \u22a5", "state_after": "no goals"}, {"tactic": "rw [hI, hJ]", "state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\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'\nI\u271d J\u271d : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nk l : \u2115\nI J : LieIdeal R L\nD : \u2115 \u2192 LieIdeal R L \u2192 LieIdeal R L := derivedSeriesOfIdeal R L\nhI : D k I = \u22a5\nhJ : D l J = \u22a5\n\u22a2 D k I + D l J \u2264 \u22a5", "state_after": "R : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\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'\nI\u271d J\u271d : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nk l : \u2115\nI J : LieIdeal R L\nD : \u2115 \u2192 LieIdeal R L \u2192 LieIdeal R L := derivedSeriesOfIdeal R L\nhI : D k I = \u22a5\nhJ : D l J = \u22a5\n\u22a2 \u22a5 + \u22a5 \u2264 \u22a5"}, {"tactic": "simp", "state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\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'\nI\u271d J\u271d : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nk l : \u2115\nI J : LieIdeal R L\nD : \u2115 \u2192 LieIdeal R L \u2192 LieIdeal R L := derivedSeriesOfIdeal R L\nhI : D k I = \u22a5\nhJ : D l J = \u22a5\n\u22a2 \u22a5 + \u22a5 \u2264 \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "full_name": "Qpf.Fix.ind_aux", "start": [299, 1], "end": [309, 20], "traced_tactics": [{"tactic": "have : Fix.mk (abs \u27e8a, fun x => \u27e6f x\u27e7\u27e9) = \u27e6Wrepr \u27e8a, f\u27e9\u27e7 := by\n  apply Quot.sound; apply Wequiv.abs'\n  rw [PFunctor.W.dest_mk, abs_map, abs_repr, \u2190 abs_map, PFunctor.map_eq]\n  conv =>\n    rhs\n    simp only [Wrepr, recF_eq, PFunctor.W.dest_mk, abs_repr, Function.comp]", "state_before": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\n\u22a2 mk (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) }) = Quotient.mk Wsetoid (WType.mk a f)", "state_after": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\nthis : mk (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) }) = Quotient.mk Wsetoid (Wrepr (WType.mk a f))\n\u22a2 mk (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) }) = Quotient.mk Wsetoid (WType.mk a f)"}, {"tactic": "rw [this]", "state_before": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\nthis : mk (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) }) = Quotient.mk Wsetoid (Wrepr (WType.mk a f))\n\u22a2 mk (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) }) = Quotient.mk Wsetoid (WType.mk a f)", "state_after": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\nthis : mk (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) }) = Quotient.mk Wsetoid (Wrepr (WType.mk a f))\n\u22a2 Quotient.mk Wsetoid (Wrepr (WType.mk a f)) = Quotient.mk Wsetoid (WType.mk a f)"}, {"tactic": "apply Quot.sound", "state_before": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\nthis : mk (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) }) = Quotient.mk Wsetoid (Wrepr (WType.mk a f))\n\u22a2 Quotient.mk Wsetoid (Wrepr (WType.mk a f)) = Quotient.mk Wsetoid (WType.mk a f)", "state_after": "case a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\nthis : mk (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) }) = Quotient.mk Wsetoid (Wrepr (WType.mk a f))\n\u22a2 Setoid.r (Wrepr (WType.mk a f)) (WType.mk a f)"}, {"tactic": "apply Wrepr_equiv", "state_before": "case a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\nthis : mk (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) }) = Quotient.mk Wsetoid (Wrepr (WType.mk a f))\n\u22a2 Setoid.r (Wrepr (WType.mk a f)) (WType.mk a f)", "state_after": "no goals"}, {"tactic": "apply Quot.sound", "state_before": "F : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\n\u22a2 mk (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) }) = Quotient.mk Wsetoid (Wrepr (WType.mk a f))", "state_after": "case a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\n\u22a2 Setoid.r (PFunctor.W.mk (fixToW <$> repr (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) })))\n    (Wrepr (WType.mk a f))"}, {"tactic": "apply Wequiv.abs'", "state_before": "case a\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\n\u22a2 Setoid.r (PFunctor.W.mk (fixToW <$> repr (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) })))\n    (Wrepr (WType.mk a f))", "state_after": "case a.h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\n\u22a2 abs\n      (PFunctor.W.dest\n        (PFunctor.W.mk (fixToW <$> repr (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) })))) =\n    abs (PFunctor.W.dest (Wrepr (WType.mk a f)))"}, {"tactic": "rw [PFunctor.W.dest_mk, abs_map, abs_repr, \u2190 abs_map, PFunctor.map_eq]", "state_before": "case a.h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\n\u22a2 abs\n      (PFunctor.W.dest\n        (PFunctor.W.mk (fixToW <$> repr (abs { fst := a, snd := fun x => Quotient.mk Wsetoid (f x) })))) =\n    abs (PFunctor.W.dest (Wrepr (WType.mk a f)))", "state_after": "case a.h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\n\u22a2 abs { fst := a, snd := fixToW \u2218 fun x => Quotient.mk Wsetoid (f x) } = abs (PFunctor.W.dest (Wrepr (WType.mk a f)))"}, {"tactic": "conv =>\n  rhs\n  simp only [Wrepr, recF_eq, PFunctor.W.dest_mk, abs_repr, Function.comp]", "state_before": "case a.h\nF : Type u \u2192 Type u\ninst\u271d : Functor F\nq : Qpf F\na : (P F).A\nf : PFunctor.B (P F) a \u2192 PFunctor.W (P F)\n\u22a2 abs { fst := a, snd := fixToW \u2218 fun x => Quotient.mk Wsetoid (f x) } = abs (PFunctor.W.dest (Wrepr (WType.mk a f)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.tendsto_atTop_add_nonneg_left'", "start": [609, 1], "end": [611, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/SchurZassenhaus.lean", "full_name": "Subgroup.exists_right_complement'_of_coprime", "start": [296, 1], "end": [313, 34], "traced_tactics": [{"tactic": "by_cases hN1 : Nat.card N = 0", "state_before": "n : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.coprime (Nat.card { x // x \u2208 N }) (index N)\n\u22a2 \u2203 H, IsComplement' N H", "state_after": "case pos\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.coprime (Nat.card { x // x \u2208 N }) (index N)\nhN1 : Nat.card { x // x \u2208 N } = 0\n\u22a2 \u2203 H, IsComplement' N H\n\ncase neg\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.coprime (Nat.card { x // x \u2208 N }) (index N)\nhN1 : \u00acNat.card { x // x \u2208 N } = 0\n\u22a2 \u2203 H, IsComplement' N H"}, {"tactic": "by_cases hN2 : N.index = 0", "state_before": "case neg\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.coprime (Nat.card { x // x \u2208 N }) (index N)\nhN1 : \u00acNat.card { x // x \u2208 N } = 0\n\u22a2 \u2203 H, IsComplement' N H", "state_after": "case pos\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.coprime (Nat.card { x // x \u2208 N }) (index N)\nhN1 : \u00acNat.card { x // x \u2208 N } = 0\nhN2 : index N = 0\n\u22a2 \u2203 H, IsComplement' N H\n\ncase neg\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.coprime (Nat.card { x // x \u2208 N }) (index N)\nhN1 : \u00acNat.card { x // x \u2208 N } = 0\nhN2 : \u00acindex N = 0\n\u22a2 \u2203 H, IsComplement' N H"}, {"tactic": "have hN3 : Nat.card G \u2260 0 := by\n  rw [\u2190 N.card_mul_index]\n  exact mul_ne_zero hN1 hN2", "state_before": "case neg\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.coprime (Nat.card { x // x \u2208 N }) (index N)\nhN1 : \u00acNat.card { x // x \u2208 N } = 0\nhN2 : \u00acindex N = 0\n\u22a2 \u2203 H, IsComplement' N H", "state_after": "case neg\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.coprime (Nat.card { x // x \u2208 N }) (index N)\nhN1 : \u00acNat.card { x // x \u2208 N } = 0\nhN2 : \u00acindex N = 0\nhN3 : Nat.card G \u2260 0\n\u22a2 \u2203 H, IsComplement' N H"}, {"tactic": "haveI := (Cardinal.lt_aleph0_iff_fintype.mp\n  (lt_of_not_ge (mt Cardinal.toNat_apply_of_aleph0_le hN3))).some", "state_before": "case neg\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.coprime (Nat.card { x // x \u2208 N }) (index N)\nhN1 : \u00acNat.card { x // x \u2208 N } = 0\nhN2 : \u00acindex N = 0\nhN3 : Nat.card G \u2260 0\n\u22a2 \u2203 H, IsComplement' N H", "state_after": "case neg\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.coprime (Nat.card { x // x \u2208 N }) (index N)\nhN1 : \u00acNat.card { x // x \u2208 N } = 0\nhN2 : \u00acindex N = 0\nhN3 : Nat.card G \u2260 0\nthis : Fintype G\n\u22a2 \u2203 H, IsComplement' N H"}, {"tactic": "apply exists_right_complement'_of_coprime_of_fintype", "state_before": "case neg\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.coprime (Nat.card { x // x \u2208 N }) (index N)\nhN1 : \u00acNat.card { x // x \u2208 N } = 0\nhN2 : \u00acindex N = 0\nhN3 : Nat.card G \u2260 0\nthis : Fintype G\n\u22a2 \u2203 H, IsComplement' N H", "state_after": "case neg.hN\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.coprime (Nat.card { x // x \u2208 N }) (index N)\nhN1 : \u00acNat.card { x // x \u2208 N } = 0\nhN2 : \u00acindex N = 0\nhN3 : Nat.card G \u2260 0\nthis : Fintype G\n\u22a2 Nat.coprime (Fintype.card { x // x \u2208 N }) (index N)"}, {"tactic": "rwa [\u2190Nat.card_eq_fintype_card]", "state_before": "case neg.hN\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.coprime (Nat.card { x // x \u2208 N }) (index N)\nhN1 : \u00acNat.card { x // x \u2208 N } = 0\nhN2 : \u00acindex N = 0\nhN3 : Nat.card G \u2260 0\nthis : Fintype G\n\u22a2 Nat.coprime (Fintype.card { x // x \u2208 N }) (index N)", "state_after": "no goals"}, {"tactic": "rw [hN1, Nat.coprime_zero_left, index_eq_one] at hN", "state_before": "case pos\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.coprime (Nat.card { x // x \u2208 N }) (index N)\nhN1 : Nat.card { x // x \u2208 N } = 0\n\u22a2 \u2203 H, IsComplement' N H", "state_after": "case pos\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : N = \u22a4\nhN1 : Nat.card { x // x \u2208 N } = 0\n\u22a2 \u2203 H, IsComplement' N H"}, {"tactic": "rw [hN]", "state_before": "case pos\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : N = \u22a4\nhN1 : Nat.card { x // x \u2208 N } = 0\n\u22a2 \u2203 H, IsComplement' N H", "state_after": "case pos\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : N = \u22a4\nhN1 : Nat.card { x // x \u2208 N } = 0\n\u22a2 \u2203 H, IsComplement' \u22a4 H"}, {"tactic": "exact \u27e8\u22a5, isComplement'_top_bot\u27e9", "state_before": "case pos\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : N = \u22a4\nhN1 : Nat.card { x // x \u2208 N } = 0\n\u22a2 \u2203 H, IsComplement' \u22a4 H", "state_after": "no goals"}, {"tactic": "rw [hN2, Nat.coprime_zero_right] at hN", "state_before": "case pos\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.coprime (Nat.card { x // x \u2208 N }) (index N)\nhN1 : \u00acNat.card { x // x \u2208 N } = 0\nhN2 : index N = 0\n\u22a2 \u2203 H, IsComplement' N H", "state_after": "case pos\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.card { x // x \u2208 N } = 1\nhN1 : \u00acNat.card { x // x \u2208 N } = 0\nhN2 : index N = 0\n\u22a2 \u2203 H, IsComplement' N H"}, {"tactic": "haveI := (Cardinal.toNat_eq_one_iff_unique.mp hN).1", "state_before": "case pos\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.card { x // x \u2208 N } = 1\nhN1 : \u00acNat.card { x // x \u2208 N } = 0\nhN2 : index N = 0\n\u22a2 \u2203 H, IsComplement' N H", "state_after": "case pos\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.card { x // x \u2208 N } = 1\nhN1 : \u00acNat.card { x // x \u2208 N } = 0\nhN2 : index N = 0\nthis : Subsingleton { x // x \u2208 N }\n\u22a2 \u2203 H, IsComplement' N H"}, {"tactic": "rw [N.eq_bot_of_subsingleton]", "state_before": "case pos\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.card { x // x \u2208 N } = 1\nhN1 : \u00acNat.card { x // x \u2208 N } = 0\nhN2 : index N = 0\nthis : Subsingleton { x // x \u2208 N }\n\u22a2 \u2203 H, IsComplement' N H", "state_after": "case pos\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.card { x // x \u2208 N } = 1\nhN1 : \u00acNat.card { x // x \u2208 N } = 0\nhN2 : index N = 0\nthis : Subsingleton { x // x \u2208 N }\n\u22a2 \u2203 H, IsComplement' \u22a5 H"}, {"tactic": "exact \u27e8\u22a4, isComplement'_bot_top\u27e9", "state_before": "case pos\nn : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.card { x // x \u2208 N } = 1\nhN1 : \u00acNat.card { x // x \u2208 N } = 0\nhN2 : index N = 0\nthis : Subsingleton { x // x \u2208 N }\n\u22a2 \u2203 H, IsComplement' \u22a5 H", "state_after": "no goals"}, {"tactic": "rw [\u2190 N.card_mul_index]", "state_before": "n : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.coprime (Nat.card { x // x \u2208 N }) (index N)\nhN1 : \u00acNat.card { x // x \u2208 N } = 0\nhN2 : \u00acindex N = 0\n\u22a2 Nat.card G \u2260 0", "state_after": "n : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.coprime (Nat.card { x // x \u2208 N }) (index N)\nhN1 : \u00acNat.card { x // x \u2208 N } = 0\nhN2 : \u00acindex N = 0\n\u22a2 Nat.card { x // x \u2208 N } * index N \u2260 0"}, {"tactic": "exact mul_ne_zero hN1 hN2", "state_before": "n : \u2115\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Normal N\nhN : Nat.coprime (Nat.card { x // x \u2208 N }) (index N)\nhN1 : \u00acNat.card { x // x \u2208 N } = 0\nhN2 : \u00acindex N = 0\n\u22a2 Nat.card { x // x \u2208 N } * index N \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.VectorMeasure.restrict_apply", "start": [699, 1], "end": [702, 18], "traced_tactics": [{"tactic": "rw [restrict, dif_pos hi]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.472690\nm inst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nM : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : TopologicalSpace M\nv : VectorMeasure \u03b1 M\ni : Set \u03b1\nhi : MeasurableSet i\nj : Set \u03b1\nhj : MeasurableSet j\n\u22a2 \u2191(restrict v i) j = \u2191v (j \u2229 i)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.472690\nm inst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nM : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : TopologicalSpace M\nv : VectorMeasure \u03b1 M\ni : Set \u03b1\nhi : MeasurableSet i\nj : Set \u03b1\nhj : MeasurableSet j\n\u22a2 \u2191{ measureOf' := fun s => if MeasurableSet s then \u2191v (s \u2229 i) else 0,\n          empty' := (_ : (if MeasurableSet \u2205 then \u2191v (\u2205 \u2229 i) else 0) = 0),\n          not_measurable' :=\n            (_ : \u2200 (i_1 : Set \u03b1), \u00acMeasurableSet i_1 \u2192 (if MeasurableSet i_1 then \u2191v (i_1 \u2229 i) else 0) = 0),\n          m_iUnion' :=\n            (_ :\n              \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984,\n                (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n                  Pairwise (Disjoint on f) \u2192\n                    HasSum (fun i_1 => (fun s => if MeasurableSet s then \u2191v (s \u2229 i) else 0) (f i_1))\n                      ((fun s => if MeasurableSet s then \u2191v (s \u2229 i) else 0) (\u22c3 (i : \u2115), f i))) }\n      j =\n    \u2191v (j \u2229 i)"}, {"tactic": "exact if_pos hj", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.472690\nm inst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\nM : Type u_2\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : TopologicalSpace M\nv : VectorMeasure \u03b1 M\ni : Set \u03b1\nhi : MeasurableSet i\nj : Set \u03b1\nhj : MeasurableSet j\n\u22a2 \u2191{ measureOf' := fun s => if MeasurableSet s then \u2191v (s \u2229 i) else 0,\n          empty' := (_ : (if MeasurableSet \u2205 then \u2191v (\u2205 \u2229 i) else 0) = 0),\n          not_measurable' :=\n            (_ : \u2200 (i_1 : Set \u03b1), \u00acMeasurableSet i_1 \u2192 (if MeasurableSet i_1 then \u2191v (i_1 \u2229 i) else 0) = 0),\n          m_iUnion' :=\n            (_ :\n              \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984,\n                (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n                  Pairwise (Disjoint on f) \u2192\n                    HasSum (fun i_1 => (fun s => if MeasurableSet s then \u2191v (s \u2229 i) else 0) (f i_1))\n                      ((fun s => if MeasurableSet s then \u2191v (s \u2229 i) else 0) (\u22c3 (i : \u2115), f i))) }\n      j =\n    \u2191v (j \u2229 i)", "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", "start": [223, 1], "end": [223, 67], "traced_tactics": [{"tactic": "simp [two_mul, sin_add]", "state_before": "\u22a2 sin (2 * \u03c0) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean", "full_name": "AffineEquiv.apply_lineMap", "start": [380, 1], "end": [382, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/FinitePresentation.lean", "full_name": "Algebra.FinitePresentation.trans", "start": [213, 1], "end": [216, 95], "traced_tactics": [{"tactic": "obtain \u27e8n, I, e, hfg\u27e9 := iff.1 hfpB", "state_before": "R : Type w\u2081\nA : Type w\u2082\nB : Type w\u2083\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nhfpA : FinitePresentation R A\nhfpB : FinitePresentation A B\n\u22a2 FinitePresentation R B", "state_after": "case intro.intro.intro\nR : Type w\u2081\nA : Type w\u2082\nB : Type w\u2083\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nhfpA : FinitePresentation R A\nhfpB : FinitePresentation A B\nn : \u2115\nI : Ideal (MvPolynomial (Fin n) A)\ne : (MvPolynomial (Fin n) A \u29f8 I) \u2243\u2090[A] B\nhfg : Ideal.FG I\n\u22a2 FinitePresentation R B"}, {"tactic": "exact equiv ((mvPolynomial_of_finitePresentation hfpA _).quotient hfg) (e.restrictScalars R)", "state_before": "case intro.intro.intro\nR : Type w\u2081\nA : Type w\u2082\nB : Type w\u2083\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsScalarTower R A B\nhfpA : FinitePresentation R A\nhfpB : FinitePresentation A B\nn : \u2115\nI : Ideal (MvPolynomial (Fin n) A)\ne : (MvPolynomial (Fin n) A \u29f8 I) \u2243\u2090[A] B\nhfg : Ideal.FG I\n\u22a2 FinitePresentation R B", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/ContinuousOn.lean", "full_name": "diff_mem_nhdsWithin_compl", "start": [104, 1], "end": [106, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/RamificationInertia.lean", "full_name": "Ideal.sum_ramification_inertia", "start": [816, 1], "end": [847, 37], "traced_tactics": [{"tactic": "set e := ramificationIdx (algebraMap R S) p", "state_before": "R : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\n\u22a2 \u2211 P in Multiset.toFinset (factors (map (algebraMap R S) p)),\n      ramificationIdx (algebraMap R S) p P * inertiaDeg (algebraMap R S) p P =\n    finrank K L", "state_after": "R : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\n\u22a2 \u2211 P in Multiset.toFinset (factors (map (algebraMap R S) p)), e P * inertiaDeg (algebraMap R S) p P = finrank K L"}, {"tactic": "set f := inertiaDeg (algebraMap R S) p", "state_before": "R : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\n\u22a2 \u2211 P in Multiset.toFinset (factors (map (algebraMap R S) p)), e P * inertiaDeg (algebraMap R S) p P = finrank K L", "state_after": "R : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\n\u22a2 \u2211 P in Multiset.toFinset (factors (map (algebraMap R S) p)), e P * f P = finrank K L"}, {"tactic": "have inj_RL : Function.Injective (algebraMap R L) := by\n  rw [IsScalarTower.algebraMap_eq R K L, RingHom.coe_comp]\n  exact (RingHom.injective _).comp (IsFractionRing.injective R K)", "state_before": "R : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\n\u22a2 \u2211 P in Multiset.toFinset (factors (map (algebraMap R S) p)), e P * f P = finrank K L", "state_after": "R : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\ninj_RL : Function.Injective \u2191(algebraMap R L)\n\u22a2 \u2211 P in Multiset.toFinset (factors (map (algebraMap R S) p)), e P * f P = finrank K L"}, {"tactic": "have inj_RS : Function.Injective (algebraMap R S) := by\n  refine Function.Injective.of_comp (show Function.Injective (algebraMap S L \u2218 _) from ?_)\n  rw [\u2190 RingHom.coe_comp, \u2190 IsScalarTower.algebraMap_eq]\n  exact inj_RL", "state_before": "R : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\ninj_RL : Function.Injective \u2191(algebraMap R L)\n\u22a2 \u2211 P in Multiset.toFinset (factors (map (algebraMap R S) p)), e P * f P = finrank K L", "state_after": "R : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\ninj_RL : Function.Injective \u2191(algebraMap R L)\ninj_RS : Function.Injective \u2191(algebraMap R S)\n\u22a2 \u2211 P in Multiset.toFinset (factors (map (algebraMap R S) p)), e P * f P = finrank K L"}, {"tactic": "calc\n  (\u2211 P in (factors (map (algebraMap R S) p)).toFinset, e P * f P) =\n      \u2211 P in (factors (map (algebraMap R S) p)).toFinset.attach,\n        finrank (R \u29f8 p) (S \u29f8 (P : Ideal S) ^ e P) := ?_\n  _ = finrank (R \u29f8 p)\n        (\u2200 P : (factors (map (algebraMap R S) p)).toFinset, S \u29f8 (P : Ideal S) ^ e P) :=\n    (finrank_pi_fintype (R \u29f8 p)).symm\n  _ = finrank (R \u29f8 p) (S \u29f8 map (algebraMap R S) p) := ?_\n  _ = finrank K L := ?_", "state_before": "R : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\ninj_RL : Function.Injective \u2191(algebraMap R L)\ninj_RS : Function.Injective \u2191(algebraMap R S)\n\u22a2 \u2211 P in Multiset.toFinset (factors (map (algebraMap R S) p)), e P * f P = finrank K L", "state_after": "case calc_1\nR : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\ninj_RL : Function.Injective \u2191(algebraMap R L)\ninj_RS : Function.Injective \u2191(algebraMap R S)\n\u22a2 \u2211 P in Multiset.toFinset (factors (map (algebraMap R S) p)), e P * f P =\n    \u2211 P in Finset.attach (Multiset.toFinset (factors (map (algebraMap R S) p))), finrank (R \u29f8 p) (S \u29f8 \u2191P ^ e \u2191P)\n\ncase calc_2\nR : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\ninj_RL : Function.Injective \u2191(algebraMap R L)\ninj_RS : Function.Injective \u2191(algebraMap R S)\n\u22a2 finrank (R \u29f8 p) ((P : { x // x \u2208 Multiset.toFinset (factors (map (algebraMap R S) p)) }) \u2192 S \u29f8 \u2191P ^ e \u2191P) =\n    finrank (R \u29f8 p) (S \u29f8 map (algebraMap R S) p)\n\ncase calc_3\nR : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\ninj_RL : Function.Injective \u2191(algebraMap R L)\ninj_RS : Function.Injective \u2191(algebraMap R S)\n\u22a2 finrank (R \u29f8 p) (S \u29f8 map (algebraMap R S) p) = finrank K L"}, {"tactic": "rw [IsScalarTower.algebraMap_eq R K L, RingHom.coe_comp]", "state_before": "R : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\n\u22a2 Function.Injective \u2191(algebraMap R L)", "state_after": "R : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\n\u22a2 Function.Injective (\u2191(algebraMap K L) \u2218 \u2191(algebraMap R K))"}, {"tactic": "exact (RingHom.injective _).comp (IsFractionRing.injective R K)", "state_before": "R : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\n\u22a2 Function.Injective (\u2191(algebraMap K L) \u2218 \u2191(algebraMap R K))", "state_after": "no goals"}, {"tactic": "refine Function.Injective.of_comp (show Function.Injective (algebraMap S L \u2218 _) from ?_)", "state_before": "R : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\ninj_RL : Function.Injective \u2191(algebraMap R L)\n\u22a2 Function.Injective \u2191(algebraMap R S)", "state_after": "R : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\ninj_RL : Function.Injective \u2191(algebraMap R L)\n\u22a2 Function.Injective (\u2191(algebraMap S L) \u2218 \u2191(algebraMap R S))"}, {"tactic": "rw [\u2190 RingHom.coe_comp, \u2190 IsScalarTower.algebraMap_eq]", "state_before": "R : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\ninj_RL : Function.Injective \u2191(algebraMap R L)\n\u22a2 Function.Injective (\u2191(algebraMap S L) \u2218 \u2191(algebraMap R S))", "state_after": "R : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\ninj_RL : Function.Injective \u2191(algebraMap R L)\n\u22a2 Function.Injective \u2191(algebraMap R L)"}, {"tactic": "exact inj_RL", "state_before": "R : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\ninj_RL : Function.Injective \u2191(algebraMap R L)\n\u22a2 Function.Injective \u2191(algebraMap R L)", "state_after": "no goals"}, {"tactic": "rw [\u2190 Finset.sum_attach]", "state_before": "case calc_1\nR : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\ninj_RL : Function.Injective \u2191(algebraMap R L)\ninj_RS : Function.Injective \u2191(algebraMap R S)\n\u22a2 \u2211 P in Multiset.toFinset (factors (map (algebraMap R S) p)), e P * f P =\n    \u2211 P in Finset.attach (Multiset.toFinset (factors (map (algebraMap R S) p))), finrank (R \u29f8 p) (S \u29f8 \u2191P ^ e \u2191P)", "state_after": "case calc_1\nR : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\ninj_RL : Function.Injective \u2191(algebraMap R L)\ninj_RS : Function.Injective \u2191(algebraMap R S)\n\u22a2 \u2211 x in Finset.attach (Multiset.toFinset (factors (map (algebraMap R S) p))), e \u2191x * f \u2191x =\n    \u2211 P in Finset.attach (Multiset.toFinset (factors (map (algebraMap R S) p))), finrank (R \u29f8 p) (S \u29f8 \u2191P ^ e \u2191P)"}, {"tactic": "refine Finset.sum_congr rfl fun P _ => ?_", "state_before": "case calc_1\nR : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\ninj_RL : Function.Injective \u2191(algebraMap R L)\ninj_RS : Function.Injective \u2191(algebraMap R S)\n\u22a2 \u2211 x in Finset.attach (Multiset.toFinset (factors (map (algebraMap R S) p))), e \u2191x * f \u2191x =\n    \u2211 P in Finset.attach (Multiset.toFinset (factors (map (algebraMap R S) p))), finrank (R \u29f8 p) (S \u29f8 \u2191P ^ e \u2191P)", "state_after": "case calc_1\nR : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP\u271d : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\ninj_RL : Function.Injective \u2191(algebraMap R L)\ninj_RS : Function.Injective \u2191(algebraMap R S)\nP : { x // x \u2208 Multiset.toFinset (factors (map (algebraMap R S) p)) }\nx\u271d : P \u2208 Finset.attach (Multiset.toFinset (factors (map (algebraMap R S) p)))\n\u22a2 e \u2191P * f \u2191P = finrank (R \u29f8 p) (S \u29f8 \u2191P ^ e \u2191P)"}, {"tactic": "rw [Factors.finrank_pow_ramificationIdx]", "state_before": "case calc_1\nR : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP\u271d : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\ninj_RL : Function.Injective \u2191(algebraMap R L)\ninj_RS : Function.Injective \u2191(algebraMap R S)\nP : { x // x \u2208 Multiset.toFinset (factors (map (algebraMap R S) p)) }\nx\u271d : P \u2208 Finset.attach (Multiset.toFinset (factors (map (algebraMap R S) p)))\n\u22a2 e \u2191P * f \u2191P = finrank (R \u29f8 p) (S \u29f8 \u2191P ^ e \u2191P)", "state_after": "no goals"}, {"tactic": "refine LinearEquiv.finrank_eq (Factors.piQuotientLinearEquiv S p ?_).symm", "state_before": "case calc_2\nR : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\ninj_RL : Function.Injective \u2191(algebraMap R L)\ninj_RS : Function.Injective \u2191(algebraMap R S)\n\u22a2 finrank (R \u29f8 p) ((P : { x // x \u2208 Multiset.toFinset (factors (map (algebraMap R S) p)) }) \u2192 S \u29f8 \u2191P ^ e \u2191P) =\n    finrank (R \u29f8 p) (S \u29f8 map (algebraMap R S) p)", "state_after": "case calc_2\nR : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\ninj_RL : Function.Injective \u2191(algebraMap R L)\ninj_RS : Function.Injective \u2191(algebraMap R S)\n\u22a2 map (algebraMap R S) p \u2260 \u22a5"}, {"tactic": "rwa [Ne.def, Ideal.map_eq_bot_iff_le_ker, (RingHom.injective_iff_ker_eq_bot _).mp inj_RS,\n  le_bot_iff]", "state_before": "case calc_2\nR : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\ninj_RL : Function.Injective \u2191(algebraMap R L)\ninj_RS : Function.Injective \u2191(algebraMap R S)\n\u22a2 map (algebraMap R S) p \u2260 \u22a5", "state_after": "no goals"}, {"tactic": "exact finrank_quotient_map p K L", "state_before": "case calc_3\nR : Type u\ninst\u271d\u00b9\u2079 : CommRing R\nS : Type v\ninst\u271d\u00b9\u2078 : CommRing S\nf\u271d : R \u2192+* S\np : Ideal R\nP : Ideal S\ninst\u271d\u00b9\u2077 : IsDomain S\ninst\u271d\u00b9\u2076 : IsDedekindDomain S\ninst\u271d\u00b9\u2075 : Algebra R S\nK : Type u_1\nL : Type u_2\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : IsDomain R\ninst\u271d\u00b9\u00b9 : IsDedekindDomain R\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra S L\ninst\u271d\u2077 : IsFractionRing S L\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : IsScalarTower R S L\ninst\u271d\u00b3 : IsScalarTower R K L\ninst\u271d\u00b2 : IsNoetherian R S\ninst\u271d\u00b9 : IsIntegralClosure S R L\ninst\u271d : IsMaximal p\nhp0 : p \u2260 \u22a5\ne : Ideal S \u2192 \u2115 := ramificationIdx (algebraMap R S) p\nf : Ideal S \u2192 [inst : IsMaximal p] \u2192 \u2115 := inertiaDeg (algebraMap R S) p\ninj_RL : Function.Injective \u2191(algebraMap R L)\ninj_RS : Function.Injective \u2191(algebraMap R S)\n\u22a2 finrank (R \u29f8 p) (S \u29f8 map (algebraMap R S) p) = finrank K L", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Lie/Normalizer.lean", "full_name": "LieSubmodule.top_lie_le_iff_le_normalizer", "start": [89, 1], "end": [90, 82], "traced_tactics": [{"tactic": "rw [lie_le_iff]", "state_before": "R : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type ?u.23856\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'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nN N\u2081 N\u2082 N' : LieSubmodule R L M\n\u22a2 \u2045\u22a4, N\u2046 \u2264 N' \u2194 N \u2264 normalizer N'", "state_after": "R : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type ?u.23856\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'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nN N\u2081 N\u2082 N' : LieSubmodule R L M\n\u22a2 (\u2200 (x : L), x \u2208 \u22a4 \u2192 \u2200 (m : M), m \u2208 N \u2192 \u2045x, m\u2046 \u2208 N') \u2194 N \u2264 normalizer N'"}, {"tactic": "tauto", "state_before": "R : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type ?u.23856\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'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nN N\u2081 N\u2082 N' : LieSubmodule R L M\n\u22a2 (\u2200 (x : L), x \u2208 \u22a4 \u2192 \u2200 (m : M), m \u2208 N \u2192 \u2045x, m\u2046 \u2208 N') \u2194 N \u2264 normalizer N'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/Adjugate.lean", "full_name": "Matrix.cramer_is_linear", "start": [86, 1], "end": [89, 38], "traced_tactics": [{"tactic": "constructor <;> intros <;> ext i", "state_before": "m : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nb : n \u2192 \u03b1\n\u22a2 IsLinearMap \u03b1 (cramerMap A)", "state_after": "case map_add.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nb x\u271d y\u271d : n \u2192 \u03b1\ni : n\n\u22a2 cramerMap A (x\u271d + y\u271d) i = (cramerMap A x\u271d + cramerMap A y\u271d) i\n\ncase map_smul.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nb : n \u2192 \u03b1\nc\u271d : \u03b1\nx\u271d : n \u2192 \u03b1\ni : n\n\u22a2 cramerMap A (c\u271d \u2022 x\u271d) i = (c\u271d \u2022 cramerMap A x\u271d) i"}, {"tactic": "apply (cramerMap_is_linear A i).1", "state_before": "case map_add.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nb x\u271d y\u271d : n \u2192 \u03b1\ni : n\n\u22a2 cramerMap A (x\u271d + y\u271d) i = (cramerMap A x\u271d + cramerMap A y\u271d) i", "state_after": "no goals"}, {"tactic": "apply (cramerMap_is_linear A i).2", "state_before": "case map_smul.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nb : n \u2192 \u03b1\nc\u271d : \u03b1\nx\u271d : n \u2192 \u03b1\ni : n\n\u22a2 cramerMap A (c\u271d \u2022 x\u271d) i = (c\u271d \u2022 cramerMap A x\u271d) i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Pi.lean", "full_name": "Filter.NeBot.coprod\u1d62", "start": [242, 1], "end": [243, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/CompareExp.lean", "full_name": "Complex.IsExpCmpFilter.isLittleO_exp_cpow", "start": [208, 1], "end": [209, 99], "traced_tactics": [{"tactic": "simpa using hl.isLittleO_cpow_mul_exp hb 0 a", "state_before": "l : Filter \u2102\nhl : IsExpCmpFilter l\na : \u2102\nb : \u211d\nhb : b < 0\n\u22a2 (fun z => exp (\u2191b * z)) =o[l] fun z => z ^ a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Equiv.lean", "full_name": "AlgEquiv.coe_trans", "start": [411, 1], "end": [412, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "full_name": "CategoryTheory.Limits.pullbackConeOfLeftIso_snd", "start": [1626, 1], "end": [1626, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecificLimits/Normed.lean", "full_name": "Monotone.cauchySeq_series_mul_of_tendsto_zero_of_bounded", "start": [582, 1], "end": [597, 60], "traced_tactics": [{"tactic": "simp_rw [Finset.sum_range_by_parts _ _ (Nat.succ _), sub_eq_add_neg, Nat.succ_sub_succ_eq_sub,\n  tsub_zero]", "state_before": "\u03b1 : Type ?u.1391284\n\u03b2 : Type ?u.1391287\n\u03b9 : Type ?u.1391290\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : \u211d\nf : \u2115 \u2192 \u211d\nz : \u2115 \u2192 E\nhfa : Monotone f\nhf0 : Tendsto f atTop (\ud835\udcdd 0)\nhgb : \u2200 (n : \u2115), \u2016\u2211 i in Finset.range n, z i\u2016 \u2264 b\n\u22a2 CauchySeq fun n => \u2211 i in Finset.range (n + 1), f i \u2022 z i", "state_after": "\u03b1 : Type ?u.1391284\n\u03b2 : Type ?u.1391287\n\u03b9 : Type ?u.1391290\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : \u211d\nf : \u2115 \u2192 \u211d\nz : \u2115 \u2192 E\nhfa : Monotone f\nhf0 : Tendsto f atTop (\ud835\udcdd 0)\nhgb : \u2200 (n : \u2115), \u2016\u2211 i in Finset.range n, z i\u2016 \u2264 b\n\u22a2 CauchySeq fun n =>\n    f n \u2022 \u2211 i in Finset.range (succ n), z i +\n      -\u2211 x in Finset.range n, (f (x + 1) + -f x) \u2022 \u2211 i in Finset.range (x + 1), z i"}, {"tactic": "apply (NormedField.tendsto_zero_smul_of_tendsto_zero_of_bounded hf0\n  \u27e8b, eventually_map.mpr <| eventually_of_forall fun n \u21a6 hgb <| n + 1\u27e9).cauchySeq.add", "state_before": "\u03b1 : Type ?u.1391284\n\u03b2 : Type ?u.1391287\n\u03b9 : Type ?u.1391290\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : \u211d\nf : \u2115 \u2192 \u211d\nz : \u2115 \u2192 E\nhfa : Monotone f\nhf0 : Tendsto f atTop (\ud835\udcdd 0)\nhgb : \u2200 (n : \u2115), \u2016\u2211 i in Finset.range n, z i\u2016 \u2264 b\n\u22a2 CauchySeq fun n =>\n    f n \u2022 \u2211 i in Finset.range (succ n), z i +\n      -\u2211 x in Finset.range n, (f (x + 1) + -f x) \u2022 \u2211 i in Finset.range (x + 1), z i", "state_after": "\u03b1 : Type ?u.1391284\n\u03b2 : Type ?u.1391287\n\u03b9 : Type ?u.1391290\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : \u211d\nf : \u2115 \u2192 \u211d\nz : \u2115 \u2192 E\nhfa : Monotone f\nhf0 : Tendsto f atTop (\ud835\udcdd 0)\nhgb : \u2200 (n : \u2115), \u2016\u2211 i in Finset.range n, z i\u2016 \u2264 b\n\u22a2 CauchySeq fun n => -\u2211 x in Finset.range n, (f (x + 1) + -f x) \u2022 \u2211 i in Finset.range (x + 1), z i"}, {"tactic": "refine' CauchySeq.neg _", "state_before": "\u03b1 : Type ?u.1391284\n\u03b2 : Type ?u.1391287\n\u03b9 : Type ?u.1391290\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : \u211d\nf : \u2115 \u2192 \u211d\nz : \u2115 \u2192 E\nhfa : Monotone f\nhf0 : Tendsto f atTop (\ud835\udcdd 0)\nhgb : \u2200 (n : \u2115), \u2016\u2211 i in Finset.range n, z i\u2016 \u2264 b\n\u22a2 CauchySeq fun n => -\u2211 x in Finset.range n, (f (x + 1) + -f x) \u2022 \u2211 i in Finset.range (x + 1), z i", "state_after": "\u03b1 : Type ?u.1391284\n\u03b2 : Type ?u.1391287\n\u03b9 : Type ?u.1391290\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : \u211d\nf : \u2115 \u2192 \u211d\nz : \u2115 \u2192 E\nhfa : Monotone f\nhf0 : Tendsto f atTop (\ud835\udcdd 0)\nhgb : \u2200 (n : \u2115), \u2016\u2211 i in Finset.range n, z i\u2016 \u2264 b\n\u22a2 CauchySeq fun n => \u2211 x in Finset.range n, (f (x + 1) + -f x) \u2022 \u2211 i in Finset.range (x + 1), z i"}, {"tactic": "refine' cauchySeq_range_of_norm_bounded _ _\n  (fun n \u21a6 _ : \u2200 n, \u2016(f (n + 1) + -f n) \u2022 (Finset.range (n + 1)).sum z\u2016 \u2264 b * |f (n + 1) - f n|)", "state_before": "\u03b1 : Type ?u.1391284\n\u03b2 : Type ?u.1391287\n\u03b9 : Type ?u.1391290\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : \u211d\nf : \u2115 \u2192 \u211d\nz : \u2115 \u2192 E\nhfa : Monotone f\nhf0 : Tendsto f atTop (\ud835\udcdd 0)\nhgb : \u2200 (n : \u2115), \u2016\u2211 i in Finset.range n, z i\u2016 \u2264 b\n\u22a2 CauchySeq fun n => \u2211 x in Finset.range n, (f (x + 1) + -f x) \u2022 \u2211 i in Finset.range (x + 1), z i", "state_after": "case refine'_1\n\u03b1 : Type ?u.1391284\n\u03b2 : Type ?u.1391287\n\u03b9 : Type ?u.1391290\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : \u211d\nf : \u2115 \u2192 \u211d\nz : \u2115 \u2192 E\nhfa : Monotone f\nhf0 : Tendsto f atTop (\ud835\udcdd 0)\nhgb : \u2200 (n : \u2115), \u2016\u2211 i in Finset.range n, z i\u2016 \u2264 b\n\u22a2 CauchySeq fun n => \u2211 i in Finset.range n, b * abs (f (i + 1) - f i)\n\ncase refine'_2\n\u03b1 : Type ?u.1391284\n\u03b2 : Type ?u.1391287\n\u03b9 : Type ?u.1391290\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : \u211d\nf : \u2115 \u2192 \u211d\nz : \u2115 \u2192 E\nhfa : Monotone f\nhf0 : Tendsto f atTop (\ud835\udcdd 0)\nhgb : \u2200 (n : \u2115), \u2016\u2211 i in Finset.range n, z i\u2016 \u2264 b\nn : \u2115\n\u22a2 \u2016(f (n + 1) + -f n) \u2022 Finset.sum (Finset.range (n + 1)) z\u2016 \u2264 b * abs (f (n + 1) - f n)"}, {"tactic": "simp_rw [abs_of_nonneg (sub_nonneg_of_le (hfa (Nat.le_succ _))), \u2190 mul_sum]", "state_before": "case refine'_1\n\u03b1 : Type ?u.1391284\n\u03b2 : Type ?u.1391287\n\u03b9 : Type ?u.1391290\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : \u211d\nf : \u2115 \u2192 \u211d\nz : \u2115 \u2192 E\nhfa : Monotone f\nhf0 : Tendsto f atTop (\ud835\udcdd 0)\nhgb : \u2200 (n : \u2115), \u2016\u2211 i in Finset.range n, z i\u2016 \u2264 b\n\u22a2 CauchySeq fun n => \u2211 i in Finset.range n, b * abs (f (i + 1) - f i)", "state_after": "case refine'_1\n\u03b1 : Type ?u.1391284\n\u03b2 : Type ?u.1391287\n\u03b9 : Type ?u.1391290\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : \u211d\nf : \u2115 \u2192 \u211d\nz : \u2115 \u2192 E\nhfa : Monotone f\nhf0 : Tendsto f atTop (\ud835\udcdd 0)\nhgb : \u2200 (n : \u2115), \u2016\u2211 i in Finset.range n, z i\u2016 \u2264 b\n\u22a2 CauchySeq fun n => b * \u2211 x in Finset.range n, (f (succ x) - f x)"}, {"tactic": "apply Real.uniformContinuous_const_mul.comp_cauchySeq", "state_before": "case refine'_1\n\u03b1 : Type ?u.1391284\n\u03b2 : Type ?u.1391287\n\u03b9 : Type ?u.1391290\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : \u211d\nf : \u2115 \u2192 \u211d\nz : \u2115 \u2192 E\nhfa : Monotone f\nhf0 : Tendsto f atTop (\ud835\udcdd 0)\nhgb : \u2200 (n : \u2115), \u2016\u2211 i in Finset.range n, z i\u2016 \u2264 b\n\u22a2 CauchySeq fun n => b * \u2211 x in Finset.range n, (f (succ x) - f x)", "state_after": "case refine'_1\n\u03b1 : Type ?u.1391284\n\u03b2 : Type ?u.1391287\n\u03b9 : Type ?u.1391290\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : \u211d\nf : \u2115 \u2192 \u211d\nz : \u2115 \u2192 E\nhfa : Monotone f\nhf0 : Tendsto f atTop (\ud835\udcdd 0)\nhgb : \u2200 (n : \u2115), \u2016\u2211 i in Finset.range n, z i\u2016 \u2264 b\n\u22a2 CauchySeq fun n => \u2211 x in Finset.range n, (f (succ x) - f x)"}, {"tactic": "simp_rw [sum_range_sub, sub_eq_add_neg]", "state_before": "case refine'_1\n\u03b1 : Type ?u.1391284\n\u03b2 : Type ?u.1391287\n\u03b9 : Type ?u.1391290\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : \u211d\nf : \u2115 \u2192 \u211d\nz : \u2115 \u2192 E\nhfa : Monotone f\nhf0 : Tendsto f atTop (\ud835\udcdd 0)\nhgb : \u2200 (n : \u2115), \u2016\u2211 i in Finset.range n, z i\u2016 \u2264 b\n\u22a2 CauchySeq fun n => \u2211 x in Finset.range n, (f (succ x) - f x)", "state_after": "case refine'_1\n\u03b1 : Type ?u.1391284\n\u03b2 : Type ?u.1391287\n\u03b9 : Type ?u.1391290\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : \u211d\nf : \u2115 \u2192 \u211d\nz : \u2115 \u2192 E\nhfa : Monotone f\nhf0 : Tendsto f atTop (\ud835\udcdd 0)\nhgb : \u2200 (n : \u2115), \u2016\u2211 i in Finset.range n, z i\u2016 \u2264 b\n\u22a2 CauchySeq fun n => f n + -f 0"}, {"tactic": "exact (Tendsto.cauchySeq hf0).add_const", "state_before": "case refine'_1\n\u03b1 : Type ?u.1391284\n\u03b2 : Type ?u.1391287\n\u03b9 : Type ?u.1391290\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : \u211d\nf : \u2115 \u2192 \u211d\nz : \u2115 \u2192 E\nhfa : Monotone f\nhf0 : Tendsto f atTop (\ud835\udcdd 0)\nhgb : \u2200 (n : \u2115), \u2016\u2211 i in Finset.range n, z i\u2016 \u2264 b\n\u22a2 CauchySeq fun n => f n + -f 0", "state_after": "no goals"}, {"tactic": "rw [norm_smul, mul_comm]", "state_before": "case refine'_2\n\u03b1 : Type ?u.1391284\n\u03b2 : Type ?u.1391287\n\u03b9 : Type ?u.1391290\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : \u211d\nf : \u2115 \u2192 \u211d\nz : \u2115 \u2192 E\nhfa : Monotone f\nhf0 : Tendsto f atTop (\ud835\udcdd 0)\nhgb : \u2200 (n : \u2115), \u2016\u2211 i in Finset.range n, z i\u2016 \u2264 b\nn : \u2115\n\u22a2 \u2016(f (n + 1) + -f n) \u2022 Finset.sum (Finset.range (n + 1)) z\u2016 \u2264 b * abs (f (n + 1) - f n)", "state_after": "case refine'_2\n\u03b1 : Type ?u.1391284\n\u03b2 : Type ?u.1391287\n\u03b9 : Type ?u.1391290\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : \u211d\nf : \u2115 \u2192 \u211d\nz : \u2115 \u2192 E\nhfa : Monotone f\nhf0 : Tendsto f atTop (\ud835\udcdd 0)\nhgb : \u2200 (n : \u2115), \u2016\u2211 i in Finset.range n, z i\u2016 \u2264 b\nn : \u2115\n\u22a2 \u2016Finset.sum (Finset.range (n + 1)) z\u2016 * \u2016f (n + 1) + -f n\u2016 \u2264 b * abs (f (n + 1) - f n)"}, {"tactic": "exact mul_le_mul_of_nonneg_right (hgb _) (abs_nonneg _)", "state_before": "case refine'_2\n\u03b1 : Type ?u.1391284\n\u03b2 : Type ?u.1391287\n\u03b9 : Type ?u.1391290\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nb : \u211d\nf : \u2115 \u2192 \u211d\nz : \u2115 \u2192 E\nhfa : Monotone f\nhf0 : Tendsto f atTop (\ud835\udcdd 0)\nhgb : \u2200 (n : \u2115), \u2016\u2211 i in Finset.range n, z i\u2016 \u2264 b\nn : \u2115\n\u22a2 \u2016Finset.sum (Finset.range (n + 1)) z\u2016 * \u2016f (n + 1) + -f n\u2016 \u2264 b * abs (f (n + 1) - f n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/Basic.lean", "full_name": "Equiv.Perm.subtypeCongr.right_apply_subtype", "start": [580, 1], "end": [581, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/FinsetOps.lean", "full_name": "Multiset.Nodup.ndinter", "start": [248, 1], "end": [249, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/LegendreSymbol/MulCharacter.lean", "full_name": "MulChar.ext_iff", "start": [179, 1], "end": [182, 14], "traced_tactics": [{"tactic": "rintro rfl a", "state_before": "R : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 \u03c7' : MulChar R R'\n\u22a2 \u03c7 = \u03c7' \u2192 \u2200 (a : R\u02e3), \u2191\u03c7 \u2191a = \u2191\u03c7' \u2191a", "state_after": "R : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\na : R\u02e3\n\u22a2 \u2191\u03c7 \u2191a = \u2191\u03c7 \u2191a"}, {"tactic": "rfl", "state_before": "R : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\na : R\u02e3\n\u22a2 \u2191\u03c7 \u2191a = \u2191\u03c7 \u2191a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Prime.lean", "full_name": "Nat.minFac_lemma", "start": [246, 1], "end": [247, 99], "traced_tactics": [{"tactic": "decide", "state_before": "n k : \u2115\nh : \u00acn < k * k\n\u22a2 0 < 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "full_name": "coe_normGroupNorm", "start": [2047, 1], "end": [2048, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Instances/EReal.lean", "full_name": "EReal.continuous_coe_ennreal_iff", "start": [126, 1], "end": [128, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fintype/Sum.lean", "full_name": "image_subtype_univ_ssubset_image_univ", "start": [63, 1], "end": [77, 25], "traced_tactics": [{"tactic": "constructor", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\nhk : k \u2208 image b univ\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\n\u22a2 image (fun i => b \u2191i) univ \u2282 image b univ", "state_after": "case left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\nhk : k \u2208 image b univ\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\n\u22a2 image (fun i => b \u2191i) univ \u2286 image b univ\n\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\nhk : k \u2208 image b univ\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\n\u22a2 \u00acimage b univ \u2286 image (fun i => b \u2191i) univ"}, {"tactic": "intro x hx", "state_before": "case left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\nhk : k \u2208 image b univ\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\n\u22a2 image (fun i => b \u2191i) univ \u2286 image b univ", "state_after": "case left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\nhk : k \u2208 image b univ\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\nx : \u03b2\nhx : x \u2208 image (fun i => b \u2191i) univ\n\u22a2 x \u2208 image b univ"}, {"tactic": "rcases mem_image.1 hx with \u27e8y, _, hy\u27e9", "state_before": "case left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\nhk : k \u2208 image b univ\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\nx : \u03b2\nhx : x \u2208 image (fun i => b \u2191i) univ\n\u22a2 x \u2208 image b univ", "state_after": "case left.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\nhk : k \u2208 image b univ\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\nx : \u03b2\nhx : x \u2208 image (fun i => b \u2191i) univ\ny : { a // p (b a) }\nleft\u271d : y \u2208 univ\nhy : b \u2191y = x\n\u22a2 x \u2208 image b univ"}, {"tactic": "exact hy \u25b8 mem_image_of_mem b (mem_univ (y : \u03b1))", "state_before": "case left.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\nhk : k \u2208 image b univ\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\nx : \u03b2\nhx : x \u2208 image (fun i => b \u2191i) univ\ny : { a // p (b a) }\nleft\u271d : y \u2208 univ\nhy : b \u2191y = x\n\u22a2 x \u2208 image b univ", "state_after": "no goals"}, {"tactic": "intro h", "state_before": "case right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\nhk : k \u2208 image b univ\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\n\u22a2 \u00acimage b univ \u2286 image (fun i => b \u2191i) univ", "state_after": "case right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\nhk : k \u2208 image b univ\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\nh : image b univ \u2286 image (fun i => b \u2191i) univ\n\u22a2 False"}, {"tactic": "rw [mem_image] at hk", "state_before": "case right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\nhk : k \u2208 image b univ\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\nh : image b univ \u2286 image (fun i => b \u2191i) univ\n\u22a2 False", "state_after": "case right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\nhk : \u2203 a, a \u2208 univ \u2227 b a = k\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\nh : image b univ \u2286 image (fun i => b \u2191i) univ\n\u22a2 False"}, {"tactic": "rcases hk with \u27e8k', _, hk'\u27e9", "state_before": "case right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\nhk : \u2203 a, a \u2208 univ \u2227 b a = k\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\nh : image b univ \u2286 image (fun i => b \u2191i) univ\n\u22a2 False", "state_after": "case right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\nh : image b univ \u2286 image (fun i => b \u2191i) univ\nk' : \u03b1\nleft\u271d : k' \u2208 univ\nhk' : b k' = k\n\u22a2 False"}, {"tactic": "subst hk'", "state_before": "case right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nk : \u03b2\nb : \u03b1 \u2192 \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nhp : \u00acp k\nh : image b univ \u2286 image (fun i => b \u2191i) univ\nk' : \u03b1\nleft\u271d : k' \u2208 univ\nhk' : b k' = k\n\u22a2 False", "state_after": "case right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nb : \u03b1 \u2192 \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nh : image b univ \u2286 image (fun i => b \u2191i) univ\nk' : \u03b1\nleft\u271d : k' \u2208 univ\nhp : \u00acp (b k')\n\u22a2 False"}, {"tactic": "have := h (mem_image_of_mem b (mem_univ k'))", "state_before": "case right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nb : \u03b1 \u2192 \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nh : image b univ \u2286 image (fun i => b \u2191i) univ\nk' : \u03b1\nleft\u271d : k' \u2208 univ\nhp : \u00acp (b k')\n\u22a2 False", "state_after": "case right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nb : \u03b1 \u2192 \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nh : image b univ \u2286 image (fun i => b \u2191i) univ\nk' : \u03b1\nleft\u271d : k' \u2208 univ\nhp : \u00acp (b k')\nthis : b k' \u2208 image (fun i => b \u2191i) univ\n\u22a2 False"}, {"tactic": "rw [mem_image] at this", "state_before": "case right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nb : \u03b1 \u2192 \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nh : image b univ \u2286 image (fun i => b \u2191i) univ\nk' : \u03b1\nleft\u271d : k' \u2208 univ\nhp : \u00acp (b k')\nthis : b k' \u2208 image (fun i => b \u2191i) univ\n\u22a2 False", "state_after": "case right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nb : \u03b1 \u2192 \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nh : image b univ \u2286 image (fun i => b \u2191i) univ\nk' : \u03b1\nleft\u271d : k' \u2208 univ\nhp : \u00acp (b k')\nthis : \u2203 a, a \u2208 univ \u2227 b \u2191a = b k'\n\u22a2 False"}, {"tactic": "rcases this with \u27e8j, _, hj'\u27e9", "state_before": "case right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nb : \u03b1 \u2192 \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nh : image b univ \u2286 image (fun i => b \u2191i) univ\nk' : \u03b1\nleft\u271d : k' \u2208 univ\nhp : \u00acp (b k')\nthis : \u2203 a, a \u2208 univ \u2227 b \u2191a = b k'\n\u22a2 False", "state_after": "case right.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nb : \u03b1 \u2192 \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nh : image b univ \u2286 image (fun i => b \u2191i) univ\nk' : \u03b1\nleft\u271d\u00b9 : k' \u2208 univ\nhp : \u00acp (b k')\nj : { a // p (b a) }\nleft\u271d : j \u2208 univ\nhj' : b \u2191j = b k'\n\u22a2 False"}, {"tactic": "exact hp (hj' \u25b8 j.2)", "state_before": "case right.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b2\nb : \u03b1 \u2192 \u03b2\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nh : image b univ \u2286 image (fun i => b \u2191i) univ\nk' : \u03b1\nleft\u271d\u00b9 : k' \u2208 univ\nhp : \u00acp (b k')\nj : { a // p (b a) }\nleft\u271d : j \u2208 univ\nhj' : b \u2191j = b k'\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/LinearPMap.lean", "full_name": "LinearPMap.sSup_apply", "start": [621, 11], "end": [626, 6], "traced_tactics": [{"tactic": "symm", "state_before": "R : Type u_3\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_1\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type ?u.406232\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\nl : E \u2192\u2097.[R] F\nhl : l \u2208 c\nx : { x // x \u2208 l.domain }\n\u22a2 \u2191(LinearPMap.sSup c hc) { val := \u2191x, property := (_ : \u2191x \u2208 (LinearPMap.sSup c hc).domain) } = \u2191l x", "state_after": "R : Type u_3\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_1\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type ?u.406232\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\nl : E \u2192\u2097.[R] F\nhl : l \u2208 c\nx : { x // x \u2208 l.domain }\n\u22a2 \u2191l x = \u2191(LinearPMap.sSup c hc) { val := \u2191x, property := (_ : \u2191x \u2208 (LinearPMap.sSup c hc).domain) }"}, {"tactic": "apply (Classical.choose_spec (sSup_aux c hc) hl).2", "state_before": "R : Type u_3\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_1\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type ?u.406232\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\nl : E \u2192\u2097.[R] F\nhl : l \u2208 c\nx : { x // x \u2208 l.domain }\n\u22a2 \u2191l x = \u2191(LinearPMap.sSup c hc) { val := \u2191x, property := (_ : \u2191x \u2208 (LinearPMap.sSup c hc).domain) }", "state_after": "case _h\nR : Type u_3\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_1\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type ?u.406232\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\nl : E \u2192\u2097.[R] F\nhl : l \u2208 c\nx : { x // x \u2208 l.domain }\n\u22a2 \u2191x = \u2191{ val := \u2191x, property := (_ : \u2191x \u2208 (LinearPMap.sSup c hc).domain) }"}, {"tactic": "rfl", "state_before": "case _h\nR : Type u_3\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_1\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type ?u.406232\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nc : Set (E \u2192\u2097.[R] F)\nhc : DirectedOn (fun x x_1 => x \u2264 x_1) c\nl : E \u2192\u2097.[R] F\nhl : l \u2208 c\nx : { x // x \u2208 l.domain }\n\u22a2 \u2191x = \u2191{ val := \u2191x, property := (_ : \u2191x \u2208 (LinearPMap.sSup c hc).domain) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "full_name": "tsub_eq_zero_iff_le", "start": [327, 1], "end": [328, 56], "traced_tactics": [{"tactic": "rw [\u2190 nonpos_iff_eq_zero, tsub_le_iff_left, add_zero]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\n\u22a2 a - b = 0 \u2194 a \u2264 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "full_name": "LinearMap.toMatrix_one", "start": [627, 1], "end": [628, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/MonoidLocalization.lean", "full_name": "Submonoid.LocalizationMap.map_mk'", "start": [1152, 1], "end": [1156, 57], "traced_tactics": [{"tactic": "rw [map, lift_mk', mul_inv_left]", "state_before": "M : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_3\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_4\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_2\ninst\u271d : CommMonoid Q\nk : LocalizationMap T Q\nx : M\ny : { x // x \u2208 S }\n\u22a2 \u2191(map f hy k) (mk' f x y) = mk' k (\u2191g x) { val := \u2191g \u2191y, property := (_ : \u2191g \u2191y \u2208 T) }", "state_after": "M : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_3\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_4\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_2\ninst\u271d : CommMonoid Q\nk : LocalizationMap T Q\nx : M\ny : { x // x \u2208 S }\n\u22a2 \u2191(MonoidHom.comp (toMap k) g) x =\n    \u2191(MonoidHom.comp (toMap k) g) \u2191y * mk' k (\u2191g x) { val := \u2191g \u2191y, property := (_ : \u2191g \u2191y \u2208 T) }"}, {"tactic": "show k.toMap (g x) = k.toMap (g y) * _", "state_before": "M : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_3\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_4\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_2\ninst\u271d : CommMonoid Q\nk : LocalizationMap T Q\nx : M\ny : { x // x \u2208 S }\n\u22a2 \u2191(MonoidHom.comp (toMap k) g) x =\n    \u2191(MonoidHom.comp (toMap k) g) \u2191y * mk' k (\u2191g x) { val := \u2191g \u2191y, property := (_ : \u2191g \u2191y \u2208 T) }", "state_after": "M : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_3\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_4\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_2\ninst\u271d : CommMonoid Q\nk : LocalizationMap T Q\nx : M\ny : { x // x \u2208 S }\n\u22a2 \u2191(toMap k) (\u2191g x) = \u2191(toMap k) (\u2191g \u2191y) * mk' k (\u2191g x) { val := \u2191g \u2191y, property := (_ : \u2191g \u2191y \u2208 T) }"}, {"tactic": "rw [mul_mk'_eq_mk'_of_mul]", "state_before": "M : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_3\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_4\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_2\ninst\u271d : CommMonoid Q\nk : LocalizationMap T Q\nx : M\ny : { x // x \u2208 S }\n\u22a2 \u2191(toMap k) (\u2191g x) = \u2191(toMap k) (\u2191g \u2191y) * mk' k (\u2191g x) { val := \u2191g \u2191y, property := (_ : \u2191g \u2191y \u2208 T) }", "state_after": "M : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_3\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_4\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_2\ninst\u271d : CommMonoid Q\nk : LocalizationMap T Q\nx : M\ny : { x // x \u2208 S }\n\u22a2 \u2191(toMap k) (\u2191g x) = mk' k (\u2191g \u2191y * \u2191g x) { val := \u2191g \u2191y, property := (_ : \u2191g \u2191y \u2208 T) }"}, {"tactic": "exact (k.mk'_mul_cancel_left (g x) \u27e8g y, hy y\u27e9).symm", "state_before": "M : Type u_1\ninst\u271d\u00b3 : CommMonoid M\nS : Submonoid M\nN : Type u_3\ninst\u271d\u00b2 : CommMonoid N\nP : Type u_4\ninst\u271d\u00b9 : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nhy : \u2200 (y : { x // x \u2208 S }), \u2191g \u2191y \u2208 T\nQ : Type u_2\ninst\u271d : CommMonoid Q\nk : LocalizationMap T Q\nx : M\ny : { x // x \u2208 S }\n\u22a2 \u2191(toMap k) (\u2191g x) = mk' k (\u2191g \u2191y * \u2191g x) { val := \u2191g \u2191y, property := (_ : \u2191g \u2191y \u2208 T) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/ENNReal.lean", "full_name": "ENNReal.exists_inv_two_pow_lt", "start": [1846, 1], "end": [1851, 21], "traced_tactics": [{"tactic": "rcases exists_inv_nat_lt ha with \u27e8n, hn\u27e9", "state_before": "\u03b1 : Type ?u.345679\n\u03b2 : Type ?u.345682\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\n\u22a2 \u2203 n, 2\u207b\u00b9 ^ n < a", "state_after": "case intro\n\u03b1 : Type ?u.345679\n\u03b2 : Type ?u.345682\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nn : \u2115\nhn : (\u2191n)\u207b\u00b9 < a\n\u22a2 \u2203 n, 2\u207b\u00b9 ^ n < a"}, {"tactic": "refine' \u27e8n, lt_trans _ hn\u27e9", "state_before": "case intro\n\u03b1 : Type ?u.345679\n\u03b2 : Type ?u.345682\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nn : \u2115\nhn : (\u2191n)\u207b\u00b9 < a\n\u22a2 \u2203 n, 2\u207b\u00b9 ^ n < a", "state_after": "case intro\n\u03b1 : Type ?u.345679\n\u03b2 : Type ?u.345682\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nn : \u2115\nhn : (\u2191n)\u207b\u00b9 < a\n\u22a2 2\u207b\u00b9 ^ n < (\u2191n)\u207b\u00b9"}, {"tactic": "rw [\u2190 ENNReal.inv_pow, ENNReal.inv_lt_inv]", "state_before": "case intro\n\u03b1 : Type ?u.345679\n\u03b2 : Type ?u.345682\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nn : \u2115\nhn : (\u2191n)\u207b\u00b9 < a\n\u22a2 2\u207b\u00b9 ^ n < (\u2191n)\u207b\u00b9", "state_after": "case intro\n\u03b1 : Type ?u.345679\n\u03b2 : Type ?u.345682\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nn : \u2115\nhn : (\u2191n)\u207b\u00b9 < a\n\u22a2 \u2191n < 2 ^ n"}, {"tactic": "norm_cast", "state_before": "case intro\n\u03b1 : Type ?u.345679\n\u03b2 : Type ?u.345682\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nn : \u2115\nhn : (\u2191n)\u207b\u00b9 < a\n\u22a2 \u2191n < 2 ^ n", "state_after": "case intro\n\u03b1 : Type ?u.345679\n\u03b2 : Type ?u.345682\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nn : \u2115\nhn : (\u2191n)\u207b\u00b9 < a\n\u22a2 n < 2 ^ n"}, {"tactic": "exact n.lt_two_pow", "state_before": "case intro\n\u03b1 : Type ?u.345679\n\u03b2 : Type ?u.345682\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 0\nn : \u2115\nhn : (\u2191n)\u207b\u00b9 < a\n\u22a2 n < 2 ^ n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/OreLocalization/Basic.lean", "full_name": "OreLocalization.universalHom_apply", "start": [813, 1], "end": [815, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Tactic/NormNum/Basic.lean", "full_name": "Mathlib.Meta.NormNum.isRat_add", "start": [179, 1], "end": [196, 73], "traced_tactics": [{"tactic": "rintro rfl \u27e8_, rfl\u27e9 \u27e8_, rfl\u27e9 (h\u2081 : na * db + nb * da = k * nc) (h\u2082 : da * db = k * dc)", "state_before": "\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\na b : \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\n\u22a2 f = HAdd.hAdd \u2192\n    IsRat a na da \u2192\n      IsRat b nb db \u2192\n        Int.add (Int.mul na \u2191db) (Int.mul nb \u2191da) = Int.mul (\u2191k) nc \u2192 Nat.mul da db = Nat.mul k dc \u2192 IsRat (f a b) nc dc", "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\n\u22a2 IsRat (\u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db) nc dc"}, {"tactic": "have : Invertible (\u2191(da * db) : \u03b1) := by simpa using invertibleMul (da:\u03b1) db", "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\n\u22a2 IsRat (\u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db) nc dc", "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis : Invertible \u2191(da * db)\n\u22a2 IsRat (\u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db) nc dc"}, {"tactic": "have := invertibleOfMul' (\u03b1 := \u03b1) h\u2082", "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis : Invertible \u2191(da * db)\n\u22a2 IsRat (\u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db) nc dc", "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\n\u22a2 IsRat (\u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db) nc dc"}, {"tactic": "use this", "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\n\u22a2 IsRat (\u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db) nc dc", "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\n\u22a2 \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191nc * \u215f\u2191dc"}, {"tactic": "have H := (Nat.cast_commute (\u03b1 := \u03b1) da db).invOf_left.invOf_right.right_comm", "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\n\u22a2 \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191nc * \u215f\u2191dc", "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\n\u22a2 \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191nc * \u215f\u2191dc"}, {"tactic": "simp only [Int.cast_add, Int.cast_mul, Int.cast_ofNat, \u2190 mul_assoc,\n  add_mul, mul_mul_invOf_self_cancel] at h\u2081", "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : (fun x => \u2191x * (\u215f\u2191da * \u215f\u2191db)) (na * \u2191db + nb * \u2191da) = (fun x => \u2191x * (\u215f\u2191da * \u215f\u2191db)) (\u2191k * nc)\n\u22a2 \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191nc * \u215f\u2191dc", "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u2191db * \u215f\u2191da * \u215f\u2191db + \u2191nb * \u215f\u2191db = \u2191k * \u2191nc * \u215f\u2191da * \u215f\u2191db\n\u22a2 \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191nc * \u215f\u2191dc"}, {"tactic": "simp [\u2190 mul_assoc, H] at h\u2081 h\u2082", "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082\u271d : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u2191db * \u215f\u2191da * \u215f\u2191db + \u2191nb * \u215f\u2191db = \u2191k * \u2191nc * \u215f\u2191da * \u215f\u2191db\nh\u2082 : (fun x => \u2191nc * \u2191x * (\u215f\u2191da * \u215f\u2191db * \u215f\u2191dc)) (da * db) = (fun x => \u2191nc * \u2191x * (\u215f\u2191da * \u215f\u2191db * \u215f\u2191dc)) (k * dc)\n\u22a2 \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191nc * \u215f\u2191dc", "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082\u271d : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191k * \u2191nc * \u215f\u2191db * \u215f\u2191da\nh\u2082 : \u2191nc * \u215f\u2191dc = \u2191nc * \u2191k * \u2191dc * \u215f\u2191db * \u215f\u2191da * \u215f\u2191dc\n\u22a2 \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191nc * \u215f\u2191dc"}, {"tactic": "rw [h\u2081, h\u2082, Nat.cast_commute]", "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082\u271d : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191k * \u2191nc * \u215f\u2191db * \u215f\u2191da\nh\u2082 : \u2191nc * \u215f\u2191dc = \u2191nc * \u2191k * \u2191dc * \u215f\u2191db * \u215f\u2191da * \u215f\u2191dc\n\u22a2 \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191nc * \u215f\u2191dc", "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082\u271d : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191k * \u2191nc * \u215f\u2191db * \u215f\u2191da\nh\u2082 : \u2191nc * \u215f\u2191dc = \u2191nc * \u2191k * \u2191dc * \u215f\u2191db * \u215f\u2191da * \u215f\u2191dc\n\u22a2 \u2191nc * \u2191k * \u215f\u2191db * \u215f\u2191da = \u2191nc * \u2191k * \u2191dc * \u215f\u2191db * \u215f\u2191da * \u215f\u2191dc"}, {"tactic": "simp only [mul_mul_invOf_self_cancel,\n  (Nat.cast_commute (\u03b1 := \u03b1) da dc).invOf_left.invOf_right.right_comm,\n  (Nat.cast_commute (\u03b1 := \u03b1) db dc).invOf_left.invOf_right.right_comm]", "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082\u271d : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191k * \u2191nc * \u215f\u2191db * \u215f\u2191da\nh\u2082 : \u2191nc * \u215f\u2191dc = \u2191nc * \u2191k * \u2191dc * \u215f\u2191db * \u215f\u2191da * \u215f\u2191dc\n\u22a2 \u2191nc * \u2191k * \u215f\u2191db * \u215f\u2191da = \u2191nc * \u2191k * \u2191dc * \u215f\u2191db * \u215f\u2191da * \u215f\u2191dc", "state_after": "no goals"}, {"tactic": "simpa using invertibleMul (da:\u03b1) db", "state_before": "\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\n\u22a2 Invertible \u2191(da * db)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Char.lean", "full_name": "String.csize_le_4", "start": [20, 1], "end": [21, 46], "traced_tactics": [{"tactic": "rcases csize_eq c with _|_|_|_ <;> simp_all", "state_before": "c : Char\n\u22a2 csize c \u2264 4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Complex/Basic.lean", "full_name": "Complex.abs_pow", "start": [1019, 1], "end": [1020, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/IsometricSMul.lean", "full_name": "IsometryEquiv.mulRight_symm", "start": [213, 1], "end": [214, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/PolynomialAlgebra.lean", "full_name": "PolyEquivTensor.toFunLinear_tmul_apply", "start": [78, 1], "end": [80, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/Basic.lean", "full_name": "Ideal.sInf_isPrime_of_isChain", "start": [463, 1], "end": [476, 77], "traced_tactics": [{"tactic": "rw [Ideal.mem_sInf] at hx e\u22a2", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\ns : Set (Ideal \u03b1)\nhs : Set.Nonempty s\nhs' : IsChain (fun x x_1 => x \u2264 x_1) s\nH : \u2200 (p : Ideal \u03b1), p \u2208 s \u2192 IsPrime p\nx\u271d y\u271d : \u03b1\ne : x\u271d * y\u271d \u2208 sInf s\nhx : \u00acx\u271d \u2208 sInf s\n\u22a2 y\u271d \u2208 sInf s", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\ns : Set (Ideal \u03b1)\nhs : Set.Nonempty s\nhs' : IsChain (fun x x_1 => x \u2264 x_1) s\nH : \u2200 (p : Ideal \u03b1), p \u2208 s \u2192 IsPrime p\nx\u271d y\u271d : \u03b1\ne : \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 x\u271d * y\u271d \u2208 I\nhx : \u00ac\u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 x\u271d \u2208 I\n\u22a2 \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 y\u271d \u2208 I"}, {"tactic": "push_neg  at hx", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\ns : Set (Ideal \u03b1)\nhs : Set.Nonempty s\nhs' : IsChain (fun x x_1 => x \u2264 x_1) s\nH : \u2200 (p : Ideal \u03b1), p \u2208 s \u2192 IsPrime p\nx\u271d y\u271d : \u03b1\ne : \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 x\u271d * y\u271d \u2208 I\nhx : \u00ac\u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 x\u271d \u2208 I\n\u22a2 \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 y\u271d \u2208 I", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\ns : Set (Ideal \u03b1)\nhs : Set.Nonempty s\nhs' : IsChain (fun x x_1 => x \u2264 x_1) s\nH : \u2200 (p : Ideal \u03b1), p \u2208 s \u2192 IsPrime p\nx\u271d y\u271d : \u03b1\ne : \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 x\u271d * y\u271d \u2208 I\nhx : Exists fun \u2983I\u2984 => I \u2208 s \u2227 \u00acx\u271d \u2208 I\n\u22a2 \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 y\u271d \u2208 I"}, {"tactic": "obtain \u27e8I, hI, hI'\u27e9 := hx", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\ns : Set (Ideal \u03b1)\nhs : Set.Nonempty s\nhs' : IsChain (fun x x_1 => x \u2264 x_1) s\nH : \u2200 (p : Ideal \u03b1), p \u2208 s \u2192 IsPrime p\nx\u271d y\u271d : \u03b1\ne : \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 x\u271d * y\u271d \u2208 I\nhx : Exists fun \u2983I\u2984 => I \u2208 s \u2227 \u00acx\u271d \u2208 I\n\u22a2 \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 y\u271d \u2208 I", "state_after": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\ns : Set (Ideal \u03b1)\nhs : Set.Nonempty s\nhs' : IsChain (fun x x_1 => x \u2264 x_1) s\nH : \u2200 (p : Ideal \u03b1), p \u2208 s \u2192 IsPrime p\nx\u271d y\u271d : \u03b1\ne : \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 x\u271d * y\u271d \u2208 I\nI : Ideal \u03b1\nhI : I \u2208 s\nhI' : \u00acx\u271d \u2208 I\n\u22a2 \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 y\u271d \u2208 I"}, {"tactic": "intro J hJ", "state_before": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\ns : Set (Ideal \u03b1)\nhs : Set.Nonempty s\nhs' : IsChain (fun x x_1 => x \u2264 x_1) s\nH : \u2200 (p : Ideal \u03b1), p \u2208 s \u2192 IsPrime p\nx\u271d y\u271d : \u03b1\ne : \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 x\u271d * y\u271d \u2208 I\nI : Ideal \u03b1\nhI : I \u2208 s\nhI' : \u00acx\u271d \u2208 I\n\u22a2 \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 y\u271d \u2208 I", "state_after": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\ns : Set (Ideal \u03b1)\nhs : Set.Nonempty s\nhs' : IsChain (fun x x_1 => x \u2264 x_1) s\nH : \u2200 (p : Ideal \u03b1), p \u2208 s \u2192 IsPrime p\nx\u271d y\u271d : \u03b1\ne : \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 x\u271d * y\u271d \u2208 I\nI : Ideal \u03b1\nhI : I \u2208 s\nhI' : \u00acx\u271d \u2208 I\nJ : Ideal \u03b1\nhJ : J \u2208 s\n\u22a2 y\u271d \u2208 J"}, {"tactic": "cases' hs'.total hI hJ with h h", "state_before": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\ns : Set (Ideal \u03b1)\nhs : Set.Nonempty s\nhs' : IsChain (fun x x_1 => x \u2264 x_1) s\nH : \u2200 (p : Ideal \u03b1), p \u2208 s \u2192 IsPrime p\nx\u271d y\u271d : \u03b1\ne : \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 x\u271d * y\u271d \u2208 I\nI : Ideal \u03b1\nhI : I \u2208 s\nhI' : \u00acx\u271d \u2208 I\nJ : Ideal \u03b1\nhJ : J \u2208 s\n\u22a2 y\u271d \u2208 J", "state_after": "case intro.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\ns : Set (Ideal \u03b1)\nhs : Set.Nonempty s\nhs' : IsChain (fun x x_1 => x \u2264 x_1) s\nH : \u2200 (p : Ideal \u03b1), p \u2208 s \u2192 IsPrime p\nx\u271d y\u271d : \u03b1\ne : \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 x\u271d * y\u271d \u2208 I\nI : Ideal \u03b1\nhI : I \u2208 s\nhI' : \u00acx\u271d \u2208 I\nJ : Ideal \u03b1\nhJ : J \u2208 s\nh : I \u2264 J\n\u22a2 y\u271d \u2208 J\n\ncase intro.intro.inr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\ns : Set (Ideal \u03b1)\nhs : Set.Nonempty s\nhs' : IsChain (fun x x_1 => x \u2264 x_1) s\nH : \u2200 (p : Ideal \u03b1), p \u2208 s \u2192 IsPrime p\nx\u271d y\u271d : \u03b1\ne : \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 x\u271d * y\u271d \u2208 I\nI : Ideal \u03b1\nhI : I \u2208 s\nhI' : \u00acx\u271d \u2208 I\nJ : Ideal \u03b1\nhJ : J \u2208 s\nh : J \u2264 I\n\u22a2 y\u271d \u2208 J"}, {"tactic": "exact h (((H I hI).mem_or_mem (e hI)).resolve_left hI')", "state_before": "case intro.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\ns : Set (Ideal \u03b1)\nhs : Set.Nonempty s\nhs' : IsChain (fun x x_1 => x \u2264 x_1) s\nH : \u2200 (p : Ideal \u03b1), p \u2208 s \u2192 IsPrime p\nx\u271d y\u271d : \u03b1\ne : \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 x\u271d * y\u271d \u2208 I\nI : Ideal \u03b1\nhI : I \u2208 s\nhI' : \u00acx\u271d \u2208 I\nJ : Ideal \u03b1\nhJ : J \u2208 s\nh : I \u2264 J\n\u22a2 y\u271d \u2208 J", "state_after": "no goals"}, {"tactic": "exact ((H J hJ).mem_or_mem (e hJ)).resolve_left fun x => hI' <| h x", "state_before": "case intro.intro.inr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI\u271d : Ideal \u03b1\na b : \u03b1\ns : Set (Ideal \u03b1)\nhs : Set.Nonempty s\nhs' : IsChain (fun x x_1 => x \u2264 x_1) s\nH : \u2200 (p : Ideal \u03b1), p \u2208 s \u2192 IsPrime p\nx\u271d y\u271d : \u03b1\ne : \u2200 \u2983I : Ideal \u03b1\u2984, I \u2208 s \u2192 x\u271d * y\u271d \u2208 I\nI : Ideal \u03b1\nhI : I \u2208 s\nhI' : \u00acx\u271d \u2208 I\nJ : Ideal \u03b1\nhJ : J \u2208 s\nh : J \u2264 I\n\u22a2 y\u271d \u2208 J", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/ContinuousAffineMap.lean", "full_name": "ContinuousAffineMap.ext", "start": [83, 1], "end": [84, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Star/Subalgebra.lean", "full_name": "StarSubalgebra.comap_mono", "start": [252, 1], "end": [254, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Symmetrized.lean", "full_name": "SymAlg.unsym_ne_one_iff", "start": [251, 1], "end": [252, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/PNat/Prime.lean", "full_name": "PNat.Coprime.pow", "start": [301, 1], "end": [302, 79], "traced_tactics": [{"tactic": "rw [\u2190 coprime_coe] at *", "state_before": "m n : \u2115+\nk l : \u2115\nh : Coprime m n\n\u22a2 coprime (\u2191m ^ k) (\u2191n ^ l)", "state_after": "m n : \u2115+\nk l : \u2115\nh : coprime \u2191m \u2191n\n\u22a2 coprime (\u2191m ^ k) (\u2191n ^ l)"}, {"tactic": "simp only [pow_coe]", "state_before": "m n : \u2115+\nk l : \u2115\nh : coprime \u2191m \u2191n\n\u22a2 coprime (\u2191m ^ k) (\u2191n ^ l)", "state_after": "m n : \u2115+\nk l : \u2115\nh : coprime \u2191m \u2191n\n\u22a2 coprime (\u2191m ^ k) (\u2191n ^ l)"}, {"tactic": "apply Nat.coprime.pow", "state_before": "m n : \u2115+\nk l : \u2115\nh : coprime \u2191m \u2191n\n\u22a2 coprime (\u2191m ^ k) (\u2191n ^ l)", "state_after": "case H1\nm n : \u2115+\nk l : \u2115\nh : coprime \u2191m \u2191n\n\u22a2 coprime \u2191m \u2191n"}, {"tactic": "apply h", "state_before": "case H1\nm n : \u2115+\nk l : \u2115\nh : coprime \u2191m \u2191n\n\u22a2 coprime \u2191m \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Homotopy/Path.lean", "full_name": "Path.Homotopy.coeFn_injective", "start": [62, 1], "end": [63, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/OmegaCompletePartialOrder.lean", "full_name": "Scott.IsOpen.inter", "start": [60, 1], "end": [61, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Bases.lean", "full_name": "Filter.HasAntitoneBasis.prod", "start": [946, 1], "end": [949, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.add_singleton_eq_iff", "start": [1043, 1], "end": [1048, 23], "traced_tactics": [{"tactic": "rw [add_comm, singleton_add]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.96470\n\u03b3 : Type ?u.96473\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Multiset \u03b1\na\u271d b : \u03b1\ns t : Multiset \u03b1\na : \u03b1\n\u22a2 s + {a} = t \u2194 a \u2208 t \u2227 s = erase t a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.96470\n\u03b3 : Type ?u.96473\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Multiset \u03b1\na\u271d b : \u03b1\ns t : Multiset \u03b1\na : \u03b1\n\u22a2 a ::\u2098 s = t \u2194 a \u2208 t \u2227 s = erase t a"}, {"tactic": "constructor", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.96470\n\u03b3 : Type ?u.96473\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Multiset \u03b1\na\u271d b : \u03b1\ns t : Multiset \u03b1\na : \u03b1\n\u22a2 a ::\u2098 s = t \u2194 a \u2208 t \u2227 s = erase t a", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type ?u.96470\n\u03b3 : Type ?u.96473\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Multiset \u03b1\na\u271d b : \u03b1\ns t : Multiset \u03b1\na : \u03b1\n\u22a2 a ::\u2098 s = t \u2192 a \u2208 t \u2227 s = erase t a\n\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type ?u.96470\n\u03b3 : Type ?u.96473\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Multiset \u03b1\na\u271d b : \u03b1\ns t : Multiset \u03b1\na : \u03b1\n\u22a2 a \u2208 t \u2227 s = erase t a \u2192 a ::\u2098 s = t"}, {"tactic": "rintro rfl", "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type ?u.96470\n\u03b3 : Type ?u.96473\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Multiset \u03b1\na\u271d b : \u03b1\ns t : Multiset \u03b1\na : \u03b1\n\u22a2 a ::\u2098 s = t \u2192 a \u2208 t \u2227 s = erase t a", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type ?u.96470\n\u03b3 : Type ?u.96473\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Multiset \u03b1\na\u271d b : \u03b1\ns : Multiset \u03b1\na : \u03b1\n\u22a2 a \u2208 a ::\u2098 s \u2227 s = erase (a ::\u2098 s) a"}, {"tactic": "exact \u27e8s.mem_cons_self a, (s.erase_cons_head a).symm\u27e9", "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type ?u.96470\n\u03b3 : Type ?u.96473\ninst\u271d : DecidableEq \u03b1\ns\u271d t : Multiset \u03b1\na\u271d b : \u03b1\ns : Multiset \u03b1\na : \u03b1\n\u22a2 a \u2208 a ::\u2098 s \u2227 s = erase (a ::\u2098 s) a", "state_after": "no goals"}, {"tactic": "rintro \u27e8h, rfl\u27e9", "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type ?u.96470\n\u03b3 : Type ?u.96473\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Multiset \u03b1\na\u271d b : \u03b1\ns t : Multiset \u03b1\na : \u03b1\n\u22a2 a \u2208 t \u2227 s = erase t a \u2192 a ::\u2098 s = t", "state_after": "case mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.96470\n\u03b3 : Type ?u.96473\ninst\u271d : DecidableEq \u03b1\ns t\u271d : Multiset \u03b1\na\u271d b : \u03b1\nt : Multiset \u03b1\na : \u03b1\nh : a \u2208 t\n\u22a2 a ::\u2098 erase t a = t"}, {"tactic": "exact cons_erase h", "state_before": "case mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.96470\n\u03b3 : Type ?u.96473\ninst\u271d : DecidableEq \u03b1\ns t\u271d : Multiset \u03b1\na\u271d b : \u03b1\nt : Multiset \u03b1\na : \u03b1\nh : a \u2208 t\n\u22a2 a ::\u2098 erase t a = t", "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.pullbackIsoPullback_hom_fst", "start": [586, 1], "end": [588, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.C_mul_X_eq_monomial", "start": [339, 1], "end": [340, 42], "traced_tactics": [{"tactic": "rw [\u2190 C_mul_X_pow_eq_monomial, pow_one]", "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na\u271d a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\ns : \u03c3\na : R\n\u22a2 \u2191C a * X s = \u2191(monomial (Finsupp.single s 1)) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "full_name": "Right.mul_lt_mul", "start": [199, 1], "end": [203, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "full_name": "complete_univ", "start": [374, 1], "end": [377, 28], "traced_tactics": [{"tactic": "rcases CompleteSpace.complete hf with \u27e8x, hx\u27e9", "state_before": "\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\u271d\n\u03b1 : Type u\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : Filter \u03b1\nhf : Cauchy f\nx\u271d : f \u2264 \ud835\udcdf univ\n\u22a2 \u2203 x, x \u2208 univ \u2227 f \u2264 \ud835\udcdd x", "state_after": "case intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\u271d\n\u03b1 : Type u\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : Filter \u03b1\nhf : Cauchy f\nx\u271d : f \u2264 \ud835\udcdf univ\nx : \u03b1\nhx : f \u2264 \ud835\udcdd x\n\u22a2 \u2203 x, x \u2208 univ \u2227 f \u2264 \ud835\udcdd x"}, {"tactic": "exact \u27e8x, mem_univ x, hx\u27e9", "state_before": "case intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : UniformSpace \u03b1\u271d\n\u03b1 : Type u\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : Filter \u03b1\nhf : Cauchy f\nx\u271d : f \u2264 \ud835\udcdf univ\nx : \u03b1\nhx : f \u2264 \ud835\udcdd x\n\u22a2 \u2203 x, x \u2208 univ \u2227 f \u2264 \ud835\udcdd x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Palindrome.lean", "full_name": "List.Palindrome.of_reverse_eq", "start": [58, 1], "end": [64, 38], "traced_tactics": [{"tactic": "refine' bidirectionalRecOn l (fun _ => Palindrome.nil) (fun a _ => Palindrome.singleton a) _", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.673\nl\u271d l : List \u03b1\n\u22a2 reverse l = l \u2192 Palindrome l", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.673\nl\u271d l : List \u03b1\n\u22a2 \u2200 (a : \u03b1) (l : List \u03b1) (b : \u03b1),\n    (reverse l = l \u2192 Palindrome l) \u2192 reverse (a :: (l ++ [b])) = a :: (l ++ [b]) \u2192 Palindrome (a :: (l ++ [b]))"}, {"tactic": "intro x l y hp hr", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.673\nl\u271d l : List \u03b1\n\u22a2 \u2200 (a : \u03b1) (l : List \u03b1) (b : \u03b1),\n    (reverse l = l \u2192 Palindrome l) \u2192 reverse (a :: (l ++ [b])) = a :: (l ++ [b]) \u2192 Palindrome (a :: (l ++ [b]))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.673\nl\u271d\u00b9 l\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\ny : \u03b1\nhp : reverse l = l \u2192 Palindrome l\nhr : reverse (x :: (l ++ [y])) = x :: (l ++ [y])\n\u22a2 Palindrome (x :: (l ++ [y]))"}, {"tactic": "rw [reverse_cons, reverse_append] at hr", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.673\nl\u271d\u00b9 l\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\ny : \u03b1\nhp : reverse l = l \u2192 Palindrome l\nhr : reverse (x :: (l ++ [y])) = x :: (l ++ [y])\n\u22a2 Palindrome (x :: (l ++ [y]))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.673\nl\u271d\u00b9 l\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\ny : \u03b1\nhp : reverse l = l \u2192 Palindrome l\nhr : reverse [y] ++ reverse l ++ [x] = x :: (l ++ [y])\n\u22a2 Palindrome (x :: (l ++ [y]))"}, {"tactic": "rw [head_eq_of_cons_eq hr]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.673\nl\u271d\u00b9 l\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\ny : \u03b1\nhp : reverse l = l \u2192 Palindrome l\nhr : reverse [y] ++ reverse l ++ [x] = x :: (l ++ [y])\n\u22a2 Palindrome (x :: (l ++ [y]))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.673\nl\u271d\u00b9 l\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\ny : \u03b1\nhp : reverse l = l \u2192 Palindrome l\nhr : reverse [y] ++ reverse l ++ [x] = x :: (l ++ [y])\n\u22a2 Palindrome (x :: (l ++ [x]))"}, {"tactic": "have : Palindrome l := hp (append_inj_left' (tail_eq_of_cons_eq hr) rfl)", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.673\nl\u271d\u00b9 l\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\ny : \u03b1\nhp : reverse l = l \u2192 Palindrome l\nhr : reverse [y] ++ reverse l ++ [x] = x :: (l ++ [y])\n\u22a2 Palindrome (x :: (l ++ [x]))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.673\nl\u271d\u00b9 l\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\ny : \u03b1\nhp : reverse l = l \u2192 Palindrome l\nhr : reverse [y] ++ reverse l ++ [x] = x :: (l ++ [y])\nthis : Palindrome l\n\u22a2 Palindrome (x :: (l ++ [x]))"}, {"tactic": "exact Palindrome.cons_concat x this", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.673\nl\u271d\u00b9 l\u271d : List \u03b1\nx : \u03b1\nl : List \u03b1\ny : \u03b1\nhp : reverse l = l \u2192 Palindrome l\nhr : reverse [y] ++ reverse l ++ [x] = x :: (l ++ [y])\nthis : Palindrome l\n\u22a2 Palindrome (x :: (l ++ [x]))", "state_after": "no goals"}]}, {"url": 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?u.639631\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nh : \u03b1 \u2243\u1d62 \u03b2\ns : Set \u03b2\n\u22a2 EMetric.diam (\u2191h \u207b\u00b9' s) = EMetric.diam s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fintype/Basic.lean", "full_name": "Fintype.rightInverse_bijInv", "start": [1149, 1], "end": [1150, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Ring/Prod.lean", "full_name": "NonUnitalRingHom.prodMap_def", "start": [170, 1], "end": [171, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": 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"PseudoMetricSpace.replaceUniformity_eq", "start": [1249, 1], "end": [1252, 6], "traced_tactics": [{"tactic": "ext", "state_before": "\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type ?u.166910\n\u03b9 : Type ?u.166913\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_1\nU : UniformSpace \u03b1\nm : PseudoMetricSpace \u03b1\nH : \ud835\udce4 \u03b1 = \ud835\udce4 \u03b1\n\u22a2 replaceUniformity m H = m", "state_after": "case h.dist.h.h\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type ?u.166910\n\u03b9 : Type ?u.166913\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_1\nU : UniformSpace \u03b1\nm : PseudoMetricSpace \u03b1\nH : \ud835\udce4 \u03b1 = \ud835\udce4 \u03b1\nx\u271d\u00b9 x\u271d : \u03b1\n\u22a2 dist x\u271d\u00b9 x\u271d = dist x\u271d\u00b9 x\u271d"}, {"tactic": "rfl", "state_before": "case h.dist.h.h\n\u03b1\u271d : Type u\n\u03b2 : Type v\nX : Type ?u.166910\n\u03b9 : Type ?u.166913\ninst\u271d : PseudoMetricSpace \u03b1\u271d\n\u03b1 : Type u_1\nU : UniformSpace \u03b1\nm : PseudoMetricSpace \u03b1\nH : \ud835\udce4 \u03b1 = \ud835\udce4 \u03b1\nx\u271d\u00b9 x\u271d : \u03b1\n\u22a2 dist x\u271d\u00b9 x\u271d = dist x\u271d\u00b9 x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/AdjoinRoot.lean", "full_name": "AdjoinRoot.induction_on", "start": [101, 1], "end": [103, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Dfinsupp.lean", "full_name": "Dfinsupp.lhom_ext", "start": [65, 1], "end": [66, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/Inverse.lean", "full_name": "HasStrictFDerivAt.to_localInverse", "start": [664, 1], "end": [667, 40], "traced_tactics": [{"tactic": "simpa [\u2190 localInverse_def] using hf", "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type ?u.747862\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\nG' : Type ?u.747965\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G'\n\u03b5 : \u211d\ninst\u271d : CompleteSpace E\nf : E \u2192 F\nf' : E \u2243L[\ud835\udd5c] F\na : E\nhf : HasStrictFDerivAt f (\u2191f') a\n\u22a2 HasStrictFDerivAt (\u2191(toLocalHomeomorph f hf)) (\u2191f') (\u2191(LocalHomeomorph.symm (toLocalHomeomorph f hf)) (f a))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": 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V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : inner x (-y) = 0\n\u22a2 angle x (x - y) = Real.arccos (\u2016x\u2016 / \u2016x - y\u2016)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Submonoid/Operations.lean", "full_name": "Submonoid.coe_prod", "start": [851, 1], "end": [853, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.LeftInvOn.mono", "start": [1068, 1], "end": [1069, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "full_name": "MeasureTheory.NullMeasurableSet.fundamentalInterior", "start": [652, 11], "end": [654, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/DirectSum/Basic.lean", "full_name": "DirectSum.toAddMonoid.unique", "start": [210, 1], "end": [214, 25], "traced_tactics": [{"tactic": "congr", "state_before": "\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\n\u03c8 : (\u2a01 (i : \u03b9), \u03b2 i) \u2192+ \u03b3\nf : \u2a01 (i : \u03b9), \u03b2 i\n\u22a2 \u2191\u03c8 f = \u2191(toAddMonoid fun i => AddMonoidHom.comp \u03c8 (of \u03b2 i)) f", "state_after": "case e_a\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\n\u03c8 : (\u2a01 (i : \u03b9), \u03b2 i) \u2192+ \u03b3\nf : \u2a01 (i : \u03b9), \u03b2 i\n\u22a2 \u03c8 = toAddMonoid fun i => AddMonoidHom.comp \u03c8 (of \u03b2 i)"}, {"tactic": "apply Dfinsupp.addHom_ext'", "state_before": "case e_a\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\n\u03c8 : (\u2a01 (i : \u03b9), \u03b2 i) \u2192+ \u03b3\nf : \u2a01 (i : \u03b9), \u03b2 i\n\u22a2 \u03c8 = toAddMonoid fun i => AddMonoidHom.comp \u03c8 (of \u03b2 i)", "state_after": "case e_a.H\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\n\u03c8 : (\u2a01 (i : \u03b9), \u03b2 i) \u2192+ \u03b3\nf : \u2a01 (i : \u03b9), \u03b2 i\n\u22a2 \u2200 (x : \u03b9),\n    AddMonoidHom.comp \u03c8 (Dfinsupp.singleAddHom (fun i => (fun i => \u03b2 i) i) x) =\n      AddMonoidHom.comp (toAddMonoid fun i => AddMonoidHom.comp \u03c8 (of \u03b2 i))\n        (Dfinsupp.singleAddHom (fun i => (fun i => \u03b2 i) i) x)"}, {"tactic": "simp [toAddMonoid, of]", "state_before": "case e_a.H\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\n\u03b3 : Type u\u2081\ninst\u271d : AddCommMonoid \u03b3\n\u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3\n\u03c8 : (\u2a01 (i : \u03b9), \u03b2 i) \u2192+ \u03b3\nf : \u2a01 (i : \u03b9), \u03b2 i\n\u22a2 \u2200 (x : \u03b9),\n    AddMonoidHom.comp \u03c8 (Dfinsupp.singleAddHom (fun i => (fun i => \u03b2 i) i) x) =\n      AddMonoidHom.comp (toAddMonoid fun i => AddMonoidHom.comp \u03c8 (of \u03b2 i))\n        (Dfinsupp.singleAddHom (fun i => (fun i => \u03b2 i) i) x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/List/Lemmas.lean", "full_name": "List.range'_append_1", "start": [1854, 9], "end": [1855, 94], "traced_tactics": [{"tactic": "simpa using range'_append s m n 1", "state_before": "s m n : Nat\n\u22a2 range' s m ++ range' (s + m) n = range' s (n + m)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/WittVector/Defs.lean", "full_name": "WittVector.constantCoeff_wittSub", "start": [300, 1], "end": [302, 57], "traced_tactics": [{"tactic": "apply constantCoeff_wittStructureInt p _ _ n", "state_before": "p : \u2115\nR : Type ?u.98408\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nn : \u2115\n\u22a2 \u2191constantCoeff (wittSub p n) = 0", "state_after": "p : \u2115\nR : Type ?u.98408\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nn : \u2115\n\u22a2 \u2191constantCoeff (X 0 - X 1) = 0"}, {"tactic": "simp only [sub_zero, RingHom.map_sub, constantCoeff_X]", "state_before": "p : \u2115\nR : Type ?u.98408\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nn : \u2115\n\u22a2 \u2191constantCoeff (X 0 - X 1) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.add_le_add_iff_left'", "start": [631, 9], "end": [632, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Topology.lean", "full_name": "Convex.combo_closure_interior_mem_interior", "start": [169, 1], "end": [173, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/String/Lemmas.lean", "full_name": "String.map_eq", "start": [740, 1], "end": [741, 39], "traced_tactics": [{"tactic": "simpa using mapAux_of_valid f [] s.1", "state_before": "f : Char \u2192 Char\ns : String\n\u22a2 map f s = { data := List.map f s.data }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/List/Lemmas.lean", "full_name": "List.infix_nil", "start": [1640, 9], "end": [1640, 98], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "Antitone.map_ciInf_of_continuousAt", "start": [2849, 1], "end": [2852, 9], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "full_name": "circleIntegral.norm_integral_lt_of_norm_le_const_of_lt", "start": [424, 1], "end": [441, 51], "traced_tactics": [{"tactic": "rw [\u2190 _root_.abs_of_pos hR, \u2190 image_circleMap_Ioc] at hlt", "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\nhlt : \u2203 z, z \u2208 sphere c R \u2227 \u2016f z\u2016 < C\n\u22a2 \u2016\u222e (z : \u2102) in C(c, R), f z\u2016 < 2 * \u03c0 * R * C", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\nhlt : \u2203 z, z \u2208 circleMap c R '' Ioc 0 (2 * \u03c0) \u2227 \u2016f z\u2016 < C\n\u22a2 \u2016\u222e (z : \u2102) in C(c, R), f z\u2016 < 2 * \u03c0 * R * C"}, {"tactic": "rcases hlt with \u27e8_, \u27e8\u03b8\u2080, hmem, rfl\u27e9, hlt\u27e9", "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\nhlt : \u2203 z, z \u2208 circleMap c R '' Ioc 0 (2 * \u03c0) \u2227 \u2016f z\u2016 < C\n\u22a2 \u2016\u222e (z : \u2102) in C(c, R), f z\u2016 < 2 * \u03c0 * R * C", "state_after": "case intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u22a2 \u2016\u222e (z : \u2102) in C(c, R), f z\u2016 < 2 * \u03c0 * R * C"}, {"tactic": "simp only [norm_smul, deriv_circleMap, norm_eq_abs, map_mul, abs_I, mul_one,\n  abs_circleMap_zero, abs_of_pos hR]", "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u22a2 (\u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, \u2016deriv (circleMap c R) \u03b8 \u2022 f (circleMap c R \u03b8)\u2016) < \u222b (x : \u211d) in 0 ..2 * \u03c0, R * C", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u22a2 (\u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, R * \u2016f (circleMap c R \u03b8)\u2016) < \u222b (x : \u211d) in 0 ..2 * \u03c0, R * C"}, {"tactic": "refine' intervalIntegral.integral_lt_integral_of_continuousOn_of_le_of_exists_lt\n    Real.two_pi_pos _ continuousOn_const (fun \u03b8 _ => _) \u27e8\u03b8\u2080, Ioc_subset_Icc_self hmem, _\u27e9", "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u22a2 (\u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, R * \u2016f (circleMap c R \u03b8)\u2016) < \u222b (x : \u211d) in 0 ..2 * \u03c0, R * C", "state_after": "case refine'_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u22a2 ContinuousOn (fun \u03b8 => R * \u2016f (circleMap c R \u03b8)\u2016) (Icc 0 (2 * \u03c0))\n\ncase refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 Ioc 0 (2 * \u03c0)\n\u22a2 R * \u2016f (circleMap c R \u03b8)\u2016 \u2264 R * C\n\ncase refine'_3\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u22a2 R * \u2016f (circleMap c R \u03b8\u2080)\u2016 < R * C"}, {"tactic": "exact continuousOn_const.mul (hc.comp (continuous_circleMap _ _).continuousOn fun \u03b8 _ =>\n  circleMap_mem_sphere _ hR.le _).norm", "state_before": "case refine'_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u22a2 ContinuousOn (fun \u03b8 => R * \u2016f (circleMap c R \u03b8)\u2016) (Icc 0 (2 * \u03c0))", "state_after": "no goals"}, {"tactic": "exact mul_le_mul_of_nonneg_left (hf _ <| circleMap_mem_sphere _ hR.le _) hR.le", "state_before": "case refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 Ioc 0 (2 * \u03c0)\n\u22a2 R * \u2016f (circleMap c R \u03b8)\u2016 \u2264 R * C", "state_after": "no goals"}, {"tactic": "exact (mul_lt_mul_left hR).2 hlt", "state_before": "case refine'_3\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u22a2 R * \u2016f (circleMap c R \u03b8\u2080)\u2016 < R * C", "state_after": "no goals"}, {"tactic": "simp [mul_assoc]", "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u22a2 (\u222b (x : \u211d) in 0 ..2 * \u03c0, R * C) = 2 * \u03c0 * R * C", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u22a2 R * (2 * (\u03c0 * C)) = 2 * (\u03c0 * (R * C))"}, {"tactic": "ring", "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\nR C : \u211d\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 \u2016f z\u2016 \u2264 C\n\u03b8\u2080 : \u211d\nhmem : \u03b8\u2080 \u2208 Ioc 0 (2 * \u03c0)\nhlt : \u2016f (circleMap c R \u03b8\u2080)\u2016 < C\n\u22a2 R * (2 * (\u03c0 * C)) = 2 * (\u03c0 * (R * C))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/RelClasses.lean", "full_name": "ssubset_of_subset_of_ne", "start": [760, 1], "end": [761, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.sInter_union_sInter", "start": [1348, 1], "end": [1350, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.coeff_zero_one", "start": [670, 1], "end": [671, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/HahnSeries.lean", "full_name": "HahnSeries.single_eq_zero_iff", "start": [200, 1], "end": [204, 44], "traced_tactics": [{"tactic": "constructor", "state_before": "\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : Zero R\na\u271d b : \u0393\nr\u271d : R\na : \u0393\nr : R\n\u22a2 \u2191(single a) r = 0 \u2194 r = 0", "state_after": "case mp\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : Zero R\na\u271d b : \u0393\nr\u271d : R\na : \u0393\nr : R\n\u22a2 \u2191(single a) r = 0 \u2192 r = 0\n\ncase mpr\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : Zero R\na\u271d b : \u0393\nr\u271d : R\na : \u0393\nr : R\n\u22a2 r = 0 \u2192 \u2191(single a) r = 0"}, {"tactic": "contrapose!", "state_before": "case mp\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : Zero R\na\u271d b : \u0393\nr\u271d : R\na : \u0393\nr : R\n\u22a2 \u2191(single a) r = 0 \u2192 r = 0", "state_after": "case mp\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : Zero R\na\u271d b : \u0393\nr\u271d : R\na : \u0393\nr : R\n\u22a2 r \u2260 0 \u2192 \u2191(single a) r \u2260 0"}, {"tactic": "exact single_ne_zero", "state_before": "case mp\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : Zero R\na\u271d b : \u0393\nr\u271d : R\na : \u0393\nr : R\n\u22a2 r \u2260 0 \u2192 \u2191(single a) r \u2260 0", "state_after": "no goals"}, {"tactic": "simp (config := { contextual := true })", "state_before": "case mpr\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : PartialOrder \u0393\ninst\u271d : Zero R\na\u271d b : \u0393\nr\u271d : R\na : \u0393\nr : R\n\u22a2 r = 0 \u2192 \u2191(single a) r = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Whiskering.lean", "full_name": "CategoryTheory.whiskerRight_id'", "start": [138, 1], "end": [139, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Complex/Basic.lean", "full_name": "Complex.abs_im_div_abs_le_one", "start": [1100, 1], "end": [1103, 75], "traced_tactics": [{"tactic": "simp [hz, zero_le_one]", "state_before": "z : \u2102\nhz : z = 0\n\u22a2 Abs.abs (z.im / \u2191abs z) \u2264 1", "state_after": "no goals"}, {"tactic": "simp_rw [_root_.abs_div, abs_abs,\ndiv_le_iff (AbsoluteValue.pos Complex.abs hz), one_mul, abs_im_le_abs]", "state_before": "z : \u2102\nhz : \u00acz = 0\n\u22a2 Abs.abs (z.im / \u2191abs z) \u2264 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Bicategory/Coherence.lean", "full_name": "CategoryTheory.FreeBicategory.normalize_naturality", "start": [164, 1], "end": [186, 14], "traced_tactics": [{"tactic": "rcases \u03b7 with \u27e8\u03b7'\u27e9", "state_before": "B : Type u\ninst\u271d : Quiver B\na b c : B\np : Path a b\nf g : Hom b c\n\u03b7 : f \u27f6 g\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 \u03b7 \u226b (normalizeIso p g).hom =\n    (normalizeIso p f).hom \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f } = { as := normalizeAux p g }))", "state_after": "case mk\nB : Type u\ninst\u271d : Quiver B\na b c : B\np : Path a b\nf g : Hom b c\n\u03b7 : f \u27f6 g\n\u03b7' : Hom\u2082 f g\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g).hom =\n    (normalizeIso p f).hom \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f } = { as := normalizeAux p g }))"}, {"tactic": "clear \u03b7", "state_before": "case mk\nB : Type u\ninst\u271d : Quiver B\na b c : B\np : Path a b\nf g : Hom b c\n\u03b7 : f \u27f6 g\n\u03b7' : Hom\u2082 f g\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g).hom =\n    (normalizeIso p f).hom \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f } = { as := normalizeAux p g }))", "state_after": "case mk\nB : Type u\ninst\u271d : Quiver B\na b c : B\np : Path a b\nf g : Hom b c\n\u03b7' : Hom\u2082 f g\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g).hom =\n    (normalizeIso p f).hom \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f } = { as := normalizeAux p g }))"}, {"tactic": "induction \u03b7' with\n| id => simp\n| vcomp \u03b7 \u03b8 ihf ihg =>\n  simp only [mk_vcomp, Bicategory.whiskerLeft_comp]\n  slice_lhs 2 3 => rw [ihg]\n  slice_lhs 1 2 => rw [ihf]\n  simp\n| whisker_left _ _ ih =>\n  dsimp\n  rw [associator_inv_naturality_right_assoc, whisker_exchange_assoc, ih]\n  simp\n| whisker_right h \u03b7' ih =>\n  dsimp\n  rw [associator_inv_naturality_middle_assoc, \u2190 comp_whiskerRight_assoc, ih, comp_whiskerRight]\n  have := dcongr_arg (fun x => (normalizeIso x h).hom) (normalizeAux_congr p (Quot.mk _ \u03b7'))\n  dsimp at this; simp [this]\n| _ => simp", "state_before": "case mk\nB : Type u\ninst\u271d : Quiver B\na b c : B\np : Path a b\nf g : Hom b c\n\u03b7' : Hom\u2082 f g\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g).hom =\n    (normalizeIso p f).hom \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f } = { as := normalizeAux p g }))", "state_after": "no goals"}, {"tactic": "simp", "state_before": "case mk.id\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.id f\u271d) \u226b (normalizeIso p f\u271d).hom =\n    (normalizeIso p f\u271d).hom \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p f\u271d }))", "state_after": "no goals"}, {"tactic": "simp only [mk_vcomp, Bicategory.whiskerLeft_comp]", "state_before": "case mk.vcomp\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.vcomp \u03b7 \u03b8) \u226b (normalizeIso p h\u271d).hom =\n    (normalizeIso p f\u271d).hom \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p h\u271d }))", "state_after": "case mk.vcomp\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n\u22a2 ((\u2191(preinclusion B)).map { as := p } \u25c1 Hom\u2082.mk \u03b7 \u226b (\u2191(preinclusion B)).map { as := p } \u25c1 Hom\u2082.mk \u03b8) \u226b\n      (normalizeIso p h\u271d).hom =\n    (normalizeIso p f\u271d).hom \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p h\u271d }))"}, {"tactic": "slice_lhs 2 3 => rw [ihg]", "state_before": "case mk.vcomp\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n\u22a2 ((\u2191(preinclusion B)).map { as := p } \u25c1 Hom\u2082.mk \u03b7 \u226b (\u2191(preinclusion B)).map { as := p } \u25c1 Hom\u2082.mk \u03b8) \u226b\n      (normalizeIso p h\u271d).hom =\n    (normalizeIso p f\u271d).hom \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p h\u271d }))", "state_after": "case mk.vcomp\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Hom\u2082.mk \u03b7 \u226b\n      (normalizeIso p g\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d })) =\n    (normalizeIso p f\u271d).hom \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p h\u271d }))"}, {"tactic": "slice_lhs 1 2 => rw [ihf]", "state_before": "case mk.vcomp\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Hom\u2082.mk \u03b7 \u226b\n      (normalizeIso p g\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d })) =\n    (normalizeIso p f\u271d).hom \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p h\u271d }))", "state_after": "case mk.vcomp\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n\u22a2 ((normalizeIso p f\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))) \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d })) =\n    (normalizeIso p f\u271d).hom \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p h\u271d }))"}, {"tactic": "simp", "state_before": "case mk.vcomp\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\nihg :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n\u22a2 ((normalizeIso p f\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))) \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d })) =\n    (normalizeIso p f\u271d).hom \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p h\u271d }))", "state_after": "no goals"}, {"tactic": "dsimp", "state_before": "case mk.whisker_left\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d h\u271d : b\u271d \u27f6 c\u271d\n\u03b7\u271d : Hom\u2082 g\u271d h\u271d\nih :\n  \u2200 (p : Path a b\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7\u271d \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.whisker_left f\u271d \u03b7\u271d) \u226b (normalizeIso p (f\u271d \u226b h\u271d)).hom =\n    (normalizeIso p (f\u271d \u226b g\u271d)).hom \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B)\n        (eqToHom (_ : { as := normalizeAux p (f\u271d \u226b g\u271d) } = { as := normalizeAux p (f\u271d \u226b h\u271d) }))", "state_after": "case mk.whisker_left\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d h\u271d : b\u271d \u27f6 c\u271d\n\u03b7\u271d : Hom\u2082 g\u271d h\u271d\nih :\n  \u2200 (p : Path a b\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7\u271d \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 f\u271d \u25c1 Hom\u2082.mk \u03b7\u271d \u226b\n      (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h\u271d).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 h\u271d \u226b (normalizeIso (normalizeAux p f\u271d) h\u271d).hom =\n    ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d g\u271d).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 g\u271d \u226b (normalizeIso (normalizeAux p f\u271d) g\u271d).hom) \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B)\n        (eqToHom (_ : { as := normalizeAux (normalizeAux p f\u271d) g\u271d } = { as := normalizeAux (normalizeAux p f\u271d) h\u271d }))"}, {"tactic": "rw [associator_inv_naturality_right_assoc, whisker_exchange_assoc, ih]", "state_before": "case mk.whisker_left\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d h\u271d : b\u271d \u27f6 c\u271d\n\u03b7\u271d : Hom\u2082 g\u271d h\u271d\nih :\n  \u2200 (p : Path a b\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7\u271d \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 f\u271d \u25c1 Hom\u2082.mk \u03b7\u271d \u226b\n      (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h\u271d).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 h\u271d \u226b (normalizeIso (normalizeAux p f\u271d) h\u271d).hom =\n    ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d g\u271d).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 g\u271d \u226b (normalizeIso (normalizeAux p f\u271d) g\u271d).hom) \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B)\n        (eqToHom (_ : { as := normalizeAux (normalizeAux p f\u271d) g\u271d } = { as := normalizeAux (normalizeAux p f\u271d) h\u271d }))", "state_after": "case mk.whisker_left\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d h\u271d : b\u271d \u27f6 c\u271d\n\u03b7\u271d : Hom\u2082 g\u271d h\u271d\nih :\n  \u2200 (p : Path a b\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7\u271d \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n\u22a2 (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d g\u271d).inv \u226b\n      (normalizeIso p f\u271d).hom \u25b7 g\u271d \u226b\n        (normalizeIso (normalizeAux p f\u271d) g\u271d).hom \u226b\n          PrelaxFunctor.map\u2082 (preinclusion B)\n            (eqToHom\n              (_ : { as := normalizeAux (normalizeAux p f\u271d) g\u271d } = { as := normalizeAux (normalizeAux p f\u271d) h\u271d })) =\n    ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d g\u271d).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 g\u271d \u226b (normalizeIso (normalizeAux p f\u271d) g\u271d).hom) \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B)\n        (eqToHom (_ : { as := normalizeAux (normalizeAux p f\u271d) g\u271d } = { as := normalizeAux (normalizeAux p f\u271d) h\u271d }))"}, {"tactic": "simp", "state_before": "case mk.whisker_left\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d h\u271d : b\u271d \u27f6 c\u271d\n\u03b7\u271d : Hom\u2082 g\u271d h\u271d\nih :\n  \u2200 (p : Path a b\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7\u271d \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p g\u271d } = { as := normalizeAux p h\u271d }))\np : Path a a\u271d\n\u22a2 (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d g\u271d).inv \u226b\n      (normalizeIso p f\u271d).hom \u25b7 g\u271d \u226b\n        (normalizeIso (normalizeAux p f\u271d) g\u271d).hom \u226b\n          PrelaxFunctor.map\u2082 (preinclusion B)\n            (eqToHom\n              (_ : { as := normalizeAux (normalizeAux p f\u271d) g\u271d } = { as := normalizeAux (normalizeAux p f\u271d) h\u271d })) =\n    ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d g\u271d).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 g\u271d \u226b (normalizeIso (normalizeAux p f\u271d) g\u271d).hom) \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B)\n        (eqToHom (_ : { as := normalizeAux (normalizeAux p f\u271d) g\u271d } = { as := normalizeAux (normalizeAux p f\u271d) h\u271d }))", "state_after": "no goals"}, {"tactic": "dsimp", "state_before": "case mk.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh : b\u271d \u27f6 c\u271d\n\u03b7' : Hom\u2082 f\u271d g\u271d\nih :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.whisker_right h \u03b7') \u226b (normalizeIso p (Hom.comp g\u271d h)).hom =\n    (normalizeIso p (Hom.comp f\u271d h)).hom \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B)\n        (eqToHom (_ : { as := normalizeAux p (Hom.comp f\u271d h) } = { as := normalizeAux p (Hom.comp g\u271d h) }))", "state_after": "case mk.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh : b\u271d \u27f6 c\u271d\n\u03b7' : Hom\u2082 f\u271d g\u271d\nih :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Hom\u2082.mk \u03b7' \u25b7 h \u226b\n      (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) g\u271d h).inv \u226b\n        (normalizeIso p g\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p g\u271d) h).hom =\n    ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p f\u271d) h).hom) \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B)\n        (eqToHom (_ : { as := normalizeAux (normalizeAux p f\u271d) h } = { as := normalizeAux (normalizeAux p g\u271d) h }))"}, {"tactic": "rw [associator_inv_naturality_middle_assoc, \u2190 comp_whiskerRight_assoc, ih, comp_whiskerRight]", "state_before": "case mk.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh : b\u271d \u27f6 c\u271d\n\u03b7' : Hom\u2082 f\u271d g\u271d\nih :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Hom\u2082.mk \u03b7' \u25b7 h \u226b\n      (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) g\u271d h).inv \u226b\n        (normalizeIso p g\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p g\u271d) h).hom =\n    ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p f\u271d) h).hom) \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B)\n        (eqToHom (_ : { as := normalizeAux (normalizeAux p f\u271d) h } = { as := normalizeAux (normalizeAux p g\u271d) h }))", "state_after": "case mk.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh : b\u271d \u27f6 c\u271d\n\u03b7' : Hom\u2082 f\u271d g\u271d\nih :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\np : Path a a\u271d\n\u22a2 (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n      ((normalizeIso p f\u271d).hom \u25b7 h \u226b\n          PrelaxFunctor.map\u2082 (preinclusion B)\n              (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d })) \u25b7\n            h) \u226b\n        (normalizeIso (normalizeAux p g\u271d) h).hom =\n    ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p f\u271d) h).hom) \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B)\n        (eqToHom (_ : { as := normalizeAux (normalizeAux p f\u271d) h } = { as := normalizeAux (normalizeAux p g\u271d) h }))"}, {"tactic": "have := dcongr_arg (fun x => (normalizeIso x h).hom) (normalizeAux_congr p (Quot.mk _ \u03b7'))", "state_before": "case mk.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh : b\u271d \u27f6 c\u271d\n\u03b7' : Hom\u2082 f\u271d g\u271d\nih :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\np : Path a a\u271d\n\u22a2 (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n      ((normalizeIso p f\u271d).hom \u25b7 h \u226b\n          PrelaxFunctor.map\u2082 (preinclusion B)\n              (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d })) \u25b7\n            h) \u226b\n        (normalizeIso (normalizeAux p g\u271d) h).hom =\n    ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p f\u271d) h).hom) \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B)\n        (eqToHom (_ : { as := normalizeAux (normalizeAux p f\u271d) h } = { as := normalizeAux (normalizeAux p g\u271d) h }))", "state_after": "case mk.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh : b\u271d \u27f6 c\u271d\n\u03b7' : Hom\u2082 f\u271d g\u271d\nih :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\np : Path a a\u271d\nthis :\n  (normalizeIso (normalizeAux p f\u271d) h).hom =\n    eqToHom\n        (_ :\n          (\u2191(preinclusion B)).map { as := normalizeAux p f\u271d } \u226b h =\n            (\u2191(preinclusion B)).map { as := normalizeAux p g\u271d } \u226b h) \u226b\n      (normalizeIso (normalizeAux p g\u271d) h).hom \u226b\n        eqToHom\n          (_ :\n            (\u2191(preinclusion B)).map { as := normalizeAux (normalizeAux p g\u271d) h } =\n              (\u2191(preinclusion B)).map { as := normalizeAux (normalizeAux p f\u271d) h })\n\u22a2 (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n      ((normalizeIso p f\u271d).hom \u25b7 h \u226b\n          PrelaxFunctor.map\u2082 (preinclusion B)\n              (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d })) \u25b7\n            h) \u226b\n        (normalizeIso (normalizeAux p g\u271d) h).hom =\n    ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p f\u271d) h).hom) \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B)\n        (eqToHom (_ : { as := normalizeAux (normalizeAux p f\u271d) h } = { as := normalizeAux (normalizeAux p g\u271d) h }))"}, {"tactic": "dsimp at this", "state_before": "case mk.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh : b\u271d \u27f6 c\u271d\n\u03b7' : Hom\u2082 f\u271d g\u271d\nih :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\np : Path a a\u271d\nthis :\n  (normalizeIso (normalizeAux p f\u271d) h).hom =\n    eqToHom\n        (_ :\n          (\u2191(preinclusion B)).map { as := normalizeAux p f\u271d } \u226b h =\n            (\u2191(preinclusion B)).map { as := normalizeAux p g\u271d } \u226b h) \u226b\n      (normalizeIso (normalizeAux p g\u271d) h).hom \u226b\n        eqToHom\n          (_ :\n            (\u2191(preinclusion B)).map { as := normalizeAux (normalizeAux p g\u271d) h } =\n              (\u2191(preinclusion B)).map { as := normalizeAux (normalizeAux p f\u271d) h })\n\u22a2 (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n      ((normalizeIso p f\u271d).hom \u25b7 h \u226b\n          PrelaxFunctor.map\u2082 (preinclusion B)\n              (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d })) \u25b7\n            h) \u226b\n        (normalizeIso (normalizeAux p g\u271d) h).hom =\n    ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p f\u271d) h).hom) \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B)\n        (eqToHom (_ : { as := normalizeAux (normalizeAux p f\u271d) h } = { as := normalizeAux (normalizeAux p g\u271d) h }))", "state_after": "case mk.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh : b\u271d \u27f6 c\u271d\n\u03b7' : Hom\u2082 f\u271d g\u271d\nih :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\np : Path a a\u271d\nthis :\n  (normalizeIso (normalizeAux p f\u271d) h).hom =\n    eqToHom\n        (_ :\n          (\u2191(preinclusion B)).map { as := normalizeAux p f\u271d } \u226b h =\n            (\u2191(preinclusion B)).map { as := normalizeAux p g\u271d } \u226b h) \u226b\n      (normalizeIso (normalizeAux p g\u271d) h).hom \u226b\n        eqToHom\n          (_ :\n            (\u2191(preinclusion B)).map { as := normalizeAux (normalizeAux p g\u271d) h } =\n              (\u2191(preinclusion B)).map { as := normalizeAux (normalizeAux p f\u271d) h })\n\u22a2 (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n      ((normalizeIso p f\u271d).hom \u25b7 h \u226b\n          PrelaxFunctor.map\u2082 (preinclusion B)\n              (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d })) \u25b7\n            h) \u226b\n        (normalizeIso (normalizeAux p g\u271d) h).hom =\n    ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p f\u271d) h).hom) \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B)\n        (eqToHom (_ : { as := normalizeAux (normalizeAux p f\u271d) h } = { as := normalizeAux (normalizeAux p g\u271d) h }))"}, {"tactic": "simp [this]", "state_before": "case mk.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh : b\u271d \u27f6 c\u271d\n\u03b7' : Hom\u2082 f\u271d g\u271d\nih :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b\n        PrelaxFunctor.map\u2082 (preinclusion B) (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d }))\np : Path a a\u271d\nthis :\n  (normalizeIso (normalizeAux p f\u271d) h).hom =\n    eqToHom\n        (_ :\n          (\u2191(preinclusion B)).map { as := normalizeAux p f\u271d } \u226b h =\n            (\u2191(preinclusion B)).map { as := normalizeAux p g\u271d } \u226b h) \u226b\n      (normalizeIso (normalizeAux p g\u271d) h).hom \u226b\n        eqToHom\n          (_ :\n            (\u2191(preinclusion B)).map { as := normalizeAux (normalizeAux p g\u271d) h } =\n              (\u2191(preinclusion B)).map { as := normalizeAux (normalizeAux p f\u271d) h })\n\u22a2 (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n      ((normalizeIso p f\u271d).hom \u25b7 h \u226b\n          PrelaxFunctor.map\u2082 (preinclusion B)\n              (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p g\u271d })) \u25b7\n            h) \u226b\n        (normalizeIso (normalizeAux p g\u271d) h).hom =\n    ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p f\u271d) h).hom) \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B)\n        (eqToHom (_ : { as := normalizeAux (normalizeAux p f\u271d) h } = { as := normalizeAux (normalizeAux p g\u271d) h }))", "state_after": "no goals"}, {"tactic": "simp", "state_before": "case mk.left_unitor_inv\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.left_unitor_inv f\u271d) \u226b (normalizeIso p (\ud835\udfd9 a\u271d \u226b f\u271d)).hom =\n    (normalizeIso p f\u271d).hom \u226b\n      PrelaxFunctor.map\u2082 (preinclusion B)\n        (eqToHom (_ : { as := normalizeAux p f\u271d } = { as := normalizeAux p (\ud835\udfd9 a\u271d \u226b f\u271d) }))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Bounds/Basic.lean", "full_name": "isGreatest_Ioc", "start": [711, 1], "end": [712, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "full_name": "Matrix.det_zero", "start": [95, 1], "end": [96, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Artinian.lean", "full_name": "isArtinian_iff_wellFounded", "start": [160, 1], "end": [162, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Regular/SMul.lean", "full_name": "Units.isSMulRegular", "start": [251, 1], "end": [252, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/QuadraticForm/Prod.lean", "full_name": "QuadraticForm.nonneg_pi_iff", "start": [168, 1], "end": [180, 49], "traced_tactics": [{"tactic": "simp_rw [pi, sum_apply, comp_apply, LinearMap.proj_apply]", "state_before": "\u03b9 : Type u_1\nR\u271d : Type ?u.75510\nM\u2081 : Type ?u.75513\nM\u2082 : Type ?u.75516\nN\u2081 : Type ?u.75519\nN\u2082 : Type ?u.75522\nM\u1d62 : \u03b9 \u2192 Type u_3\nN\u1d62 : \u03b9 \u2192 Type ?u.75532\ninst\u271d\u00b9\u2075 : Semiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid N\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\u2082\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2081\ninst\u271d\u2079 : Module R\u271d M\u2082\ninst\u271d\u2078 : Module R\u271d N\u2081\ninst\u271d\u2077 : Module R\u271d N\u2082\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i)\ninst\u271d\u00b2 : Fintype \u03b9\nR : Type u_2\ninst\u271d\u00b9 : OrderedRing R\ninst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i)\nQ : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i)\n\u22a2 (\u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2191(pi Q) x) \u2194 \u2200 (i : \u03b9) (x : M\u1d62 i), 0 \u2264 \u2191(Q i) x", "state_after": "\u03b9 : Type u_1\nR\u271d : Type ?u.75510\nM\u2081 : Type ?u.75513\nM\u2082 : Type ?u.75516\nN\u2081 : Type ?u.75519\nN\u2082 : Type ?u.75522\nM\u1d62 : \u03b9 \u2192 Type u_3\nN\u1d62 : \u03b9 \u2192 Type ?u.75532\ninst\u271d\u00b9\u2075 : Semiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid N\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\u2082\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2081\ninst\u271d\u2079 : Module R\u271d M\u2082\ninst\u271d\u2078 : Module R\u271d N\u2081\ninst\u271d\u2077 : Module R\u271d N\u2082\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i)\ninst\u271d\u00b2 : Fintype \u03b9\nR : Type u_2\ninst\u271d\u00b9 : OrderedRing R\ninst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i)\nQ : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i)\n\u22a2 (\u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1)) \u2194 \u2200 (i : \u03b9) (x : M\u1d62 i), 0 \u2264 \u2191(Q i) x"}, {"tactic": "dsimp only", "state_before": "\u03b9 : Type u_1\nR\u271d : Type ?u.75510\nM\u2081 : Type ?u.75513\nM\u2082 : Type ?u.75516\nN\u2081 : Type ?u.75519\nN\u2082 : Type ?u.75522\nM\u1d62 : \u03b9 \u2192 Type u_3\nN\u1d62 : \u03b9 \u2192 Type ?u.75532\ninst\u271d\u00b9\u2075 : Semiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid N\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\u2082\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2081\ninst\u271d\u2079 : Module R\u271d M\u2082\ninst\u271d\u2078 : Module R\u271d N\u2081\ninst\u271d\u2077 : Module R\u271d N\u2082\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i)\ninst\u271d\u00b2 : Fintype \u03b9\nR : Type u_2\ninst\u271d\u00b9 : OrderedRing R\ninst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i)\nQ : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i)\n\u22a2 (\u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1)) \u2194 \u2200 (i : \u03b9) (x : M\u1d62 i), 0 \u2264 \u2191(Q i) x", "state_after": "\u03b9 : Type u_1\nR\u271d : Type ?u.75510\nM\u2081 : Type ?u.75513\nM\u2082 : Type ?u.75516\nN\u2081 : Type ?u.75519\nN\u2082 : Type ?u.75522\nM\u1d62 : \u03b9 \u2192 Type u_3\nN\u1d62 : \u03b9 \u2192 Type ?u.75532\ninst\u271d\u00b9\u2075 : Semiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid N\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\u2082\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2081\ninst\u271d\u2079 : Module R\u271d M\u2082\ninst\u271d\u2078 : Module R\u271d N\u2081\ninst\u271d\u2077 : Module R\u271d N\u2082\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i)\ninst\u271d\u00b2 : Fintype \u03b9\nR : Type u_2\ninst\u271d\u00b9 : OrderedRing R\ninst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i)\nQ : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i)\n\u22a2 (\u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1)) \u2194 \u2200 (i : \u03b9) (x : M\u1d62 i), 0 \u2264 \u2191(Q i) x"}, {"tactic": "constructor", "state_before": "\u03b9 : Type u_1\nR\u271d : Type ?u.75510\nM\u2081 : Type ?u.75513\nM\u2082 : Type ?u.75516\nN\u2081 : Type ?u.75519\nN\u2082 : Type ?u.75522\nM\u1d62 : \u03b9 \u2192 Type u_3\nN\u1d62 : \u03b9 \u2192 Type ?u.75532\ninst\u271d\u00b9\u2075 : Semiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid N\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\u2082\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2081\ninst\u271d\u2079 : Module R\u271d M\u2082\ninst\u271d\u2078 : Module R\u271d N\u2081\ninst\u271d\u2077 : Module R\u271d N\u2082\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i)\ninst\u271d\u00b2 : Fintype \u03b9\nR : Type u_2\ninst\u271d\u00b9 : OrderedRing R\ninst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i)\nQ : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i)\n\u22a2 (\u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1)) \u2194 \u2200 (i : \u03b9) (x : M\u1d62 i), 0 \u2264 \u2191(Q i) x", "state_after": "case mp\n\u03b9 : Type u_1\nR\u271d : Type ?u.75510\nM\u2081 : Type ?u.75513\nM\u2082 : Type ?u.75516\nN\u2081 : Type ?u.75519\nN\u2082 : Type ?u.75522\nM\u1d62 : \u03b9 \u2192 Type u_3\nN\u1d62 : \u03b9 \u2192 Type ?u.75532\ninst\u271d\u00b9\u2075 : Semiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid N\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\u2082\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2081\ninst\u271d\u2079 : Module R\u271d M\u2082\ninst\u271d\u2078 : Module R\u271d N\u2081\ninst\u271d\u2077 : Module R\u271d N\u2082\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i)\ninst\u271d\u00b2 : Fintype \u03b9\nR : Type u_2\ninst\u271d\u00b9 : OrderedRing R\ninst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i)\nQ : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i)\n\u22a2 (\u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1)) \u2192 \u2200 (i : \u03b9) (x : M\u1d62 i), 0 \u2264 \u2191(Q i) x\n\ncase mpr\n\u03b9 : Type u_1\nR\u271d : Type ?u.75510\nM\u2081 : Type ?u.75513\nM\u2082 : Type ?u.75516\nN\u2081 : Type ?u.75519\nN\u2082 : Type ?u.75522\nM\u1d62 : \u03b9 \u2192 Type u_3\nN\u1d62 : \u03b9 \u2192 Type ?u.75532\ninst\u271d\u00b9\u2075 : Semiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid N\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\u2082\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2081\ninst\u271d\u2079 : Module R\u271d M\u2082\ninst\u271d\u2078 : Module R\u271d N\u2081\ninst\u271d\u2077 : Module R\u271d N\u2082\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i)\ninst\u271d\u00b2 : Fintype \u03b9\nR : Type u_2\ninst\u271d\u00b9 : OrderedRing R\ninst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i)\nQ : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i)\n\u22a2 (\u2200 (i : \u03b9) (x : M\u1d62 i), 0 \u2264 \u2191(Q i) x) \u2192 \u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1)"}, {"tactic": "intro h i x", "state_before": "case mp\n\u03b9 : Type u_1\nR\u271d : Type ?u.75510\nM\u2081 : Type ?u.75513\nM\u2082 : Type ?u.75516\nN\u2081 : Type ?u.75519\nN\u2082 : Type ?u.75522\nM\u1d62 : \u03b9 \u2192 Type u_3\nN\u1d62 : \u03b9 \u2192 Type ?u.75532\ninst\u271d\u00b9\u2075 : Semiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid N\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\u2082\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2081\ninst\u271d\u2079 : Module R\u271d M\u2082\ninst\u271d\u2078 : Module R\u271d N\u2081\ninst\u271d\u2077 : Module R\u271d N\u2082\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i)\ninst\u271d\u00b2 : Fintype \u03b9\nR : Type u_2\ninst\u271d\u00b9 : OrderedRing R\ninst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i)\nQ : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i)\n\u22a2 (\u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1)) \u2192 \u2200 (i : \u03b9) (x : M\u1d62 i), 0 \u2264 \u2191(Q i) x", "state_after": "case mp\n\u03b9 : Type u_1\nR\u271d : Type ?u.75510\nM\u2081 : Type ?u.75513\nM\u2082 : Type ?u.75516\nN\u2081 : Type ?u.75519\nN\u2082 : Type ?u.75522\nM\u1d62 : \u03b9 \u2192 Type u_3\nN\u1d62 : \u03b9 \u2192 Type ?u.75532\ninst\u271d\u00b9\u2075 : Semiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid N\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\u2082\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2081\ninst\u271d\u2079 : Module R\u271d M\u2082\ninst\u271d\u2078 : Module R\u271d N\u2081\ninst\u271d\u2077 : Module R\u271d N\u2082\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i)\ninst\u271d\u00b2 : Fintype \u03b9\nR : Type u_2\ninst\u271d\u00b9 : OrderedRing R\ninst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i)\nQ : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i)\nh : \u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1)\ni : \u03b9\nx : M\u1d62 i\n\u22a2 0 \u2264 \u2191(Q i) x"}, {"tactic": "classical\nconvert h (Pi.single i x) using 1\nrw [Finset.sum_eq_single_of_mem i (Finset.mem_univ _) fun j _ hji => ?_, Pi.single_eq_same]\nrw [Pi.single_eq_of_ne hji, map_zero]", "state_before": "case mp\n\u03b9 : Type u_1\nR\u271d : Type ?u.75510\nM\u2081 : Type ?u.75513\nM\u2082 : Type ?u.75516\nN\u2081 : Type ?u.75519\nN\u2082 : Type ?u.75522\nM\u1d62 : \u03b9 \u2192 Type u_3\nN\u1d62 : \u03b9 \u2192 Type ?u.75532\ninst\u271d\u00b9\u2075 : Semiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid N\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\u2082\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2081\ninst\u271d\u2079 : Module R\u271d M\u2082\ninst\u271d\u2078 : Module R\u271d N\u2081\ninst\u271d\u2077 : Module R\u271d N\u2082\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i)\ninst\u271d\u00b2 : Fintype \u03b9\nR : Type u_2\ninst\u271d\u00b9 : OrderedRing R\ninst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i)\nQ : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i)\nh : \u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1)\ni : \u03b9\nx : M\u1d62 i\n\u22a2 0 \u2264 \u2191(Q i) x", "state_after": "no goals"}, {"tactic": "convert h (Pi.single i x) using 1", "state_before": "case mp\n\u03b9 : Type u_1\nR\u271d : Type ?u.75510\nM\u2081 : Type ?u.75513\nM\u2082 : Type ?u.75516\nN\u2081 : Type ?u.75519\nN\u2082 : Type ?u.75522\nM\u1d62 : \u03b9 \u2192 Type u_3\nN\u1d62 : \u03b9 \u2192 Type ?u.75532\ninst\u271d\u00b9\u2075 : Semiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid N\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\u2082\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2081\ninst\u271d\u2079 : Module R\u271d M\u2082\ninst\u271d\u2078 : Module R\u271d N\u2081\ninst\u271d\u2077 : Module R\u271d N\u2082\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i)\ninst\u271d\u00b2 : Fintype \u03b9\nR : Type u_2\ninst\u271d\u00b9 : OrderedRing R\ninst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i)\nQ : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i)\nh : \u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1)\ni : \u03b9\nx : M\u1d62 i\n\u22a2 0 \u2264 \u2191(Q i) x", "state_after": "case h.e'_4\n\u03b9 : Type u_1\nR\u271d : Type ?u.75510\nM\u2081 : Type ?u.75513\nM\u2082 : Type ?u.75516\nN\u2081 : Type ?u.75519\nN\u2082 : Type ?u.75522\nM\u1d62 : \u03b9 \u2192 Type u_3\nN\u1d62 : \u03b9 \u2192 Type ?u.75532\ninst\u271d\u00b9\u2075 : Semiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid N\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\u2082\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2081\ninst\u271d\u2079 : Module R\u271d M\u2082\ninst\u271d\u2078 : Module R\u271d N\u2081\ninst\u271d\u2077 : Module R\u271d N\u2082\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i)\ninst\u271d\u00b2 : Fintype \u03b9\nR : Type u_2\ninst\u271d\u00b9 : OrderedRing R\ninst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i)\nQ : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i)\nh : \u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1)\ni : \u03b9\nx : M\u1d62 i\n\u22a2 \u2191(Q i) x = \u2211 x_1 : \u03b9, \u2191(Q x_1) (Pi.single i x x_1)"}, {"tactic": "rw [Finset.sum_eq_single_of_mem i (Finset.mem_univ _) fun j _ hji => ?_, Pi.single_eq_same]", "state_before": "case h.e'_4\n\u03b9 : Type u_1\nR\u271d : Type ?u.75510\nM\u2081 : Type ?u.75513\nM\u2082 : Type ?u.75516\nN\u2081 : Type ?u.75519\nN\u2082 : Type ?u.75522\nM\u1d62 : \u03b9 \u2192 Type u_3\nN\u1d62 : \u03b9 \u2192 Type ?u.75532\ninst\u271d\u00b9\u2075 : Semiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid N\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\u2082\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2081\ninst\u271d\u2079 : Module R\u271d M\u2082\ninst\u271d\u2078 : Module R\u271d N\u2081\ninst\u271d\u2077 : Module R\u271d N\u2082\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i)\ninst\u271d\u00b2 : Fintype \u03b9\nR : Type u_2\ninst\u271d\u00b9 : OrderedRing R\ninst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i)\nQ : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i)\nh : \u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1)\ni : \u03b9\nx : M\u1d62 i\n\u22a2 \u2191(Q i) x = \u2211 x_1 : \u03b9, \u2191(Q x_1) (Pi.single i x x_1)", "state_after": "\u03b9 : Type u_1\nR\u271d : Type ?u.75510\nM\u2081 : Type ?u.75513\nM\u2082 : Type ?u.75516\nN\u2081 : Type ?u.75519\nN\u2082 : Type ?u.75522\nM\u1d62 : \u03b9 \u2192 Type u_3\nN\u1d62 : \u03b9 \u2192 Type ?u.75532\ninst\u271d\u00b9\u2075 : Semiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid N\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\u2082\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2081\ninst\u271d\u2079 : Module R\u271d M\u2082\ninst\u271d\u2078 : Module R\u271d N\u2081\ninst\u271d\u2077 : Module R\u271d N\u2082\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i)\ninst\u271d\u00b2 : Fintype \u03b9\nR : Type u_2\ninst\u271d\u00b9 : OrderedRing R\ninst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i)\nQ : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i)\nh : \u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1)\ni : \u03b9\nx : M\u1d62 i\nj : \u03b9\nx\u271d : j \u2208 Finset.univ\nhji : j \u2260 i\n\u22a2 \u2191(Q j) (Pi.single i x j) = 0"}, {"tactic": "rw [Pi.single_eq_of_ne hji, map_zero]", "state_before": "\u03b9 : Type u_1\nR\u271d : Type ?u.75510\nM\u2081 : Type ?u.75513\nM\u2082 : Type ?u.75516\nN\u2081 : Type ?u.75519\nN\u2082 : Type ?u.75522\nM\u1d62 : \u03b9 \u2192 Type u_3\nN\u1d62 : \u03b9 \u2192 Type ?u.75532\ninst\u271d\u00b9\u2075 : Semiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid N\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\u2082\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2081\ninst\u271d\u2079 : Module R\u271d M\u2082\ninst\u271d\u2078 : Module R\u271d N\u2081\ninst\u271d\u2077 : Module R\u271d N\u2082\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i)\ninst\u271d\u00b2 : Fintype \u03b9\nR : Type u_2\ninst\u271d\u00b9 : OrderedRing R\ninst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i)\nQ : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i)\nh : \u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1)\ni : \u03b9\nx : M\u1d62 i\nj : \u03b9\nx\u271d : j \u2208 Finset.univ\nhji : j \u2260 i\n\u22a2 \u2191(Q j) (Pi.single i x j) = 0", "state_after": "no goals"}, {"tactic": "rintro h x", "state_before": "case mpr\n\u03b9 : Type u_1\nR\u271d : Type ?u.75510\nM\u2081 : Type ?u.75513\nM\u2082 : Type ?u.75516\nN\u2081 : Type ?u.75519\nN\u2082 : Type ?u.75522\nM\u1d62 : \u03b9 \u2192 Type u_3\nN\u1d62 : \u03b9 \u2192 Type ?u.75532\ninst\u271d\u00b9\u2075 : Semiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid N\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\u2082\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2081\ninst\u271d\u2079 : Module R\u271d M\u2082\ninst\u271d\u2078 : Module R\u271d N\u2081\ninst\u271d\u2077 : Module R\u271d N\u2082\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i)\ninst\u271d\u00b2 : Fintype \u03b9\nR : Type u_2\ninst\u271d\u00b9 : OrderedRing R\ninst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i)\nQ : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i)\n\u22a2 (\u2200 (i : \u03b9) (x : M\u1d62 i), 0 \u2264 \u2191(Q i) x) \u2192 \u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1)", "state_after": "case mpr\n\u03b9 : Type u_1\nR\u271d : Type ?u.75510\nM\u2081 : Type ?u.75513\nM\u2082 : Type ?u.75516\nN\u2081 : Type ?u.75519\nN\u2082 : Type ?u.75522\nM\u1d62 : \u03b9 \u2192 Type u_3\nN\u1d62 : \u03b9 \u2192 Type ?u.75532\ninst\u271d\u00b9\u2075 : Semiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid N\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\u2082\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2081\ninst\u271d\u2079 : Module R\u271d M\u2082\ninst\u271d\u2078 : Module R\u271d N\u2081\ninst\u271d\u2077 : Module R\u271d N\u2082\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i)\ninst\u271d\u00b2 : Fintype \u03b9\nR : Type u_2\ninst\u271d\u00b9 : OrderedRing R\ninst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i)\nQ : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i)\nh : \u2200 (i : \u03b9) (x : M\u1d62 i), 0 \u2264 \u2191(Q i) x\nx : (i : \u03b9) \u2192 M\u1d62 i\n\u22a2 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1)"}, {"tactic": "exact Finset.sum_nonneg fun i _ => h i (x i)", "state_before": "case mpr\n\u03b9 : Type u_1\nR\u271d : Type ?u.75510\nM\u2081 : Type ?u.75513\nM\u2082 : Type ?u.75516\nN\u2081 : Type ?u.75519\nN\u2082 : Type ?u.75522\nM\u1d62 : \u03b9 \u2192 Type u_3\nN\u1d62 : \u03b9 \u2192 Type ?u.75532\ninst\u271d\u00b9\u2075 : Semiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid N\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid N\u2082\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2081\ninst\u271d\u2079 : Module R\u271d M\u2082\ninst\u271d\u2078 : Module R\u271d N\u2081\ninst\u271d\u2077 : Module R\u271d N\u2082\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i)\ninst\u271d\u00b2 : Fintype \u03b9\nR : Type u_2\ninst\u271d\u00b9 : OrderedRing R\ninst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i)\nQ : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i)\nh : \u2200 (i : \u03b9) (x : M\u1d62 i), 0 \u2264 \u2191(Q i) x\nx : (i : \u03b9) \u2192 M\u1d62 i\n\u22a2 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Homeomorph.lean", "full_name": "Homeomorph.preimage_interior", "start": [351, 1], "end": [352, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Basic.lean", "full_name": "congr_refl_right", "start": [534, 1], "end": [535, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/TrivSqZeroExt.lean", "full_name": "TrivSqZeroExt.hasSum_expSeries_of_smul_comm", "start": [85, 1], "end": [93, 67], "traced_tactics": [{"tactic": "simpa only [inl_fst_add_inr_snd_eq] using\n  (hasSum_inl _ <| hasSum_fst_expSeries \ud835\udd5c x h).add\n    (hasSum_inr _ <| hasSum_snd_expSeries_of_smul_comm \ud835\udd5c x hx h)", "state_before": "\ud835\udd5c : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u00b9\u2076 : TopologicalSpace R\ninst\u271d\u00b9\u2075 : TopologicalSpace M\ninst\u271d\u00b9\u2074 : Field \ud835\udd5c\ninst\u271d\u00b9\u00b3 : CharZero \ud835\udd5c\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Algebra \ud835\udd5c R\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u2077 : SMulCommClass R R\u1d50\u1d52\u1d56 M\ninst\u271d\u2076 : Module \ud835\udd5c M\ninst\u271d\u2075 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2074 : IsScalarTower \ud835\udd5c R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b3 : TopologicalRing R\ninst\u271d\u00b2 : TopologicalAddGroup M\ninst\u271d\u00b9 : ContinuousSMul R M\ninst\u271d : ContinuousSMul R\u1d50\u1d52\u1d56 M\nx : tsze R M\nhx : MulOpposite.op (fst x) \u2022 snd x = fst x \u2022 snd x\ne : R\nh : HasSum (fun n => \u2191(expSeries \ud835\udd5c R n) fun x_1 => fst x) e\n\u22a2 HasSum (fun n => \u2191(expSeries \ud835\udd5c (tsze R M) n) fun x_1 => x) (inl e + inr (e \u2022 snd x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Dfinsupp/Order.lean", "full_name": "Dfinsupp.le_iff", "start": [170, 1], "end": [171, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Types.lean", "full_name": "CategoryTheory.types_comp", "start": [69, 1], "end": [70, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "full_name": "Subalgebra.coe_pow", "start": [418, 11], "end": [419, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/Hermitian.lean", "full_name": "Matrix.IsHermitian.submatrix", "start": [89, 1], "end": [90, 90], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/PartitionOfUnity.lean", "full_name": "BumpCovering.point_finite", "start": [243, 11], "end": [244, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/MvPolynomial/Expand.lean", "full_name": "MvPolynomial.expand_bind\u2081", "start": [76, 1], "end": [78, 46], "traced_tactics": [{"tactic": "rw [\u2190 AlgHom.comp_apply, expand_comp_bind\u2081]", "state_before": "\u03c3 : Type u_3\n\u03c4 : Type u_1\nR : Type u_2\nS : Type ?u.145830\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\np : \u2115\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\n\u22a2 \u2191(expand p) (\u2191(bind\u2081 f) \u03c6) = \u2191(bind\u2081 fun i => \u2191(expand p) (f i)) \u03c6", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Constructions.lean", "full_name": "closedEmbedding_subtype_val", "start": [1006, 1], "end": [1008, 62], "traced_tactics": [{"tactic": "rwa [Subtype.range_coe_subtype]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type ?u.154850\n\u03b4 : Type ?u.154853\n\u03b5 : Type ?u.154856\n\u03b6 : Type ?u.154859\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\np : \u03b1 \u2192 Prop\nh : IsClosed {a | p a}\n\u22a2 IsClosed (range Subtype.val)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Function/Basic.lean", "full_name": "Function.curry_apply", "start": [812, 1], "end": [813, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Monotone/Basic.lean", "full_name": "StrictAnti.isMax_of_apply", "start": [582, 1], "end": [585, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/MonoCoprod.lean", "full_name": "CategoryTheory.Limits.MonoCoprod.binaryCofan_inr", "start": [61, 1], "end": [67, 30], "traced_tactics": [{"tactic": "haveI hc' : IsColimit (BinaryCofan.mk c.inr c.inl) :=\n  BinaryCofan.IsColimit.mk _ (fun f\u2081 f\u2082 => hc.desc (BinaryCofan.mk f\u2082 f\u2081))\n    (by aesop_cat) (by aesop_cat)\n    (fun f\u2081 f\u2082 m h\u2081 h\u2082 => BinaryCofan.IsColimit.hom_ext hc (by aesop_cat) (by aesop_cat))", "state_before": "C : Type u_1\ninst\u271d\u00b9 : Category C\nA B : C\ninst\u271d : MonoCoprod C\nc : BinaryCofan A B\nhc : IsColimit c\n\u22a2 Mono (BinaryCofan.inr c)", "state_after": "C : Type u_1\ninst\u271d\u00b9 : Category C\nA B : C\ninst\u271d : MonoCoprod C\nc : BinaryCofan A B\nhc : IsColimit c\nhc' : IsColimit (BinaryCofan.mk (BinaryCofan.inr c) (BinaryCofan.inl c))\n\u22a2 Mono (BinaryCofan.inr c)"}, {"tactic": "exact binaryCofan_inl _ hc'", "state_before": "C : Type u_1\ninst\u271d\u00b9 : Category C\nA B : C\ninst\u271d : MonoCoprod C\nc : BinaryCofan A B\nhc : IsColimit c\nhc' : IsColimit (BinaryCofan.mk (BinaryCofan.inr c) (BinaryCofan.inl c))\n\u22a2 Mono (BinaryCofan.inr c)", "state_after": "no goals"}, {"tactic": "aesop_cat", "state_before": "C : Type u_1\ninst\u271d\u00b9 : Category C\nA B : C\ninst\u271d : MonoCoprod C\nc : BinaryCofan A B\nhc : IsColimit c\n\u22a2 \u2200 {T : C} (f : (pair A B).obj { as := WalkingPair.right } \u27f6 T) (g : (pair A B).obj { as := WalkingPair.left } \u27f6 T),\n    BinaryCofan.inl (BinaryCofan.mk (BinaryCofan.inr c) (BinaryCofan.inl c)) \u226b\n        (fun {T} f\u2081 f\u2082 => IsColimit.desc hc (BinaryCofan.mk f\u2082 f\u2081)) f g =\n      f", "state_after": "no goals"}, {"tactic": "aesop_cat", "state_before": "C : Type u_1\ninst\u271d\u00b9 : Category C\nA B : C\ninst\u271d : MonoCoprod C\nc : BinaryCofan A B\nhc : IsColimit c\n\u22a2 \u2200 {T : C} (f : (pair A B).obj { as := WalkingPair.right } \u27f6 T) (g : (pair A B).obj { as := WalkingPair.left } \u27f6 T),\n    BinaryCofan.inr (BinaryCofan.mk (BinaryCofan.inr c) (BinaryCofan.inl c)) \u226b\n        (fun {T} f\u2081 f\u2082 => IsColimit.desc hc (BinaryCofan.mk f\u2082 f\u2081)) f g =\n      g", "state_after": "no goals"}, {"tactic": "aesop_cat", "state_before": "C : Type u_1\ninst\u271d\u00b9 : Category C\nA B : C\ninst\u271d : MonoCoprod C\nc : BinaryCofan A B\nhc : IsColimit c\nT\u271d : C\nf\u2081 : (pair A B).obj { as := WalkingPair.right } \u27f6 T\u271d\nf\u2082 : (pair A B).obj { as := WalkingPair.left } \u27f6 T\u271d\nm : (BinaryCofan.mk (BinaryCofan.inr c) (BinaryCofan.inl c)).pt \u27f6 T\u271d\nh\u2081 : BinaryCofan.inl (BinaryCofan.mk (BinaryCofan.inr c) (BinaryCofan.inl c)) \u226b m = f\u2081\nh\u2082 : BinaryCofan.inr (BinaryCofan.mk (BinaryCofan.inr c) (BinaryCofan.inl c)) \u226b m = f\u2082\n\u22a2 BinaryCofan.inl c \u226b m = BinaryCofan.inl c \u226b (fun {T} f\u2081 f\u2082 => IsColimit.desc hc (BinaryCofan.mk f\u2082 f\u2081)) f\u2081 f\u2082", "state_after": "no goals"}, {"tactic": "aesop_cat", "state_before": "C : Type u_1\ninst\u271d\u00b9 : Category C\nA B : C\ninst\u271d : MonoCoprod C\nc : BinaryCofan A B\nhc : IsColimit c\nT\u271d : C\nf\u2081 : (pair A B).obj { as := WalkingPair.right } \u27f6 T\u271d\nf\u2082 : (pair A B).obj { as := WalkingPair.left } \u27f6 T\u271d\nm : (BinaryCofan.mk (BinaryCofan.inr c) (BinaryCofan.inl c)).pt \u27f6 T\u271d\nh\u2081 : BinaryCofan.inl (BinaryCofan.mk (BinaryCofan.inr c) (BinaryCofan.inl c)) \u226b m = f\u2081\nh\u2082 : BinaryCofan.inr (BinaryCofan.mk (BinaryCofan.inr c) (BinaryCofan.inl c)) \u226b m = f\u2082\n\u22a2 BinaryCofan.inr c \u226b m = BinaryCofan.inr c \u226b (fun {T} f\u2081 f\u2082 => IsColimit.desc hc (BinaryCofan.mk f\u2082 f\u2081)) f\u2081 f\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Instances/ENNReal.lean", "full_name": "ENNReal.tsum_mul_left", "start": [900, 11], "end": [907, 30], "traced_tactics": [{"tactic": "by_cases hf : \u2200 i, f i = 0", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.252239\n\u03b3 : Type ?u.252242\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\nf g : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 (\u2211' (i : \u03b1), a * f i) = a * \u2211' (i : \u03b1), f i", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.252239\n\u03b3 : Type ?u.252242\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\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b1), f i = 0\n\u22a2 (\u2211' (i : \u03b1), a * f i) = a * \u2211' (i : \u03b1), f i\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.252239\n\u03b3 : Type ?u.252242\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\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u00ac\u2200 (i : \u03b1), f i = 0\n\u22a2 (\u2211' (i : \u03b1), a * f i) = a * \u2211' (i : \u03b1), f i"}, {"tactic": "simp [hf]", "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.252239\n\u03b3 : Type ?u.252242\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\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (i : \u03b1), f i = 0\n\u22a2 (\u2211' (i : \u03b1), a * f i) = a * \u2211' (i : \u03b1), f i", "state_after": "no goals"}, {"tactic": "rw [\u2190 ENNReal.tsum_eq_zero] at hf", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.252239\n\u03b3 : Type ?u.252242\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\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u00ac\u2200 (i : \u03b1), f i = 0\n\u22a2 (\u2211' (i : \u03b1), a * f i) = a * \u2211' (i : \u03b1), f i", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.252239\n\u03b3 : Type ?u.252242\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\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u00ac(\u2211' (i : \u03b1), f i) = 0\n\u22a2 (\u2211' (i : \u03b1), a * f i) = a * \u2211' (i : \u03b1), f i"}, {"tactic": "have : Tendsto (fun s : Finset \u03b1 => \u2211 j in s, a * f j) atTop (\ud835\udcdd (a * \u2211' i, f i)) := by\n  simp only [\u2190 Finset.mul_sum]\n  exact ENNReal.Tendsto.const_mul ENNReal.summable.hasSum (Or.inl hf)", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.252239\n\u03b3 : Type ?u.252242\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\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u00ac(\u2211' (i : \u03b1), f i) = 0\n\u22a2 (\u2211' (i : \u03b1), a * f i) = a * \u2211' (i : \u03b1), f i", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.252239\n\u03b3 : Type ?u.252242\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\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u00ac(\u2211' (i : \u03b1), f i) = 0\nthis : Tendsto (fun s => \u2211 j in s, a * f j) atTop (\ud835\udcdd (a * \u2211' (i : \u03b1), f i))\n\u22a2 (\u2211' (i : \u03b1), a * f i) = a * \u2211' (i : \u03b1), f i"}, {"tactic": "exact HasSum.tsum_eq this", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.252239\n\u03b3 : Type ?u.252242\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\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u00ac(\u2211' (i : \u03b1), f i) = 0\nthis : Tendsto (fun s => \u2211 j in s, a * f j) atTop (\ud835\udcdd (a * \u2211' (i : \u03b1), f i))\n\u22a2 (\u2211' (i : \u03b1), a * f i) = a * \u2211' (i : \u03b1), f i", "state_after": "no goals"}, {"tactic": "simp only [\u2190 Finset.mul_sum]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.252239\n\u03b3 : Type ?u.252242\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\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u00ac(\u2211' (i : \u03b1), f i) = 0\n\u22a2 Tendsto (fun s => \u2211 j in s, a * f j) atTop (\ud835\udcdd (a * \u2211' (i : \u03b1), f i))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.252239\n\u03b3 : Type ?u.252242\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\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u00ac(\u2211' (i : \u03b1), f i) = 0\n\u22a2 Tendsto (fun s => a * \u2211 x in s, f x) atTop (\ud835\udcdd (a * \u2211' (x : \u03b1), f x))"}, {"tactic": "exact ENNReal.Tendsto.const_mul ENNReal.summable.hasSum (Or.inl hf)", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.252239\n\u03b3 : Type ?u.252242\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\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u00ac(\u2211' (i : \u03b1), f i) = 0\n\u22a2 Tendsto (fun s => a * \u2211 x in s, f x) atTop (\ud835\udcdd (a * \u2211' (x : \u03b1), f x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Conj.lean", "full_name": "CategoryTheory.Iso.homCongr_comp", "start": [57, 1], "end": [58, 84], "traced_tactics": [{"tactic": "simp", "state_before": "C : Type u\ninst\u271d : Category C\nX Y Z X\u2081 Y\u2081 Z\u2081 : C\n\u03b1 : X \u2245 X\u2081\n\u03b2 : Y \u2245 Y\u2081\n\u03b3 : Z \u2245 Z\u2081\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 \u2191(homCongr \u03b1 \u03b3) (f \u226b g) = \u2191(homCongr \u03b1 \u03b2) f \u226b \u2191(homCongr \u03b2 \u03b3) g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "full_name": "Polynomial.not_isUnit_X", "start": [1272, 1], "end": [1276, 9], "traced_tactics": [{"tactic": "conv at hgf => change g * monomial 1 1 = 1", "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Nontrivial R\np q : R[X]\nx\u271d : IsUnit X\ng : R[X]\n_hfg : \u2191(monomial 1) 1 * g = 1\nhgf : g * \u2191(monomial 1) 1 = 1\n\u22a2 0 = 1", "state_after": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Nontrivial R\np q : R[X]\nx\u271d : IsUnit X\ng : R[X]\n_hfg : \u2191(monomial 1) 1 * g = 1\nhgf : g * \u2191(monomial 1) 1 = 1\n\u22a2 0 = 1"}, {"tactic": "rw [\u2190 coeff_one_zero, \u2190 hgf]", "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Nontrivial R\np q : R[X]\nx\u271d : IsUnit X\ng : R[X]\n_hfg : \u2191(monomial 1) 1 * g = 1\nhgf : g * \u2191(monomial 1) 1 = 1\n\u22a2 0 = 1", "state_after": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Nontrivial R\np q : R[X]\nx\u271d : IsUnit X\ng : R[X]\n_hfg : \u2191(monomial 1) 1 * g = 1\nhgf : g * \u2191(monomial 1) 1 = 1\n\u22a2 0 = coeff (g * \u2191(monomial 1) 1) 0"}, {"tactic": "simp", "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Nontrivial R\np q : R[X]\nx\u271d : IsUnit X\ng : R[X]\n_hfg : \u2191(monomial 1) 1 * g = 1\nhgf : g * \u2191(monomial 1) 1 = 1\n\u22a2 0 = coeff (g * \u2191(monomial 1) 1) 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "mem_interior_iff_mem_nhds", "start": [897, 1], "end": [898, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Pointwise/BigOperators.lean", "full_name": "Set.multiset_prod_subset_multiset_prod", "start": [140, 1], "end": [144, 44], "traced_tactics": [{"tactic": "induction t using Quotient.inductionOn", "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type ?u.79804\nF : Type ?u.79807\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nt : Multiset \u03b9\nf\u2081 f\u2082 : \u03b9 \u2192 Set \u03b1\nhf : \u2200 (i : \u03b9), i \u2208 t \u2192 f\u2081 i \u2286 f\u2082 i\n\u22a2 Multiset.prod (Multiset.map f\u2081 t) \u2286 Multiset.prod (Multiset.map f\u2082 t)", "state_after": "case h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type ?u.79804\nF : Type ?u.79807\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf\u2081 f\u2082 : \u03b9 \u2192 Set \u03b1\na\u271d : List \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 Quotient.mk (List.isSetoid \u03b9) a\u271d \u2192 f\u2081 i \u2286 f\u2082 i\n\u22a2 Multiset.prod (Multiset.map f\u2081 (Quotient.mk (List.isSetoid \u03b9) a\u271d)) \u2286\n    Multiset.prod (Multiset.map f\u2082 (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.79804\nF : Type ?u.79807\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf\u2081 f\u2082 : \u03b9 \u2192 Set \u03b1\na\u271d : List \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 Quotient.mk (List.isSetoid \u03b9) a\u271d \u2192 f\u2081 i \u2286 f\u2082 i\n\u22a2 Multiset.prod (Multiset.map f\u2081 (Quotient.mk (List.isSetoid \u03b9) a\u271d)) \u2286\n    Multiset.prod (Multiset.map f\u2082 (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.79804\nF : Type ?u.79807\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf\u2081 f\u2082 : \u03b9 \u2192 Set \u03b1\na\u271d : List \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 Quotient.mk (List.isSetoid \u03b9) a\u271d \u2192 f\u2081 i \u2286 f\u2082 i\n\u22a2 List.prod (List.map f\u2081 a\u271d) \u2286 List.prod (List.map f\u2082 a\u271d)"}, {"tactic": "exact list_prod_subset_list_prod _ _ _ hf", "state_before": "case h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type ?u.79804\nF : Type ?u.79807\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf\u2081 f\u2082 : \u03b9 \u2192 Set \u03b1\na\u271d : List \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 Quotient.mk (List.isSetoid \u03b9) a\u271d \u2192 f\u2081 i \u2286 f\u2082 i\n\u22a2 List.prod (List.map f\u2081 a\u271d) \u2286 List.prod (List.map f\u2082 a\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Maps.lean", "full_name": "IsOpenMap.image_mem_nhds", "start": [358, 1], "end": [360, 92], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Interval.lean", "full_name": "NonemptyInterval.toProd_one", "start": [55, 1], "end": [56, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Data/Nat/Basic.lean", "full_name": "Nat.mul_eq", "start": [85, 9], "end": [85, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "Multiset.stronglyMeasurable_prod", "start": [575, 1], "end": [578, 76], "traced_tactics": [{"tactic": "simpa only [\u2190 Pi.multiset_prod_apply] using s.stronglyMeasurable_prod' hs", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.127112\n\u03b3 : Type ?u.127115\n\u03b9 : Type ?u.127118\ninst\u271d\u00b3 : Countable \u03b9\nf g : \u03b1 \u2192 \u03b2\nM : Type u_2\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : ContinuousMul M\nm : MeasurableSpace \u03b1\ns : Multiset (\u03b1 \u2192 M)\nhs : \u2200 (f : \u03b1 \u2192 M), f \u2208 s \u2192 StronglyMeasurable f\n\u22a2 StronglyMeasurable fun x => Multiset.prod (Multiset.map (fun f => f x) s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.nonempty_def", "start": [455, 1], "end": [456, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snorm'_const_smul", "start": [1554, 1], "end": [1560, 99], "traced_tactics": [{"tactic": "obtain rfl | hc := eq_or_ne c 0", "state_before": "\u03b1 : Type u_1\nE : Type ?u.5959548\nF : Type u_2\nG : Type ?u.5959554\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\nf : \u03b1 \u2192 F\nc : \ud835\udd5c\nhq_pos : 0 < q\n\u22a2 snorm' (c \u2022 f) q \u03bc = \u2016c\u2016\u208a \u2022 snorm' f q \u03bc", "state_after": "case inl\n\u03b1 : Type u_1\nE : Type ?u.5959548\nF : Type u_2\nG : Type ?u.5959554\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\n\u22a2 snorm' (0 \u2022 f) q \u03bc = \u20160\u2016\u208a \u2022 snorm' f q \u03bc\n\ncase inr\n\u03b1 : Type u_1\nE : Type ?u.5959548\nF : Type u_2\nG : Type ?u.5959554\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\nf : \u03b1 \u2192 F\nc : \ud835\udd5c\nhq_pos : 0 < q\nhc : c \u2260 0\n\u22a2 snorm' (c \u2022 f) q \u03bc = \u2016c\u2016\u208a \u2022 snorm' f q \u03bc"}, {"tactic": "refine' le_antisymm (snorm'_const_smul_le _ _ hq_pos) _", "state_before": "case inr\n\u03b1 : Type u_1\nE : Type ?u.5959548\nF : Type u_2\nG : Type ?u.5959554\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\nf : \u03b1 \u2192 F\nc : \ud835\udd5c\nhq_pos : 0 < q\nhc : c \u2260 0\n\u22a2 snorm' (c \u2022 f) q \u03bc = \u2016c\u2016\u208a \u2022 snorm' f q \u03bc", "state_after": "case inr\n\u03b1 : Type u_1\nE : Type ?u.5959548\nF : Type u_2\nG : Type ?u.5959554\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\nf : \u03b1 \u2192 F\nc : \ud835\udd5c\nhq_pos : 0 < q\nhc : c \u2260 0\n\u22a2 \u2016c\u2016\u208a \u2022 snorm' f q \u03bc \u2264 snorm' (c \u2022 f) q \u03bc"}, {"tactic": "have : snorm' _ q \u03bc \u2264 _ := snorm'_const_smul_le c\u207b\u00b9 (c \u2022 f) hq_pos", "state_before": "case inr\n\u03b1 : Type u_1\nE : Type ?u.5959548\nF : Type u_2\nG : Type ?u.5959554\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\nf : \u03b1 \u2192 F\nc : \ud835\udd5c\nhq_pos : 0 < q\nhc : c \u2260 0\n\u22a2 \u2016c\u2016\u208a \u2022 snorm' f q \u03bc \u2264 snorm' (c \u2022 f) q \u03bc", "state_after": "case inr\n\u03b1 : Type u_1\nE : Type ?u.5959548\nF : Type u_2\nG : Type ?u.5959554\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\nf : \u03b1 \u2192 F\nc : \ud835\udd5c\nhq_pos : 0 < q\nhc : c \u2260 0\nthis : snorm' (c\u207b\u00b9 \u2022 c \u2022 f) q \u03bc \u2264 \u2016c\u207b\u00b9\u2016\u208a \u2022 snorm' (c \u2022 f) q \u03bc\n\u22a2 \u2016c\u2016\u208a \u2022 snorm' f q \u03bc \u2264 snorm' (c \u2022 f) q \u03bc"}, {"tactic": "rwa [inv_smul_smul\u2080 hc, nnnorm_inv, ENNReal.le_inv_smul_iff (nnnorm_ne_zero_iff.mpr hc)] at this", "state_before": "case inr\n\u03b1 : Type u_1\nE : Type ?u.5959548\nF : Type u_2\nG : Type ?u.5959554\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\nf : \u03b1 \u2192 F\nc : \ud835\udd5c\nhq_pos : 0 < q\nhc : c \u2260 0\nthis : snorm' (c\u207b\u00b9 \u2022 c \u2022 f) q \u03bc \u2264 \u2016c\u207b\u00b9\u2016\u208a \u2022 snorm' (c \u2022 f) q \u03bc\n\u22a2 \u2016c\u2016\u208a \u2022 snorm' f q \u03bc \u2264 snorm' (c \u2022 f) q \u03bc", "state_after": "no goals"}, {"tactic": "simp [snorm', hq_pos]", "state_before": "case inl\n\u03b1 : Type u_1\nE : Type ?u.5959548\nF : Type u_2\nG : Type ?u.5959554\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b3 : MulActionWithZero \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c F\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\n\u22a2 snorm' (0 \u2022 f) q \u03bc = \u20160\u2016\u208a \u2022 snorm' f q \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/GroupAction/SubMulAction.lean", "full_name": "SubMulAction.copy_eq", "start": [151, 1], "end": [152, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "isOpen_setOf_eventually_nhds", "start": [1222, 1], "end": [1223, 51], "traced_tactics": [{"tactic": "simp only [\u2190 interior_setOf_eq, isOpen_interior]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\na : \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\np\u271d p\u2081 p\u2082 : \u03b1 \u2192 Prop\ninst\u271d : TopologicalSpace \u03b1\np : \u03b1 \u2192 Prop\n\u22a2 IsOpen {x | \u2200\u1da0 (y : \u03b1) in \ud835\udcdd x, p y}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/IsEmpty.lean", "full_name": "wellFounded_of_isEmpty", "start": [199, 1], "end": [200, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBNode.foldr_eq_foldr_toList", "start": [343, 1], "end": [344, 45], "traced_tactics": [{"tactic": "induction t generalizing init <;> simp [*]", "state_before": "\u03b1 : Type u_1\n\u03b1\u271d : Type u_2\nf : \u03b1 \u2192 \u03b1\u271d \u2192 \u03b1\u271d\ninit : \u03b1\u271d\nt : RBNode \u03b1\n\u22a2 foldr f t init = List.foldr f init (toList t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Homology/ShortExact/Preadditive.lean", "full_name": "CategoryTheory.Split.map", "start": [148, 1], "end": [155, 38], "traced_tactics": [{"tactic": "obtain \u27e8\u03c6, \u03c7, h1, h2, h3, h4, h5\u27e9 := h", "state_before": "\ud835\udc9c\u271d : Type ?u.45480\ninst\u271d\u2078 : Category \ud835\udc9c\u271d\nA\u271d B\u271d C\u271d A' B' C' : \ud835\udc9c\u271d\nf\u271d : A\u271d \u27f6 B\u271d\ng\u271d : B\u271d \u27f6 C\u271d\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\ninst\u271d\u2077 : Preadditive \ud835\udc9c\u271d\ninst\u271d\u2076 : HasKernels \ud835\udc9c\u271d\ninst\u271d\u2075 : HasImages \ud835\udc9c\u271d\n\ud835\udc9c : Type u_1\n\u212c : Type u_2\ninst\u271d\u2074 : Category \ud835\udc9c\ninst\u271d\u00b3 : Preadditive \ud835\udc9c\ninst\u271d\u00b2 : Category \u212c\ninst\u271d\u00b9 : Preadditive \u212c\nF : \ud835\udc9c \u2964 \u212c\ninst\u271d : Functor.Additive F\nA B C : \ud835\udc9c\nf : A \u27f6 B\ng : B \u27f6 C\nh : Split f g\n\u22a2 Split (F.map f) (F.map g)", "state_after": "case mk.intro.intro.intro.intro.intro.intro\n\ud835\udc9c\u271d : Type ?u.45480\ninst\u271d\u2078 : Category \ud835\udc9c\u271d\nA\u271d B\u271d C\u271d A' B' C' : \ud835\udc9c\u271d\nf\u271d : A\u271d \u27f6 B\u271d\ng\u271d : B\u271d \u27f6 C\u271d\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\ninst\u271d\u2077 : Preadditive \ud835\udc9c\u271d\ninst\u271d\u2076 : HasKernels \ud835\udc9c\u271d\ninst\u271d\u2075 : HasImages \ud835\udc9c\u271d\n\ud835\udc9c : Type u_1\n\u212c : Type u_2\ninst\u271d\u2074 : Category \ud835\udc9c\ninst\u271d\u00b3 : Preadditive \ud835\udc9c\ninst\u271d\u00b2 : Category \u212c\ninst\u271d\u00b9 : Preadditive \u212c\nF : \ud835\udc9c \u2964 \u212c\ninst\u271d : Functor.Additive F\nA B C : \ud835\udc9c\nf : A \u27f6 B\ng : B \u27f6 C\n\u03c6 : B \u27f6 A\n\u03c7 : C \u27f6 B\nh1 : f \u226b \u03c6 = \ud835\udfd9 A\nh2 : \u03c7 \u226b g = \ud835\udfd9 C\nh3 : f \u226b g = 0\nh4 : \u03c7 \u226b \u03c6 = 0\nh5 : \u03c6 \u226b f + g \u226b \u03c7 = \ud835\udfd9 B\n\u22a2 Split (F.map f) (F.map g)"}, {"tactic": "refine \u27e8\u27e8F.map \u03c6, F.map \u03c7, ?_\u27e9\u27e9", "state_before": "case mk.intro.intro.intro.intro.intro.intro\n\ud835\udc9c\u271d : Type ?u.45480\ninst\u271d\u2078 : Category \ud835\udc9c\u271d\nA\u271d B\u271d C\u271d A' B' C' : \ud835\udc9c\u271d\nf\u271d : A\u271d \u27f6 B\u271d\ng\u271d : B\u271d \u27f6 C\u271d\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\ninst\u271d\u2077 : Preadditive \ud835\udc9c\u271d\ninst\u271d\u2076 : HasKernels \ud835\udc9c\u271d\ninst\u271d\u2075 : HasImages \ud835\udc9c\u271d\n\ud835\udc9c : Type u_1\n\u212c : Type u_2\ninst\u271d\u2074 : Category \ud835\udc9c\ninst\u271d\u00b3 : Preadditive \ud835\udc9c\ninst\u271d\u00b2 : Category \u212c\ninst\u271d\u00b9 : Preadditive \u212c\nF : \ud835\udc9c \u2964 \u212c\ninst\u271d : Functor.Additive F\nA B C : \ud835\udc9c\nf : A \u27f6 B\ng : B \u27f6 C\n\u03c6 : B \u27f6 A\n\u03c7 : C \u27f6 B\nh1 : f \u226b \u03c6 = \ud835\udfd9 A\nh2 : \u03c7 \u226b g = \ud835\udfd9 C\nh3 : f \u226b g = 0\nh4 : \u03c7 \u226b \u03c6 = 0\nh5 : \u03c6 \u226b f + g \u226b \u03c7 = \ud835\udfd9 B\n\u22a2 Split (F.map f) (F.map g)", "state_after": "case mk.intro.intro.intro.intro.intro.intro\n\ud835\udc9c\u271d : Type ?u.45480\ninst\u271d\u2078 : Category \ud835\udc9c\u271d\nA\u271d B\u271d C\u271d A' B' C' : \ud835\udc9c\u271d\nf\u271d : A\u271d \u27f6 B\u271d\ng\u271d : B\u271d \u27f6 C\u271d\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\ninst\u271d\u2077 : Preadditive \ud835\udc9c\u271d\ninst\u271d\u2076 : HasKernels \ud835\udc9c\u271d\ninst\u271d\u2075 : HasImages \ud835\udc9c\u271d\n\ud835\udc9c : Type u_1\n\u212c : Type u_2\ninst\u271d\u2074 : Category \ud835\udc9c\ninst\u271d\u00b3 : Preadditive \ud835\udc9c\ninst\u271d\u00b2 : Category \u212c\ninst\u271d\u00b9 : Preadditive \u212c\nF : \ud835\udc9c \u2964 \u212c\ninst\u271d : Functor.Additive F\nA B C : \ud835\udc9c\nf : A \u27f6 B\ng : B \u27f6 C\n\u03c6 : B \u27f6 A\n\u03c7 : C \u27f6 B\nh1 : f \u226b \u03c6 = \ud835\udfd9 A\nh2 : \u03c7 \u226b g = \ud835\udfd9 C\nh3 : f \u226b g = 0\nh4 : \u03c7 \u226b \u03c6 = 0\nh5 : \u03c6 \u226b f + g \u226b \u03c7 = \ud835\udfd9 B\n\u22a2 F.map f \u226b F.map \u03c6 = \ud835\udfd9 (F.obj A) \u2227\n    F.map \u03c7 \u226b F.map g = \ud835\udfd9 (F.obj C) \u2227\n      F.map f \u226b F.map g = 0 \u2227 F.map \u03c7 \u226b F.map \u03c6 = 0 \u2227 F.map \u03c6 \u226b F.map f + F.map g \u226b F.map \u03c7 = \ud835\udfd9 (F.obj B)"}, {"tactic": "simp only [\u2190 F.map_comp, \u2190 F.map_id]", "state_before": "case mk.intro.intro.intro.intro.intro.intro\n\ud835\udc9c\u271d : Type ?u.45480\ninst\u271d\u2078 : Category \ud835\udc9c\u271d\nA\u271d B\u271d C\u271d A' B' C' : \ud835\udc9c\u271d\nf\u271d : A\u271d \u27f6 B\u271d\ng\u271d : B\u271d \u27f6 C\u271d\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\ninst\u271d\u2077 : Preadditive \ud835\udc9c\u271d\ninst\u271d\u2076 : HasKernels \ud835\udc9c\u271d\ninst\u271d\u2075 : HasImages \ud835\udc9c\u271d\n\ud835\udc9c : Type u_1\n\u212c : Type u_2\ninst\u271d\u2074 : Category \ud835\udc9c\ninst\u271d\u00b3 : Preadditive \ud835\udc9c\ninst\u271d\u00b2 : Category \u212c\ninst\u271d\u00b9 : Preadditive \u212c\nF : \ud835\udc9c \u2964 \u212c\ninst\u271d : Functor.Additive F\nA B C : \ud835\udc9c\nf : A \u27f6 B\ng : B \u27f6 C\n\u03c6 : B \u27f6 A\n\u03c7 : C \u27f6 B\nh1 : f \u226b \u03c6 = \ud835\udfd9 A\nh2 : \u03c7 \u226b g = \ud835\udfd9 C\nh3 : f \u226b g = 0\nh4 : \u03c7 \u226b \u03c6 = 0\nh5 : \u03c6 \u226b f + g \u226b \u03c7 = \ud835\udfd9 B\n\u22a2 F.map f \u226b F.map \u03c6 = \ud835\udfd9 (F.obj A) \u2227\n    F.map \u03c7 \u226b F.map g = \ud835\udfd9 (F.obj C) \u2227\n      F.map f \u226b F.map g = 0 \u2227 F.map \u03c7 \u226b F.map \u03c6 = 0 \u2227 F.map \u03c6 \u226b F.map f + F.map g \u226b F.map \u03c7 = \ud835\udfd9 (F.obj B)", "state_after": "case mk.intro.intro.intro.intro.intro.intro\n\ud835\udc9c\u271d : Type ?u.45480\ninst\u271d\u2078 : Category \ud835\udc9c\u271d\nA\u271d B\u271d C\u271d A' B' C' : \ud835\udc9c\u271d\nf\u271d : A\u271d \u27f6 B\u271d\ng\u271d : B\u271d \u27f6 C\u271d\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\ninst\u271d\u2077 : Preadditive \ud835\udc9c\u271d\ninst\u271d\u2076 : HasKernels \ud835\udc9c\u271d\ninst\u271d\u2075 : HasImages \ud835\udc9c\u271d\n\ud835\udc9c : Type u_1\n\u212c : Type u_2\ninst\u271d\u2074 : Category \ud835\udc9c\ninst\u271d\u00b3 : Preadditive \ud835\udc9c\ninst\u271d\u00b2 : Category \u212c\ninst\u271d\u00b9 : Preadditive \u212c\nF : \ud835\udc9c \u2964 \u212c\ninst\u271d : Functor.Additive F\nA B C : \ud835\udc9c\nf : A \u27f6 B\ng : B \u27f6 C\n\u03c6 : B \u27f6 A\n\u03c7 : C \u27f6 B\nh1 : f \u226b \u03c6 = \ud835\udfd9 A\nh2 : \u03c7 \u226b g = \ud835\udfd9 C\nh3 : f \u226b g = 0\nh4 : \u03c7 \u226b \u03c6 = 0\nh5 : \u03c6 \u226b f + g \u226b \u03c7 = \ud835\udfd9 B\n\u22a2 F.map (f \u226b \u03c6) = F.map (\ud835\udfd9 A) \u2227\n    F.map (\u03c7 \u226b g) = F.map (\ud835\udfd9 C) \u2227 F.map (f \u226b g) = 0 \u2227 F.map (\u03c7 \u226b \u03c6) = 0 \u2227 F.map (\u03c6 \u226b f) + F.map (g \u226b \u03c7) = F.map (\ud835\udfd9 B)"}, {"tactic": "rw [\u2190 F.map_add]", "state_before": "case mk.intro.intro.intro.intro.intro.intro\n\ud835\udc9c\u271d : Type ?u.45480\ninst\u271d\u2078 : Category \ud835\udc9c\u271d\nA\u271d B\u271d C\u271d A' B' C' : \ud835\udc9c\u271d\nf\u271d : A\u271d \u27f6 B\u271d\ng\u271d : B\u271d \u27f6 C\u271d\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\ninst\u271d\u2077 : Preadditive \ud835\udc9c\u271d\ninst\u271d\u2076 : HasKernels \ud835\udc9c\u271d\ninst\u271d\u2075 : HasImages \ud835\udc9c\u271d\n\ud835\udc9c : Type u_1\n\u212c : Type u_2\ninst\u271d\u2074 : Category \ud835\udc9c\ninst\u271d\u00b3 : Preadditive \ud835\udc9c\ninst\u271d\u00b2 : Category \u212c\ninst\u271d\u00b9 : Preadditive \u212c\nF : \ud835\udc9c \u2964 \u212c\ninst\u271d : Functor.Additive F\nA B C : \ud835\udc9c\nf : A \u27f6 B\ng : B \u27f6 C\n\u03c6 : B \u27f6 A\n\u03c7 : C \u27f6 B\nh1 : f \u226b \u03c6 = \ud835\udfd9 A\nh2 : \u03c7 \u226b g = \ud835\udfd9 C\nh3 : f \u226b g = 0\nh4 : \u03c7 \u226b \u03c6 = 0\nh5 : \u03c6 \u226b f + g \u226b \u03c7 = \ud835\udfd9 B\n\u22a2 F.map (f \u226b \u03c6) = F.map (\ud835\udfd9 A) \u2227\n    F.map (\u03c7 \u226b g) = F.map (\ud835\udfd9 C) \u2227 F.map (f \u226b g) = 0 \u2227 F.map (\u03c7 \u226b \u03c6) = 0 \u2227 F.map (\u03c6 \u226b f) + F.map (g \u226b \u03c7) = F.map (\ud835\udfd9 B)", "state_after": "case mk.intro.intro.intro.intro.intro.intro\n\ud835\udc9c\u271d : Type ?u.45480\ninst\u271d\u2078 : Category \ud835\udc9c\u271d\nA\u271d B\u271d C\u271d A' B' C' : \ud835\udc9c\u271d\nf\u271d : A\u271d \u27f6 B\u271d\ng\u271d : B\u271d \u27f6 C\u271d\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\ninst\u271d\u2077 : Preadditive \ud835\udc9c\u271d\ninst\u271d\u2076 : HasKernels \ud835\udc9c\u271d\ninst\u271d\u2075 : HasImages \ud835\udc9c\u271d\n\ud835\udc9c : Type u_1\n\u212c : Type u_2\ninst\u271d\u2074 : Category \ud835\udc9c\ninst\u271d\u00b3 : Preadditive \ud835\udc9c\ninst\u271d\u00b2 : Category \u212c\ninst\u271d\u00b9 : Preadditive \u212c\nF : \ud835\udc9c \u2964 \u212c\ninst\u271d : Functor.Additive F\nA B C : \ud835\udc9c\nf : A \u27f6 B\ng : B \u27f6 C\n\u03c6 : B \u27f6 A\n\u03c7 : C \u27f6 B\nh1 : f \u226b \u03c6 = \ud835\udfd9 A\nh2 : \u03c7 \u226b g = \ud835\udfd9 C\nh3 : f \u226b g = 0\nh4 : \u03c7 \u226b \u03c6 = 0\nh5 : \u03c6 \u226b f + g \u226b \u03c7 = \ud835\udfd9 B\n\u22a2 F.map (f \u226b \u03c6) = F.map (\ud835\udfd9 A) \u2227\n    F.map (\u03c7 \u226b g) = F.map (\ud835\udfd9 C) \u2227 F.map (f \u226b g) = 0 \u2227 F.map (\u03c7 \u226b \u03c6) = 0 \u2227 F.map (\u03c6 \u226b f + g \u226b \u03c7) = F.map (\ud835\udfd9 B)"}, {"tactic": "simp only [F.map_zero, *, true_and]", "state_before": "case mk.intro.intro.intro.intro.intro.intro\n\ud835\udc9c\u271d : Type ?u.45480\ninst\u271d\u2078 : Category \ud835\udc9c\u271d\nA\u271d B\u271d C\u271d A' B' C' : \ud835\udc9c\u271d\nf\u271d : A\u271d \u27f6 B\u271d\ng\u271d : B\u271d \u27f6 C\u271d\nf' : A' \u27f6 B'\ng' : B' \u27f6 C'\ninst\u271d\u2077 : Preadditive \ud835\udc9c\u271d\ninst\u271d\u2076 : HasKernels \ud835\udc9c\u271d\ninst\u271d\u2075 : HasImages \ud835\udc9c\u271d\n\ud835\udc9c : Type u_1\n\u212c : Type u_2\ninst\u271d\u2074 : Category \ud835\udc9c\ninst\u271d\u00b3 : Preadditive \ud835\udc9c\ninst\u271d\u00b2 : Category \u212c\ninst\u271d\u00b9 : Preadditive \u212c\nF : \ud835\udc9c \u2964 \u212c\ninst\u271d : Functor.Additive F\nA B C : \ud835\udc9c\nf : A \u27f6 B\ng : B \u27f6 C\n\u03c6 : B \u27f6 A\n\u03c7 : C \u27f6 B\nh1 : f \u226b \u03c6 = \ud835\udfd9 A\nh2 : \u03c7 \u226b g = \ud835\udfd9 C\nh3 : f \u226b g = 0\nh4 : \u03c7 \u226b \u03c6 = 0\nh5 : \u03c6 \u226b f + g \u226b \u03c7 = \ud835\udfd9 B\n\u22a2 F.map (f \u226b \u03c6) = F.map (\ud835\udfd9 A) \u2227\n    F.map (\u03c7 \u226b g) = F.map (\ud835\udfd9 C) \u2227 F.map (f \u226b g) = 0 \u2227 F.map (\u03c7 \u226b \u03c6) = 0 \u2227 F.map (\u03c6 \u226b f + g \u226b \u03c7) = F.map (\ud835\udfd9 B)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/Cotangent.lean", "full_name": "Ideal.cotangent_subsingleton_iff", "start": [93, 1], "end": [101, 88], "traced_tactics": [{"tactic": "constructor", "state_before": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\n\u22a2 Subsingleton (Cotangent I) \u2194 IsIdempotentElem I", "state_after": "case mp\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\n\u22a2 Subsingleton (Cotangent I) \u2192 IsIdempotentElem I\n\ncase mpr\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\n\u22a2 IsIdempotentElem I \u2192 Subsingleton (Cotangent I)"}, {"tactic": "intro H", "state_before": "case mp\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\n\u22a2 Subsingleton (Cotangent I) \u2192 IsIdempotentElem I", "state_after": "case mp\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\nH : Subsingleton (Cotangent I)\n\u22a2 IsIdempotentElem I"}, {"tactic": "refine' (pow_two I).symm.trans (le_antisymm (Ideal.pow_le_self two_ne_zero) _)", "state_before": "case mp\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\nH : Subsingleton (Cotangent I)\n\u22a2 IsIdempotentElem I", "state_after": "case mp\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\nH : Subsingleton (Cotangent I)\n\u22a2 I \u2264 I ^ 2"}, {"tactic": "exact fun x hx => (I.toCotangent_eq_zero \u27e8x, hx\u27e9).mp (Subsingleton.elim _ _)", "state_before": "case mp\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\nH : Subsingleton (Cotangent I)\n\u22a2 I \u2264 I ^ 2", "state_after": "no goals"}, {"tactic": "exact fun e =>\n  \u27e8fun x y =>\n    Quotient.inductionOn\u2082' x y fun x y =>\n      I.toCotangent_eq.mpr <| ((pow_two I).trans e).symm \u25b8 I.sub_mem x.prop y.prop\u27e9", "state_before": "case mpr\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\n\u22a2 IsIdempotentElem I \u2192 Subsingleton (Cotangent I)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Perm/Sign.lean", "full_name": "Equiv.Perm.sign_prod_list_swap", "start": [617, 1], "end": [624, 66], "traced_tactics": [{"tactic": "have h\u2081 : l.map sign = List.replicate l.length (-1) :=\n  List.eq_replicate.2\n    \u27e8by simp, fun u hu =>\n      let \u27e8g, hg\u27e9 := List.mem_map.1 hu\n      hg.2 \u25b8 (hl _ hg.1).sign_eq\u27e9", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nl : List (Perm \u03b1)\nhl : \u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g\n\u22a2 \u2191sign (List.prod l) = (-1) ^ List.length l", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nl : List (Perm \u03b1)\nhl : \u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g\nh\u2081 : List.map (\u2191sign) l = List.replicate (List.length l) (-1)\n\u22a2 \u2191sign (List.prod l) = (-1) ^ List.length l"}, {"tactic": "rw [\u2190 List.prod_replicate, \u2190 h\u2081, List.prod_hom _ (@sign \u03b1 _ _)]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nl : List (Perm \u03b1)\nhl : \u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g\nh\u2081 : List.map (\u2191sign) l = List.replicate (List.length l) (-1)\n\u22a2 \u2191sign (List.prod l) = (-1) ^ List.length l", "state_after": "no goals"}, {"tactic": "simp", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nl : List (Perm \u03b1)\nhl : \u2200 (g : Perm \u03b1), g \u2208 l \u2192 IsSwap g\n\u22a2 List.length (List.map (\u2191sign) l) = List.length l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.card_Iio_eq_card_Iic_sub_one", "start": [735, 1], "end": [736, 57], "traced_tactics": [{"tactic": "rw [Iic_eq_cons_Iio, card_cons, add_tsub_cancel_right]", "state_before": "\u03b9 : Type ?u.120186\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\na : \u03b1\n\u22a2 card (Iio a) = card (Iic a) - 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/Minpoly/Field.lean", "full_name": "minpoly.eq_of_irreducible_of_monic", "start": [114, 1], "end": [119, 96], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/BilinearMap.lean", "full_name": "LinearMap.compl\u2081\u2082_apply", "start": [351, 1], "end": [352, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/Cardinality.lean", "full_name": "Cardinal.mk_Ioo_real", "start": [267, 1], "end": [275, 48], "traced_tactics": [{"tactic": "refine' le_antisymm (mk_real \u25b8 mk_set_le _) _", "state_before": "c : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh : a < b\n\u22a2 (#\u2191(Ioo a b)) = \ud835\udd20", "state_after": "c : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh : a < b\n\u22a2 \ud835\udd20 \u2264 (#\u2191(Ioo a b))"}, {"tactic": "have h1 : (#(fun x => x - a) '' Ioo a b) \u2264 (#Ioo a b) := mk_image_le", "state_before": "c : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh : a < b\n\u22a2 \ud835\udd20 \u2264 (#\u2191(Ioo a b))", "state_after": "c : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh : a < b\nh1 : (#\u2191((fun x => x - a) '' Ioo a b)) \u2264 (#\u2191(Ioo a b))\n\u22a2 \ud835\udd20 \u2264 (#\u2191(Ioo a b))"}, {"tactic": "refine' le_trans _ h1", "state_before": "c : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh : a < b\nh1 : (#\u2191((fun x => x - a) '' Ioo a b)) \u2264 (#\u2191(Ioo a b))\n\u22a2 \ud835\udd20 \u2264 (#\u2191(Ioo a b))", "state_after": "c : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh : a < b\nh1 : (#\u2191((fun x => x - a) '' Ioo a b)) \u2264 (#\u2191(Ioo a b))\n\u22a2 \ud835\udd20 \u2264 (#\u2191((fun x => x - a) '' Ioo a b))"}, {"tactic": "rw [image_sub_const_Ioo, sub_self]", "state_before": "c : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh : a < b\nh1 : (#\u2191((fun x => x - a) '' Ioo a b)) \u2264 (#\u2191(Ioo a b))\n\u22a2 \ud835\udd20 \u2264 (#\u2191((fun x => x - a) '' Ioo a b))", "state_after": "c : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh : a < b\nh1 : (#\u2191((fun x => x - a) '' Ioo a b)) \u2264 (#\u2191(Ioo a b))\n\u22a2 \ud835\udd20 \u2264 (#\u2191(Ioo 0 (b - a)))"}, {"tactic": "replace h := sub_pos_of_lt h", "state_before": "c : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh : a < b\nh1 : (#\u2191((fun x => x - a) '' Ioo a b)) \u2264 (#\u2191(Ioo a b))\n\u22a2 \ud835\udd20 \u2264 (#\u2191(Ioo 0 (b - a)))", "state_after": "c : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh1 : (#\u2191((fun x => x - a) '' Ioo a b)) \u2264 (#\u2191(Ioo a b))\nh : 0 < b - a\n\u22a2 \ud835\udd20 \u2264 (#\u2191(Ioo 0 (b - a)))"}, {"tactic": "have h2 : (#Inv.inv '' Ioo 0 (b - a)) \u2264 (#Ioo 0 (b - a)) := mk_image_le", "state_before": "c : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh1 : (#\u2191((fun x => x - a) '' Ioo a b)) \u2264 (#\u2191(Ioo a b))\nh : 0 < b - a\n\u22a2 \ud835\udd20 \u2264 (#\u2191(Ioo 0 (b - a)))", "state_after": "c : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh1 : (#\u2191((fun x => x - a) '' Ioo a b)) \u2264 (#\u2191(Ioo a b))\nh : 0 < b - a\nh2 : (#\u2191(Inv.inv '' Ioo 0 (b - a))) \u2264 (#\u2191(Ioo 0 (b - a)))\n\u22a2 \ud835\udd20 \u2264 (#\u2191(Ioo 0 (b - a)))"}, {"tactic": "refine' le_trans _ h2", "state_before": "c : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh1 : (#\u2191((fun x => x - a) '' Ioo a b)) \u2264 (#\u2191(Ioo a b))\nh : 0 < b - a\nh2 : (#\u2191(Inv.inv '' Ioo 0 (b - a))) \u2264 (#\u2191(Ioo 0 (b - a)))\n\u22a2 \ud835\udd20 \u2264 (#\u2191(Ioo 0 (b - a)))", "state_after": "c : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh1 : (#\u2191((fun x => x - a) '' Ioo a b)) \u2264 (#\u2191(Ioo a b))\nh : 0 < b - a\nh2 : (#\u2191(Inv.inv '' Ioo 0 (b - a))) \u2264 (#\u2191(Ioo 0 (b - a)))\n\u22a2 \ud835\udd20 \u2264 (#\u2191(Inv.inv '' Ioo 0 (b - a)))"}, {"tactic": "rw [image_inv, inv_Ioo_0_left h, mk_Ioi_real]", "state_before": "c : \u211d\nf g : \u2115 \u2192 Bool\nn : \u2115\na b : \u211d\nh1 : (#\u2191((fun x => x - a) '' Ioo a b)) \u2264 (#\u2191(Ioo a b))\nh : 0 < b - a\nh2 : (#\u2191(Inv.inv '' Ioo 0 (b - a))) \u2264 (#\u2191(Ioo 0 (b - a)))\n\u22a2 \ud835\udd20 \u2264 (#\u2191(Inv.inv '' Ioo 0 (b - a)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/DedekindDomain/Ideal.lean", "full_name": "Ideal.inf_eq_mul_of_coprime", "start": [882, 1], "end": [883, 94], "traced_tactics": [{"tactic": "rw [\u2190 associated_iff_eq.mp (gcd_mul_lcm I J), lcm_eq_inf I J, gcd_eq_sup, coprime, top_mul]", "state_before": "R : Type ?u.899540\nA : Type u_1\nK : Type ?u.899546\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : IsDomain A\ninst\u271d\u00b2 : IsDedekindDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nI J : Ideal A\ncoprime : I \u2294 J = \u22a4\n\u22a2 I \u2293 J = I * J", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/UpperLower/Basic.lean", "full_name": "isUpperSet_iUnion", "start": [129, 1], "end": [130, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "full_name": "MeasureTheory.Measure.haar.index_pos", "start": [204, 1], "end": [212, 39], "traced_tactics": [{"tactic": "unfold index", "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 0 < index (\u2191K) V", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 0 < sInf (Finset.card '' {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V})"}, {"tactic": "rw [Nat.sInf_def, Nat.find_pos, mem_image]", "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 0 < sInf (Finset.card '' {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V})", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 \u00ac\u2203 x, x \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V} \u2227 Finset.card x = 0\n\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 \u2203 n, n \u2208 Finset.card '' {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}"}, {"tactic": "rintro \u27e8t, h1t, h2t\u27e9", "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 \u00ac\u2203 x, x \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V} \u2227 Finset.card x = 0", "state_after": "case intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nt : Finset G\nh1t : t \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\nh2t : Finset.card t = 0\n\u22a2 False"}, {"tactic": "rw [Finset.card_eq_zero] at h2t", "state_before": "case intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nt : Finset G\nh1t : t \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\nh2t : Finset.card t = 0\n\u22a2 False", "state_after": "case intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nt : Finset G\nh1t : t \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\nh2t : t = \u2205\n\u22a2 False"}, {"tactic": "subst h2t", "state_before": "case intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nt : Finset G\nh1t : t \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\nh2t : t = \u2205\n\u22a2 False", "state_after": "case intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\n\u22a2 False"}, {"tactic": "obtain \u27e8g, hg\u27e9 := K.interior_nonempty", "state_before": "case intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\n\u22a2 False", "state_after": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\ng : G\nhg : g \u2208 interior \u2191K\n\u22a2 False"}, {"tactic": "show g \u2208 (\u2205 : Set G)", "state_before": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\ng : G\nhg : g \u2208 interior \u2191K\n\u22a2 False", "state_after": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\ng : G\nhg : g \u2208 interior \u2191K\n\u22a2 g \u2208 \u2205"}, {"tactic": "convert h1t (interior_subset hg)", "state_before": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\ng : G\nhg : g \u2208 interior \u2191K\n\u22a2 g \u2208 \u2205", "state_after": "case h.e'_5\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\ng : G\nhg : g \u2208 interior \u2191K\n\u22a2 \u2205 = \u22c3 (g : G) (_ : g \u2208 \u2205), (fun h => g * h) \u207b\u00b9' V"}, {"tactic": "symm", "state_before": "case h.e'_5\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\ng : G\nhg : g \u2208 interior \u2191K\n\u22a2 \u2205 = \u22c3 (g : G) (_ : g \u2208 \u2205), (fun h => g * h) \u207b\u00b9' V", "state_after": "case h.e'_5\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\ng : G\nhg : g \u2208 interior \u2191K\n\u22a2 (\u22c3 (g : G) (_ : g \u2208 \u2205), (fun h => g * h) \u207b\u00b9' V) = \u2205"}, {"tactic": "simp only [Finset.not_mem_empty, iUnion_of_empty, iUnion_empty]", "state_before": "case h.e'_5\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\nh1t : \u2205 \u2208 {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}\ng : G\nhg : g \u2208 interior \u2191K\n\u22a2 (\u22c3 (g : G) (_ : g \u2208 \u2205), (fun h => g * h) \u207b\u00b9' V) = \u2205", "state_after": "no goals"}, {"tactic": "exact index_defined K.isCompact hV", "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : Set.Nonempty (interior V)\n\u22a2 \u2203 n, n \u2208 Finset.card '' {t | \u2191K \u2286 \u22c3 (g : G) (_ : g \u2208 t), (fun h => g * h) \u207b\u00b9' V}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean", "full_name": "ContinuousLinearEquiv.coord_toSpanNonzeroSingleton", "start": [313, 1], "end": [315, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "full_name": "WithTop.add_eq_top", "start": [154, 1], "end": [155, 77], "traced_tactics": [{"tactic": "cases a <;> cases b <;> simp [none_eq_top, some_eq_coe, \u2190 WithTop.coe_add]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\na b c d : WithTop \u03b1\nx y : \u03b1\n\u22a2 a + b = \u22a4 \u2194 a = \u22a4 \u2228 b = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Spectral/Hom.lean", "full_name": "SpectralMap.coe_comp_continuousMap'", "start": [195, 1], "end": [197, 31], "traced_tactics": [{"tactic": "simp only [@coe_comp]", "state_before": "F : Type ?u.9868\n\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.9880\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf : SpectralMap \u03b2 \u03b3\ng : SpectralMap \u03b1 \u03b2\n\u22a2 ContinuousMap.mk \u2191(comp f g) = ContinuousMap.comp (ContinuousMap.mk \u2191f) (ContinuousMap.mk \u2191g)", "state_after": "F : Type ?u.9868\n\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.9880\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf : SpectralMap \u03b2 \u03b3\ng : SpectralMap \u03b1 \u03b2\n\u22a2 ContinuousMap.mk (\u2191f \u2218 \u2191g) = ContinuousMap.comp (ContinuousMap.mk \u2191f) (ContinuousMap.mk \u2191g)"}, {"tactic": "rfl", "state_before": "F : Type ?u.9868\n\u03b1 : Type u_3\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.9880\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf : SpectralMap \u03b2 \u03b3\ng : SpectralMap \u03b1 \u03b2\n\u22a2 ContinuousMap.mk (\u2191f \u2218 \u2191g) = ContinuousMap.comp (ContinuousMap.mk \u2191f) (ContinuousMap.mk \u2191g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "full_name": "Subalgebra.gc_map_comap", "start": [519, 1], "end": [519, 97], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Sum/Basic.lean", "full_name": "Sum.getRight_eq_none_iff", "start": [113, 9], "end": [114, 61], "traced_tactics": [{"tactic": "cases x <;> simp only [getRight, isLeft, eq_self_iff_true]", "state_before": "\u03b1 : Type u\n\u03b1' : Type w\n\u03b2 : Type v\n\u03b2' : Type x\n\u03b3 : Type ?u.4983\n\u03b4 : Type ?u.4986\nx y : \u03b1 \u2295 \u03b2\n\u22a2 getRight x = none \u2194 isLeft x = true", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.coeff_X_mul", "start": [724, 1], "end": [726, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Support.lean", "full_name": "Function.mulSupport_comp_subset", "start": [202, 1], "end": [203, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Trunc.lift_mk", "start": [487, 11], "end": [488, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Core.lean", "full_name": "Quot.indepCoherent", "start": [1252, 11], "end": [1256, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.bot_eq_empty", "start": [647, 1], "end": [648, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.mem_of_mem_filter", "start": [3539, 1], "end": [3540, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/Fermat4.lean", "full_name": "Fermat42.ne_zero", "start": [60, 1], "end": [64, 98], "traced_tactics": [{"tactic": "apply ne_zero_pow two_ne_zero _", "state_before": "a b c : \u2124\nh : Fermat42 a b c\n\u22a2 c \u2260 0", "state_after": "a b c : \u2124\nh : Fermat42 a b c\n\u22a2 c ^ 2 \u2260 0"}, {"tactic": "apply ne_of_gt", "state_before": "a b c : \u2124\nh : Fermat42 a b c\n\u22a2 c ^ 2 \u2260 0", "state_after": "case h\na b c : \u2124\nh : Fermat42 a b c\n\u22a2 0 < c ^ 2"}, {"tactic": "rw [\u2190 h.2.2, (by ring : a ^ 4 + b ^ 4 = (a ^ 2) ^ 2 + (b ^ 2) ^ 2)]", "state_before": "case h\na b c : \u2124\nh : Fermat42 a b c\n\u22a2 0 < c ^ 2", "state_after": "case h\na b c : \u2124\nh : Fermat42 a b c\n\u22a2 0 < (a ^ 2) ^ 2 + (b ^ 2) ^ 2"}, {"tactic": "exact\n  add_pos (sq_pos_of_ne_zero _ (pow_ne_zero 2 h.1)) (sq_pos_of_ne_zero _ (pow_ne_zero 2 h.2.1))", "state_before": "case h\na b c : \u2124\nh : Fermat42 a b c\n\u22a2 0 < (a ^ 2) ^ 2 + (b ^ 2) ^ 2", "state_after": "no goals"}, {"tactic": "ring", "state_before": "a b c : \u2124\nh : Fermat42 a b c\n\u22a2 a ^ 4 + b ^ 4 = (a ^ 2) ^ 2 + (b ^ 2) ^ 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Prime.lean", "full_name": "Nat.eq_one_iff_not_exists_prime_dvd", "start": [726, 1], "end": [727, 58], "traced_tactics": [{"tactic": "simpa using not_iff_not.mpr ne_one_iff_exists_prime_dvd", "state_before": "n : \u2115\n\u22a2 n = 1 \u2194 \u2200 (p : \u2115), Prime p \u2192 \u00acp \u2223 n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Digits.lean", "full_name": "Nat.eleven_dvd_iff", "start": [610, 1], "end": [614, 10], "traced_tactics": [{"tactic": "have t := dvd_iff_dvd_ofDigits 11 10 (-1 : \u2124) (by norm_num) n", "state_before": "n : \u2115\n\u22a2 11 \u2223 n \u2194 11 \u2223 List.alternatingSum (List.map (fun n => \u2191n) (digits 10 n))", "state_after": "n : \u2115\nt : 11 \u2223 n \u2194 \u219111 \u2223 ofDigits (-1) (digits 10 n)\n\u22a2 11 \u2223 n \u2194 11 \u2223 List.alternatingSum (List.map (fun n => \u2191n) (digits 10 n))"}, {"tactic": "rw [ofDigits_neg_one] at t", "state_before": "n : \u2115\nt : 11 \u2223 n \u2194 \u219111 \u2223 ofDigits (-1) (digits 10 n)\n\u22a2 11 \u2223 n \u2194 11 \u2223 List.alternatingSum (List.map (fun n => \u2191n) (digits 10 n))", "state_after": "n : \u2115\nt : 11 \u2223 n \u2194 \u219111 \u2223 List.alternatingSum (List.map (fun n => \u2191n) (digits 10 n))\n\u22a2 11 \u2223 n \u2194 11 \u2223 List.alternatingSum (List.map (fun n => \u2191n) (digits 10 n))"}, {"tactic": "exact t", "state_before": "n : \u2115\nt : 11 \u2223 n \u2194 \u219111 \u2223 List.alternatingSum (List.map (fun n => \u2191n) (digits 10 n))\n\u22a2 11 \u2223 n \u2194 11 \u2223 List.alternatingSum (List.map (fun n => \u2191n) (digits 10 n))", "state_after": "no goals"}, {"tactic": "norm_num", "state_before": "n : \u2115\n\u22a2 \u219111 \u2223 \u219110 - -1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/Hyperreal.lean", "full_name": "Hyperreal.infiniteNeg_def", "start": [434, 1], "end": [434, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/LocalHomeomorph.lean", "full_name": "LocalHomeomorph.map_nhdsWithin_preimage_eq", "start": [390, 1], "end": [393, 49], "traced_tactics": [{"tactic": "rw [e.map_nhdsWithin_eq hx, e.image_source_inter_eq', e.target_inter_inv_preimage_preimage,\n  e.nhdsWithin_target_inter (e.map_source hx)]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.32937\n\u03b4 : Type ?u.32940\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\ne\u271d : LocalHomeomorph \u03b1 \u03b2\ne' : LocalHomeomorph \u03b2 \u03b3\ne : LocalHomeomorph \u03b1 \u03b2\nx : \u03b1\nhx : x \u2208 e.source\ns : Set \u03b2\n\u22a2 map (\u2191e) (\ud835\udcdd[\u2191e \u207b\u00b9' s] x) = \ud835\udcdd[s] \u2191e x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean", "full_name": "HomogeneousLocalization.NumDenSameDeg.den_one", "start": [138, 1], "end": [139, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/TypeVec.lean", "full_name": "TypeVec.const_append1", "start": [441, 1], "end": [443, 32], "traced_tactics": [{"tactic": "ext i : 1", "state_before": "n\u271d : \u2115\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nx : \u03b3\nn : \u2115\n\u03b1 : TypeVec n\n\u22a2 TypeVec.const x (\u03b1 ::: \u03b2) = (TypeVec.const x \u03b1 ::: fun x_1 => x)", "state_after": "case a\nn\u271d : \u2115\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nx : \u03b3\nn : \u2115\n\u03b1 : TypeVec n\ni : Fin2 (n + 1)\n\u22a2 TypeVec.const x (\u03b1 ::: \u03b2) i = (TypeVec.const x \u03b1 ::: fun x_1 => x) i"}, {"tactic": "cases i <;> rfl", "state_before": "case a\nn\u271d : \u2115\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nx : \u03b3\nn : \u2115\n\u03b1 : TypeVec n\ni : Fin2 (n + 1)\n\u22a2 TypeVec.const x (\u03b1 ::: \u03b2) i = (TypeVec.const x \u03b1 ::: fun x_1 => x) i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Div.lean", "full_name": "Polynomial.rootMultiplicity_eq_multiplicity", "start": [550, 1], "end": [553, 75], "traced_tactics": [{"tactic": "simp [multiplicity, rootMultiplicity, Part.Dom]", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d : CommRing R\np\u271d q p : R[X]\na : R\n\u22a2 rootMultiplicity a p =\n    if h0 : p = 0 then 0 else Part.get (multiplicity (X - \u2191C a) p) (_ : multiplicity.Finite (X - \u2191C a) p)", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d : CommRing R\np\u271d q p : R[X]\na : R\n\u22a2 (if h : p = 0 then 0 else Nat.find (_ : multiplicity.Finite (X - \u2191C a) p)) =\n    if h : p = 0 then 0 else Nat.find (_ : (PartENat.find fun n => \u00ac(X - \u2191C a) ^ (n + 1) \u2223 p).Dom)"}, {"tactic": "congr", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d : CommRing R\np\u271d q p : R[X]\na : R\n\u22a2 (if h : p = 0 then 0 else Nat.find (_ : multiplicity.Finite (X - \u2191C a) p)) =\n    if h : p = 0 then 0 else Nat.find (_ : (PartENat.find fun n => \u00ac(X - \u2191C a) ^ (n + 1) \u2223 p).Dom)", "state_after": "case e_e\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d : CommRing R\np\u271d q p : R[X]\na : R\n\u22a2 (fun h => Nat.find (_ : multiplicity.Finite (X - \u2191C a) p)) = fun h =>\n    Nat.find (_ : (PartENat.find fun n => \u00ac(X - \u2191C a) ^ (n + 1) \u2223 p).Dom)"}, {"tactic": "funext", "state_before": "case e_e\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d : CommRing R\np\u271d q p : R[X]\na : R\n\u22a2 (fun h => Nat.find (_ : multiplicity.Finite (X - \u2191C a) p)) = fun h =>\n    Nat.find (_ : (PartENat.find fun n => \u00ac(X - \u2191C a) ^ (n + 1) \u2223 p).Dom)", "state_after": "case e_e.h\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d : CommRing R\np\u271d q p : R[X]\na : R\nx\u271d : \u00acp = 0\n\u22a2 Nat.find (_ : multiplicity.Finite (X - \u2191C a) p) =\n    Nat.find (_ : (PartENat.find fun n => \u00ac(X - \u2191C a) ^ (n + 1) \u2223 p).Dom)"}, {"tactic": "congr", "state_before": "case e_e.h\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d : CommRing R\np\u271d q p : R[X]\na : R\nx\u271d : \u00acp = 0\n\u22a2 Nat.find (_ : multiplicity.Finite (X - \u2191C a) p) =\n    Nat.find (_ : (PartENat.find fun n => \u00ac(X - \u2191C a) ^ (n + 1) \u2223 p).Dom)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Matrix/Basic.lean", "full_name": "Matrix.conjTranspose_mul", "start": [2240, 1], "end": [2242, 36], "traced_tactics": [{"tactic": "simp [mul_apply]", "state_before": "l : Type u_3\nm : Type u_2\nn : Type u_1\no : Type ?u.1044104\nm' : o \u2192 Type ?u.1044109\nn' : o \u2192 Type ?u.1044114\nR : Type ?u.1044117\nS : Type ?u.1044120\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type ?u.1044127\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : NonUnitalSemiring \u03b1\ninst\u271d : StarRing \u03b1\nM : Matrix m n \u03b1\nN : Matrix n l \u03b1\n\u22a2 \u2200 (i : l) (j : m), (M \u2b1d N)\u1d34 i j = (N\u1d34 \u2b1d M\u1d34) i j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Antichain.lean", "full_name": "IsAntichain.image_compl", "start": [196, 1], "end": [198, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Derivative.lean", "full_name": "Polynomial.derivative_monomial", "start": [91, 1], "end": [94, 7], "traced_tactics": [{"tactic": "rw [derivative_apply, sum_monomial_index, C_mul_X_pow_eq_monomial]", "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na\u271d b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\na : R\nn : \u2115\n\u22a2 \u2191derivative (\u2191(monomial n) a) = \u2191(monomial (n - 1)) (a * \u2191n)", "state_after": "case hf\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na\u271d b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\na : R\nn : \u2115\n\u22a2 \u2191C (0 * \u2191n) * X ^ (n - 1) = 0"}, {"tactic": "simp", "state_before": "case hf\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na\u271d b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\na : R\nn : \u2115\n\u22a2 \u2191C (0 * \u2191n) * X ^ (n - 1) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/RatFunc.lean", "full_name": "RatFunc.coe_sub", "start": [1715, 1], "end": [1716, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Monoidal/End.lean", "full_name": "CategoryTheory.left_unitality_app", "start": [186, 1], "end": [190, 24], "traced_tactics": [{"tactic": "have := congr_app (F.toLaxMonoidalFunctor.left_unitality n) X", "state_before": "C : Type u\ninst\u271d\u00b2 : Category C\nM : Type u_2\ninst\u271d\u00b9 : Category M\ninst\u271d : MonoidalCategory M\nF : MonoidalFunctor M (C \u2964 C)\nn : M\nX : C\n\u22a2 (F.obj n).map (F.\u03b5.app X) \u226b (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ M) n).app X \u226b (F.map (\u03bb_ n).hom).app X =\n    \ud835\udfd9 ((F.obj n).obj ((\ud835\udfd9_ (C \u2964 C)).obj X))", "state_after": "C : Type u\ninst\u271d\u00b2 : Category C\nM : Type u_2\ninst\u271d\u00b9 : Category M\ninst\u271d : MonoidalCategory M\nF : MonoidalFunctor M (C \u2964 C)\nn : M\nX : C\nthis :\n  (\u03bb_ (F.obj n)).hom.app X =\n    ((F.\u03b5 \u2297 \ud835\udfd9 (F.obj n)) \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ M) n \u226b F.map (\u03bb_ n).hom).app X\n\u22a2 (F.obj n).map (F.\u03b5.app X) \u226b (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ M) n).app X \u226b (F.map (\u03bb_ n).hom).app X =\n    \ud835\udfd9 ((F.obj n).obj ((\ud835\udfd9_ (C \u2964 C)).obj X))"}, {"tactic": "dsimp [endofunctorMonoidalCategory] at this", "state_before": "C : Type u\ninst\u271d\u00b2 : Category C\nM : Type u_2\ninst\u271d\u00b9 : Category M\ninst\u271d : MonoidalCategory M\nF : MonoidalFunctor M (C \u2964 C)\nn : M\nX : C\nthis :\n  (\u03bb_ (F.obj n)).hom.app X =\n    ((F.\u03b5 \u2297 \ud835\udfd9 (F.obj n)) \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ M) n \u226b F.map (\u03bb_ n).hom).app X\n\u22a2 (F.obj n).map (F.\u03b5.app X) \u226b (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ M) n).app X \u226b (F.map (\u03bb_ n).hom).app X =\n    \ud835\udfd9 ((F.obj n).obj ((\ud835\udfd9_ (C \u2964 C)).obj X))", "state_after": "C : Type u\ninst\u271d\u00b2 : Category C\nM : Type u_2\ninst\u271d\u00b9 : Category M\ninst\u271d : MonoidalCategory M\nF : MonoidalFunctor M (C \u2964 C)\nn : M\nX : C\nthis :\n  \ud835\udfd9 ((F.obj n).obj X) =\n    (\ud835\udfd9 ((F.obj n).obj ((\ud835\udfd9_ (C \u2964 C)).obj X)) \u226b (F.obj n).map (F.\u03b5.app X)) \u226b\n      (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ M) n).app X \u226b (F.map (\u03bb_ n).hom).app X\n\u22a2 (F.obj n).map (F.\u03b5.app X) \u226b (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ M) n).app X \u226b (F.map (\u03bb_ n).hom).app X =\n    \ud835\udfd9 ((F.obj n).obj ((\ud835\udfd9_ (C \u2964 C)).obj X))"}, {"tactic": "simpa using this.symm", "state_before": "C : Type u\ninst\u271d\u00b2 : Category C\nM : Type u_2\ninst\u271d\u00b9 : Category M\ninst\u271d : MonoidalCategory M\nF : MonoidalFunctor M (C \u2964 C)\nn : M\nX : C\nthis :\n  \ud835\udfd9 ((F.obj n).obj X) =\n    (\ud835\udfd9 ((F.obj n).obj ((\ud835\udfd9_ (C \u2964 C)).obj X)) \u226b (F.obj n).map (F.\u03b5.app X)) \u226b\n      (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ M) n).app X \u226b (F.map (\u03bb_ n).hom).app X\n\u22a2 (F.obj n).map (F.\u03b5.app X) \u226b (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (\ud835\udfd9_ M) n).app X \u226b (F.map (\u03bb_ n).hom).app X =\n    \ud835\udfd9 ((F.obj n).obj ((\ud835\udfd9_ (C \u2964 C)).obj X))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Option.lean", "full_name": "Option.mem_toFinset", "start": [54, 1], "end": [55, 29], "traced_tactics": [{"tactic": "cases o <;> simp [eq_comm]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.398\na : \u03b1\no : Option \u03b1\n\u22a2 a \u2208 toFinset o \u2194 a \u2208 o", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Basic.lean", "full_name": "Multiset.finset_sum_eq_sup_iff_disjoint", "start": [2110, 1], "end": [2124, 98], "traced_tactics": [{"tactic": "induction' i using Finset.cons_induction_on with z i hz hr", "state_before": "\u03b9 : Type ?u.903903\n\u03b2\u271d : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\n\u03b2 : Type u_1\ni : Finset \u03b2\nf : \u03b2 \u2192 Multiset \u03b1\n\u22a2 Finset.sum i f = Finset.sup i f \u2194 \u2200 (x : \u03b2), x \u2208 i \u2192 \u2200 (y : \u03b2), y \u2208 i \u2192 x \u2260 y \u2192 Disjoint (f x) (f y)", "state_after": "case h\u2081\n\u03b9 : Type ?u.903903\n\u03b2\u271d : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\n\u03b2 : Type u_1\nf : \u03b2 \u2192 Multiset \u03b1\n\u22a2 Finset.sum \u2205 f = Finset.sup \u2205 f \u2194 \u2200 (x : \u03b2), x \u2208 \u2205 \u2192 \u2200 (y : \u03b2), y \u2208 \u2205 \u2192 x \u2260 y \u2192 Disjoint (f x) (f y)\n\ncase h\u2082\n\u03b9 : Type ?u.903903\n\u03b2\u271d : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\n\u03b2 : Type u_1\nf : \u03b2 \u2192 Multiset \u03b1\nz : \u03b2\ni : Finset \u03b2\nhz : \u00acz \u2208 i\nhr : Finset.sum i f = Finset.sup i f \u2194 \u2200 (x : \u03b2), x \u2208 i \u2192 \u2200 (y : \u03b2), y \u2208 i \u2192 x \u2260 y \u2192 Disjoint (f x) (f y)\n\u22a2 Finset.sum (Finset.cons z i hz) f = Finset.sup (Finset.cons z i hz) f \u2194\n    \u2200 (x : \u03b2), x \u2208 Finset.cons z i hz \u2192 \u2200 (y : \u03b2), y \u2208 Finset.cons z i hz \u2192 x \u2260 y \u2192 Disjoint (f x) (f y)"}, {"tactic": "simp only [Finset.not_mem_empty, IsEmpty.forall_iff, imp_true_iff, Finset.sum_empty,\n  Finset.sup_empty, bot_eq_zero, eq_self_iff_true]", "state_before": "case h\u2081\n\u03b9 : Type ?u.903903\n\u03b2\u271d : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\n\u03b2 : Type u_1\nf : \u03b2 \u2192 Multiset \u03b1\n\u22a2 Finset.sum \u2205 f = Finset.sup \u2205 f \u2194 \u2200 (x : \u03b2), x \u2208 \u2205 \u2192 \u2200 (y : \u03b2), y \u2208 \u2205 \u2192 x \u2260 y \u2192 Disjoint (f x) (f y)", "state_after": "no goals"}, {"tactic": "simp_rw [Finset.sum_cons hz, Finset.sup_cons, Finset.mem_cons, Multiset.sup_eq_union,\n  forall_eq_or_imp, Ne.def, IsEmpty.forall_iff, true_and_iff,\n  imp_and, forall_and, \u2190 hr, @eq_comm _ z]", "state_before": "case h\u2082\n\u03b9 : Type ?u.903903\n\u03b2\u271d : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\n\u03b2 : Type u_1\nf : \u03b2 \u2192 Multiset \u03b1\nz : \u03b2\ni : Finset \u03b2\nhz : \u00acz \u2208 i\nhr : Finset.sum i f = Finset.sup i f \u2194 \u2200 (x : \u03b2), x \u2208 i \u2192 \u2200 (y : \u03b2), y \u2208 i \u2192 x \u2260 y \u2192 Disjoint (f x) (f y)\n\u22a2 Finset.sum (Finset.cons z i hz) f = Finset.sup (Finset.cons z i hz) f \u2194\n    \u2200 (x : \u03b2), x \u2208 Finset.cons z i hz \u2192 \u2200 (y : \u03b2), y \u2208 Finset.cons z i hz \u2192 x \u2260 y \u2192 Disjoint (f x) (f y)", "state_after": "case h\u2082\n\u03b9 : Type ?u.903903\n\u03b2\u271d : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\n\u03b2 : Type u_1\nf : \u03b2 \u2192 Multiset \u03b1\nz : \u03b2\ni : Finset \u03b2\nhz : \u00acz \u2208 i\nhr : Finset.sum i f = Finset.sup i f \u2194 \u2200 (x : \u03b2), x \u2208 i \u2192 \u2200 (y : \u03b2), y \u2208 i \u2192 x \u2260 y \u2192 Disjoint (f x) (f y)\n\u22a2 f z + \u2211 x in i, f x = f z \u222a Finset.sup i f \u2194\n    (\u2200 (a : \u03b2), a \u2208 i \u2192 \u00aca = z \u2192 Disjoint (f z) (f a)) \u2227\n      (\u2200 (x : \u03b2), x \u2208 i \u2192 \u00acx = z \u2192 Disjoint (f x) (f z)) \u2227 Finset.sum i f = Finset.sup i f"}, {"tactic": "have := fun x (H : x \u2208 i) => ne_of_mem_of_not_mem H hz", "state_before": "case h\u2082\n\u03b9 : Type ?u.903903\n\u03b2\u271d : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\n\u03b2 : Type u_1\nf : \u03b2 \u2192 Multiset \u03b1\nz : \u03b2\ni : Finset \u03b2\nhz : \u00acz \u2208 i\nhr : Finset.sum i f = Finset.sup i f \u2194 \u2200 (x : \u03b2), x \u2208 i \u2192 \u2200 (y : \u03b2), y \u2208 i \u2192 x \u2260 y \u2192 Disjoint (f x) (f y)\n\u22a2 f z + \u2211 x in i, f x = f z \u222a Finset.sup i f \u2194\n    (\u2200 (a : \u03b2), a \u2208 i \u2192 \u00aca = z \u2192 Disjoint (f z) (f a)) \u2227\n      (\u2200 (x : \u03b2), x \u2208 i \u2192 \u00acx = z \u2192 Disjoint (f x) (f z)) \u2227 Finset.sum i f = Finset.sup i f", "state_after": "case h\u2082\n\u03b9 : Type ?u.903903\n\u03b2\u271d : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\n\u03b2 : Type u_1\nf : \u03b2 \u2192 Multiset \u03b1\nz : \u03b2\ni : Finset \u03b2\nhz : \u00acz \u2208 i\nhr : Finset.sum i f = Finset.sup i f \u2194 \u2200 (x : \u03b2), x \u2208 i \u2192 \u2200 (y : \u03b2), y \u2208 i \u2192 x \u2260 y \u2192 Disjoint (f x) (f y)\nthis : \u2200 (x : \u03b2), x \u2208 i \u2192 x \u2260 z\n\u22a2 f z + \u2211 x in i, f x = f z \u222a Finset.sup i f \u2194\n    (\u2200 (a : \u03b2), a \u2208 i \u2192 \u00aca = z \u2192 Disjoint (f z) (f a)) \u2227\n      (\u2200 (x : \u03b2), x \u2208 i \u2192 \u00acx = z \u2192 Disjoint (f x) (f z)) \u2227 Finset.sum i f = Finset.sup i f"}, {"tactic": "simp (config := { contextual := true }) only [this, not_false_iff, true_imp_iff]", "state_before": "case h\u2082\n\u03b9 : Type ?u.903903\n\u03b2\u271d : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\n\u03b2 : Type u_1\nf : \u03b2 \u2192 Multiset \u03b1\nz : \u03b2\ni : Finset \u03b2\nhz : \u00acz \u2208 i\nhr : Finset.sum i f = Finset.sup i f \u2194 \u2200 (x : \u03b2), x \u2208 i \u2192 \u2200 (y : \u03b2), y \u2208 i \u2192 x \u2260 y \u2192 Disjoint (f x) (f y)\nthis : \u2200 (x : \u03b2), x \u2208 i \u2192 x \u2260 z\n\u22a2 f z + \u2211 x in i, f x = f z \u222a Finset.sup i f \u2194\n    (\u2200 (a : \u03b2), a \u2208 i \u2192 \u00aca = z \u2192 Disjoint (f z) (f a)) \u2227\n      (\u2200 (x : \u03b2), x \u2208 i \u2192 \u00acx = z \u2192 Disjoint (f x) (f z)) \u2227 Finset.sum i f = Finset.sup i f", "state_after": "case h\u2082\n\u03b9 : Type ?u.903903\n\u03b2\u271d : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\n\u03b2 : Type u_1\nf : \u03b2 \u2192 Multiset \u03b1\nz : \u03b2\ni : Finset \u03b2\nhz : \u00acz \u2208 i\nhr : Finset.sum i f = Finset.sup i f \u2194 \u2200 (x : \u03b2), x \u2208 i \u2192 \u2200 (y : \u03b2), y \u2208 i \u2192 x \u2260 y \u2192 Disjoint (f x) (f y)\nthis : \u2200 (x : \u03b2), x \u2208 i \u2192 x \u2260 z\n\u22a2 f z + \u2211 x in i, f x = f z \u222a Finset.sup i f \u2194\n    (\u2200 (a : \u03b2), a \u2208 i \u2192 Disjoint (f z) (f a)) \u2227\n      (\u2200 (x : \u03b2), x \u2208 i \u2192 Disjoint (f x) (f z)) \u2227 Finset.sum i f = Finset.sup i f"}, {"tactic": "simp_rw [\u2190 disjoint_finset_sum_left, \u2190 disjoint_finset_sum_right, disjoint_comm, \u2190 and_assoc,\n  and_self_iff]", "state_before": "case h\u2082\n\u03b9 : Type ?u.903903\n\u03b2\u271d : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\n\u03b2 : Type u_1\nf : \u03b2 \u2192 Multiset \u03b1\nz : \u03b2\ni : Finset \u03b2\nhz : \u00acz \u2208 i\nhr : Finset.sum i f = Finset.sup i f \u2194 \u2200 (x : \u03b2), x \u2208 i \u2192 \u2200 (y : \u03b2), y \u2208 i \u2192 x \u2260 y \u2192 Disjoint (f x) (f y)\nthis : \u2200 (x : \u03b2), x \u2208 i \u2192 x \u2260 z\n\u22a2 f z + \u2211 x in i, f x = f z \u222a Finset.sup i f \u2194\n    (\u2200 (a : \u03b2), a \u2208 i \u2192 Disjoint (f z) (f a)) \u2227\n      (\u2200 (x : \u03b2), x \u2208 i \u2192 Disjoint (f x) (f z)) \u2227 Finset.sum i f = Finset.sup i f", "state_after": "case h\u2082\n\u03b9 : Type ?u.903903\n\u03b2\u271d : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\n\u03b2 : Type u_1\nf : \u03b2 \u2192 Multiset \u03b1\nz : \u03b2\ni : Finset \u03b2\nhz : \u00acz \u2208 i\nhr : Finset.sum i f = Finset.sup i f \u2194 \u2200 (x : \u03b2), x \u2208 i \u2192 \u2200 (y : \u03b2), y \u2208 i \u2192 x \u2260 y \u2192 Disjoint (f x) (f y)\nthis : \u2200 (x : \u03b2), x \u2208 i \u2192 x \u2260 z\n\u22a2 f z + \u2211 x in i, f x = f z \u222a Finset.sup i f \u2194 Disjoint (f z) (\u2211 x in i, f x) \u2227 Finset.sum i f = Finset.sup i f"}, {"tactic": "exact add_eq_union_left_of_le (Finset.sup_le fun x hx => le_sum_of_mem (mem_map_of_mem f hx))", "state_before": "case h\u2082\n\u03b9 : Type ?u.903903\n\u03b2\u271d : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\ninst\u271d : DecidableEq \u03b1\n\u03b2 : Type u_1\nf : \u03b2 \u2192 Multiset \u03b1\nz : \u03b2\ni : Finset \u03b2\nhz : \u00acz \u2208 i\nhr : Finset.sum i f = Finset.sup i f \u2194 \u2200 (x : \u03b2), x \u2208 i \u2192 \u2200 (y : \u03b2), y \u2208 i \u2192 x \u2260 y \u2192 Disjoint (f x) (f y)\nthis : \u2200 (x : \u03b2), x \u2208 i \u2192 x \u2260 z\n\u22a2 f z + \u2211 x in i, f x = f z \u222a Finset.sup i f \u2194 Disjoint (f z) (\u2211 x in i, f x) \u2227 Finset.sum i f = Finset.sup i f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Int/Basic.lean", "full_name": "Int.Prime.dvd_pow'", "start": [287, 1], "end": [290, 31], "traced_tactics": [{"tactic": "rw [Int.coe_nat_dvd_left]", "state_before": "n : \u2124\nk p : \u2115\nhp : Nat.Prime p\nh : \u2191p \u2223 n ^ k\n\u22a2 \u2191p \u2223 n", "state_after": "n : \u2124\nk p : \u2115\nhp : Nat.Prime p\nh : \u2191p \u2223 n ^ k\n\u22a2 p \u2223 natAbs n"}, {"tactic": "exact Int.Prime.dvd_pow hp h", "state_before": "n : \u2124\nk p : \u2115\nhp : Nat.Prime p\nh : \u2191p \u2223 n ^ k\n\u22a2 p \u2223 natAbs n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.neBot_of_comap", "start": [2210, 1], "end": [2214, 18], "traced_tactics": [{"tactic": "rw [neBot_iff] at *", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.252476\n\u03b9 : Sort x\nf 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\nh : NeBot (comap m g)\n\u22a2 NeBot g", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.252476\n\u03b9 : Sort x\nf 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\nh : comap m g \u2260 \u22a5\n\u22a2 g \u2260 \u22a5"}, {"tactic": "contrapose! h", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.252476\n\u03b9 : Sort x\nf 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\nh : comap m g \u2260 \u22a5\n\u22a2 g \u2260 \u22a5", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.252476\n\u03b9 : Sort x\nf 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\nh : g = \u22a5\n\u22a2 comap m g = \u22a5"}, {"tactic": "rw [h]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.252476\n\u03b9 : Sort x\nf 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\nh : g = \u22a5\n\u22a2 comap m g = \u22a5", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.252476\n\u03b9 : Sort x\nf 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\nh : g = \u22a5\n\u22a2 comap m \u22a5 = \u22a5"}, {"tactic": "exact comap_bot", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.252476\n\u03b9 : Sort x\nf 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\nh : g = \u22a5\n\u22a2 comap m \u22a5 = \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Spectrum.lean", "full_name": "spectrum.map_inv", "start": [375, 11], "end": [383, 66], "traced_tactics": [{"tactic": "refine' Set.eq_of_subset_of_subset (fun k hk => _) fun k hk => _", "state_before": "\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\n\u22a2 (\u03c3 \u2191a)\u207b\u00b9 = \u03c3 \u2191a\u207b\u00b9", "state_after": "case refine'_1\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\nhk : k \u2208 (\u03c3 \u2191a)\u207b\u00b9\n\u22a2 k \u2208 \u03c3 \u2191a\u207b\u00b9\n\ncase refine'_2\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\nhk : k \u2208 \u03c3 \u2191a\u207b\u00b9\n\u22a2 k \u2208 (\u03c3 \u2191a)\u207b\u00b9"}, {"tactic": "rw [Set.mem_inv] at hk", "state_before": "case refine'_1\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\nhk : k \u2208 (\u03c3 \u2191a)\u207b\u00b9\n\u22a2 k \u2208 \u03c3 \u2191a\u207b\u00b9", "state_after": "case refine'_1\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\nhk : k\u207b\u00b9 \u2208 \u03c3 \u2191a\n\u22a2 k \u2208 \u03c3 \u2191a\u207b\u00b9"}, {"tactic": "have : k \u2260 0 := by simpa only [inv_inv] using inv_ne_zero (ne_zero_of_mem_of_unit hk)", "state_before": "case refine'_1\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\nhk : k\u207b\u00b9 \u2208 \u03c3 \u2191a\n\u22a2 k \u2208 \u03c3 \u2191a\u207b\u00b9", "state_after": "case refine'_1\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\nhk : k\u207b\u00b9 \u2208 \u03c3 \u2191a\nthis : k \u2260 0\n\u22a2 k \u2208 \u03c3 \u2191a\u207b\u00b9"}, {"tactic": "lift k to \ud835\udd5c\u02e3 using isUnit_iff_ne_zero.mpr this", "state_before": "case refine'_1\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\nhk : k\u207b\u00b9 \u2208 \u03c3 \u2191a\nthis : k \u2260 0\n\u22a2 k \u2208 \u03c3 \u2191a\u207b\u00b9", "state_after": "case refine'_1.intro\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\u02e3\nhk : (\u2191k)\u207b\u00b9 \u2208 \u03c3 \u2191a\nthis : \u2191k \u2260 0\n\u22a2 \u2191k \u2208 \u03c3 \u2191a\u207b\u00b9"}, {"tactic": "rw [\u2190 Units.val_inv_eq_inv_val k] at hk", "state_before": "case refine'_1.intro\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\u02e3\nhk : (\u2191k)\u207b\u00b9 \u2208 \u03c3 \u2191a\nthis : \u2191k \u2260 0\n\u22a2 \u2191k \u2208 \u03c3 \u2191a\u207b\u00b9", "state_after": "case refine'_1.intro\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\u02e3\nhk : \u2191k\u207b\u00b9 \u2208 \u03c3 \u2191a\nthis : \u2191k \u2260 0\n\u22a2 \u2191k \u2208 \u03c3 \u2191a\u207b\u00b9"}, {"tactic": "exact inv_mem_iff.mp hk", "state_before": "case refine'_1.intro\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\u02e3\nhk : \u2191k\u207b\u00b9 \u2208 \u03c3 \u2191a\nthis : \u2191k \u2260 0\n\u22a2 \u2191k \u2208 \u03c3 \u2191a\u207b\u00b9", "state_after": "no goals"}, {"tactic": "simpa only [inv_inv] using inv_ne_zero (ne_zero_of_mem_of_unit hk)", "state_before": "\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\nhk : k\u207b\u00b9 \u2208 \u03c3 \u2191a\n\u22a2 k \u2260 0", "state_after": "no goals"}, {"tactic": "lift k to \ud835\udd5c\u02e3 using isUnit_iff_ne_zero.mpr (ne_zero_of_mem_of_unit hk)", "state_before": "case refine'_2\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\nhk : k \u2208 \u03c3 \u2191a\u207b\u00b9\n\u22a2 k \u2208 (\u03c3 \u2191a)\u207b\u00b9", "state_after": "case refine'_2.intro\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\u02e3\nhk : \u2191k \u2208 \u03c3 \u2191a\u207b\u00b9\n\u22a2 \u2191k \u2208 (\u03c3 \u2191a)\u207b\u00b9"}, {"tactic": "simpa only [Units.val_inv_eq_inv_val] using inv_mem_iff.mp hk", "state_before": "case refine'_2.intro\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\u02e3\nhk : \u2191k \u2208 \u03c3 \u2191a\u207b\u00b9\n\u22a2 \u2191k \u2208 (\u03c3 \u2191a)\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/IsROrC/Basic.lean", "full_name": "IsROrC.normSq_zero", "start": [474, 1], "end": [475, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "full_name": "CategoryTheory.Limits.PushoutCocone.epi_inr_of_is_pushout_of_epi", "start": [909, 1], "end": [911, 93], "traced_tactics": [{"tactic": "simp [\u2190 cancel_epi f, t.condition_assoc, i]", "state_before": "C : Type u\ninst\u271d\u00b2 : Category C\nD : Type u\u2082\ninst\u271d\u00b9 : Category D\nW\u271d X Y Z : C\nf : X \u27f6 Y\ng : X \u27f6 Z\nt : PushoutCocone f g\nht : IsColimit t\ninst\u271d : Epi f\nW : C\nh k : t.pt \u27f6 W\ni : inr t \u226b h = inr t \u226b k\n\u22a2 inl t \u226b h = inl t \u226b k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Parity.lean", "full_name": "Nat.odd_mul", "start": [161, 1], "end": [161, 76], "traced_tactics": [{"tactic": "simp [not_or, even_mul]", "state_before": "m n : \u2115\n\u22a2 Odd (m * n) \u2194 Odd m \u2227 Odd n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Group/Defs.lean", "full_name": "div_le_div_flip", "start": [1073, 1], "end": [1076, 27], "traced_tactics": [{"tactic": "rw [div_eq_mul_inv b, mul_comm]", "state_before": "\u03b1\u271d : Type u\ninst\u271d\u2075 : Group \u03b1\u271d\ninst\u271d\u2074 : LinearOrder \u03b1\u271d\ninst\u271d\u00b3 : CovariantClass \u03b1\u271d \u03b1\u271d (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na\u271d b\u271d c : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a / b \u2264 b / a \u2194 a \u2264 b", "state_after": "\u03b1\u271d : Type u\ninst\u271d\u2075 : Group \u03b1\u271d\ninst\u271d\u2074 : LinearOrder \u03b1\u271d\ninst\u271d\u00b3 : CovariantClass \u03b1\u271d \u03b1\u271d (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na\u271d b\u271d c : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a / b \u2264 a\u207b\u00b9 * b \u2194 a \u2264 b"}, {"tactic": "exact div_le_inv_mul_iff", "state_before": "\u03b1\u271d : Type u\ninst\u271d\u2075 : Group \u03b1\u271d\ninst\u271d\u2074 : LinearOrder \u03b1\u271d\ninst\u271d\u00b3 : CovariantClass \u03b1\u271d \u03b1\u271d (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na\u271d b\u271d c : \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a / b \u2264 a\u207b\u00b9 * b \u2194 a \u2264 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "full_name": "Pmf.toOuterMeasure_apply_singleton", "start": [179, 1], "end": [182, 61], "traced_tactics": [{"tactic": "refine' (p.toOuterMeasure_apply {a}).trans ((tsum_eq_single a fun b hb => _).trans _)", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.24121\n\u03b3 : Type ?u.24124\np : Pmf \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 \u2191(toOuterMeasure p) {a} = \u2191p a", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type ?u.24121\n\u03b3 : Type ?u.24124\np : Pmf \u03b1\ns t : Set \u03b1\na b : \u03b1\nhb : b \u2260 a\n\u22a2 Set.indicator {a} (\u2191p) b = 0\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type ?u.24121\n\u03b3 : Type ?u.24124\np : Pmf \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 Set.indicator {a} (\u2191p) a = \u2191p a"}, {"tactic": "exact ite_eq_right_iff.2 fun hb' => False.elim <| hb hb'", "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type ?u.24121\n\u03b3 : Type ?u.24124\np : Pmf \u03b1\ns t : Set \u03b1\na b : \u03b1\nhb : b \u2260 a\n\u22a2 Set.indicator {a} (\u2191p) b = 0", "state_after": "no goals"}, {"tactic": "exact ite_eq_left_iff.2 fun ha' => False.elim <| ha' rfl", "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type ?u.24121\n\u03b3 : Type ?u.24124\np : Pmf \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 Set.indicator {a} (\u2191p) a = \u2191p a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Rat/Lemmas.lean", "full_name": "Rat.mkRat_add_mkRat", "start": [195, 1], "end": [197, 101], "traced_tactics": [{"tactic": "rw [\u2190 normalize_eq_mkRat z\u2081, \u2190 normalize_eq_mkRat z\u2082, normalize_add_normalize, normalize_eq_mkRat]", "state_before": "n\u2081 n\u2082 : Int\nd\u2081 d\u2082 : Nat\nz\u2081 : d\u2081 \u2260 0\nz\u2082 : d\u2082 \u2260 0\n\u22a2 mkRat n\u2081 d\u2081 + mkRat n\u2082 d\u2082 = mkRat (n\u2081 * \u2191d\u2082 + n\u2082 * \u2191d\u2081) (d\u2081 * d\u2082)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Image.lean", "full_name": "Finset.not_mem_map_subtype_of_not_property", "start": [705, 1], "end": [707, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean", "full_name": "HasStrictDerivAt.sub", "start": [309, 1], "end": [311, 50], "traced_tactics": [{"tactic": "simpa only [sub_eq_add_neg] using hf.add hg.neg", "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' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhf : HasStrictDerivAt f f' x\nhg : HasStrictDerivAt g g' x\n\u22a2 HasStrictDerivAt (fun x => f x - g x) (f' - g') x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Sites/Plus.lean", "full_name": "CategoryTheory.GrothendieckTopology.plusMap_id", "start": [171, 1], "end": [177, 7], "traced_tactics": [{"tactic": "ext : 2", "state_before": "C : Type u\ninst\u271d\u00b3 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP : C\u1d52\u1d56 \u2964 D\n\u22a2 plusMap J (\ud835\udfd9 P) = \ud835\udfd9 (plusObj J P)", "state_after": "case w.h\nC : Type u\ninst\u271d\u00b3 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP : C\u1d52\u1d56 \u2964 D\nx\u271d : C\u1d52\u1d56\n\u22a2 (plusMap J (\ud835\udfd9 P)).app x\u271d = (\ud835\udfd9 (plusObj J P)).app x\u271d"}, {"tactic": "dsimp only [plusMap, plusObj]", "state_before": "case w.h\nC : Type u\ninst\u271d\u00b3 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP : C\u1d52\u1d56 \u2964 D\nx\u271d : C\u1d52\u1d56\n\u22a2 (plusMap J (\ud835\udfd9 P)).app x\u271d = (\ud835\udfd9 (plusObj J P)).app x\u271d", "state_after": "case w.h\nC : Type u\ninst\u271d\u00b3 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP : C\u1d52\u1d56 \u2964 D\nx\u271d : C\u1d52\u1d56\n\u22a2 colimMap (diagramNatTrans J (\ud835\udfd9 P) x\u271d.unop) =\n    (\ud835\udfd9\n          (Functor.mk\n            { obj := fun X => colimit (diagram J P X.unop),\n              map := fun {X Y} f =>\n                colimMap (diagramPullback J P f.unop) \u226b\n                  colimit.pre (diagram J P Y.unop) (Functor.op (pullback J f.unop)) })).app\n      x\u271d"}, {"tactic": "rw [J.diagramNatTrans_id, NatTrans.id_app]", "state_before": "case w.h\nC : Type u\ninst\u271d\u00b3 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP : C\u1d52\u1d56 \u2964 D\nx\u271d : C\u1d52\u1d56\n\u22a2 colimMap (diagramNatTrans J (\ud835\udfd9 P) x\u271d.unop) =\n    (\ud835\udfd9\n          (Functor.mk\n            { obj := fun X => colimit (diagram J P X.unop),\n              map := fun {X Y} f =>\n                colimMap (diagramPullback J P f.unop) \u226b\n                  colimit.pre (diagram J P Y.unop) (Functor.op (pullback J f.unop)) })).app\n      x\u271d", "state_after": "case w.h\nC : Type u\ninst\u271d\u00b3 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP : C\u1d52\u1d56 \u2964 D\nx\u271d : C\u1d52\u1d56\n\u22a2 colimMap (\ud835\udfd9 (diagram J P x\u271d.unop)) =\n    \ud835\udfd9\n      ((Functor.mk\n            { obj := fun X => colimit (diagram J P X.unop),\n              map := fun {X Y} f =>\n                colimMap (diagramPullback J P f.unop) \u226b\n                  colimit.pre (diagram J P Y.unop) (Functor.op (pullback J f.unop)) }).obj\n        x\u271d)"}, {"tactic": "ext", "state_before": "case w.h\nC : Type u\ninst\u271d\u00b3 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP : C\u1d52\u1d56 \u2964 D\nx\u271d : C\u1d52\u1d56\n\u22a2 colimMap (\ud835\udfd9 (diagram J P x\u271d.unop)) =\n    \ud835\udfd9\n      ((Functor.mk\n            { obj := fun X => colimit (diagram J P X.unop),\n              map := fun {X Y} f =>\n                colimMap (diagramPullback J P f.unop) \u226b\n                  colimit.pre (diagram J P Y.unop) (Functor.op (pullback J f.unop)) }).obj\n        x\u271d)", "state_after": "case w.h.w\nC : Type u\ninst\u271d\u00b3 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP : C\u1d52\u1d56 \u2964 D\nx\u271d : C\u1d52\u1d56\nj\u271d : (Cover J x\u271d.unop)\u1d52\u1d56\n\u22a2 colimit.\u03b9 (diagram J P x\u271d.unop) j\u271d \u226b colimMap (\ud835\udfd9 (diagram J P x\u271d.unop)) =\n    colimit.\u03b9 (diagram J P x\u271d.unop) j\u271d \u226b\n      \ud835\udfd9\n        ((Functor.mk\n              { obj := fun X => colimit (diagram J P X.unop),\n                map := fun {X Y} f =>\n                  colimMap (diagramPullback J P f.unop) \u226b\n                    colimit.pre (diagram J P Y.unop) (Functor.op (pullback J f.unop)) }).obj\n          x\u271d)"}, {"tactic": "dsimp", "state_before": "case w.h.w\nC : Type u\ninst\u271d\u00b3 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP : C\u1d52\u1d56 \u2964 D\nx\u271d : C\u1d52\u1d56\nj\u271d : (Cover J x\u271d.unop)\u1d52\u1d56\n\u22a2 colimit.\u03b9 (diagram J P x\u271d.unop) j\u271d \u226b colimMap (\ud835\udfd9 (diagram J P x\u271d.unop)) =\n    colimit.\u03b9 (diagram J P x\u271d.unop) j\u271d \u226b\n      \ud835\udfd9\n        ((Functor.mk\n              { obj := fun X => colimit (diagram J P X.unop),\n                map := fun {X Y} f =>\n                  colimMap (diagramPullback J P f.unop) \u226b\n                    colimit.pre (diagram J P Y.unop) (Functor.op (pullback J f.unop)) }).obj\n          x\u271d)", "state_after": "case w.h.w\nC : Type u\ninst\u271d\u00b3 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP : C\u1d52\u1d56 \u2964 D\nx\u271d : C\u1d52\u1d56\nj\u271d : (Cover J x\u271d.unop)\u1d52\u1d56\n\u22a2 colimit.\u03b9 (diagram J P x\u271d.unop) j\u271d \u226b colimMap (\ud835\udfd9 (diagram J P x\u271d.unop)) =\n    colimit.\u03b9 (diagram J P x\u271d.unop) j\u271d \u226b \ud835\udfd9 (colimit (diagram J P x\u271d.unop))"}, {"tactic": "simp", "state_before": "case w.h.w\nC : Type u\ninst\u271d\u00b3 : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP : C\u1d52\u1d56 \u2964 D\nx\u271d : C\u1d52\u1d56\nj\u271d : (Cover J x\u271d.unop)\u1d52\u1d56\n\u22a2 colimit.\u03b9 (diagram J P x\u271d.unop) j\u271d \u226b colimMap (\ud835\udfd9 (diagram J P x\u271d.unop)) =\n    colimit.\u03b9 (diagram J P x\u271d.unop) j\u271d \u226b \ud835\udfd9 (colimit (diagram J P x\u271d.unop))", "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.BinaryBicone.fstKernelFork_\u03b9", "start": [1608, 1], "end": [1608, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/ValuedField.lean", "full_name": "Valued.closure_coe_completion_v_lt", "start": [327, 1], "end": [353, 35], "traced_tactics": [{"tactic": "ext x", "state_before": "K : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\n\u22a2 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) = {x | \u2191extensionValuation x < \u2191\u03b3}", "state_after": "case h\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u22a2 x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 x \u2208 {x | \u2191extensionValuation x < \u2191\u03b3}"}, {"tactic": "let \u03b3\u2080 := extensionValuation x", "state_before": "case h\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u22a2 x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 x \u2208 {x | \u2191extensionValuation x < \u2191\u03b3}", "state_after": "case h\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\n\u22a2 x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 x \u2208 {x | \u2191extensionValuation x < \u2191\u03b3}"}, {"tactic": "intro h", "state_before": "case h\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\n\u22a2 \u03b3\u2080 \u2260 0 \u2192 (x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3)", "state_after": "case h\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\n\u22a2 x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3"}, {"tactic": "have h\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x :=\n  continuous_extension.continuousAt.preimage_mem_nhds\n    (WithZeroTopology.singleton_mem_nhds_of_ne_zero h)", "state_before": "case h\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\n\u22a2 x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3", "state_after": "case h\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\n\u22a2 x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3"}, {"tactic": "rw [mem_closure_iff_nhds']", "state_before": "case h\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\n\u22a2 x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3", "state_after": "case h\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\n\u22a2 (\u2200 (t : Set (hat K)), t \u2208 \ud835\udcdd x \u2192 \u2203 y, \u2191y \u2208 t) \u2194 \u03b3\u2080 < \u2191\u03b3"}, {"tactic": "refine' \u27e8fun hx => _, fun hx s hs => _\u27e9", "state_before": "case h\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\n\u22a2 (\u2200 (t : Set (hat K)), t \u2208 \ud835\udcdd x \u2192 \u2203 y, \u2191y \u2208 t) \u2194 \u03b3\u2080 < \u2191\u03b3", "state_after": "case h.refine'_1\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u2200 (t : Set (hat K)), t \u2208 \ud835\udcdd x \u2192 \u2203 y, \u2191y \u2208 t\n\u22a2 \u03b3\u2080 < \u2191\u03b3\n\ncase h.refine'_2\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u03b3\u2080 < \u2191\u03b3\ns : Set (hat K)\nhs : s \u2208 \ud835\udcdd x\n\u22a2 \u2203 y, \u2191y \u2208 s"}, {"tactic": "cases' eq_or_ne \u03b3\u2080 0 with h h", "state_before": "K : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nthis : \u03b3\u2080 \u2260 0 \u2192 (x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3)\n\u22a2 x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 x \u2208 {x | \u2191extensionValuation x < \u2191\u03b3}", "state_after": "case inl\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nthis : \u03b3\u2080 \u2260 0 \u2192 (x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3)\nh : \u03b3\u2080 = 0\n\u22a2 x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 x \u2208 {x | \u2191extensionValuation x < \u2191\u03b3}\n\ncase inr\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nthis : \u03b3\u2080 \u2260 0 \u2192 (x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3)\nh : \u03b3\u2080 \u2260 0\n\u22a2 x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 x \u2208 {x | \u2191extensionValuation x < \u2191\u03b3}"}, {"tactic": "simp only [h, (Valuation.zero_iff _).mp h, mem_setOf_eq, Valuation.map_zero, Units.zero_lt,\n  iff_true_iff]", "state_before": "case inl\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nthis : \u03b3\u2080 \u2260 0 \u2192 (x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3)\nh : \u03b3\u2080 = 0\n\u22a2 x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 x \u2208 {x | \u2191extensionValuation x < \u2191\u03b3}", "state_after": "case inl\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nthis : \u03b3\u2080 \u2260 0 \u2192 (x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3)\nh : \u03b3\u2080 = 0\n\u22a2 0 \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3})"}, {"tactic": "apply subset_closure", "state_before": "case inl\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nthis : \u03b3\u2080 \u2260 0 \u2192 (x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3)\nh : \u03b3\u2080 = 0\n\u22a2 0 \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3})", "state_after": "case inl.a\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nthis : \u03b3\u2080 \u2260 0 \u2192 (x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3)\nh : \u03b3\u2080 = 0\n\u22a2 0 \u2208 \u2191K '' {x | \u2191v x < \u2191\u03b3}"}, {"tactic": "exact \u27e80, by simp only [mem_setOf_eq, Valuation.map_zero, Units.zero_lt, true_and_iff]; rfl \u27e9", "state_before": "case inl.a\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nthis : \u03b3\u2080 \u2260 0 \u2192 (x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3)\nh : \u03b3\u2080 = 0\n\u22a2 0 \u2208 \u2191K '' {x | \u2191v x < \u2191\u03b3}", "state_after": "no goals"}, {"tactic": "simp only [mem_setOf_eq, Valuation.map_zero, Units.zero_lt, true_and_iff]", "state_before": "K : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nthis : \u03b3\u2080 \u2260 0 \u2192 (x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3)\nh : \u03b3\u2080 = 0\n\u22a2 0 \u2208 {x | \u2191v x < \u2191\u03b3} \u2227 \u2191K 0 = 0", "state_after": "K : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nthis : \u03b3\u2080 \u2260 0 \u2192 (x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3)\nh : \u03b3\u2080 = 0\n\u22a2 \u2191K 0 = 0"}, {"tactic": "rfl", "state_before": "K : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nthis : \u03b3\u2080 \u2260 0 \u2192 (x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3)\nh : \u03b3\u2080 = 0\n\u22a2 \u2191K 0 = 0", "state_after": "no goals"}, {"tactic": "exact this h", "state_before": "case inr\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nthis : \u03b3\u2080 \u2260 0 \u2192 (x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3)\nh : \u03b3\u2080 \u2260 0\n\u22a2 x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 x \u2208 {x | \u2191extensionValuation x < \u2191\u03b3}", "state_after": "no goals"}, {"tactic": "obtain \u27e8\u27e8-, y, hy\u2081 : v y < (\u03b3 : \u0393\u2080), rfl\u27e9, hy\u2082\u27e9 := hx _ h\u03b3\u2080", "state_before": "case h.refine'_1\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u2200 (t : Set (hat K)), t \u2208 \ud835\udcdd x \u2192 \u2203 y, \u2191y \u2208 t\n\u22a2 \u03b3\u2080 < \u2191\u03b3", "state_after": "case h.refine'_1.intro.mk.intro.intro\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u2200 (t : Set (hat K)), t \u2208 \ud835\udcdd x \u2192 \u2203 y, \u2191y \u2208 t\ny : K\nhy\u2081 : \u2191v y < \u2191\u03b3\nhy\u2082 : \u2191{ val := \u2191K y, property := (_ : \u2203 a, a \u2208 {x | \u2191v x < \u2191\u03b3} \u2227 \u2191K a = \u2191K y) } \u2208 extension \u207b\u00b9' {\u03b3\u2080}\n\u22a2 \u03b3\u2080 < \u2191\u03b3"}, {"tactic": "replace hy\u2082 : v y = \u03b3\u2080", "state_before": "case h.refine'_1.intro.mk.intro.intro\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u2200 (t : Set (hat K)), t \u2208 \ud835\udcdd x \u2192 \u2203 y, \u2191y \u2208 t\ny : K\nhy\u2081 : \u2191v y < \u2191\u03b3\nhy\u2082 : \u2191{ val := \u2191K y, property := (_ : \u2203 a, a \u2208 {x | \u2191v x < \u2191\u03b3} \u2227 \u2191K a = \u2191K y) } \u2208 extension \u207b\u00b9' {\u03b3\u2080}\n\u22a2 \u03b3\u2080 < \u2191\u03b3", "state_after": "case hy\u2082\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u2200 (t : Set (hat K)), t \u2208 \ud835\udcdd x \u2192 \u2203 y, \u2191y \u2208 t\ny : K\nhy\u2081 : \u2191v y < \u2191\u03b3\nhy\u2082 : \u2191{ val := \u2191K y, property := (_ : \u2203 a, a \u2208 {x | \u2191v x < \u2191\u03b3} \u2227 \u2191K a = \u2191K y) } \u2208 extension \u207b\u00b9' {\u03b3\u2080}\n\u22a2 \u2191v y = \u03b3\u2080\n\ncase h.refine'_1.intro.mk.intro.intro\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u2200 (t : Set (hat K)), t \u2208 \ud835\udcdd x \u2192 \u2203 y, \u2191y \u2208 t\ny : K\nhy\u2081 : \u2191v y < \u2191\u03b3\nhy\u2082 : \u2191v y = \u03b3\u2080\n\u22a2 \u03b3\u2080 < \u2191\u03b3"}, {"tactic": "rwa [\u2190 hy\u2082]", "state_before": "case h.refine'_1.intro.mk.intro.intro\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u2200 (t : Set (hat K)), t \u2208 \ud835\udcdd x \u2192 \u2203 y, \u2191y \u2208 t\ny : K\nhy\u2081 : \u2191v y < \u2191\u03b3\nhy\u2082 : \u2191v y = \u03b3\u2080\n\u22a2 \u03b3\u2080 < \u2191\u03b3", "state_after": "no goals"}, {"tactic": "simpa using hy\u2082", "state_before": "case hy\u2082\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u2200 (t : Set (hat K)), t \u2208 \ud835\udcdd x \u2192 \u2203 y, \u2191y \u2208 t\ny : K\nhy\u2081 : \u2191v y < \u2191\u03b3\nhy\u2082 : \u2191{ val := \u2191K y, property := (_ : \u2203 a, a \u2208 {x | \u2191v x < \u2191\u03b3} \u2227 \u2191K a = \u2191K y) } \u2208 extension \u207b\u00b9' {\u03b3\u2080}\n\u22a2 \u2191v y = \u03b3\u2080", "state_after": "no goals"}, {"tactic": "obtain \u27e8y, hy\u2081, hy\u2082\u27e9 := Completion.denseRange_coe.mem_nhds (inter_mem h\u03b3\u2080 hs)", "state_before": "case h.refine'_2\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u03b3\u2080 < \u2191\u03b3\ns : Set (hat K)\nhs : s \u2208 \ud835\udcdd x\n\u22a2 \u2203 y, \u2191y \u2208 s", "state_after": "case h.refine'_2.intro.intro\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u03b3\u2080 < \u2191\u03b3\ns : Set (hat K)\nhs : s \u2208 \ud835\udcdd x\ny : K\nhy\u2081 : \u2191K y \u2208 extension \u207b\u00b9' {\u03b3\u2080}\nhy\u2082 : \u2191K y \u2208 s\n\u22a2 \u2203 y, \u2191y \u2208 s"}, {"tactic": "replace hy\u2081 : v y = \u03b3\u2080", "state_before": "case h.refine'_2.intro.intro\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u03b3\u2080 < \u2191\u03b3\ns : Set (hat K)\nhs : s \u2208 \ud835\udcdd x\ny : K\nhy\u2081 : \u2191K y \u2208 extension \u207b\u00b9' {\u03b3\u2080}\nhy\u2082 : \u2191K y \u2208 s\n\u22a2 \u2203 y, \u2191y \u2208 s", "state_after": "case hy\u2081\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u03b3\u2080 < \u2191\u03b3\ns : Set (hat K)\nhs : s \u2208 \ud835\udcdd x\ny : K\nhy\u2081 : \u2191K y \u2208 extension \u207b\u00b9' {\u03b3\u2080}\nhy\u2082 : \u2191K y \u2208 s\n\u22a2 \u2191v y = \u03b3\u2080\n\ncase h.refine'_2.intro.intro\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u03b3\u2080 < \u2191\u03b3\ns : Set (hat K)\nhs : s \u2208 \ud835\udcdd x\ny : K\nhy\u2082 : \u2191K y \u2208 s\nhy\u2081 : \u2191v y = \u03b3\u2080\n\u22a2 \u2203 y, \u2191y \u2208 s"}, {"tactic": "rw [\u2190 hy\u2081] at hx", "state_before": "case h.refine'_2.intro.intro\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u03b3\u2080 < \u2191\u03b3\ns : Set (hat K)\nhs : s \u2208 \ud835\udcdd x\ny : K\nhy\u2082 : \u2191K y \u2208 s\nhy\u2081 : \u2191v y = \u03b3\u2080\n\u22a2 \u2203 y, \u2191y \u2208 s", "state_after": "case h.refine'_2.intro.intro\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\ns : Set (hat K)\nhs : s \u2208 \ud835\udcdd x\ny : K\nhx : \u2191v y < \u2191\u03b3\nhy\u2082 : \u2191K y \u2208 s\nhy\u2081 : \u2191v y = \u03b3\u2080\n\u22a2 \u2203 y, \u2191y \u2208 s"}, {"tactic": "exact \u27e8\u27e8y, \u27e8y, hx, rfl\u27e9\u27e9, hy\u2082\u27e9", "state_before": "case h.refine'_2.intro.intro\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\ns : Set (hat K)\nhs : s \u2208 \ud835\udcdd x\ny : K\nhx : \u2191v y < \u2191\u03b3\nhy\u2082 : \u2191K y \u2208 s\nhy\u2081 : \u2191v y = \u03b3\u2080\n\u22a2 \u2203 y, \u2191y \u2208 s", "state_after": "no goals"}, {"tactic": "simpa using hy\u2081", "state_before": "case hy\u2081\nK : Type u_2\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_1\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u03b3\u2080 < \u2191\u03b3\ns : Set (hat K)\nhs : s \u2208 \ud835\udcdd x\ny : K\nhy\u2081 : \u2191K y \u2208 extension \u207b\u00b9' {\u03b3\u2080}\nhy\u2082 : \u2191K y \u2208 s\n\u22a2 \u2191v y = \u03b3\u2080", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/HasseDeriv.lean", "full_name": "Polynomial.hasseDeriv_C", "start": [132, 1], "end": [134, 48], "traced_tactics": [{"tactic": "rw [\u2190 monomial_zero_left, hasseDeriv_monomial, Nat.choose_eq_zero_of_lt hk, Nat.cast_zero,\n  MulZeroClass.zero_mul, monomial_zero_right]", "state_before": "R : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nr : R\nhk : 0 < k\n\u22a2 \u2191(hasseDeriv k) (\u2191C r) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Int/Lemmas.lean", "full_name": "Int.subNatNat_of_le", "start": [130, 1], "end": [131, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Finrank.lean", "full_name": "finrank_span_le_card", "start": [314, 1], "end": [315, 65], "traced_tactics": [{"tactic": "simpa using rank_span_le (K := K) s", "state_before": "K : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ns : Set V\ninst\u271d : Fintype \u2191s\n\u22a2 Module.rank K { x // x \u2208 span K s } \u2264 \u2191(Finset.card (Set.toFinset s))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/PNat/Prime.lean", "full_name": "Nat.Primes.coe_pnat_injective", "start": [38, 1], "end": [39, 31], "traced_tactics": [{"tactic": "injection h", "state_before": "p q : Primes\nh : \u2191p = \u2191q\n\u22a2 \u2191p = \u2191q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Matrix/Basic.lean", "full_name": "Matrix.diagonal_map", "start": [509, 1], "end": [513, 25], "traced_tactics": [{"tactic": "ext", "state_before": "l : Type ?u.70038\nm : Type ?u.70041\nn : Type u_1\no : Type ?u.70047\nm' : o \u2192 Type ?u.70052\nn' : o \u2192 Type ?u.70057\nR : Type ?u.70060\nS : Type ?u.70063\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type ?u.70070\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : Zero \u03b2\nf : \u03b1 \u2192 \u03b2\nh : f 0 = 0\nd : n \u2192 \u03b1\n\u22a2 map (diagonal d) f = diagonal fun m => f (d m)", "state_after": "case a.h\nl : Type ?u.70038\nm : Type ?u.70041\nn : Type u_1\no : Type ?u.70047\nm' : o \u2192 Type ?u.70052\nn' : o \u2192 Type ?u.70057\nR : Type ?u.70060\nS : Type ?u.70063\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type ?u.70070\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : Zero \u03b2\nf : \u03b1 \u2192 \u03b2\nh : f 0 = 0\nd : n \u2192 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\u03b1\ninst\u271d : Zero \u03b2\nf : \u03b1 \u2192 \u03b2\nh : f 0 = 0\nd : n \u2192 \u03b1\ni\u271d x\u271d : n\n\u22a2 f (if i\u271d = x\u271d then d i\u271d else 0) = if i\u271d = x\u271d then f (d i\u271d) else 0"}, {"tactic": "split_ifs <;> simp [h]", "state_before": "case a.h\nl : Type ?u.70038\nm : Type ?u.70041\nn : Type u_1\no : Type ?u.70047\nm' : o \u2192 Type ?u.70052\nn' : o \u2192 Type ?u.70057\nR : Type ?u.70060\nS : Type ?u.70063\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type ?u.70070\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : Zero \u03b2\nf : \u03b1 \u2192 \u03b2\nh : f 0 = 0\nd : n \u2192 \u03b1\ni\u271d x\u271d : n\n\u22a2 f (if i\u271d = x\u271d then d i\u271d else 0) = if i\u271d = x\u271d then f (d i\u271d) else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": 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"end": [68, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasurableEmbedding.snorm_map_measure", "start": [929, 1], "end": [938, 8], "traced_tactics": [{"tactic": "by_cases hp_zero : p = 0", "state_before": "\u03b1 : Type u_1\nE : Type ?u.3101853\nF : Type u_3\nG : Type ?u.3101859\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\n\u03b2 : Type u_2\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type ?u.3101853\nF : Type u_3\nG : Type ?u.3101859\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\n\u03b2 : Type u_2\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : p = 0\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc\n\ncase neg\n\u03b1 : Type u_1\nE : Type ?u.3101853\nF : Type u_3\nG : Type ?u.3101859\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\n\u03b2 : Type u_2\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc"}, {"tactic": "by_cases hp : p = \u221e", "state_before": "case neg\n\u03b1 : Type u_1\nE : Type ?u.3101853\nF : Type u_3\nG : Type ?u.3101859\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\n\u03b2 : Type u_2\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type ?u.3101853\nF : Type u_3\nG : Type ?u.3101859\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\n\u03b2 : Type u_2\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : p = \u22a4\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc\n\ncase neg\n\u03b1 : Type u_1\nE : Type ?u.3101853\nF : Type u_3\nG : Type ?u.3101859\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\n\u03b2 : Type u_2\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : \u00acp = \u22a4\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc"}, {"tactic": "simp only [hp_zero, snorm_exponent_zero]", "state_before": "case pos\n\u03b1 : Type u_1\nE : Type ?u.3101853\nF : Type u_3\nG : Type ?u.3101859\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\n\u03b2 : Type u_2\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : p = 0\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc", "state_after": "no goals"}, {"tactic": "simp_rw [hp, snorm_exponent_top]", "state_before": "case pos\n\u03b1 : Type u_1\nE : Type ?u.3101853\nF : Type u_3\nG : Type ?u.3101859\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\n\u03b2 : Type u_2\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : p = \u22a4\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type ?u.3101853\nF : Type u_3\nG : Type ?u.3101859\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\n\u03b2 : Type u_2\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : p = \u22a4\n\u22a2 snormEssSup g (Measure.map f \u03bc) = snormEssSup (g \u2218 f) \u03bc"}, {"tactic": "exact hf.essSup_map_measure", "state_before": "case pos\n\u03b1 : Type u_1\nE : Type ?u.3101853\nF : Type u_3\nG : Type ?u.3101859\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\n\u03b2 : Type u_2\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : p = \u22a4\n\u22a2 snormEssSup g (Measure.map f \u03bc) = snormEssSup (g \u2218 f) \u03bc", "state_after": "no goals"}, {"tactic": "simp_rw [snorm_eq_lintegral_rpow_nnnorm hp_zero hp]", "state_before": "case neg\n\u03b1 : Type u_1\nE : Type ?u.3101853\nF : Type u_3\nG : Type ?u.3101859\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\n\u03b2 : Type u_2\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : \u00acp = \u22a4\n\u22a2 snorm g p (Measure.map f \u03bc) = snorm (g \u2218 f) p \u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type ?u.3101853\nF : Type u_3\nG : Type ?u.3101859\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\n\u03b2 : Type u_2\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : \u00acp = \u22a4\n\u22a2 (\u222b\u207b (x : \u03b2), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202Measure.map f \u03bc) ^ (1 / ENNReal.toReal p) =\n    (\u222b\u207b (x : \u03b1), \u2191\u2016(g \u2218 f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)"}, {"tactic": "rw [hf.lintegral_map]", "state_before": "case neg\n\u03b1 : Type u_1\nE : Type ?u.3101853\nF : Type u_3\nG : Type ?u.3101859\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\n\u03b2 : Type u_2\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : \u00acp = \u22a4\n\u22a2 (\u222b\u207b (x : \u03b2), \u2191\u2016g x\u2016\u208a ^ ENNReal.toReal p \u2202Measure.map f \u03bc) ^ (1 / ENNReal.toReal p) =\n    (\u222b\u207b (x : \u03b1), \u2191\u2016(g \u2218 f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type ?u.3101853\nF : Type u_3\nG : Type ?u.3101859\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\n\u03b2 : Type u_2\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : \u00acp = \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016g (f a)\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) =\n    (\u222b\u207b (x : \u03b1), \u2191\u2016(g \u2218 f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)"}, {"tactic": "rfl", "state_before": "case neg\n\u03b1 : Type u_1\nE : Type ?u.3101853\nF : Type u_3\nG : Type ?u.3101859\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\n\u03b2 : Type u_2\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nhf : MeasurableEmbedding f\nhp_zero : \u00acp = 0\nhp : \u00acp = \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016g (f a)\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) =\n    (\u222b\u207b (x : \u03b1), \u2191\u2016(g \u2218 f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Basic.lean", "full_name": "Inducing.isSeparable_preimage", "start": [1937, 11], "end": [1944, 48], "traced_tactics": [{"tactic": "have : SeparableSpace s := hs.separableSpace", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.307583\n\u03b9 : Type ?u.307586\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ninst\u271d : TopologicalSpace \u03b2\nhf : Inducing f\ns : Set \u03b1\nhs : IsSeparable s\n\u22a2 IsSeparable (f \u207b\u00b9' s)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.307583\n\u03b9 : Type ?u.307586\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ninst\u271d : TopologicalSpace \u03b2\nhf : Inducing f\ns : Set \u03b1\nhs : IsSeparable s\nthis : SeparableSpace \u2191s\n\u22a2 IsSeparable (f \u207b\u00b9' s)"}, {"tactic": "have : SecondCountableTopology s := UniformSpace.secondCountable_of_separable _", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.307583\n\u03b9 : Type ?u.307586\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ninst\u271d : TopologicalSpace \u03b2\nhf : Inducing f\ns : Set \u03b1\nhs : IsSeparable s\nthis : SeparableSpace \u2191s\n\u22a2 IsSeparable (f \u207b\u00b9' s)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.307583\n\u03b9 : Type ?u.307586\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ninst\u271d : TopologicalSpace \u03b2\nhf : Inducing f\ns : Set \u03b1\nhs : IsSeparable s\nthis\u271d : SeparableSpace \u2191s\nthis : SecondCountableTopology \u2191s\n\u22a2 IsSeparable (f \u207b\u00b9' s)"}, {"tactic": "have : Inducing ((mapsTo_preimage f s).restrict _ _ _) :=\n  (hf.comp inducing_subtype_val).codRestrict _", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.307583\n\u03b9 : Type ?u.307586\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ninst\u271d : TopologicalSpace \u03b2\nhf : Inducing f\ns : Set \u03b1\nhs : IsSeparable s\nthis\u271d : SeparableSpace \u2191s\nthis : SecondCountableTopology \u2191s\n\u22a2 IsSeparable (f \u207b\u00b9' s)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.307583\n\u03b9 : Type ?u.307586\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ninst\u271d : TopologicalSpace \u03b2\nhf : Inducing f\ns : Set \u03b1\nhs : IsSeparable s\nthis\u271d\u00b9 : SeparableSpace \u2191s\nthis\u271d : SecondCountableTopology \u2191s\nthis : Inducing (MapsTo.restrict f (f \u207b\u00b9' s) s (_ : MapsTo f (f \u207b\u00b9' s) s))\n\u22a2 IsSeparable (f \u207b\u00b9' s)"}, {"tactic": "have := this.secondCountableTopology", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.307583\n\u03b9 : Type ?u.307586\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ninst\u271d : TopologicalSpace \u03b2\nhf : Inducing f\ns : Set \u03b1\nhs : IsSeparable s\nthis\u271d\u00b9 : SeparableSpace \u2191s\nthis\u271d : SecondCountableTopology \u2191s\nthis : Inducing (MapsTo.restrict f (f \u207b\u00b9' s) s (_ : MapsTo f (f \u207b\u00b9' s) s))\n\u22a2 IsSeparable (f \u207b\u00b9' s)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.307583\n\u03b9 : Type ?u.307586\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ninst\u271d : TopologicalSpace \u03b2\nhf : Inducing f\ns : Set \u03b1\nhs : IsSeparable s\nthis\u271d\u00b2 : SeparableSpace \u2191s\nthis\u271d\u00b9 : SecondCountableTopology \u2191s\nthis\u271d : Inducing (MapsTo.restrict f (f \u207b\u00b9' s) s (_ : MapsTo f (f \u207b\u00b9' s) s))\nthis : SecondCountableTopology \u2191(f \u207b\u00b9' s)\n\u22a2 IsSeparable (f \u207b\u00b9' s)"}, {"tactic": "exact isSeparable_of_separableSpace_subtype _", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.307583\n\u03b9 : Type ?u.307586\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\nf : \u03b2 \u2192 \u03b1\ninst\u271d : TopologicalSpace \u03b2\nhf : Inducing f\ns : Set \u03b1\nhs : IsSeparable s\nthis\u271d\u00b2 : SeparableSpace \u2191s\nthis\u271d\u00b9 : SecondCountableTopology \u2191s\nthis\u271d : Inducing (MapsTo.restrict f (f \u207b\u00b9' s) s (_ : MapsTo f (f \u207b\u00b9' s) s))\nthis : SecondCountableTopology \u2191(f \u207b\u00b9' s)\n\u22a2 IsSeparable (f \u207b\u00b9' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/FiniteDimension.lean", "full_name": "exists_norm_le_le_norm_sub_of_finset", "start": [397, 1], "end": [417, 62], "traced_tactics": [{"tactic": "let F := Submodule.span \ud835\udd5c (s : Set E)", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\n\u22a2 \u2203 x, \u2016x\u2016 \u2264 R \u2227 \u2200 (y : E), y \u2208 s \u2192 1 \u2264 \u2016y - x\u2016", "state_after": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\n\u22a2 \u2203 x, \u2016x\u2016 \u2264 R \u2227 \u2200 (y : E), y \u2208 s \u2192 1 \u2264 \u2016y - x\u2016"}, {"tactic": "haveI : FiniteDimensional \ud835\udd5c F :=\n  Module.finite_def.2\n    ((Submodule.fg_top _).2 (Submodule.fg_def.2 \u27e8s, Finset.finite_toSet _, rfl\u27e9))", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\n\u22a2 \u2203 x, \u2016x\u2016 \u2264 R \u2227 \u2200 (y : E), y \u2208 s \u2192 1 \u2264 \u2016y - x\u2016", "state_after": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\n\u22a2 \u2203 x, \u2016x\u2016 \u2264 R \u2227 \u2200 (y : E), y \u2208 s \u2192 1 \u2264 \u2016y - x\u2016"}, {"tactic": "have Fclosed : IsClosed (F : Set E) := Submodule.closed_of_finiteDimensional _", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\n\u22a2 \u2203 x, \u2016x\u2016 \u2264 R \u2227 \u2200 (y : E), y \u2208 s \u2192 1 \u2264 \u2016y - x\u2016", "state_after": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\n\u22a2 \u2203 x, \u2016x\u2016 \u2264 R \u2227 \u2200 (y : E), y \u2208 s \u2192 1 \u2264 \u2016y - x\u2016"}, {"tactic": "have : \u2203 x, x \u2209 F := by\n  contrapose! h\n  have : (\u22a4 : Submodule \ud835\udd5c E) = F := by\n    ext x\n    simp [h]\n  have : FiniteDimensional \ud835\udd5c (\u22a4 : Submodule \ud835\udd5c E) := by rwa [this]\n  refine' Module.finite_def.2 ((Submodule.fg_top _).1 (Module.finite_def.1 this))", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\n\u22a2 \u2203 x, \u2016x\u2016 \u2264 R \u2227 \u2200 (y : E), y \u2208 s \u2192 1 \u2264 \u2016y - x\u2016", "state_after": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nthis : \u2203 x, \u00acx \u2208 F\n\u22a2 \u2203 x, \u2016x\u2016 \u2264 R \u2227 \u2200 (y : E), y \u2208 s \u2192 1 \u2264 \u2016y - x\u2016"}, {"tactic": "obtain \u27e8x, xR, hx\u27e9 : \u2203 x : E, \u2016x\u2016 \u2264 R \u2227 \u2200 y : E, y \u2208 F \u2192 1 \u2264 \u2016x - y\u2016 :=\n  riesz_lemma_of_norm_lt hc hR Fclosed this", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nthis : \u2203 x, \u00acx \u2208 F\n\u22a2 \u2203 x, \u2016x\u2016 \u2264 R \u2227 \u2200 (y : E), y \u2208 s \u2192 1 \u2264 \u2016y - x\u2016", "state_after": "case intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nthis : \u2203 x, \u00acx \u2208 F\nx : E\nxR : \u2016x\u2016 \u2264 R\nhx : \u2200 (y : E), y \u2208 F \u2192 1 \u2264 \u2016x - y\u2016\n\u22a2 \u2203 x, \u2016x\u2016 \u2264 R \u2227 \u2200 (y : E), y \u2208 s \u2192 1 \u2264 \u2016y - x\u2016"}, {"tactic": "have hx' : \u2200 y : E, y \u2208 F \u2192 1 \u2264 \u2016y - x\u2016 := by\n  intro y hy\n  rw [\u2190 norm_neg]\n  simpa using hx y hy", "state_before": "case intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nthis : \u2203 x, \u00acx \u2208 F\nx : E\nxR : \u2016x\u2016 \u2264 R\nhx : \u2200 (y : E), y \u2208 F \u2192 1 \u2264 \u2016x - y\u2016\n\u22a2 \u2203 x, \u2016x\u2016 \u2264 R \u2227 \u2200 (y : E), y \u2208 s \u2192 1 \u2264 \u2016y - x\u2016", "state_after": "case intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nthis : \u2203 x, \u00acx \u2208 F\nx : E\nxR : \u2016x\u2016 \u2264 R\nhx : \u2200 (y : E), y \u2208 F \u2192 1 \u2264 \u2016x - y\u2016\nhx' : \u2200 (y : E), y \u2208 F \u2192 1 \u2264 \u2016y - x\u2016\n\u22a2 \u2203 x, \u2016x\u2016 \u2264 R \u2227 \u2200 (y : E), y \u2208 s \u2192 1 \u2264 \u2016y - x\u2016"}, {"tactic": "exact \u27e8x, xR, fun y hy => hx' _ (Submodule.subset_span hy)\u27e9", "state_before": "case intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nthis : \u2203 x, \u00acx \u2208 F\nx : E\nxR : \u2016x\u2016 \u2264 R\nhx : \u2200 (y : E), y \u2208 F \u2192 1 \u2264 \u2016x - y\u2016\nhx' : \u2200 (y : E), y \u2208 F \u2192 1 \u2264 \u2016y - x\u2016\n\u22a2 \u2203 x, \u2016x\u2016 \u2264 R \u2227 \u2200 (y : E), y \u2208 s \u2192 1 \u2264 \u2016y - x\u2016", "state_after": "no goals"}, {"tactic": "contrapose! h", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\n\u22a2 \u2203 x, \u00acx \u2208 F", "state_after": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nh : \u2200 (x : E), x \u2208 Submodule.span \ud835\udd5c \u2191s\n\u22a2 FiniteDimensional \ud835\udd5c E"}, {"tactic": "have : (\u22a4 : Submodule \ud835\udd5c E) = F := by\n  ext x\n  simp [h]", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nh : \u2200 (x : E), x \u2208 Submodule.span \ud835\udd5c \u2191s\n\u22a2 FiniteDimensional \ud835\udd5c E", "state_after": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nh : \u2200 (x : E), x \u2208 Submodule.span \ud835\udd5c \u2191s\nthis : \u22a4 = F\n\u22a2 FiniteDimensional \ud835\udd5c E"}, {"tactic": "have : FiniteDimensional \ud835\udd5c (\u22a4 : Submodule \ud835\udd5c E) := by rwa [this]", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nh : \u2200 (x : E), x \u2208 Submodule.span \ud835\udd5c \u2191s\nthis : \u22a4 = F\n\u22a2 FiniteDimensional \ud835\udd5c E", "state_after": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d\u00b9 : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nh : \u2200 (x : E), x \u2208 Submodule.span \ud835\udd5c \u2191s\nthis\u271d : \u22a4 = F\nthis : FiniteDimensional \ud835\udd5c { x // x \u2208 \u22a4 }\n\u22a2 FiniteDimensional \ud835\udd5c E"}, {"tactic": "refine' Module.finite_def.2 ((Submodule.fg_top _).1 (Module.finite_def.1 this))", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d\u00b9 : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nh : \u2200 (x : E), x \u2208 Submodule.span \ud835\udd5c \u2191s\nthis\u271d : \u22a4 = F\nthis : FiniteDimensional \ud835\udd5c { x // x \u2208 \u22a4 }\n\u22a2 FiniteDimensional \ud835\udd5c E", "state_after": "no goals"}, {"tactic": "ext x", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nh : \u2200 (x : E), x \u2208 Submodule.span \ud835\udd5c \u2191s\n\u22a2 \u22a4 = F", "state_after": "case h\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nh : \u2200 (x : E), x \u2208 Submodule.span \ud835\udd5c \u2191s\nx : E\n\u22a2 x \u2208 \u22a4 \u2194 x \u2208 F"}, {"tactic": "simp [h]", "state_before": "case h\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nh : \u2200 (x : E), x \u2208 Submodule.span \ud835\udd5c \u2191s\nx : E\n\u22a2 x \u2208 \u22a4 \u2194 x \u2208 F", "state_after": "no goals"}, {"tactic": "rwa [this]", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nh : \u2200 (x : E), x \u2208 Submodule.span \ud835\udd5c \u2191s\nthis : \u22a4 = F\n\u22a2 FiniteDimensional \ud835\udd5c { x // x \u2208 \u22a4 }", "state_after": "no goals"}, {"tactic": "intro y hy", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nthis : \u2203 x, \u00acx \u2208 F\nx : E\nxR : \u2016x\u2016 \u2264 R\nhx : \u2200 (y : E), y \u2208 F \u2192 1 \u2264 \u2016x - y\u2016\n\u22a2 \u2200 (y : E), y \u2208 F \u2192 1 \u2264 \u2016y - x\u2016", "state_after": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nthis : \u2203 x, \u00acx \u2208 F\nx : E\nxR : \u2016x\u2016 \u2264 R\nhx : \u2200 (y : E), y \u2208 F \u2192 1 \u2264 \u2016x - y\u2016\ny : E\nhy : y \u2208 F\n\u22a2 1 \u2264 \u2016y - x\u2016"}, {"tactic": "rw [\u2190 norm_neg]", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nthis : \u2203 x, \u00acx \u2208 F\nx : E\nxR : \u2016x\u2016 \u2264 R\nhx : \u2200 (y : E), y \u2208 F \u2192 1 \u2264 \u2016x - y\u2016\ny : E\nhy : y \u2208 F\n\u22a2 1 \u2264 \u2016y - x\u2016", "state_after": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nthis : \u2203 x, \u00acx \u2208 F\nx : E\nxR : \u2016x\u2016 \u2264 R\nhx : \u2200 (y : E), y \u2208 F \u2192 1 \u2264 \u2016x - y\u2016\ny : E\nhy : y \u2208 F\n\u22a2 1 \u2264 \u2016-(y - x)\u2016"}, {"tactic": "simpa using hx y hy", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nthis : \u2203 x, \u00acx \u2208 F\nx : E\nxR : \u2016x\u2016 \u2264 R\nhx : \u2200 (y : E), y \u2208 F \u2192 1 \u2264 \u2016x - y\u2016\ny : E\nhy : y \u2208 F\n\u22a2 1 \u2264 \u2016-(y - x)\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Control/Fold.lean", "full_name": "Traversable.toList_map", "start": [370, 1], "end": [372, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/Multiplicity.lean", "full_name": "padicValNat.pow_two_sub_pow", "start": [384, 1], "end": [394, 80], "traced_tactics": [{"tactic": "simp only [\u2190 PartENat.natCast_inj, Nat.cast_add]", "state_before": "R : Type ?u.877564\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 padicValNat 2 (x ^ n - y ^ n) + 1 = padicValNat 2 (x + y) + padicValNat 2 (x - y) + padicValNat 2 n", "state_after": "R : Type ?u.877564\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 \u2191(padicValNat 2 (x ^ n - y ^ n)) + \u21911 = \u2191(padicValNat 2 (x + y)) + \u2191(padicValNat 2 (x - y)) + \u2191(padicValNat 2 n)"}, {"tactic": "iterate 4 rw [padicValNat_def, PartENat.natCast_get]", "state_before": "R : Type ?u.877564\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 \u2191(padicValNat 2 (x ^ n - y ^ n)) + \u21911 = \u2191(padicValNat 2 (x + y)) + \u2191(padicValNat 2 (x - y)) + \u2191(padicValNat 2 n)", "state_after": "R : Type ?u.877564\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 multiplicity 2 (x ^ n - y ^ n) + \u21911 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 n\n\nR : Type ?u.877564\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 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IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))"}, {"tactic": "refine' (hf.mono _).intervalIntegrable", "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))", "state_after": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 uIcc (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1)) \u2286 Icc x\u2080 (x\u2080 + \u2191a)"}, {"tactic": "rw [uIcc_of_le]", "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 uIcc (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1)) \u2286 Icc x\u2080 (x\u2080 + \u2191a)", "state_after": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 Icc (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1)) \u2286 Icc x\u2080 (x\u2080 + \u2191a)\n\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 x\u2080 + \u2191k \u2264 x\u2080 + \u2191(k + 1)"}, {"tactic": "apply Icc_subset_Icc", "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 Icc (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1)) \u2286 Icc x\u2080 (x\u2080 + \u2191a)", "state_after": "case h\u2081\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 x\u2080 \u2264 x\u2080 + \u2191k\n\ncase h\u2082\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 x\u2080 + \u2191(k + 1) \u2264 x\u2080 + \u2191a"}, {"tactic": "simp only [le_add_iff_nonneg_right, Nat.cast_nonneg]", "state_before": "case h\u2081\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 x\u2080 \u2264 x\u2080 + \u2191k", "state_after": "no goals"}, {"tactic": "simp only [add_le_add_iff_left, Nat.cast_le, Nat.succ_le_of_lt hk]", "state_before": "case h\u2082\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 x\u2080 + \u2191(k + 1) \u2264 x\u2080 + \u2191a", "state_after": "no goals"}, {"tactic": "simp only [add_le_add_iff_left, Nat.cast_le, Nat.le_succ]", "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nk : \u2115\nhk : k < a\n\u22a2 x\u2080 + \u2191k \u2264 x\u2080 + \u2191(k + 1)", "state_after": "no goals"}, {"tactic": "simp", "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\n\u22a2 \u2211 i in Finset.range a, f (x\u2080 + \u2191(i + 1)) =\n    \u2211 i in Finset.range a, \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f (x\u2080 + \u2191(i + 1))", "state_after": "no goals"}, {"tactic": "apply Finset.sum_le_sum fun i hi => ?_", "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\n\u22a2 (\u2211 i in Finset.range a, \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f (x\u2080 + \u2191(i + 1))) \u2264\n    \u2211 i in Finset.range a, \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f x", "state_after": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\n\u22a2 (\u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f (x\u2080 + \u2191(i + 1))) \u2264 \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f x"}, {"tactic": "have ia : i + 1 \u2264 a := Finset.mem_range.1 hi", "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\n\u22a2 (\u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f (x\u2080 + \u2191(i + 1))) \u2264 \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f x", "state_after": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i + 1 \u2264 a\n\u22a2 (\u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f (x\u2080 + \u2191(i + 1))) \u2264 \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f x"}, {"tactic": "refine' intervalIntegral.integral_mono_on (by simp) (by simp) (hint _ ia) fun x hx => _", "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i + 1 \u2264 a\n\u22a2 (\u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f (x\u2080 + \u2191(i + 1))) \u2264 \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f x", "state_after": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i + 1 \u2264 a\nx : \u211d\nhx : x \u2208 Icc (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))\n\u22a2 f (x\u2080 + \u2191(i + 1)) \u2264 f x"}, {"tactic": "apply hf _ _ hx.2", "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i + 1 \u2264 a\nx : \u211d\nhx : x \u2208 Icc (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))\n\u22a2 f (x\u2080 + \u2191(i + 1)) \u2264 f x", "state_after": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i + 1 \u2264 a\nx : \u211d\nhx : x \u2208 Icc (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))\n\u22a2 x \u2208 Icc x\u2080 (x\u2080 + \u2191a)\n\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i + 1 \u2264 a\nx : \u211d\nhx : x \u2208 Icc (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))\n\u22a2 x\u2080 + \u2191(i + 1) \u2208 Icc x\u2080 (x\u2080 + \u2191a)"}, {"tactic": "simp", "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i + 1 \u2264 a\n\u22a2 x\u2080 + \u2191i \u2264 x\u2080 + \u2191(i + 1)", "state_after": "no goals"}, {"tactic": "simp", "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i + 1 \u2264 a\n\u22a2 IntervalIntegrable (fun x => f (x\u2080 + \u2191(i + 1))) volume (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))", "state_after": "no goals"}, {"tactic": "refine' mem_Icc.2 \u27e8le_trans ((le_add_iff_nonneg_right _).2 (Nat.cast_nonneg _)) hx.1,\n  le_trans hx.2 _\u27e9", "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i + 1 \u2264 a\nx : \u211d\nhx : x \u2208 Icc (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))\n\u22a2 x \u2208 Icc x\u2080 (x\u2080 + \u2191a)", "state_after": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i + 1 \u2264 a\nx : \u211d\nhx : x \u2208 Icc (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))\n\u22a2 x\u2080 + \u2191(i + 1) \u2264 x\u2080 + \u2191a"}, {"tactic": "simp only [Nat.cast_le, add_le_add_iff_left, ia]", "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i + 1 \u2264 a\nx : \u211d\nhx : x \u2208 Icc (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))\n\u22a2 x\u2080 + \u2191(i + 1) \u2264 x\u2080 + \u2191a", "state_after": "no goals"}, {"tactic": "refine' mem_Icc.2 \u27e8(le_add_iff_nonneg_right _).2 (Nat.cast_nonneg _), _\u27e9", "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i + 1 \u2264 a\nx : \u211d\nhx : x \u2208 Icc (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))\n\u22a2 x\u2080 + \u2191(i + 1) \u2208 Icc x\u2080 (x\u2080 + \u2191a)", "state_after": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i + 1 \u2264 a\nx : \u211d\nhx : x \u2208 Icc (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))\n\u22a2 x\u2080 + \u2191(i + 1) \u2264 x\u2080 + \u2191a"}, {"tactic": "simp only [add_le_add_iff_left, Nat.cast_le, ia]", "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\ni : \u2115\nhi : i \u2208 Finset.range a\nia : i + 1 \u2264 a\nx : \u211d\nhx : x \u2208 Icc (x\u2080 + \u2191i) (x\u2080 + \u2191(i + 1))\n\u22a2 x\u2080 + \u2191(i + 1) \u2264 x\u2080 + \u2191a", "state_after": "no goals"}, {"tactic": "convert intervalIntegral.sum_integral_adjacent_intervals hint", "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\n\u22a2 (\u2211 i in Finset.range a, \u222b (x : \u211d) in x\u2080 + \u2191i..x\u2080 + \u2191(i + 1), f x) = \u222b (x : \u211d) in x\u2080..x\u2080 + \u2191a, f x", "state_after": "case h.e'_3.h.e'_6\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\n\u22a2 x\u2080 = x\u2080 + \u21910"}, {"tactic": "simp only [Nat.cast_zero, add_zero]", "state_before": "case h.e'_3.h.e'_6\nx\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhf : AntitoneOn f (Icc x\u2080 (x\u2080 + \u2191a))\nhint : \u2200 (k : \u2115), k < a \u2192 IntervalIntegrable f volume (x\u2080 + \u2191k) (x\u2080 + \u2191(k + 1))\n\u22a2 x\u2080 = x\u2080 + \u21910", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/AlgebraMap.lean", "full_name": "Polynomial.eval_mul_X_sub_C", "start": [486, 1], "end": [502, 24], "traced_tactics": [{"tactic": "simp only [eval, eval\u2082_eq_sum, RingHom.id_apply]", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1586223\nB' : Type ?u.1586226\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommSemiring A'\ninst\u271d\u00b9 : Semiring B'\ninst\u271d : Ring R\np : R[X]\nr : R\n\u22a2 eval r (p * (X - \u2191C r)) = 0", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1586223\nB' : Type ?u.1586226\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommSemiring A'\ninst\u271d\u00b9 : Semiring B'\ninst\u271d : Ring R\np : R[X]\nr : R\n\u22a2 (sum (p * (X - \u2191C r)) fun e a => a * r ^ e) = 0"}, {"tactic": "have bound :=\n  calc\n    (p * (X - C r)).natDegree \u2264 p.natDegree + (X - C r).natDegree := natDegree_mul_le\n    _ \u2264 p.natDegree + 1 := (add_le_add_left (natDegree_X_sub_C_le _) _)\n    _ < p.natDegree + 2 := lt_add_one _", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1586223\nB' : Type ?u.1586226\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommSemiring A'\ninst\u271d\u00b9 : Semiring B'\ninst\u271d : Ring R\np : R[X]\nr : R\n\u22a2 (sum (p * (X - \u2191C r)) fun e a => a * r ^ e) = 0", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1586223\nB' : Type ?u.1586226\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommSemiring A'\ninst\u271d\u00b9 : Semiring B'\ninst\u271d : Ring R\np : R[X]\nr : R\nbound : natDegree (p * (X - \u2191C r)) < natDegree p + 2\n\u22a2 (sum (p * (X - \u2191C r)) fun e a => a * r ^ e) = 0"}, {"tactic": "rw [sum_over_range' _ _ (p.natDegree + 2) bound]", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1586223\nB' : Type ?u.1586226\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommSemiring A'\ninst\u271d\u00b9 : Semiring B'\ninst\u271d : Ring R\np : R[X]\nr : R\nbound : natDegree (p * (X - \u2191C r)) < natDegree p + 2\n\u22a2 (sum (p * (X - \u2191C r)) fun e a => a * r ^ e) = 0", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1586223\nB' : Type ?u.1586226\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommSemiring A'\ninst\u271d\u00b9 : Semiring B'\ninst\u271d : Ring R\np : R[X]\nr : R\nbound : natDegree (p * (X - \u2191C r)) < natDegree p + 2\n\u22a2 \u2211 a in range (natDegree p + 2), coeff (p * (X - \u2191C r)) a * r ^ a = 0\n\nR : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1586223\nB' : Type ?u.1586226\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommSemiring A'\ninst\u271d\u00b9 : Semiring B'\ninst\u271d : Ring R\np : R[X]\nr : R\nbound : natDegree (p * (X - \u2191C r)) < natDegree p + 2\n\u22a2 \u2200 (n : \u2115), 0 * r ^ n = 0"}, {"tactic": "swap", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1586223\nB' : Type ?u.1586226\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommSemiring A'\ninst\u271d\u00b9 : Semiring B'\ninst\u271d : Ring R\np : R[X]\nr : R\nbound : natDegree (p * (X - \u2191C r)) < natDegree p + 2\n\u22a2 \u2211 a in range (natDegree p + 2), coeff (p * (X - \u2191C r)) a * r ^ a = 0\n\nR : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1586223\nB' : Type ?u.1586226\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommSemiring A'\ninst\u271d\u00b9 : Semiring B'\ninst\u271d : Ring R\np : R[X]\nr : R\nbound : natDegree (p * (X - \u2191C r)) < natDegree p + 2\n\u22a2 \u2200 (n : \u2115), 0 * r ^ n = 0", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1586223\nB' : Type ?u.1586226\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommSemiring A'\ninst\u271d\u00b9 : Semiring B'\ninst\u271d : Ring R\np : R[X]\nr : R\nbound : natDegree (p * (X - \u2191C r)) < natDegree p + 2\n\u22a2 \u2200 (n : \u2115), 0 * r ^ n = 0\n\nR : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1586223\nB' : Type ?u.1586226\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommSemiring A'\ninst\u271d\u00b9 : Semiring B'\ninst\u271d : Ring R\np : R[X]\nr : R\nbound : natDegree (p * (X - \u2191C r)) < natDegree p + 2\n\u22a2 \u2211 a in range (natDegree p + 2), coeff (p * (X - \u2191C r)) a * r ^ a = 0"}, {"tactic": "rw [sum_range_succ']", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1586223\nB' : Type ?u.1586226\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommSemiring A'\ninst\u271d\u00b9 : Semiring B'\ninst\u271d : Ring R\np : R[X]\nr : R\nbound : natDegree (p * (X - \u2191C r)) < natDegree p + 2\n\u22a2 \u2211 a in range (natDegree p + 2), coeff (p * (X - \u2191C r)) a * r ^ a = 0", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1586223\nB' : Type ?u.1586226\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommSemiring A'\ninst\u271d\u00b9 : Semiring B'\ninst\u271d : Ring R\np : R[X]\nr : R\nbound : natDegree (p * (X - \u2191C r)) < natDegree p + 2\n\u22a2 \u2211 k in range (natDegree p + 1), coeff (p * (X - \u2191C r)) (k + 1) * r ^ (k + 1) + coeff (p * (X - \u2191C r)) 0 * r ^ 0 = 0"}, {"tactic": "conv_lhs =>\n  congr\n  arg 2\n  simp [coeff_mul_X_sub_C, sub_mul, mul_assoc, \u2190 pow_succ]", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1586223\nB' : Type ?u.1586226\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommSemiring A'\ninst\u271d\u00b9 : Semiring B'\ninst\u271d : Ring R\np : R[X]\nr : R\nbound : natDegree (p * (X - \u2191C r)) < natDegree p + 2\n\u22a2 \u2211 k in range (natDegree p + 1), coeff (p * (X - \u2191C r)) (k + 1) * r ^ (k + 1) + coeff (p * (X - \u2191C r)) 0 * r ^ 0 = 0", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1586223\nB' : Type ?u.1586226\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommSemiring A'\ninst\u271d\u00b9 : Semiring B'\ninst\u271d : Ring R\np : R[X]\nr : R\nbound : natDegree (p * (X - \u2191C r)) < natDegree p + 2\n\u22a2 \u2211 k in range (natDegree p + 1), (coeff p k * r ^ (k + 1) - coeff p (k + 1) * r ^ (k + 1 + 1)) +\n      coeff (p * (X - \u2191C r)) 0 * r ^ 0 =\n    0"}, {"tactic": "rw [sum_range_sub']", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1586223\nB' : Type ?u.1586226\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommSemiring A'\ninst\u271d\u00b9 : Semiring B'\ninst\u271d : Ring R\np : R[X]\nr : R\nbound : natDegree (p * (X - \u2191C r)) < natDegree p + 2\n\u22a2 \u2211 k in range (natDegree p + 1), (coeff p k * r ^ (k + 1) - coeff p (k + 1) * r ^ (k + 1 + 1)) +\n      coeff (p * (X - \u2191C r)) 0 * r ^ 0 =\n    0", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1586223\nB' : Type ?u.1586226\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommSemiring A'\ninst\u271d\u00b9 : Semiring B'\ninst\u271d : Ring R\np : R[X]\nr : R\nbound : natDegree (p * (X - \u2191C r)) < natDegree p + 2\n\u22a2 coeff p 0 * r ^ (0 + 1) - coeff p (natDegree p + 1) * r ^ (natDegree p + 1 + 1) + coeff (p * (X - \u2191C r)) 0 * r ^ 0 = 0"}, {"tactic": "simp [coeff_monomial]", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1586223\nB' : Type ?u.1586226\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommSemiring A'\ninst\u271d\u00b9 : Semiring B'\ninst\u271d : Ring R\np : R[X]\nr : R\nbound : natDegree (p * (X - \u2191C r)) < natDegree p + 2\n\u22a2 coeff p 0 * r ^ (0 + 1) - coeff p (natDegree p + 1) * r ^ (natDegree p + 1 + 1) + coeff (p * (X - \u2191C r)) 0 * r ^ 0 = 0", "state_after": "no goals"}, {"tactic": "simp", "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1586223\nB' : Type ?u.1586226\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommSemiring A'\ninst\u271d\u00b9 : Semiring B'\ninst\u271d : Ring R\np : R[X]\nr : R\nbound : natDegree (p * (X - \u2191C r)) < natDegree p + 2\n\u22a2 \u2200 (n : \u2115), 0 * r ^ n = 0", "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.corec_def", "start": [578, 1], "end": [586, 10], "traced_tactics": [{"tactic": "dsimp only [M.corec, M.mk]", "state_before": "F : PFunctor\nX\u271d : Type ?u.29252\nf\u271d : X\u271d \u2192 Obj F X\u271d\nX : Type u\nf : X \u2192 Obj F X\nx\u2080 : X\n\u22a2 M.corec f x\u2080 = M.mk (M.corec f <$> f x\u2080)", "state_after": "F : PFunctor\nX\u271d : Type ?u.29252\nf\u271d : X\u271d \u2192 Obj F X\u271d\nX : Type u\nf : X \u2192 Obj F X\nx\u2080 : X\n\u22a2 { approx := sCorec f x\u2080, consistent := (_ : \u2200 (n : \u2115), Agree (sCorec f x\u2080 n) (sCorec f x\u2080 (succ n))) } =\n    {\n      approx :=\n        Approx.sMk\n          ((fun i =>\n              { approx := sCorec f i, consistent := (_ : \u2200 (n : \u2115), Agree (sCorec f i n) (sCorec f i (succ n))) }) <$>\n            f x\u2080),\n      consistent :=\n        (_ :\n          AllAgree\n            (Approx.sMk\n              ((fun i =>\n                  { approx := sCorec f i,\n                    consistent := (_ : \u2200 (n : \u2115), Agree (sCorec f i n) (sCorec f i (succ n))) }) <$>\n                f x\u2080))) }"}, {"tactic": "congr with n", "state_before": "F : PFunctor\nX\u271d : Type ?u.29252\nf\u271d : X\u271d \u2192 Obj F X\u271d\nX : Type u\nf : X \u2192 Obj F X\nx\u2080 : X\n\u22a2 { approx := sCorec f x\u2080, consistent := (_ : \u2200 (n : \u2115), Agree (sCorec f x\u2080 n) (sCorec f x\u2080 (succ n))) } =\n    {\n      approx :=\n        Approx.sMk\n          ((fun i =>\n              { approx := sCorec f i, consistent := (_ : \u2200 (n : \u2115), Agree (sCorec f i n) (sCorec f i (succ n))) }) <$>\n            f x\u2080),\n      consistent :=\n        (_ :\n          AllAgree\n            (Approx.sMk\n              ((fun i =>\n                  { approx := sCorec f i,\n                    consistent := (_ : \u2200 (n : \u2115), Agree (sCorec f i n) (sCorec f i (succ n))) }) <$>\n                f x\u2080))) }", "state_after": "case e_approx.h\nF : PFunctor\nX\u271d : Type ?u.29252\nf\u271d : X\u271d \u2192 Obj F X\u271d\nX : Type u\nf : X \u2192 Obj F X\nx\u2080 : X\nn : \u2115\n\u22a2 sCorec f x\u2080 n =\n    Approx.sMk\n      ((fun i =>\n          { approx := sCorec f i, consistent := (_ : \u2200 (n : \u2115), Agree (sCorec f i n) (sCorec f i (succ n))) }) <$>\n        f x\u2080)\n      n"}, {"tactic": "cases' n with n", "state_before": "case e_approx.h\nF : PFunctor\nX\u271d : Type ?u.29252\nf\u271d : X\u271d \u2192 Obj F X\u271d\nX : Type u\nf : X \u2192 Obj F X\nx\u2080 : X\nn : \u2115\n\u22a2 sCorec f x\u2080 n =\n    Approx.sMk\n      ((fun i =>\n          { approx := sCorec f i, consistent := (_ : \u2200 (n : \u2115), Agree (sCorec f i n) (sCorec f i (succ n))) }) <$>\n        f x\u2080)\n      n", "state_after": "case e_approx.h.zero\nF : PFunctor\nX\u271d : Type ?u.29252\nf\u271d : X\u271d \u2192 Obj F X\u271d\nX : Type u\nf : X \u2192 Obj F X\nx\u2080 : X\n\u22a2 sCorec f x\u2080 zero =\n    Approx.sMk\n      ((fun i =>\n          { approx := sCorec f i, consistent := (_ : \u2200 (n : \u2115), Agree (sCorec f i n) (sCorec f i (succ n))) }) <$>\n        f x\u2080)\n      zero\n\ncase e_approx.h.succ\nF : PFunctor\nX\u271d : Type ?u.29252\nf\u271d : X\u271d \u2192 Obj F X\u271d\nX : Type u\nf : X \u2192 Obj F X\nx\u2080 : X\nn : \u2115\n\u22a2 sCorec f x\u2080 (succ n) =\n    Approx.sMk\n      ((fun i =>\n          { approx := sCorec f i, consistent := (_ : \u2200 (n : \u2115), Agree (sCorec f i n) (sCorec f i (succ n))) }) <$>\n        f x\u2080)\n      (succ n)"}, {"tactic": "dsimp only [sCorec, Approx.sMk]", "state_before": "case e_approx.h.zero\nF : PFunctor\nX\u271d : Type ?u.29252\nf\u271d : X\u271d \u2192 Obj F X\u271d\nX : Type u\nf : X \u2192 Obj F X\nx\u2080 : X\n\u22a2 sCorec f x\u2080 zero =\n    Approx.sMk\n      ((fun i =>\n          { approx := sCorec f i, consistent := (_ : \u2200 (n : \u2115), Agree (sCorec f i n) (sCorec f i (succ n))) }) <$>\n        f x\u2080)\n      zero", "state_after": "no goals"}, {"tactic": "dsimp only [sCorec, Approx.sMk]", "state_before": "case e_approx.h.succ\nF : PFunctor\nX\u271d : Type ?u.29252\nf\u271d : X\u271d \u2192 Obj F X\u271d\nX : Type u\nf : X \u2192 Obj F X\nx\u2080 : X\nn : \u2115\n\u22a2 sCorec f x\u2080 (succ n) =\n    Approx.sMk\n      ((fun i =>\n          { approx := sCorec f i, consistent := (_ : \u2200 (n : \u2115), Agree (sCorec f i n) (sCorec f i (succ n))) }) <$>\n        f x\u2080)\n      (succ n)", "state_after": "case e_approx.h.succ\nF : PFunctor\nX\u271d : Type ?u.29252\nf\u271d : X\u271d \u2192 Obj F X\u271d\nX : Type u\nf : X \u2192 Obj F X\nx\u2080 : X\nn : \u2115\n\u22a2 (CofixA.intro (f x\u2080).fst fun i => sCorec f (Sigma.snd (f x\u2080) i) n) =\n    CofixA.intro\n      ((fun i =>\n            { approx := sCorec f i, consistent := (_ : \u2200 (n : \u2115), Agree (sCorec f i n) (sCorec f i (succ n))) }) <$>\n          f x\u2080).fst\n      fun i =>\n      MIntl.approx\n        (Sigma.snd\n          ((fun i =>\n              { approx := sCorec f i, consistent := (_ : \u2200 (n : \u2115), Agree (sCorec f i n) (sCorec f i (succ n))) }) <$>\n            f x\u2080)\n          i)\n        n"}, {"tactic": "cases h : f x\u2080", "state_before": "case e_approx.h.succ\nF : PFunctor\nX\u271d : Type ?u.29252\nf\u271d : X\u271d \u2192 Obj F X\u271d\nX : Type u\nf : X \u2192 Obj F X\nx\u2080 : X\nn : \u2115\n\u22a2 (CofixA.intro (f x\u2080).fst fun i => sCorec f (Sigma.snd (f x\u2080) i) n) =\n    CofixA.intro\n      ((fun i =>\n            { approx := sCorec f i, consistent := (_ : \u2200 (n : \u2115), Agree (sCorec f i n) (sCorec f i (succ n))) }) <$>\n          f x\u2080).fst\n      fun i =>\n      MIntl.approx\n        (Sigma.snd\n          ((fun i =>\n              { approx := sCorec f i, consistent := (_ : \u2200 (n : \u2115), Agree (sCorec f i n) (sCorec f i (succ n))) }) <$>\n            f x\u2080)\n          i)\n        n", "state_after": "case e_approx.h.succ.mk\nF : PFunctor\nX\u271d : Type ?u.29252\nf\u271d : X\u271d \u2192 Obj F X\u271d\nX : Type u\nf : X \u2192 Obj F X\nx\u2080 : X\nn : \u2115\nfst\u271d : F.A\nsnd\u271d : B F fst\u271d \u2192 X\nh : f x\u2080 = { fst := fst\u271d, snd := snd\u271d }\n\u22a2 (CofixA.intro { fst := fst\u271d, snd := snd\u271d }.fst fun i => sCorec f (Sigma.snd { fst := fst\u271d, snd := snd\u271d } i) n) =\n    CofixA.intro\n      ((fun i =>\n            { approx := sCorec f i, consistent := (_ : \u2200 (n : \u2115), Agree (sCorec f i n) (sCorec f i (succ n))) }) <$>\n          { fst := fst\u271d, snd := snd\u271d }).fst\n      fun i =>\n      MIntl.approx\n        (Sigma.snd\n          ((fun i =>\n              { approx := sCorec f i, consistent := (_ : \u2200 (n : \u2115), Agree (sCorec f i n) (sCorec f i (succ n))) }) <$>\n            { fst := fst\u271d, snd := snd\u271d })\n          i)\n        n"}, {"tactic": "congr", "state_before": "case e_approx.h.succ.mk\nF : PFunctor\nX\u271d : Type ?u.29252\nf\u271d : X\u271d \u2192 Obj F X\u271d\nX : Type u\nf : X \u2192 Obj F X\nx\u2080 : X\nn : \u2115\nfst\u271d : F.A\nsnd\u271d : B F fst\u271d \u2192 X\nh : f x\u2080 = { fst := fst\u271d, snd := snd\u271d }\n\u22a2 (CofixA.intro fst\u271d fun i => sCorec f (snd\u271d i) n) =\n    CofixA.intro fst\u271d fun i =>\n      MIntl.approx\n        (((fun i =>\n              { approx := sCorec f i, consistent := (_ : \u2200 (n : \u2115), Agree (sCorec f i n) (sCorec f i (succ n))) }) \u2218\n            snd\u271d)\n          i)\n        n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.mem_iff_nthLe", "start": [1266, 1], "end": [1267, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "full_name": "Subgroup.map_eq_range_iff", "start": [3095, 1], "end": [3096, 69], "traced_tactics": [{"tactic": "rw [f.range_eq_map, map_eq_map_iff, codisjoint_iff, top_sup_eq]", "state_before": "G : Type u_1\nG' : Type ?u.592426\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group G'\nA : Type ?u.592435\ninst\u271d\u00b9 : AddGroup A\nN : Type u_2\ninst\u271d : Group N\nf\u271d f : G \u2192* N\nH : Subgroup G\n\u22a2 map f H = range f \u2194 Codisjoint H (ker f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Hom/Open.lean", "full_name": "ContinuousOpenMap.comp_id", "start": [148, 1], "end": [149, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/Adjoin.lean", "full_name": "IntermediateField.coe_sInf", "start": [145, 1], "end": [147, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.Lp.simpleFunc.denseEmbedding", "start": [770, 11], "end": [786, 33], "traced_tactics": [{"tactic": "borelize E", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.1653495\n\u03b9 : Type ?u.1653498\nE : Type u_2\nF : Type ?u.1653504\n\ud835\udd5c : Type ?u.1653507\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\n\u22a2 DenseEmbedding Subtype.val", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.1653495\n\u03b9 : Type ?u.1653498\nE : Type u_2\nF : Type ?u.1653504\n\ud835\udd5c : Type ?u.1653507\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 DenseEmbedding Subtype.val"}, {"tactic": "apply simpleFunc.uniformEmbedding.denseEmbedding", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.1653495\n\u03b9 : Type ?u.1653498\nE : Type u_2\nF : Type ?u.1653504\n\ud835\udd5c : Type ?u.1653507\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 DenseEmbedding Subtype.val", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.1653495\n\u03b9 : Type ?u.1653498\nE : Type u_2\nF : Type ?u.1653504\n\ud835\udd5c : Type ?u.1653507\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 DenseRange Subtype.val"}, {"tactic": "intro f", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.1653495\n\u03b9 : Type ?u.1653498\nE : Type u_2\nF : Type ?u.1653504\n\ud835\udd5c : Type ?u.1653507\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 DenseRange Subtype.val", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.1653495\n\u03b9 : Type ?u.1653498\nE : Type u_2\nF : Type ?u.1653504\n\ud835\udd5c : Type ?u.1653507\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\n\u22a2 f \u2208 closure (Set.range Subtype.val)"}, {"tactic": "rw [mem_closure_iff_seq_limit]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.1653495\n\u03b9 : Type ?u.1653498\nE : Type u_2\nF : Type ?u.1653504\n\ud835\udd5c : Type ?u.1653507\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\n\u22a2 f \u2208 closure (Set.range Subtype.val)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.1653495\n\u03b9 : Type ?u.1653498\nE : Type u_2\nF : Type ?u.1653504\n\ud835\udd5c : Type ?u.1653507\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range Subtype.val) \u2227 Tendsto x atTop (\ud835\udcdd f)"}, {"tactic": "have hfi' : Mem\u2112p f p \u03bc := Lp.mem\u2112p f", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.1653495\n\u03b9 : Type ?u.1653498\nE : Type u_2\nF : Type ?u.1653504\n\ud835\udd5c : Type ?u.1653507\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range Subtype.val) \u2227 Tendsto x atTop (\ud835\udcdd f)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.1653495\n\u03b9 : Type ?u.1653498\nE : Type u_2\nF : Type ?u.1653504\n\ud835\udd5c : Type ?u.1653507\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\nhfi' : Mem\u2112p (\u2191\u2191f) p\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range Subtype.val) \u2227 Tendsto x atTop (\ud835\udcdd f)"}, {"tactic": "haveI : SeparableSpace (range f \u222a {0} : Set E) :=\n  (Lp.stronglyMeasurable f).separableSpace_range_union_singleton", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.1653495\n\u03b9 : Type ?u.1653498\nE : Type u_2\nF : Type ?u.1653504\n\ud835\udd5c : Type ?u.1653507\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\nhfi' : Mem\u2112p (\u2191\u2191f) p\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range Subtype.val) \u2227 Tendsto x atTop (\ud835\udcdd f)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.1653495\n\u03b9 : Type ?u.1653498\nE : Type u_2\nF : Type ?u.1653504\n\ud835\udd5c : Type ?u.1653507\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\nhfi' : Mem\u2112p (\u2191\u2191f) p\nthis : SeparableSpace \u2191(Set.range \u2191\u2191f \u222a {0})\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range Subtype.val) \u2227 Tendsto x atTop (\ud835\udcdd f)"}, {"tactic": "refine'\n  \u27e8fun n =>\n    toLp\n      (SimpleFunc.approxOn f (Lp.stronglyMeasurable f).measurable (range f \u222a {0}) 0 _ n)\n      (SimpleFunc.mem\u2112p_approxOn_range (Lp.stronglyMeasurable f).measurable hfi' n),\n    fun n => mem_range_self _, _\u27e9", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.1653495\n\u03b9 : Type ?u.1653498\nE : Type u_2\nF : Type ?u.1653504\n\ud835\udd5c : Type ?u.1653507\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\nhfi' : Mem\u2112p (\u2191\u2191f) p\nthis : SeparableSpace \u2191(Set.range \u2191\u2191f \u222a {0})\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range Subtype.val) \u2227 Tendsto x atTop (\ud835\udcdd f)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.1653495\n\u03b9 : Type ?u.1653498\nE : Type u_2\nF : Type ?u.1653504\n\ud835\udd5c : Type ?u.1653507\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\nhfi' : Mem\u2112p (\u2191\u2191f) p\nthis : SeparableSpace \u2191(Set.range \u2191\u2191f \u222a {0})\n\u22a2 Tendsto\n    (fun n =>\n      \u2191(toLp (SimpleFunc.approxOn \u2191\u2191f (_ : Measurable \u2191\u2191f) (Set.range \u2191\u2191f \u222a {0}) 0 (_ : 0 \u2208 Set.range \u2191\u2191f \u222a {0}) n)\n          (_ :\n            Mem\u2112p\n              (\u2191(SimpleFunc.approxOn \u2191\u2191f (_ : Measurable \u2191\u2191f) (Set.range \u2191\u2191f \u222a {0}) 0 (_ : 0 \u2208 Set.range \u2191\u2191f \u222a {0}) n))\n              p)))\n    atTop (\ud835\udcdd f)"}, {"tactic": "convert SimpleFunc.tendsto_approxOn_range_Lp hp_ne_top (Lp.stronglyMeasurable f).measurable hfi'", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.1653495\n\u03b9 : Type ?u.1653498\nE : Type u_2\nF : Type ?u.1653504\n\ud835\udd5c : Type ?u.1653507\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\nhfi' : Mem\u2112p (\u2191\u2191f) p\nthis : SeparableSpace \u2191(Set.range \u2191\u2191f \u222a {0})\n\u22a2 Tendsto\n    (fun n =>\n      \u2191(toLp (SimpleFunc.approxOn \u2191\u2191f (_ : Measurable \u2191\u2191f) (Set.range \u2191\u2191f \u222a {0}) 0 (_ : 0 \u2208 Set.range \u2191\u2191f \u222a {0}) n)\n          (_ :\n            Mem\u2112p\n              (\u2191(SimpleFunc.approxOn \u2191\u2191f (_ : Measurable \u2191\u2191f) (Set.range \u2191\u2191f \u222a {0}) 0 (_ : 0 \u2208 Set.range \u2191\u2191f \u222a {0}) n))\n              p)))\n    atTop (\ud835\udcdd f)", "state_after": "case h.e'_5.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type ?u.1653495\n\u03b9 : Type ?u.1653498\nE : Type u_2\nF : Type ?u.1653504\n\ud835\udd5c : Type ?u.1653507\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\nhfi' : Mem\u2112p (\u2191\u2191f) p\nthis : SeparableSpace \u2191(Set.range \u2191\u2191f \u222a {0})\n\u22a2 f = Mem\u2112p.toLp (\u2191\u2191f) hfi'"}, {"tactic": "rw [toLp_coeFn f (Lp.mem\u2112p f)]", "state_before": "case h.e'_5.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type ?u.1653495\n\u03b9 : Type ?u.1653498\nE : Type u_2\nF : Type ?u.1653504\n\ud835\udd5c : Type ?u.1653507\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\nhfi' : Mem\u2112p (\u2191\u2191f) p\nthis : SeparableSpace \u2191(Set.range \u2191\u2191f \u222a {0})\n\u22a2 f = Mem\u2112p.toLp (\u2191\u2191f) hfi'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Finsupp.lean", "full_name": "Finsupp.total_equivMapDomain", "start": [648, 1], "end": [650, 52], "traced_tactics": [{"tactic": "rw [equivMapDomain_eq_mapDomain, total_mapDomain]", "state_before": "\u03b1 : Type u_1\nM : Type ?u.371094\nN : Type ?u.371097\nP : Type ?u.371100\nR : Type u_3\nS : Type ?u.371106\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_2\nM' : Type u_4\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'\nf : \u03b1 \u2243 \u03b1'\nl : \u03b1 \u2192\u2080 R\n\u22a2 \u2191(Finsupp.total \u03b1' M' R v') (equivMapDomain f l) = \u2191(Finsupp.total \u03b1 M' R (v' \u2218 \u2191f)) l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Normed/Order/Lattice.lean", "full_name": "norm_sup_sub_sup_le_norm", "start": [190, 1], "end": [191, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": 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exponent G = iSup orderOf"}, {"tactic": "exact exponent_eq_iSup_orderOf orderOf_pos", "state_before": "G : Type u\ninst\u271d\u00b9 : CancelCommMonoid G\ninst\u271d : Fintype G\n\u22a2 exponent G = iSup orderOf", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Dual.lean", "full_name": "Module.mapEvalEquiv_apply", "start": [620, 1], "end": [622, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fin/Interval.lean", "full_name": "Fin.Ioc_eq_finset_subtype", "start": [50, 1], "end": [51, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AdicCompletion.lean", "full_name": "adicCompletion.of_apply", "start": [221, 1], "end": [222, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Module/Zlattice.lean", "full_name": "Zspan.exist_unique_vadd_mem_fundamentalDomain", "start": [220, 1], "end": [225, 50], "traced_tactics": [{"tactic": "cases nonempty_fintype \u03b9", "state_before": "E : Type u_2\n\u03b9 : Type u_1\nK : Type u_3\ninst\u271d\u2074 : NormedLinearOrderedField K\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace K E\nb : Basis \u03b9 K E\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Finite \u03b9\nx : E\n\u22a2 \u2203! v, v +\u1d65 x \u2208 fundamentalDomain b", "state_after": "case intro\nE : Type u_2\n\u03b9 : Type u_1\nK : Type u_3\ninst\u271d\u2074 : NormedLinearOrderedField K\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace K E\nb : Basis \u03b9 K E\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Finite \u03b9\nx : E\nval\u271d : Fintype \u03b9\n\u22a2 \u2203! v, v +\u1d65 x \u2208 fundamentalDomain b"}, {"tactic": "refine \u27e8-floor b x, ?_, fun y h => ?_\u27e9", "state_before": "case intro\nE : Type u_2\n\u03b9 : Type u_1\nK : Type u_3\ninst\u271d\u2074 : NormedLinearOrderedField K\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace K E\nb : Basis \u03b9 K E\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Finite \u03b9\nx : E\nval\u271d : Fintype \u03b9\n\u22a2 \u2203! v, v +\u1d65 x \u2208 fundamentalDomain b", "state_after": "case intro.refine_1\nE : Type u_2\n\u03b9 : Type u_1\nK : Type u_3\ninst\u271d\u2074 : NormedLinearOrderedField K\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace K E\nb : Basis \u03b9 K E\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Finite \u03b9\nx : E\nval\u271d : Fintype \u03b9\n\u22a2 (fun v => v +\u1d65 x \u2208 fundamentalDomain b) (-floor b x)\n\ncase intro.refine_2\nE : Type u_2\n\u03b9 : Type u_1\nK : Type u_3\ninst\u271d\u2074 : NormedLinearOrderedField K\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace K E\nb : Basis \u03b9 K E\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Finite \u03b9\nx : E\nval\u271d : Fintype \u03b9\ny : { x // x \u2208 span \u2124 (Set.range \u2191b) }\nh : (fun v => v +\u1d65 x \u2208 fundamentalDomain b) y\n\u22a2 y = -floor b x"}, {"tactic": "exact (vadd_mem_fundamentalDomain b (-floor b x) x).mpr rfl", "state_before": "case intro.refine_1\nE : Type u_2\n\u03b9 : Type u_1\nK : Type u_3\ninst\u271d\u2074 : NormedLinearOrderedField K\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace K E\nb : Basis \u03b9 K E\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Finite \u03b9\nx : E\nval\u271d : Fintype \u03b9\n\u22a2 (fun v => v +\u1d65 x \u2208 fundamentalDomain b) (-floor b x)", "state_after": "no goals"}, {"tactic": "exact (vadd_mem_fundamentalDomain b y x).mp h", "state_before": "case intro.refine_2\nE : Type u_2\n\u03b9 : Type u_1\nK : Type u_3\ninst\u271d\u2074 : NormedLinearOrderedField K\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace K E\nb : Basis \u03b9 K E\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Finite \u03b9\nx : E\nval\u271d : Fintype \u03b9\ny : { x // x \u2208 span \u2124 (Set.range \u2191b) }\nh : (fun v => v +\u1d65 x \u2208 fundamentalDomain b) y\n\u22a2 y = -floor b x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "full_name": "norm_toAdd", "start": [2212, 1], "end": [2213, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Subring/Pointwise.lean", "full_name": "Subring.pointwise_smul_le_pointwise_smul_iff", "start": [127, 1], "end": [128, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Image.lean", "full_name": "Finset.map_nonempty", "start": [246, 1], "end": [247, 74], "traced_tactics": [{"tactic": "rw [nonempty_iff_ne_empty, nonempty_iff_ne_empty, Ne.def, map_eq_empty]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.45024\nf : \u03b1 \u21aa \u03b2\ns : Finset \u03b1\n\u22a2 Finset.Nonempty (map f s) \u2194 Finset.Nonempty s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Finsupp.lean", "full_name": "Finsupp.domLCongr_single", "start": [799, 1], "end": [801, 7], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b1 : Type ?u.477990\nM : Type u_3\nN : Type ?u.477996\nP : Type ?u.477999\nR : Type u_4\nS : Type ?u.478005\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : Semiring S\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : AddCommMonoid P\ninst\u271d : Module R P\n\u03b1\u2081 : Type u_1\n\u03b1\u2082 : Type u_2\ne : \u03b1\u2081 \u2243 \u03b1\u2082\ni : \u03b1\u2081\nm : M\n\u22a2 \u2191(Finsupp.domLCongr e) (single i m) = single (\u2191e i) m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Operations.lean", "full_name": "Submodule.one_eq_span_one_set", "start": [117, 1], "end": [118, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Analytic/Basic.lean", "full_name": "AnalyticAt.add", "start": [550, 1], "end": [553, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Int/DivMod.lean", "full_name": "Int.add_emod", "start": [427, 1], "end": [428, 36], "traced_tactics": [{"tactic": "rw [add_emod_emod, emod_add_emod]", "state_before": "a b n : Int\n\u22a2 (a + b) % n = (a % n + b % n) % n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.mk_coe_finset", "start": [1319, 1], "end": [1319, 87], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b1\u271d \u03b2 \u03b1 : Type u\ns : Finset \u03b1\n\u22a2 (#{ x // x \u2208 s }) = \u2191(Finset.card s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Sheaves/Stalks.lean", "full_name": "TopCat.Presheaf.mono_of_stalk_mono", "start": [514, 1], "end": [520, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/RelIso/Group.lean", "full_name": "RelIso.apply_inv_self", "start": [52, 1], "end": [53, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/ContinuousOn.lean", "full_name": "mem_nhdsWithin_of_mem_nhds", "start": [148, 1], "end": [149, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/PiTensorProduct.lean", "full_name": "PiTensorProduct.reindex_tprod", "start": [474, 1], "end": [477, 26], "traced_tactics": [{"tactic": "dsimp [reindex]", "state_before": "\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type ?u.429705\nR : Type u_3\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type ?u.429714\nR\u2082 : Type ?u.429717\ns : \u03b9 \u2192 Type ?u.429722\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_4\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type ?u.430041\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type ?u.430175\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ne : \u03b9 \u2243 \u03b9\u2082\nf : \u03b9 \u2192 M\n\u22a2 \u2191(reindex R M e) (\u2191(tprod R) f) = \u2a02\u209c[R] (i : \u03b9\u2082), f (\u2191e.symm i)", "state_after": "\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type ?u.429705\nR : Type u_3\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type ?u.429714\nR\u2082 : Type ?u.429717\ns : \u03b9 \u2192 Type ?u.429722\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_4\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type ?u.430041\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type ?u.430175\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ne : \u03b9 \u2243 \u03b9\u2082\nf : \u03b9 \u2192 M\n\u22a2 \u2191(\u2191lift (domDomCongr e.symm (tprod R))) (\u2191(tprod R) f) = \u2a02\u209c[R] (i : \u03b9\u2082), f (\u2191e.symm i)"}, {"tactic": "exact liftAux_tprod _ f", "state_before": "\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type ?u.429705\nR : Type u_3\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type ?u.429714\nR\u2082 : Type ?u.429717\ns : \u03b9 \u2192 Type ?u.429722\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_4\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type ?u.430041\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type ?u.430175\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\ne : \u03b9 \u2243 \u03b9\u2082\nf : \u03b9 \u2192 M\n\u22a2 \u2191(\u2191lift (domDomCongr e.symm (tprod R))) (\u2191(tprod R) f) = \u2a02\u209c[R] (i : \u03b9\u2082), f (\u2191e.symm i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Int.ceil_mono", "start": [1139, 1], "end": [1140, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Degree/Lemmas.lean", "full_name": "Polynomial.natDegree_bit0", "start": [257, 1], "end": [258, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": 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"5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Hom/Ring.lean", "full_name": "NonUnitalRingHom.coe_mulHom_mk", "start": [143, 1], "end": [145, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/BoolIndicator.lean", "full_name": "Set.preimage_boolIndicator_false", "start": [44, 1], "end": [45, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/FiberBundle/Trivialization.lean", "full_name": "Trivialization.symm_apply_mk_proj", "start": [417, 1], "end": [418, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Tactic/Abel.lean", "full_name": "Mathlib.Tactic.Abel.term_eq", "start": [396, 1], "end": [396, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Seq/Computation.lean", "full_name": "Computation.promises_congr", "start": [1012, 1], "end": [1013, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "full_name": "AffineMap.lineMap_mem_affineSpan_pair", "start": [1310, 1], "end": [1312, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Deprecated/Subgroup.lean", "full_name": "IsGroupHom.image_subgroup", "start": [403, 1], "end": [411, 20], "traced_tactics": [{"tactic": "simp [eq\u2081, eq\u2082, hf.map_mul]", "state_before": "G : Type u_1\nH : Type u_2\nA : Type ?u.86268\na a\u2081\u271d a\u2082\u271d b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set G\nhs : IsSubgroup s\na\u2081 a\u2082 : H\nx\u271d\u00b9 : a\u2081 \u2208 f '' s\nx\u271d : a\u2082 \u2208 f '' s\nb\u2081 : G\nhb\u2081 : b\u2081 \u2208 s\neq\u2081 : f b\u2081 = a\u2081\nb\u2082 : G\nhb\u2082 : b\u2082 \u2208 s\neq\u2082 : f b\u2082 = a\u2082\n\u22a2 f (b\u2081 * b\u2082) = a\u2081 * a\u2082", "state_after": "no goals"}, {"tactic": "rw [hf.map_inv]", "state_before": "G : Type u_1\nH : Type u_2\nA : Type ?u.86268\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set G\nhs : IsSubgroup s\na : H\nx\u271d : a \u2208 f '' s\nb : G\nhb : b \u2208 s\nEq : f b = a\n\u22a2 f b\u207b\u00b9 = a\u207b\u00b9", "state_after": "G : Type u_1\nH : Type u_2\nA : Type ?u.86268\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set G\nhs : IsSubgroup s\na : H\nx\u271d : a \u2208 f '' s\nb : G\nhb : b \u2208 s\nEq : f b = a\n\u22a2 (f b)\u207b\u00b9 = a\u207b\u00b9"}, {"tactic": "simp [*]", "state_before": "G : Type u_1\nH : Type u_2\nA : Type ?u.86268\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set G\nhs : IsSubgroup s\na : H\nx\u271d : a \u2208 f '' s\nb : G\nhb : b \u2208 s\nEq : f b = a\n\u22a2 (f b)\u207b\u00b9 = a\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.reverse_take", "start": [2296, 1], "end": [2311, 8], "traced_tactics": [{"tactic": "induction' xs with xs_hd xs_tl xs_ih generalizing n <;>\n  simp only [reverse_cons, drop, reverse_nil, zero_tsub, length, take_nil]", "state_before": "\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn : \u2115\nh : n \u2264 length xs\n\u22a2 take n (reverse xs) = reverse (drop (length xs - n) xs)", "state_after": "case cons\n\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn\u271d : \u2115\nh\u271d : n\u271d \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nn : \u2115\nh : n \u2264 length (xs_hd :: xs_tl)\n\u22a2 take n (reverse xs_tl ++ [xs_hd]) = reverse (drop (length xs_tl + 1 - n) (xs_hd :: xs_tl))"}, {"tactic": "cases' h.lt_or_eq_dec with h' h'", "state_before": "case cons\n\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn\u271d : \u2115\nh\u271d : n\u271d \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nn : \u2115\nh : n \u2264 length (xs_hd :: xs_tl)\n\u22a2 take n (reverse xs_tl ++ [xs_hd]) = reverse (drop (length xs_tl + 1 - n) (xs_hd :: xs_tl))", "state_after": "case cons.inl\n\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn\u271d : \u2115\nh\u271d : n\u271d \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nn : \u2115\nh : n \u2264 length (xs_hd :: xs_tl)\nh' : n < length (xs_hd :: xs_tl)\n\u22a2 take n (reverse xs_tl ++ [xs_hd]) = reverse (drop (length xs_tl + 1 - n) (xs_hd :: xs_tl))\n\ncase cons.inr\n\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn\u271d : \u2115\nh\u271d : n\u271d \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nn : \u2115\nh : n \u2264 length (xs_hd :: xs_tl)\nh' : n = length (xs_hd :: xs_tl)\n\u22a2 take n (reverse xs_tl ++ [xs_hd]) = reverse (drop (length xs_tl + 1 - n) (xs_hd :: xs_tl))"}, {"tactic": "replace h' := le_of_succ_le_succ h'", "state_before": "case cons.inl\n\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn\u271d : \u2115\nh\u271d : n\u271d \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nn : \u2115\nh : n \u2264 length (xs_hd :: xs_tl)\nh' : n < length (xs_hd :: xs_tl)\n\u22a2 take n (reverse xs_tl ++ [xs_hd]) = reverse (drop (length xs_tl + 1 - n) (xs_hd :: xs_tl))", "state_after": "case cons.inl\n\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn\u271d : \u2115\nh\u271d : n\u271d \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nn : \u2115\nh : n \u2264 length (xs_hd :: xs_tl)\nh' : n \u2264 length xs_tl\n\u22a2 take n (reverse xs_tl ++ [xs_hd]) = reverse (drop (length xs_tl + 1 - n) (xs_hd :: xs_tl))"}, {"tactic": "rw [take_append_of_le_length, xs_ih _ h']", "state_before": "case cons.inl\n\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn\u271d : \u2115\nh\u271d : n\u271d \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nn : \u2115\nh : n \u2264 length (xs_hd :: xs_tl)\nh' : n \u2264 length xs_tl\n\u22a2 take n (reverse xs_tl ++ [xs_hd]) = reverse (drop (length xs_tl + 1 - n) (xs_hd :: xs_tl))", "state_after": "case cons.inl\n\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn\u271d : \u2115\nh\u271d : n\u271d \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nn : \u2115\nh : n \u2264 length (xs_hd :: xs_tl)\nh' : n \u2264 length xs_tl\n\u22a2 reverse (drop (length xs_tl - n) xs_tl) = reverse (drop (length xs_tl + 1 - n) (xs_hd :: xs_tl))\n\ncase cons.inl\n\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn\u271d : \u2115\nh\u271d : n\u271d \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nn : \u2115\nh : n \u2264 length (xs_hd :: xs_tl)\nh' : n \u2264 length xs_tl\n\u22a2 n \u2264 length (reverse xs_tl)"}, {"tactic": "rw [show xs_tl.length + 1 - n = succ (xs_tl.length - n) from _, drop]", "state_before": "case cons.inl\n\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn\u271d : \u2115\nh\u271d : n\u271d \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nn : \u2115\nh : n \u2264 length (xs_hd :: xs_tl)\nh' : n \u2264 length xs_tl\n\u22a2 reverse (drop (length xs_tl - n) xs_tl) = reverse (drop (length xs_tl + 1 - n) (xs_hd :: xs_tl))\n\ncase cons.inl\n\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn\u271d : \u2115\nh\u271d : n\u271d \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nn : \u2115\nh : n \u2264 length (xs_hd :: xs_tl)\nh' : n \u2264 length xs_tl\n\u22a2 n \u2264 length (reverse xs_tl)", "state_after": "\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn\u271d : \u2115\nh\u271d : n\u271d \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nn : \u2115\nh : n \u2264 length (xs_hd :: xs_tl)\nh' : n \u2264 length xs_tl\n\u22a2 length xs_tl + 1 - n = succ (length xs_tl - n)\n\ncase cons.inl\n\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn\u271d : \u2115\nh\u271d : n\u271d \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nn : \u2115\nh : n \u2264 length (xs_hd :: xs_tl)\nh' : n \u2264 length xs_tl\n\u22a2 n \u2264 length (reverse xs_tl)"}, {"tactic": "rwa [succ_eq_add_one, \u2190 @tsub_add_eq_add_tsub]", "state_before": "\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn\u271d : \u2115\nh\u271d : n\u271d \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nn : \u2115\nh : n \u2264 length (xs_hd :: xs_tl)\nh' : n \u2264 length xs_tl\n\u22a2 length xs_tl + 1 - n = succ (length xs_tl - n)", "state_after": "no goals"}, {"tactic": "rwa [length_reverse]", "state_before": "case cons.inl\n\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn\u271d : \u2115\nh\u271d : n\u271d \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nn : \u2115\nh : n \u2264 length (xs_hd :: xs_tl)\nh' : n \u2264 length xs_tl\n\u22a2 n \u2264 length (reverse xs_tl)", "state_after": "no goals"}, {"tactic": "subst h'", "state_before": "case cons.inr\n\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn\u271d : \u2115\nh\u271d : n\u271d \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nn : \u2115\nh : n \u2264 length (xs_hd :: xs_tl)\nh' : n = length (xs_hd :: xs_tl)\n\u22a2 take n (reverse xs_tl ++ [xs_hd]) = reverse (drop (length xs_tl + 1 - n) (xs_hd :: xs_tl))", "state_after": "case cons.inr\n\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn : \u2115\nh\u271d : n \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nh : length (xs_hd :: xs_tl) \u2264 length (xs_hd :: xs_tl)\n\u22a2 take (length (xs_hd :: xs_tl)) (reverse xs_tl ++ [xs_hd]) =\n    reverse (drop (length xs_tl + 1 - length (xs_hd :: xs_tl)) (xs_hd :: xs_tl))"}, {"tactic": "rw [length, tsub_self, drop]", "state_before": "case cons.inr\n\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn : \u2115\nh\u271d : n \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nh : length (xs_hd :: xs_tl) \u2264 length (xs_hd :: xs_tl)\n\u22a2 take (length (xs_hd :: xs_tl)) (reverse xs_tl ++ [xs_hd]) =\n    reverse (drop (length xs_tl + 1 - length (xs_hd :: xs_tl)) (xs_hd :: xs_tl))", "state_after": "case cons.inr\n\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn : \u2115\nh\u271d : n \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nh : length (xs_hd :: xs_tl) \u2264 length (xs_hd :: xs_tl)\n\u22a2 take (length xs_tl + 1) (reverse xs_tl ++ [xs_hd]) = reverse (xs_hd :: xs_tl)"}, {"tactic": "suffices xs_tl.length + 1 = (xs_tl.reverse ++ [xs_hd]).length by\n  rw [this, take_length, reverse_cons]", "state_before": "case cons.inr\n\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn : \u2115\nh\u271d : n \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nh : length (xs_hd :: xs_tl) \u2264 length (xs_hd :: xs_tl)\n\u22a2 take (length xs_tl + 1) (reverse xs_tl ++ [xs_hd]) = reverse (xs_hd :: xs_tl)", "state_after": "case cons.inr\n\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn : \u2115\nh\u271d : n \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nh : length (xs_hd :: xs_tl) \u2264 length (xs_hd :: xs_tl)\n\u22a2 length xs_tl + 1 = length (reverse xs_tl ++ [xs_hd])"}, {"tactic": "rw [length_append, length_reverse]", "state_before": "case cons.inr\n\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn : \u2115\nh\u271d : n \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nh : length (xs_hd :: xs_tl) \u2264 length (xs_hd :: xs_tl)\n\u22a2 length xs_tl + 1 = length (reverse xs_tl ++ [xs_hd])", "state_after": "case cons.inr\n\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn : \u2115\nh\u271d : n \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nh : length (xs_hd :: xs_tl) \u2264 length (xs_hd :: xs_tl)\n\u22a2 length xs_tl + 1 = length xs_tl + length [xs_hd]"}, {"tactic": "rfl", "state_before": "case cons.inr\n\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn : \u2115\nh\u271d : n \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nh : length (xs_hd :: xs_tl) \u2264 length (xs_hd :: xs_tl)\n\u22a2 length xs_tl + 1 = length xs_tl + length [xs_hd]", "state_after": "no goals"}, {"tactic": "rw [this, take_length, reverse_cons]", "state_before": "\u03b9 : Type ?u.221288\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\nxs : List \u03b1\nn : \u2115\nh\u271d : n \u2264 length xs\nxs_hd : \u03b1\nxs_tl : List \u03b1\nxs_ih : \u2200 (n : \u2115), n \u2264 length xs_tl \u2192 take n (reverse xs_tl) = reverse (drop (length xs_tl - n) xs_tl)\nh : length (xs_hd :: xs_tl) \u2264 length (xs_hd :: xs_tl)\nthis : length xs_tl + 1 = length (reverse xs_tl ++ [xs_hd])\n\u22a2 take (length xs_tl + 1) (reverse xs_tl ++ [xs_hd]) = reverse (xs_hd :: xs_tl)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GroupPower/Order.lean", "full_name": "pow_abs", "start": [630, 1], "end": [631, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/Star/Basic.lean", "full_name": "CstarRing.norm_self_mul_star", "start": [121, 1], "end": [123, 44], "traced_tactics": [{"tactic": "nth_rw 1 [\u2190 star_star x]", "state_before": "\ud835\udd5c : Type ?u.32816\nE : Type u_1\n\u03b1 : Type ?u.32822\ninst\u271d\u00b2 : NonUnitalNormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\n\u22a2 \u2016x * x\u22c6\u2016 = \u2016x\u2016 * \u2016x\u2016", "state_after": "\ud835\udd5c : Type ?u.32816\nE : Type u_1\n\u03b1 : Type ?u.32822\ninst\u271d\u00b2 : NonUnitalNormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\n\u22a2 \u2016x\u22c6\u22c6 * x\u22c6\u2016 = \u2016x\u2016 * \u2016x\u2016"}, {"tactic": "simp only [norm_star_mul_self, norm_star]", "state_before": "\ud835\udd5c : Type ?u.32816\nE : Type u_1\n\u03b1 : Type ?u.32822\ninst\u271d\u00b2 : NonUnitalNormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\n\u22a2 \u2016x\u22c6\u22c6 * x\u22c6\u2016 = \u2016x\u2016 * \u2016x\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/WithOne/Basic.lean", "full_name": "WithOne.map_id", "start": [116, 1], "end": [118, 45], "traced_tactics": [{"tactic": "ext x", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : Mul \u03b2\ninst\u271d : Mul \u03b3\n\u22a2 map (MulHom.id \u03b1) = MonoidHom.id (WithOne \u03b1)", "state_after": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : Mul \u03b2\ninst\u271d : Mul \u03b3\nx : WithOne \u03b1\n\u22a2 \u2191(map (MulHom.id \u03b1)) x = \u2191(MonoidHom.id (WithOne \u03b1)) x"}, {"tactic": "induction x using WithOne.cases_on <;> rfl", "state_before": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : Mul \u03b1\ninst\u271d\u00b9 : Mul \u03b2\ninst\u271d : Mul \u03b3\nx : WithOne \u03b1\n\u22a2 \u2191(map (MulHom.id \u03b1)) x = \u2191(MonoidHom.id (WithOne \u03b1)) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Commute.lean", "full_name": "Commute.units_inv_right_iff", "start": [228, 1], "end": [229, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Denumerable.lean", "full_name": "Denumerable.prod_nat_ofNat", "start": [181, 1], "end": [181, 67], "traced_tactics": [{"tactic": "funext", "state_before": "\u03b1 : Type ?u.32247\n\u03b2 : Type ?u.32250\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\n\u22a2 ofNat (\u2115 \u00d7 \u2115) = unpair", "state_after": "case h\n\u03b1 : Type ?u.32247\n\u03b2 : Type ?u.32250\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nx\u271d : \u2115\n\u22a2 ofNat (\u2115 \u00d7 \u2115) x\u271d = unpair x\u271d"}, {"tactic": "simp", "state_before": "case h\n\u03b1 : Type ?u.32247\n\u03b2 : Type ?u.32250\ninst\u271d\u00b9 : Denumerable \u03b1\ninst\u271d : Denumerable \u03b2\nx\u271d : \u2115\n\u22a2 ofNat (\u2115 \u00d7 \u2115) x\u271d = unpair x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Preserves/Shapes/Kernels.lean", "full_name": "CategoryTheory.Limits.kernel_map_comp_preserves_kernel_iso_inv", "start": [123, 1], "end": [130, 63], "traced_tactics": [{"tactic": "rw [\u2190 G.map_comp, hpq, G.map_comp]", "state_before": "C : Type u\u2081\ninst\u271d\u00b9\u2070 : Category C\ninst\u271d\u2079 : HasZeroMorphisms C\nD : Type u\u2082\ninst\u271d\u2078 : Category D\ninst\u271d\u2077 : HasZeroMorphisms D\nG : C \u2964 D\ninst\u271d\u2076 : Functor.PreservesZeroMorphisms G\nX Y Z : C\nf : X \u27f6 Y\nh : Z \u27f6 X\nw : h \u226b f = 0\ninst\u271d\u2075 : HasKernel f\ninst\u271d\u2074 : HasKernel (G.map f)\ninst\u271d\u00b3 : PreservesLimit (parallelPair f 0) G\nX' Y' : C\ng : X' \u27f6 Y'\ninst\u271d\u00b2 : HasKernel g\ninst\u271d\u00b9 : HasKernel (G.map g)\ninst\u271d : PreservesLimit (parallelPair g 0) G\np : X \u27f6 X'\nq : Y \u27f6 Y'\nhpq : f \u226b q = p \u226b g\n\u22a2 G.map f \u226b G.map q = G.map p \u226b G.map g", "state_after": "no goals"}, {"tactic": "rw [Iso.comp_inv_eq, Category.assoc, PreservesKernel.iso_hom, Iso.eq_inv_comp,\n  PreservesKernel.iso_hom, kernelComparison_comp_kernel_map]", "state_before": "C : Type u\u2081\ninst\u271d\u00b9\u2070 : Category C\ninst\u271d\u2079 : HasZeroMorphisms C\nD : Type u\u2082\ninst\u271d\u2078 : Category D\ninst\u271d\u2077 : HasZeroMorphisms D\nG : C \u2964 D\ninst\u271d\u2076 : Functor.PreservesZeroMorphisms G\nX Y Z : C\nf : X \u27f6 Y\nh : Z \u27f6 X\nw : h \u226b f = 0\ninst\u271d\u2075 : HasKernel f\ninst\u271d\u2074 : HasKernel (G.map f)\ninst\u271d\u00b3 : PreservesLimit (parallelPair f 0) G\nX' Y' : C\ng : X' \u27f6 Y'\ninst\u271d\u00b2 : HasKernel g\ninst\u271d\u00b9 : HasKernel (G.map g)\ninst\u271d : PreservesLimit (parallelPair g 0) G\np : X \u27f6 X'\nq : Y \u27f6 Y'\nhpq : f \u226b q = p \u226b g\n\u22a2 kernel.map (G.map f) (G.map g) (G.map p) (G.map q) (_ : G.map f \u226b G.map q = G.map p \u226b G.map g) \u226b\n      (PreservesKernel.iso G g).inv =\n    (PreservesKernel.iso G f).inv \u226b G.map (kernel.map f g p q hpq)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Parity.lean", "full_name": "Nat.Odd.sub_odd", "start": [150, 1], "end": [152, 53], "traced_tactics": [{"tactic": "simp only [even_sub' h, *]", "state_before": "m n : \u2115\nhm : Odd m\nhn : Odd n\nh : n \u2264 m\n\u22a2 Even (m - n)", "state_after": "no goals"}, {"tactic": "simp only [tsub_eq_zero_iff_le.mpr h, even_zero]", "state_before": "m n : \u2115\nhm : Odd m\nhn : Odd n\nh : m \u2264 n\n\u22a2 Even (m - n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Prime.lean", "full_name": "Nat.Prime.dvd_factorial", "start": [661, 1], "end": [667, 94], "traced_tactics": [{"tactic": "rw [factorial_succ, hp.dvd_mul, Prime.dvd_factorial hp]", "state_before": "n p : \u2115\nhp : Prime p\n\u22a2 p \u2223 (n + 1)! \u2194 p \u2264 n + 1", "state_after": "n p : \u2115\nhp : Prime p\n\u22a2 p \u2223 n + 1 \u2228 p \u2264 n \u2194 p \u2264 n + 1"}, {"tactic": "exact\n  \u27e8fun h => h.elim (le_of_dvd (succ_pos _)) le_succ_of_le, fun h =>\n    (_root_.lt_or_eq_of_le h).elim (Or.inr \u2218 le_of_lt_succ) fun h => Or.inl <| by rw [h]\u27e9", "state_before": "n p : \u2115\nhp : Prime p\n\u22a2 p \u2223 n + 1 \u2228 p \u2264 n \u2194 p \u2264 n + 1", "state_after": "no goals"}, {"tactic": "rw [h]", "state_before": "n p : \u2115\nhp : Prime p\nh\u271d : p \u2264 n + 1\nh : p = n + 1\n\u22a2 p \u2223 n + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/WithBot.lean", "full_name": "WithBot.unbot'_eq_unbot'_iff", "start": [142, 1], "end": [144, 63], "traced_tactics": [{"tactic": "induction y using recBotCoe <;> simp [unbot'_eq_iff, or_comm]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.4891\n\u03b3 : Type ?u.4894\n\u03b4 : Type ?u.4897\na b d : \u03b1\nx y : WithBot \u03b1\n\u22a2 unbot' d x = unbot' d y \u2194 x = y \u2228 x = \u2191d \u2227 y = \u22a5 \u2228 x = \u22a5 \u2227 y = \u2191d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/RingDivision.lean", "full_name": "Polynomial.mem_nthRoots", "start": [779, 1], "end": [781, 25], "traced_tactics": [{"tactic": "rw [nthRoots, mem_roots (X_pow_sub_C_ne_zero hn a), IsRoot.def, eval_sub, eval_C, eval_pow,\n  eval_X, sub_eq_zero]", "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]\nn : \u2115\nhn : 0 < n\na x : R\n\u22a2 x \u2208 nthRoots n a \u2194 x ^ n = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Configuration.lean", "full_name": "Configuration.ProjectivePlane.lineCount_eq", "start": [449, 1], "end": [455, 44], "traced_tactics": [{"tactic": "classical\n  obtain \u27e8q, -, -, l, -, -, -, -, h, -\u27e9 := Classical.choose_spec (@exists_config P L _ _)\n  cases nonempty_fintype { l : L // q \u2208 l }\n  rw [order, lineCount_eq_lineCount L p q, lineCount_eq_lineCount L (Classical.choose _) q,\n    lineCount, Nat.card_eq_fintype_card, Nat.sub_add_cancel]\n  exact Fintype.card_pos_iff.mpr \u27e8\u27e8l, h\u27e9\u27e9", "state_before": "P : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np : P\n\u22a2 lineCount L p = order P L + 1", "state_after": "no goals"}, {"tactic": "obtain \u27e8q, -, -, l, -, -, -, -, h, -\u27e9 := Classical.choose_spec (@exists_config P L _ _)", "state_before": "P : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np : P\n\u22a2 lineCount L p = order P L + 1", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\nl : L\nh : q \u2208 l\n\u22a2 lineCount L p = order P L + 1"}, {"tactic": "cases nonempty_fintype { l : L // q \u2208 l }", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\nl : L\nh : q \u2208 l\n\u22a2 lineCount L p = order P L + 1", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\nl : L\nh : q \u2208 l\nval\u271d : Fintype { l // q \u2208 l }\n\u22a2 lineCount L p = order P L + 1"}, {"tactic": "rw [order, lineCount_eq_lineCount L p q, lineCount_eq_lineCount L (Classical.choose _) q,\n  lineCount, Nat.card_eq_fintype_card, Nat.sub_add_cancel]", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\nl : L\nh : q \u2208 l\nval\u271d : Fintype { l // q \u2208 l }\n\u22a2 lineCount L p = order P L + 1", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\nl : L\nh : q \u2208 l\nval\u271d : Fintype { l // q \u2208 l }\n\u22a2 1 \u2264 Fintype.card { l // q \u2208 l }"}, {"tactic": "exact Fintype.card_pos_iff.mpr \u27e8\u27e8l, h\u27e9\u27e9", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\nl : L\nh : q \u2208 l\nval\u271d : Fintype { l // q \u2208 l }\n\u22a2 1 \u2264 Fintype.card { l // q \u2208 l }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/Operations.lean", "full_name": "Ideal.span_singleton_mul_left_inj", "start": [584, 1], "end": [586, 63], "traced_tactics": [{"tactic": "simp only [le_antisymm_iff, span_singleton_mul_left_mono hx]", "state_before": "R : Type u\n\u03b9 : Type ?u.241212\ninst\u271d\u00b9 : CommSemiring R\nI J K L : Ideal R\ninst\u271d : IsDomain R\nx : R\nhx : x \u2260 0\n\u22a2 I * span {x} = J * span {x} \u2194 I = J", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Star.lean", "full_name": "StarConvex.linear_preimage", "start": [218, 1], "end": [222, 24], "traced_tactics": [{"tactic": "intro y hy a b ha hb hab", "state_before": "\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_1\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y z : E\ns\u271d : Set E\ns : Set F\nf : E \u2192\u2097[\ud835\udd5c] F\nhs : StarConvex \ud835\udd5c (\u2191f x) s\n\u22a2 StarConvex \ud835\udd5c x (\u2191f \u207b\u00b9' s)", "state_after": "\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_1\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y\u271d z : E\ns\u271d : Set E\ns : Set F\nf : E \u2192\u2097[\ud835\udd5c] F\nhs : StarConvex \ud835\udd5c (\u2191f x) s\ny : E\nhy : y \u2208 \u2191f \u207b\u00b9' s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 x + b \u2022 y \u2208 \u2191f \u207b\u00b9' s"}, {"tactic": "rw [mem_preimage, f.map_add, f.map_smul, f.map_smul]", "state_before": "\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_1\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y\u271d z : E\ns\u271d : Set E\ns : Set F\nf : E \u2192\u2097[\ud835\udd5c] F\nhs : StarConvex \ud835\udd5c (\u2191f x) s\ny : E\nhy : y \u2208 \u2191f \u207b\u00b9' s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 x + b \u2022 y \u2208 \u2191f \u207b\u00b9' s", "state_after": "\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_1\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y\u271d z : E\ns\u271d : Set E\ns : Set F\nf : E \u2192\u2097[\ud835\udd5c] F\nhs : StarConvex \ud835\udd5c (\u2191f x) s\ny : E\nhy : y \u2208 \u2191f \u207b\u00b9' s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 \u2191f x + b \u2022 \u2191f y \u2208 s"}, {"tactic": "exact hs hy ha hb hab", "state_before": "\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_1\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y\u271d z : E\ns\u271d : Set E\ns : Set F\nf : E \u2192\u2097[\ud835\udd5c] F\nhs : StarConvex \ud835\udd5c (\u2191f x) s\ny : E\nhy : y \u2208 \u2191f \u207b\u00b9' s\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 \u2191f x + b \u2022 \u2191f y \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/AddTorsorBases.lean", "full_name": "AffineBasis.centroid_mem_interior_convexHull", "start": [143, 1], "end": [147, 83], "traced_tactics": [{"tactic": "haveI := b.nonempty", "state_before": "V : Type u_2\nP : Type ?u.93147\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : AffineBasis \u03b9 \u211d V\n\u22a2 Finset.centroid \u211d Finset.univ \u2191b \u2208 interior (\u2191(convexHull \u211d).toOrderHom (range \u2191b))", "state_after": "V : Type u_2\nP : Type ?u.93147\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : AffineBasis \u03b9 \u211d V\nthis : Nonempty \u03b9\n\u22a2 Finset.centroid \u211d Finset.univ \u2191b \u2208 interior (\u2191(convexHull \u211d).toOrderHom (range \u2191b))"}, {"tactic": "simp only [b.interior_convexHull, mem_setOf_eq, b.coord_apply_centroid (Finset.mem_univ _),\n  inv_pos, Nat.cast_pos, Finset.card_pos, Finset.univ_nonempty, forall_true_iff]", "state_before": "V : Type u_2\nP : Type ?u.93147\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : AffineBasis \u03b9 \u211d V\nthis : Nonempty \u03b9\n\u22a2 Finset.centroid \u211d Finset.univ \u2191b \u2208 interior (\u2191(convexHull \u211d).toOrderHom (range \u2191b))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/BoxIntegral/Partition/Basic.lean", "full_name": "BoxIntegral.Prepartition.distortion_le_of_mem", "start": [678, 1], "end": [679, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Bounds/Basic.lean", "full_name": "bddBelow_union", "start": [435, 1], "end": [437, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/Combination.lean", "full_name": "mem_vectorSpan_iff_eq_weightedVSub", "start": [1047, 1], "end": [1085, 45], "traced_tactics": [{"tactic": "constructor", "state_before": "k : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\n\u22a2 v \u2208 vectorSpan k (Set.range p) \u2194 \u2203 s w x, v = \u2191(Finset.weightedVSub s p) w", "state_after": "case mp\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\n\u22a2 v \u2208 vectorSpan k (Set.range p) \u2192 \u2203 s w x, v = \u2191(Finset.weightedVSub s p) w\n\ncase mpr\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\n\u22a2 (\u2203 s w x, v = \u2191(Finset.weightedVSub s p) w) \u2192 v \u2208 vectorSpan k (Set.range p)"}, {"tactic": "rcases isEmpty_or_nonempty \u03b9 with (h\u03b9 | \u27e8\u27e8i0\u27e9\u27e9)", "state_before": "case mp\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\n\u22a2 v \u2208 vectorSpan k (Set.range p) \u2192 \u2203 s w x, v = \u2191(Finset.weightedVSub s p) w", "state_after": "case mp.inl\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\nh\u03b9 : IsEmpty \u03b9\n\u22a2 v \u2208 vectorSpan k (Set.range p) \u2192 \u2203 s w x, v = \u2191(Finset.weightedVSub s p) w\n\ncase mp.inr.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\n\u22a2 v \u2208 vectorSpan k (Set.range p) \u2192 \u2203 s w x, v = \u2191(Finset.weightedVSub s p) w"}, {"tactic": "swap", "state_before": "case mp.inl\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\nh\u03b9 : IsEmpty \u03b9\n\u22a2 v \u2208 vectorSpan k (Set.range p) \u2192 \u2203 s w x, v = \u2191(Finset.weightedVSub s p) w\n\ncase mp.inr.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\n\u22a2 v \u2208 vectorSpan k (Set.range p) \u2192 \u2203 s w x, v = \u2191(Finset.weightedVSub s p) w", "state_after": "case mp.inr.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\n\u22a2 v \u2208 vectorSpan k (Set.range p) \u2192 \u2203 s w x, v = \u2191(Finset.weightedVSub s p) w\n\ncase mp.inl\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\nh\u03b9 : IsEmpty \u03b9\n\u22a2 v \u2208 vectorSpan k (Set.range p) \u2192 \u2203 s w x, v = \u2191(Finset.weightedVSub s p) w"}, {"tactic": "rw [vectorSpan_range_eq_span_range_vsub_right k p i0, \u2190 Set.image_univ,\n  Finsupp.mem_span_image_iff_total]", "state_before": "case mp.inr.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\n\u22a2 v \u2208 vectorSpan k (Set.range p) \u2192 \u2203 s w x, v = \u2191(Finset.weightedVSub s p) w", "state_after": "case mp.inr.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\n\u22a2 (\u2203 l, l \u2208 Finsupp.supported k k Set.univ \u2227 \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v) \u2192\n    \u2203 s w x, v = \u2191(Finset.weightedVSub s p) w"}, {"tactic": "rintro \u27e8l, _, hv\u27e9", "state_before": "case mp.inr.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\n\u22a2 (\u2203 l, l \u2208 Finsupp.supported k k Set.univ \u2227 \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v) \u2192\n    \u2203 s w x, v = \u2191(Finset.weightedVSub s p) w", "state_after": "case mp.inr.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\n\u22a2 \u2203 s w x, v = \u2191(Finset.weightedVSub s p) w"}, {"tactic": "use insert i0 l.support", "state_before": "case mp.inr.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\n\u22a2 \u2203 s w x, v = \u2191(Finset.weightedVSub s p) w", "state_after": "case mp.inr.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\n\u22a2 \u2203 w x, v = \u2191(Finset.weightedVSub (insert i0 l.support) p) w"}, {"tactic": "set w :=\n  (l : \u03b9 \u2192 k) - Function.update (Function.const \u03b9 0 : \u03b9 \u2192 k) i0 (\u2211 i in l.support, l i) with\n  hwdef", "state_before": "case mp.inr.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\n\u22a2 \u2203 w x, v = \u2191(Finset.weightedVSub (insert i0 l.support) p) w", "state_after": "case mp.inr.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\n\u22a2 \u2203 w x, v = \u2191(Finset.weightedVSub (insert i0 l.support) p) w"}, {"tactic": "use w", "state_before": "case mp.inr.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\n\u22a2 \u2203 w x, v = \u2191(Finset.weightedVSub (insert i0 l.support) p) w", "state_after": "case mp.inr.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\n\u22a2 \u2203 x, v = \u2191(Finset.weightedVSub (insert i0 l.support) p) w"}, {"tactic": "have hw : (\u2211 i in insert i0 l.support, w i) = 0 := by\n  rw [hwdef]\n  simp_rw [Pi.sub_apply, Finset.sum_sub_distrib,\n    Finset.sum_update_of_mem (Finset.mem_insert_self _ _),\n    Finset.sum_insert_of_eq_zero_if_not_mem Finsupp.not_mem_support_iff.1]\n  simp only [Finsupp.mem_support_iff, ne_eq, Finset.mem_insert, true_or, not_true,\n    Function.const_apply, Finset.sum_const_zero, add_zero, sub_self]", "state_before": "case mp.inr.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\n\u22a2 \u2203 x, v = \u2191(Finset.weightedVSub (insert i0 l.support) p) w", "state_after": "case mp.inr.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhw : \u2211 i in insert i0 l.support, w i = 0\n\u22a2 \u2203 x, v = \u2191(Finset.weightedVSub (insert i0 l.support) p) w"}, {"tactic": "use hw", "state_before": "case mp.inr.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhw : \u2211 i in insert i0 l.support, w i = 0\n\u22a2 \u2203 x, v = \u2191(Finset.weightedVSub (insert i0 l.support) p) w", "state_after": "case mp.inr.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhw : \u2211 i in insert i0 l.support, w i = 0\n\u22a2 v = \u2191(Finset.weightedVSub (insert i0 l.support) p) w"}, {"tactic": "have hz : w i0 \u2022 (p i0 -\u1d65 p i0 : V) = 0 := (vsub_self (p i0)).symm \u25b8 smul_zero _", "state_before": "case mp.inr.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhw : \u2211 i in insert i0 l.support, w i = 0\n\u22a2 v = \u2191(Finset.weightedVSub (insert i0 l.support) p) w", "state_after": "case mp.inr.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhw : \u2211 i in insert i0 l.support, w i = 0\nhz : w i0 \u2022 (p i0 -\u1d65 p i0) = 0\n\u22a2 v = \u2191(Finset.weightedVSub (insert i0 l.support) p) w"}, {"tactic": "change (fun i => w i \u2022 (p i -\u1d65 p i0 : V)) i0 = 0 at hz", "state_before": "case mp.inr.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhw : \u2211 i in insert i0 l.support, w i = 0\nhz : w i0 \u2022 (p i0 -\u1d65 p i0) = 0\n\u22a2 v = \u2191(Finset.weightedVSub (insert i0 l.support) p) w", "state_after": "case mp.inr.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhw : \u2211 i in insert i0 l.support, w i = 0\nhz : (fun i => w i \u2022 (p i -\u1d65 p i0)) i0 = 0\n\u22a2 v = \u2191(Finset.weightedVSub (insert i0 l.support) p) w"}, {"tactic": "rw [Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero _ w p hw (p i0),\n  Finset.weightedVSubOfPoint_apply, \u2190 hv, Finsupp.total_apply,\n  @Finset.sum_insert_zero _ _ l.support i0 _ _ _ hz]", "state_before": "case mp.inr.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhw : \u2211 i in insert i0 l.support, w i = 0\nhz : (fun i => w i \u2022 (p i -\u1d65 p i0)) i0 = 0\n\u22a2 v = \u2191(Finset.weightedVSub (insert i0 l.support) p) w", "state_after": "case mp.inr.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhw : \u2211 i in insert i0 l.support, w i = 0\nhz : (fun i => w i \u2022 (p i -\u1d65 p i0)) i0 = 0\n\u22a2 (Finsupp.sum l fun i a => a \u2022 (p i -\u1d65 p i0)) = \u2211 x in l.support, w x \u2022 (p x -\u1d65 p i0)"}, {"tactic": "change (\u2211 i in l.support, l i \u2022 _) = _", "state_before": "case mp.inr.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhw : \u2211 i in insert i0 l.support, w i = 0\nhz : (fun i => w i \u2022 (p i -\u1d65 p i0)) i0 = 0\n\u22a2 (Finsupp.sum l fun i a => a \u2022 (p i -\u1d65 p i0)) = \u2211 x in l.support, w x \u2022 (p x -\u1d65 p i0)", "state_after": "case mp.inr.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhw : \u2211 i in insert i0 l.support, w i = 0\nhz : (fun i => w i \u2022 (p i -\u1d65 p i0)) i0 = 0\n\u22a2 \u2211 i in l.support, \u2191l i \u2022 (p i -\u1d65 p i0) = \u2211 x in l.support, w x \u2022 (p x -\u1d65 p i0)"}, {"tactic": "congr with i", "state_before": "case mp.inr.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhw : \u2211 i in insert i0 l.support, w i = 0\nhz : (fun i => w i \u2022 (p i -\u1d65 p i0)) i0 = 0\n\u22a2 \u2211 i in l.support, \u2191l i \u2022 (p i -\u1d65 p i0) = \u2211 x in l.support, w x \u2022 (p x -\u1d65 p i0)", "state_after": "case mp.inr.intro.intro.intro.e_f.h\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhw : \u2211 i in insert i0 l.support, w i = 0\nhz : (fun i => w i \u2022 (p i -\u1d65 p i0)) i0 = 0\ni : \u03b9\n\u22a2 \u2191l i \u2022 (p i -\u1d65 p i0) = w i \u2022 (p i -\u1d65 p i0)"}, {"tactic": "by_cases h : i = i0", "state_before": "case mp.inr.intro.intro.intro.e_f.h\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhw : \u2211 i in insert i0 l.support, w i = 0\nhz : (fun i => w i \u2022 (p i -\u1d65 p i0)) i0 = 0\ni : \u03b9\n\u22a2 \u2191l i \u2022 (p i -\u1d65 p i0) = w i \u2022 (p i -\u1d65 p i0)", "state_after": "case pos\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhw : \u2211 i in insert i0 l.support, w i = 0\nhz : (fun i => w i \u2022 (p i -\u1d65 p i0)) i0 = 0\ni : \u03b9\nh : i = i0\n\u22a2 \u2191l i \u2022 (p i -\u1d65 p i0) = w i \u2022 (p i -\u1d65 p i0)\n\ncase neg\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhw : \u2211 i in insert i0 l.support, w i = 0\nhz : (fun i => w i \u2022 (p i -\u1d65 p i0)) i0 = 0\ni : \u03b9\nh : \u00aci = i0\n\u22a2 \u2191l i \u2022 (p i -\u1d65 p i0) = w i \u2022 (p i -\u1d65 p i0)"}, {"tactic": "rw [hwdef]", "state_before": "k : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\n\u22a2 \u2211 i in insert i0 l.support, w i = 0", "state_after": "k : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\n\u22a2 \u2211 i in insert i0 l.support, (\u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)) i = 0"}, {"tactic": "simp_rw [Pi.sub_apply, Finset.sum_sub_distrib,\n  Finset.sum_update_of_mem (Finset.mem_insert_self _ _),\n  Finset.sum_insert_of_eq_zero_if_not_mem Finsupp.not_mem_support_iff.1]", "state_before": "k : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\n\u22a2 \u2211 i in insert i0 l.support, (\u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)) i = 0", "state_after": "k : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\n\u22a2 \u2211 x in l.support, \u2191l x - (\u2211 x in l.support, \u2191l x + \u2211 x in insert i0 l.support \\ {i0}, Function.const \u03b9 0 x) = 0"}, {"tactic": "simp only [Finsupp.mem_support_iff, ne_eq, Finset.mem_insert, true_or, not_true,\n  Function.const_apply, Finset.sum_const_zero, add_zero, sub_self]", "state_before": "k : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\n\u22a2 \u2211 x in l.support, \u2191l x - (\u2211 x in l.support, \u2191l x + \u2211 x in insert i0 l.support \\ {i0}, Function.const \u03b9 0 x) = 0", "state_after": "no goals"}, {"tactic": "simp [h]", "state_before": "case pos\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhw : \u2211 i in insert i0 l.support, w i = 0\nhz : (fun i => w i \u2022 (p i -\u1d65 p i0)) i0 = 0\ni : \u03b9\nh : i = i0\n\u22a2 \u2191l i \u2022 (p i -\u1d65 p i0) = w i \u2022 (p i -\u1d65 p i0)", "state_after": "no goals"}, {"tactic": "simp [hwdef, h]", "state_before": "case neg\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\ni0 : \u03b9\nl : \u03b9 \u2192\u2080 k\nleft\u271d : l \u2208 Finsupp.supported k k Set.univ\nhv : \u2191(Finsupp.total \u03b9 V k fun i => p i -\u1d65 p i0) l = v\nw : (a : \u03b9) \u2192 (fun x => k) a := \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhwdef : w = \u2191l - Function.update (Function.const \u03b9 0) i0 (\u2211 i in l.support, \u2191l i)\nhw : \u2211 i in insert i0 l.support, w i = 0\nhz : (fun i => w i \u2022 (p i -\u1d65 p i0)) i0 = 0\ni : \u03b9\nh : \u00aci = i0\n\u22a2 \u2191l i \u2022 (p i -\u1d65 p i0) = w i \u2022 (p i -\u1d65 p i0)", "state_after": "no goals"}, {"tactic": "rw [Set.range_eq_empty, vectorSpan_empty, Submodule.mem_bot]", "state_before": "case mp.inl\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\nh\u03b9 : IsEmpty \u03b9\n\u22a2 v \u2208 vectorSpan k (Set.range p) \u2192 \u2203 s w x, v = \u2191(Finset.weightedVSub s p) w", "state_after": "case mp.inl\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\nh\u03b9 : IsEmpty \u03b9\n\u22a2 v = 0 \u2192 \u2203 s w x, v = \u2191(Finset.weightedVSub s p) w"}, {"tactic": "rintro rfl", "state_before": "case mp.inl\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\nh\u03b9 : IsEmpty \u03b9\n\u22a2 v = 0 \u2192 \u2203 s w x, v = \u2191(Finset.weightedVSub s p) w", "state_after": "case mp.inl\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\np : \u03b9 \u2192 P\nh\u03b9 : IsEmpty \u03b9\n\u22a2 \u2203 s w x, 0 = \u2191(Finset.weightedVSub s p) w"}, {"tactic": "use \u2205", "state_before": "case mp.inl\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\np : \u03b9 \u2192 P\nh\u03b9 : IsEmpty \u03b9\n\u22a2 \u2203 s w x, 0 = \u2191(Finset.weightedVSub s p) w", "state_after": "case mp.inl\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\np : \u03b9 \u2192 P\nh\u03b9 : IsEmpty \u03b9\n\u22a2 \u2203 w x, 0 = \u2191(Finset.weightedVSub \u2205 p) w"}, {"tactic": "simp", "state_before": "case mp.inl\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\np : \u03b9 \u2192 P\nh\u03b9 : IsEmpty \u03b9\n\u22a2 \u2203 w x, 0 = \u2191(Finset.weightedVSub \u2205 p) w", "state_after": "no goals"}, {"tactic": "rintro \u27e8s, w, hw, rfl\u27e9", "state_before": "case mpr\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\nv : V\np : \u03b9 \u2192 P\n\u22a2 (\u2203 s w x, v = \u2191(Finset.weightedVSub s p) w) \u2192 v \u2208 vectorSpan k (Set.range p)", "state_after": "case mpr.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\np : \u03b9 \u2192 P\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\n\u22a2 \u2191(Finset.weightedVSub s p) w \u2208 vectorSpan k (Set.range p)"}, {"tactic": "exact weightedVSub_mem_vectorSpan hw p", "state_before": "case mpr.intro.intro.intro\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_1\np : \u03b9 \u2192 P\ns : Finset \u03b9\nw : \u03b9 \u2192 k\nhw : \u2211 i in s, w i = 0\n\u22a2 \u2191(Finset.weightedVSub s p) w \u2208 vectorSpan k (Set.range p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/LiminfLimsup.lean", "full_name": "Filter.blimsup_or_eq_sup", "start": [981, 1], "end": [985, 98], "traced_tactics": [{"tactic": "refine' le_antisymm _ blimsup_sup_le_or", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.162092\n\u03b9 : Type ?u.162095\ninst\u271d : CompleteDistribLattice \u03b1\nf : Filter \u03b2\np q : \u03b2 \u2192 Prop\nu : \u03b2 \u2192 \u03b1\n\u22a2 (blimsup u f fun x => p x \u2228 q x) = blimsup u f p \u2294 blimsup u f q", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.162092\n\u03b9 : Type ?u.162095\ninst\u271d : CompleteDistribLattice \u03b1\nf : Filter \u03b2\np q : \u03b2 \u2192 Prop\nu : \u03b2 \u2192 \u03b1\n\u22a2 (blimsup u f fun x => p x \u2228 q x) \u2264 blimsup u f p \u2294 blimsup u f q"}, {"tactic": "simp only [blimsup_eq, sInf_sup_eq, sup_sInf_eq, le_iInf\u2082_iff, mem_setOf_eq]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.162092\n\u03b9 : Type ?u.162095\ninst\u271d : CompleteDistribLattice \u03b1\nf : Filter \u03b2\np q : \u03b2 \u2192 Prop\nu : \u03b2 \u2192 \u03b1\n\u22a2 (blimsup u f fun x => p x \u2228 q x) \u2264 blimsup u f p \u2294 blimsup u f q", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.162092\n\u03b9 : Type ?u.162095\ninst\u271d : CompleteDistribLattice \u03b1\nf : Filter \u03b2\np q : \u03b2 \u2192 Prop\nu : \u03b2 \u2192 \u03b1\n\u22a2 \u2200 (i : \u03b1),\n    (\u2200\u1da0 (x : \u03b2) in f, q x \u2192 u x \u2264 i) \u2192\n      \u2200 (i_1 : \u03b1), (\u2200\u1da0 (x : \u03b2) in f, p x \u2192 u x \u2264 i_1) \u2192 sInf {a | \u2200\u1da0 (x : \u03b2) in f, p x \u2228 q x \u2192 u x \u2264 a} \u2264 i_1 \u2294 i"}, {"tactic": "refine' fun a' ha' a ha => sInf_le ((ha.and ha').mono fun b h hb => _)", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.162092\n\u03b9 : Type ?u.162095\ninst\u271d : CompleteDistribLattice \u03b1\nf : Filter \u03b2\np q : \u03b2 \u2192 Prop\nu : \u03b2 \u2192 \u03b1\n\u22a2 \u2200 (i : \u03b1),\n    (\u2200\u1da0 (x : \u03b2) in f, q x \u2192 u x \u2264 i) \u2192\n      \u2200 (i_1 : \u03b1), (\u2200\u1da0 (x : \u03b2) in f, p x \u2192 u x \u2264 i_1) \u2192 sInf {a | \u2200\u1da0 (x : \u03b2) in f, p x \u2228 q x \u2192 u x \u2264 a} \u2264 i_1 \u2294 i", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.162092\n\u03b9 : Type ?u.162095\ninst\u271d : CompleteDistribLattice \u03b1\nf : Filter \u03b2\np q : \u03b2 \u2192 Prop\nu : \u03b2 \u2192 \u03b1\na' : \u03b1\nha' : \u2200\u1da0 (x : \u03b2) in f, q x \u2192 u x \u2264 a'\na : \u03b1\nha : \u2200\u1da0 (x : \u03b2) in f, p x \u2192 u x \u2264 a\nb : \u03b2\nh : (p b \u2192 u b \u2264 a) \u2227 (q b \u2192 u b \u2264 a')\nhb : p b \u2228 q b\n\u22a2 u b \u2264 a \u2294 a'"}, {"tactic": "exact Or.elim hb (fun hb => le_sup_of_le_left <| h.1 hb) fun hb => le_sup_of_le_right <| h.2 hb", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.162092\n\u03b9 : Type ?u.162095\ninst\u271d : CompleteDistribLattice \u03b1\nf : Filter \u03b2\np q : \u03b2 \u2192 Prop\nu : \u03b2 \u2192 \u03b1\na' : \u03b1\nha' : \u2200\u1da0 (x : \u03b2) in f, q x \u2192 u x \u2264 a'\na : \u03b1\nha : \u2200\u1da0 (x : \u03b2) in f, p x \u2192 u x \u2264 a\nb : \u03b2\nh : (p b \u2192 u b \u2264 a) \u2227 (q b \u2192 u b \u2264 a')\nhb : p b \u2228 q b\n\u22a2 u b \u2264 a \u2294 a'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Submonoid/Basic.lean", "full_name": "Submonoid.mem_mk", "start": [179, 1], "end": [180, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/UpperLower/Basic.lean", "full_name": "lowerClosure_mono", "start": [1386, 1], "end": [1387, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Monotone/Basic.lean", "full_name": "Nat.rel_of_forall_rel_succ_of_le_of_lt", "start": [967, 1], "end": [971, 70], "traced_tactics": [{"tactic": "induction' hbc with k b_lt_k r_b_k", "state_before": "\u03b9 : Type ?u.39250\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.39259\n\u03c0 : \u03b9 \u2192 Type ?u.39264\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsTrans \u03b2 r\nf : \u2115 \u2192 \u03b2\na : \u2115\nh : \u2200 (n : \u2115), a \u2264 n \u2192 r (f n) (f (n + 1))\nb c : \u2115\nhab : a \u2264 b\nhbc : b < c\n\u22a2 r (f b) (f c)", "state_after": "case refl\n\u03b9 : Type ?u.39250\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.39259\n\u03c0 : \u03b9 \u2192 Type ?u.39264\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsTrans \u03b2 r\nf : \u2115 \u2192 \u03b2\na : \u2115\nh : \u2200 (n : \u2115), a \u2264 n \u2192 r (f n) (f (n + 1))\nb c : \u2115\nhab : a \u2264 b\n\u22a2 r (f b) (f (succ b))\n\ncase step\n\u03b9 : Type ?u.39250\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.39259\n\u03c0 : \u03b9 \u2192 Type ?u.39264\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsTrans \u03b2 r\nf : \u2115 \u2192 \u03b2\na : \u2115\nh : \u2200 (n : \u2115), a \u2264 n \u2192 r (f n) (f (n + 1))\nb c : \u2115\nhab : a \u2264 b\nk : \u2115\nb_lt_k : Nat.le (succ b) k\nr_b_k : r (f b) (f k)\n\u22a2 r (f b) (f (succ k))"}, {"tactic": "exacts [h _ hab, _root_.trans r_b_k (h _ (hab.trans_lt b_lt_k).le)]", "state_before": "case refl\n\u03b9 : Type ?u.39250\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.39259\n\u03c0 : \u03b9 \u2192 Type ?u.39264\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsTrans \u03b2 r\nf : \u2115 \u2192 \u03b2\na : \u2115\nh : \u2200 (n : \u2115), a \u2264 n \u2192 r (f n) (f (n + 1))\nb c : \u2115\nhab : a \u2264 b\n\u22a2 r (f b) (f (succ b))\n\ncase step\n\u03b9 : Type ?u.39250\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.39259\n\u03c0 : \u03b9 \u2192 Type ?u.39264\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\nr : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsTrans \u03b2 r\nf : \u2115 \u2192 \u03b2\na : \u2115\nh : \u2200 (n : \u2115), a \u2264 n \u2192 r (f n) (f (n + 1))\nb c : \u2115\nhab : a \u2264 b\nk : \u2115\nb_lt_k : Nat.le (succ b) k\nr_b_k : r (f b) (f k)\n\u22a2 r (f b) (f (succ k))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/PolarCoord.lean", "full_name": "hasFDerivAt_polarCoord_symm", "start": [102, 1], "end": [110, 58], "traced_tactics": [{"tactic": "rw [Matrix.toLin_finTwoProd_toContinuousLinearMap]", "state_before": "p : \u211d \u00d7 \u211d\n\u22a2 HasFDerivAt (\u2191(LocalHomeomorph.symm polarCoord))\n    (\u2191LinearMap.toContinuousLinearMap\n      (\u2191(Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (\u2191Matrix.of ![![cos p.snd, -p.fst * sin p.snd], ![sin p.snd, p.fst * cos p.snd]])))\n    p", "state_after": "p : \u211d \u00d7 \u211d\n\u22a2 HasFDerivAt (\u2191(LocalHomeomorph.symm polarCoord))\n    (ContinuousLinearMap.prod\n      (cos p.snd \u2022 ContinuousLinearMap.fst \u211d \u211d \u211d + (-p.fst * sin p.snd) \u2022 ContinuousLinearMap.snd \u211d \u211d \u211d)\n      (sin p.snd \u2022 ContinuousLinearMap.fst \u211d \u211d \u211d + (p.fst * cos p.snd) \u2022 ContinuousLinearMap.snd \u211d \u211d \u211d))\n    p"}, {"tactic": "convert HasFDerivAt.prod (\ud835\udd5c := \u211d)\n  (hasFDerivAt_fst.mul ((hasDerivAt_cos p.2).comp_hasFDerivAt p hasFDerivAt_snd))\n  (hasFDerivAt_fst.mul ((hasDerivAt_sin p.2).comp_hasFDerivAt p hasFDerivAt_snd)) using 2 <;>\nsimp [smul_smul, add_comm, neg_mul, neg_smul, smul_neg]", "state_before": "p : \u211d \u00d7 \u211d\n\u22a2 HasFDerivAt (\u2191(LocalHomeomorph.symm polarCoord))\n    (ContinuousLinearMap.prod\n      (cos p.snd \u2022 ContinuousLinearMap.fst \u211d \u211d \u211d + (-p.fst * sin p.snd) \u2022 ContinuousLinearMap.snd \u211d \u211d \u211d)\n      (sin p.snd \u2022 ContinuousLinearMap.fst \u211d \u211d \u211d + (p.fst * cos p.snd) \u2022 ContinuousLinearMap.snd \u211d \u211d \u211d))\n    p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Permutation.lean", "full_name": "List.map_map_permutations'Aux", "start": [133, 1], "end": [135, 59], "traced_tactics": [{"tactic": "induction' ts with a ts ih <;> [rfl; (simp [\u2190 ih]; rfl)]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nt : \u03b1\nts : List \u03b1\n\u22a2 map (map f) (permutations'Aux t ts) = permutations'Aux (f t) (map f ts)", "state_after": "no goals"}, {"tactic": "simp [\u2190 ih]", "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nt a : \u03b1\nts : List \u03b1\nih : map (map f) (permutations'Aux t ts) = permutations'Aux (f t) (map f ts)\n\u22a2 map (map f) (permutations'Aux t (a :: ts)) = permutations'Aux (f t) (map f (a :: ts))", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nt a : \u03b1\nts : List \u03b1\nih : map (map f) (permutations'Aux t ts) = permutations'Aux (f t) (map f ts)\n\u22a2 map (map f \u2218 cons a) (permutations'Aux t ts) = map (cons (f a) \u2218 map f) (permutations'Aux t ts)"}, {"tactic": "rfl", "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nt a : \u03b1\nts : List \u03b1\nih : map (map f) (permutations'Aux t ts) = permutations'Aux (f t) (map f ts)\n\u22a2 map (map f \u2218 cons a) (permutations'Aux t ts) = map (cons (f a) \u2218 map f) (permutations'Aux t ts)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/Sqrt.lean", "full_name": "Real.sqrt_eq_iff_mul_self_eq_of_pos", "start": [223, 1], "end": [224, 36], "traced_tactics": [{"tactic": "simp [sqrt_eq_cases, h.ne', h.le]", "state_before": "x y : \u211d\nh : 0 < y\n\u22a2 sqrt x = y \u2194 y * y = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "full_name": "Cardinal.countable_iff_lt_aleph_one", "start": [351, 1], "end": [352, 63], "traced_tactics": [{"tactic": "rw [\u2190 succ_aleph0, lt_succ_iff, le_aleph0_iff_set_countable]", "state_before": "\u03b1 : Type u_1\ns : Set \u03b1\n\u22a2 Set.Countable s \u2194 (#\u2191s) < aleph 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Monotone/Basic.lean", "full_name": "antitoneOn_toDual_comp_iff", "start": [176, 1], "end": [177, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "full_name": "BoxIntegral.Box.exists_seq_mono_tendsto", "start": [475, 1], "end": [485, 66], "traced_tactics": [{"tactic": "choose a b ha_anti hb_mono ha_mem hb_mem hab ha_tendsto hb_tendsto using\n  fun i \u21a6 exists_seq_strictAnti_strictMono_tendsto (I.lower_lt_upper i)", "state_before": "\u03b9 : Type u_1\nI\u271d J : Box \u03b9\nx y : \u03b9 \u2192 \u211d\nI 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(lower \u2218 \u2191J) atTop (\ud835\udcdd I.lower) \u2227 Tendsto (upper \u2218 \u2191J) atTop (\ud835\udcdd I.upper)"}, {"tactic": "exact\n  \u27e8\u27e8fun k \u21a6 \u27e8flip a k, flip b k, fun i \u21a6 hab _ _ _\u27e9, fun k l hkl \u21a6\n      le_iff_bounds.2 \u27e8fun i \u21a6 (ha_anti i).antitone hkl, fun i \u21a6 (hb_mono i).monotone hkl\u27e9\u27e9,\n    fun n x hx i _ \u21a6 \u27e8(ha_mem _ _).1.trans_le (hx.1 _), (hx.2 _).trans_lt (hb_mem _ _).2\u27e9,\n    tendsto_pi_nhds.2 ha_tendsto, tendsto_pi_nhds.2 hb_tendsto\u27e9", "state_before": "\u03b9 : Type u_1\nI\u271d J : Box \u03b9\nx y : \u03b9 \u2192 \u211d\nI : Box \u03b9\na b : \u03b9 \u2192 \u2115 \u2192 \u211d\nha_anti : \u2200 (i : \u03b9), StrictAnti (a i)\nhb_mono : \u2200 (i : \u03b9), StrictMono (b i)\nha_mem : \u2200 (i : \u03b9) (k : \u2115), a i k \u2208 Ioo (lower I i) (upper I i)\nhb_mem : \u2200 (i : \u03b9) (l : \u2115), b i l \u2208 Ioo (lower I i) (upper I i)\nhab : \u2200 (i : \u03b9) (k l : \u2115), a i 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"https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/GroupAction/Prod.lean", "full_name": "Prod.pow_def", "start": [111, 1], "end": [112, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/ContDiff.lean", "full_name": "ContinuousLinearEquiv.contDiffWithinAt_comp_iff", "start": [471, 1], "end": [478, 39], "traced_tactics": [{"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.555631\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\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ne : G \u2243L[\ud835\udd5c] E\n\u22a2 ContDiffWithinAt \ud835\udd5c n (f \u2218 \u2191e) (\u2191e \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm e) x) \u2194 ContDiffWithinAt \ud835\udd5c n f s x", "state_after": "case mp\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.555631\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\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ne : G \u2243L[\ud835\udd5c] E\n\u22a2 ContDiffWithinAt \ud835\udd5c n (f \u2218 \u2191e) (\u2191e \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm e) x) \u2192 ContDiffWithinAt \ud835\udd5c n f s x\n\ncase mpr\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.555631\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\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ne : G \u2243L[\ud835\udd5c] E\n\u22a2 ContDiffWithinAt \ud835\udd5c n f s x \u2192 ContDiffWithinAt \ud835\udd5c n (f \u2218 \u2191e) (\u2191e \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm e) x)"}, {"tactic": "intro H", "state_before": "case mp\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.555631\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\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ne : G \u2243L[\ud835\udd5c] E\n\u22a2 ContDiffWithinAt \ud835\udd5c n (f \u2218 \u2191e) (\u2191e \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm e) x) \u2192 ContDiffWithinAt \ud835\udd5c n f s x", "state_after": "case mp\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.555631\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\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ne : G \u2243L[\ud835\udd5c] E\nH : ContDiffWithinAt \ud835\udd5c n (f \u2218 \u2191e) (\u2191e \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm e) x)\n\u22a2 ContDiffWithinAt \ud835\udd5c n f s x"}, {"tactic": "intro H", "state_before": "case mpr\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.555631\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\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ne : G \u2243L[\ud835\udd5c] E\n\u22a2 ContDiffWithinAt \ud835\udd5c n f s x \u2192 ContDiffWithinAt \ud835\udd5c n (f \u2218 \u2191e) (\u2191e \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm e) x)", "state_after": "case mpr\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.555631\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\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ne : G \u2243L[\ud835\udd5c] E\nH : ContDiffWithinAt \ud835\udd5c n f s x\n\u22a2 ContDiffWithinAt \ud835\udd5c n (f \u2218 \u2191e) (\u2191e \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm e) x)"}, {"tactic": "rw [\u2190 e.apply_symm_apply x, \u2190 e.coe_coe] at H", "state_before": "case mpr\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.555631\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\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ne : G \u2243L[\ud835\udd5c] E\nH : ContDiffWithinAt \ud835\udd5c n f s x\n\u22a2 ContDiffWithinAt \ud835\udd5c n (f \u2218 \u2191e) (\u2191e \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm e) x)", "state_after": "case mpr\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.555631\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\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ne : G \u2243L[\ud835\udd5c] E\nH : ContDiffWithinAt \ud835\udd5c n f s (\u2191\u2191e (\u2191(ContinuousLinearEquiv.symm e) x))\n\u22a2 ContDiffWithinAt \ud835\udd5c n (f \u2218 \u2191e) (\u2191e \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm e) x)"}, {"tactic": "exact H.comp_continuousLinearMap _", "state_before": "case mpr\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.555631\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\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ne : G \u2243L[\ud835\udd5c] E\nH : ContDiffWithinAt \ud835\udd5c n f s (\u2191\u2191e (\u2191(ContinuousLinearEquiv.symm e) x))\n\u22a2 ContDiffWithinAt \ud835\udd5c n (f \u2218 \u2191e) (\u2191e \u207b\u00b9' s) (\u2191(ContinuousLinearEquiv.symm e) x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/PFunctor/Multivariate/Basic.lean", "full_name": "MvPFunctor.const.get_mk", "start": [110, 1], "end": [110, 90], "traced_tactics": [{"tactic": "rfl", "state_before": "n m : \u2115\nP : MvPFunctor n\nA : Type u\n\u03b1 \u03b2 : TypeVec n\nx : A\n\u22a2 get (mk n x) = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/FiniteType.lean", "full_name": "Algebra.FiniteType.self", "start": [77, 1], "end": [78, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Regular/SMul.lean", "full_name": "IsSMulRegular.zero_iff_subsingleton", "start": [196, 1], "end": [197, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/Padics/Hensel.lean", "full_name": "newton_seq_norm_eq", "start": [261, 9], "end": [265, 34], "traced_tactics": [{"tactic": "rw [newton_seq_gen, newton_seq_gen, newton_seq_aux, ih_n]", "state_before": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn : \u2115\n\u22a2 \u2016newton_seq (n + 1) - newton_seq n\u2016 =\n    \u2016Polynomial.eval (newton_seq n) F\u2016 / \u2016Polynomial.eval (newton_seq n) (\u2191Polynomial.derivative F)\u2016", "state_after": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn : \u2115\n\u22a2 \u2016\u2191(let_fun h1 :=\n            (_ :\n              \u2016\u2191(Polynomial.eval (\u2191(newton_seq_aux hnorm n)) F) /\n                    \u2191(Polynomial.eval (\u2191(newton_seq_aux hnorm n)) (\u2191Polynomial.derivative F))\u2016 \u2264\n                1);\n          let z1 :=\n            {\n              val :=\n                \u2191(Polynomial.eval (\u2191(newton_seq_aux hnorm n)) F) /\n                  \u2191(Polynomial.eval (\u2191(newton_seq_aux hnorm n)) (\u2191Polynomial.derivative F)),\n              property := h1 };\n          let z' := \u2191(newton_seq_aux hnorm n) - z1;\n          { val := z', property := (_ : ih_gen (n + 1) z') }) -\n        \u2191(newton_seq_aux hnorm n)\u2016 =\n    \u2016Polynomial.eval (\u2191(newton_seq_aux hnorm n)) F\u2016 /\n      \u2016Polynomial.eval (\u2191(newton_seq_aux hnorm n)) (\u2191Polynomial.derivative F)\u2016"}, {"tactic": "simp [sub_eq_add_neg, add_comm]", "state_before": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nF : Polynomial \u2124_[p]\na : \u2124_[p]\nhnorm : \u2016Polynomial.eval a F\u2016 < \u2016Polynomial.eval a (\u2191Polynomial.derivative F)\u2016 ^ 2\nhnsol : Polynomial.eval a F \u2260 0\nn : \u2115\n\u22a2 \u2016\u2191(let_fun h1 :=\n            (_ :\n              \u2016\u2191(Polynomial.eval (\u2191(newton_seq_aux hnorm n)) F) /\n                    \u2191(Polynomial.eval (\u2191(newton_seq_aux hnorm n)) (\u2191Polynomial.derivative F))\u2016 \u2264\n                1);\n          let z1 :=\n            {\n              val :=\n                \u2191(Polynomial.eval (\u2191(newton_seq_aux hnorm n)) F) /\n                  \u2191(Polynomial.eval (\u2191(newton_seq_aux hnorm n)) (\u2191Polynomial.derivative F)),\n              property := h1 };\n          let z' := \u2191(newton_seq_aux hnorm n) - z1;\n          { val := z', property := (_ : ih_gen (n + 1) z') }) -\n        \u2191(newton_seq_aux hnorm n)\u2016 =\n    \u2016Polynomial.eval (\u2191(newton_seq_aux hnorm n)) F\u2016 /\n      \u2016Polynomial.eval (\u2191(newton_seq_aux hnorm n)) (\u2191Polynomial.derivative F)\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean", "full_name": "HasStrictDerivAt.clm_comp", "start": [350, 1], "end": [354, 32], "traced_tactics": [{"tactic": "have := (hc.hasStrictFDerivAt.clm_comp hd.hasStrictFDerivAt).hasStrictDerivAt", "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\nG : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nc : \ud835\udd5c \u2192 F \u2192L[\ud835\udd5c] G\nc' : F \u2192L[\ud835\udd5c] G\nd : \ud835\udd5c \u2192 E \u2192L[\ud835\udd5c] F\nd' : E \u2192L[\ud835\udd5c] F\nu : \ud835\udd5c \u2192 F\nu' : F\nhc : HasStrictDerivAt c c' x\nhd : HasStrictDerivAt d d' x\n\u22a2 HasStrictDerivAt (fun y => comp (c y) (d y)) (comp c' (d x) + comp (c x) d') x", "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\nG : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nc : \ud835\udd5c \u2192 F \u2192L[\ud835\udd5c] G\nc' : F \u2192L[\ud835\udd5c] G\nd : \ud835\udd5c \u2192 E \u2192L[\ud835\udd5c] F\nd' : E \u2192L[\ud835\udd5c] F\nu : \ud835\udd5c \u2192 F\nu' : F\nhc : HasStrictDerivAt c c' x\nhd : HasStrictDerivAt d d' x\nthis :\n  HasStrictDerivAt (fun y => comp (c y) (d y))\n    (\u2191(comp (\u2191(compL \ud835\udd5c E F G) (c x)) (smulRight 1 d') +\n          comp (\u2191(ContinuousLinearMap.flip (compL \ud835\udd5c E F G)) (d x)) (smulRight 1 c'))\n      1)\n    x\n\u22a2 HasStrictDerivAt (fun y => comp (c y) (d y)) (comp c' (d x) + comp (c x) d') x"}, {"tactic": "rwa [add_apply, comp_apply, comp_apply, smulRight_apply, smulRight_apply, one_apply, one_smul,\n  one_smul, add_comm] at this", "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\nG : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nc : \ud835\udd5c \u2192 F \u2192L[\ud835\udd5c] G\nc' : F \u2192L[\ud835\udd5c] G\nd : \ud835\udd5c \u2192 E \u2192L[\ud835\udd5c] F\nd' : E \u2192L[\ud835\udd5c] F\nu : \ud835\udd5c \u2192 F\nu' : F\nhc : HasStrictDerivAt c c' x\nhd : HasStrictDerivAt d d' x\nthis :\n  HasStrictDerivAt (fun y => comp (c y) (d y))\n    (\u2191(comp (\u2191(compL \ud835\udd5c E F G) (c x)) (smulRight 1 d') +\n          comp (\u2191(ContinuousLinearMap.flip (compL \ud835\udd5c E F G)) (d x)) (smulRight 1 c'))\n      1)\n    x\n\u22a2 HasStrictDerivAt (fun y => comp (c y) (d y)) (comp c' (d x) + comp (c x) d') x", "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_mem_smul", "start": [104, 1], "end": [105, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/RamificationInertia.lean", "full_name": "Ideal.le_pow_of_le_ramificationIdx", "start": [127, 1], "end": [130, 30], "traced_tactics": [{"tactic": "contrapose! hn", "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\nS : Type v\ninst\u271d : CommRing S\nf : R \u2192+* S\np : Ideal R\nP : Ideal S\nn : \u2115\nhn : n \u2264 ramificationIdx f p P\n\u22a2 map f p \u2264 P ^ n", "state_after": "R : Type u\ninst\u271d\u00b9 : CommRing R\nS : Type v\ninst\u271d : CommRing S\nf : R \u2192+* S\np : Ideal R\nP : Ideal S\nn : \u2115\nhn : \u00acmap f p \u2264 P ^ n\n\u22a2 ramificationIdx f p P < n"}, {"tactic": "exact ramificationIdx_lt hn", "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\nS : Type v\ninst\u271d : CommRing S\nf : R \u2192+* S\np : Ideal R\nP : Ideal S\nn : \u2115\nhn : \u00acmap f p \u2264 P ^ n\n\u22a2 ramificationIdx f p P < n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/LocalEquiv.lean", "full_name": "LocalEquiv.EqOnSource.restr", "start": [875, 1], "end": [881, 20], "traced_tactics": [{"tactic": "constructor", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.57579\n\u03b4 : Type ?u.57582\ne\u271d : LocalEquiv \u03b1 \u03b2\ne'\u271d : LocalEquiv \u03b2 \u03b3\ne e' : LocalEquiv \u03b1 \u03b2\nhe : e \u2248 e'\ns : Set \u03b1\n\u22a2 LocalEquiv.restr e s \u2248 LocalEquiv.restr e' s", "state_after": "case left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.57579\n\u03b4 : Type ?u.57582\ne\u271d : LocalEquiv \u03b1 \u03b2\ne'\u271d : LocalEquiv \u03b2 \u03b3\ne e' : LocalEquiv \u03b1 \u03b2\nhe : e \u2248 e'\ns : Set \u03b1\n\u22a2 (LocalEquiv.restr e s).source = (LocalEquiv.restr e' s).source\n\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.57579\n\u03b4 : Type ?u.57582\ne\u271d : LocalEquiv \u03b1 \u03b2\ne'\u271d : LocalEquiv \u03b2 \u03b3\ne e' : LocalEquiv \u03b1 \u03b2\nhe : e \u2248 e'\ns : Set \u03b1\n\u22a2 EqOn (\u2191(LocalEquiv.restr e s)) (\u2191(LocalEquiv.restr e' s)) (LocalEquiv.restr e s).source"}, {"tactic": "simp [he.1]", "state_before": "case left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.57579\n\u03b4 : Type ?u.57582\ne\u271d : LocalEquiv \u03b1 \u03b2\ne'\u271d : LocalEquiv \u03b2 \u03b3\ne e' : LocalEquiv \u03b1 \u03b2\nhe : e \u2248 e'\ns : Set \u03b1\n\u22a2 (LocalEquiv.restr e s).source = (LocalEquiv.restr e' s).source", "state_after": "no goals"}, {"tactic": "intro x hx", "state_before": "case right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.57579\n\u03b4 : Type ?u.57582\ne\u271d : LocalEquiv \u03b1 \u03b2\ne'\u271d : LocalEquiv \u03b2 \u03b3\ne e' : LocalEquiv \u03b1 \u03b2\nhe : e \u2248 e'\ns : Set \u03b1\n\u22a2 EqOn (\u2191(LocalEquiv.restr e s)) (\u2191(LocalEquiv.restr e' s)) (LocalEquiv.restr e s).source", "state_after": "case right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.57579\n\u03b4 : Type ?u.57582\ne\u271d : LocalEquiv \u03b1 \u03b2\ne'\u271d : LocalEquiv \u03b2 \u03b3\ne e' : LocalEquiv \u03b1 \u03b2\nhe : e \u2248 e'\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 (LocalEquiv.restr e s).source\n\u22a2 \u2191(LocalEquiv.restr e s) x = \u2191(LocalEquiv.restr e' s) x"}, {"tactic": "simp only [mem_inter_iff, restr_source] at hx", "state_before": "case right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.57579\n\u03b4 : Type ?u.57582\ne\u271d : LocalEquiv \u03b1 \u03b2\ne'\u271d : LocalEquiv \u03b2 \u03b3\ne e' : LocalEquiv \u03b1 \u03b2\nhe : e \u2248 e'\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 (LocalEquiv.restr e s).source\n\u22a2 \u2191(LocalEquiv.restr e s) x = \u2191(LocalEquiv.restr e' s) x", "state_after": "case right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.57579\n\u03b4 : Type ?u.57582\ne\u271d : LocalEquiv \u03b1 \u03b2\ne'\u271d : LocalEquiv \u03b2 \u03b3\ne e' : LocalEquiv \u03b1 \u03b2\nhe : e \u2248 e'\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 e.source \u2227 x \u2208 s\n\u22a2 \u2191(LocalEquiv.restr e s) x = \u2191(LocalEquiv.restr e' s) x"}, {"tactic": "exact he.2 hx.1", "state_before": "case right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.57579\n\u03b4 : Type ?u.57582\ne\u271d : LocalEquiv \u03b1 \u03b2\ne'\u271d : LocalEquiv \u03b2 \u03b3\ne e' : LocalEquiv \u03b1 \u03b2\nhe : e \u2248 e'\ns : Set \u03b1\nx : \u03b1\nhx : x \u2208 e.source \u2227 x \u2208 s\n\u22a2 \u2191(LocalEquiv.restr e s) x = \u2191(LocalEquiv.restr e' s) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.integrable_map_equiv", "start": [619, 1], "end": [622, 32], "traced_tactics": [{"tactic": "simp_rw [\u2190 mem\u2112p_one_iff_integrable]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_3\n\u03b3 : Type ?u.859671\n\u03b4 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2243\u1d50 \u03b4\ng : \u03b4 \u2192 \u03b2\n\u22a2 Integrable g \u2194 Integrable (g \u2218 \u2191f)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_3\n\u03b3 : Type ?u.859671\n\u03b4 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2243\u1d50 \u03b4\ng : \u03b4 \u2192 \u03b2\n\u22a2 Mem\u2112p g 1 \u2194 Mem\u2112p (g \u2218 \u2191f) 1"}, {"tactic": "exact f.mem\u2112p_map_measure_iff", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_3\n\u03b3 : Type ?u.859671\n\u03b4 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2243\u1d50 \u03b4\ng : \u03b4 \u2192 \u03b2\n\u22a2 Mem\u2112p g 1 \u2194 Mem\u2112p (g \u2218 \u2191f) 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Heyting/Basic.lean", "full_name": "IsCompl.eq_hnot", "start": [1056, 1], "end": [1057, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Nilpotent.lean", "full_name": "IsNilpotent.neg", "start": [50, 1], "end": [53, 42], "traced_tactics": [{"tactic": "obtain \u27e8n, hn\u27e9 := h", "state_before": "R S : Type u\nx y : R\ninst\u271d : Ring R\nh : IsNilpotent x\n\u22a2 IsNilpotent (-x)", "state_after": "case intro\nR S : Type u\nx y : R\ninst\u271d : Ring R\nn : \u2115\nhn : x ^ n = 0\n\u22a2 IsNilpotent (-x)"}, {"tactic": "use n", "state_before": "case intro\nR S : Type u\nx y : R\ninst\u271d : Ring R\nn : \u2115\nhn : x ^ n = 0\n\u22a2 IsNilpotent (-x)", "state_after": "case intro\nR S : Type u\nx y : R\ninst\u271d : Ring R\nn : \u2115\nhn : x ^ n = 0\n\u22a2 (-x) ^ n = 0"}, {"tactic": "rw [neg_pow, hn, MulZeroClass.mul_zero]", "state_before": "case intro\nR S : Type u\nx y : R\ninst\u271d : Ring R\nn : \u2115\nhn : x ^ n = 0\n\u22a2 (-x) ^ n = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.mem_of_mem_inter_right", "start": [1578, 1], "end": [1579, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/ToIntervalMod.lean", "full_name": "AddCommGroup.modEq_iff_toIocMod_eq_right", "start": [600, 1], "end": [604, 61], "traced_tactics": [{"tactic": "refine' modEq_iff_eq_add_zsmul.trans \u27e8_, fun h => \u27e8toIocDiv hp a b + 1, _\u27e9\u27e9", "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\n\u22a2 a \u2261 b [PMOD p] \u2194 toIocMod hp a b = a + p", "state_after": "case refine'_1\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nh\u03b1 : Archimedean \u03b1\np : \u03b1\nhp : 0 < p\na b c : \u03b1\nn : \u2124\n\u22a2 (\u2203 z, b = a + z \u2022 p) \u2192 toIocMod hp a b = a + p\n\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nh\u03b1 : Archimedean \u03b1\np : \u03b1\nhp : 0 < p\na b c : \u03b1\nn : \u2124\nh : toIocMod hp a b = a + p\n\u22a2 b = a + (toIocDiv hp a b + 1) \u2022 p"}, {"tactic": "rintro \u27e8z, rfl\u27e9", "state_before": "case refine'_1\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nh\u03b1 : Archimedean \u03b1\np : \u03b1\nhp : 0 < p\na b c : \u03b1\nn : \u2124\n\u22a2 (\u2203 z, b = a + z \u2022 p) \u2192 toIocMod hp a b = a + p", "state_after": "case refine'_1.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nh\u03b1 : Archimedean \u03b1\np : \u03b1\nhp : 0 < p\na c : \u03b1\nn z : \u2124\n\u22a2 toIocMod hp a (a + z \u2022 p) = a + p"}, {"tactic": "rw [toIocMod_add_zsmul, toIocMod_apply_left]", "state_before": "case refine'_1.intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nh\u03b1 : Archimedean \u03b1\np : \u03b1\nhp : 0 < p\na c : \u03b1\nn z : \u2124\n\u22a2 toIocMod hp a (a + z \u2022 p) = a + p", "state_after": "no goals"}, {"tactic": "rwa [add_one_zsmul, add_left_comm, \u2190 sub_eq_iff_eq_add']", "state_before": "case refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nh\u03b1 : Archimedean \u03b1\np : \u03b1\nhp : 0 < p\na b c : \u03b1\nn : \u2124\nh : toIocMod hp a b = a + p\n\u22a2 b = a + (toIocDiv hp a b + 1) \u2022 p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/Operations.lean", "full_name": "Ideal.isPrimary_inf", "start": [1868, 1], "end": [1877, 24], "traced_tactics": [{"tactic": "rw [radical_inf, hij, inf_idem]", "state_before": "R : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical (I \u2293 J)", "state_after": "R : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical J"}, {"tactic": "cases' hi.2 hxyi with hxi hyi", "state_before": "R : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical J", "state_after": "case inl\nR : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhxi : x \u2208 I\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical J\n\ncase inr\nR : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhyi : y \u2208 radical I\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical J"}, {"tactic": "cases' hj.2 hxyj with hxj hyj", "state_before": "case inl\nR : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhxi : x \u2208 I\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical J\n\ncase inr\nR : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhyi : y \u2208 radical I\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical J", "state_after": "case inl.inl\nR : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhxi : x \u2208 I\nhxj : x \u2208 J\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical J\n\ncase inl.inr\nR : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhxi : x \u2208 I\nhyj : y \u2208 radical J\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical J\n\ncase inr\nR : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhyi : y \u2208 radical I\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical J"}, {"tactic": "exact Or.inl \u27e8hxi, hxj\u27e9", "state_before": "case inl.inl\nR : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhxi : x \u2208 I\nhxj : x \u2208 J\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical J", "state_after": "no goals"}, {"tactic": "exact Or.inr hyj", "state_before": "case inl.inr\nR : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhxi : x \u2208 I\nhyj : y \u2208 radical J\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical J", "state_after": "no goals"}, {"tactic": "rw [hij] at hyi", "state_before": "case inr\nR : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhyi : y \u2208 radical I\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical J", "state_after": "case inr\nR : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhyi : y \u2208 radical J\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical J"}, {"tactic": "exact Or.inr hyi", "state_before": "case inr\nR : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhyi : y \u2208 radical J\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical J", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "full_name": "Finsupp.liftAddHom_comp_single", "start": [493, 1], "end": [495, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Manifold/LocalInvariantProperties.lean", "full_name": "StructureGroupoid.LocalInvariantProp.liftPropAt_of_liftPropWithinAt", "start": [376, 1], "end": [378, 58], "traced_tactics": [{"tactic": "rwa [\u2190 univ_inter s, hG.liftPropWithinAt_inter hs] at h", "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type ?u.36556\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace 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only [one_pow, mul_one, eval_monomial, map_monomial]", "state_before": "case h_monomial\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r\u271d : R[X]\ninst\u271d : Semiring S\nf\u271d f : R \u2192+* S\nn : \u2115\nr : R\n\u22a2 eval 1 (map f (\u2191(monomial n) r)) = \u2191f (eval 1 (\u2191(monomial n) r))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "full_name": "is_simple_module_of_finrank_eq_one", "start": [1333, 1], "end": [1340, 80], "traced_tactics": [{"tactic": "haveI := nontrivial_of_finrank_eq_succ h", "state_before": "K : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nA : Type u_1\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Module A V\ninst\u271d\u00b9 : SMul K A\ninst\u271d : IsScalarTower K A V\nh : finrank K V = 1\n\u22a2 IsSimpleOrder (Submodule A V)", "state_after": "K : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nA : Type u_1\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Module A V\ninst\u271d\u00b9 : SMul K A\ninst\u271d : IsScalarTower K A V\nh : finrank K V = 1\nthis : Nontrivial V\n\u22a2 IsSimpleOrder (Submodule A V)"}, {"tactic": "refine' \u27e8fun S => or_iff_not_imp_left.2 fun hn => _\u27e9", "state_before": "K : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nA : Type u_1\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Module A V\ninst\u271d\u00b9 : SMul K A\ninst\u271d : IsScalarTower K A V\nh : finrank K V = 1\nthis : Nontrivial V\n\u22a2 IsSimpleOrder (Submodule A V)", "state_after": "K : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nA : Type u_1\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Module A V\ninst\u271d\u00b9 : SMul K A\ninst\u271d : IsScalarTower K A V\nh : finrank K V = 1\nthis : Nontrivial V\nS : Submodule A V\nhn : \u00acS = \u22a5\n\u22a2 S = \u22a4"}, {"tactic": "rw [\u2190 restrictScalars_inj K] at hn\u22a2", "state_before": "K : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nA : Type u_1\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Module A V\ninst\u271d\u00b9 : SMul K A\ninst\u271d : IsScalarTower K A V\nh : finrank K V = 1\nthis : Nontrivial V\nS : Submodule A V\nhn : \u00acS = \u22a5\n\u22a2 S = \u22a4", "state_after": "K : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nA : Type u_1\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Module A V\ninst\u271d\u00b9 : SMul K A\ninst\u271d : IsScalarTower K A V\nh : finrank K V = 1\nthis : Nontrivial V\nS : Submodule A V\nhn : \u00acrestrictScalars K S = restrictScalars K \u22a5\n\u22a2 restrictScalars K S = restrictScalars K \u22a4"}, {"tactic": "haveI : FiniteDimensional _ _ := finiteDimensional_of_finrank_eq_succ h", "state_before": "K : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nA : Type u_1\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Module A V\ninst\u271d\u00b9 : SMul K A\ninst\u271d : IsScalarTower K A V\nh : finrank K V = 1\nthis : Nontrivial V\nS : Submodule A V\nhn : \u00acrestrictScalars K S = restrictScalars K \u22a5\n\u22a2 restrictScalars K S = restrictScalars K \u22a4", "state_after": "K : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nA : Type u_1\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Module A V\ninst\u271d\u00b9 : SMul K A\ninst\u271d : IsScalarTower K A V\nh : finrank K V = 1\nthis\u271d : Nontrivial V\nS : Submodule A V\nhn : \u00acrestrictScalars K S = restrictScalars K \u22a5\nthis : FiniteDimensional K V\n\u22a2 restrictScalars K S = restrictScalars K \u22a4"}, {"tactic": "refine' Submodule.eq_top_of_finrank_eq ((Submodule.finrank_le _).antisymm _)", "state_before": "K : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nA : Type u_1\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Module A V\ninst\u271d\u00b9 : SMul K A\ninst\u271d : IsScalarTower K A V\nh : finrank K V = 1\nthis\u271d : Nontrivial V\nS : Submodule A V\nhn : \u00acrestrictScalars K S = restrictScalars K \u22a5\nthis : FiniteDimensional K V\n\u22a2 restrictScalars K S = restrictScalars K \u22a4", "state_after": "K : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nA : Type u_1\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Module A V\ninst\u271d\u00b9 : SMul K A\ninst\u271d : IsScalarTower K A V\nh : finrank K V = 1\nthis\u271d : Nontrivial V\nS : Submodule A V\nhn : \u00acrestrictScalars K S = restrictScalars K \u22a5\nthis : FiniteDimensional K V\n\u22a2 finrank K V \u2264 finrank K { x // x \u2208 restrictScalars K S }"}, {"tactic": "simpa only [h, finrank_bot] using Submodule.finrank_strictMono (Ne.bot_lt hn)", "state_before": "K : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nA : Type u_1\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Module A V\ninst\u271d\u00b9 : SMul K A\ninst\u271d : IsScalarTower K A V\nh : finrank K V = 1\nthis\u271d : Nontrivial V\nS : Submodule A V\nhn : \u00acrestrictScalars K S = restrictScalars K \u22a5\nthis : FiniteDimensional K V\n\u22a2 finrank K V \u2264 finrank K { x // x \u2208 restrictScalars K S }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Hom.lean", "full_name": "AlgHom.coe_id", "start": [312, 1], "end": [313, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.inter_iInter", "start": [547, 1], "end": [548, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/List/Lemmas.lean", "full_name": "List.mem_or_eq_of_mem_set", "start": [848, 1], "end": [851, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/PiSystem.lean", "full_name": "MeasurableSpace.DynkinSystem.has_iUnion", "start": [575, 1], "end": [581, 51], "traced_tactics": [{"tactic": "cases nonempty_encodable \u03b2", "state_before": "\u03b1 : Type u_2\nd : DynkinSystem \u03b1\n\u03b2 : Type u_1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhd : Pairwise (Disjoint on f)\nh : \u2200 (i : \u03b2), Has d (f i)\n\u22a2 Has d (\u22c3 (i : \u03b2), f i)", "state_after": "case intro\n\u03b1 : Type u_2\nd : DynkinSystem \u03b1\n\u03b2 : Type u_1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhd : Pairwise (Disjoint on f)\nh : \u2200 (i : \u03b2), Has d (f i)\nval\u271d : Encodable \u03b2\n\u22a2 Has d (\u22c3 (i : \u03b2), f i)"}, {"tactic": "rw [\u2190 Encodable.iUnion_decode\u2082]", "state_before": "case intro\n\u03b1 : Type u_2\nd : DynkinSystem \u03b1\n\u03b2 : Type u_1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhd : Pairwise (Disjoint on f)\nh : \u2200 (i : \u03b2), Has d (f i)\nval\u271d : Encodable \u03b2\n\u22a2 Has d (\u22c3 (i : \u03b2), f i)", "state_after": "case intro\n\u03b1 : Type u_2\nd : DynkinSystem \u03b1\n\u03b2 : Type u_1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhd : Pairwise (Disjoint on f)\nh : \u2200 (i : \u03b2), Has d (f i)\nval\u271d : Encodable \u03b2\n\u22a2 Has d (\u22c3 (i : \u2115) (b : \u03b2) (_ : b \u2208 Encodable.decode\u2082 \u03b2 i), f b)"}, {"tactic": "exact\n  d.has_iUnion_nat (Encodable.iUnion_decode\u2082_disjoint_on hd) fun n =>\n    Encodable.iUnion_decode\u2082_cases d.has_empty h", "state_before": "case intro\n\u03b1 : Type u_2\nd : DynkinSystem \u03b1\n\u03b2 : Type u_1\ninst\u271d : Countable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhd : Pairwise (Disjoint on f)\nh : \u2200 (i : \u03b2), Has d (f i)\nval\u271d : Encodable \u03b2\n\u22a2 Has d (\u22c3 (i : \u2115) (b : \u03b2) (_ : b \u2208 Encodable.decode\u2082 \u03b2 i), f b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/ZeroObjects.lean", "full_name": "CategoryTheory.Functor.isZero", "start": [151, 1], "end": [168, 31], "traced_tactics": [{"tactic": "constructor <;> intro G <;> refine' \u27e8\u27e8\u27e8_\u27e9, _\u27e9\u27e9", "state_before": "C : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\n\u22a2 IsZero F", "state_after": "case unique_to.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\n\u22a2 F \u27f6 G\n\ncase unique_to.refine'_2\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\n\u22a2 \u2200 (a : F \u27f6 G), a = default\n\ncase unique_from.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\n\u22a2 G \u27f6 F\n\ncase unique_from.refine'_2\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\n\u22a2 \u2200 (a : G \u27f6 F), a = default"}, {"tactic": "refine'\n  { app := fun X => (hF _).to_ _\n    naturality := _ }", "state_before": "case unique_to.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\n\u22a2 F \u27f6 G", "state_after": "case unique_to.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\n\u22a2 \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y),\n    F.map f \u226b (fun X => IsZero.to_ (_ : IsZero (F.obj X)) (G.obj X)) Y =\n      (fun X => IsZero.to_ (_ : IsZero (F.obj X)) (G.obj X)) X \u226b G.map f"}, {"tactic": "intros", "state_before": "case unique_to.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\n\u22a2 \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y),\n    F.map f \u226b (fun X => IsZero.to_ (_ : IsZero (F.obj X)) (G.obj X)) Y =\n      (fun X => IsZero.to_ (_ : IsZero (F.obj X)) (G.obj X)) X \u226b G.map f", "state_after": "case unique_to.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 F.map f\u271d \u226b (fun X => IsZero.to_ (_ : IsZero (F.obj X)) (G.obj X)) Y\u271d =\n    (fun X => IsZero.to_ (_ : IsZero (F.obj X)) (G.obj X)) X\u271d \u226b G.map f\u271d"}, {"tactic": "exact (hF _).eq_of_src _ _", "state_before": "case unique_to.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 F.map f\u271d \u226b (fun X => IsZero.to_ (_ : IsZero (F.obj X)) (G.obj X)) Y\u271d =\n    (fun X => IsZero.to_ (_ : IsZero (F.obj X)) (G.obj X)) X\u271d \u226b G.map f\u271d", "state_after": "no goals"}, {"tactic": "intro f", "state_before": "case unique_to.refine'_2\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\n\u22a2 \u2200 (a : F \u27f6 G), a = default", "state_after": "case unique_to.refine'_2\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\nf : F \u27f6 G\n\u22a2 f = default"}, {"tactic": "ext", "state_before": "case unique_to.refine'_2\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\nf : F \u27f6 G\n\u22a2 f = default", "state_after": "case unique_to.refine'_2.w.h\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\nf : F \u27f6 G\nx\u271d : C\n\u22a2 f.app x\u271d = default.app x\u271d"}, {"tactic": "apply (hF _).eq_of_src _ _", "state_before": "case unique_to.refine'_2.w.h\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\nf : F \u27f6 G\nx\u271d : C\n\u22a2 f.app x\u271d = default.app x\u271d", "state_after": "no goals"}, {"tactic": "refine'\n  { app := fun X => (hF _).from_ _\n    naturality := _ }", "state_before": "case unique_from.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\n\u22a2 G \u27f6 F", "state_after": "case unique_from.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\n\u22a2 \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y),\n    G.map f \u226b (fun X => IsZero.from_ (_ : IsZero (F.obj X)) (G.obj X)) Y =\n      (fun X => IsZero.from_ (_ : IsZero (F.obj X)) (G.obj X)) X \u226b F.map f"}, {"tactic": "intros", "state_before": "case unique_from.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\n\u22a2 \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y),\n    G.map f \u226b (fun X => IsZero.from_ (_ : IsZero (F.obj X)) (G.obj X)) Y =\n      (fun X => IsZero.from_ (_ : IsZero (F.obj X)) (G.obj X)) X \u226b F.map f", "state_after": "case unique_from.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 G.map f\u271d \u226b (fun X => IsZero.from_ (_ : IsZero (F.obj X)) (G.obj X)) Y\u271d =\n    (fun X => IsZero.from_ (_ : IsZero (F.obj X)) (G.obj X)) X\u271d \u226b F.map f\u271d"}, {"tactic": "exact (hF _).eq_of_tgt _ _", "state_before": "case unique_from.refine'_1\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\nX\u271d Y\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 G.map f\u271d \u226b (fun X => IsZero.from_ (_ : IsZero (F.obj X)) (G.obj X)) Y\u271d =\n    (fun X => IsZero.from_ (_ : IsZero (F.obj X)) (G.obj X)) X\u271d \u226b F.map f\u271d", "state_after": "no goals"}, {"tactic": "intro f", "state_before": "case unique_from.refine'_2\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\n\u22a2 \u2200 (a : G \u27f6 F), a = default", "state_after": "case unique_from.refine'_2\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\nf : G \u27f6 F\n\u22a2 f = default"}, {"tactic": "ext", "state_before": "case unique_from.refine'_2\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\nf : G \u27f6 F\n\u22a2 f = default", "state_after": "case unique_from.refine'_2.w.h\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\nf : G \u27f6 F\nx\u271d : C\n\u22a2 f.app x\u271d = default.app x\u271d"}, {"tactic": "apply (hF _).eq_of_tgt _ _", "state_before": "case unique_from.refine'_2.w.h\nC : Type u\ninst\u271d\u00b9 : Category C\nD : Type u'\ninst\u271d : Category D\nF : C \u2964 D\nhF : \u2200 (X : C), IsZero (F.obj X)\nG : C \u2964 D\nf : G \u27f6 F\nx\u271d : C\n\u22a2 f.app x\u271d = default.app x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.iSup_approx_apply", "start": [859, 1], "end": [872, 19], "traced_tactics": [{"tactic": "refine' le_antisymm (iSup_le fun n => _) (iSup_le fun k => iSup_le fun hk => _)", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.829816\n\u03b4 : Type ?u.829819\ninst\u271d\u2076 : MeasurableSpace \u03b1\nK : Type ?u.829825\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : OrderClosedTopology \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ni : \u2115 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : Measurable f\nh_zero : 0 = \u22a5\n\u22a2 (\u2a06 (n : \u2115), \u2191(approx i f n) a) = \u2a06 (k : \u2115) (_ : i k \u2264 f a), i k", "state_after": "case refine'_1\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.829816\n\u03b4 : Type ?u.829819\ninst\u271d\u2076 : MeasurableSpace \u03b1\nK : Type ?u.829825\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : OrderClosedTopology \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ni : \u2115 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : Measurable f\nh_zero : 0 = \u22a5\nn : \u2115\n\u22a2 \u2191(approx i f n) a \u2264 \u2a06 (k : \u2115) (_ : i k \u2264 f a), i k\n\ncase refine'_2\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.829816\n\u03b4 : Type ?u.829819\ninst\u271d\u2076 : MeasurableSpace \u03b1\nK : Type ?u.829825\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : OrderClosedTopology \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ni : \u2115 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : Measurable f\nh_zero : 0 = \u22a5\nk : \u2115\nhk : i k \u2264 f a\n\u22a2 i k \u2264 \u2a06 (n : \u2115), \u2191(approx i f n) a"}, {"tactic": "rw [approx_apply a hf, h_zero]", "state_before": "case refine'_1\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.829816\n\u03b4 : Type ?u.829819\ninst\u271d\u2076 : MeasurableSpace \u03b1\nK : Type ?u.829825\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : OrderClosedTopology \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ni : \u2115 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : Measurable f\nh_zero : 0 = \u22a5\nn : \u2115\n\u22a2 \u2191(approx i f n) a \u2264 \u2a06 (k : \u2115) (_ : i k \u2264 f a), i k", "state_after": "case refine'_1\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.829816\n\u03b4 : Type ?u.829819\ninst\u271d\u2076 : MeasurableSpace \u03b1\nK : Type ?u.829825\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : OrderClosedTopology \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ni : \u2115 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : Measurable f\nh_zero : 0 = \u22a5\nn : \u2115\n\u22a2 (Finset.sup (Finset.range n) fun k => if i k \u2264 f a then i k else \u22a5) \u2264 \u2a06 (k : \u2115) (_ : i k \u2264 f a), i k"}, {"tactic": "refine' Finset.sup_le fun k _ => _", "state_before": "case refine'_1\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.829816\n\u03b4 : Type ?u.829819\ninst\u271d\u2076 : MeasurableSpace \u03b1\nK : Type ?u.829825\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : OrderClosedTopology \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ni : \u2115 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : Measurable f\nh_zero : 0 = \u22a5\nn : \u2115\n\u22a2 (Finset.sup (Finset.range n) fun k => if i k \u2264 f a then i k else \u22a5) \u2264 \u2a06 (k : \u2115) (_ : i k \u2264 f a), i k", "state_after": "case refine'_1\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.829816\n\u03b4 : Type ?u.829819\ninst\u271d\u2076 : MeasurableSpace \u03b1\nK : Type ?u.829825\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : OrderClosedTopology \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ni : \u2115 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : Measurable f\nh_zero : 0 = \u22a5\nn k : \u2115\nx\u271d : k \u2208 Finset.range n\n\u22a2 (if i k \u2264 f a then i k else \u22a5) \u2264 \u2a06 (k : \u2115) (_ : i k \u2264 f a), i k"}, {"tactic": "split_ifs with h", "state_before": "case refine'_1\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.829816\n\u03b4 : Type ?u.829819\ninst\u271d\u2076 : MeasurableSpace \u03b1\nK : Type ?u.829825\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : OrderClosedTopology \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ni : \u2115 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : Measurable f\nh_zero : 0 = \u22a5\nn k : \u2115\nx\u271d : k \u2208 Finset.range n\n\u22a2 (if i k \u2264 f a then i k else \u22a5) \u2264 \u2a06 (k : \u2115) (_ : i k \u2264 f a), i k", "state_after": "case refine'_1.inl\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.829816\n\u03b4 : Type ?u.829819\ninst\u271d\u2076 : MeasurableSpace \u03b1\nK : Type ?u.829825\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : OrderClosedTopology \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ni : \u2115 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : Measurable f\nh_zero : 0 = \u22a5\nn k : \u2115\nx\u271d : k \u2208 Finset.range n\nh : i k \u2264 f a\n\u22a2 i k \u2264 \u2a06 (k : \u2115) (_ : i k \u2264 f a), i k\n\ncase refine'_1.inr\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.829816\n\u03b4 : Type ?u.829819\ninst\u271d\u2076 : MeasurableSpace \u03b1\nK : Type ?u.829825\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : OrderClosedTopology \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ni : \u2115 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : Measurable f\nh_zero : 0 = \u22a5\nn k : \u2115\nx\u271d : k \u2208 Finset.range n\nh : \u00aci k \u2264 f a\n\u22a2 \u22a5 \u2264 \u2a06 (k : \u2115) (_ : i k \u2264 f a), i k"}, {"tactic": "exact le_iSup_of_le k (le_iSup (fun _ : i k \u2264 f a => i k) h)", "state_before": "case refine'_1.inl\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.829816\n\u03b4 : Type ?u.829819\ninst\u271d\u2076 : MeasurableSpace \u03b1\nK : Type ?u.829825\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : OrderClosedTopology \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ni : \u2115 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : Measurable f\nh_zero : 0 = \u22a5\nn k : \u2115\nx\u271d : k \u2208 Finset.range n\nh : i k \u2264 f a\n\u22a2 i k \u2264 \u2a06 (k : \u2115) (_ : i k \u2264 f a), i k\n\ncase refine'_1.inr\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.829816\n\u03b4 : Type ?u.829819\ninst\u271d\u2076 : MeasurableSpace \u03b1\nK : Type ?u.829825\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : OrderClosedTopology \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ni : \u2115 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : Measurable f\nh_zero : 0 = \u22a5\nn k : \u2115\nx\u271d : k \u2208 Finset.range n\nh : \u00aci k \u2264 f a\n\u22a2 \u22a5 \u2264 \u2a06 (k : \u2115) (_ : i k \u2264 f a), i k", "state_after": "case refine'_1.inr\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.829816\n\u03b4 : Type ?u.829819\ninst\u271d\u2076 : MeasurableSpace \u03b1\nK : Type ?u.829825\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : OrderClosedTopology \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ni : \u2115 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : Measurable f\nh_zero : 0 = \u22a5\nn k : \u2115\nx\u271d : k \u2208 Finset.range n\nh : \u00aci k \u2264 f a\n\u22a2 \u22a5 \u2264 \u2a06 (k : \u2115) (_ : i k \u2264 f a), i k"}, {"tactic": "exact bot_le", "state_before": "case refine'_1.inr\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.829816\n\u03b4 : Type ?u.829819\ninst\u271d\u2076 : MeasurableSpace \u03b1\nK : Type ?u.829825\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : OrderClosedTopology \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ni : \u2115 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : Measurable f\nh_zero : 0 = \u22a5\nn k : \u2115\nx\u271d : k \u2208 Finset.range n\nh : \u00aci k \u2264 f a\n\u22a2 \u22a5 \u2264 \u2a06 (k : \u2115) (_ : i k \u2264 f a), i k", "state_after": "no goals"}, {"tactic": "refine' le_iSup_of_le (k + 1) _", "state_before": "case refine'_2\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.829816\n\u03b4 : Type ?u.829819\ninst\u271d\u2076 : MeasurableSpace \u03b1\nK : Type ?u.829825\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : OrderClosedTopology \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ni : \u2115 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : Measurable f\nh_zero : 0 = \u22a5\nk : \u2115\nhk : i k \u2264 f a\n\u22a2 i k \u2264 \u2a06 (n : \u2115), \u2191(approx i f n) a", "state_after": "case refine'_2\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.829816\n\u03b4 : Type ?u.829819\ninst\u271d\u2076 : MeasurableSpace \u03b1\nK : Type ?u.829825\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : OrderClosedTopology \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ni : \u2115 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : Measurable f\nh_zero : 0 = \u22a5\nk : \u2115\nhk : i k \u2264 f a\n\u22a2 i k \u2264 \u2191(approx i f (k + 1)) a"}, {"tactic": "rw [approx_apply a hf]", "state_before": "case refine'_2\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.829816\n\u03b4 : Type ?u.829819\ninst\u271d\u2076 : MeasurableSpace \u03b1\nK : Type ?u.829825\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : OrderClosedTopology \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ni : \u2115 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : Measurable f\nh_zero : 0 = \u22a5\nk : \u2115\nhk : i k \u2264 f a\n\u22a2 i k \u2264 \u2191(approx i f (k + 1)) a", "state_after": "case refine'_2\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.829816\n\u03b4 : Type ?u.829819\ninst\u271d\u2076 : MeasurableSpace \u03b1\nK : Type ?u.829825\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : OrderClosedTopology \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ni : \u2115 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : Measurable f\nh_zero : 0 = \u22a5\nk : \u2115\nhk : i k \u2264 f a\n\u22a2 i k \u2264 Finset.sup (Finset.range (k + 1)) fun k => if i k \u2264 f a then i k else 0"}, {"tactic": "have : k \u2208 Finset.range (k + 1) := Finset.mem_range.2 (Nat.lt_succ_self _)", "state_before": "case refine'_2\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.829816\n\u03b4 : Type ?u.829819\ninst\u271d\u2076 : MeasurableSpace \u03b1\nK : Type ?u.829825\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : OrderClosedTopology \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ni : \u2115 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : Measurable f\nh_zero : 0 = \u22a5\nk : \u2115\nhk : i k \u2264 f a\n\u22a2 i k \u2264 Finset.sup (Finset.range (k + 1)) fun k => if i k \u2264 f a then i k else 0", "state_after": "case refine'_2\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.829816\n\u03b4 : Type ?u.829819\ninst\u271d\u2076 : MeasurableSpace \u03b1\nK : Type ?u.829825\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : OrderClosedTopology \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ni : \u2115 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : Measurable f\nh_zero : 0 = \u22a5\nk : \u2115\nhk : i k \u2264 f a\nthis : k \u2208 Finset.range (k + 1)\n\u22a2 i k \u2264 Finset.sup (Finset.range (k + 1)) fun k => if i k \u2264 f a then i k else 0"}, {"tactic": "refine' le_trans (le_of_eq _) (Finset.le_sup this)", "state_before": "case refine'_2\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.829816\n\u03b4 : Type ?u.829819\ninst\u271d\u2076 : MeasurableSpace \u03b1\nK : Type ?u.829825\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : CompleteLattice \u03b2\ninst\u271d\u00b3 : OrderClosedTopology \u03b2\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : OpensMeasurableSpace \u03b2\ni : \u2115 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nhf : Measurable f\nh_zero : 0 = 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"https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicGeometry/SheafedSpace.lean", "full_name": "AlgebraicGeometry.SheafedSpace.\u0393_def", "start": [198, 1], "end": [199, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Pointwise.lean", "full_name": "Filter.NeBot.of_mul_left", "start": [341, 1], "end": [342, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "full_name": "ball_eq'", "start": [606, 1], "end": [607, 46], "traced_tactics": [{"tactic": "simp [dist_eq_norm_div]", "state_before": "\ud835\udcd5 : Type ?u.89542\n\ud835\udd5c : Type ?u.89545\n\u03b1 : Type ?u.89548\n\u03b9 : Type ?u.89551\n\u03ba : Type ?u.89554\nE : Type u_1\nF : Type ?u.89560\nG : Type ?u.89563\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\ny : E\n\u03b5 : \u211d\na : E\n\u22a2 a \u2208 ball y \u03b5 \u2194 a \u2208 {x | \u2016x / y\u2016 < \u03b5}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Derivation/Basic.lean", "full_name": "Derivation.zero_apply", "start": [196, 1], "end": [197, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "full_name": "CategoryTheory.Limits.HasLimit.ofConesIso", "start": [337, 1], "end": [339, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", 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S\nthis : IsLocalRingHom f \u2194 Ideal.comap f (maximalIdeal S) = maximalIdeal R\n\u22a2 IsLocalRingHom f \u2194 \u2191(PrimeSpectrum.comap f) (closedPoint S) = closedPoint R", "state_after": "R : Type u\nS\u271d : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : LocalRing R\nS : Type v\ninst\u271d\u00b9 : CommRing S\ninst\u271d : LocalRing S\nf : R \u2192+* S\nthis : IsLocalRingHom f \u2194 Ideal.comap f (maximalIdeal S) = maximalIdeal R\n\u22a2 Ideal.comap f (maximalIdeal S) = maximalIdeal R \u2194\n    (\u2191(PrimeSpectrum.comap f) (closedPoint S)).asIdeal = (closedPoint R).asIdeal"}, {"tactic": "rfl", "state_before": "R : Type u\nS\u271d : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\u271d\ninst\u271d\u00b2 : LocalRing R\nS : Type v\ninst\u271d\u00b9 : CommRing S\ninst\u271d : LocalRing S\nf : R \u2192+* S\nthis : IsLocalRingHom f \u2194 Ideal.comap f (maximalIdeal S) = maximalIdeal R\n\u22a2 Ideal.comap f (maximalIdeal S) = maximalIdeal R \u2194\n    (\u2191(PrimeSpectrum.comap f) (closedPoint S)).asIdeal = (closedPoint R).asIdeal", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Rat/Defs.lean", "full_name": "Rat.inv_def'", "start": [193, 1], "end": [193, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Sheaves/Presheaf.lean", "full_name": "TopCat.Presheaf.pushforwardToOfIso_app", "start": [444, 1], "end": [449, 75], "traced_tactics": [{"tactic": "simp [Opens.map, Set.preimage_preimage]", "state_before": "C : Type u\ninst\u271d : Category C\nX Y : TopCat\nH\u2081 : X \u2245 Y\n\u2131 : Presheaf C Y\n\ud835\udca2 : Presheaf C X\nH\u2082 : \u2131 \u27f6 H\u2081.hom _* \ud835\udca2\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 (Functor.op (Opens.map H\u2081.hom)).obj ((Opens.map H\u2081.inv).obj U.unop).op = U", "state_after": "no goals"}, {"tactic": "simp [pushforwardToOfIso, Equivalence.toAdjunction, CategoryStruct.comp]", "state_before": "C : Type u\ninst\u271d : Category C\nX Y : TopCat\nH\u2081 : X \u2245 Y\n\u2131 : Presheaf C Y\n\ud835\udca2 : Presheaf C X\nH\u2082 : \u2131 \u27f6 H\u2081.hom _* \ud835\udca2\nU : (Opens \u2191X)\u1d52\u1d56\n\u22a2 (pushforwardToOfIso H\u2081 H\u2082).app U =\n    H\u2082.app ((Opens.map H\u2081.inv).obj U.unop).op \u226b\n      \ud835\udca2.map\n        (eqToHom\n          (_ :\n            {\n                  carrier :=\n                    (forget TopCat).map H\u2081.hom \u207b\u00b9'\n                      \u2191{ carrier := (forget TopCat).map H\u2081.inv \u207b\u00b9' \u2191U.unop,\n                              is_open' := (_ : IsOpen (\u2191H\u2081.inv \u207b\u00b9' \u2191U.unop)) }.op.unop,\n                  is_open' :=\n                    (_ :\n                      IsOpen\n                        (\u2191H\u2081.hom \u207b\u00b9'\n                          \u2191{ carrier := (forget TopCat).map H\u2081.inv \u207b\u00b9' \u2191U.unop,\n                                  is_open' := (_ : IsOpen (\u2191H\u2081.inv \u207b\u00b9' \u2191U.unop)) }.op.unop)) }.op =\n              U))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Localization/AtPrime.lean", "full_name": "Localization.le_comap_primeCompl_iff", "start": [207, 1], "end": [211, 48], "traced_tactics": [{"tactic": "contrapose! hx", "state_before": "R : Type u_2\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type ?u.85926\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_1\ninst\u271d : CommSemiring P\nI : Ideal R\nhI : Ideal.IsPrime I\nJ : Ideal P\nhJ : Ideal.IsPrime J\nf : R \u2192+* P\nh : Ideal.primeCompl I \u2264 Submonoid.comap f (Ideal.primeCompl J)\nx : R\nhx : x \u2208 Ideal.comap f J\n\u22a2 x \u2208 I", "state_after": "R : Type u_2\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type ?u.85926\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_1\ninst\u271d : CommSemiring P\nI : Ideal R\nhI : Ideal.IsPrime I\nJ : Ideal P\nhJ : Ideal.IsPrime J\nf : R \u2192+* P\nh : Ideal.primeCompl I \u2264 Submonoid.comap f (Ideal.primeCompl J)\nx : R\nhx : \u00acx \u2208 I\n\u22a2 \u00acx \u2208 Ideal.comap f J"}, {"tactic": "exact h hx", "state_before": "R : Type u_2\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type ?u.85926\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_1\ninst\u271d : CommSemiring P\nI : Ideal R\nhI : Ideal.IsPrime I\nJ : Ideal P\nhJ : Ideal.IsPrime J\nf : R \u2192+* P\nh : Ideal.primeCompl I \u2264 Submonoid.comap f (Ideal.primeCompl J)\nx : R\nhx : \u00acx \u2208 I\n\u22a2 \u00acx \u2208 Ideal.comap f J", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.disjoint_atTop_principal_Iic", "start": [88, 1], "end": [91, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Basic.lean", "full_name": "Set.Iio_union_Icc_eq_Iic", "start": [1368, 1], "end": [1370, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/ToIntervalMod.lean", "full_name": "toIocMod_add_zsmul'", "start": [423, 1], "end": [425, 63], "traced_tactics": [{"tactic": "simp only [toIocMod, toIocDiv_add_zsmul', sub_smul, sub_add]", "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\nm : \u2124\n\u22a2 toIocMod hp (a + m \u2022 p) b = toIocMod hp a b + m \u2022 p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Torsion.lean", "full_name": "IsTorsion.quotient_iff", "start": [126, 1], "end": [128, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Order/IntermediateValue.lean", "full_name": "isConnected_Ioi", "start": [473, 1], "end": [474, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Generator.lean", "full_name": "CategoryTheory.IsDetecting.isSeparating", "start": [153, 1], "end": [156, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "full_name": "Cardinal.add_mk_eq_max", "start": [740, 1], "end": [741, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Subsemiring/Basic.lean", "full_name": "RingHom.eqOn_sclosure", "start": [1202, 1], "end": [1203, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean", "full_name": "EuclideanGeometry.dist_eq_add_dist_iff_angle_eq_pi", "start": [228, 1], "end": [233, 39], "traced_tactics": [{"tactic": "rw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, dist_eq_norm_vsub V, \u2190 vsub_sub_vsub_cancel_right]", "state_before": "V : Type u_2\nP : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\nhp1p2 : p1 \u2260 p2\nhp3p2 : p3 \u2260 p2\n\u22a2 dist p1 p3 = dist p1 p2 + dist p3 p2 \u2194 \u2220 p1 p2 p3 = \u03c0", "state_after": "V : Type u_2\nP : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\nhp1p2 : p1 \u2260 p2\nhp3p2 : p3 \u2260 p2\n\u22a2 \u2016p1 -\u1d65 ?p3 - (p3 -\u1d65 ?p3)\u2016 = \u2016p1 -\u1d65 p2\u2016 + \u2016p3 -\u1d65 p2\u2016 \u2194 \u2220 p1 p2 p3 = \u03c0\n\ncase p3\nV : Type u_2\nP : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\nhp1p2 : p1 \u2260 p2\nhp3p2 : p3 \u2260 p2\n\u22a2 P"}, {"tactic": "exact\n  norm_sub_eq_add_norm_iff_angle_eq_pi (fun he => hp1p2 (vsub_eq_zero_iff_eq.1 he)) fun he =>\n    hp3p2 (vsub_eq_zero_iff_eq.1 he)", "state_before": "V : Type u_2\nP : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\nhp1p2 : p1 \u2260 p2\nhp3p2 : p3 \u2260 p2\n\u22a2 \u2016p1 -\u1d65 ?p3 - (p3 -\u1d65 ?p3)\u2016 = \u2016p1 -\u1d65 p2\u2016 + \u2016p3 -\u1d65 p2\u2016 \u2194 \u2220 p1 p2 p3 = \u03c0\n\ncase p3\nV : Type u_2\nP : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\nhp1p2 : p1 \u2260 p2\nhp3p2 : p3 \u2260 p2\n\u22a2 P", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean", "full_name": "CircleDeg1Lift.semiconj_of_group_action_of_forall_translationNumber_eq", "start": [969, 1], "end": [1000, 17], "traced_tactics": [{"tactic": "have : \u2200 x, BddAbove (range fun g => f\u2082 g\u207b\u00b9 (f\u2081 g x)) := by\n  refine' fun x => \u27e8x + 2, _\u27e9\n  rintro _ \u27e8g, rfl\u27e9\n  have : \u03c4 (f\u2082 g\u207b\u00b9) = -\u03c4 (f\u2082 g) := by\n    rw [\u2190 MonoidHom.coe_toHomUnits, MonoidHom.map_inv, translationNumber_units_inv,\n      MonoidHom.coe_toHomUnits]\n  calc\n    f\u2082 g\u207b\u00b9 (f\u2081 g x) \u2264 f\u2082 g\u207b\u00b9 (x + \u03c4 (f\u2081 g) + 1) :=\n      mono _ (map_lt_add_translationNumber_add_one _ _).le\n    _ = f\u2082 g\u207b\u00b9 (x + \u03c4 (f\u2082 g)) + 1 := by rw [h, map_add_one]\n    _ \u2264 x + \u03c4 (f\u2082 g) + \u03c4 (f\u2082 g\u207b\u00b9) + 1 + 1 :=\n      add_le_add_right (map_lt_add_translationNumber_add_one _ _).le _\n    _ = x + 2 := by simp [this, add_assoc, one_add_one_eq_two]", "state_before": "f g : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\n\u22a2 \u2203 F, \u2200 (g : G), Semiconj \u2191F \u2191(\u2191f\u2081 g) \u2191(\u2191f\u2082 g)", "state_after": "f g : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nthis : \u2200 (x : \u211d), BddAbove (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))\n\u22a2 \u2203 F, \u2200 (g : G), Semiconj \u2191F \u2191(\u2191f\u2081 g) \u2191(\u2191f\u2082 g)"}, {"tactic": "set F\u2081 := toOrderIso.comp f\u2081.toHomUnits", "state_before": "f g : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nthis : \u2200 (x : \u211d), BddAbove (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))\n\u22a2 \u2203 F, \u2200 (g : G), Semiconj \u2191F \u2191(\u2191f\u2081 g) \u2191(\u2191f\u2082 g)", "state_after": "f g : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nthis : \u2200 (x : \u211d), BddAbove (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))\nF\u2081 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2081)\n\u22a2 \u2203 F, \u2200 (g : G), Semiconj \u2191F \u2191(\u2191f\u2081 g) \u2191(\u2191f\u2082 g)"}, {"tactic": "set F\u2082 := toOrderIso.comp f\u2082.toHomUnits", "state_before": "f g : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nthis : \u2200 (x : \u211d), BddAbove (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))\nF\u2081 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2081)\n\u22a2 \u2203 F, \u2200 (g : G), Semiconj \u2191F \u2191(\u2191f\u2081 g) \u2191(\u2191f\u2082 g)", "state_after": "f g : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nthis : \u2200 (x : \u211d), BddAbove (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))\nF\u2081 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2081)\nF\u2082 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2082)\n\u22a2 \u2203 F, \u2200 (g : G), Semiconj \u2191F \u2191(\u2191f\u2081 g) \u2191(\u2191f\u2082 g)"}, {"tactic": "have hF\u2081 : \u2200 g, \u21d1(F\u2081 g) = f\u2081 g := fun _ => rfl", "state_before": "f g : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nthis : \u2200 (x : \u211d), BddAbove (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))\nF\u2081 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2081)\nF\u2082 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2082)\n\u22a2 \u2203 F, \u2200 (g : G), Semiconj \u2191F \u2191(\u2191f\u2081 g) \u2191(\u2191f\u2082 g)", "state_after": "f g : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nthis : \u2200 (x : \u211d), BddAbove (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))\nF\u2081 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2081)\nF\u2082 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2082)\nhF\u2081 : \u2200 (g : G), \u2191(\u2191F\u2081 g) = \u2191(\u2191f\u2081 g)\n\u22a2 \u2203 F, \u2200 (g : G), Semiconj \u2191F \u2191(\u2191f\u2081 g) \u2191(\u2191f\u2082 g)"}, {"tactic": "have hF\u2082 : \u2200 g, \u21d1(F\u2082 g) = f\u2082 g := fun _ => rfl", "state_before": "f g : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nthis : \u2200 (x : \u211d), BddAbove (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))\nF\u2081 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2081)\nF\u2082 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2082)\nhF\u2081 : \u2200 (g : G), \u2191(\u2191F\u2081 g) = \u2191(\u2191f\u2081 g)\n\u22a2 \u2203 F, \u2200 (g : G), Semiconj \u2191F \u2191(\u2191f\u2081 g) \u2191(\u2191f\u2082 g)", "state_after": "f g : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nthis : \u2200 (x : \u211d), BddAbove (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))\nF\u2081 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2081)\nF\u2082 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2082)\nhF\u2081 : \u2200 (g : G), \u2191(\u2191F\u2081 g) = \u2191(\u2191f\u2081 g)\nhF\u2082 : \u2200 (g : G), \u2191(\u2191F\u2082 g) = \u2191(\u2191f\u2082 g)\n\u22a2 \u2203 F, \u2200 (g : G), Semiconj \u2191F \u2191(\u2191f\u2081 g) \u2191(\u2191f\u2082 g)"}, {"tactic": "refine' \u27e8\u27e8\u27e8_, fun x y hxy => _\u27e9, fun x => _\u27e9, csSup_div_semiconj F\u2082 F\u2081 fun x => _\u27e9 <;>\n  simp only [hF\u2081, hF\u2082, \u2190 map_inv, coe_mk]", "state_before": "f g : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nthis : \u2200 (x : \u211d), BddAbove (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))\nF\u2081 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2081)\nF\u2082 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2082)\nhF\u2081 : \u2200 (g : G), \u2191(\u2191F\u2081 g) = \u2191(\u2191f\u2081 g)\nhF\u2082 : \u2200 (g : G), \u2191(\u2191F\u2082 g) = \u2191(\u2191f\u2082 g)\n\u22a2 \u2203 F, \u2200 (g : G), Semiconj \u2191F \u2191(\u2191f\u2081 g) \u2191(\u2191f\u2082 g)", "state_after": "case refine'_1\nf g : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nthis : \u2200 (x : \u211d), BddAbove (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))\nF\u2081 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2081)\nF\u2082 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2082)\nhF\u2081 : \u2200 (g : G), \u2191(\u2191F\u2081 g) = \u2191(\u2191f\u2081 g)\nhF\u2082 : \u2200 (g : G), \u2191(\u2191F\u2082 g) = \u2191(\u2191f\u2082 g)\nx y : \u211d\nhxy : x \u2264 y\n\u22a2 sSup (range fun g' => \u2191(\u2191f\u2082 g'\u207b\u00b9) (\u2191(\u2191f\u2081 g') x)) \u2264 sSup (range fun g' => \u2191(\u2191f\u2082 g'\u207b\u00b9) (\u2191(\u2191f\u2081 g') y))\n\ncase refine'_2\nf g : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nthis : \u2200 (x : \u211d), BddAbove (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))\nF\u2081 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2081)\nF\u2082 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2082)\nhF\u2081 : \u2200 (g : G), \u2191(\u2191F\u2081 g) = \u2191(\u2191f\u2081 g)\nhF\u2082 : \u2200 (g : G), \u2191(\u2191F\u2082 g) = \u2191(\u2191f\u2082 g)\nx : \u211d\n\u22a2 sSup (range fun g' => \u2191(\u2191f\u2082 g'\u207b\u00b9) (\u2191(\u2191f\u2081 g') (x + 1))) = sSup (range fun g' => \u2191(\u2191f\u2082 g'\u207b\u00b9) (\u2191(\u2191f\u2081 g') x)) + 1\n\ncase refine'_3\nf g : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nthis : \u2200 (x : \u211d), BddAbove (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))\nF\u2081 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2081)\nF\u2082 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2082)\nhF\u2081 : \u2200 (g : G), \u2191(\u2191F\u2081 g) = \u2191(\u2191f\u2081 g)\nhF\u2082 : \u2200 (g : G), \u2191(\u2191F\u2082 g) = \u2191(\u2191f\u2082 g)\nx : \u211d\n\u22a2 BddAbove (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))"}, {"tactic": "refine' fun x => \u27e8x + 2, _\u27e9", "state_before": "f g : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\n\u22a2 \u2200 (x : \u211d), BddAbove (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))", "state_after": "f g : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nx : \u211d\n\u22a2 x + 2 \u2208 upperBounds (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))"}, {"tactic": "rintro _ \u27e8g, rfl\u27e9", "state_before": "f g : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nx : \u211d\n\u22a2 x + 2 \u2208 upperBounds (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))", "state_after": "case intro\nf g\u271d : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nx : \u211d\ng : G\n\u22a2 (fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x)) g \u2264 x + 2"}, {"tactic": "have : \u03c4 (f\u2082 g\u207b\u00b9) = -\u03c4 (f\u2082 g) := by\n  rw [\u2190 MonoidHom.coe_toHomUnits, MonoidHom.map_inv, translationNumber_units_inv,\n    MonoidHom.coe_toHomUnits]", "state_before": "case intro\nf g\u271d : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nx : \u211d\ng : G\n\u22a2 (fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x)) g \u2264 x + 2", "state_after": "case intro\nf g\u271d : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nx : \u211d\ng : G\nthis : \u03c4 (\u2191f\u2082 g\u207b\u00b9) = -\u03c4 (\u2191f\u2082 g)\n\u22a2 (fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x)) g \u2264 x + 2"}, {"tactic": "calc\n  f\u2082 g\u207b\u00b9 (f\u2081 g x) \u2264 f\u2082 g\u207b\u00b9 (x + \u03c4 (f\u2081 g) + 1) :=\n    mono _ (map_lt_add_translationNumber_add_one _ _).le\n  _ = f\u2082 g\u207b\u00b9 (x + \u03c4 (f\u2082 g)) + 1 := by rw [h, map_add_one]\n  _ \u2264 x + \u03c4 (f\u2082 g) + \u03c4 (f\u2082 g\u207b\u00b9) + 1 + 1 :=\n    add_le_add_right (map_lt_add_translationNumber_add_one _ _).le _\n  _ = x + 2 := by simp [this, add_assoc, one_add_one_eq_two]", "state_before": "case intro\nf g\u271d : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nx : \u211d\ng : G\nthis : \u03c4 (\u2191f\u2082 g\u207b\u00b9) = -\u03c4 (\u2191f\u2082 g)\n\u22a2 (fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x)) g \u2264 x + 2", "state_after": "no goals"}, {"tactic": "rw [\u2190 MonoidHom.coe_toHomUnits, MonoidHom.map_inv, translationNumber_units_inv,\n  MonoidHom.coe_toHomUnits]", "state_before": "f g\u271d : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nx : \u211d\ng : G\n\u22a2 \u03c4 (\u2191f\u2082 g\u207b\u00b9) = -\u03c4 (\u2191f\u2082 g)", "state_after": "no goals"}, {"tactic": "rw [h, map_add_one]", "state_before": "f g\u271d : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nx : \u211d\ng : G\nthis : \u03c4 (\u2191f\u2082 g\u207b\u00b9) = -\u03c4 (\u2191f\u2082 g)\n\u22a2 \u2191(\u2191f\u2082 g\u207b\u00b9) (x + \u03c4 (\u2191f\u2081 g) + 1) = \u2191(\u2191f\u2082 g\u207b\u00b9) (x + \u03c4 (\u2191f\u2082 g)) + 1", "state_after": "no goals"}, {"tactic": "simp [this, add_assoc, one_add_one_eq_two]", "state_before": "f g\u271d : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nx : \u211d\ng : G\nthis : \u03c4 (\u2191f\u2082 g\u207b\u00b9) = -\u03c4 (\u2191f\u2082 g)\n\u22a2 x + \u03c4 (\u2191f\u2082 g) + \u03c4 (\u2191f\u2082 g\u207b\u00b9) + 1 + 1 = x + 2", "state_after": "no goals"}, {"tactic": "exact ciSup_mono (this y) fun g => mono _ (mono _ hxy)", "state_before": "case refine'_1\nf g : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nthis : \u2200 (x : \u211d), BddAbove (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))\nF\u2081 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2081)\nF\u2082 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2082)\nhF\u2081 : \u2200 (g : G), \u2191(\u2191F\u2081 g) = \u2191(\u2191f\u2081 g)\nhF\u2082 : \u2200 (g : G), \u2191(\u2191F\u2082 g) = \u2191(\u2191f\u2082 g)\nx y : \u211d\nhxy : x \u2264 y\n\u22a2 sSup (range fun g' => \u2191(\u2191f\u2082 g'\u207b\u00b9) (\u2191(\u2191f\u2081 g') x)) \u2264 sSup (range fun g' => \u2191(\u2191f\u2082 g'\u207b\u00b9) (\u2191(\u2191f\u2081 g') y))", "state_after": "no goals"}, {"tactic": "simp only [map_add_one]", "state_before": "case refine'_2\nf g : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nthis : \u2200 (x : \u211d), BddAbove (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))\nF\u2081 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2081)\nF\u2082 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2082)\nhF\u2081 : \u2200 (g : G), \u2191(\u2191F\u2081 g) = \u2191(\u2191f\u2081 g)\nhF\u2082 : \u2200 (g : G), \u2191(\u2191F\u2082 g) = \u2191(\u2191f\u2082 g)\nx : \u211d\n\u22a2 sSup (range fun g' => \u2191(\u2191f\u2082 g'\u207b\u00b9) (\u2191(\u2191f\u2081 g') (x + 1))) = sSup (range fun g' => \u2191(\u2191f\u2082 g'\u207b\u00b9) (\u2191(\u2191f\u2081 g') x)) + 1", "state_after": "case refine'_2\nf g : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nthis : \u2200 (x : \u211d), BddAbove (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))\nF\u2081 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2081)\nF\u2082 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2082)\nhF\u2081 : \u2200 (g : G), \u2191(\u2191F\u2081 g) = \u2191(\u2191f\u2081 g)\nhF\u2082 : \u2200 (g : G), \u2191(\u2191F\u2082 g) = \u2191(\u2191f\u2082 g)\nx : \u211d\n\u22a2 sSup (range fun g' => \u2191(\u2191f\u2082 g'\u207b\u00b9) (\u2191(\u2191f\u2081 g') x) + 1) = sSup (range fun g' => \u2191(\u2191f\u2082 g'\u207b\u00b9) (\u2191(\u2191f\u2081 g') x)) + 1"}, {"tactic": "exact (Monotone.map_ciSup_of_continuousAt (continuousAt_id.add continuousAt_const)\n  (monotone_id.add_const (1 : \u211d)) (this x)).symm", "state_before": "case refine'_2\nf g : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nthis : \u2200 (x : \u211d), BddAbove (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))\nF\u2081 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2081)\nF\u2082 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2082)\nhF\u2081 : \u2200 (g : G), \u2191(\u2191F\u2081 g) = \u2191(\u2191f\u2081 g)\nhF\u2082 : \u2200 (g : G), \u2191(\u2191F\u2082 g) = \u2191(\u2191f\u2082 g)\nx : \u211d\n\u22a2 sSup (range fun g' => \u2191(\u2191f\u2082 g'\u207b\u00b9) (\u2191(\u2191f\u2081 g') x) + 1) = sSup (range fun g' => \u2191(\u2191f\u2082 g'\u207b\u00b9) (\u2191(\u2191f\u2081 g') x)) + 1", "state_after": "no goals"}, {"tactic": "exact this x", "state_before": "case refine'_3\nf g : CircleDeg1Lift\nG : Type u_1\ninst\u271d : Group G\nf\u2081 f\u2082 : G \u2192* CircleDeg1Lift\nh : \u2200 (g : G), \u03c4 (\u2191f\u2081 g) = \u03c4 (\u2191f\u2082 g)\nthis : \u2200 (x : \u211d), BddAbove (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))\nF\u2081 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2081)\nF\u2082 : G \u2192* \u211d \u2243o \u211d := MonoidHom.comp toOrderIso (MonoidHom.toHomUnits f\u2082)\nhF\u2081 : \u2200 (g : G), \u2191(\u2191F\u2081 g) = \u2191(\u2191f\u2081 g)\nhF\u2082 : \u2200 (g : G), \u2191(\u2191F\u2082 g) = \u2191(\u2191f\u2082 g)\nx : \u211d\n\u22a2 BddAbove (range fun g => \u2191(\u2191f\u2082 g\u207b\u00b9) (\u2191(\u2191f\u2081 g) x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "Set.Finite.nullMeasurableSet_biInter", "start": [390, 1], "end": [392, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Lattice.lean", "full_name": "List.forall_mem_union", "start": [121, 1], "end": [122, 44], "traced_tactics": [{"tactic": "simp only [mem_union, or_imp, forall_and]", "state_before": "\u03b1 : Type u_1\nl l\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Prop\na : \u03b1\ninst\u271d : DecidableEq \u03b1\n\u22a2 (\u2200 (x : \u03b1), x \u2208 l\u2081 \u222a l\u2082 \u2192 p x) \u2194 (\u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 p x) \u2227 \u2200 (x : \u03b1), x \u2208 l\u2082 \u2192 p x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/LocallyFinite.lean", "full_name": "Finset.map_subtype_embedding_Ioi", "start": [1334, 1], "end": [1336, 74], "traced_tactics": [{"tactic": "rw [subtype_Ioi_eq]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.223996\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrderTop \u03b1\na : Subtype p\nhp : \u2200 \u2983a x : \u03b1\u2984, a \u2264 x \u2192 p a \u2192 p x\n\u22a2 map (Embedding.subtype p) (Ioi a) = Ioi \u2191a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.223996\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrderTop \u03b1\na : Subtype p\nhp : \u2200 \u2983a x : \u03b1\u2984, a \u2264 x \u2192 p a \u2192 p x\n\u22a2 map (Embedding.subtype p) (Finset.subtype p (Ioi \u2191a)) = Ioi \u2191a"}, {"tactic": "exact Finset.subtype_map_of_mem fun x hx => hp (mem_Ioi.1 hx).le a.prop", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.223996\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrderTop \u03b1\na : Subtype p\nhp : \u2200 \u2983a x : \u03b1\u2984, a \u2264 x \u2192 p a \u2192 p x\n\u22a2 map (Embedding.subtype p) (Finset.subtype p (Ioi \u2191a)) = Ioi \u2191a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.set_lintegral_withDensity_eq_set_lintegral_mul_non_measurable", "start": [1861, 1], "end": [1865, 98], "traced_tactics": [{"tactic": "rw [restrict_withDensity hs, lintegral_withDensity_eq_lintegral_mul_non_measurable _ f_meas hf]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.1811286\n\u03b3 : Type ?u.1811289\n\u03b4 : Type ?u.1811292\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nf_meas : Measurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, f x < \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1) in s, g a \u2202Measure.withDensity \u03bc f) = \u222b\u207b (a : \u03b1) in s, (f * g) a \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/Partrec.lean", "full_name": "Computable.vector_toList", "start": [373, 1], "end": [374, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/Projection.lean", "full_name": "OrthogonalFamily.isInternal_iff_of_isComplete", "start": [1241, 1], "end": [1246, 51], "traced_tactics": [{"tactic": "haveI : CompleteSpace (\u21a5(iSup V)) := hc.completeSpace_coe", "state_before": "\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type ?u.1082736\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nhc : IsComplete \u2191(iSup V)\n\u22a2 DirectSum.IsInternal V \u2194 (iSup V)\u15ee = \u22a5", "state_after": "\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type ?u.1082736\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nhc : IsComplete \u2191(iSup V)\nthis : CompleteSpace { x // x \u2208 iSup V }\n\u22a2 DirectSum.IsInternal V \u2194 (iSup V)\u15ee = \u22a5"}, {"tactic": "simp only [DirectSum.isInternal_submodule_iff_independent_and_iSup_eq_top, hV.independent,\n  true_and_iff, Submodule.orthogonal_eq_bot_iff]", "state_before": "\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type ?u.1082736\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_1\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nhc : IsComplete \u2191(iSup V)\nthis : CompleteSpace { x // x \u2208 iSup V }\n\u22a2 DirectSum.IsInternal V \u2194 (iSup V)\u15ee = \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Finiteness.lean", "full_name": "Subalgebra.fg_bot_toSubmodule", "start": [150, 1], "end": [152, 43], "traced_tactics": [{"tactic": "simp [Algebra.toSubmodule_bot]", "state_before": "R\u271d : Type ?u.53562\nM : Type ?u.53565\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R\u271d M\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\n\u22a2 span R \u2191{1} = \u2191Subalgebra.toSubmodule \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.Subsingleton.induction_on", "start": [2369, 1], "end": [2372, 20], "traced_tactics": [{"tactic": "rcases hs.eq_empty_or_singleton with (rfl | \u27e8x, rfl\u27e9)", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 u : Set \u03b1\np : Set \u03b1 \u2192 Prop\nhs : Set.Subsingleton s\nhe : p \u2205\nh\u2081 : \u2200 (x : \u03b1), p {x}\n\u22a2 p s", "state_after": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns\u2081 s\u2082 t t\u2081 t\u2082 u : Set \u03b1\np : Set \u03b1 \u2192 Prop\nhe : p \u2205\nh\u2081 : \u2200 (x : \u03b1), p {x}\nhs : Set.Subsingleton \u2205\n\u22a2 p \u2205\n\ncase inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns\u2081 s\u2082 t t\u2081 t\u2082 u : Set \u03b1\np : Set \u03b1 \u2192 Prop\nhe : p \u2205\nh\u2081 : \u2200 (x : \u03b1), p {x}\nx : \u03b1\nhs : Set.Subsingleton {x}\n\u22a2 p {x}"}, {"tactic": "exacts [he, h\u2081 _]", "state_before": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns\u2081 s\u2082 t t\u2081 t\u2082 u : Set \u03b1\np : Set \u03b1 \u2192 Prop\nhe : p \u2205\nh\u2081 : \u2200 (x : \u03b1), p {x}\nhs : Set.Subsingleton \u2205\n\u22a2 p \u2205\n\ncase inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns\u2081 s\u2082 t t\u2081 t\u2082 u : Set \u03b1\np : Set \u03b1 \u2192 Prop\nhe : p \u2205\nh\u2081 : \u2200 (x : \u03b1), p {x}\nx : \u03b1\nhs : Set.Subsingleton {x}\n\u22a2 p {x}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "full_name": "finrank_eq_zero_of_rank_eq_zero", "start": [652, 1], "end": [655, 20], "traced_tactics": [{"tactic": "convert finrank_eq_rank K V", "state_before": "K : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nh : Module.rank K V = 0\n\u22a2 finrank K V = 0", "state_after": "case a\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nh : Module.rank K V = 0\n\u22a2 finrank K V = 0 \u2194 \u2191(finrank K V) = Module.rank K V"}, {"tactic": "rw [h]", "state_before": "case a\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nh : Module.rank K V = 0\n\u22a2 finrank K V = 0 \u2194 \u2191(finrank K V) = Module.rank K V", "state_after": "case a\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nh : Module.rank K V = 0\n\u22a2 finrank K V = 0 \u2194 \u2191(finrank K V) = 0"}, {"tactic": "norm_cast", "state_before": "case a\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nh : Module.rank K V = 0\n\u22a2 finrank K V = 0 \u2194 \u2191(finrank K V) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean", "full_name": "Gamma1_mem'", "start": [149, 1], "end": [150, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/lpSpace.lean", "full_name": "lp.infty_coeFn_pow", "start": [905, 1], "end": [906, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Basic.lean", "full_name": "SimpleGraph.commonNeighbors_symm", "start": [1020, 1], "end": [1021, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Span.lean", "full_name": "Submodule.mem_iSup_of_chain", "start": [337, 1], "end": [338, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "borel_eq_generateFrom_of_subbasis", "start": [80, 1], "end": [95, 95], "traced_tactics": [{"tactic": "rw [hs] at hu", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nhu : TopologicalSpace.IsOpen u\n\u22a2 MeasurableSet u", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nhu : TopologicalSpace.IsOpen u\n\u22a2 MeasurableSet u"}, {"tactic": "induction hu", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nhu : TopologicalSpace.IsOpen u\n\u22a2 MeasurableSet u", "state_after": "case basic\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d\u00b9 t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu s\u271d : Set \u03b1\na\u271d : s\u271d \u2208 s\n\u22a2 MeasurableSet s\u271d\n\ncase univ\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\n\u22a2 MeasurableSet univ\n\ncase inter\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d\u00b9 t\u271d\u00b9 u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu s\u271d t\u271d : Set \u03b1\na\u271d\u00b9 : GenerateOpen s s\u271d\na\u271d : GenerateOpen s t\u271d\na_ih\u271d\u00b9 : MeasurableSet s\u271d\na_ih\u271d : MeasurableSet t\u271d\n\u22a2 MeasurableSet (s\u271d \u2229 t\u271d)\n\ncase sUnion\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 GenerateOpen s s_1\na_ih\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 MeasurableSet s_1\n\u22a2 MeasurableSet (\u22c3\u2080 S\u271d)"}, {"tactic": "case basic u hu => exact GenerateMeasurable.basic u hu", "state_before": "case basic\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d\u00b9 t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu s\u271d : Set \u03b1\na\u271d : s\u271d \u2208 s\n\u22a2 MeasurableSet s\u271d\n\ncase univ\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\n\u22a2 MeasurableSet univ\n\ncase inter\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d\u00b9 t\u271d\u00b9 u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu s\u271d t\u271d : Set \u03b1\na\u271d\u00b9 : GenerateOpen s s\u271d\na\u271d : GenerateOpen s t\u271d\na_ih\u271d\u00b9 : MeasurableSet s\u271d\na_ih\u271d : MeasurableSet t\u271d\n\u22a2 MeasurableSet (s\u271d \u2229 t\u271d)\n\ncase sUnion\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 GenerateOpen s s_1\na_ih\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 MeasurableSet s_1\n\u22a2 MeasurableSet (\u22c3\u2080 S\u271d)", "state_after": "case univ\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\n\u22a2 MeasurableSet univ\n\ncase inter\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d\u00b9 t\u271d\u00b9 u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu s\u271d t\u271d : Set \u03b1\na\u271d\u00b9 : GenerateOpen s s\u271d\na\u271d : GenerateOpen s t\u271d\na_ih\u271d\u00b9 : MeasurableSet s\u271d\na_ih\u271d : MeasurableSet t\u271d\n\u22a2 MeasurableSet (s\u271d \u2229 t\u271d)\n\ncase sUnion\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 GenerateOpen s s_1\na_ih\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 MeasurableSet s_1\n\u22a2 MeasurableSet (\u22c3\u2080 S\u271d)"}, {"tactic": "case univ => exact @MeasurableSet.univ \u03b1 (generateFrom s)", "state_before": "case univ\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\n\u22a2 MeasurableSet univ\n\ncase inter\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d\u00b9 t\u271d\u00b9 u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu s\u271d t\u271d : Set \u03b1\na\u271d\u00b9 : GenerateOpen s s\u271d\na\u271d : GenerateOpen s t\u271d\na_ih\u271d\u00b9 : MeasurableSet s\u271d\na_ih\u271d : MeasurableSet t\u271d\n\u22a2 MeasurableSet (s\u271d \u2229 t\u271d)\n\ncase sUnion\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 GenerateOpen s s_1\na_ih\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 MeasurableSet s_1\n\u22a2 MeasurableSet (\u22c3\u2080 S\u271d)", "state_after": "case inter\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d\u00b9 t\u271d\u00b9 u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu s\u271d t\u271d : Set \u03b1\na\u271d\u00b9 : GenerateOpen s s\u271d\na\u271d : GenerateOpen s t\u271d\na_ih\u271d\u00b9 : MeasurableSet s\u271d\na_ih\u271d : MeasurableSet t\u271d\n\u22a2 MeasurableSet (s\u271d \u2229 t\u271d)\n\ncase sUnion\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 GenerateOpen s s_1\na_ih\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 MeasurableSet s_1\n\u22a2 MeasurableSet (\u22c3\u2080 S\u271d)"}, {"tactic": "case inter s\u2081 s\u2082 _ _ hs\u2081 hs\u2082 => exact @MeasurableSet.inter \u03b1 (generateFrom s) _ _ hs\u2081 hs\u2082", "state_before": "case inter\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d\u00b9 t\u271d\u00b9 u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu s\u271d t\u271d : Set \u03b1\na\u271d\u00b9 : GenerateOpen s s\u271d\na\u271d : GenerateOpen s t\u271d\na_ih\u271d\u00b9 : MeasurableSet s\u271d\na_ih\u271d : MeasurableSet t\u271d\n\u22a2 MeasurableSet (s\u271d \u2229 t\u271d)\n\ncase sUnion\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 GenerateOpen s s_1\na_ih\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 MeasurableSet s_1\n\u22a2 MeasurableSet (\u22c3\u2080 S\u271d)", "state_after": "case sUnion\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 GenerateOpen s s_1\na_ih\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 MeasurableSet s_1\n\u22a2 MeasurableSet (\u22c3\u2080 S\u271d)"}, {"tactic": "case\n  sUnion f hf ih =>\n  rcases isOpen_sUnion_countable f (by rwa [hs]) with \u27e8v, hv, vf, vu\u27e9\n  rw [\u2190 vu]\n  exact @MeasurableSet.sUnion \u03b1 (generateFrom s) _ hv fun x xv => ih _ (vf xv)", "state_before": "case sUnion\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nS\u271d : Set (Set \u03b1)\na\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 GenerateOpen s s_1\na_ih\u271d : \u2200 (s_1 : Set \u03b1), s_1 \u2208 S\u271d \u2192 MeasurableSet s_1\n\u22a2 MeasurableSet (\u22c3\u2080 S\u271d)", "state_after": "no goals"}, {"tactic": "exact GenerateMeasurable.basic u hu", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d\u00b9 : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu\u271d u : Set \u03b1\nhu : u \u2208 s\n\u22a2 MeasurableSet u", "state_after": "no goals"}, {"tactic": "exact @MeasurableSet.univ \u03b1 (generateFrom s)", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\n\u22a2 MeasurableSet univ", "state_after": "no goals"}, {"tactic": "exact @MeasurableSet.inter \u03b1 (generateFrom s) _ _ hs\u2081 hs\u2082", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu s\u2081 s\u2082 : Set \u03b1\na\u271d\u00b9 : GenerateOpen s s\u2081\na\u271d : GenerateOpen s s\u2082\nhs\u2081 : MeasurableSet s\u2081\nhs\u2082 : MeasurableSet s\u2082\n\u22a2 MeasurableSet (s\u2081 \u2229 s\u2082)", "state_after": "no goals"}, {"tactic": "rcases isOpen_sUnion_countable f (by rwa [hs]) with \u27e8v, hv, vf, vu\u27e9", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nf : Set (Set \u03b1)\nhf : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 GenerateOpen s s_1\nih : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 MeasurableSet s_1\n\u22a2 MeasurableSet (\u22c3\u2080 f)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nf : Set (Set \u03b1)\nhf : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 GenerateOpen s s_1\nih : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 MeasurableSet s_1\nv : Set (Set \u03b1)\nhv : Set.Countable v\nvf : v \u2286 f\nvu : \u22c3\u2080 v = \u22c3\u2080 f\n\u22a2 MeasurableSet (\u22c3\u2080 f)"}, {"tactic": "rw [\u2190 vu]", "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nf : Set (Set \u03b1)\nhf : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 GenerateOpen s s_1\nih : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 MeasurableSet s_1\nv : Set (Set \u03b1)\nhv : Set.Countable v\nvf : v \u2286 f\nvu : \u22c3\u2080 v = \u22c3\u2080 f\n\u22a2 MeasurableSet (\u22c3\u2080 f)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nf : Set (Set \u03b1)\nhf : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 GenerateOpen s s_1\nih : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 MeasurableSet s_1\nv : Set (Set \u03b1)\nhv : Set.Countable v\nvf : v \u2286 f\nvu : \u22c3\u2080 v = \u22c3\u2080 f\n\u22a2 MeasurableSet (\u22c3\u2080 v)"}, {"tactic": "exact @MeasurableSet.sUnion \u03b1 (generateFrom s) _ hv fun x xv => ih _ (vf xv)", "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nf : Set (Set \u03b1)\nhf : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 GenerateOpen s s_1\nih : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 MeasurableSet s_1\nv : Set (Set \u03b1)\nhv : Set.Countable v\nvf : v \u2286 f\nvu : \u22c3\u2080 v = \u22c3\u2080 f\n\u22a2 MeasurableSet (\u22c3\u2080 v)", "state_after": "no goals"}, {"tactic": "rwa [hs]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nf : Set (Set \u03b1)\nhf : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 GenerateOpen s s_1\nih : \u2200 (s_1 : Set \u03b1), s_1 \u2208 f \u2192 MeasurableSet s_1\n\u22a2 \u2200 (s : Set \u03b1), s \u2208 f \u2192 IsOpen s", "state_after": "no goals"}, {"tactic": "rw [hs]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nhu : u \u2208 s\n\u22a2 TopologicalSpace.IsOpen u", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nhu : u \u2208 s\n\u22a2 TopologicalSpace.IsOpen u"}, {"tactic": "exact GenerateOpen.basic _ hu", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.27507\n\u03b3 : Type ?u.27510\n\u03b3\u2082 : Type ?u.27513\n\u03b4 : Type ?u.27516\n\u03b9 : Sort y\ns\u271d t\u271d u\u271d : Set \u03b1\ns : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nhs : t = TopologicalSpace.generateFrom s\nu : Set \u03b1\nhu : u \u2208 s\n\u22a2 TopologicalSpace.IsOpen u", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "full_name": "cthickening_cthickening", "start": [315, 1], "end": [319, 30], "traced_tactics": [{"tactic": "simp_rw [mem_cthickening_iff, ENNReal.ofReal_add h\u03b5 h\u03b4, infEdist_cthickening]", "state_before": "\ud835\udd5c : Type ?u.231373\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 \u2264 \u03b4\ns : Set E\nx : E\n\u22a2 x \u2208 cthickening (\u03b5 + \u03b4) s \u2192 x \u2208 cthickening \u03b5 (cthickening \u03b4 s)", "state_after": "\ud835\udd5c : Type ?u.231373\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 \u2264 \u03b4\ns : Set E\nx : E\n\u22a2 infEdist x s \u2264 ENNReal.ofReal \u03b5 + ENNReal.ofReal \u03b4 \u2192 infEdist x s - ENNReal.ofReal \u03b4 \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "exact tsub_le_iff_right.2", "state_before": "\ud835\udd5c : Type ?u.231373\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 \u2264 \u03b4\ns : Set E\nx : E\n\u22a2 infEdist x s \u2264 ENNReal.ofReal \u03b5 + ENNReal.ofReal \u03b4 \u2192 infEdist x s - ENNReal.ofReal \u03b4 \u2264 ENNReal.ofReal \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.sdiff_insert_of_not_mem", "start": [2191, 1], "end": [2194, 49], "traced_tactics": [{"tactic": "refine' Subset.antisymm (sdiff_subset_sdiff (Subset.refl _) (subset_insert _ _)) fun y hy => _", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.242065\n\u03b3 : Type ?u.242068\ninst\u271d : DecidableEq \u03b1\ns t\u271d u v : Finset \u03b1\na b x : \u03b1\nh : \u00acx \u2208 s\nt : Finset \u03b1\n\u22a2 s \\ insert x t = s \\ t", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.242065\n\u03b3 : Type ?u.242068\ninst\u271d : DecidableEq \u03b1\ns t\u271d u v : Finset \u03b1\na b x : \u03b1\nh : \u00acx \u2208 s\nt : Finset \u03b1\ny : \u03b1\nhy : y \u2208 s \\ t\n\u22a2 y \u2208 s \\ insert x t"}, {"tactic": "simp only [mem_sdiff, mem_insert, not_or] at hy\u22a2", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.242065\n\u03b3 : Type ?u.242068\ninst\u271d : DecidableEq \u03b1\ns t\u271d u v : Finset \u03b1\na b x : \u03b1\nh : \u00acx \u2208 s\nt : Finset \u03b1\ny : \u03b1\nhy : y \u2208 s \\ t\n\u22a2 y \u2208 s \\ insert x t", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.242065\n\u03b3 : Type ?u.242068\ninst\u271d : DecidableEq \u03b1\ns t\u271d u v : Finset \u03b1\na b x : \u03b1\nh : \u00acx \u2208 s\nt : Finset \u03b1\ny : \u03b1\nhy : y \u2208 s \u2227 \u00acy \u2208 t\n\u22a2 y \u2208 s \u2227 \u00acy = x \u2227 \u00acy \u2208 t"}, {"tactic": "exact \u27e8hy.1, fun hxy => h <| hxy \u25b8 hy.1, hy.2\u27e9", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.242065\n\u03b3 : Type ?u.242068\ninst\u271d : DecidableEq \u03b1\ns t\u271d u v : Finset \u03b1\na b x : \u03b1\nh : \u00acx \u2208 s\nt : Finset \u03b1\ny : \u03b1\nhy : y \u2208 s \u2227 \u00acy \u2208 t\n\u22a2 y \u2208 s \u2227 \u00acy = x \u2227 \u00acy \u2208 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.eventually_sup", "start": [1213, 1], "end": [1215, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/EuclideanDomain/Basic.lean", "full_name": "EuclideanDomain.mul_div_assoc", "start": [102, 1], "end": [107, 73], "traced_tactics": [{"tactic": "by_cases hz : z = 0", "state_before": "R : Type u\ninst\u271d : EuclideanDomain R\nx y z : R\nh : z \u2223 y\n\u22a2 x * y / z = x * (y / z)", "state_after": "case pos\nR : Type u\ninst\u271d : EuclideanDomain R\nx y z : R\nh : z \u2223 y\nhz : z = 0\n\u22a2 x * y / z = x * (y / z)\n\ncase neg\nR : Type u\ninst\u271d : EuclideanDomain R\nx y z : R\nh : z \u2223 y\nhz : \u00acz = 0\n\u22a2 x * y / z = x * (y / z)"}, {"tactic": "rcases h with \u27e8p, rfl\u27e9", "state_before": "case neg\nR : Type u\ninst\u271d : EuclideanDomain R\nx y z : R\nh : z \u2223 y\nhz : \u00acz = 0\n\u22a2 x * y / z = x * (y / z)", "state_after": "case neg.intro\nR : Type u\ninst\u271d : EuclideanDomain R\nx z : R\nhz : \u00acz = 0\np : R\n\u22a2 x * (z * p) / z = x * (z * p / z)"}, {"tactic": "rw [mul_div_cancel_left _ hz, mul_left_comm, mul_div_cancel_left _ hz]", "state_before": "case neg.intro\nR : Type u\ninst\u271d : EuclideanDomain R\nx z : R\nhz : \u00acz = 0\np : R\n\u22a2 x * (z * p) / z = x * (z * p / z)", "state_after": "no goals"}, {"tactic": "subst hz", "state_before": "case pos\nR : Type u\ninst\u271d : EuclideanDomain R\nx y z : R\nh : z \u2223 y\nhz : z = 0\n\u22a2 x * y / z = x * (y / z)", "state_after": "case pos\nR : Type u\ninst\u271d : EuclideanDomain R\nx y : R\nh : 0 \u2223 y\n\u22a2 x * y / 0 = x * (y / 0)"}, {"tactic": "rw [div_zero, div_zero, mul_zero]", "state_before": "case pos\nR : Type u\ninst\u271d : EuclideanDomain R\nx y : R\nh : 0 \u2223 y\n\u22a2 x * y / 0 = x * (y / 0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/RamificationInertia.lean", "full_name": "Ideal.le_pow_ramificationIdx", "start": [133, 1], "end": [134, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean", "full_name": "AlgebraicTopology.map_alternatingFaceMapComplex", "start": [195, 1], "end": [214, 10], "traced_tactics": [{"tactic": "apply CategoryTheory.Functor.ext", "state_before": "C : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\n\u22a2 alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115) =\n    (SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D", "state_after": "case h_map\nC : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\n\u22a2 autoParam\n    (\u2200 (X Y : SimplicialObject C) (f : X \u27f6 Y),\n      (alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map f =\n        eqToHom (_ : ?F.obj X = ?G.obj X) \u226b\n          ((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).map f \u226b\n            eqToHom\n              (_ :\n                ((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj Y =\n                  (alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj Y))\n    _auto\u271d\n\ncase h_obj\nC : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\n\u22a2 \u2200 (X : SimplicialObject C),\n    (alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj X =\n      ((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj X"}, {"tactic": "intro X Y f", "state_before": "case h_map\nC : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\n\u22a2 autoParam\n    (\u2200 (X Y : SimplicialObject C) (f : X \u27f6 Y),\n      (alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map f =\n        eqToHom (_ : ?F.obj X = ?G.obj X) \u226b\n          ((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).map f \u226b\n            eqToHom\n              (_ :\n                ((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj Y =\n                  (alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj Y))\n    _auto\u271d", "state_after": "case h_map\nC : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\nX Y : SimplicialObject C\nf : X \u27f6 Y\n\u22a2 (alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map f =\n    eqToHom (_ : ?F.obj X = ?G.obj X) \u226b\n      ((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).map f \u226b\n        eqToHom\n          (_ :\n            ((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj Y =\n              (alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj Y)"}, {"tactic": "ext n", "state_before": "case h_map\nC : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\nX Y : SimplicialObject C\nf : X \u27f6 Y\n\u22a2 (alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map f =\n    eqToHom (_ : ?F.obj X = ?G.obj X) \u226b\n      ((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).map f \u226b\n        eqToHom\n          (_ :\n            ((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj Y =\n              (alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj Y)", "state_after": "case h_map.h\nC : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\nX Y : SimplicialObject C\nf : X \u27f6 Y\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f ((alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map f)\n      n =\n    HomologicalComplex.Hom.f\n      (eqToHom (_ : ?F.obj X = ?G.obj X) \u226b\n        ((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).map f \u226b\n          eqToHom\n            (_ :\n              ((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj Y =\n                (alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj Y))\n      n"}, {"tactic": "simp only [Functor.comp_map, HomologicalComplex.comp_f, alternatingFaceMapComplex_map_f,\n  Functor.mapHomologicalComplex_map_f, HomologicalComplex.eqToHom_f, eqToHom_refl, comp_id,\n  id_comp, SimplicialObject.whiskering_obj_map_app]", "state_before": "case h_map.h\nC : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\nX Y : SimplicialObject C\nf : X \u27f6 Y\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f ((alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map f)\n      n =\n    HomologicalComplex.Hom.f\n      (eqToHom (_ : ?F.obj X = ?G.obj X) \u226b\n        ((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).map f \u226b\n          eqToHom\n            (_ :\n              ((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj Y =\n                (alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj Y))\n      n", "state_after": "no goals"}, {"tactic": "intro X", "state_before": "case h_obj\nC : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\n\u22a2 \u2200 (X : SimplicialObject C),\n    (alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj X =\n      ((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj X", "state_after": "case h_obj\nC : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\nX : SimplicialObject C\n\u22a2 (alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj X =\n    ((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj X"}, {"tactic": "apply HomologicalComplex.ext", "state_before": "case h_obj\nC : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\nX : SimplicialObject C\n\u22a2 (alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj X =\n    ((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj X", "state_after": "case h_obj.h_d\nC : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\nX : SimplicialObject C\n\u22a2 \u2200 (i j : \u2115),\n    ComplexShape.Rel (ComplexShape.down \u2115) i j \u2192\n      HomologicalComplex.d ((alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj X)\n            i j \u226b\n          eqToHom\n            (_ :\n              HomologicalComplex.X\n                  ((alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj X) j =\n                HomologicalComplex.X (((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj X)\n                  j) =\n        eqToHom\n            (_ :\n              HomologicalComplex.X\n                  ((alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj X) i =\n                HomologicalComplex.X (((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj X)\n                  i) \u226b\n          HomologicalComplex.d (((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj X) i j\n\ncase h_obj.h_X\nC : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\nX : SimplicialObject C\n\u22a2 ((alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj X).X =\n    (((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj X).X"}, {"tactic": "rintro i j (rfl : j + 1 = i)", "state_before": "case h_obj.h_d\nC : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\nX : SimplicialObject C\n\u22a2 \u2200 (i j : \u2115),\n    ComplexShape.Rel (ComplexShape.down \u2115) i j \u2192\n      HomologicalComplex.d ((alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj X)\n            i j \u226b\n          eqToHom\n            (_ :\n              HomologicalComplex.X\n                  ((alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj X) j =\n                HomologicalComplex.X (((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj X)\n                  j) =\n        eqToHom\n            (_ :\n              HomologicalComplex.X\n                  ((alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj X) i =\n                HomologicalComplex.X (((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj X)\n                  i) \u226b\n          HomologicalComplex.d (((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj X) i j", "state_after": "case h_obj.h_d\nC : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\nX : SimplicialObject C\nj : \u2115\n\u22a2 HomologicalComplex.d ((alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj X)\n        (j + 1) j \u226b\n      eqToHom\n        (_ :\n          HomologicalComplex.X\n              ((alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj X) j =\n            HomologicalComplex.X (((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj X) j) =\n    eqToHom\n        (_ :\n          HomologicalComplex.X\n              ((alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj X) (j + 1) =\n            HomologicalComplex.X (((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj X)\n              (j + 1)) \u226b\n      HomologicalComplex.d (((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj X) (j + 1) j"}, {"tactic": "dsimp only [Functor.comp_obj]", "state_before": "case h_obj.h_d\nC : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\nX : SimplicialObject C\nj : \u2115\n\u22a2 HomologicalComplex.d ((alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj X)\n        (j + 1) j \u226b\n      eqToHom\n        (_ :\n          HomologicalComplex.X\n              ((alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj X) j =\n            HomologicalComplex.X (((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj X) j) =\n    eqToHom\n        (_ :\n          HomologicalComplex.X\n              ((alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj X) (j + 1) =\n            HomologicalComplex.X (((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj X)\n              (j + 1)) \u226b\n      HomologicalComplex.d (((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj X) (j + 1) j", "state_after": "case h_obj.h_d\nC : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\nX : SimplicialObject C\nj : \u2115\n\u22a2 HomologicalComplex.d\n        ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj ((alternatingFaceMapComplex C).obj X)) (j + 1) j \u226b\n      eqToHom\n        (_ :\n          HomologicalComplex.X\n              ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj ((alternatingFaceMapComplex C).obj X)) j =\n            HomologicalComplex.X ((alternatingFaceMapComplex D).obj (((SimplicialObject.whiskering C D).obj F).obj X))\n              j) =\n    eqToHom\n        (_ :\n          HomologicalComplex.X\n              ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj ((alternatingFaceMapComplex C).obj X))\n              (j + 1) =\n            HomologicalComplex.X ((alternatingFaceMapComplex D).obj (((SimplicialObject.whiskering C D).obj F).obj X))\n              (j + 1)) \u226b\n      HomologicalComplex.d ((alternatingFaceMapComplex D).obj (((SimplicialObject.whiskering C D).obj F).obj X)) (j + 1)\n        j"}, {"tactic": "simp only [Functor.mapHomologicalComplex_obj_d, alternatingFaceMapComplex_obj_d,\n  eqToHom_refl, id_comp, comp_id, AlternatingFaceMapComplex.objD, Functor.map_sum,\n  Functor.map_zsmul]", "state_before": "case h_obj.h_d\nC : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\nX : SimplicialObject C\nj : \u2115\n\u22a2 HomologicalComplex.d\n        ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj ((alternatingFaceMapComplex C).obj X)) (j + 1) j \u226b\n      eqToHom\n        (_ :\n          HomologicalComplex.X\n              ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj ((alternatingFaceMapComplex C).obj X)) j =\n            HomologicalComplex.X ((alternatingFaceMapComplex D).obj (((SimplicialObject.whiskering C D).obj F).obj X))\n              j) =\n    eqToHom\n        (_ :\n          HomologicalComplex.X\n              ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj ((alternatingFaceMapComplex C).obj X))\n              (j + 1) =\n            HomologicalComplex.X ((alternatingFaceMapComplex D).obj (((SimplicialObject.whiskering C D).obj F).obj X))\n              (j + 1)) \u226b\n      HomologicalComplex.d ((alternatingFaceMapComplex D).obj (((SimplicialObject.whiskering C D).obj F).obj X)) (j + 1)\n        j", "state_after": "case h_obj.h_d\nC : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\nX : SimplicialObject C\nj : \u2115\n\u22a2 \u2211 x : Fin (j + 2), (-1) ^ \u2191x \u2022 F.map (SimplicialObject.\u03b4 X x) =\n    \u2211 i : Fin (j + 2), (-1) ^ \u2191i \u2022 SimplicialObject.\u03b4 (((SimplicialObject.whiskering C D).obj F).obj X) i"}, {"tactic": "rfl", "state_before": "case h_obj.h_d\nC : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\nX : SimplicialObject C\nj : \u2115\n\u22a2 \u2211 x : Fin (j + 2), (-1) ^ \u2191x \u2022 F.map (SimplicialObject.\u03b4 X x) =\n    \u2211 i : Fin (j + 2), (-1) ^ \u2191i \u2022 SimplicialObject.\u03b4 (((SimplicialObject.whiskering C D).obj F).obj X) i", "state_after": "no goals"}, {"tactic": "ext n", "state_before": "case h_obj.h_X\nC : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\nX : SimplicialObject C\n\u22a2 ((alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj X).X =\n    (((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj X).X", "state_after": "case h_obj.h_X.h\nC : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\nX : SimplicialObject C\nn : \u2115\n\u22a2 HomologicalComplex.X ((alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj X) n =\n    HomologicalComplex.X (((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj X) n"}, {"tactic": "rfl", "state_before": "case h_obj.h_X.h\nC : Type u_4\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : Preadditive C\nD : Type u_1\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : Preadditive D\nF : C \u2964 D\ninst\u271d : Functor.Additive F\nX : SimplicialObject C\nn : \u2115\n\u22a2 HomologicalComplex.X ((alternatingFaceMapComplex C \u22d9 Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj X) n =\n    HomologicalComplex.X (((SimplicialObject.whiskering C D).obj F \u22d9 alternatingFaceMapComplex D).obj X) n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/AffineIsometry.lean", "full_name": "AffineIsometry.ext", "start": [96, 1], "end": [97, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/Partrec.lean", "full_name": "Computable.nat_casesOn", "start": [673, 1], "end": [676, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Inseparable.lean", "full_name": "inseparable_iff_forall_open", "start": [286, 1], "end": [288, 14], "traced_tactics": [{"tactic": "simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, \u2190 forall_and, \u2190 iff_def,\n  Iff.comm]", "state_before": "X : Type u_1\nY : Type ?u.28194\nZ : Type ?u.28197\n\u03b1 : Type ?u.28200\n\u03b9 : Type ?u.28203\n\u03c0 : \u03b9 \u2192 Type ?u.28208\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\n\u22a2 (x ~\u1d62 y) \u2194 \u2200 (s : Set X), IsOpen s \u2192 (x \u2208 s \u2194 y \u2208 s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.le_comap_top", "start": [2192, 1], "end": [2194, 15], "traced_tactics": [{"tactic": "rw [comap_top]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.251116\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\nl : Filter \u03b1\n\u22a2 l \u2264 comap f \u22a4", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.251116\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\nl : Filter \u03b1\n\u22a2 l \u2264 \u22a4"}, {"tactic": "exact le_top", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.251116\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\nl : Filter \u03b1\n\u22a2 l \u2264 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Fin.lean", "full_name": "Fin.prod_univ_three", "start": [118, 1], "end": [120, 6], "traced_tactics": [{"tactic": "rw [prod_univ_castSucc, prod_univ_two]", "state_before": "\u03b1 : Type ?u.18255\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : Fin 3 \u2192 \u03b2\n\u22a2 \u220f i : Fin 3, f i = f 0 * f 1 * f 2", "state_after": "\u03b1 : Type ?u.18255\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : Fin 3 \u2192 \u03b2\n\u22a2 f (\u2191castSucc 0) * f (\u2191castSucc 1) * f (last 2) = f 0 * f 1 * f 2"}, {"tactic": "rfl", "state_before": "\u03b1 : Type ?u.18255\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : Fin 3 \u2192 \u03b2\n\u22a2 f (\u2191castSucc 0) * f (\u2191castSucc 1) * f (last 2) = f 0 * f 1 * f 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Function/Basic.lean", "full_name": "IsSymmOp.flip_eq", "start": [1058, 1], "end": [1059, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/Combination.lean", "full_name": "Finset.sum_centroidWeightsIndicator_eq_one_of_nonempty", "start": [925, 1], "end": [928, 53], "traced_tactics": [{"tactic": "rw [sum_centroidWeightsIndicator]", "state_before": "k : Type u_1\nV : Type ?u.516833\nP : Type ?u.516836\ninst\u271d\u2075 : DivisionRing k\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module k V\ninst\u271d\u00b2 : AffineSpace V P\n\u03b9 : Type u_2\ns : Finset \u03b9\n\u03b9\u2082 : Type ?u.517355\ns\u2082 : Finset \u03b9\u2082\ninst\u271d\u00b9 : CharZero k\ninst\u271d : Fintype \u03b9\nh : Finset.Nonempty s\n\u22a2 \u2211 i : \u03b9, centroidWeightsIndicator k s i = 1", "state_after": "k : Type u_1\nV : Type ?u.516833\nP : Type ?u.516836\ninst\u271d\u2075 : DivisionRing k\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module k V\ninst\u271d\u00b2 : AffineSpace V P\n\u03b9 : Type u_2\ns : Finset \u03b9\n\u03b9\u2082 : Type ?u.517355\ns\u2082 : Finset \u03b9\u2082\ninst\u271d\u00b9 : CharZero k\ninst\u271d : Fintype \u03b9\nh : Finset.Nonempty s\n\u22a2 \u2211 i in s, centroidWeights k s i = 1"}, {"tactic": "exact s.sum_centroidWeights_eq_one_of_nonempty k h", "state_before": "k : Type u_1\nV : Type ?u.516833\nP : Type ?u.516836\ninst\u271d\u2075 : DivisionRing k\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module k V\ninst\u271d\u00b2 : AffineSpace V P\n\u03b9 : Type u_2\ns : Finset \u03b9\n\u03b9\u2082 : Type ?u.517355\ns\u2082 : Finset \u03b9\u2082\ninst\u271d\u00b9 : CharZero k\ninst\u271d : Fintype \u03b9\nh : Finset.Nonempty s\n\u22a2 \u2211 i in s, centroidWeights k s i = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/QuotientGroup.lean", "full_name": "QuotientGroup.mk_inv", "start": [167, 1], "end": [168, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Dual.lean", "full_name": "Basis.eval_range", "start": [474, 1], "end": [478, 90], "traced_tactics": [{"tactic": "classical\n  cases nonempty_fintype \u03b9\n  rw [\u2190 b.toDual_toDual, range_comp, b.toDual_range, Submodule.map_top, toDual_range _]", "state_before": "R : Type u\nM : Type v\nK : Type u\u2081\nV : Type u\u2082\n\u03b9\u271d : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : DecidableEq \u03b9\u271d\nb\u271d : Basis \u03b9\u271d R M\n\u03b9 : Type u_1\ninst\u271d : _root_.Finite \u03b9\nb : Basis \u03b9 R M\n\u22a2 range (Dual.eval R M) = \u22a4", "state_after": "no goals"}, {"tactic": "cases nonempty_fintype \u03b9", "state_before": "R : Type u\nM : Type v\nK : Type u\u2081\nV : Type u\u2082\n\u03b9\u271d : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : DecidableEq \u03b9\u271d\nb\u271d : Basis \u03b9\u271d R M\n\u03b9 : Type u_1\ninst\u271d : _root_.Finite \u03b9\nb : Basis \u03b9 R M\n\u22a2 range (Dual.eval R M) = \u22a4", "state_after": "case intro\nR : Type u\nM : Type v\nK : Type u\u2081\nV : Type u\u2082\n\u03b9\u271d : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : DecidableEq \u03b9\u271d\nb\u271d : Basis \u03b9\u271d R M\n\u03b9 : Type u_1\ninst\u271d : _root_.Finite \u03b9\nb : Basis \u03b9 R M\nval\u271d : Fintype \u03b9\n\u22a2 range (Dual.eval R M) = \u22a4"}, {"tactic": "rw [\u2190 b.toDual_toDual, range_comp, b.toDual_range, Submodule.map_top, toDual_range _]", "state_before": "case intro\nR : Type u\nM : Type v\nK : Type u\u2081\nV : Type u\u2082\n\u03b9\u271d : Type w\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : DecidableEq \u03b9\u271d\nb\u271d : Basis \u03b9\u271d R M\n\u03b9 : Type u_1\ninst\u271d : _root_.Finite \u03b9\nb : Basis \u03b9 R M\nval\u271d : Fintype \u03b9\n\u22a2 range (Dual.eval R M) = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Basic.lean", "full_name": "Directed.convex_iUnion", "start": [124, 1], "end": [131, 44], "traced_tactics": [{"tactic": "rintro x hx y hy a b ha hb hab", "state_before": "\ud835\udd5c : Type u_3\nE : Type u_2\nF : Type ?u.7918\n\u03b2 : Type ?u.7921\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c F\ns\u271d : Set E\nx : E\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set E\nhdir : Directed (fun x x_1 => x \u2286 x_1) s\nhc : \u2200 \u2983i : \u03b9\u2984, Convex \ud835\udd5c (s i)\n\u22a2 Convex \ud835\udd5c (\u22c3 (i : \u03b9), s i)", "state_after": "\ud835\udd5c : Type u_3\nE : Type u_2\nF : Type ?u.7918\n\u03b2 : Type ?u.7921\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c F\ns\u271d : Set E\nx\u271d : E\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set E\nhdir : Directed (fun x x_1 => x \u2286 x_1) s\nhc : \u2200 \u2983i : \u03b9\u2984, Convex \ud835\udd5c (s i)\nx : E\nhx : x \u2208 \u22c3 (i : \u03b9), s i\ny : E\nhy : y \u2208 \u22c3 (i : \u03b9), s i\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 x + b \u2022 y \u2208 \u22c3 (i : \u03b9), s i"}, {"tactic": "rw [mem_iUnion] at hx hy\u22a2", "state_before": "\ud835\udd5c : Type u_3\nE : Type u_2\nF : Type ?u.7918\n\u03b2 : Type ?u.7921\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c F\ns\u271d : Set E\nx\u271d : E\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set E\nhdir : Directed (fun x x_1 => x \u2286 x_1) s\nhc : \u2200 \u2983i : \u03b9\u2984, Convex \ud835\udd5c (s i)\nx : E\nhx : x \u2208 \u22c3 (i : \u03b9), s i\ny : E\nhy : y \u2208 \u22c3 (i : \u03b9), s i\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 x + b \u2022 y \u2208 \u22c3 (i : \u03b9), s i", "state_after": "\ud835\udd5c : Type u_3\nE : Type u_2\nF : Type ?u.7918\n\u03b2 : Type ?u.7921\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c F\ns\u271d : Set E\nx\u271d : E\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set E\nhdir : Directed (fun x x_1 => x \u2286 x_1) s\nhc : \u2200 \u2983i : \u03b9\u2984, Convex \ud835\udd5c (s i)\nx : E\nhx : \u2203 i, x \u2208 s i\ny : E\nhy : \u2203 i, y \u2208 s i\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 \u2203 i, a \u2022 x + b \u2022 y \u2208 s i"}, {"tactic": "obtain \u27e8i, hx\u27e9 := hx", "state_before": "\ud835\udd5c : Type u_3\nE : Type u_2\nF : Type ?u.7918\n\u03b2 : Type ?u.7921\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c F\ns\u271d : Set E\nx\u271d : E\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set E\nhdir : Directed (fun x x_1 => x \u2286 x_1) s\nhc : \u2200 \u2983i : \u03b9\u2984, Convex \ud835\udd5c (s i)\nx : E\nhx : \u2203 i, x \u2208 s i\ny : E\nhy : \u2203 i, y \u2208 s i\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 \u2203 i, a \u2022 x + b \u2022 y \u2208 s i", "state_after": "case intro\n\ud835\udd5c : Type u_3\nE : Type u_2\nF : Type ?u.7918\n\u03b2 : Type ?u.7921\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c F\ns\u271d : Set E\nx\u271d : E\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set E\nhdir : Directed (fun x x_1 => x \u2286 x_1) s\nhc : \u2200 \u2983i : \u03b9\u2984, Convex \ud835\udd5c (s i)\nx y : E\nhy : \u2203 i, y \u2208 s i\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\ni : \u03b9\nhx : x \u2208 s i\n\u22a2 \u2203 i, a \u2022 x + b \u2022 y \u2208 s i"}, {"tactic": "obtain \u27e8j, hy\u27e9 := hy", "state_before": "case intro\n\ud835\udd5c : Type u_3\nE : Type u_2\nF : Type ?u.7918\n\u03b2 : Type ?u.7921\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c F\ns\u271d : Set E\nx\u271d : E\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set E\nhdir : Directed (fun x x_1 => x \u2286 x_1) s\nhc : \u2200 \u2983i : \u03b9\u2984, Convex \ud835\udd5c (s i)\nx y : E\nhy : \u2203 i, y \u2208 s i\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\ni : \u03b9\nhx : x \u2208 s i\n\u22a2 \u2203 i, a \u2022 x + b \u2022 y \u2208 s i", "state_after": "case intro.intro\n\ud835\udd5c : Type u_3\nE : Type u_2\nF : Type ?u.7918\n\u03b2 : Type ?u.7921\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c F\ns\u271d : Set E\nx\u271d : E\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set E\nhdir : Directed (fun x x_1 => x \u2286 x_1) s\nhc : \u2200 \u2983i : \u03b9\u2984, Convex \ud835\udd5c (s i)\nx y : E\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\ni : \u03b9\nhx : x \u2208 s i\nj : \u03b9\nhy : y \u2208 s j\n\u22a2 \u2203 i, a \u2022 x + b \u2022 y \u2208 s i"}, {"tactic": "obtain \u27e8k, hik, hjk\u27e9 := hdir i j", "state_before": "case intro.intro\n\ud835\udd5c : Type u_3\nE : Type u_2\nF : Type ?u.7918\n\u03b2 : Type ?u.7921\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c F\ns\u271d : Set E\nx\u271d : E\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set E\nhdir : Directed (fun x x_1 => x \u2286 x_1) s\nhc : \u2200 \u2983i : \u03b9\u2984, Convex \ud835\udd5c (s i)\nx y : E\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\ni : \u03b9\nhx : x \u2208 s i\nj : \u03b9\nhy : y \u2208 s j\n\u22a2 \u2203 i, a \u2022 x + b \u2022 y \u2208 s i", "state_after": "case intro.intro.intro.intro\n\ud835\udd5c : Type u_3\nE : Type u_2\nF : Type ?u.7918\n\u03b2 : Type ?u.7921\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c F\ns\u271d : Set E\nx\u271d : E\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set E\nhdir : Directed (fun x x_1 => x \u2286 x_1) s\nhc : \u2200 \u2983i : \u03b9\u2984, Convex \ud835\udd5c (s i)\nx y : E\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\ni : \u03b9\nhx : x \u2208 s i\nj : \u03b9\nhy : y \u2208 s j\nk : \u03b9\nhik : s i \u2286 s k\nhjk : s j \u2286 s k\n\u22a2 \u2203 i, a \u2022 x + b \u2022 y \u2208 s i"}, {"tactic": "exact \u27e8k, hc (hik hx) (hjk hy) ha hb hab\u27e9", "state_before": "case intro.intro.intro.intro\n\ud835\udd5c : Type u_3\nE : Type u_2\nF : Type ?u.7918\n\u03b2 : Type ?u.7921\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c F\ns\u271d : Set E\nx\u271d : E\n\u03b9 : Sort u_1\ns : \u03b9 \u2192 Set E\nhdir : Directed (fun x x_1 => x \u2286 x_1) s\nhc : \u2200 \u2983i : \u03b9\u2984, Convex \ud835\udd5c (s i)\nx y : E\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\ni : \u03b9\nhx : x \u2208 s i\nj : \u03b9\nhy : y \u2208 s j\nk : \u03b9\nhik : s i \u2286 s k\nhjk : s j \u2286 s k\n\u22a2 \u2203 i, a \u2022 x + b \u2022 y \u2208 s i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "IsOpen.div_left", "start": [1335, 1], "end": [1337, 60], "traced_tactics": [{"tactic": "rw [\u2190 iUnion_div_left_image]", "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 : TopologicalGroup \u03b1\ns t : Set \u03b1\nht : IsOpen t\n\u22a2 IsOpen (s / t)", "state_after": "\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 : TopologicalGroup \u03b1\ns t : Set \u03b1\nht : IsOpen t\n\u22a2 IsOpen (\u22c3 (a : \u03b1) (_ : a \u2208 s), (fun x x_1 => x / x_1) a '' t)"}, {"tactic": "exact isOpen_biUnion fun a _ => isOpenMap_div_left a t ht", "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 : TopologicalGroup \u03b1\ns t : Set \u03b1\nht : IsOpen t\n\u22a2 IsOpen (\u22c3 (a : \u03b1) (_ : a \u2208 s), (fun x x_1 => x / x_1) a '' t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Cast/Field.lean", "full_name": "Nat.one_div_pos_of_nat", "start": [67, 1], "end": [69, 23], "traced_tactics": [{"tactic": "rw [one_div]", "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nn : \u2115\n\u22a2 0 < 1 / (\u2191n + 1)", "state_after": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nn : \u2115\n\u22a2 0 < (\u2191n + 1)\u207b\u00b9"}, {"tactic": "exact inv_pos_of_nat", "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nn : \u2115\n\u22a2 0 < (\u2191n + 1)\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Nonneg/Ring.lean", "full_name": "Nonneg.mk_add_mk", "start": [131, 1], "end": [134, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Sym/Sym2.lean", "full_name": "Sym2.coe_lift_symm_apply", "start": [199, 1], "end": [201, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Complex/Basic.lean", "full_name": "Complex.equivRealProd_apply_le", "start": [208, 1], "end": [209, 53], "traced_tactics": [{"tactic": "simp [Prod.norm_def, abs_re_le_abs, abs_im_le_abs]", "state_before": "E : Type ?u.223982\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\n\u22a2 \u2016\u2191equivRealProd z\u2016 \u2264 \u2191abs z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.case_strongInductionOn", "start": [827, 1], "end": [831, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.one_subset", "start": [97, 1], "end": [98, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "full_name": "Set.image_neg_Ici", "start": [291, 1], "end": [291, 63], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedAddCommGroup \u03b1\na b c : \u03b1\n\u22a2 Neg.neg '' Ici a = Iic (-a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Int/DivMod.lean", "full_name": "Int.ediv_eq_iff_eq_mul_left", "start": [767, 11], "end": [769, 61], "traced_tactics": [{"tactic": "rw [Int.mul_comm]", "state_before": "a b c : Int\nH : b \u2260 0\nH' : b \u2223 a\n\u22a2 a / b = c \u2194 a = c * b", "state_after": "a b c : Int\nH : b \u2260 0\nH' : b \u2223 a\n\u22a2 a / b = c \u2194 a = b * c"}, {"tactic": "exact Int.ediv_eq_iff_eq_mul_right H H'", "state_before": "a b c : Int\nH : b \u2260 0\nH' : b \u2223 a\n\u22a2 a / b = c \u2194 a = b * c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "full_name": "image_parallelepiped", "start": [56, 1], "end": [60, 93], "traced_tactics": [{"tactic": "simp only [parallelepiped, \u2190 image_comp]", "state_before": "\u03b9 : Type u_3\n\u03b9' : Type ?u.24139\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nf : E \u2192\u2097[\u211d] F\nv : \u03b9 \u2192 E\n\u22a2 \u2191f '' parallelepiped v = parallelepiped (\u2191f \u2218 v)", "state_after": "\u03b9 : Type u_3\n\u03b9' : Type ?u.24139\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nf : E \u2192\u2097[\u211d] F\nv : \u03b9 \u2192 E\n\u22a2 ((fun a => \u2191f a) \u2218 fun t => \u2211 i : \u03b9, t i \u2022 v i) '' Icc 0 1 = (fun a => \u2211 x : \u03b9, a x \u2022 (\u2191f \u2218 v) x) '' Icc 0 1"}, {"tactic": "congr 1 with t", "state_before": "\u03b9 : Type u_3\n\u03b9' : Type ?u.24139\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nf : E \u2192\u2097[\u211d] F\nv : \u03b9 \u2192 E\n\u22a2 ((fun a => \u2191f a) \u2218 fun t => \u2211 i : \u03b9, t i \u2022 v i) '' Icc 0 1 = (fun a => \u2211 x : \u03b9, a x \u2022 (\u2191f \u2218 v) x) '' Icc 0 1", "state_after": "case h\n\u03b9 : Type u_3\n\u03b9' : Type ?u.24139\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nf : E \u2192\u2097[\u211d] F\nv : \u03b9 \u2192 E\nt : F\n\u22a2 t \u2208 ((fun a => \u2191f a) \u2218 fun t => \u2211 i : \u03b9, t i \u2022 v i) '' Icc 0 1 \u2194 t \u2208 (fun a => \u2211 x : \u03b9, a x \u2022 (\u2191f \u2218 v) x) '' Icc 0 1"}, {"tactic": "simp only [Function.comp_apply, LinearMap.map_sum, LinearMap.map_smul\u209b\u2097, RingHom.id_apply]", "state_before": "case h\n\u03b9 : Type u_3\n\u03b9' : Type ?u.24139\nE : Type u_1\nF : Type u_2\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nf : E \u2192\u2097[\u211d] F\nv : \u03b9 \u2192 E\nt : F\n\u22a2 t \u2208 ((fun a => \u2191f a) \u2218 fun t => \u2211 i : \u03b9, t i \u2022 v i) '' Icc 0 1 \u2194 t \u2208 (fun a => \u2211 x : \u03b9, a x \u2022 (\u2191f \u2218 v) x) '' Icc 0 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean", "full_name": "Real.le_sin_mul", "start": [235, 1], "end": [238, 72], "traced_tactics": [{"tactic": "simpa [mul_comm x] using\n  strictConcaveOn_sin_Icc.concaveOn.2 \u27e8le_rfl, pi_pos.le\u27e9\n    \u27e8pi_div_two_pos.le, half_le_self pi_pos.le\u27e9 (sub_nonneg.2 hx') hx", "state_before": "x : \u211d\nhx : 0 \u2264 x\nhx' : x \u2264 1\n\u22a2 x \u2264 sin (\u03c0 / 2 * x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "full_name": "Qpf.recF_eq'", "start": [179, 1], "end": [181, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/Subfield.lean", "full_name": "Subfield.closure_le", "start": [719, 1], "end": [720, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Module/Injective.lean", "full_name": "Module.Baer.ExtensionOfMaxAdjoin.extendIdealTo_eq", "start": [353, 1], "end": [358, 83], "traced_tactics": [{"tactic": "simp only [ExtensionOfMaxAdjoin.extendIdealTo_is_extension i f h _ _ hr,\n  ExtensionOfMaxAdjoin.idealTo, LinearMap.coe_mk, Subtype.coe_mk, AddHom.coe_mk]", "state_before": "R : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nr : R\nhr : r \u2022 y \u2208 (extensionOfMax i f).toLinearPMap.domain\n\u22a2 \u2191(extendIdealTo i f h y) r = \u2191(extensionOfMax i f).toLinearPMap { val := r \u2022 y, property := hr }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Instances/Matrix.lean", "full_name": "Continuous.matrix_fromBlocks", "start": [244, 1], "end": [249, 83], "traced_tactics": [{"tactic": "rintro (i | i) (j | j) <;> refine' Continuous.matrix_elem _ i j <;> assumption", "state_before": "X : Type u_6\n\u03b1 : Type ?u.35471\nl : Type u_2\nm : Type u_4\nn : Type u_1\np : Type u_5\nS : Type ?u.35486\nR : Type u_3\nm' : l \u2192 Type ?u.35494\nn' : l \u2192 Type ?u.35499\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace R\nA : X \u2192 Matrix n l R\nB : X \u2192 Matrix n m R\nC : X \u2192 Matrix p l R\nD : X \u2192 Matrix p m R\nhA : Continuous A\nhB : Continuous B\nhC : Continuous C\nhD : Continuous D\n\u22a2 \u2200 (i : n \u2295 p) (j : l \u2295 m), Continuous fun a => fromBlocks (A a) (B a) (C a) (D a) i j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.Ico_subset_Iic_self", "start": [442, 1], "end": [443, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Equiv.lean", "full_name": "AlgEquiv.refl_toAlgHom", "start": [299, 1], "end": [300, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/GroupWithZero.lean", "full_name": "Filter.tendsto_mul_iff_of_ne_zero", "start": [169, 1], "end": [175, 69], "traced_tactics": [{"tactic": "refine' \u27e8fun hfg => _, fun hf => hf.mul hg\u27e9", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.26017\nG\u2080 : Type u_1\ninst\u271d\u2074 : GroupWithZero G\u2080\ninst\u271d\u00b3 : TopologicalSpace G\u2080\ninst\u271d\u00b2 : HasContinuousInv\u2080 G\u2080\ninst\u271d\u00b9 : ContinuousMul G\u2080\nf\u271d g\u271d : \u03b1 \u2192 G\u2080\ninst\u271d : T1Space G\u2080\nf g : \u03b1 \u2192 G\u2080\nl : Filter \u03b1\nx y : G\u2080\nhg : Tendsto g l (\ud835\udcdd y)\nhy : y \u2260 0\n\u22a2 Tendsto (fun n => f n * g n) l (\ud835\udcdd (x * y)) \u2194 Tendsto f l (\ud835\udcdd x)", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type ?u.26017\nG\u2080 : Type u_1\ninst\u271d\u2074 : GroupWithZero G\u2080\ninst\u271d\u00b3 : TopologicalSpace G\u2080\ninst\u271d\u00b2 : HasContinuousInv\u2080 G\u2080\ninst\u271d\u00b9 : ContinuousMul G\u2080\nf\u271d g\u271d : \u03b1 \u2192 G\u2080\ninst\u271d : T1Space G\u2080\nf g : \u03b1 \u2192 G\u2080\nl : Filter \u03b1\nx y : G\u2080\nhg : Tendsto g l (\ud835\udcdd y)\nhy : y \u2260 0\nhfg : Tendsto (fun n => f n * g n) l (\ud835\udcdd (x * y))\n\u22a2 Tendsto f l (\ud835\udcdd x)"}, {"tactic": "rw [\u2190 mul_div_cancel x hy]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.26017\nG\u2080 : Type u_1\ninst\u271d\u2074 : GroupWithZero G\u2080\ninst\u271d\u00b3 : TopologicalSpace G\u2080\ninst\u271d\u00b2 : HasContinuousInv\u2080 G\u2080\ninst\u271d\u00b9 : ContinuousMul G\u2080\nf\u271d g\u271d : \u03b1 \u2192 G\u2080\ninst\u271d : T1Space G\u2080\nf g : \u03b1 \u2192 G\u2080\nl : Filter \u03b1\nx y : G\u2080\nhg : Tendsto g l (\ud835\udcdd y)\nhy : y \u2260 0\nhfg : Tendsto (fun n => f n * g n) l (\ud835\udcdd (x * y))\n\u22a2 Tendsto f l (\ud835\udcdd x)", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type ?u.26017\nG\u2080 : Type u_1\ninst\u271d\u2074 : GroupWithZero G\u2080\ninst\u271d\u00b3 : TopologicalSpace G\u2080\ninst\u271d\u00b2 : HasContinuousInv\u2080 G\u2080\ninst\u271d\u00b9 : ContinuousMul G\u2080\nf\u271d g\u271d : \u03b1 \u2192 G\u2080\ninst\u271d : T1Space G\u2080\nf g : \u03b1 \u2192 G\u2080\nl : Filter \u03b1\nx y : G\u2080\nhg : Tendsto g l (\ud835\udcdd y)\nhy : y \u2260 0\nhfg : Tendsto (fun n => f n * g n) l (\ud835\udcdd (x * y))\n\u22a2 Tendsto f l (\ud835\udcdd (x * y / y))"}, {"tactic": "refine' Tendsto.congr' _ (hfg.div hg hy)", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.26017\nG\u2080 : Type u_1\ninst\u271d\u2074 : GroupWithZero G\u2080\ninst\u271d\u00b3 : TopologicalSpace G\u2080\ninst\u271d\u00b2 : HasContinuousInv\u2080 G\u2080\ninst\u271d\u00b9 : ContinuousMul G\u2080\nf\u271d g\u271d : \u03b1 \u2192 G\u2080\ninst\u271d : T1Space G\u2080\nf g : \u03b1 \u2192 G\u2080\nl : Filter \u03b1\nx y : G\u2080\nhg : Tendsto g l (\ud835\udcdd y)\nhy : y \u2260 0\nhfg : Tendsto (fun n => f n * g n) l (\ud835\udcdd (x * y))\n\u22a2 Tendsto f l (\ud835\udcdd (x * y / y))", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type ?u.26017\nG\u2080 : Type u_1\ninst\u271d\u2074 : GroupWithZero G\u2080\ninst\u271d\u00b3 : TopologicalSpace G\u2080\ninst\u271d\u00b2 : HasContinuousInv\u2080 G\u2080\ninst\u271d\u00b9 : ContinuousMul G\u2080\nf\u271d g\u271d : \u03b1 \u2192 G\u2080\ninst\u271d : T1Space G\u2080\nf g : \u03b1 \u2192 G\u2080\nl : Filter \u03b1\nx y : G\u2080\nhg : Tendsto g l (\ud835\udcdd y)\nhy : y \u2260 0\nhfg : Tendsto (fun n => f n * g n) l (\ud835\udcdd (x * y))\n\u22a2 (fun n => f n * g n) / g =\u1da0[l] f"}, {"tactic": "refine' (hg.eventually_ne hy).mono fun n hn => mul_div_cancel _ hn", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.26017\nG\u2080 : Type u_1\ninst\u271d\u2074 : GroupWithZero G\u2080\ninst\u271d\u00b3 : TopologicalSpace G\u2080\ninst\u271d\u00b2 : HasContinuousInv\u2080 G\u2080\ninst\u271d\u00b9 : ContinuousMul G\u2080\nf\u271d g\u271d : \u03b1 \u2192 G\u2080\ninst\u271d : T1Space G\u2080\nf g : \u03b1 \u2192 G\u2080\nl : Filter \u03b1\nx y : G\u2080\nhg : Tendsto g l (\ud835\udcdd y)\nhy : y \u2260 0\nhfg : Tendsto (fun n => f n * g n) l (\ud835\udcdd (x * y))\n\u22a2 (fun n => f n * g n) / g =\u1da0[l] f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/List.lean", "full_name": "Vector.continuous_removeNth", "start": [226, 1], "end": [228, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "full_name": "Matrix.TransvectionStruct.prod_mul_reverse_inv_prod", "start": [230, 1], "end": [239, 44], "traced_tactics": [{"tactic": "induction' L with t L IH", "state_before": "n : Type u_1\np : Type ?u.42156\nR : Type u\u2082\n\ud835\udd5c : Type ?u.42161\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\nL : List (TransvectionStruct n R)\n\u22a2 List.prod (List.map toMatrix L) \u2b1d List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L)) = 1", "state_after": "case nil\nn : Type u_1\np : Type ?u.42156\nR : Type u\u2082\n\ud835\udd5c : Type ?u.42161\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\n\u22a2 List.prod (List.map toMatrix []) \u2b1d List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse [])) = 1\n\ncase cons\nn : Type u_1\np : Type ?u.42156\nR : Type u\u2082\n\ud835\udd5c : Type ?u.42161\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\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : List.prod (List.map toMatrix L) \u2b1d List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L)) = 1\n\u22a2 List.prod (List.map toMatrix (t :: L)) \u2b1d\n      List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse (t :: L))) =\n    1"}, {"tactic": "simp", "state_before": "case nil\nn : Type u_1\np : Type ?u.42156\nR : Type u\u2082\n\ud835\udd5c : Type ?u.42161\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\n\u22a2 List.prod (List.map toMatrix []) \u2b1d List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse [])) = 1", "state_after": "no goals"}, {"tactic": "suffices\n  t.toMatrix \u2b1d\n        ((L.map toMatrix).prod \u2b1d (L.reverse.map (toMatrix \u2218 TransvectionStruct.inv)).prod) \u2b1d\n      t.inv.toMatrix = 1\n  by simpa [Matrix.mul_assoc]", "state_before": "case cons\nn : Type u_1\np : Type ?u.42156\nR : Type u\u2082\n\ud835\udd5c : Type ?u.42161\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\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : List.prod (List.map toMatrix L) \u2b1d List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L)) = 1\n\u22a2 List.prod (List.map toMatrix (t :: L)) \u2b1d\n      List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse (t :: L))) =\n    1", "state_after": "case cons\nn : Type u_1\np : Type ?u.42156\nR : Type u\u2082\n\ud835\udd5c : Type ?u.42161\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\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : List.prod (List.map toMatrix L) \u2b1d List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L)) = 1\n\u22a2 toMatrix t \u2b1d\n        (List.prod (List.map toMatrix L) \u2b1d List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L))) \u2b1d\n      toMatrix (TransvectionStruct.inv t) =\n    1"}, {"tactic": "simp_rw [IH, Matrix.mul_one, t.mul_inv]", "state_before": "case cons\nn : Type u_1\np : Type ?u.42156\nR : Type u\u2082\n\ud835\udd5c : Type ?u.42161\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\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : List.prod (List.map toMatrix L) \u2b1d List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L)) = 1\n\u22a2 toMatrix t \u2b1d\n        (List.prod (List.map toMatrix L) \u2b1d List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L))) \u2b1d\n      toMatrix (TransvectionStruct.inv t) =\n    1", "state_after": "no goals"}, {"tactic": "simpa [Matrix.mul_assoc]", "state_before": "n : Type u_1\np : Type ?u.42156\nR : Type u\u2082\n\ud835\udd5c : Type ?u.42161\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\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : List.prod (List.map toMatrix L) \u2b1d List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L)) = 1\nthis :\n  toMatrix t \u2b1d\n        (List.prod (List.map toMatrix L) \u2b1d List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L))) \u2b1d\n      toMatrix (TransvectionStruct.inv t) =\n    1\n\u22a2 List.prod (List.map toMatrix (t :: L)) \u2b1d\n      List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse (t :: L))) =\n    1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Complex/ReImTopology.lean", "full_name": "Complex.isOpenMap_im", "start": [60, 1], "end": [61, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Disjoint.lean", "full_name": "iUnion_Iic_eq_Iio_iSup", "start": [236, 1], "end": [238, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/HahnSeries.lean", "full_name": "HahnSeries.single_ne_zero", "start": [195, 1], "end": [196, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/LocalEquiv.lean", "full_name": "LocalEquiv.exists_mem_target", "start": [343, 1], "end": [344, 47], "traced_tactics": [{"tactic": "rw [\u2190 image_source_eq_target, bex_image_iff]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.25497\n\u03b4 : Type ?u.25500\ne : LocalEquiv \u03b1 \u03b2\ne' : LocalEquiv \u03b2 \u03b3\np : \u03b2 \u2192 Prop\n\u22a2 (\u2203 y, y \u2208 e.target \u2227 p y) \u2194 \u2203 x, x \u2208 e.source \u2227 p (\u2191e x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean", "full_name": "Continuous.rpow", "start": [306, 1], "end": [308, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/ModelTheory/Substructures.lean", "full_name": "FirstOrder.Language.Substructure.monotone_map", "start": [494, 1], "end": [495, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Quiver/Path.lean", "full_name": "Quiver.Path.toList_comp", "start": [175, 1], "end": [177, 52], "traced_tactics": [{"tactic": "simp", "state_before": "V : Type u\ninst\u271d : Quiver V\na b c d : V\np : Path a b\n\u22a2 toList (comp p nil) = toList nil ++ toList p", "state_after": "no goals"}, {"tactic": "simp [toList_comp]", "state_before": "V : Type u\ninst\u271d : Quiver V\na b c d\u271d : V\np : Path a b\nx\u271d d : V\nq : Path b d\na\u271d : d \u27f6 x\u271d\n\u22a2 toList (comp p (cons q a\u271d)) = toList (cons q a\u271d) ++ toList p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "full_name": "tsupport_fderiv_subset", "start": [1159, 1], "end": [1160, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Bounds/Basic.lean", "full_name": "IsLeast.unique", "start": [1072, 1], "end": [1073, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/TensorProduct.lean", "full_name": "TensorProduct.rid_tmul", "start": [677, 1], "end": [678, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Group/Prod.lean", "full_name": "MeasureTheory.quasiMeasurePreserving_div", "start": [477, 1], "end": [480, 94], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/FractionalIdeal.lean", "full_name": "IsFractional.sup", "start": [423, 1], "end": [433, 42], "traced_tactics": [{"tactic": "rcases mem_sup.mp hb with \u27e8bI, hbI, bJ, hbJ, rfl\u27e9", "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 : Submodule R P\naI : R\nhaI : aI \u2208 S\nhI : \u2200 (b : P), b \u2208 I \u2192 IsInteger R (aI \u2022 b)\naJ : R\nhaJ : aJ \u2208 S\nhJ : \u2200 (b : P), b \u2208 J \u2192 IsInteger R (aJ \u2022 b)\nb : P\nhb : b \u2208 I \u2294 J\n\u22a2 IsInteger R ((aI * aJ) \u2022 b)", "state_after": "case intro.intro.intro.intro\nR : 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 : Submodule R P\naI : R\nhaI : aI \u2208 S\nhI : \u2200 (b : P), b \u2208 I \u2192 IsInteger R (aI \u2022 b)\naJ : R\nhaJ : aJ \u2208 S\nhJ : \u2200 (b : P), b \u2208 J \u2192 IsInteger R (aJ \u2022 b)\nbI : P\nhbI : bI \u2208 I\nbJ : P\nhbJ : bJ \u2208 J\nhb : bI + bJ \u2208 I \u2294 J\n\u22a2 IsInteger R ((aI * aJ) \u2022 (bI + bJ))"}, {"tactic": "rw [smul_add]", "state_before": "case intro.intro.intro.intro\nR : 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 : Submodule R P\naI : R\nhaI : aI \u2208 S\nhI : \u2200 (b : P), b \u2208 I \u2192 IsInteger R (aI \u2022 b)\naJ : R\nhaJ : aJ \u2208 S\nhJ : \u2200 (b : P), b \u2208 J \u2192 IsInteger R (aJ \u2022 b)\nbI : P\nhbI : bI \u2208 I\nbJ : P\nhbJ : bJ \u2208 J\nhb : bI + bJ \u2208 I \u2294 J\n\u22a2 IsInteger R ((aI * aJ) \u2022 (bI + bJ))", "state_after": "case intro.intro.intro.intro\nR : 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 : Submodule R P\naI : R\nhaI : aI \u2208 S\nhI : \u2200 (b : P), b \u2208 I \u2192 IsInteger R (aI \u2022 b)\naJ : R\nhaJ : aJ \u2208 S\nhJ : \u2200 (b : P), b \u2208 J \u2192 IsInteger R (aJ \u2022 b)\nbI : P\nhbI : bI \u2208 I\nbJ : P\nhbJ : bJ \u2208 J\nhb : bI + bJ \u2208 I \u2294 J\n\u22a2 IsInteger R ((aI * aJ) \u2022 bI + (aI * aJ) \u2022 bJ)"}, {"tactic": "apply isInteger_add", "state_before": "case intro.intro.intro.intro\nR : 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 : Submodule R P\naI : R\nhaI : aI \u2208 S\nhI : \u2200 (b : P), b \u2208 I \u2192 IsInteger R (aI \u2022 b)\naJ : R\nhaJ : aJ \u2208 S\nhJ : \u2200 (b : P), b \u2208 J \u2192 IsInteger R (aJ \u2022 b)\nbI : P\nhbI : bI \u2208 I\nbJ : P\nhbJ : bJ \u2208 J\nhb : bI + bJ \u2208 I \u2294 J\n\u22a2 IsInteger R ((aI * aJ) \u2022 bI + (aI * aJ) \u2022 bJ)", "state_after": "case intro.intro.intro.intro.ha\nR : 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 : Submodule R P\naI : R\nhaI : aI \u2208 S\nhI : \u2200 (b : P), b \u2208 I \u2192 IsInteger R (aI \u2022 b)\naJ : R\nhaJ : aJ \u2208 S\nhJ : \u2200 (b : P), b \u2208 J \u2192 IsInteger R (aJ \u2022 b)\nbI : P\nhbI : bI \u2208 I\nbJ : P\nhbJ : bJ \u2208 J\nhb : bI + bJ \u2208 I \u2294 J\n\u22a2 IsInteger R ((aI * aJ) \u2022 bI)\n\ncase intro.intro.intro.intro.hb\nR : 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 : Submodule R P\naI : R\nhaI : aI \u2208 S\nhI : \u2200 (b : P), b \u2208 I \u2192 IsInteger R (aI \u2022 b)\naJ : R\nhaJ : aJ \u2208 S\nhJ : \u2200 (b : P), b \u2208 J \u2192 IsInteger R (aJ \u2022 b)\nbI : P\nhbI : bI \u2208 I\nbJ : P\nhbJ : bJ \u2208 J\nhb : bI + bJ \u2208 I \u2294 J\n\u22a2 IsInteger R ((aI * aJ) \u2022 bJ)"}, {"tactic": "rw [mul_smul, smul_comm]", "state_before": "case intro.intro.intro.intro.ha\nR : 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 : Submodule R P\naI : R\nhaI : aI \u2208 S\nhI : \u2200 (b : P), b \u2208 I \u2192 IsInteger R (aI \u2022 b)\naJ : R\nhaJ : aJ \u2208 S\nhJ : \u2200 (b : P), b \u2208 J \u2192 IsInteger R (aJ \u2022 b)\nbI : P\nhbI : bI \u2208 I\nbJ : P\nhbJ : bJ \u2208 J\nhb : bI + bJ \u2208 I \u2294 J\n\u22a2 IsInteger R ((aI * aJ) \u2022 bI)", "state_after": "case intro.intro.intro.intro.ha\nR : 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 : Submodule R P\naI : R\nhaI : aI \u2208 S\nhI : \u2200 (b : P), b \u2208 I \u2192 IsInteger R (aI \u2022 b)\naJ : R\nhaJ : aJ \u2208 S\nhJ : \u2200 (b : P), b \u2208 J \u2192 IsInteger R (aJ \u2022 b)\nbI : P\nhbI : bI \u2208 I\nbJ : P\nhbJ : bJ \u2208 J\nhb : bI + bJ \u2208 I \u2294 J\n\u22a2 IsInteger R (aJ \u2022 aI \u2022 bI)"}, {"tactic": "exact isInteger_smul (hI bI hbI)", "state_before": "case intro.intro.intro.intro.ha\nR : 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 : Submodule R P\naI : R\nhaI : aI \u2208 S\nhI : \u2200 (b : P), b \u2208 I \u2192 IsInteger R (aI \u2022 b)\naJ : R\nhaJ : aJ \u2208 S\nhJ : \u2200 (b : P), b \u2208 J \u2192 IsInteger R (aJ \u2022 b)\nbI : P\nhbI : bI \u2208 I\nbJ : P\nhbJ : bJ \u2208 J\nhb : bI + bJ \u2208 I \u2294 J\n\u22a2 IsInteger R (aJ \u2022 aI \u2022 bI)", "state_after": "no goals"}, {"tactic": "rw [mul_smul]", "state_before": "case intro.intro.intro.intro.hb\nR : 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 : Submodule R P\naI : R\nhaI : aI \u2208 S\nhI : \u2200 (b : P), b \u2208 I \u2192 IsInteger R (aI \u2022 b)\naJ : R\nhaJ : aJ \u2208 S\nhJ : \u2200 (b : P), b \u2208 J \u2192 IsInteger R (aJ \u2022 b)\nbI : P\nhbI : bI \u2208 I\nbJ : P\nhbJ : bJ \u2208 J\nhb : bI + bJ \u2208 I \u2294 J\n\u22a2 IsInteger R ((aI * aJ) \u2022 bJ)", "state_after": "case intro.intro.intro.intro.hb\nR : 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 : Submodule R P\naI : R\nhaI : aI \u2208 S\nhI : \u2200 (b : P), b \u2208 I \u2192 IsInteger R (aI \u2022 b)\naJ : R\nhaJ : aJ \u2208 S\nhJ : \u2200 (b : P), b \u2208 J \u2192 IsInteger R (aJ \u2022 b)\nbI : P\nhbI : bI \u2208 I\nbJ : P\nhbJ : bJ \u2208 J\nhb : bI + bJ \u2208 I \u2294 J\n\u22a2 IsInteger R (aI \u2022 aJ \u2022 bJ)"}, {"tactic": "exact isInteger_smul (hJ bJ hbJ)", "state_before": "case intro.intro.intro.intro.hb\nR : 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 : Submodule R P\naI : R\nhaI : aI \u2208 S\nhI : \u2200 (b : P), b \u2208 I \u2192 IsInteger R (aI \u2022 b)\naJ : R\nhaJ : aJ \u2208 S\nhJ : \u2200 (b : P), b \u2208 J \u2192 IsInteger R (aJ \u2022 b)\nbI : P\nhbI : bI \u2208 I\nbJ : P\nhbJ : bJ \u2208 J\nhb : bI + bJ \u2208 I \u2294 J\n\u22a2 IsInteger R (aI \u2022 aJ \u2022 bJ)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "IsGLB.exists_seq_antitone_tendsto", "start": [2165, 1], "end": [2168, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Connected.lean", "full_name": "locallyConnectedSpace_iff_open_connected_subsets", "start": [1120, 1], "end": [1131, 87], "traced_tactics": [{"tactic": "simp_rw [locallyConnectedSpace_iff_open_connected_basis]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.112377\n\u03c0 : \u03b9 \u2192 Type ?u.112382\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 LocallyConnectedSpace \u03b1 \u2194 \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x \u2208 V \u2227 IsConnected V", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.112377\n\u03c0 : \u03b9 \u2192 Type ?u.112382\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 (\u2200 (x : \u03b1), Filter.HasBasis (\ud835\udcdd x) (fun s => IsOpen s \u2227 x \u2208 s \u2227 IsConnected s) id) \u2194\n    \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x \u2208 V \u2227 IsConnected V"}, {"tactic": "refine forall_congr' fun _ => ?_", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.112377\n\u03c0 : \u03b9 \u2192 Type ?u.112382\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 (\u2200 (x : \u03b1), Filter.HasBasis (\ud835\udcdd x) (fun s => IsOpen s \u2227 x \u2208 s \u2227 IsConnected s) id) \u2194\n    \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x \u2208 V \u2227 IsConnected V", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.112377\n\u03c0 : \u03b9 \u2192 Type ?u.112382\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx\u271d : \u03b1\n\u22a2 Filter.HasBasis (\ud835\udcdd x\u271d) (fun s => IsOpen s \u2227 x\u271d \u2208 s \u2227 IsConnected s) id \u2194\n    \u2200 (U : Set \u03b1), U \u2208 \ud835\udcdd x\u271d \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x\u271d \u2208 V \u2227 IsConnected V"}, {"tactic": "constructor", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.112377\n\u03c0 : \u03b9 \u2192 Type ?u.112382\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx\u271d : \u03b1\n\u22a2 Filter.HasBasis (\ud835\udcdd x\u271d) (fun s => IsOpen s \u2227 x\u271d \u2208 s \u2227 IsConnected s) id \u2194\n    \u2200 (U : Set \u03b1), U \u2208 \ud835\udcdd x\u271d \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x\u271d \u2208 V \u2227 IsConnected V", "state_after": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.112377\n\u03c0 : \u03b9 \u2192 Type ?u.112382\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx\u271d : \u03b1\n\u22a2 Filter.HasBasis (\ud835\udcdd x\u271d) (fun s => IsOpen s \u2227 x\u271d \u2208 s \u2227 IsConnected s) id \u2192\n    \u2200 (U : Set \u03b1), U \u2208 \ud835\udcdd x\u271d \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x\u271d \u2208 V \u2227 IsConnected V\n\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.112377\n\u03c0 : \u03b9 \u2192 Type ?u.112382\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx\u271d : \u03b1\n\u22a2 (\u2200 (U : Set \u03b1), U \u2208 \ud835\udcdd x\u271d \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x\u271d \u2208 V \u2227 IsConnected V) \u2192\n    Filter.HasBasis (\ud835\udcdd x\u271d) (fun s => IsOpen s \u2227 x\u271d \u2208 s \u2227 IsConnected s) id"}, {"tactic": "intro h U hU", "state_before": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.112377\n\u03c0 : \u03b9 \u2192 Type ?u.112382\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx\u271d : \u03b1\n\u22a2 Filter.HasBasis (\ud835\udcdd x\u271d) (fun s => IsOpen s \u2227 x\u271d \u2208 s \u2227 IsConnected s) id \u2192\n    \u2200 (U : Set \u03b1), U \u2208 \ud835\udcdd x\u271d \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x\u271d \u2208 V \u2227 IsConnected V", "state_after": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.112377\n\u03c0 : \u03b9 \u2192 Type ?u.112382\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx\u271d : \u03b1\nh : Filter.HasBasis (\ud835\udcdd x\u271d) (fun s => IsOpen s \u2227 x\u271d \u2208 s \u2227 IsConnected s) id\nU : Set \u03b1\nhU : U \u2208 \ud835\udcdd x\u271d\n\u22a2 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x\u271d \u2208 V \u2227 IsConnected V"}, {"tactic": "rcases h.mem_iff.mp hU with \u27e8V, hV, hVU\u27e9", "state_before": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.112377\n\u03c0 : \u03b9 \u2192 Type ?u.112382\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx\u271d : \u03b1\nh : Filter.HasBasis (\ud835\udcdd x\u271d) (fun s => IsOpen s \u2227 x\u271d \u2208 s \u2227 IsConnected s) id\nU : Set \u03b1\nhU : U \u2208 \ud835\udcdd x\u271d\n\u22a2 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x\u271d \u2208 V \u2227 IsConnected V", "state_after": "case mp.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.112377\n\u03c0 : \u03b9 \u2192 Type ?u.112382\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx\u271d : \u03b1\nh : Filter.HasBasis (\ud835\udcdd x\u271d) (fun s => IsOpen s \u2227 x\u271d \u2208 s \u2227 IsConnected s) id\nU : Set \u03b1\nhU : U \u2208 \ud835\udcdd x\u271d\nV : Set \u03b1\nhV : IsOpen V \u2227 x\u271d \u2208 V \u2227 IsConnected V\nhVU : id V \u2286 U\n\u22a2 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x\u271d \u2208 V \u2227 IsConnected V"}, {"tactic": "exact \u27e8V, hVU, hV\u27e9", "state_before": "case mp.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.112377\n\u03c0 : \u03b9 \u2192 Type ?u.112382\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx\u271d : \u03b1\nh : Filter.HasBasis (\ud835\udcdd x\u271d) (fun s => IsOpen s \u2227 x\u271d \u2208 s \u2227 IsConnected s) id\nU : Set \u03b1\nhU : U \u2208 \ud835\udcdd x\u271d\nV : Set \u03b1\nhV : IsOpen V \u2227 x\u271d \u2208 V \u2227 IsConnected V\nhVU : id V \u2286 U\n\u22a2 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x\u271d \u2208 V \u2227 IsConnected V", "state_after": "no goals"}, {"tactic": "exact fun h => \u27e8fun U => \u27e8fun hU =>\n  let \u27e8V, hVU, hV\u27e9 := h U hU\n  \u27e8V, hV, hVU\u27e9, fun \u27e8V, \u27e8hV, hxV, _\u27e9, hVU\u27e9 => mem_nhds_iff.mpr \u27e8V, hVU, hV, hxV\u27e9\u27e9\u27e9", "state_before": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.112377\n\u03c0 : \u03b9 \u2192 Type ?u.112382\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx\u271d : \u03b1\n\u22a2 (\u2200 (U : Set \u03b1), U \u2208 \ud835\udcdd x\u271d \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x\u271d \u2208 V \u2227 IsConnected V) \u2192\n    Filter.HasBasis (\ud835\udcdd x\u271d) (fun s => IsOpen s \u2227 x\u271d \u2208 s \u2227 IsConnected s) id", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/UniformConvergence.lean", "full_name": "tendsto_prod_principal_iff", "start": [333, 1], "end": [336, 32], "traced_tactics": [{"tactic": "rw [tendstoUniformlyOn_iff_tendstoUniformlyOnFilter]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : 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\nc : \u03b2\n\u22a2 Tendsto (\u21bfF) (p \u00d7\u02e2 \ud835\udcdf s) (\ud835\udcdd c) \u2194 TendstoUniformlyOn F (fun x => c) p s", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : 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\nc : \u03b2\n\u22a2 Tendsto (\u21bfF) (p \u00d7\u02e2 \ud835\udcdf s) (\ud835\udcdd c) \u2194 TendstoUniformlyOnFilter F (fun x => c) p (\ud835\udcdf s)"}, {"tactic": "exact tendsto_prod_filter_iff", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : 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\nc : \u03b2\n\u22a2 Tendsto (\u21bfF) (p \u00d7\u02e2 \ud835\udcdf s) (\ud835\udcdd c) \u2194 TendstoUniformlyOnFilter F (fun x => c) p (\ud835\udcdf s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Sigma.lean", "full_name": "List.dlookup_isSome", "start": [192, 1], "end": [198, 31], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\n\u22a2 Option.isSome (dlookup a []) = true \u2194 a \u2208 keys []", "state_after": "no goals"}, {"tactic": "by_cases h : a = a'", "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 a' : \u03b1\nb : \u03b2 a'\nl : List (Sigma \u03b2)\n\u22a2 Option.isSome (dlookup a ({ fst := a', snd := b } :: l)) = true \u2194 a \u2208 keys ({ fst := a', snd := b } :: l)", "state_after": "case pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a'\nl : List (Sigma \u03b2)\nh : a = a'\n\u22a2 Option.isSome (dlookup a ({ fst := a', snd := b } :: l)) = true \u2194 a \u2208 keys ({ fst := a', snd := b } :: l)\n\ncase neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a'\nl : List (Sigma \u03b2)\nh : \u00aca = a'\n\u22a2 Option.isSome (dlookup a ({ fst := a', snd := b } :: l)) = true \u2194 a \u2208 keys ({ fst := a', snd := b } :: l)"}, {"tactic": "subst a'", "state_before": "case pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a'\nl : List (Sigma \u03b2)\nh : a = a'\n\u22a2 Option.isSome (dlookup a ({ fst := a', snd := b } :: l)) = true \u2194 a \u2208 keys ({ fst := a', snd := b } :: l)", "state_after": "case pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nb : \u03b2 a\n\u22a2 Option.isSome (dlookup a ({ fst := a, snd := b } :: l)) = true \u2194 a \u2208 keys ({ fst := a, snd := b } :: l)"}, {"tactic": "simp", "state_before": "case pos\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nb : \u03b2 a\n\u22a2 Option.isSome (dlookup a ({ fst := a, snd := b } :: l)) = true \u2194 a \u2208 keys ({ fst := a, snd := b } :: l)", "state_after": "no goals"}, {"tactic": "simp [h, dlookup_isSome]", "state_before": "case neg\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na a' : \u03b1\nb : \u03b2 a'\nl : List (Sigma \u03b2)\nh : \u00aca = a'\n\u22a2 Option.isSome (dlookup a ({ fst := a', snd := b } :: l)) = true \u2194 a \u2208 keys ({ fst := a', snd := b } :: l)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicTopology/DoldKan/NReflectsIso.lean", "full_name": "AlgebraicTopology.DoldKan.compatibility_N\u2082_N\u2081_karoubi", "start": [69, 1], "end": [93, 66], "traced_tactics": [{"tactic": "refine' CategoryTheory.Functor.ext (fun P => _) fun P Q f => _", "state_before": "C : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\n\u22a2 N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor =\n    karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n      N\u2081 \u22d9\n        (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n          Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)", "state_after": "case refine'_1\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\n\u22a2 (N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P =\n    (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n          N\u2081 \u22d9\n            (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n              Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n      P\n\ncase refine'_2\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP Q : Karoubi (SimplicialObject C)\nf : P \u27f6 Q\n\u22a2 (N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).map f =\n    eqToHom\n        (_ :\n          (N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P =\n            (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                  N\u2081 \u22d9\n                    (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                      Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n              P) \u226b\n      (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n              N\u2081 \u22d9\n                (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                  Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).map\n          f \u226b\n        eqToHom\n          (_ :\n            (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                    N\u2081 \u22d9\n                      (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                        Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n                Q =\n              (N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj Q)"}, {"tactic": "refine' HomologicalComplex.ext _ _", "state_before": "case refine'_1\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\n\u22a2 (N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P =\n    (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n          N\u2081 \u22d9\n            (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n              Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n      P", "state_after": "case refine'_1.refine'_1\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\n\u22a2 ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P).X =\n    ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n            N\u2081 \u22d9\n              (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n        P).X\n\ncase refine'_1.refine'_2\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\n\u22a2 \u2200 (i j : \u2115),\n    ComplexShape.Rel (ComplexShape.down \u2115) i j \u2192\n      HomologicalComplex.d ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) i j \u226b\n          eqToHom\n            (_ :\n              HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) j =\n                HomologicalComplex.X\n                  ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                        N\u2081 \u22d9\n                          (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                            Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n                              (ComplexShape.down \u2115)).obj\n                    P)\n                  j) =\n        eqToHom\n            (_ :\n              HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) i =\n                HomologicalComplex.X\n                  ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                        N\u2081 \u22d9\n                          (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                            Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n                              (ComplexShape.down \u2115)).obj\n                    P)\n                  i) \u226b\n          HomologicalComplex.d\n            ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                  N\u2081 \u22d9\n                    (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                      Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n              P)\n            i j"}, {"tactic": "ext n", "state_before": "case refine'_1.refine'_1\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\n\u22a2 ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P).X =\n    ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n            N\u2081 \u22d9\n              (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n        P).X", "state_after": "case refine'_1.refine'_1.h.h_p\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : \u2115\n\u22a2 (HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) n).p \u226b\n      eqToHom ?refine'_1.refine'_1.h.h_X =\n    eqToHom ?refine'_1.refine'_1.h.h_X \u226b\n      (HomologicalComplex.X\n          ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                N\u2081 \u22d9\n                  (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                    Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n            P)\n          n).p\n\ncase refine'_1.refine'_1.h.h_X\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : \u2115\n\u22a2 (HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) n).X =\n    (HomologicalComplex.X\n        ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n              N\u2081 \u22d9\n                (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                  Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n          P)\n        n).X"}, {"tactic": ". rfl", "state_before": "case refine'_1.refine'_1.h.h_X\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : \u2115\n\u22a2 (HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) n).X =\n    (HomologicalComplex.X\n        ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n              N\u2081 \u22d9\n                (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                  Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n          P)\n        n).X", "state_after": "no goals"}, {"tactic": "dsimp", "state_before": "case refine'_1.refine'_1.h.h_p\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : \u2115\n\u22a2 (HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) n).p \u226b\n      eqToHom ?refine'_1.refine'_1.h.h_X =\n    eqToHom ?refine'_1.refine'_1.h.h_X \u226b\n      (HomologicalComplex.X\n          ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                N\u2081 \u22d9\n                  (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                    Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n            P)\n          n).p", "state_after": "case refine'_1.refine'_1.h.h_p\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : \u2115\n\u22a2 (HomologicalComplex.Hom.f PInfty n \u226b P.p.app [n].op) \u226b eqToHom ?refine'_1.refine'_1.h.h_X =\n    eqToHom ?refine'_1.refine'_1.h.h_X \u226b (HomologicalComplex.Hom.f PInfty n).f"}, {"tactic": "simp only [karoubi_PInfty_f, comp_id, PInfty_f_naturality, id_comp, eqToHom_refl]", "state_before": "case refine'_1.refine'_1.h.h_p\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : \u2115\n\u22a2 (HomologicalComplex.Hom.f PInfty n \u226b P.p.app [n].op) \u226b eqToHom ?refine'_1.refine'_1.h.h_X =\n    eqToHom ?refine'_1.refine'_1.h.h_X \u226b (HomologicalComplex.Hom.f PInfty n).f", "state_after": "no goals"}, {"tactic": "rfl", "state_before": "case refine'_1.refine'_1.h.h_X\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : \u2115\n\u22a2 (HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) n).X =\n    (HomologicalComplex.X\n        ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n              N\u2081 \u22d9\n                (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                  Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n          P)\n        n).X", "state_after": "no goals"}, {"tactic": "rintro _ n (rfl : n + 1 = _)", "state_before": "case refine'_1.refine'_2\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\n\u22a2 \u2200 (i j : \u2115),\n    ComplexShape.Rel (ComplexShape.down \u2115) i j \u2192\n      HomologicalComplex.d ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) i j \u226b\n          eqToHom\n            (_ :\n              HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) j =\n                HomologicalComplex.X\n                  ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                        N\u2081 \u22d9\n                          (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                            Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n                              (ComplexShape.down \u2115)).obj\n                    P)\n                  j) =\n        eqToHom\n            (_ :\n              HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) i =\n                HomologicalComplex.X\n                  ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                        N\u2081 \u22d9\n                          (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                            Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n                              (ComplexShape.down \u2115)).obj\n                    P)\n                  i) \u226b\n          HomologicalComplex.d\n            ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                  N\u2081 \u22d9\n                    (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                      Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n              P)\n            i j", "state_after": "case refine'_1.refine'_2\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : \u2115\n\u22a2 HomologicalComplex.d ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) (n + 1) n \u226b\n      eqToHom\n        (_ :\n          HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) n =\n            HomologicalComplex.X\n              ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                    N\u2081 \u22d9\n                      (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                        Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n                P)\n              n) =\n    eqToHom\n        (_ :\n          HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) (n + 1) =\n            HomologicalComplex.X\n              ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                    N\u2081 \u22d9\n                      (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                        Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n                P)\n              (n + 1)) \u226b\n      HomologicalComplex.d\n        ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n              N\u2081 \u22d9\n                (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                  Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n          P)\n        (n + 1) n"}, {"tactic": "ext", "state_before": "case refine'_1.refine'_2\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : \u2115\n\u22a2 HomologicalComplex.d ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) (n + 1) n \u226b\n      eqToHom\n        (_ :\n          HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) n =\n            HomologicalComplex.X\n              ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                    N\u2081 \u22d9\n                      (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                        Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n                P)\n              n) =\n    eqToHom\n        (_ :\n          HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) (n + 1) =\n            HomologicalComplex.X\n              ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                    N\u2081 \u22d9\n                      (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                        Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n                P)\n              (n + 1)) \u226b\n      HomologicalComplex.d\n        ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n              N\u2081 \u22d9\n                (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                  Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n          P)\n        (n + 1) n", "state_after": "case refine'_1.refine'_2.h\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : \u2115\n\u22a2 (HomologicalComplex.d ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) (n + 1) n \u226b\n        eqToHom\n          (_ :\n            HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) n =\n              HomologicalComplex.X\n                ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                      N\u2081 \u22d9\n                        (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                          Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n                            (ComplexShape.down \u2115)).obj\n                  P)\n                n)).f =\n    (eqToHom\n          (_ :\n            HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) (n + 1) =\n              HomologicalComplex.X\n                ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                      N\u2081 \u22d9\n                        (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                          Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n                            (ComplexShape.down \u2115)).obj\n                  P)\n                (n + 1)) \u226b\n        HomologicalComplex.d\n          ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                N\u2081 \u22d9\n                  (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                    Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n            P)\n          (n + 1) n).f"}, {"tactic": "have h := (AlternatingFaceMapComplex.map P.p).comm (n + 1) n", "state_before": "case refine'_1.refine'_2.h\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : \u2115\n\u22a2 (HomologicalComplex.d ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) (n + 1) n \u226b\n        eqToHom\n          (_ :\n            HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) n =\n              HomologicalComplex.X\n                ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                      N\u2081 \u22d9\n                        (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                          Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n                            (ComplexShape.down \u2115)).obj\n                  P)\n                n)).f =\n    (eqToHom\n          (_ :\n            HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) (n + 1) =\n              HomologicalComplex.X\n                ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                      N\u2081 \u22d9\n                        (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                          Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n                            (ComplexShape.down \u2115)).obj\n                  P)\n                (n + 1)) \u226b\n        HomologicalComplex.d\n          ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                N\u2081 \u22d9\n                  (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                    Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n            P)\n          (n + 1) n).f", "state_after": "case refine'_1.refine'_2.h\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : \u2115\nh :\n  HomologicalComplex.Hom.f (AlternatingFaceMapComplex.map P.p) (n + 1) \u226b\n      HomologicalComplex.d (AlternatingFaceMapComplex.obj P.X) (n + 1) n =\n    HomologicalComplex.d (AlternatingFaceMapComplex.obj P.X) (n + 1) n \u226b\n      HomologicalComplex.Hom.f (AlternatingFaceMapComplex.map P.p) n\n\u22a2 (HomologicalComplex.d ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) (n + 1) n \u226b\n        eqToHom\n          (_ :\n            HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) n =\n              HomologicalComplex.X\n                ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                      N\u2081 \u22d9\n                        (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                          Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n                            (ComplexShape.down \u2115)).obj\n                  P)\n                n)).f =\n    (eqToHom\n          (_ :\n            HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) (n + 1) =\n              HomologicalComplex.X\n                ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                      N\u2081 \u22d9\n                        (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                          Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n                            (ComplexShape.down \u2115)).obj\n                  P)\n                (n + 1)) \u226b\n        HomologicalComplex.d\n          ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                N\u2081 \u22d9\n                  (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                    Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n            P)\n          (n + 1) n).f"}, {"tactic": "dsimp [N\u2082, karoubiChainComplexEquivalence,\n  KaroubiHomologicalComplexEquivalence.Functor.obj] at h \u22a2", "state_before": "case refine'_1.refine'_2.h\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : \u2115\nh :\n  HomologicalComplex.Hom.f (AlternatingFaceMapComplex.map P.p) (n + 1) \u226b\n      HomologicalComplex.d (AlternatingFaceMapComplex.obj P.X) (n + 1) n =\n    HomologicalComplex.d (AlternatingFaceMapComplex.obj P.X) (n + 1) n \u226b\n      HomologicalComplex.Hom.f (AlternatingFaceMapComplex.map P.p) n\n\u22a2 (HomologicalComplex.d ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) (n + 1) n \u226b\n        eqToHom\n          (_ :\n            HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) n =\n              HomologicalComplex.X\n                ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                      N\u2081 \u22d9\n                        (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                          Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n                            (ComplexShape.down \u2115)).obj\n                  P)\n                n)).f =\n    (eqToHom\n          (_ :\n            HomologicalComplex.X ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P) (n + 1) =\n              HomologicalComplex.X\n                ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                      N\u2081 \u22d9\n                        (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                          Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n                            (ComplexShape.down \u2115)).obj\n                  P)\n                (n + 1)) \u226b\n        HomologicalComplex.d\n          ((karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                N\u2081 \u22d9\n                  (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                    Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n            P)\n          (n + 1) n).f", "state_after": "case refine'_1.refine'_2.h\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : \u2115\nh :\n  P.p.app [n + 1].op \u226b HomologicalComplex.d (AlternatingFaceMapComplex.obj P.X) (n + 1) n =\n    HomologicalComplex.d (AlternatingFaceMapComplex.obj P.X) (n + 1) n \u226b P.p.app [n].op\n\u22a2 ((HomologicalComplex.Hom.f PInfty (n + 1) \u226b P.p.app [n + 1].op) \u226b\n        HomologicalComplex.d (AlternatingFaceMapComplex.obj P.X) (n + 1) n) \u226b\n      (eqToHom\n          (_ :\n            Karoubi.mk (P.X.obj [n].op) (HomologicalComplex.Hom.f PInfty n \u226b P.p.app [n].op) =\n              HomologicalComplex.X\n                ((Functor.mapHomologicalComplex (KaroubiKaroubi.inverse C) (ComplexShape.down \u2115)).obj\n                  (HomologicalComplex.mk\n                    (fun n =>\n                      Karoubi.mk ((KaroubiFunctorCategoryEmbedding.obj P).obj [n].op)\n                        (HomologicalComplex.Hom.f PInfty n))\n                    fun i j =>\n                    Karoubi.Hom.mk\n                      (HomologicalComplex.Hom.f PInfty i \u226b\n                        HomologicalComplex.d (AlternatingFaceMapComplex.obj (KaroubiFunctorCategoryEmbedding.obj P)) i\n                          j)))\n                n)).f =\n    (eqToHom\n          (_ :\n            Karoubi.mk (P.X.obj [n + 1].op) (HomologicalComplex.Hom.f PInfty (n + 1) \u226b P.p.app [n + 1].op) =\n              HomologicalComplex.X\n                ((Functor.mapHomologicalComplex (KaroubiKaroubi.inverse C) (ComplexShape.down \u2115)).obj\n                  (HomologicalComplex.mk\n                    (fun n =>\n                      Karoubi.mk ((KaroubiFunctorCategoryEmbedding.obj P).obj [n].op)\n                        (HomologicalComplex.Hom.f PInfty n))\n                    fun i j =>\n                    Karoubi.Hom.mk\n                      (HomologicalComplex.Hom.f PInfty i \u226b\n                        HomologicalComplex.d (AlternatingFaceMapComplex.obj (KaroubiFunctorCategoryEmbedding.obj P)) i\n                          j)))\n                (n + 1))).f \u226b\n      (HomologicalComplex.Hom.f PInfty (n + 1)).f \u226b\n        (HomologicalComplex.d (AlternatingFaceMapComplex.obj (KaroubiFunctorCategoryEmbedding.obj P)) (n + 1) n).f"}, {"tactic": "simp only [assoc, Karoubi.eqToHom_f, eqToHom_refl, comp_id,\n  karoubi_alternatingFaceMapComplex_d, karoubi_PInfty_f,\n  \u2190 HomologicalComplex.Hom.comm_assoc, \u2190 h, app_idem_assoc]", "state_before": "case refine'_1.refine'_2.h\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : \u2115\nh :\n  P.p.app [n + 1].op \u226b HomologicalComplex.d (AlternatingFaceMapComplex.obj P.X) (n + 1) n =\n    HomologicalComplex.d (AlternatingFaceMapComplex.obj P.X) (n + 1) n \u226b P.p.app [n].op\n\u22a2 ((HomologicalComplex.Hom.f PInfty (n + 1) \u226b P.p.app [n + 1].op) \u226b\n        HomologicalComplex.d (AlternatingFaceMapComplex.obj P.X) (n + 1) n) \u226b\n      (eqToHom\n          (_ :\n            Karoubi.mk (P.X.obj [n].op) (HomologicalComplex.Hom.f PInfty n \u226b P.p.app [n].op) =\n              HomologicalComplex.X\n                ((Functor.mapHomologicalComplex (KaroubiKaroubi.inverse C) (ComplexShape.down \u2115)).obj\n                  (HomologicalComplex.mk\n                    (fun n =>\n                      Karoubi.mk ((KaroubiFunctorCategoryEmbedding.obj P).obj [n].op)\n                        (HomologicalComplex.Hom.f PInfty n))\n                    fun i j =>\n                    Karoubi.Hom.mk\n                      (HomologicalComplex.Hom.f PInfty i \u226b\n                        HomologicalComplex.d (AlternatingFaceMapComplex.obj (KaroubiFunctorCategoryEmbedding.obj P)) i\n                          j)))\n                n)).f =\n    (eqToHom\n          (_ :\n            Karoubi.mk (P.X.obj [n + 1].op) (HomologicalComplex.Hom.f PInfty (n + 1) \u226b P.p.app [n + 1].op) =\n              HomologicalComplex.X\n                ((Functor.mapHomologicalComplex (KaroubiKaroubi.inverse C) (ComplexShape.down \u2115)).obj\n                  (HomologicalComplex.mk\n                    (fun n =>\n                      Karoubi.mk ((KaroubiFunctorCategoryEmbedding.obj P).obj [n].op)\n                        (HomologicalComplex.Hom.f PInfty n))\n                    fun i j =>\n                    Karoubi.Hom.mk\n                      (HomologicalComplex.Hom.f PInfty i \u226b\n                        HomologicalComplex.d (AlternatingFaceMapComplex.obj (KaroubiFunctorCategoryEmbedding.obj P)) i\n                          j)))\n                (n + 1))).f \u226b\n      (HomologicalComplex.Hom.f PInfty (n + 1)).f \u226b\n        (HomologicalComplex.d (AlternatingFaceMapComplex.obj (KaroubiFunctorCategoryEmbedding.obj P)) (n + 1) n).f", "state_after": "no goals"}, {"tactic": "ext n", "state_before": "case refine'_2\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP Q : Karoubi (SimplicialObject C)\nf : P \u27f6 Q\n\u22a2 (N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).map f =\n    eqToHom\n        (_ :\n          (N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P =\n            (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                  N\u2081 \u22d9\n                    (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                      Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n              P) \u226b\n      (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n              N\u2081 \u22d9\n                (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                  Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).map\n          f \u226b\n        eqToHom\n          (_ :\n            (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                    N\u2081 \u22d9\n                      (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                        Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).obj\n                Q =\n              (N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj Q)", "state_after": "case refine'_2.h.h\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP Q : Karoubi (SimplicialObject C)\nf : P \u27f6 Q\nn : \u2115\n\u22a2 (HomologicalComplex.Hom.f ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).map f) n).f =\n    (HomologicalComplex.Hom.f\n        (eqToHom\n            (_ :\n              (N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P =\n                (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                      N\u2081 \u22d9\n                        (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                          Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n                            (ComplexShape.down \u2115)).obj\n                  P) \u226b\n          (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                  N\u2081 \u22d9\n                    (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                      Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).map\n              f \u226b\n            eqToHom\n              (_ :\n                (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                        N\u2081 \u22d9\n                          (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                            Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n                              (ComplexShape.down \u2115)).obj\n                    Q =\n                  (N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj Q))\n        n).f"}, {"tactic": "dsimp [KaroubiKaroubi.inverse, Functor.mapHomologicalComplex]", "state_before": "case refine'_2.h.h\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP Q : Karoubi (SimplicialObject C)\nf : P \u27f6 Q\nn : \u2115\n\u22a2 (HomologicalComplex.Hom.f ((N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).map f) n).f =\n    (HomologicalComplex.Hom.f\n        (eqToHom\n            (_ :\n              (N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj P =\n                (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                      N\u2081 \u22d9\n                        (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                          Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n                            (ComplexShape.down \u2115)).obj\n                  P) \u226b\n          (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                  N\u2081 \u22d9\n                    (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                      Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse (ComplexShape.down \u2115)).map\n              f \u226b\n            eqToHom\n              (_ :\n                (karoubiFunctorCategoryEmbedding SimplexCategory\u1d52\u1d56 C \u22d9\n                        N\u2081 \u22d9\n                          (karoubiChainComplexEquivalence (Karoubi C) \u2115).functor \u22d9\n                            Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse\n                              (ComplexShape.down \u2115)).obj\n                    Q =\n                  (N\u2082 \u22d9 (karoubiChainComplexEquivalence C \u2115).functor).obj Q))\n        n).f", "state_after": "case refine'_2.h.h\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP Q : Karoubi (SimplicialObject C)\nf : P \u27f6 Q\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f PInfty n \u226b f.f.app [n].op =\n    (HomologicalComplex.Hom.f\n          (eqToHom\n            (_ :\n              (karoubiChainComplexEquivalence C \u2115).functor.obj (N\u2082.obj P) =\n                HomologicalComplex.mk (fun i => Karoubi.mk (P.X.obj [i].op) (HomologicalComplex.Hom.f PInfty i).f)\n                  fun i j =>\n                  Karoubi.Hom.mk\n                    (HomologicalComplex.d\n                          ((karoubiChainComplexEquivalence (Karoubi C) \u2115).functor.obj\n                            (N\u2081.obj (KaroubiFunctorCategoryEmbedding.obj P)))\n                          i j).f.f))\n          n).f \u226b\n      ((HomologicalComplex.Hom.f PInfty n).f \u226b f.f.app [n].op) \u226b\n        (HomologicalComplex.Hom.f\n            (eqToHom\n              (_ :\n                (HomologicalComplex.mk (fun i => Karoubi.mk (Q.X.obj [i].op) (HomologicalComplex.Hom.f PInfty i).f)\n                    fun i j =>\n                    Karoubi.Hom.mk\n                      (HomologicalComplex.d\n                            ((karoubiChainComplexEquivalence (Karoubi C) \u2115).functor.obj\n                              (N\u2081.obj (KaroubiFunctorCategoryEmbedding.obj Q)))\n                            i j).f.f) =\n                  (karoubiChainComplexEquivalence C \u2115).functor.obj (N\u2082.obj Q)))\n            n).f"}, {"tactic": "simp only [karoubi_PInfty_f, HomologicalComplex.eqToHom_f, Karoubi.eqToHom_f,\n  assoc, comp_id, PInfty_f_naturality, app_p_comp,\n  karoubiChainComplexEquivalence_functor_obj_X_p, N\u2082_obj_p_f, eqToHom_refl,\n  PInfty_f_naturality_assoc, app_comp_p, PInfty_f_idem_assoc]", "state_before": "case refine'_2.h.h\nC : Type u_1\ninst\u271d\u00b9 : Category C\ninst\u271d : Preadditive C\nP Q : Karoubi (SimplicialObject C)\nf : P \u27f6 Q\nn : \u2115\n\u22a2 HomologicalComplex.Hom.f PInfty n \u226b f.f.app [n].op =\n    (HomologicalComplex.Hom.f\n          (eqToHom\n            (_ :\n              (karoubiChainComplexEquivalence C \u2115).functor.obj (N\u2082.obj P) =\n                HomologicalComplex.mk (fun i => Karoubi.mk (P.X.obj [i].op) (HomologicalComplex.Hom.f PInfty i).f)\n                  fun i j =>\n                  Karoubi.Hom.mk\n                    (HomologicalComplex.d\n                          ((karoubiChainComplexEquivalence (Karoubi C) \u2115).functor.obj\n                            (N\u2081.obj (KaroubiFunctorCategoryEmbedding.obj P)))\n                          i j).f.f))\n          n).f \u226b\n      ((HomologicalComplex.Hom.f PInfty n).f \u226b f.f.app [n].op) \u226b\n        (HomologicalComplex.Hom.f\n            (eqToHom\n              (_ :\n                (HomologicalComplex.mk (fun i => Karoubi.mk (Q.X.obj [i].op) (HomologicalComplex.Hom.f PInfty i).f)\n                    fun i j =>\n                    Karoubi.Hom.mk\n                      (HomologicalComplex.d\n                            ((karoubiChainComplexEquivalence (Karoubi C) \u2115).functor.obj\n                              (N\u2081.obj (KaroubiFunctorCategoryEmbedding.obj Q)))\n                            i j).f.f) =\n                  (karoubiChainComplexEquivalence C \u2115).functor.obj (N\u2082.obj Q)))\n            n).f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Maps.lean", "full_name": "Embedding.continuous", "start": [249, 1], "end": [250, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Category/GroupCat/Biproducts.lean", "full_name": "AddCommGroupCat.binaryProductLimitCone_cone_\u03c0_app_left", "start": [62, 1], "end": [64, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/TypeVec.lean", "full_name": "TypeVec.subtypeVal_nil", "start": [654, 1], "end": [656, 25], "traced_tactics": [{"tactic": "rintro \u27e8\u27e9", "state_before": "n : \u2115\n\u03b1 : TypeVec 0\nps : \u03b1 \u27f9 repeat 0 Prop\n\u22a2 \u2200 (x : Fin2 0), subtypeVal ps x = nilFun x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean", "full_name": "NNReal.eventually_pow_one_div_le", "start": [453, 1], "end": [459, 84], "traced_tactics": [{"tactic": "obtain \u27e8m, hm\u27e9 := add_one_pow_unbounded_of_pos x (tsub_pos_of_lt hy)", "state_before": "x y : \u211d\u22650\nhy : 1 < y\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, x ^ (1 / \u2191n) \u2264 y", "state_after": "case intro\nx y : \u211d\u22650\nhy : 1 < y\nm : \u2115\nhm : x < (y - 1 + 1) ^ m\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, x ^ (1 / \u2191n) \u2264 y"}, {"tactic": "rw [tsub_add_cancel_of_le hy.le] at hm", "state_before": "case intro\nx y : \u211d\u22650\nhy : 1 < y\nm : \u2115\nhm : x < (y - 1 + 1) ^ m\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, x ^ (1 / \u2191n) \u2264 y", "state_after": "case intro\nx y : \u211d\u22650\nhy : 1 < y\nm : \u2115\nhm : x < y ^ m\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, x ^ (1 / \u2191n) \u2264 y"}, {"tactic": "refine' eventually_atTop.2 \u27e8m + 1, fun n hn => _\u27e9", "state_before": "case intro\nx y : \u211d\u22650\nhy : 1 < y\nm : \u2115\nhm : x < y ^ m\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, x ^ (1 / \u2191n) \u2264 y", "state_after": "case intro\nx y : \u211d\u22650\nhy : 1 < y\nm : \u2115\nhm : x < y ^ m\nn : \u2115\nhn : n \u2265 m + 1\n\u22a2 x ^ (1 / \u2191n) \u2264 y"}, {"tactic": "simpa only [NNReal.rpow_one_div_le_iff (Nat.cast_pos.2 <| m.succ_pos.trans_le hn),\n  NNReal.rpow_nat_cast] using hm.le.trans (pow_le_pow hy.le (m.le_succ.trans hn))", "state_before": "case intro\nx y : \u211d\u22650\nhy : 1 < y\nm : \u2115\nhm : x < y ^ m\nn : \u2115\nhn : n \u2265 m + 1\n\u22a2 x ^ (1 / \u2191n) \u2264 y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/PerfectClosure.lean", "full_name": "PerfectClosure.R.sound", "start": [326, 1], "end": [333, 16], "traced_tactics": [{"tactic": "subst H", "state_before": "K : Type u\ninst\u271d\u00b2 : CommRing K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP K p\nm n : \u2115\nx y : K\nH : (\u2191(frobenius K p)^[m]) x = y\n\u22a2 mk K p (n, x) = mk K p (m + n, y)", "state_after": "K : Type u\ninst\u271d\u00b2 : CommRing K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP K p\nm n : \u2115\nx : K\n\u22a2 mk K p (n, x) = mk K p (m + n, (\u2191(frobenius K p)^[m]) x)"}, {"tactic": "induction' m with m ih", "state_before": "K : Type u\ninst\u271d\u00b2 : CommRing K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP K p\nm n : \u2115\nx : K\n\u22a2 mk K p (n, x) = mk K p (m + n, (\u2191(frobenius K p)^[m]) x)", "state_after": "case zero\nK : Type u\ninst\u271d\u00b2 : CommRing K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP K p\nn : \u2115\nx : K\n\u22a2 mk K p (n, x) = mk K p (Nat.zero + n, (\u2191(frobenius K p)^[Nat.zero]) x)\n\ncase succ\nK : Type u\ninst\u271d\u00b2 : CommRing K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP K p\nn : \u2115\nx : K\nm : \u2115\nih : mk K p (n, x) = mk K p (m + n, (\u2191(frobenius K p)^[m]) x)\n\u22a2 mk K p (n, x) = mk K p (Nat.succ m + n, (\u2191(frobenius K p)^[Nat.succ m]) x)"}, {"tactic": "rw [ih, Nat.succ_add, iterate_succ']", "state_before": "case succ\nK : Type u\ninst\u271d\u00b2 : CommRing K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP K p\nn : \u2115\nx : K\nm : \u2115\nih : mk K p (n, x) = mk K p (m + n, (\u2191(frobenius K p)^[m]) x)\n\u22a2 mk K p (n, x) = mk K p (Nat.succ m + n, (\u2191(frobenius K p)^[Nat.succ m]) x)", "state_after": "case succ\nK : Type u\ninst\u271d\u00b2 : CommRing K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP K p\nn : \u2115\nx : K\nm : \u2115\nih : mk K p (n, x) = mk K p (m + n, (\u2191(frobenius K p)^[m]) x)\n\u22a2 mk K p (m + n, (\u2191(frobenius K p)^[m]) x) = mk K p (Nat.succ (m + n), (\u2191(frobenius K p) \u2218 \u2191(frobenius K p)^[m]) x)"}, {"tactic": "apply Quot.sound", "state_before": "case succ\nK : Type u\ninst\u271d\u00b2 : CommRing K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP K p\nn : \u2115\nx : K\nm : \u2115\nih : mk K p (n, x) = mk K p (m + n, (\u2191(frobenius K p)^[m]) x)\n\u22a2 mk K p (m + n, (\u2191(frobenius K p)^[m]) x) = mk K p (Nat.succ (m + n), (\u2191(frobenius K p) \u2218 \u2191(frobenius K p)^[m]) x)", "state_after": "case succ.a\nK : Type u\ninst\u271d\u00b2 : CommRing K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP K p\nn : \u2115\nx : K\nm : \u2115\nih : mk K p (n, x) = mk K p (m + n, (\u2191(frobenius K p)^[m]) x)\n\u22a2 R K p (m + n, (\u2191(frobenius K p)^[m]) x) (Nat.succ (m + n), (\u2191(frobenius K p) \u2218 \u2191(frobenius K p)^[m]) x)"}, {"tactic": "apply R.intro", "state_before": "case succ.a\nK : Type u\ninst\u271d\u00b2 : CommRing K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP K p\nn : \u2115\nx : K\nm : \u2115\nih : mk K p (n, x) = mk K p (m + n, (\u2191(frobenius K p)^[m]) x)\n\u22a2 R K p (m + n, (\u2191(frobenius K p)^[m]) x) (Nat.succ (m + n), (\u2191(frobenius K p) \u2218 \u2191(frobenius K p)^[m]) x)", "state_after": "no goals"}, {"tactic": "simp only [Nat.zero_eq, zero_add, iterate_zero_apply]", "state_before": "case zero\nK : Type u\ninst\u271d\u00b2 : CommRing K\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP K p\nn : \u2115\nx : K\n\u22a2 mk K p (n, x) = mk K p (Nat.zero + n, (\u2191(frobenius K p)^[Nat.zero]) x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/MinMax.lean", "full_name": "le_max_of_le_right", "start": [87, 1], "end": [88, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Nat/Gcd.lean", "full_name": "Nat.coprime_mul_iff_right", "start": [337, 1], "end": [338, 79], "traced_tactics": [{"tactic": "rw [@coprime_comm k, @coprime_comm k, @coprime_comm k, coprime_mul_iff_left]", "state_before": "k m n : Nat\n\u22a2 coprime k (m * n) \u2194 coprime k m \u2227 coprime k n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Module/LinearMap.lean", "full_name": "LinearMap.add_apply", "start": [885, 1], "end": [886, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/FreeMonoid/Basic.lean", "full_name": "FreeMonoid.lift_comp_of", "start": [257, 1], "end": [257, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/GDelta.lean", "full_name": "IsG\u03b4.union", "start": [109, 1], "end": [115, 40], "traced_tactics": [{"tactic": "rcases hs with \u27e8S, Sopen, Scount, rfl\u27e9", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.6391\n\u03b3 : Type ?u.6394\n\u03b9 : Type ?u.6397\ninst\u271d : TopologicalSpace \u03b1\ns t : Set \u03b1\nhs : IsG\u03b4 s\nht : IsG\u03b4 t\n\u22a2 IsG\u03b4 (s \u222a t)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.6391\n\u03b3 : Type ?u.6394\n\u03b9 : Type ?u.6397\ninst\u271d : TopologicalSpace \u03b1\nt : Set \u03b1\nht : IsG\u03b4 t\nS : Set (Set \u03b1)\nSopen : \u2200 (t : Set \u03b1), t \u2208 S \u2192 IsOpen t\nScount : Set.Countable S\n\u22a2 IsG\u03b4 (\u22c2\u2080 S \u222a t)"}, {"tactic": "rcases ht with \u27e8T, Topen, Tcount, rfl\u27e9", "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.6391\n\u03b3 : Type ?u.6394\n\u03b9 : Type ?u.6397\ninst\u271d : TopologicalSpace \u03b1\nt : Set \u03b1\nht : IsG\u03b4 t\nS : Set (Set \u03b1)\nSopen : \u2200 (t : Set \u03b1), t \u2208 S \u2192 IsOpen t\nScount : Set.Countable S\n\u22a2 IsG\u03b4 (\u22c2\u2080 S \u222a t)", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.6391\n\u03b3 : Type ?u.6394\n\u03b9 : Type ?u.6397\ninst\u271d : TopologicalSpace \u03b1\nS : Set (Set \u03b1)\nSopen : \u2200 (t : Set \u03b1), t \u2208 S \u2192 IsOpen t\nScount : Set.Countable S\nT : Set (Set \u03b1)\nTopen : \u2200 (t : Set \u03b1), t \u2208 T \u2192 IsOpen t\nTcount : Set.Countable T\n\u22a2 IsG\u03b4 (\u22c2\u2080 S \u222a \u22c2\u2080 T)"}, {"tactic": "rw [sInter_union_sInter]", "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.6391\n\u03b3 : Type ?u.6394\n\u03b9 : Type ?u.6397\ninst\u271d : TopologicalSpace \u03b1\nS : Set (Set \u03b1)\nSopen : \u2200 (t : Set \u03b1), t \u2208 S \u2192 IsOpen t\nScount : Set.Countable S\nT : Set (Set \u03b1)\nTopen : \u2200 (t : Set \u03b1), t \u2208 T \u2192 IsOpen t\nTcount : Set.Countable T\n\u22a2 IsG\u03b4 (\u22c2\u2080 S \u222a \u22c2\u2080 T)", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.6391\n\u03b3 : Type ?u.6394\n\u03b9 : Type ?u.6397\ninst\u271d : TopologicalSpace \u03b1\nS : Set (Set \u03b1)\nSopen : \u2200 (t : Set \u03b1), t \u2208 S \u2192 IsOpen t\nScount : Set.Countable S\nT : Set (Set \u03b1)\nTopen : \u2200 (t : Set \u03b1), t \u2208 T \u2192 IsOpen t\nTcount : Set.Countable T\n\u22a2 IsG\u03b4 (\u22c2 (p : Set \u03b1 \u00d7 Set \u03b1) (_ : p \u2208 S \u00d7\u02e2 T), p.fst \u222a p.snd)"}, {"tactic": "apply isG\u03b4_biInter_of_open (Scount.prod Tcount)", "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.6391\n\u03b3 : Type ?u.6394\n\u03b9 : Type ?u.6397\ninst\u271d : TopologicalSpace \u03b1\nS : Set (Set \u03b1)\nSopen : \u2200 (t : Set \u03b1), t \u2208 S \u2192 IsOpen t\nScount : Set.Countable S\nT : Set (Set \u03b1)\nTopen : \u2200 (t : Set \u03b1), t \u2208 T \u2192 IsOpen t\nTcount : Set.Countable T\n\u22a2 IsG\u03b4 (\u22c2 (p : Set \u03b1 \u00d7 Set \u03b1) (_ : p \u2208 S \u00d7\u02e2 T), p.fst \u222a p.snd)", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.6391\n\u03b3 : Type ?u.6394\n\u03b9 : Type ?u.6397\ninst\u271d : TopologicalSpace \u03b1\nS : Set (Set \u03b1)\nSopen : \u2200 (t : Set \u03b1), t \u2208 S \u2192 IsOpen t\nScount : Set.Countable S\nT : Set (Set \u03b1)\nTopen : \u2200 (t : Set \u03b1), t \u2208 T \u2192 IsOpen t\nTcount : Set.Countable T\n\u22a2 \u2200 (i : Set \u03b1 \u00d7 Set \u03b1), i \u2208 S \u00d7\u02e2 T \u2192 IsOpen (i.fst \u222a i.snd)"}, {"tactic": "rintro \u27e8a, b\u27e9 \u27e8ha, hb\u27e9", "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.6391\n\u03b3 : Type ?u.6394\n\u03b9 : Type ?u.6397\ninst\u271d : TopologicalSpace \u03b1\nS : Set (Set \u03b1)\nSopen : \u2200 (t : Set \u03b1), t \u2208 S \u2192 IsOpen t\nScount : Set.Countable S\nT : Set (Set \u03b1)\nTopen : \u2200 (t : Set \u03b1), t \u2208 T \u2192 IsOpen t\nTcount : Set.Countable T\n\u22a2 \u2200 (i : Set \u03b1 \u00d7 Set \u03b1), i \u2208 S \u00d7\u02e2 T \u2192 IsOpen (i.fst \u222a i.snd)", "state_after": "case intro.intro.intro.intro.intro.intro.mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.6391\n\u03b3 : Type ?u.6394\n\u03b9 : Type ?u.6397\ninst\u271d : TopologicalSpace \u03b1\nS : Set (Set \u03b1)\nSopen : \u2200 (t : Set \u03b1), t \u2208 S \u2192 IsOpen t\nScount : Set.Countable S\nT : Set (Set \u03b1)\nTopen : \u2200 (t : Set \u03b1), t \u2208 T \u2192 IsOpen t\nTcount : Set.Countable T\na b : Set \u03b1\nha : (a, b).fst \u2208 S\nhb : (a, b).snd \u2208 T\n\u22a2 IsOpen ((a, b).fst \u222a (a, b).snd)"}, {"tactic": "exact (Sopen a ha).union (Topen b hb)", "state_before": "case intro.intro.intro.intro.intro.intro.mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.6391\n\u03b3 : Type ?u.6394\n\u03b9 : Type ?u.6397\ninst\u271d : TopologicalSpace \u03b1\nS : Set (Set \u03b1)\nSopen : \u2200 (t : Set \u03b1), t \u2208 S \u2192 IsOpen t\nScount : Set.Countable S\nT : Set (Set \u03b1)\nTopen : \u2200 (t : Set \u03b1), t \u2208 T \u2192 IsOpen t\nTcount : Set.Countable T\na b : Set \u03b1\nha : (a, b).fst \u2208 S\nhb : (a, b).snd \u2208 T\n\u22a2 IsOpen ((a, b).fst \u222a (a, b).snd)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "full_name": "FiniteDimensional.finiteDimensional_iff_of_rank_eq_nsmul", "start": [220, 1], "end": [224, 48], "traced_tactics": [{"tactic": "simp only [FiniteDimensional, \u2190 IsNoetherian.iff_fg, IsNoetherian.iff_rank_lt_aleph0, hVW,\n  Cardinal.nsmul_lt_aleph0_iff_of_ne_zero hn]", "state_before": "K : Type u\nV : Type v\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b3 : AddCommGroup V\u2082\ninst\u271d\u00b2 : Module K V\u2082\nW : Type v\ninst\u271d\u00b9 : AddCommGroup W\ninst\u271d : Module K W\nn : \u2115\nhn : n \u2260 0\nhVW : Module.rank K V = n \u2022 Module.rank K W\n\u22a2 FiniteDimensional K V \u2194 FiniteDimensional K W", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/AList.lean", "full_name": "AList.perm_replace", "start": [205, 1], "end": [207, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Monic.lean", "full_name": "Polynomial.Monic.map", "start": [68, 1], "end": [76, 81], "traced_tactics": [{"tactic": "unfold Monic", "state_before": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nhp : Monic p\n\u22a2 Monic (Polynomial.map f p)", "state_after": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nhp : Monic p\n\u22a2 Polynomial.leadingCoeff (Polynomial.map f p) = 1"}, {"tactic": "nontriviality", "state_before": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nhp : Monic p\n\u22a2 Polynomial.leadingCoeff (Polynomial.map f p) = 1", "state_after": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nhp : Monic p\n\u271d : Nontrivial S\n\u22a2 Polynomial.leadingCoeff (Polynomial.map f p) = 1"}, {"tactic": "have : f p.leadingCoeff \u2260 0 := by\n  rw [show _ = _ from hp, f.map_one]\n  exact one_ne_zero", "state_before": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nhp : Monic p\n\u271d : Nontrivial S\n\u22a2 Polynomial.leadingCoeff (Polynomial.map f p) = 1", "state_after": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nhp : Monic p\n\u271d : Nontrivial S\nthis : \u2191f (Polynomial.leadingCoeff p) \u2260 0\n\u22a2 Polynomial.leadingCoeff (Polynomial.map f p) = 1"}, {"tactic": "rw [Polynomial.leadingCoeff, coeff_map]", "state_before": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nhp : Monic p\n\u271d : Nontrivial S\nthis : \u2191f (Polynomial.leadingCoeff p) \u2260 0\n\u22a2 Polynomial.leadingCoeff (Polynomial.map f p) = 1", "state_after": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nhp : Monic p\n\u271d : Nontrivial S\nthis : \u2191f (Polynomial.leadingCoeff p) \u2260 0\n\u22a2 \u2191f (coeff p (natDegree (Polynomial.map f p))) = 1"}, {"tactic": "suffices p.coeff (p.map f).natDegree = 1 by simp [this]", "state_before": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nhp : Monic p\n\u271d : Nontrivial S\nthis : \u2191f (Polynomial.leadingCoeff p) \u2260 0\n\u22a2 \u2191f (coeff p (natDegree (Polynomial.map f p))) = 1", "state_after": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nhp : Monic p\n\u271d : Nontrivial S\nthis : \u2191f (Polynomial.leadingCoeff p) \u2260 0\n\u22a2 coeff p (natDegree (Polynomial.map f p)) = 1"}, {"tactic": "rwa [natDegree_eq_of_degree_eq (degree_map_eq_of_leadingCoeff_ne_zero f this)]", "state_before": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nhp : Monic p\n\u271d : Nontrivial S\nthis : \u2191f (Polynomial.leadingCoeff p) \u2260 0\n\u22a2 coeff p (natDegree (Polynomial.map f p)) = 1", "state_after": "no goals"}, {"tactic": "rw [show _ = _ from hp, f.map_one]", "state_before": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nhp : Monic p\n\u271d : Nontrivial S\n\u22a2 \u2191f (Polynomial.leadingCoeff p) \u2260 0", "state_after": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nhp : Monic p\n\u271d : Nontrivial S\n\u22a2 1 \u2260 0"}, {"tactic": "exact one_ne_zero", "state_before": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nhp : Monic p\n\u271d : Nontrivial S\n\u22a2 1 \u2260 0", "state_after": "no goals"}, {"tactic": "simp [this]", "state_before": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Semiring S\nf : R \u2192+* S\nhp : Monic p\n\u271d : Nontrivial S\nthis\u271d : \u2191f (Polynomial.leadingCoeff p) \u2260 0\nthis : coeff p (natDegree (Polynomial.map f p)) = 1\n\u22a2 \u2191f (coeff p (natDegree (Polynomial.map f p))) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Composition.lean", "full_name": "CompositionAsSet.length_lt_card_boundaries", "start": [894, 1], "end": [896, 21], "traced_tactics": [{"tactic": "rw [c.card_boundaries_eq_succ_length]", "state_before": "n : \u2115\nc : CompositionAsSet n\n\u22a2 length c < Finset.card c.boundaries", "state_after": "n : \u2115\nc : CompositionAsSet n\n\u22a2 length c < length c + 1"}, {"tactic": "exact lt_add_one _", "state_before": "n : \u2115\nc : CompositionAsSet n\n\u22a2 length c < length c + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Bounded.lean", "full_name": "Set.bounded_ge_inter_not_ge", "start": [395, 1], "end": [397, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Dfinsupp/Basic.lean", "full_name": "Dfinsupp.induction", "start": [956, 11], "end": [988, 18], "traced_tactics": [{"tactic": "cases' f with f s", "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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\n\u22a2 p f", "state_after": "case mk'\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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf : (i : \u03b9) \u2192 \u03b2 i\ns : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\n\u22a2 p { toFun := f, support' := s }"}, {"tactic": "induction' s using Trunc.induction_on with s", "state_before": "case mk'\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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf : (i : \u03b9) \u2192 \u03b2 i\ns : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\n\u22a2 p { toFun := f, support' := s }", "state_after": "case mk'.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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf : (i : \u03b9) \u2192 \u03b2 i\ns : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\n\u22a2 p { toFun := f, support' := Trunc.mk s }"}, {"tactic": "cases' s with s H", "state_before": "case mk'.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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf : (i : \u03b9) \u2192 \u03b2 i\ns : { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 }\n\u22a2 p { toFun := f, support' := Trunc.mk s }", "state_after": "case mk'.h.mk\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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf : (i : \u03b9) \u2192 \u03b2 i\ns : Multiset \u03b9\nH : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0\n\u22a2 p { toFun := f, support' := Trunc.mk { val := s, property := H } }"}, {"tactic": "induction' s using Multiset.induction_on with i s ih generalizing f", "state_before": "case mk'.h.mk\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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf : (i : \u03b9) \u2192 \u03b2 i\ns : Multiset \u03b9\nH : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0\n\u22a2 p { toFun := f, support' := Trunc.mk { val := s, property := H } }", "state_after": "case mk'.h.mk.empty\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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s \u2228 f\u271d i = 0\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i : \u03b9), i \u2208 0 \u2228 f i = 0\n\u22a2 p { toFun := f, support' := Trunc.mk { val := 0, property := H } }\n\ncase mk'.h.mk.cons\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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\n\u22a2 p { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }"}, {"tactic": "have H3 : single i _ + _ = (\u27e8f, Trunc.mk \u27e8i ::\u2098 s, H\u27e9\u27e9 : \u03a0\u2080 i, \u03b2 i) := single_add_erase _ _", "state_before": "case mk'.h.mk.cons\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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\n\u22a2 p { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }", "state_after": "case mk'.h.mk.cons\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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\nH3 :\n  single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n      erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } =\n    { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\n\u22a2 p { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }"}, {"tactic": "rw [\u2190 H3]", "state_before": "case mk'.h.mk.cons\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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\nH3 :\n  single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n      erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } =\n    { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\n\u22a2 p { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }", "state_after": "case mk'.h.mk.cons\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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\nH3 :\n  single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n      erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } =\n    { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\n\u22a2 p\n    (single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n      erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })"}, {"tactic": "change p (single i (f i) + _)", "state_before": "case mk'.h.mk.cons\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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\nH3 :\n  single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n      erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } =\n    { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\n\u22a2 p\n    (single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n      erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })", "state_after": "case mk'.h.mk.cons\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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\nH3 :\n  single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n      erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } =\n    { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\n\u22a2 p (single i (f i) + erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })"}, {"tactic": "cases' Classical.em (f i = 0) with h h", "state_before": "case mk'.h.mk.cons\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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\nH3 :\n  single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n      erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } =\n    { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\n\u22a2 p (single i (f i) + erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })", "state_after": "case mk'.h.mk.cons.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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\nH3 :\n  single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n      erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } =\n    { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\nh : f i = 0\n\u22a2 p (single i (f i) + erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\n\ncase mk'.h.mk.cons.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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\nH3 :\n  single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n      erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } =\n    { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\nh : \u00acf i = 0\n\u22a2 p (single i (f i) + erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })"}, {"tactic": "refine' ha _ _ _ _ h H2", "state_before": "case mk'.h.mk.cons.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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\nH3 :\n  single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n      erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } =\n    { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\nh : \u00acf i = 0\n\u22a2 p (single i (f i) + erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })", "state_after": "case mk'.h.mk.cons.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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\nH3 :\n  single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n      erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } =\n    { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\nh : \u00acf i = 0\n\u22a2 \u2191(erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }) i = 0"}, {"tactic": "rw [erase_same]", "state_before": "case mk'.h.mk.cons.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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\nH3 :\n  single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n      erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } =\n    { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\nh : \u00acf i = 0\n\u22a2 \u2191(erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }) i = 0", "state_after": "no goals"}, {"tactic": "have : f = 0 := funext fun i => (H i).resolve_left (Multiset.not_mem_zero _)", "state_before": "case mk'.h.mk.empty\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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s \u2228 f\u271d i = 0\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i : \u03b9), i \u2208 0 \u2228 f i = 0\n\u22a2 p { toFun := f, support' := Trunc.mk { val := 0, property := H } }", "state_after": "case mk'.h.mk.empty\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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s \u2228 f\u271d i = 0\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i : \u03b9), i \u2208 0 \u2228 f i = 0\nthis : f = 0\n\u22a2 p { toFun := f, support' := Trunc.mk { val := 0, property := H } }"}, {"tactic": "subst this", "state_before": "case mk'.h.mk.empty\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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s \u2228 f\u271d i = 0\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i : \u03b9), i \u2208 0 \u2228 f i = 0\nthis : f = 0\n\u22a2 p { toFun := f, support' := Trunc.mk { val := 0, property := H } }", "state_after": "case mk'.h.mk.empty\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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf : (i : \u03b9) \u2192 \u03b2 i\ns : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0\nH : \u2200 (i : \u03b9), i \u2208 0 \u2228 OfNat.ofNat 0 i = 0\n\u22a2 p { toFun := 0, support' := Trunc.mk { val := 0, property := H } }"}, {"tactic": "exact h0", "state_before": "case mk'.h.mk.empty\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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf : (i : \u03b9) \u2192 \u03b2 i\ns : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0\nH : \u2200 (i : \u03b9), i \u2208 0 \u2228 OfNat.ofNat 0 i = 0\n\u22a2 p { toFun := 0, support' := Trunc.mk { val := 0, property := H } }", "state_after": "no goals"}, {"tactic": "dsimp only [erase, Trunc.map, Trunc.bind, Trunc.liftOn, Trunc.lift_mk,\n  Function.comp, Subtype.coe_mk]", "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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\n\u22a2 p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })", "state_after": "\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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\n\u22a2 p\n    { toFun := fun j => if j = i then 0 else f j,\n      support' :=\n        Trunc.mk\n          { val := i ::\u2098 s,\n            property :=\n              (_ :\n                \u2200 (j : \u03b9),\n                  j \u2208 \u2191{ val := i ::\u2098 s, property := H } \u2228\n                    (fun j =>\n                          if j = i then 0\n                          else toFun { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } j)\n                        j =\n                      0) } }"}, {"tactic": "have H3 : \u2200 aux, (\u27e8fun j : \u03b9 => ite (j = i) 0 (f j), Trunc.mk \u27e8i ::\u2098 s, aux\u27e9\u27e9 : \u03a0\u2080 i, \u03b2 i) =\n    \u27e8fun j : \u03b9 => ite (j = i) 0 (f j), Trunc.mk \u27e8s, H2\u27e9\u27e9 :=\n  fun _ \u21a6 ext fun _ => rfl", "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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : \u2200 (j : \u03b9), j \u2208 s \u2228 (if j = i then 0 else f j) = 0\n\u22a2 p\n    { toFun := fun j => if j = i then 0 else f j,\n      support' :=\n        Trunc.mk\n          { val := i ::\u2098 s,\n            property :=\n              (_ :\n                \u2200 (j : \u03b9),\n                  j \u2208 \u2191{ val := i ::\u2098 s, property := H } \u2228\n                    (fun j =>\n                          if j = i then 0\n                          else toFun { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } j)\n                        j =\n                      0) } }", "state_after": "\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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : \u2200 (j : \u03b9), j \u2208 s \u2228 (if j = i then 0 else f j) = 0\nH3 :\n  \u2200 (aux : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 (fun j => if j = i then 0 else f j) i_1 = 0),\n    { toFun := fun j => if j = i then 0 else f j, support' := Trunc.mk { val := i ::\u2098 s, property := aux } } =\n      { toFun := fun j => if j = i then 0 else f j, support' := Trunc.mk { val := s, property := H2 } }\n\u22a2 p\n    { toFun := fun j => if j = i then 0 else f j,\n      support' :=\n        Trunc.mk\n          { val := i ::\u2098 s,\n            property :=\n              (_ :\n                \u2200 (j : \u03b9),\n                  j \u2208 \u2191{ val := i ::\u2098 s, property := H } \u2228\n                    (fun j =>\n                          if j = i then 0\n                          else toFun { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } j)\n                        j =\n                      0) } }"}, {"tactic": "rw [H3]", "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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : \u2200 (j : \u03b9), j \u2208 s \u2228 (if j = i then 0 else f j) = 0\nH3 :\n  \u2200 (aux : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 (fun j => if j = i then 0 else f j) i_1 = 0),\n    { toFun := fun j => if j = i then 0 else f j, support' := Trunc.mk { val := i ::\u2098 s, property := aux } } =\n      { toFun := fun j => if j = i then 0 else f j, support' := Trunc.mk { val := s, property := H2 } }\n\u22a2 p\n    { toFun := fun j => if j = i then 0 else f j,\n      support' :=\n        Trunc.mk\n          { val := i ::\u2098 s,\n            property :=\n              (_ :\n                \u2200 (j : \u03b9),\n                  j \u2208 \u2191{ val := i ::\u2098 s, property := H } \u2228\n                    (fun j =>\n                          if j = i then 0\n                          else toFun { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } j)\n                        j =\n                      0) } }", "state_after": "\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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : \u2200 (j : \u03b9), j \u2208 s \u2228 (if j = i then 0 else f j) = 0\nH3 :\n  \u2200 (aux : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 (fun j => if j = i then 0 else f j) i_1 = 0),\n    { toFun := fun j => if j = i then 0 else f j, support' := Trunc.mk { val := i ::\u2098 s, property := aux } } =\n      { toFun := fun j => if j = i then 0 else f j, support' := Trunc.mk { val := s, property := H2 } }\n\u22a2 p { toFun := fun j => if j = i then 0 else f j, support' := Trunc.mk { val := s, property := H2 } }"}, {"tactic": "apply ih", "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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : \u2200 (j : \u03b9), j \u2208 s \u2228 (if j = i then 0 else f j) = 0\nH3 :\n  \u2200 (aux : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 (fun j => if j = i then 0 else f j) i_1 = 0),\n    { toFun := fun j => if j = i then 0 else f j, support' := Trunc.mk { val := i ::\u2098 s, property := aux } } =\n      { toFun := fun j => if j = i then 0 else f j, support' := Trunc.mk { val := s, property := H2 } }\n\u22a2 p { toFun := fun j => if j = i then 0 else f j, support' := Trunc.mk { val := s, property := H2 } }", "state_after": "no goals"}, {"tactic": "intro 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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\n\u22a2 \u2200 (j : \u03b9), j \u2208 s \u2228 (if j = i then 0 else f j) = 0", "state_after": "\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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\n\u22a2 j \u2208 s \u2228 (if j = i then 0 else f j) = 0"}, {"tactic": "cases' H j with H2 H2", "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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\n\u22a2 j \u2208 s \u2228 (if j = i then 0 else f j) = 0", "state_after": "case 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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : j \u2208 i ::\u2098 s\n\u22a2 j \u2208 s \u2228 (if j = i then 0 else f j) = 0\n\ncase 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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : f j = 0\n\u22a2 j \u2208 s \u2228 (if j = i then 0 else f j) = 0"}, {"tactic": "right", "state_before": "case 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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : f j = 0\n\u22a2 j \u2208 s \u2228 (if j = i then 0 else f j) = 0", "state_after": "case inr.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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : f j = 0\n\u22a2 (if j = i then 0 else f j) = 0"}, {"tactic": "split_ifs <;> [rfl; exact H2]", "state_before": "case inr.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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : f j = 0\n\u22a2 (if j = i then 0 else f j) = 0", "state_after": "no goals"}, {"tactic": "cases' Multiset.mem_cons.1 H2 with H3 H3", "state_before": "case 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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : j \u2208 i ::\u2098 s\n\u22a2 j \u2208 s \u2228 (if j = i then 0 else f j) = 0", "state_after": "case inl.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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : j \u2208 i ::\u2098 s\nH3 : j = i\n\u22a2 j \u2208 s \u2228 (if j = i then 0 else f j) = 0\n\ncase inl.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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : j \u2208 i ::\u2098 s\nH3 : j \u2208 s\n\u22a2 j \u2208 s \u2228 (if j = i then 0 else f j) = 0"}, {"tactic": "right", "state_before": "case inl.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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : j \u2208 i ::\u2098 s\nH3 : j = i\n\u22a2 j \u2208 s \u2228 (if j = i then 0 else f j) = 0", "state_after": "case inl.inl.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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : j \u2208 i ::\u2098 s\nH3 : j = i\n\u22a2 (if j = i then 0 else f j) = 0"}, {"tactic": "exact if_pos H3", "state_before": "case inl.inl.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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : j \u2208 i ::\u2098 s\nH3 : j = i\n\u22a2 (if j = i then 0 else f j) = 0", "state_after": "no goals"}, {"tactic": "left", "state_before": "case inl.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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : j \u2208 i ::\u2098 s\nH3 : j \u2208 s\n\u22a2 j \u2208 s \u2228 (if j = i then 0 else f j) = 0", "state_after": "case inl.inr.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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : j \u2208 i ::\u2098 s\nH3 : j \u2208 s\n\u22a2 j \u2208 s"}, {"tactic": "exact H3", "state_before": "case inl.inr.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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nj : \u03b9\nH2 : j \u2208 i ::\u2098 s\nH3 : j \u2208 s\n\u22a2 j \u2208 s", "state_after": "no goals"}, {"tactic": "rw [h, single_zero, zero_add]", "state_before": "case mk'.h.mk.cons.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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\nH3 :\n  single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n      erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } =\n    { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\nh : f i = 0\n\u22a2 p (single i (f i) + erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })", "state_after": "case mk'.h.mk.cons.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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\nH3 :\n  single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n      erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } =\n    { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\nh : f i = 0\n\u22a2 p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })"}, {"tactic": "exact H2", "state_before": "case mk'.h.mk.cons.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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\np : (\u03a0\u2080 (i : \u03b9), \u03b2 i) \u2192 Prop\nh0 : p 0\nha : \u2200 (i : \u03b9) (b : \u03b2 i) (f : \u03a0\u2080 (i : \u03b9), \u03b2 i), \u2191f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)\nf\u271d : (i : \u03b9) \u2192 \u03b2 i\ns\u271d : Multiset \u03b9\nH\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2228 f\u271d i = 0\ni : \u03b9\ns : Multiset \u03b9\nih :\n  \u2200 (f : (i : \u03b9) \u2192 \u03b2 i) (H : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0),\n    p { toFun := f, support' := Trunc.mk { val := s, property := H } }\nf : (i : \u03b9) \u2192 \u03b2 i\nH : \u2200 (i_1 : \u03b9), i_1 \u2208 i ::\u2098 s \u2228 f i_1 = 0\nH2 : p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })\nH3 :\n  single i (\u2191{ toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } i) +\n      erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } } =\n    { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } }\nh : f i = 0\n\u22a2 p (erase i { toFun := f, support' := Trunc.mk { val := i ::\u2098 s, property := H } })", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finsupp/Defs.lean", "full_name": "Finsupp.mapRange_sub", "start": [1258, 1], "end": [1261, 61], "traced_tactics": [{"tactic": "simp only [hf', sub_apply, mapRange_apply]", "state_before": "\u03b1 : Type u_3\n\u03b2 : Type ?u.443969\n\u03b3 : Type ?u.443972\n\u03b9 : Type ?u.443975\nM : Type ?u.443978\nM' : Type ?u.443981\nN : Type ?u.443984\nP : Type ?u.443987\nG : Type u_1\nH : Type u_2\nR : Type ?u.443996\nS : Type ?u.443999\ninst\u271d\u00b9 : SubNegZeroMonoid G\ninst\u271d : SubNegZeroMonoid H\nf : G \u2192 H\nhf : f 0 = 0\nhf' : \u2200 (x y : G), f (x - y) = f x - f y\nv\u2081 v\u2082 : \u03b1 \u2192\u2080 G\nx\u271d : \u03b1\n\u22a2 \u2191(mapRange f hf (v\u2081 - v\u2082)) x\u271d = \u2191(mapRange f hf v\u2081 - mapRange f hf v\u2082) x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "full_name": "PiNat.disjoint_cylinder_of_longestPrefix_lt", "start": [560, 1], "end": [566, 26], "traced_tactics": [{"tactic": "contrapose! hn", "state_before": "E : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nn : \u2115\nhn : longestPrefix x s < n\n\u22a2 Disjoint s (cylinder x n)", "state_after": "E : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nn : \u2115\nhn : \u00acDisjoint s (cylinder x n)\n\u22a2 n \u2264 longestPrefix x s"}, {"tactic": "rcases not_disjoint_iff_nonempty_inter.1 hn with \u27e8y, ys, hy\u27e9", "state_before": "E : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nn : \u2115\nhn : \u00acDisjoint s (cylinder x n)\n\u22a2 n \u2264 longestPrefix x s", "state_after": "case intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nn : \u2115\nhn : \u00acDisjoint s (cylinder x n)\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : y \u2208 cylinder x n\n\u22a2 n \u2264 longestPrefix x s"}, {"tactic": "apply le_trans _ (firstDiff_le_longestPrefix hs hx ys)", "state_before": "case intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nn : \u2115\nhn : \u00acDisjoint s (cylinder x n)\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : y \u2208 cylinder x n\n\u22a2 n \u2264 longestPrefix x s", "state_after": "E : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nn : \u2115\nhn : \u00acDisjoint s (cylinder x n)\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : y \u2208 cylinder x n\n\u22a2 n \u2264 firstDiff x y"}, {"tactic": "apply (mem_cylinder_iff_le_firstDiff (ne_of_mem_of_not_mem ys hx).symm _).1", "state_before": "E : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nn : \u2115\nhn : \u00acDisjoint s (cylinder x n)\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : y \u2208 cylinder x n\n\u22a2 n \u2264 firstDiff x y", "state_after": "E : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nn : \u2115\nhn : \u00acDisjoint s (cylinder x n)\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : y \u2208 cylinder x n\n\u22a2 x \u2208 cylinder y n"}, {"tactic": "rwa [mem_cylinder_comm]", "state_before": "E : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nn : \u2115\nhn : \u00acDisjoint s (cylinder x n)\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : y \u2208 cylinder x n\n\u22a2 x \u2208 cylinder y n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Lie/Submodule.lean", "full_name": "LieModuleHom.ker_coeSubmodule", "start": [1173, 1], "end": [1174, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Polynomial/BigOperators.lean", "full_name": "Polynomial.natDegree_multiset_prod'", "start": [171, 1], "end": [179, 34], "traced_tactics": [{"tactic": "revert h", "state_before": "R : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0\n\u22a2 natDegree (prod t) = Multiset.sum (Multiset.map (fun f => natDegree f) t)", "state_after": "R : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\n\u22a2 prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0 \u2192\n    natDegree (prod t) = Multiset.sum (Multiset.map (fun f => natDegree f) t)"}, {"tactic": "refine' Multiset.induction_on t _ fun a t ih ht => _", "state_before": "R : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\n\u22a2 prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0 \u2192\n    natDegree (prod t) = Multiset.sum (Multiset.map (fun f => natDegree f) t)", "state_after": "case refine'_1\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\n\u22a2 prod (Multiset.map (fun f => leadingCoeff f) 0) \u2260 0 \u2192\n    natDegree (prod 0) = Multiset.sum (Multiset.map (fun f => natDegree f) 0)\n\ncase refine'_2\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\na : R[X]\nt : Multiset R[X]\nih :\n  prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0 \u2192\n    natDegree (prod t) = Multiset.sum (Multiset.map (fun f => natDegree f) t)\nht : prod (Multiset.map (fun f => leadingCoeff f) (a ::\u2098 t)) \u2260 0\n\u22a2 natDegree (prod (a ::\u2098 t)) = Multiset.sum (Multiset.map (fun f => natDegree f) (a ::\u2098 t))"}, {"tactic": "rw [Multiset.map_cons, Multiset.prod_cons] at ht\u22a2", "state_before": "case refine'_2\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\na : R[X]\nt : Multiset R[X]\nih :\n  prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0 \u2192\n    natDegree (prod t) = Multiset.sum (Multiset.map (fun f => natDegree f) t)\nht : prod (Multiset.map (fun f => leadingCoeff f) (a ::\u2098 t)) \u2260 0\n\u22a2 natDegree (prod (a ::\u2098 t)) = Multiset.sum (Multiset.map (fun f => natDegree f) (a ::\u2098 t))", "state_after": "case refine'_2\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\na : R[X]\nt : Multiset R[X]\nih :\n  prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0 \u2192\n    natDegree (prod t) = Multiset.sum (Multiset.map (fun f => natDegree f) t)\nht : leadingCoeff a * prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0\n\u22a2 natDegree (a * prod t) = Multiset.sum (natDegree a ::\u2098 Multiset.map (fun f => natDegree f) t)"}, {"tactic": "rw [Multiset.sum_cons, Polynomial.natDegree_mul', ih]", "state_before": "case refine'_2\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\na : R[X]\nt : Multiset R[X]\nih :\n  prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0 \u2192\n    natDegree (prod t) = Multiset.sum (Multiset.map (fun f => natDegree f) t)\nht : leadingCoeff a * prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0\n\u22a2 natDegree (a * prod t) = Multiset.sum (natDegree a ::\u2098 Multiset.map (fun f => natDegree f) t)", "state_after": "case refine'_2\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\na : R[X]\nt : Multiset R[X]\nih :\n  prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0 \u2192\n    natDegree (prod t) = Multiset.sum (Multiset.map (fun f => natDegree f) t)\nht : leadingCoeff a * prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0\n\u22a2 prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0\n\ncase refine'_2\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\na : R[X]\nt : Multiset R[X]\nih :\n  prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0 \u2192\n    natDegree (prod t) = Multiset.sum (Multiset.map (fun f => natDegree f) t)\nht : leadingCoeff a * prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0\n\u22a2 leadingCoeff a * leadingCoeff (prod t) \u2260 0"}, {"tactic": "simp", "state_before": "case refine'_1\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\n\u22a2 prod (Multiset.map (fun f => leadingCoeff f) 0) \u2260 0 \u2192\n    natDegree (prod 0) = Multiset.sum (Multiset.map (fun f => natDegree f) 0)", "state_after": "no goals"}, {"tactic": "apply right_ne_zero_of_mul ht", "state_before": "case refine'_2\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\na : R[X]\nt : Multiset R[X]\nih :\n  prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0 \u2192\n    natDegree (prod t) = Multiset.sum (Multiset.map (fun f => natDegree f) t)\nht : leadingCoeff a * prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0\n\u22a2 prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0", "state_after": "no goals"}, {"tactic": "rwa [Polynomial.leadingCoeff_multiset_prod']", "state_before": "case refine'_2\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\na : R[X]\nt : Multiset R[X]\nih :\n  prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0 \u2192\n    natDegree (prod t) = Multiset.sum (Multiset.map (fun f => natDegree f) t)\nht : leadingCoeff a * prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0\n\u22a2 leadingCoeff a * leadingCoeff (prod t) \u2260 0", "state_after": "case refine'_2.h\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\na : R[X]\nt : Multiset R[X]\nih :\n  prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0 \u2192\n    natDegree (prod t) = Multiset.sum (Multiset.map (fun f => natDegree f) t)\nht : leadingCoeff a * prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0\n\u22a2 prod (Multiset.map leadingCoeff t) \u2260 0"}, {"tactic": "apply right_ne_zero_of_mul ht", "state_before": "case refine'_2.h\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt\u271d : Multiset R[X]\na : R[X]\nt : Multiset R[X]\nih :\n  prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0 \u2192\n    natDegree (prod t) = Multiset.sum (Multiset.map (fun f => natDegree f) t)\nht : leadingCoeff a * prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0\n\u22a2 prod (Multiset.map leadingCoeff t) \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Setoid/Basic.lean", "full_name": "Setoid.bot_def", "start": [193, 1], "end": [194, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Heyting/Basic.lean", "full_name": "sup_sdiff_distrib", "start": [643, 1], "end": [644, 83], "traced_tactics": [{"tactic": "simp_rw [sdiff_le_iff, sup_le_iff, sdiff_le_iff]", "state_before": "\u03b9 : Type ?u.136072\n\u03b1 : Type u_1\n\u03b2 : Type ?u.136078\ninst\u271d : GeneralizedCoheytingAlgebra \u03b1\na\u271d b\u271d c\u271d d\u271d a b c d : \u03b1\n\u22a2 (a \u2294 b) \\ c \u2264 d \u2194 a \\ c \u2294 b \\ c \u2264 d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Subobject/WellPowered.lean", "full_name": "CategoryTheory.wellPowered_of_equiv", "start": [83, 1], "end": [85, 78], "traced_tactics": [{"tactic": "infer_instance", "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category C\nD : Type u\u2082\ninst\u271d\u00b9 : Category D\ne : C \u224c D\ninst\u271d : WellPowered C\nX : D\n\u22a2 EssentiallySmall (MonoOver ((Equivalence.symm e).functor.obj X))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/BilinearForm.lean", "full_name": "BilinForm.sub_right", "start": [140, 1], "end": [141, 60], "traced_tactics": [{"tactic": "rw [sub_eq_add_neg, sub_eq_add_neg, add_right, neg_right]", "state_before": "R : Type ?u.61548\nM : Type ?u.61551\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_1\nM\u2081 : Type u_2\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type ?u.62199\nM\u2082 : Type ?u.62202\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type ?u.62389\nM\u2083 : Type ?u.62392\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type ?u.62980\nK : Type ?u.62983\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nx y z : M\u2081\n\u22a2 bilin B\u2081 x (y - z) = bilin B\u2081 x y - bilin B\u2081 x z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.getLast_append_singleton", "start": [705, 1], "end": [707, 29], "traced_tactics": [{"tactic": "simp only [getLast_append]", "state_before": "\u03b9 : Type ?u.30072\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 getLast (l ++ [a]) (_ : l ++ [a] \u2260 []) = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Logic.lean", "full_name": "and_congr_left_eq", "start": [174, 1], "end": [175, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean", "full_name": "Real.arcsin_eq_of_sin_eq", "start": [108, 1], "end": [111, 70], "traced_tactics": [{"tactic": "subst y", "state_before": "x y : \u211d\nh\u2081 : sin x = y\nh\u2082 : x \u2208 Icc (-(\u03c0 / 2)) (\u03c0 / 2)\n\u22a2 arcsin y = x", "state_after": "x : \u211d\nh\u2082 : x \u2208 Icc (-(\u03c0 / 2)) (\u03c0 / 2)\n\u22a2 arcsin (sin x) = x"}, {"tactic": "exact injOn_sin (arcsin_mem_Icc _) h\u2082 (sin_arcsin' (sin_mem_Icc x))", "state_before": "x : \u211d\nh\u2082 : x \u2208 Icc (-(\u03c0 / 2)) (\u03c0 / 2)\n\u22a2 arcsin (sin x) = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Order/Lattice.lean", "full_name": "Filter.Tendsto.sup_right_nhds", "start": [115, 1], "end": [118, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.range_familyOfBFamily'", "start": [1185, 1], "end": [1191, 47], "traced_tactics": [{"tactic": "refine' Set.ext fun a => \u27e8_, _\u27e9", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.275577\n\u03b3 : Type ?u.275580\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b9 : Type u\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ninst\u271d : IsWellOrder \u03b9 r\no : Ordinal\nho : type r = o\nf : (a : Ordinal) \u2192 a < o \u2192 \u03b1\n\u22a2 range (familyOfBFamily' r ho f) = brange o f", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type ?u.275577\n\u03b3 : Type ?u.275580\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b9 : Type u\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ninst\u271d : IsWellOrder \u03b9 r\no : Ordinal\nho : type r = o\nf : (a : Ordinal) \u2192 a < o \u2192 \u03b1\na : \u03b1\n\u22a2 a \u2208 range (familyOfBFamily' r ho f) \u2192 a \u2208 brange o f\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type ?u.275577\n\u03b3 : Type ?u.275580\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b9 : Type u\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ninst\u271d : IsWellOrder \u03b9 r\no : Ordinal\nho : type r = o\nf : (a : Ordinal) \u2192 a < o \u2192 \u03b1\na : \u03b1\n\u22a2 a \u2208 brange o f \u2192 a \u2208 range (familyOfBFamily' r ho f)"}, {"tactic": "rintro \u27e8b, rfl\u27e9", "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type ?u.275577\n\u03b3 : Type ?u.275580\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b9 : Type u\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ninst\u271d : IsWellOrder \u03b9 r\no : Ordinal\nho : type r = o\nf : (a : Ordinal) \u2192 a < o \u2192 \u03b1\na : \u03b1\n\u22a2 a \u2208 range (familyOfBFamily' r ho f) \u2192 a \u2208 brange o f", "state_after": "case refine'_1.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.275577\n\u03b3 : Type ?u.275580\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b9 : Type u\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ninst\u271d : IsWellOrder \u03b9 r\no : Ordinal\nho : type r = o\nf : (a : Ordinal) \u2192 a < o \u2192 \u03b1\nb : \u03b9\n\u22a2 familyOfBFamily' r ho f b \u2208 brange o f"}, {"tactic": "apply mem_brange_self", "state_before": "case refine'_1.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.275577\n\u03b3 : Type ?u.275580\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b9 : Type u\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ninst\u271d : IsWellOrder \u03b9 r\no : Ordinal\nho : type r = o\nf : (a : Ordinal) \u2192 a < o \u2192 \u03b1\nb : \u03b9\n\u22a2 familyOfBFamily' r ho f b \u2208 brange o f", "state_after": "no goals"}, {"tactic": "rintro \u27e8i, hi, rfl\u27e9", "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type ?u.275577\n\u03b3 : Type ?u.275580\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b9 : Type u\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ninst\u271d : IsWellOrder \u03b9 r\no : Ordinal\nho : type r = o\nf : (a : Ordinal) \u2192 a < o \u2192 \u03b1\na : \u03b1\n\u22a2 a \u2208 brange o f \u2192 a \u2208 range (familyOfBFamily' r ho f)", "state_after": "case refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.275577\n\u03b3 : Type ?u.275580\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b9 : Type u\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ninst\u271d : IsWellOrder \u03b9 r\no : Ordinal\nho : type r = o\nf : (a : Ordinal) \u2192 a < o \u2192 \u03b1\ni : Ordinal\nhi : i < o\n\u22a2 f i hi \u2208 range (familyOfBFamily' r ho f)"}, {"tactic": "exact \u27e8_, familyOfBFamily'_enum _ _ _ _ _\u27e9", "state_before": "case refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.275577\n\u03b3 : Type ?u.275580\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b9 : Type u\nr : \u03b9 \u2192 \u03b9 \u2192 Prop\ninst\u271d : IsWellOrder \u03b9 r\no : Ordinal\nho : type r = o\nf : (a : Ordinal) \u2192 a < o \u2192 \u03b1\ni : Ordinal\nhi : i < o\n\u22a2 f i hi \u2208 range (familyOfBFamily' r ho f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Parity.lean", "full_name": "Nat.bit1_div_bit0", "start": [259, 1], "end": [260, 63], "traced_tactics": [{"tactic": "rw [bit0_eq_two_mul, \u2190 Nat.div_div_eq_div_mul, bit1_div_two]", "state_before": "m n : \u2115\n\u22a2 bit1 n / bit0 m = n / m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finsupp/Basic.lean", "full_name": "Finsupp.filter_eq_indicator", "start": [899, 1], "end": [900, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Nat.ceil_add_nat", "start": [467, 1], "end": [474, 91], "traced_tactics": [{"tactic": "rw [\u2190 not_lt, \u2190 not_lt, not_iff_not, lt_ceil]", "state_before": "F : Type ?u.86599\n\u03b1 : Type u_1\n\u03b2 : Type ?u.86605\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn\u271d : \u2115\nha : 0 \u2264 a\nn b : \u2115\n\u22a2 \u2308a + \u2191n\u2309\u208a \u2264 b \u2194 \u2308a\u2309\u208a + n \u2264 b", "state_after": "F : Type ?u.86599\n\u03b1 : Type u_1\n\u03b2 : Type ?u.86605\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn\u271d : \u2115\nha : 0 \u2264 a\nn b : \u2115\n\u22a2 \u2191b < a + \u2191n \u2194 b < \u2308a\u2309\u208a + n"}, {"tactic": "obtain hb | hb := le_or_lt n b", "state_before": "F : Type ?u.86599\n\u03b1 : Type u_1\n\u03b2 : Type ?u.86605\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn\u271d : \u2115\nha : 0 \u2264 a\nn b : \u2115\n\u22a2 \u2191b < a + \u2191n \u2194 b < \u2308a\u2309\u208a + n", "state_after": "case inl\nF : Type ?u.86599\n\u03b1 : Type u_1\n\u03b2 : Type ?u.86605\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn\u271d : \u2115\nha : 0 \u2264 a\nn b : \u2115\nhb : n \u2264 b\n\u22a2 \u2191b < a + \u2191n \u2194 b < \u2308a\u2309\u208a + n\n\ncase inr\nF : Type ?u.86599\n\u03b1 : Type u_1\n\u03b2 : Type ?u.86605\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn\u271d : \u2115\nha : 0 \u2264 a\nn b : \u2115\nhb : b < n\n\u22a2 \u2191b < a + \u2191n \u2194 b < \u2308a\u2309\u208a + n"}, {"tactic": "obtain \u27e8d, rfl\u27e9 := exists_add_of_le hb", "state_before": "case inl\nF : Type ?u.86599\n\u03b1 : Type u_1\n\u03b2 : Type ?u.86605\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn\u271d : \u2115\nha : 0 \u2264 a\nn b : \u2115\nhb : n \u2264 b\n\u22a2 \u2191b < a + \u2191n \u2194 b < \u2308a\u2309\u208a + n", "state_after": "case inl.intro\nF : Type ?u.86599\n\u03b1 : Type u_1\n\u03b2 : Type ?u.86605\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn\u271d : \u2115\nha : 0 \u2264 a\nn d : \u2115\nhb : n \u2264 n + d\n\u22a2 \u2191(n + d) < a + \u2191n \u2194 n + d < \u2308a\u2309\u208a + n"}, {"tactic": "rw [Nat.cast_add, add_comm n, add_comm (n : \u03b1), add_lt_add_iff_right, add_lt_add_iff_right,\n  lt_ceil]", "state_before": "case inl.intro\nF : Type ?u.86599\n\u03b1 : Type u_1\n\u03b2 : Type ?u.86605\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn\u271d : \u2115\nha : 0 \u2264 a\nn d : \u2115\nhb : n \u2264 n + d\n\u22a2 \u2191(n + d) < a + \u2191n \u2194 n + d < \u2308a\u2309\u208a + n", "state_after": "no goals"}, {"tactic": "exact iff_of_true (lt_add_of_nonneg_of_lt ha <| cast_lt.2 hb) (lt_add_left _ _ _ hb)", 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1], "end": [62, 6], "traced_tactics": [{"tactic": "letI := f.toAlgebra", "state_before": "R A B : CommRingCat\nf : R \u27f6 A\ng : R \u27f6 B\n\u22a2 A \u27f6 (pushoutCocone f g).pt", "state_after": "R A B : CommRingCat\nf : R \u27f6 A\ng : R \u27f6 B\nthis : Algebra \u2191R \u2191A := RingHom.toAlgebra f\n\u22a2 A \u27f6 (pushoutCocone f g).pt"}, {"tactic": "letI := g.toAlgebra", "state_before": "R A B : CommRingCat\nf : R \u27f6 A\ng : R \u27f6 B\nthis : Algebra \u2191R \u2191A := RingHom.toAlgebra f\n\u22a2 A \u27f6 (pushoutCocone f g).pt", "state_after": "R A B : CommRingCat\nf : R \u27f6 A\ng : R \u27f6 B\nthis\u271d : Algebra \u2191R \u2191A := RingHom.toAlgebra f\nthis : Algebra \u2191R \u2191B := RingHom.toAlgebra g\n\u22a2 A \u27f6 (pushoutCocone f g).pt"}, {"tactic": "exact Algebra.TensorProduct.includeLeft.toRingHom", "state_before": "R A B : CommRingCat\nf : R \u27f6 A\ng : R \u27f6 B\nthis\u271d : Algebra \u2191R \u2191A := RingHom.toAlgebra f\nthis : Algebra \u2191R 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"Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Ico_ae_eq_Ioc'", "start": [2995, 1], "end": [2996, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/FilterBasis.lean", "full_name": "GroupFilterBasis.N_one", "start": [161, 1], "end": [162, 34], "traced_tactics": [{"tactic": "simp only [N, one_mul, map_id']", "state_before": "G : Type u\ninst\u271d : Group G\nB\u271d B : GroupFilterBasis G\n\u22a2 N B 1 = FilterBasis.filter toFilterBasis", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean", "full_name": "HasDerivAt.arctan", "start": [139, 1], "end": [141, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": 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e'", "state_before": "\u03b1 : Type u_1\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b1\nl\u2082 r\u2081 r\u2082 : List \u03b1\ne : b :: l\u2082 ++ r\u2082 = a :: l\u2081 ++ r\u2081\nll : length (a :: l\u2081) \u2264 length (b :: l\u2082)\n\u22a2 a :: l\u2081 <+: b :: l\u2082", "state_after": "\u03b1 : Type u_1\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b1\nl\u2082 r\u2081 r\u2082 : List \u03b1\nll : length (a :: l\u2081) \u2264 length (b :: l\u2082)\nhead_eq\u271d : b = a\ne' : List.append l\u2082 r\u2082 = List.append l\u2081 r\u2081\n\u22a2 a :: l\u2081 <+: b :: l\u2082"}, {"tactic": "subst b", "state_before": "\u03b1 : Type u_1\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b1\nl\u2082 r\u2081 r\u2082 : List \u03b1\nll : length (a :: l\u2081) \u2264 length (b :: l\u2082)\nhead_eq\u271d : b = a\ne' : List.append l\u2082 r\u2082 = List.append l\u2081 r\u2081\n\u22a2 a :: l\u2081 <+: b :: l\u2082", "state_after": "\u03b1 : Type u_1\na : \u03b1\nl\u2081 l\u2082 r\u2081 r\u2082 : List 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[494, 1], "end": [495, 87], "traced_tactics": [{"tactic": "simp only [mem_iff, star_mul, star_star, mem_iff.mp hx, neg_mul, mul_neg, mul_assoc]", "state_before": "R : Type u_1\nA : Type ?u.111600\ninst\u271d\u00b9 : Ring R\ninst\u271d : StarRing R\nx : R\nhx : x \u2208 skewAdjoint R\nz : R\n\u22a2 star z * x * z \u2208 skewAdjoint R", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Star/SelfAdjoint.lean", "full_name": "IsSelfAdjoint.conjugate", "start": [159, 1], "end": [160, 76], "traced_tactics": [{"tactic": "simp only [isSelfAdjoint_iff, star_mul, star_star, mul_assoc, hx.star_eq]", "state_before": "R : Type u_1\nA : Type ?u.18062\ninst\u271d\u00b9 : Semigroup R\ninst\u271d : StarSemigroup R\nx : R\nhx : IsSelfAdjoint x\nz : R\n\u22a2 IsSelfAdjoint (z * x * star z)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/PiLp.lean", "full_name": "PiLp.nndist_equiv_symm_single_same", "start": [857, 1], "end": [861, 30], "traced_tactics": [{"tactic": "rw [nndist_eq_nnnorm, nndist_eq_nnnorm, \u2190 equiv_symm_sub, \u2190 Pi.single_sub,\n  nnnorm_equiv_symm_single]", "state_before": "p : \u211d\u22650\u221e\n\ud835\udd5c : Type ?u.488127\n\ud835\udd5c' : Type ?u.488130\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type ?u.488138\n\u03b2 : \u03b9 \u2192 Type u_2\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fact (1 \u2264 p)\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : NormedField \ud835\udd5c'\ninst\u271d\u2075 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\nx y : PiLp p \u03b2\nx' y' : (i : \u03b9) \u2192 \u03b2 i\ni\u271d : \u03b9\n\u03b9' : Type ?u.488339\ninst\u271d\u00b3 : 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DecidableEq \u03b9''\ni : \u03b9\nj : \u03b9''\n\u22a2 (toMatrix b \u2191b' \u2b1d toMatrix b' b'') i j = toMatrix b b'' i j"}, {"tactic": "simp only [Matrix.mul_apply, Basis.toMatrix_apply, Basis.sum_repr_mul_repr]", "state_before": "case a.h\n\u03b9 : Type u_3\n\u03b9' : Type u_2\n\u03ba : Type ?u.623919\n\u03ba' : Type ?u.623922\nR : Type u_4\nM : Type u_5\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\nR\u2082 : Type ?u.624115\nM\u2082 : Type ?u.624118\ninst\u271d\u2075 : CommRing R\u2082\ninst\u271d\u2074 : AddCommGroup M\u2082\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ne : Basis \u03b9 R M\nv : \u03b9' \u2192 M\ni\u271d : \u03b9\nj\u271d : \u03b9'\nN : Type ?u.624863\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : Module R N\nb : Basis \u03b9 R M\nb' : Basis \u03b9' R M\nc : Basis \u03ba R N\nc' : Basis \u03ba' R N\nf : M \u2192\u2097[R] N\n\u03b9'' : Type u_1\ninst\u271d : Fintype \u03b9'\nb'' : \u03b9'' \u2192 M\nthis\u271d\u00b9 : DecidableEq \u03b9\nthis\u271d : DecidableEq \u03b9'\nthis : DecidableEq \u03b9''\ni : \u03b9\nj : \u03b9''\n\u22a2 (toMatrix b \u2191b' \u2b1d toMatrix b' b'') i j = toMatrix b b'' i j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fintype/Fin.lean", "full_name": "Fin.Ioi_zero_eq_map", "start": [34, 1], "end": [44, 21], "traced_tactics": [{"tactic": "ext i", "state_before": "\u03b1 : Type ?u.5462\n\u03b2 : Type ?u.5465\nn : \u2115\n\u22a2 Ioi 0 = map (succEmbedding n).toEmbedding univ", "state_after": "case a\n\u03b1 : Type ?u.5462\n\u03b2 : Type ?u.5465\nn : \u2115\ni : Fin (Nat.succ n)\n\u22a2 i \u2208 Ioi 0 \u2194 i \u2208 map (succEmbedding n).toEmbedding univ"}, {"tactic": "simp only [mem_Ioi, mem_map, mem_univ, Function.Embedding.coeFn_mk, exists_true_left]", "state_before": "case a\n\u03b1 : Type ?u.5462\n\u03b2 : Type ?u.5465\nn : \u2115\ni : Fin (Nat.succ n)\n\u22a2 i \u2208 Ioi 0 \u2194 i \u2208 map (succEmbedding n).toEmbedding univ", "state_after": "case a\n\u03b1 : Type ?u.5462\n\u03b2 : Type ?u.5465\nn : \u2115\ni : Fin (Nat.succ n)\n\u22a2 0 < i \u2194 \u2203 a, True \u2227 \u2191(succEmbedding n).toEmbedding a = i"}, {"tactic": "constructor", "state_before": "case a\n\u03b1 : Type ?u.5462\n\u03b2 : Type ?u.5465\nn : \u2115\ni : Fin (Nat.succ n)\n\u22a2 0 < i \u2194 \u2203 a, True \u2227 \u2191(succEmbedding n).toEmbedding a = i", "state_after": "case a.mp\n\u03b1 : Type ?u.5462\n\u03b2 : Type ?u.5465\nn : \u2115\ni : Fin (Nat.succ n)\n\u22a2 0 < i \u2192 \u2203 a, True \u2227 \u2191(succEmbedding n).toEmbedding a = i\n\ncase a.mpr\n\u03b1 : Type ?u.5462\n\u03b2 : Type ?u.5465\nn : \u2115\ni : Fin (Nat.succ n)\n\u22a2 (\u2203 a, True \u2227 \u2191(succEmbedding n).toEmbedding a = i) \u2192 0 < i"}, {"tactic": "refine' cases _ _ i", "state_before": "case a.mp\n\u03b1 : Type ?u.5462\n\u03b2 : Type ?u.5465\nn : \u2115\ni : Fin (Nat.succ n)\n\u22a2 0 < i \u2192 \u2203 a, True \u2227 \u2191(succEmbedding n).toEmbedding a = i", "state_after": "case a.mp.refine'_1\n\u03b1 : Type ?u.5462\n\u03b2 : Type ?u.5465\nn : \u2115\ni : Fin (Nat.succ n)\n\u22a2 0 < 0 \u2192 \u2203 a, True \u2227 \u2191(succEmbedding n).toEmbedding a = 0\n\ncase a.mp.refine'_2\n\u03b1 : Type ?u.5462\n\u03b2 : Type ?u.5465\nn : \u2115\ni : Fin (Nat.succ n)\n\u22a2 \u2200 (i : Fin n), 0 < succ i \u2192 \u2203 a, True \u2227 \u2191(succEmbedding n).toEmbedding a = succ i"}, {"tactic": "rintro \u27e8\u27e8\u27e9\u27e9", "state_before": "case a.mp.refine'_1\n\u03b1 : Type ?u.5462\n\u03b2 : Type ?u.5465\nn : \u2115\ni : Fin (Nat.succ n)\n\u22a2 0 < 0 \u2192 \u2203 a, True \u2227 \u2191(succEmbedding n).toEmbedding a = 0", "state_after": "no goals"}, {"tactic": "intro j _", "state_before": "case a.mp.refine'_2\n\u03b1 : Type ?u.5462\n\u03b2 : Type ?u.5465\nn : \u2115\ni : Fin (Nat.succ n)\n\u22a2 \u2200 (i : Fin n), 0 < succ i \u2192 \u2203 a, True \u2227 \u2191(succEmbedding n).toEmbedding a = succ i", "state_after": "case a.mp.refine'_2\n\u03b1 : Type ?u.5462\n\u03b2 : Type ?u.5465\nn : \u2115\ni : Fin (Nat.succ n)\nj : Fin n\na\u271d : 0 < succ j\n\u22a2 \u2203 a, True \u2227 \u2191(succEmbedding n).toEmbedding a = succ j"}, {"tactic": "use j", "state_before": "case a.mp.refine'_2\n\u03b1 : Type ?u.5462\n\u03b2 : Type ?u.5465\nn : \u2115\ni : Fin (Nat.succ n)\nj : Fin n\na\u271d : 0 < succ j\n\u22a2 \u2203 a, True \u2227 \u2191(succEmbedding n).toEmbedding a = succ j", "state_after": "case a.mp.refine'_2\n\u03b1 : Type ?u.5462\n\u03b2 : Type ?u.5465\nn : \u2115\ni : Fin (Nat.succ n)\nj : Fin n\na\u271d : 0 < succ j\n\u22a2 True \u2227 \u2191(succEmbedding n).toEmbedding j = succ j"}, {"tactic": "simp only [val_succEmbedding, and_self, RelEmbedding.coe_toEmbedding]", "state_before": "case a.mp.refine'_2\n\u03b1 : Type ?u.5462\n\u03b2 : Type ?u.5465\nn : \u2115\ni : Fin (Nat.succ n)\nj : Fin n\na\u271d : 0 < succ j\n\u22a2 True \u2227 \u2191(succEmbedding n).toEmbedding j = succ j", "state_after": "no goals"}, {"tactic": "rintro \u27e8i, _, rfl\u27e9", "state_before": "case a.mpr\n\u03b1 : Type ?u.5462\n\u03b2 : Type ?u.5465\nn : \u2115\ni : Fin (Nat.succ n)\n\u22a2 (\u2203 a, True \u2227 \u2191(succEmbedding n).toEmbedding a = i) \u2192 0 < i", "state_after": "case a.mpr.intro.intro\n\u03b1 : Type ?u.5462\n\u03b2 : Type ?u.5465\nn : \u2115\ni : Fin n\nleft\u271d : True\n\u22a2 0 < \u2191(succEmbedding n).toEmbedding i"}, {"tactic": "exact succ_pos _", "state_before": "case a.mpr.intro.intro\n\u03b1 : Type ?u.5462\n\u03b2 : Type ?u.5465\nn : \u2115\ni : Fin n\nleft\u271d : True\n\u22a2 0 < \u2191(succEmbedding n).toEmbedding i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.apply_piecewise\u2082", "start": [1477, 1], "end": [1481, 39], "traced_tactics": [{"tactic": "by_cases hx : x \u2208 s <;> simp [hx]", "state_before": "\u03b1 : Type u_3\n\u03b2 : Type ?u.91797\n\u03b3 : Type ?u.91800\n\u03b9 : Sort ?u.91803\n\u03c0 : \u03b1 \u2192 Type ?u.91808\n\u03b4 : \u03b1 \u2192 Sort u_4\ns : Set \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\n\u03b4' : \u03b1 \u2192 Sort u_1\n\u03b4'' : \u03b1 \u2192 Sort u_2\nf' g' : (i : \u03b1) \u2192 \u03b4' i\nh : (i : \u03b1) \u2192 \u03b4 i \u2192 \u03b4' i \u2192 \u03b4'' i\nx : \u03b1\n\u22a2 h x (piecewise s f g x) (piecewise s f' g' x) = piecewise s (fun x => h x (f x) (f' x)) (fun x => h x (g x) (g' x)) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Seq/WSeq.lean", "full_name": "Stream'.WSeq.toList_cons", "start": [1334, 1], "end": [1335, 75], "traced_tactics": [{"tactic": "unfold toList", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : WSeq \u03b1\n\u22a2 Computation.destruct (toList (cons a s)) = Sum.inr (List.cons a <$> toList s)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : WSeq \u03b1\n\u22a2 Computation.destruct\n      (Computation.corec\n        (fun x =>\n          match x with\n          | (l, s) =>\n            match Seq.destruct s with\n            | none => Sum.inl (List.reverse l)\n            | some (none, s') => Sum.inr (l, s')\n            | some (some a, s') => Sum.inr (a :: l, s'))\n        ([], cons a s)) =\n    Sum.inr\n      (List.cons a <$>\n        Computation.corec\n          (fun x =>\n            match x with\n            | (l, s) =>\n              match Seq.destruct s with\n              | none => Sum.inl (List.reverse l)\n              | some (none, s') => Sum.inr (l, s')\n              | some (some a, s') => Sum.inr (a :: l, s'))\n          ([], s))"}, {"tactic": "simp", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : WSeq \u03b1\n\u22a2 Computation.destruct\n      (Computation.corec\n        (fun x =>\n          match x with\n          | (l, s) =>\n            match Seq.destruct s with\n            | none => Sum.inl (List.reverse l)\n            | some (none, s') => Sum.inr (l, s')\n            | some (some a, s') => Sum.inr (a :: l, s'))\n        ([], cons a s)) =\n    Sum.inr\n      (List.cons a <$>\n        Computation.corec\n          (fun x =>\n            match x with\n            | (l, s) =>\n              match Seq.destruct s with\n              | none => Sum.inl (List.reverse l)\n              | some (none, s') => Sum.inr (l, s')\n              | some (some a, s') => Sum.inr (a :: l, s'))\n          ([], s))", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : WSeq \u03b1\n\u22a2 Computation.corec\n      (fun x =>\n        match Seq.destruct x.snd with\n        | none => Sum.inl (List.reverse x.fst)\n        | some (none, s') => Sum.inr (x.fst, s')\n        | some (some a, s') => Sum.inr (a :: x.fst, s'))\n      ([a], s) =\n    List.cons a <$>\n      Computation.corec\n        (fun x =>\n          match Seq.destruct x.snd with\n          | none => Sum.inl (List.reverse x.fst)\n          | some (none, s') => Sum.inr (x.fst, s')\n          | some (some a, s') => Sum.inr (a :: x.fst, s'))\n        ([], s)"}, {"tactic": "rw [toList'_map]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : WSeq \u03b1\n\u22a2 Computation.corec\n      (fun x =>\n        match Seq.destruct x.snd with\n        | none => Sum.inl (List.reverse x.fst)\n        | some (none, s') => Sum.inr (x.fst, s')\n        | some (some a, s') => Sum.inr (a :: x.fst, s'))\n      ([a], s) =\n    List.cons a <$>\n      Computation.corec\n        (fun x =>\n          match Seq.destruct x.snd with\n          | none => Sum.inl (List.reverse x.fst)\n          | some (none, s') => Sum.inr (x.fst, s')\n          | some (some a, s') => Sum.inr (a :: x.fst, s'))\n        ([], s)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : WSeq \u03b1\n\u22a2 (fun x x_1 => x ++ x_1) (List.reverse [a]) <$> toList s =\n    List.cons a <$>\n      Computation.corec\n        (fun x =>\n          match Seq.destruct x.snd with\n          | none => Sum.inl (List.reverse x.fst)\n          | some (none, s') => Sum.inr (x.fst, s')\n          | some (some a, s') => Sum.inr (a :: x.fst, s'))\n        ([], s)"}, {"tactic": "simp", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : WSeq \u03b1\n\u22a2 (fun x x_1 => x ++ x_1) (List.reverse [a]) <$> toList s =\n    List.cons a <$>\n      Computation.corec\n        (fun x =>\n          match Seq.destruct x.snd with\n          | none => Sum.inl (List.reverse x.fst)\n          | some (none, s') => Sum.inr (x.fst, s')\n          | some (some a, s') => Sum.inr (a :: x.fst, s'))\n        ([], s)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : WSeq \u03b1\n\u22a2 (fun x => a :: x) <$> toList s =\n    List.cons a <$>\n      Computation.corec\n        (fun x =>\n          match Seq.destruct x.snd with\n          | none => Sum.inl (List.reverse x.fst)\n          | some (none, s') => Sum.inr (x.fst, s')\n          | some (some a, s') => Sum.inr (a :: x.fst, s'))\n        ([], s)"}, {"tactic": "rfl", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\ns : WSeq \u03b1\n\u22a2 (fun x => a :: x) <$> toList s =\n    List.cons a <$>\n      Computation.corec\n        (fun x =>\n          match Seq.destruct x.snd with\n          | none => Sum.inl (List.reverse x.fst)\n          | some (none, s') => Sum.inr (x.fst, s')\n          | some (some a, s') => Sum.inr (a :: x.fst, s'))\n        ([], s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/Separable.lean", "full_name": "Polynomial.nodup_roots", "start": [267, 1], "end": [268, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/CharZero/Lemmas.lean", "full_name": "half_add_self", "start": [163, 1], "end": [163, 99], "traced_tactics": [{"tactic": "rw [\u2190 mul_two, mul_div_cancel a two_ne_zero]", "state_before": "R : Type u_1\ninst\u271d\u00b9 : DivisionRing R\ninst\u271d : CharZero R\na : R\n\u22a2 (a + a) / 2 = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.setToFun_congr_left", "start": [1302, 1], "end": [1307, 34], "traced_tactics": [{"tactic": "by_cases hf : Integrable f \u03bc", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1331504\nG : Type ?u.1331507\n\ud835\udd5c : Type ?u.1331510\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf : \u03b1 \u2192 E\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T' hT' f", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1331504\nG : Type ?u.1331507\n\ud835\udd5c : Type ?u.1331510\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T' hT' f\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1331504\nG : Type ?u.1331507\n\ud835\udd5c : Type ?u.1331510\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf : \u03b1 \u2192 E\nhf : \u00acIntegrable f\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T' hT' f"}, {"tactic": "simp_rw [setToFun_eq _ hf, L1.setToL1_congr_left T T' hT hT' h]", "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1331504\nG : Type ?u.1331507\n\ud835\udd5c : Type ?u.1331510\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf\u271d g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : T = T'\nf : \u03b1 \u2192 E\nhf : Integrable f\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T' hT' f", "state_after": "no goals"}, {"tactic": "simp_rw [setToFun_undef _ hf]", "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1331504\nG : Type ?u.1331507\n\ud835\udd5c : Type ?u.1331510\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace 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u 0 = x"}, {"tactic": "rfl", "state_before": "\u03b1 : Type u\nm n o : \u2115\nm' : Type ?u.22598\nn' : Type ?u.22601\no' : Type ?u.22604\nx : \u03b1\nu : Fin 0 \u2192 \u03b1\n\u22a2 vecCons x u 0 = x", "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_one_iff_mem_unitary", "start": [1082, 1], "end": [1084, 12], "traced_tactics": [{"tactic": "rw [unitary.mem_iff_self_mul_star, \u2190 norm_eq_mul_conj]", "state_before": "R : Type\ninst\u271d : CommRing R\nd : \u2124\na : \u2124\u221ad\n\u22a2 norm a = 1 \u2194 a \u2208 unitary (\u2124\u221ad)", "state_after": "R : Type\ninst\u271d : CommRing R\nd : \u2124\na : \u2124\u221ad\n\u22a2 norm a = 1 \u2194 \u2191(norm a) = 1"}, {"tactic": "norm_cast", "state_before": "R : Type\ninst\u271d : CommRing R\nd : \u2124\na : \u2124\u221ad\n\u22a2 norm a = 1 \u2194 \u2191(norm a) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Young/YoungDiagram.lean", "full_name": "YoungDiagram.mem_col_iff", "start": [351, 1], "end": [352, 13], "traced_tactics": [{"tactic": "simp [col]", "state_before": "\u03bc : YoungDiagram\nj : \u2115\nc : \u2115 \u00d7 \u2115\n\u22a2 c \u2208 col \u03bc j \u2194 c \u2208 \u03bc \u2227 c.snd = j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Localization/Basic.lean", "full_name": "IsLocalization.isDomain_of_le_nonZeroDivisors", "start": [1247, 1], "end": [1253, 52], "traced_tactics": [{"tactic": "apply @NoZeroDivisors.to_isDomain _ _ (id _) (id _)", "state_before": "R : Type ?u.3159419\ninst\u271d\u00b9\u2070 : CommRing R\nM\u271d : Submonoid R\nS : Type u_2\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\nP : Type ?u.3159848\ninst\u271d\u2077 : CommRing P\nK : Type ?u.3159854\ninst\u271d\u2076 : IsLocalization M\u271d S\nQ : Type ?u.3160035\ninst\u271d\u2075 : CommRing Q\ng : R \u2192+* P\ninst\u271d\u2074 : Algebra P Q\nA : Type u_1\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A S\nM : Submonoid A\ninst\u271d : IsLocalization M S\nhM : M \u2264 nonZeroDivisors A\n\u22a2 IsDomain S", "state_after": "R : Type ?u.3159419\ninst\u271d\u00b9\u2070 : CommRing R\nM\u271d : Submonoid R\nS : Type u_2\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\nP : Type ?u.3159848\ninst\u271d\u2077 : CommRing P\nK : Type ?u.3159854\ninst\u271d\u2076 : IsLocalization M\u271d S\nQ : Type ?u.3160035\ninst\u271d\u2075 : CommRing Q\ng : R \u2192+* P\ninst\u271d\u2074 : Algebra P Q\nA : Type u_1\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A S\nM : Submonoid A\ninst\u271d : IsLocalization M S\nhM : M \u2264 nonZeroDivisors A\n\u22a2 Nontrivial S\n\nR : Type ?u.3159419\ninst\u271d\u00b9\u2070 : CommRing R\nM\u271d : Submonoid R\nS : Type u_2\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\nP : Type ?u.3159848\ninst\u271d\u2077 : CommRing P\nK : Type ?u.3159854\ninst\u271d\u2076 : IsLocalization M\u271d S\nQ : Type ?u.3160035\ninst\u271d\u2075 : CommRing Q\ng : R \u2192+* P\ninst\u271d\u2074 : Algebra P Q\nA : Type u_1\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A S\nM : Submonoid A\ninst\u271d : IsLocalization M S\nhM : M \u2264 nonZeroDivisors A\n\u22a2 NoZeroDivisors S"}, {"tactic": "exact\n  \u27e8\u27e8(algebraMap A S) 0, (algebraMap A S) 1, fun h =>\n      zero_ne_one (IsLocalization.injective S hM h)\u27e9\u27e9", "state_before": "R : Type ?u.3159419\ninst\u271d\u00b9\u2070 : CommRing R\nM\u271d : Submonoid R\nS : Type u_2\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\nP : Type ?u.3159848\ninst\u271d\u2077 : CommRing P\nK : Type ?u.3159854\ninst\u271d\u2076 : IsLocalization M\u271d S\nQ : Type ?u.3160035\ninst\u271d\u2075 : CommRing Q\ng : R \u2192+* P\ninst\u271d\u2074 : Algebra P Q\nA : Type u_1\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A S\nM : Submonoid A\ninst\u271d : IsLocalization M S\nhM : M \u2264 nonZeroDivisors A\n\u22a2 Nontrivial S", "state_after": "no goals"}, {"tactic": "exact noZeroDivisors_of_le_nonZeroDivisors _ hM", "state_before": "R : Type ?u.3159419\ninst\u271d\u00b9\u2070 : CommRing R\nM\u271d : Submonoid R\nS : Type u_2\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\nP : Type ?u.3159848\ninst\u271d\u2077 : CommRing P\nK : Type ?u.3159854\ninst\u271d\u2076 : IsLocalization M\u271d S\nQ : Type ?u.3160035\ninst\u271d\u2075 : CommRing Q\ng : R \u2192+* P\ninst\u271d\u2074 : Algebra P Q\nA : Type u_1\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A S\nM : Submonoid A\ninst\u271d : IsLocalization M S\nhM : M \u2264 nonZeroDivisors A\n\u22a2 NoZeroDivisors S", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "iInf_mono'", "start": [862, 1], "end": [863, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean", "full_name": "AffineIndependent.finrank_vectorSpan", "start": [124, 1], "end": [128, 46], "traced_tactics": [{"tactic": "rw [\u2190 Finset.card_univ] at hc", "state_before": "k : Type u_2\nV : Type u_3\nP : Type u_4\n\u03b9 : Type u_1\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\nhi : AffineIndependent k p\nn : \u2115\nhc : Fintype.card \u03b9 = n + 1\n\u22a2 finrank k { x // x \u2208 vectorSpan k (Set.range p) } = n", "state_after": "k : Type u_2\nV : Type u_3\nP : Type u_4\n\u03b9 : Type u_1\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\nhi : AffineIndependent k p\nn : \u2115\nhc : Finset.card Finset.univ = n + 1\n\u22a2 finrank k { x // x \u2208 vectorSpan k (Set.range p) } = n"}, {"tactic": "rw [\u2190 Set.image_univ, \u2190 Finset.coe_univ, \u2190 Finset.coe_image]", "state_before": "k : Type u_2\nV : Type u_3\nP : Type u_4\n\u03b9 : Type u_1\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\nhi : AffineIndependent k p\nn : \u2115\nhc : Finset.card Finset.univ = n + 1\n\u22a2 finrank k { x // x \u2208 vectorSpan k (Set.range p) } = n", "state_after": "k : Type u_2\nV : Type u_3\nP : Type u_4\n\u03b9 : Type u_1\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\nhi : AffineIndependent k p\nn : \u2115\nhc : Finset.card Finset.univ = n + 1\n\u22a2 finrank k { x // x \u2208 vectorSpan k \u2191(Finset.image p Finset.univ) } = n"}, {"tactic": "exact hi.finrank_vectorSpan_image_finset hc", "state_before": "k : Type u_2\nV : Type u_3\nP : Type u_4\n\u03b9 : Type u_1\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\nhi : AffineIndependent k p\nn : \u2115\nhc : Finset.card Finset.univ = n + 1\n\u22a2 finrank k { x // x \u2208 vectorSpan k \u2191(Finset.image p Finset.univ) } = n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Subring/Basic.lean", "full_name": "Subring.centralizer_le", "start": [873, 1], "end": [874, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Chain.lean", "full_name": "List.Chain'.infix", "start": [309, 1], "end": [311, 41], "traced_tactics": [{"tactic": "rcases h' with \u27e8l\u2082, l\u2083, rfl\u27e9", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\nh : Chain' R l\nh' : l\u2081 <:+: l\n\u22a2 Chain' R l\u2081", "state_after": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u2081 l\u2082\u271d : List \u03b1\na b : \u03b1\nl\u2082 l\u2083 : List \u03b1\nh : Chain' R (l\u2082 ++ l\u2081 ++ l\u2083)\n\u22a2 Chain' R l\u2081"}, {"tactic": "exact h.left_of_append.right_of_append", "state_before": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u2081 l\u2082\u271d : List \u03b1\na b : \u03b1\nl\u2082 l\u2083 : List \u03b1\nh : Chain' R (l\u2082 ++ l\u2081 ++ l\u2083)\n\u22a2 Chain' R l\u2081", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Prod.lean", "full_name": "MulHom.coe_fst", "start": [323, 1], "end": [324, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "Monotone.intervalIntegrable", "start": [392, 1], "end": [394, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.le_map_of_right_inverse", "start": [2860, 1], "end": [2863, 20], "traced_tactics": [{"tactic": "rw [\u2190 @map_id _ g, \u2190 map_congr h\u2081, \u2190 map_map]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.308339\n\u03b9 : Sort x\nmab : \u03b1 \u2192 \u03b2\nmba : \u03b2 \u2192 \u03b1\nf : Filter \u03b1\ng : Filter \u03b2\nh\u2081 : mab \u2218 mba =\u1da0[g] id\nh\u2082 : Tendsto mba g f\n\u22a2 g \u2264 map mab f", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.308339\n\u03b9 : Sort x\nmab : \u03b1 \u2192 \u03b2\nmba : \u03b2 \u2192 \u03b1\nf : Filter \u03b1\ng : Filter \u03b2\nh\u2081 : mab \u2218 mba =\u1da0[g] id\nh\u2082 : Tendsto mba g f\n\u22a2 map mab (map mba g) \u2264 map mab f"}, {"tactic": "exact map_mono h\u2082", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.308339\n\u03b9 : Sort x\nmab : \u03b1 \u2192 \u03b2\nmba : \u03b2 \u2192 \u03b1\nf : Filter \u03b1\ng : Filter \u03b2\nh\u2081 : mab \u2218 mba =\u1da0[g] id\nh\u2082 : Tendsto mba g f\n\u22a2 map mab (map mba g) \u2264 map mab f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Tactic/PushNeg.lean", "full_name": "Mathlib.Tactic.PushNeg.not_exists_eq", "start": [24, 1], "end": [24, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Additive/SalemSpencer.lean", "full_name": "addSalemSpencer_frontier", "start": [288, 1], "end": [296, 26], "traced_tactics": [{"tactic": "intro a b c ha hb hc habc", "state_before": "F : Type ?u.111497\n\u03b1 : Type ?u.111500\n\u03b2 : Type ?u.111503\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs\u2080 : IsClosed s\nhs\u2081 : StrictConvex \ud835\udd5c s\n\u22a2 AddSalemSpencer (frontier s)", "state_after": "F : Type ?u.111497\n\u03b1 : Type ?u.111500\n\u03b2 : Type ?u.111503\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs\u2080 : IsClosed s\nhs\u2081 : StrictConvex \ud835\udd5c s\na b c : E\nha : a \u2208 frontier s\nhb : b \u2208 frontier s\nhc : c \u2208 frontier s\nhabc : a + b = c + c\n\u22a2 a = b"}, {"tactic": "obtain rfl : (1 / 2 : \ud835\udd5c) \u2022 a + (1 / 2 : \ud835\udd5c) \u2022 b = c := by\n  rwa [\u2190 smul_add, one_div, inv_smul_eq_iff\u2080 (show (2 : \ud835\udd5c) \u2260 0 by norm_num), two_smul]", "state_before": "F : Type ?u.111497\n\u03b1 : Type ?u.111500\n\u03b2 : Type ?u.111503\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs\u2080 : IsClosed s\nhs\u2081 : StrictConvex \ud835\udd5c s\na b c : E\nha : a \u2208 frontier s\nhb : b \u2208 frontier s\nhc : c \u2208 frontier s\nhabc : a + b = c + c\n\u22a2 a = b", "state_after": "F : Type ?u.111497\n\u03b1 : Type ?u.111500\n\u03b2 : Type ?u.111503\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs\u2080 : IsClosed s\nhs\u2081 : StrictConvex \ud835\udd5c s\na b : E\nha : a \u2208 frontier s\nhb : b \u2208 frontier s\nhc : (1 / 2) \u2022 a + (1 / 2) \u2022 b \u2208 frontier s\nhabc : a + b = (1 / 2) \u2022 a + (1 / 2) \u2022 b + ((1 / 2) \u2022 a + (1 / 2) \u2022 b)\n\u22a2 a = b"}, {"tactic": "exact\n  hs\u2081.eq (hs\u2080.frontier_subset ha) (hs\u2080.frontier_subset hb) one_half_pos one_half_pos\n    (add_halves _) hc.2", "state_before": "F : Type ?u.111497\n\u03b1 : Type ?u.111500\n\u03b2 : Type ?u.111503\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs\u2080 : IsClosed s\nhs\u2081 : StrictConvex \ud835\udd5c s\na b : E\nha : a \u2208 frontier s\nhb : b \u2208 frontier s\nhc : (1 / 2) \u2022 a + (1 / 2) \u2022 b \u2208 frontier s\nhabc : a + b = (1 / 2) \u2022 a + (1 / 2) \u2022 b + ((1 / 2) \u2022 a + (1 / 2) \u2022 b)\n\u22a2 a = b", "state_after": "no goals"}, {"tactic": "rwa [\u2190 smul_add, one_div, inv_smul_eq_iff\u2080 (show (2 : \ud835\udd5c) \u2260 0 by norm_num), two_smul]", "state_before": "F : Type ?u.111497\n\u03b1 : Type ?u.111500\n\u03b2 : Type ?u.111503\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs\u2080 : IsClosed s\nhs\u2081 : StrictConvex \ud835\udd5c s\na b c : E\nha : a \u2208 frontier s\nhb : b \u2208 frontier s\nhc : c \u2208 frontier s\nhabc : a + b = c + c\n\u22a2 (1 / 2) \u2022 a + (1 / 2) \u2022 b = c", "state_after": "no goals"}, {"tactic": "norm_num", "state_before": "F : Type ?u.111497\n\u03b1 : Type ?u.111500\n\u03b2 : Type ?u.111503\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nhs\u2080 : IsClosed s\nhs\u2081 : StrictConvex \ud835\udd5c s\na b c : E\nha : a \u2208 frontier s\nhb : b \u2208 frontier s\nhc : c \u2208 frontier s\nhabc : a + b = c + c\n\u22a2 2 \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/ModEq.lean", "full_name": "Nat.ModEq.add", "start": [141, 11], "end": [143, 30], "traced_tactics": [{"tactic": "rw [modEq_iff_dvd, Int.ofNat_add, Int.ofNat_add, add_sub_add_comm]", "state_before": "m n a b c d : \u2115\nh\u2081 : a \u2261 b [MOD n]\nh\u2082 : c \u2261 d [MOD n]\n\u22a2 a + c \u2261 b + d [MOD n]", "state_after": "m n a b c d : \u2115\nh\u2081 : a \u2261 b [MOD n]\nh\u2082 : c \u2261 d [MOD n]\n\u22a2 \u2191n \u2223 \u2191b - \u2191a + (\u2191d - \u2191c)"}, {"tactic": "exact dvd_add h\u2081.dvd h\u2082.dvd", "state_before": "m n a b c d : \u2115\nh\u2081 : a \u2261 b [MOD n]\nh\u2082 : c \u2261 d [MOD n]\n\u22a2 \u2191n \u2223 \u2191b - \u2191a + (\u2191d - \u2191c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/EMetricSpace.lean", "full_name": "uniformity_edist", "start": [1095, 1], "end": [1096, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Generator.lean", "full_name": "CategoryTheory.isCodetecting_unop_iff", "start": [147, 1], "end": [148, 41], "traced_tactics": [{"tactic": "rw [\u2190 isDetecting_op_iff, Set.unop_op]", "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category C\nD : Type u\u2082\ninst\u271d : Category D\n\ud835\udca2 : Set C\u1d52\u1d56\n\u22a2 IsCodetecting (Set.unop \ud835\udca2) \u2194 IsDetecting \ud835\udca2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Quaternion.lean", "full_name": "Cardinal.mk_quaternion_of_infinite", "start": [1462, 1], "end": [1463, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/Basic.lean", "full_name": "Equiv.sumAssoc_symm_apply_inr_inr", "start": [382, 1], "end": [383, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Adjoin/FG.lean", "full_name": "Subalgebra.fg_bot", "start": [108, 1], "end": [109, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.setOf_inter_eq_sep", "start": [1034, 1], "end": [1035, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/NNReal.lean", "full_name": "NNReal.mul_sup", "start": [557, 1], "end": [558, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/CancelLeads.lean", "full_name": "Polynomial.natDegree_cancelLeads_lt_of_natDegree_le_natDegree", "start": [89, 1], "end": [91, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Basic.lean", "full_name": "dist_triangle4", "start": [211, 1], "end": [214, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "full_name": "Pmf.toMeasure_eq_iff_eq_toPmf", "start": [391, 1], "end": [392, 86], "traced_tactics": [{"tactic": "rw [\u2190 toMeasure_inj, Measure.toPmf_toMeasure]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.246280\n\u03b3 : Type ?u.246283\ninst\u271d\u2074 : Countable \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSingletonClass \u03b1\np : Pmf \u03b1\n\u03bc\u271d : MeasureTheory.Measure \u03b1\ninst\u271d\u00b9 : IsProbabilityMeasure \u03bc\u271d\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : IsProbabilityMeasure \u03bc\n\u22a2 toMeasure p = \u03bc \u2194 p = Measure.toPmf \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Chain.lean", "full_name": "List.chain'_iff_pairwise", "start": [247, 1], "end": [249, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Span.lean", "full_name": "LinearMap.map_le_map_iff'", "start": [892, 1], "end": [893, 48], "traced_tactics": [{"tactic": "rw [LinearMap.map_le_map_iff, hf, sup_bot_eq]", "state_before": "R : Type u_2\nR\u2082 : Type u_3\nK : Type ?u.313356\nM : Type u_1\nM\u2082 : Type u_4\nV : Type ?u.313365\nS : Type ?u.313368\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring R\u2082\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nF : Type u_5\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\nhf : ker f = \u22a5\np p' : Submodule R M\n\u22a2 map f p \u2264 map f p' \u2194 p \u2264 p'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Ncard.lean", "full_name": "Set.ncard_add_ncard_compl", "start": [568, 1], "end": [570, 93], "traced_tactics": [{"tactic": "rw [\u2190 ncard_univ, \u2190 ncard_union_eq (@disjoint_compl_right _ _ s) hs hsc, union_compl_self]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.130552\ns\u271d t : Set \u03b1\na b x y : \u03b1\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nhs : autoParam (Set.Finite s) _auto\u271d\nhsc : autoParam (Set.Finite (s\u1d9c)) _auto\u271d\n\u22a2 ncard s + ncard (s\u1d9c) = Nat.card \u03b1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.range_piecewise", "start": [1501, 1], "end": [1505, 64], "traced_tactics": [{"tactic": "ext y", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.95004\n\u03b9 : Sort ?u.95007\n\u03c0 : \u03b1 \u2192 Type ?u.95012\n\u03b4 : \u03b1 \u2192 Sort ?u.95017\ns : Set \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\nf g : \u03b1 \u2192 \u03b2\n\u22a2 range (piecewise s f g) = f '' s \u222a g '' s\u1d9c", "state_after": "case h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.95004\n\u03b9 : Sort ?u.95007\n\u03c0 : \u03b1 \u2192 Type ?u.95012\n\u03b4 : \u03b1 \u2192 Sort ?u.95017\ns : Set \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\nf g : \u03b1 \u2192 \u03b2\ny : \u03b2\n\u22a2 y \u2208 range (piecewise s f g) \u2194 y \u2208 f '' s \u222a g '' s\u1d9c"}, {"tactic": "constructor", "state_before": "case h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.95004\n\u03b9 : Sort ?u.95007\n\u03c0 : \u03b1 \u2192 Type ?u.95012\n\u03b4 : \u03b1 \u2192 Sort ?u.95017\ns : Set \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\nf g : \u03b1 \u2192 \u03b2\ny : \u03b2\n\u22a2 y \u2208 range (piecewise s f g) \u2194 y \u2208 f '' s \u222a g '' s\u1d9c", "state_after": "case h.mp\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.95004\n\u03b9 : Sort ?u.95007\n\u03c0 : \u03b1 \u2192 Type ?u.95012\n\u03b4 : \u03b1 \u2192 Sort ?u.95017\ns : Set \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\nf g : \u03b1 \u2192 \u03b2\ny : \u03b2\n\u22a2 y \u2208 range (piecewise s f g) \u2192 y \u2208 f '' s \u222a g '' s\u1d9c\n\ncase h.mpr\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.95004\n\u03b9 : Sort ?u.95007\n\u03c0 : \u03b1 \u2192 Type ?u.95012\n\u03b4 : \u03b1 \u2192 Sort ?u.95017\ns : Set \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\nf g : \u03b1 \u2192 \u03b2\ny : \u03b2\n\u22a2 y \u2208 f '' s \u222a g '' s\u1d9c \u2192 y \u2208 range (piecewise s f g)"}, {"tactic": "rintro \u27e8x, rfl\u27e9", "state_before": "case h.mp\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.95004\n\u03b9 : Sort ?u.95007\n\u03c0 : \u03b1 \u2192 Type ?u.95012\n\u03b4 : \u03b1 \u2192 Sort ?u.95017\ns : Set \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\nf g : \u03b1 \u2192 \u03b2\ny : \u03b2\n\u22a2 y \u2208 range (piecewise s f g) \u2192 y \u2208 f '' s \u222a g '' s\u1d9c", "state_after": "case h.mp.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.95004\n\u03b9 : Sort ?u.95007\n\u03c0 : \u03b1 \u2192 Type ?u.95012\n\u03b4 : \u03b1 \u2192 Sort ?u.95017\ns : Set \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\nf g : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 piecewise s f g x \u2208 f '' s \u222a g '' s\u1d9c"}, {"tactic": "by_cases h : x \u2208 s <;> [left; right] <;> use x <;> simp [h]", "state_before": "case h.mp.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.95004\n\u03b9 : Sort ?u.95007\n\u03c0 : \u03b1 \u2192 Type ?u.95012\n\u03b4 : \u03b1 \u2192 Sort ?u.95017\ns : Set \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\nf g : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 piecewise s f g x \u2208 f '' s \u222a g '' s\u1d9c", "state_after": "no goals"}, {"tactic": "rintro (\u27e8x, hx, rfl\u27e9 | \u27e8x, hx, rfl\u27e9) <;> use x <;> simp_all", "state_before": "case h.mpr\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.95004\n\u03b9 : Sort ?u.95007\n\u03c0 : \u03b1 \u2192 Type ?u.95012\n\u03b4 : \u03b1 \u2192 Sort ?u.95017\ns : Set \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\nf g : \u03b1 \u2192 \u03b2\ny : \u03b2\n\u22a2 y \u2208 f '' s \u222a g '' s\u1d9c \u2192 y \u2208 range (piecewise s f g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "full_name": "Besicovitch.ae_tendsto_measure_inter_div_of_measurableSet", "start": [1173, 1], "end": [1181, 39], "traced_tactics": [{"tactic": "filter_upwards [VitaliFamily.ae_tendsto_measure_inter_div_of_measurableSet\n    (Besicovitch.vitaliFamily \u03bc) hs]", "state_before": "\u03b1 : Type ?u.1286254\ninst\u271d\u00b9\u2070 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2079 : SecondCountableTopology \u03b1\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : OpensMeasurableSpace \u03b1\ninst\u271d\u2076 : HasBesicovitchCovering \u03b1\ninst\u271d\u2075 : MetricSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b2\n\u03bc : MeasureTheory.Measure \u03b2\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202\u03bc, Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (indicator s 1 x))", "state_after": "case h\n\u03b1 : Type ?u.1286254\ninst\u271d\u00b9\u2070 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2079 : SecondCountableTopology \u03b1\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : OpensMeasurableSpace \u03b1\ninst\u271d\u2076 : HasBesicovitchCovering \u03b1\ninst\u271d\u2075 : MetricSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b2\n\u03bc : MeasureTheory.Measure \u03b2\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2200 (a : \u03b2),\n    Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) a)\n        (\ud835\udcdd (indicator s 1 a)) \u2192\n      Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall a r) / \u2191\u2191\u03bc (closedBall a r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (indicator s 1 a))"}, {"tactic": "intro x hx", "state_before": "case h\n\u03b1 : Type ?u.1286254\ninst\u271d\u00b9\u2070 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2079 : SecondCountableTopology \u03b1\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : OpensMeasurableSpace \u03b1\ninst\u271d\u2076 : HasBesicovitchCovering \u03b1\ninst\u271d\u2075 : MetricSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b2\n\u03bc : MeasureTheory.Measure \u03b2\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u2200 (a : \u03b2),\n    Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) a)\n        (\ud835\udcdd (indicator s 1 a)) \u2192\n      Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall a r) / \u2191\u2191\u03bc (closedBall a r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (indicator s 1 a))", "state_after": "case h\n\u03b1 : Type ?u.1286254\ninst\u271d\u00b9\u2070 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2079 : SecondCountableTopology \u03b1\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : OpensMeasurableSpace \u03b1\ninst\u271d\u2076 : HasBesicovitchCovering \u03b1\ninst\u271d\u2075 : MetricSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b2\n\u03bc : MeasureTheory.Measure \u03b2\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b2\nhs : MeasurableSet s\nx : \u03b2\nhx : Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x) (\ud835\udcdd (indicator s 1 x))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (indicator s 1 x))"}, {"tactic": "exact hx.comp (tendsto_filterAt \u03bc x)", "state_before": "case h\n\u03b1 : Type ?u.1286254\ninst\u271d\u00b9\u2070 : MetricSpace \u03b1\n\u03b2 : Type u\ninst\u271d\u2079 : SecondCountableTopology \u03b1\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : OpensMeasurableSpace \u03b1\ninst\u271d\u2076 : HasBesicovitchCovering \u03b1\ninst\u271d\u2075 : MetricSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : BorelSpace \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : HasBesicovitchCovering \u03b2\n\u03bc : MeasureTheory.Measure \u03b2\ninst\u271d : IsLocallyFiniteMeasure \u03bc\ns : Set \u03b2\nhs : MeasurableSet s\nx : \u03b2\nhx : Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (VitaliFamily.filterAt (Besicovitch.vitaliFamily \u03bc) x) (\ud835\udcdd (indicator s 1 x))\n\u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (indicator s 1 x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Rotate.lean", "full_name": "List.rotate_eq_iff", "start": [339, 1], "end": [347, 33], "traced_tactics": [{"tactic": "rw [\u2190 @rotate_eq_rotate _ l _ n, rotate_rotate, \u2190 rotate_mod l', add_mod]", "state_before": "\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\n\u22a2 rotate l n = l' \u2194 l = rotate l' (length l' - n % length l')", "state_after": "\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\n\u22a2 rotate l n = l' \u2194 rotate l n = rotate l' (((length l' - n % length l') % length l' + n % length l') % length l')"}, {"tactic": "cases' l'.length.zero_le.eq_or_lt with hl hl", "state_before": "\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\n\u22a2 rotate l n = l' \u2194 rotate l n = rotate l' (((length l' - n % length l') % length l' + n % length l') % length l')", "state_after": "case inl\n\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\nhl : 0 = length l'\n\u22a2 rotate l n = l' \u2194 rotate l n = rotate l' (((length l' - n % length l') % length l' + n % length l') % length l')\n\ncase inr\n\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\nhl : 0 < length l'\n\u22a2 rotate l n = l' \u2194 rotate l n = rotate l' (((length l' - n % length l') % length l' + n % length l') % length l')"}, {"tactic": "rw [eq_nil_of_length_eq_zero hl.symm, rotate_nil, rotate_eq_nil_iff]", "state_before": "case inl\n\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\nhl : 0 = length l'\n\u22a2 rotate l n = l' \u2194 rotate l n = rotate l' (((length l' - n % length l') % length l' + n % length l') % length l')", "state_after": "no goals"}, {"tactic": "cases' (Nat.zero_le (n % l'.length)).eq_or_lt with hn hn", "state_before": "case inr\n\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\nhl : 0 < length l'\n\u22a2 rotate l n = l' \u2194 rotate l n = rotate l' (((length l' - n % length l') % length l' + n % length l') % length l')", "state_after": "case inr.inl\n\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\nhl : 0 < length l'\nhn : 0 = n % length l'\n\u22a2 rotate l n = l' \u2194 rotate l n = rotate l' (((length l' - n % length l') % length l' + n % length l') % length l')\n\ncase inr.inr\n\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\nhl : 0 < length l'\nhn : 0 < n % length l'\n\u22a2 rotate l n = l' \u2194 rotate l n = rotate l' (((length l' - n % length l') % length l' + n % length l') % length l')"}, {"tactic": "simp [\u2190 hn]", "state_before": "case inr.inl\n\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\nhl : 0 < length l'\nhn : 0 = n % length l'\n\u22a2 rotate l n = l' \u2194 rotate l n = rotate l' (((length l' - n % length l') % length l' + n % length l') % length l')", "state_after": "no goals"}, {"tactic": "rw [mod_eq_of_lt (tsub_lt_self hl hn), tsub_add_cancel_of_le, mod_self, rotate_zero]", "state_before": "case inr.inr\n\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\nhl : 0 < length l'\nhn : 0 < n % length l'\n\u22a2 rotate l n = l' \u2194 rotate l n = rotate l' (((length l' - n % length l') % length l' + n % length l') % length l')", "state_after": "case inr.inr\n\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\nhl : 0 < length l'\nhn : 0 < n % length l'\n\u22a2 n % length l' \u2264 length l'"}, {"tactic": "exact (Nat.mod_lt _ hl).le", "state_before": "case inr.inr\n\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\nhl : 0 < length l'\nhn : 0 < n % length l'\n\u22a2 n % length l' \u2264 length l'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.integral_deriv_comp_mul_deriv", "start": [1514, 1], "end": [1518, 70], "traced_tactics": [{"tactic": "simpa [mul_comm] using integral_deriv_comp_smul_deriv hf hg hf' hg'", "state_before": "\u03b9 : Type ?u.1960831\n\ud835\udd5c : Type ?u.1960834\nE : Type ?u.1960837\nF : Type ?u.1960840\nA : Type ?u.1960843\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d\u00b9 : \u211d \u2192 E\ng'\u271d g\u271d \u03c6 : \u211d \u2192 \u211d\nf\u271d f'\u271d : \u211d \u2192 E\na b : \u211d\nf f' g g' : \u211d \u2192 \u211d\nhf : \u2200 (x : \u211d), x \u2208 [[a, b]] \u2192 HasDerivAt f (f' x) x\nhg : \u2200 (x : \u211d), x \u2208 [[a, b]] \u2192 HasDerivAt g (g' (f x)) (f x)\nhf' : ContinuousOn f' [[a, b]]\nhg' : Continuous g'\n\u22a2 (\u222b (x : \u211d) in a..b, (g' \u2218 f) x * f' x) = (g \u2218 f) b - (g \u2218 f) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "Filter.Tendsto.zpow", "start": [527, 1], "end": [529, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/RingQuot.lean", "full_name": "RingQuot.mul_quot", "start": [242, 1], "end": [245, 6], "traced_tactics": [{"tactic": "show mul r _ _ = _", "state_before": "R : Type u\u2081\ninst\u271d\u00b3 : Semiring R\nS : Type u\u2082\ninst\u271d\u00b2 : CommSemiring S\nA : Type u\u2083\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra S A\nr : R \u2192 R \u2192 Prop\na b : R\n\u22a2 { toQuot := Quot.mk (Rel r) a } * { 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DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\ni : m\nj : n\n\u22a2 (\u2211 c : n, if i = i \u2227 c = j then x i c else 0) = x i j", "state_after": "no goals"}, {"tactic": "intro j' hj'", "state_before": "case a.h.refine_1\nl : Type ?u.8855\nm : Type u_1\nn : Type u_2\nR : Type ?u.8864\n\u03b1 : Type u_3\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\ni : m\nj : n\n\u22a2 \u2200 (x_1 : m), x_1 \u2260 i \u2192 Finset.sum Finset.univ (fun j => stdBasisMatrix x_1 j (x x_1 j)) i j = 0", "state_after": "case a.h.refine_1\nl : Type ?u.8855\nm : Type u_1\nn : Type u_2\nR : Type ?u.8864\n\u03b1 : Type u_3\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\ni : m\nj : n\nj' : m\nhj' : j' \u2260 i\n\u22a2 Finset.sum Finset.univ (fun j => stdBasisMatrix j' j (x j' j)) i j = 0"}, {"tactic": "simp only [stdBasisMatrix]", "state_before": "case a.h.refine_1\nl : Type ?u.8855\nm : Type u_1\nn : Type u_2\nR : Type ?u.8864\n\u03b1 : Type u_3\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\ni : m\nj : n\nj' : m\nhj' : j' \u2260 i\n\u22a2 Finset.sum Finset.univ (fun j => stdBasisMatrix j' j (x j' j)) i j = 0", "state_after": "case a.h.refine_1\nl : Type ?u.8855\nm : Type u_1\nn : Type u_2\nR : Type ?u.8864\n\u03b1 : Type u_3\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\ni : m\nj : n\nj' : m\nhj' : j' \u2260 i\n\u22a2 Finset.sum Finset.univ (fun x_1 i' j'_1 => if j' = i' \u2227 x_1 = j'_1 then x j' x_1 else 0) i j = 0"}, {"tactic": "rw [Fintype.sum_apply, Fintype.sum_apply]", "state_before": "case a.h.refine_1\nl : Type ?u.8855\nm : Type u_1\nn : Type u_2\nR : Type ?u.8864\n\u03b1 : Type u_3\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\ni : m\nj : n\nj' : m\nhj' : j' \u2260 i\n\u22a2 Finset.sum Finset.univ (fun x_1 i' j'_1 => if j' = i' \u2227 x_1 = j'_1 then x j' x_1 else 0) i j = 0", "state_after": "case a.h.refine_1\nl : Type ?u.8855\nm : Type u_1\nn : Type u_2\nR : Type ?u.8864\n\u03b1 : Type u_3\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\ni : m\nj : n\nj' : m\nhj' : j' \u2260 i\n\u22a2 (\u2211 c : n, if j' = i \u2227 c = j then x j' c else 0) = 0"}, {"tactic": "simp [hj']", "state_before": "case a.h.refine_1\nl : Type ?u.8855\nm : Type u_1\nn : Type u_2\nR : Type ?u.8864\n\u03b1 : Type u_3\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\ni : m\nj : n\nj' : m\nhj' : j' \u2260 i\n\u22a2 (\u2211 c : n, if j' = i \u2227 c = j then x j' c else 0) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Bases.lean", "full_name": "Filter.HasBasis.forall_mem_mem", "start": [808, 1], "end": [811, 80], "traced_tactics": [{"tactic": "simp only [h.mem_iff, exists_imp, and_imp]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.66654\n\u03b3 : Type ?u.66657\n\u03b9 : Sort u_2\n\u03b9' : Sort ?u.66663\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nh : HasBasis l p s\nx : \u03b1\n\u22a2 (\u2200 (t : Set \u03b1), t \u2208 l \u2192 x \u2208 t) \u2194 \u2200 (i : \u03b9), p i \u2192 x \u2208 s i", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.66654\n\u03b3 : Type ?u.66657\n\u03b9 : Sort u_2\n\u03b9' : Sort ?u.66663\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nh : HasBasis l p s\nx : \u03b1\n\u22a2 (\u2200 (t : Set \u03b1) (x_1 : \u03b9), p x_1 \u2192 s x_1 \u2286 t \u2192 x \u2208 t) \u2194 \u2200 (i : \u03b9), p i \u2192 x \u2208 s i"}, {"tactic": "exact \u27e8fun h i hi => h (s i) i hi Subset.rfl, fun h t i hi ht => ht (h i hi)\u27e9", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.66654\n\u03b3 : Type ?u.66657\n\u03b9 : Sort u_2\n\u03b9' : Sort ?u.66663\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nh : HasBasis l p s\nx : \u03b1\n\u22a2 (\u2200 (t : Set \u03b1) (x_1 : \u03b9), p x_1 \u2192 s x_1 \u2286 t \u2192 x \u2208 t) \u2194 \u2200 (i : \u03b9), p i \u2192 x \u2208 s i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Interval.lean", "full_name": "Interval.bot_sub", "start": [387, 1], "end": [388, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "exists_Icc_mem_subset_of_mem_nhdsWithin_Iic", "start": [1261, 1], "end": [1264, 75], "traced_tactics": [{"tactic": "simpa only [dual_Icc, toDual.surjective.exists] using\n  @exists_Icc_mem_subset_of_mem_nhdsWithin_Ici \u03b1\u1d52\u1d48 _ _ _ (toDual a) _ hs", "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 : \u03b1\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd[Iic a] a\n\u22a2 \u2203 b, b \u2264 a \u2227 Icc b a \u2208 \ud835\udcdd[Iic a] a \u2227 Icc b a \u2286 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/OrderSynonym.lean", "full_name": "ofLex_mul", "start": [259, 1], "end": [259, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "full_name": "tsum_op", "start": [1378, 1], "end": [1382, 94], "traced_tactics": [{"tactic": "by_cases h : Summable f", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.806829\n\u03b4 : Type ?u.806832\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf : \u03b2 \u2192 \u03b1\na : \u03b1\ninst\u271d : T2Space \u03b1\n\u22a2 (\u2211' (x : \u03b2), op (f x)) = op (\u2211' (x : \u03b2), f x)", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.806829\n\u03b4 : Type ?u.806832\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf : \u03b2 \u2192 \u03b1\na : \u03b1\ninst\u271d : T2Space \u03b1\nh : Summable f\n\u22a2 (\u2211' (x : \u03b2), op (f x)) = op (\u2211' (x : \u03b2), f x)\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.806829\n\u03b4 : Type ?u.806832\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf : \u03b2 \u2192 \u03b1\na : \u03b1\ninst\u271d : T2Space \u03b1\nh : \u00acSummable f\n\u22a2 (\u2211' (x : \u03b2), op (f x)) = op (\u2211' (x : \u03b2), f x)"}, {"tactic": "exact h.hasSum.op.tsum_eq", "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.806829\n\u03b4 : Type ?u.806832\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf : \u03b2 \u2192 \u03b1\na : \u03b1\ninst\u271d : T2Space \u03b1\nh : Summable f\n\u22a2 (\u2211' (x : \u03b2), op (f x)) = op (\u2211' (x : \u03b2), f x)", "state_after": "no goals"}, {"tactic": "have ho := summable_op.not.mpr h", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.806829\n\u03b4 : Type ?u.806832\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf : \u03b2 \u2192 \u03b1\na : \u03b1\ninst\u271d : T2Space \u03b1\nh : \u00acSummable f\n\u22a2 (\u2211' (x : \u03b2), op (f x)) = op (\u2211' (x : \u03b2), f x)", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.806829\n\u03b4 : Type ?u.806832\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf : \u03b2 \u2192 \u03b1\na : \u03b1\ninst\u271d : T2Space \u03b1\nh : \u00acSummable f\nho : \u00acSummable fun a => op (f a)\n\u22a2 (\u2211' (x : \u03b2), op (f x)) = op (\u2211' (x : \u03b2), f x)"}, {"tactic": "rw [tsum_eq_zero_of_not_summable h, tsum_eq_zero_of_not_summable ho, MulOpposite.op_zero]", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.806829\n\u03b4 : Type ?u.806832\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\nf : \u03b2 \u2192 \u03b1\na : \u03b1\ninst\u271d : T2Space \u03b1\nh : \u00acSummable f\nho : \u00acSummable fun a => op (f a)\n\u22a2 (\u2211' (x : \u03b2), op (f x)) = op (\u2211' (x : \u03b2), f x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Topology.lean", "full_name": "Convex.strictConvex'", "start": [252, 11], "end": [264, 56], "traced_tactics": [{"tactic": "refine' strictConvex_iff_openSegment_subset.2 _", "state_before": "\u03b9 : Type ?u.145602\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\nh : Set.Pairwise (s \\ interior s) fun x y => \u2203 c, \u2191(lineMap x y) c \u2208 interior s\n\u22a2 StrictConvex \ud835\udd5c s", "state_after": "\u03b9 : Type ?u.145602\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\nh : Set.Pairwise (s \\ interior s) fun x y => \u2203 c, \u2191(lineMap x y) c \u2208 interior s\n\u22a2 Set.Pairwise s fun x y => openSegment \ud835\udd5c x y \u2286 interior s"}, {"tactic": "intro x hx y hy hne", "state_before": "\u03b9 : Type ?u.145602\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\nh : Set.Pairwise (s \\ interior s) fun x y => \u2203 c, \u2191(lineMap x y) c \u2208 interior s\n\u22a2 Set.Pairwise s fun x y => openSegment \ud835\udd5c x y \u2286 interior s", "state_after": "\u03b9 : Type ?u.145602\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\nh : Set.Pairwise (s \\ interior s) fun x y => \u2203 c, \u2191(lineMap x y) c \u2208 interior s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhne : x \u2260 y\n\u22a2 openSegment \ud835\udd5c x y \u2286 interior s"}, {"tactic": "by_cases hx' : x \u2208 interior s", "state_before": "\u03b9 : Type ?u.145602\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\nh : Set.Pairwise (s \\ interior s) fun x y => \u2203 c, \u2191(lineMap x y) c \u2208 interior s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhne : x \u2260 y\n\u22a2 openSegment \ud835\udd5c x y \u2286 interior s", "state_after": "case pos\n\u03b9 : Type ?u.145602\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\nh : Set.Pairwise (s \\ interior s) fun x y => \u2203 c, \u2191(lineMap x y) c \u2208 interior s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhne : x \u2260 y\nhx' : x \u2208 interior s\n\u22a2 openSegment \ud835\udd5c x y \u2286 interior s\n\ncase neg\n\u03b9 : Type ?u.145602\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\nh : Set.Pairwise (s \\ interior s) fun x y => \u2203 c, \u2191(lineMap x y) c \u2208 interior s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhne : x \u2260 y\nhx' : \u00acx \u2208 interior s\n\u22a2 openSegment \ud835\udd5c x y \u2286 interior s"}, {"tactic": "by_cases hy' : y \u2208 interior s", "state_before": "case neg\n\u03b9 : Type ?u.145602\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\nh : Set.Pairwise (s \\ interior s) fun x y => \u2203 c, \u2191(lineMap x y) c \u2208 interior s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhne : x \u2260 y\nhx' : \u00acx \u2208 interior s\n\u22a2 openSegment \ud835\udd5c x y \u2286 interior s", "state_after": "case pos\n\u03b9 : Type ?u.145602\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\nh : Set.Pairwise (s \\ interior s) fun x y => \u2203 c, \u2191(lineMap x y) c \u2208 interior s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhne : x \u2260 y\nhx' : \u00acx \u2208 interior s\nhy' : y \u2208 interior s\n\u22a2 openSegment \ud835\udd5c x y \u2286 interior s\n\ncase neg\n\u03b9 : Type ?u.145602\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\nh : Set.Pairwise (s \\ interior s) fun x y => \u2203 c, \u2191(lineMap x y) c \u2208 interior s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhne : x \u2260 y\nhx' : \u00acx \u2208 interior s\nhy' : \u00acy \u2208 interior s\n\u22a2 openSegment \ud835\udd5c x y \u2286 interior s"}, {"tactic": "rcases h \u27e8hx, hx'\u27e9 \u27e8hy, hy'\u27e9 hne with \u27e8c, hc\u27e9", "state_before": "case neg\n\u03b9 : Type ?u.145602\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\nh : Set.Pairwise (s \\ interior s) fun x y => \u2203 c, \u2191(lineMap x y) c \u2208 interior s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhne : x \u2260 y\nhx' : \u00acx \u2208 interior s\nhy' : \u00acy \u2208 interior s\n\u22a2 openSegment \ud835\udd5c x y \u2286 interior s", "state_after": "case neg.intro\n\u03b9 : Type ?u.145602\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\nh : Set.Pairwise (s \\ interior s) fun x y => \u2203 c, \u2191(lineMap x y) c \u2208 interior s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhne : x \u2260 y\nhx' : \u00acx \u2208 interior s\nhy' : \u00acy \u2208 interior s\nc : \ud835\udd5c\nhc : \u2191(lineMap x y) c \u2208 interior s\n\u22a2 openSegment \ud835\udd5c x y \u2286 interior s"}, {"tactic": "refine' (openSegment_subset_union x y \u27e8c, rfl\u27e9).trans (insert_subset.2 \u27e8hc, union_subset _ _\u27e9)", "state_before": "case neg.intro\n\u03b9 : Type ?u.145602\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\nh : Set.Pairwise (s \\ interior s) fun x y => \u2203 c, \u2191(lineMap x y) c \u2208 interior s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhne : x \u2260 y\nhx' : \u00acx \u2208 interior s\nhy' : \u00acy \u2208 interior s\nc : \ud835\udd5c\nhc : \u2191(lineMap x y) c \u2208 interior s\n\u22a2 openSegment \ud835\udd5c x y \u2286 interior s", "state_after": "case neg.intro.refine'_1\n\u03b9 : Type ?u.145602\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\nh : Set.Pairwise (s \\ interior s) fun x y => \u2203 c, \u2191(lineMap x y) c \u2208 interior s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhne : x \u2260 y\nhx' : \u00acx \u2208 interior s\nhy' : \u00acy \u2208 interior s\nc : \ud835\udd5c\nhc : \u2191(lineMap x y) c \u2208 interior s\n\u22a2 openSegment \ud835\udd5c x (\u2191(lineMap x y) c) \u2286 interior s\n\ncase neg.intro.refine'_2\n\u03b9 : Type ?u.145602\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\nh : Set.Pairwise (s \\ interior s) fun x y => \u2203 c, \u2191(lineMap x y) c \u2208 interior s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhne : x \u2260 y\nhx' : \u00acx \u2208 interior s\nhy' : \u00acy \u2208 interior s\nc : \ud835\udd5c\nhc : \u2191(lineMap x y) c \u2208 interior s\n\u22a2 openSegment \ud835\udd5c (\u2191(lineMap x y) c) y \u2286 interior s"}, {"tactic": "exacts [hs.openSegment_self_interior_subset_interior hx hc,\n  hs.openSegment_interior_self_subset_interior hc hy]", "state_before": "case neg.intro.refine'_1\n\u03b9 : Type ?u.145602\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\nh : Set.Pairwise (s \\ interior s) fun x y => \u2203 c, \u2191(lineMap x y) c \u2208 interior s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhne : x \u2260 y\nhx' : \u00acx \u2208 interior s\nhy' : \u00acy \u2208 interior s\nc : \ud835\udd5c\nhc : \u2191(lineMap x y) c \u2208 interior s\n\u22a2 openSegment \ud835\udd5c x (\u2191(lineMap x y) c) \u2286 interior s\n\ncase neg.intro.refine'_2\n\u03b9 : Type ?u.145602\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\nh : Set.Pairwise (s \\ interior s) fun x y => \u2203 c, \u2191(lineMap x y) c \u2208 interior s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhne : x \u2260 y\nhx' : \u00acx \u2208 interior s\nhy' : \u00acy \u2208 interior s\nc : \ud835\udd5c\nhc : \u2191(lineMap x y) c \u2208 interior s\n\u22a2 openSegment \ud835\udd5c (\u2191(lineMap x y) c) y \u2286 interior s", "state_after": "no goals"}, {"tactic": "exact hs.openSegment_interior_self_subset_interior hx' hy", "state_before": "case pos\n\u03b9 : Type ?u.145602\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\nh : Set.Pairwise (s \\ interior s) fun x y => \u2203 c, \u2191(lineMap x y) c \u2208 interior s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhne : x \u2260 y\nhx' : x \u2208 interior s\n\u22a2 openSegment \ud835\udd5c x y \u2286 interior s", "state_after": "no goals"}, {"tactic": "exact hs.openSegment_self_interior_subset_interior hx hy'", "state_before": "case pos\n\u03b9 : Type ?u.145602\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\nh : Set.Pairwise (s \\ interior s) fun x y => \u2203 c, \u2191(lineMap x y) c \u2208 interior s\nx : E\nhx : x \u2208 s\ny : E\nhy : y \u2208 s\nhne : x \u2260 y\nhx' : \u00acx \u2208 interior s\nhy' : y \u2208 interior s\n\u22a2 openSegment \ud835\udd5c x y \u2286 interior s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/TensorProduct.lean", "full_name": "TensorProduct.assoc_symm_tmul", "start": [714, 1], "end": [716, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "full_name": "circleIntegral.integral_sub_inv_smul_sub_smul", "start": [371, 1], "end": [377, 51], "traced_tactics": [{"tactic": "rcases eq_or_ne R 0 with (rfl | hR)", "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc w : \u2102\nR : \u211d\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) = \u222e (z : \u2102) in C(c, R), f z", "state_after": "case inl\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc w : \u2102\n\u22a2 (\u222e (z : \u2102) in C(c, 0), (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) = \u222e (z : \u2102) in C(c, 0), f z\n\ncase inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc w : \u2102\nR : \u211d\nhR : R \u2260 0\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) = \u222e (z : \u2102) in C(c, R), f z"}, {"tactic": "have : (circleMap c R \u207b\u00b9' {w}).Countable := (countable_singleton _).preimage_circleMap c hR", "state_before": "case inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc w : \u2102\nR : \u211d\nhR : R \u2260 0\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) = \u222e (z : \u2102) in C(c, R), f z", "state_after": "case inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc w : \u2102\nR : \u211d\nhR : R \u2260 0\nthis : Set.Countable (circleMap c R \u207b\u00b9' {w})\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) = \u222e (z : \u2102) in C(c, R), f z"}, {"tactic": "refine' intervalIntegral.integral_congr_ae ((this.ae_not_mem _).mono fun \u03b8 h\u03b8 _' => _)", "state_before": "case inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc w : \u2102\nR : \u211d\nhR : R \u2260 0\nthis : Set.Countable (circleMap c R \u207b\u00b9' {w})\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) = \u222e (z : \u2102) in C(c, R), f z", "state_after": "case inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc w : \u2102\nR : \u211d\nhR : R \u2260 0\nthis : Set.Countable (circleMap c R \u207b\u00b9' {w})\n\u03b8 : \u211d\nh\u03b8 : \u00ac\u03b8 \u2208 circleMap c R \u207b\u00b9' {w}\n_' : \u03b8 \u2208 \u0399 0 (2 * \u03c0)\n\u22a2 deriv (circleMap c R) \u03b8 \u2022 (fun z => (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) (circleMap c R \u03b8) =\n    deriv (circleMap c R) \u03b8 \u2022 (fun z => f z) (circleMap c R \u03b8)"}, {"tactic": "change circleMap c R \u03b8 \u2260 w at h\u03b8", "state_before": "case inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc w : \u2102\nR : \u211d\nhR : R \u2260 0\nthis : Set.Countable (circleMap c R \u207b\u00b9' {w})\n\u03b8 : \u211d\nh\u03b8 : \u00ac\u03b8 \u2208 circleMap c R \u207b\u00b9' {w}\n_' : \u03b8 \u2208 \u0399 0 (2 * \u03c0)\n\u22a2 deriv (circleMap c R) \u03b8 \u2022 (fun z => (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) (circleMap c R \u03b8) =\n    deriv (circleMap c R) \u03b8 \u2022 (fun z => f z) (circleMap c R \u03b8)", "state_after": "case inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc w : \u2102\nR : \u211d\nhR : R \u2260 0\nthis : Set.Countable (circleMap c R \u207b\u00b9' {w})\n\u03b8 : \u211d\n_' : \u03b8 \u2208 \u0399 0 (2 * \u03c0)\nh\u03b8 : circleMap c R \u03b8 \u2260 w\n\u22a2 deriv (circleMap c R) \u03b8 \u2022 (fun z => (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) (circleMap c R \u03b8) =\n    deriv (circleMap c R) \u03b8 \u2022 (fun z => f z) (circleMap c R \u03b8)"}, {"tactic": "simp only [inv_smul_smul\u2080 (sub_ne_zero.2 <| h\u03b8)]", "state_before": "case inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc w : \u2102\nR : \u211d\nhR : R \u2260 0\nthis : Set.Countable (circleMap c R \u207b\u00b9' {w})\n\u03b8 : \u211d\n_' : \u03b8 \u2208 \u0399 0 (2 * \u03c0)\nh\u03b8 : circleMap c R \u03b8 \u2260 w\n\u22a2 deriv (circleMap c R) \u03b8 \u2022 (fun z => (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) (circleMap c R \u03b8) =\n    deriv (circleMap c R) \u03b8 \u2022 (fun z => f z) (circleMap c R \u03b8)", "state_after": "no goals"}, {"tactic": "simp only [integral_radius_zero]", "state_before": "case inl\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc w : \u2102\n\u22a2 (\u222e (z : \u2102) in C(c, 0), (z - w)\u207b\u00b9 \u2022 (z - w) \u2022 f z) = \u222e (z : \u2102) in C(c, 0), f z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/LiminfLimsup.lean", "full_name": "Filter.liminf_const_top", "start": [656, 1], "end": [657, 30], "traced_tactics": []}, {"url": 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{"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Hom/Basic.lean", "full_name": "OrderIso.surjective", "start": [821, 11], "end": [822, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "full_name": "Asymptotics.IsLittleO.const_mul_right'", "start": [1551, 1], "end": [1553, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/LegendreSymbol/MulCharacter.lean", "full_name": "MulChar.inv_mul", "start": [378, 1], "end": [384, 21], "traced_tactics": [{"tactic": "ext x", "state_before": "R : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\n\u22a2 \u03c7\u207b\u00b9 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R\u02e3\n\u22a2 Ring.inverse (\u2191\u03c7 \u2191x) * \u2191\u03c7 \u2191x = \u21911 \u2191x", "state_after": "case h\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\nx : R\u02e3\n\u22a2 1 = \u21911 \u2191x"}, {"tactic": "rw [one_apply_coe]", "state_before": "case h\nR : Type u\ninst\u271d\u00b9 : CommMonoid R\nR' : Type v\ninst\u271d : CommMonoidWithZero R'\n\u03c7 : MulChar R R'\nx : R\u02e3\n\u22a2 1 = \u21911 \u2191x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Init/Data/Nat/Basic.lean", "full_name": "Nat.bit0_succ_eq", "start": [14, 11], "end": [15, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Subgraph.lean", "full_name": 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Type ?u.1417145\nQ : Type ?u.1417148\ninst\u271d\u00b2\u2074 : AddCommMonoid P\ninst\u271d\u00b2\u00b3 : Module R P\ninst\u271d\u00b2\u00b2 : AddCommMonoid Q\ninst\u271d\u00b2\u00b9 : Module S Q\nT : Type ?u.1417463\nO : Type ?u.1417466\ninst\u271d\u00b2\u2070 : CommRing T\ninst\u271d\u00b9\u2079 : Algebra R T\ninst\u271d\u00b9\u2078 : Algebra S T\ninst\u271d\u00b9\u2077 : IsScalarTower R S T\ninst\u271d\u00b9\u2076 : AddCommMonoid O\ninst\u271d\u00b9\u2075 : Module R O\ninst\u271d\u00b9\u2074 : Module S O\ninst\u271d\u00b9\u00b3 : Module T O\ninst\u271d\u00b9\u00b2 : IsScalarTower S T O\ninst\u271d\u00b9\u00b9 : IsScalarTower R S O\ninst\u271d\u00b9\u2070 : IsScalarTower R T O\nR' : Type u_2\nS' : Type u_3\ninst\u271d\u2079 : CommRing R'\ninst\u271d\u2078 : CommRing S'\ninst\u271d\u2077 : Algebra R R'\ninst\u271d\u2076 : Algebra S S'\ninst\u271d\u2075 : Algebra R' S'\ninst\u271d\u2074 : Algebra R S'\ninst\u271d\u00b3 : IsScalarTower R R' S'\ninst\u271d\u00b2 : IsScalarTower R S 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(evaluation K C).obj j) j\u271d"}, {"tactic": "simp", "state_before": "case w\nC : Type u\ninst\u271d\u2074 : Category C\nD : Type u'\ninst\u271d\u00b3 : Category D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category J\nK : Type u\u2082\ninst\u271d\u00b9 : Category K\ninst\u271d : HasLimitsOfShape J C\ni j : K\nF : J \u2964 K \u2964 C\nf : i \u27f6 j\nj\u271d : J\n\u22a2 ((limit F).map f \u226b (limitObjIsoLimitCompEvaluation F j).hom) \u226b limit.\u03c0 (F \u22d9 (evaluation K C).obj j) j\u271d =\n    ((limitObjIsoLimitCompEvaluation F i).hom \u226b limMap (whiskerLeft F ((evaluation K C).map f))) \u226b\n      limit.\u03c0 (F \u22d9 (evaluation K C).obj j) j\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.Icc_diff_Ioo_self", "start": [593, 1], "end": [593, 93], "traced_tactics": [{"tactic": "simp [\u2190 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"5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/RingDivision.lean", "full_name": "Polynomial.Monic.not_irreducible_iff_exists_add_mul_eq_coeff", "start": [313, 1], "end": [335, 32], "traced_tactics": [{"tactic": "cases subsingleton_or_nontrivial R", "state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\n\u22a2 \u00acIrreducible p \u2194 \u2203 c\u2081 c\u2082, coeff p 0 = c\u2081 * c\u2082 \u2227 coeff p 1 = c\u2081 + c\u2082", "state_after": "case inl\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Subsingleton R\n\u22a2 \u00acIrreducible p \u2194 \u2203 c\u2081 c\u2082, coeff p 0 = c\u2081 * c\u2082 \u2227 coeff p 1 = c\u2081 + c\u2082\n\ncase inr\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\n\u22a2 \u00acIrreducible p \u2194 \u2203 c\u2081 c\u2082, coeff p 0 = c\u2081 * c\u2082 \u2227 coeff p 1 = c\u2081 + c\u2082"}, {"tactic": "rw [hm.irreducible_iff_natDegree', and_iff_right, hnd]", "state_before": "case inr\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\n\u22a2 \u00acIrreducible p \u2194 \u2203 c\u2081 c\u2082, coeff p 0 = c\u2081 * c\u2082 \u2227 coeff p 1 = c\u2081 + c\u2082", "state_after": "case inr\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\n\u22a2 (\u00ac\u2200 (f g : R[X]), Monic f \u2192 Monic g \u2192 f * g = p \u2192 \u00acnatDegree g \u2208 Ioc 0 (2 / 2)) \u2194\n    \u2203 c\u2081 c\u2082, coeff p 0 = c\u2081 * c\u2082 \u2227 coeff p 1 = c\u2081 + c\u2082\n\ncase inr\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\n\u22a2 p \u2260 1"}, {"tactic": "push_neg", "state_before": "case inr\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\n\u22a2 (\u00ac\u2200 (f g : R[X]), Monic f \u2192 Monic g \u2192 f * g = p \u2192 \u00acnatDegree g \u2208 Ioc 0 (2 / 2)) \u2194\n    \u2203 c\u2081 c\u2082, coeff p 0 = c\u2081 * c\u2082 \u2227 coeff p 1 = c\u2081 + c\u2082\n\ncase inr\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\n\u22a2 p \u2260 1", "state_after": "case inr\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\n\u22a2 (\u2203 f g, Monic f \u2227 Monic g \u2227 f * g = p \u2227 natDegree g \u2208 Ioc 0 (2 / 2)) \u2194\n    \u2203 c\u2081 c\u2082, coeff p 0 = c\u2081 * c\u2082 \u2227 coeff p 1 = c\u2081 + c\u2082\n\ncase inr\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\n\u22a2 p \u2260 1"}, {"tactic": "constructor", "state_before": "case inr\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\n\u22a2 (\u2203 f g, Monic f \u2227 Monic g \u2227 f * g = p \u2227 natDegree g \u2208 Ioc 0 (2 / 2)) \u2194\n    \u2203 c\u2081 c\u2082, coeff p 0 = c\u2081 * c\u2082 \u2227 coeff p 1 = c\u2081 + c\u2082\n\ncase inr\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\n\u22a2 p \u2260 1", "state_after": "case inr.mp\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\n\u22a2 (\u2203 f g, Monic f \u2227 Monic g \u2227 f * g = p \u2227 natDegree g \u2208 Ioc 0 (2 / 2)) \u2192\n    \u2203 c\u2081 c\u2082, coeff p 0 = c\u2081 * c\u2082 \u2227 coeff p 1 = c\u2081 + c\u2082\n\ncase inr.mpr\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\n\u22a2 (\u2203 c\u2081 c\u2082, coeff p 0 = c\u2081 * c\u2082 \u2227 coeff p 1 = c\u2081 + c\u2082) \u2192\n    \u2203 f g, Monic f \u2227 Monic g \u2227 f * g = p \u2227 natDegree g \u2208 Ioc 0 (2 / 2)\n\ncase inr\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\n\u22a2 p \u2260 1"}, {"tactic": "simp [natDegree_of_subsingleton] at hnd", "state_before": "case inl\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Subsingleton R\n\u22a2 \u00acIrreducible p \u2194 \u2203 c\u2081 c\u2082, coeff p 0 = c\u2081 * c\u2082 \u2227 coeff p 1 = c\u2081 + c\u2082", "state_after": "no goals"}, {"tactic": "rintro \u27e8a, b, ha, hb, rfl, hdb\u27e9", "state_before": "case inr.mp\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\n\u22a2 (\u2203 f g, Monic f \u2227 Monic g \u2227 f * g = p \u2227 natDegree g \u2208 Ioc 0 (2 / 2)) \u2192\n    \u2203 c\u2081 c\u2082, coeff p 0 = c\u2081 * c\u2082 \u2227 coeff p 1 = c\u2081 + c\u2082", "state_after": "case inr.mp.intro.intro.intro.intro.intro\nR : Type u\nS : Type v\nT : Type w\na\u271d b\u271d : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nq : R[X]\nh\u271d : Nontrivial R\na b : R[X]\nha : Monic a\nhb : Monic b\nhdb : natDegree b \u2208 Ioc 0 (2 / 2)\nhm : Monic (a * b)\nhnd : natDegree (a * b) = 2\n\u22a2 \u2203 c\u2081 c\u2082, coeff (a * b) 0 = c\u2081 * c\u2082 \u2227 coeff (a * b) 1 = c\u2081 + c\u2082"}, {"tactic": "simp only [zero_lt_two, Nat.div_self, ge_iff_le,\n  Nat.Ioc_succ_singleton, zero_add, mem_singleton] at hdb", "state_before": "case inr.mp.intro.intro.intro.intro.intro\nR : Type u\nS : Type v\nT : Type w\na\u271d b\u271d : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nq : R[X]\nh\u271d : Nontrivial R\na b : R[X]\nha : Monic a\nhb : Monic b\nhdb : natDegree b \u2208 Ioc 0 (2 / 2)\nhm : Monic (a * b)\nhnd : natDegree (a * b) = 2\n\u22a2 \u2203 c\u2081 c\u2082, coeff (a * b) 0 = c\u2081 * c\u2082 \u2227 coeff (a * b) 1 = c\u2081 + c\u2082", "state_after": "case inr.mp.intro.intro.intro.intro.intro\nR : Type u\nS : Type v\nT : Type w\na\u271d b\u271d : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nq : R[X]\nh\u271d : Nontrivial R\na b : R[X]\nha : Monic a\nhb : Monic b\nhm : Monic (a * b)\nhnd : natDegree (a * b) = 2\nhdb : natDegree b = 1\n\u22a2 \u2203 c\u2081 c\u2082, coeff (a * b) 0 = c\u2081 * c\u2082 \u2227 coeff (a * b) 1 = c\u2081 + c\u2082"}, {"tactic": "have hda := hnd", "state_before": "case inr.mp.intro.intro.intro.intro.intro\nR : Type u\nS : Type v\nT : Type w\na\u271d b\u271d : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nq : R[X]\nh\u271d : Nontrivial R\na b : R[X]\nha : Monic a\nhb : Monic b\nhm : Monic (a * b)\nhnd : natDegree (a * b) = 2\nhdb : natDegree b = 1\n\u22a2 \u2203 c\u2081 c\u2082, coeff (a * b) 0 = c\u2081 * c\u2082 \u2227 coeff (a * b) 1 = c\u2081 + c\u2082", "state_after": "case inr.mp.intro.intro.intro.intro.intro\nR : Type u\nS : Type v\nT : Type w\na\u271d b\u271d : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nq : R[X]\nh\u271d : Nontrivial R\na b : R[X]\nha : Monic a\nhb : Monic b\nhm : Monic (a * b)\nhnd : natDegree (a * b) = 2\nhdb : natDegree b = 1\nhda : natDegree (a * b) = 2\n\u22a2 \u2203 c\u2081 c\u2082, coeff (a * b) 0 = c\u2081 * c\u2082 \u2227 coeff (a * b) 1 = c\u2081 + c\u2082"}, {"tactic": "rw [ha.natDegree_mul hb, hdb] at hda", "state_before": "case inr.mp.intro.intro.intro.intro.intro\nR : Type u\nS : Type v\nT : Type w\na\u271d b\u271d : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nq : R[X]\nh\u271d : Nontrivial R\na b : R[X]\nha : Monic a\nhb : Monic b\nhm : Monic (a * b)\nhnd : natDegree (a * b) = 2\nhdb : natDegree b = 1\nhda : natDegree (a * b) = 2\n\u22a2 \u2203 c\u2081 c\u2082, coeff (a * b) 0 = c\u2081 * c\u2082 \u2227 coeff (a * b) 1 = c\u2081 + c\u2082", "state_after": "case inr.mp.intro.intro.intro.intro.intro\nR : Type u\nS : Type v\nT : Type w\na\u271d b\u271d : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nq : R[X]\nh\u271d : Nontrivial R\na b : R[X]\nha : Monic a\nhb : Monic b\nhm : Monic (a * b)\nhnd : natDegree (a * b) = 2\nhdb : natDegree b = 1\nhda : natDegree a + 1 = 2\n\u22a2 \u2203 c\u2081 c\u2082, coeff (a * b) 0 = c\u2081 * c\u2082 \u2227 coeff (a * b) 1 = c\u2081 + c\u2082"}, {"tactic": "use a.coeff 0, b.coeff 0, mul_coeff_zero a b", "state_before": "case inr.mp.intro.intro.intro.intro.intro\nR : Type u\nS : Type v\nT : Type w\na\u271d b\u271d : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nq : R[X]\nh\u271d : Nontrivial R\na b : R[X]\nha : Monic a\nhb : Monic b\nhm : Monic (a * b)\nhnd : natDegree (a * b) = 2\nhdb : natDegree b = 1\nhda : natDegree a + 1 = 2\n\u22a2 \u2203 c\u2081 c\u2082, coeff (a * b) 0 = c\u2081 * c\u2082 \u2227 coeff (a * b) 1 = c\u2081 + c\u2082", "state_after": "case inr.mp.intro.intro.intro.intro.intro\nR : Type u\nS : Type v\nT : Type w\na\u271d b\u271d : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nq : R[X]\nh\u271d : Nontrivial R\na b : R[X]\nha : Monic a\nhb : Monic b\nhm : Monic (a * b)\nhnd : natDegree (a * b) = 2\nhdb : natDegree b = 1\nhda : natDegree a + 1 = 2\n\u22a2 coeff (a * b) 1 = coeff a 0 + coeff b 0"}, {"tactic": "simpa only [nextCoeff, hnd, add_right_cancel hda, hdb] using ha.nextCoeff_mul hb", "state_before": "case inr.mp.intro.intro.intro.intro.intro\nR : Type u\nS : Type v\nT : Type w\na\u271d b\u271d : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nq : R[X]\nh\u271d : Nontrivial R\na b : R[X]\nha : Monic a\nhb : Monic b\nhm : Monic (a * b)\nhnd : natDegree (a * b) = 2\nhdb : natDegree b = 1\nhda : natDegree a + 1 = 2\n\u22a2 coeff (a * b) 1 = coeff a 0 + coeff b 0", "state_after": "no goals"}, {"tactic": "rintro \u27e8c\u2081, c\u2082, hmul, hadd\u27e9", "state_before": "case inr.mpr\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\n\u22a2 (\u2203 c\u2081 c\u2082, coeff p 0 = c\u2081 * c\u2082 \u2227 coeff p 1 = c\u2081 + c\u2082) \u2192\n    \u2203 f g, Monic f \u2227 Monic g \u2227 f * g = p \u2227 natDegree g \u2208 Ioc 0 (2 / 2)", "state_after": "case inr.mpr.intro.intro.intro\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\nc\u2081 c\u2082 : R\nhmul : coeff p 0 = c\u2081 * c\u2082\nhadd : coeff p 1 = c\u2081 + c\u2082\n\u22a2 \u2203 f g, Monic f \u2227 Monic g \u2227 f * g = p \u2227 natDegree g \u2208 Ioc 0 (2 / 2)"}, {"tactic": "refine\n  \u27e8X + C c\u2081, X + C c\u2082, monic_X_add_C _, monic_X_add_C _, ?_, ?_ \u27e9", "state_before": "case inr.mpr.intro.intro.intro\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\nc\u2081 c\u2082 : R\nhmul : coeff p 0 = c\u2081 * c\u2082\nhadd : coeff p 1 = c\u2081 + c\u2082\n\u22a2 \u2203 f g, Monic f \u2227 Monic g \u2227 f * g = p \u2227 natDegree g \u2208 Ioc 0 (2 / 2)", "state_after": "case inr.mpr.intro.intro.intro.refine_1\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\nc\u2081 c\u2082 : R\nhmul : coeff p 0 = c\u2081 * c\u2082\nhadd : coeff p 1 = c\u2081 + c\u2082\n\u22a2 (X + \u2191C c\u2081) * (X + \u2191C c\u2082) = p\n\ncase inr.mpr.intro.intro.intro.refine_2\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\nc\u2081 c\u2082 : R\nhmul : coeff p 0 = c\u2081 * c\u2082\nhadd : coeff p 1 = c\u2081 + c\u2082\n\u22a2 natDegree (X + \u2191C c\u2082) \u2208 Ioc 0 (2 / 2)"}, {"tactic": "rw [p.as_sum_range_C_mul_X_pow, hnd, Finset.sum_range_succ, Finset.sum_range_succ,\n  Finset.sum_range_one, \u2190 hnd, hm.coeff_natDegree, hnd, hmul, hadd, C_mul, C_add, C_1]", "state_before": "case inr.mpr.intro.intro.intro.refine_1\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\nc\u2081 c\u2082 : R\nhmul : coeff p 0 = c\u2081 * c\u2082\nhadd : coeff p 1 = c\u2081 + c\u2082\n\u22a2 (X + \u2191C c\u2081) * (X + \u2191C c\u2082) = p", "state_after": "case inr.mpr.intro.intro.intro.refine_1\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\nc\u2081 c\u2082 : R\nhmul : coeff p 0 = c\u2081 * c\u2082\nhadd : coeff p 1 = c\u2081 + c\u2082\n\u22a2 (X + \u2191C c\u2081) * (X + \u2191C c\u2082) = \u2191C c\u2081 * \u2191C c\u2082 * X ^ 0 + (\u2191C c\u2081 + \u2191C c\u2082) * X ^ 1 + 1 * X ^ 2"}, {"tactic": "ring", "state_before": "case inr.mpr.intro.intro.intro.refine_1\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\nc\u2081 c\u2082 : R\nhmul : coeff p 0 = c\u2081 * c\u2082\nhadd : coeff p 1 = c\u2081 + c\u2082\n\u22a2 (X + \u2191C c\u2081) * (X + \u2191C c\u2082) = \u2191C c\u2081 * \u2191C c\u2082 * X ^ 0 + (\u2191C c\u2081 + \u2191C c\u2082) * X ^ 1 + 1 * X ^ 2", "state_after": "no goals"}, {"tactic": "rw [mem_Ioc, natDegree_X_add_C _]", "state_before": "case inr.mpr.intro.intro.intro.refine_2\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\nc\u2081 c\u2082 : R\nhmul : coeff p 0 = c\u2081 * c\u2082\nhadd : coeff p 1 = c\u2081 + c\u2082\n\u22a2 natDegree (X + \u2191C c\u2082) \u2208 Ioc 0 (2 / 2)", "state_after": "case inr.mpr.intro.intro.intro.refine_2\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\nc\u2081 c\u2082 : R\nhmul : coeff p 0 = c\u2081 * c\u2082\nhadd : coeff p 1 = c\u2081 + c\u2082\n\u22a2 0 < 1 \u2227 1 \u2264 2 / 2"}, {"tactic": "simp", "state_before": "case inr.mpr.intro.intro.intro.refine_2\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\nc\u2081 c\u2082 : R\nhmul : coeff p 0 = c\u2081 * c\u2082\nhadd : coeff p 1 = c\u2081 + c\u2082\n\u22a2 0 < 1 \u2227 1 \u2264 2 / 2", "state_after": "no goals"}, {"tactic": "rintro rfl", "state_before": "case inr\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\nhm : Monic p\nhnd : natDegree p = 2\nh\u271d : Nontrivial R\n\u22a2 p \u2260 1", "state_after": "case inr\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nq : R[X]\nh\u271d : Nontrivial R\nhm : Monic 1\nhnd : natDegree 1 = 2\n\u22a2 False"}, {"tactic": "simp [natDegree_one] at hnd", "state_before": "case inr\nR : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nq : R[X]\nh\u271d : Nontrivial R\nhm : Monic 1\nhnd : natDegree 1 = 2\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Max.lean", "full_name": "isMax_toDual_iff", "start": [247, 1], "end": [248, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/Orthogonal.lean", "full_name": "Submodule.IsOrtho.comap_iff", "start": [409, 1], "end": [413, 86], "traced_tactics": [{"tactic": "have hf : \u2200 p : Submodule \ud835\udd5c F, (p.comap f).map f.toLinearIsometry = p :=\n  map_comap_eq_of_surjective f.surjective", "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\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\nf : E \u2243\u2097\u1d62[\ud835\udd5c] F\nU V : Submodule \ud835\udd5c F\nh : Submodule.comap f U \u27c2 Submodule.comap f V\n\u22a2 U \u27c2 V", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\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\nf : E \u2243\u2097\u1d62[\ud835\udd5c] F\nU V : Submodule \ud835\udd5c F\nh : Submodule.comap f U \u27c2 Submodule.comap f V\nhf : \u2200 (p : Submodule \ud835\udd5c F), Submodule.map (LinearIsometryEquiv.toLinearIsometry f) (Submodule.comap f p) = p\n\u22a2 U \u27c2 V"}, {"tactic": "simpa only [hf] using h.map f.toLinearIsometry", "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\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\nf : E \u2243\u2097\u1d62[\ud835\udd5c] F\nU V : Submodule \ud835\udd5c F\nh : Submodule.comap f U \u27c2 Submodule.comap f V\nhf : \u2200 (p : Submodule \ud835\udd5c F), Submodule.map (LinearIsometryEquiv.toLinearIsometry f) (Submodule.comap f p) = p\n\u22a2 U \u27c2 V", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.compl_univ_iff", "start": [1683, 1], "end": [1684, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Ring/BooleanRing.lean", "full_name": "BooleanRing.inf_assoc", "start": [213, 1], "end": [215, 7], "traced_tactics": [{"tactic": "ring", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.13707\n\u03b3 : Type ?u.13710\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\na b c : \u03b1\n\u22a2 a * b * c = a * (b * c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Covering/LiminfLimsup.lean", "full_name": "blimsup_cthickening_ae_eq_blimsup_thickening", "start": [233, 1], "end": [244, 68], "traced_tactics": [{"tactic": "refine' eventuallyLE_antisymm_iff.mpr \u27e8_, HasSubset.Subset.eventuallyLE (_ : _ \u2264 _)\u27e9", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\n\u22a2 blimsup (fun i => cthickening (r i) (s i)) atTop p =\u1d50[\u03bc] blimsup (fun i => thickening (r i) (s i)) atTop p", "state_after": "case refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\n\u22a2 blimsup (fun i => cthickening (r i) (s i)) atTop p \u2264\u1d50[\u03bc] blimsup (fun i => thickening (r i) (s i)) atTop p\n\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\n\u22a2 blimsup (fun i => thickening (r i) (s i)) atTop p \u2264 blimsup (fun i => cthickening (r i) (s i)) atTop p"}, {"tactic": "rw [eventuallyLE_congr (blimsup_cthickening_mul_ae_eq \u03bc p s (@one_half_pos \u211d _) r hr).symm\n  EventuallyEq.rfl]", "state_before": "case refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\n\u22a2 blimsup (fun i => cthickening (r i) (s i)) atTop p \u2264\u1d50[\u03bc] blimsup (fun i => thickening (r i) (s i)) atTop p", "state_after": "case refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\n\u22a2 blimsup (fun i => cthickening (1 / 2 * r i) (s i)) atTop p \u2264\u1d50[\u03bc] blimsup (fun i => thickening (r i) (s i)) atTop p"}, {"tactic": "apply HasSubset.Subset.eventuallyLE", "state_before": "case refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\n\u22a2 blimsup (fun i => cthickening (1 / 2 * r i) (s i)) atTop p \u2264\u1d50[\u03bc] blimsup (fun i => thickening (r i) (s i)) atTop p", "state_after": "case refine'_1.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\n\u22a2 blimsup (fun i => cthickening (1 / 2 * r i) (s i)) atTop p \u2286 blimsup (fun i => thickening (r i) (s i)) atTop p"}, {"tactic": "change _ \u2264 _", "state_before": "case refine'_1.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\n\u22a2 blimsup (fun i => cthickening (1 / 2 * r i) (s i)) atTop p \u2286 blimsup (fun i => thickening (r i) (s i)) atTop p", "state_after": "case refine'_1.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\n\u22a2 blimsup (fun i => cthickening (1 / 2 * r i) (s i)) atTop p \u2264 blimsup (fun i => thickening (r i) (s i)) atTop p"}, {"tactic": "refine' mono_blimsup' (hr'.mono fun i hi pi => cthickening_subset_thickening' (hi pi) _ (s i))", "state_before": "case refine'_1.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\n\u22a2 blimsup (fun i => cthickening (1 / 2 * r i) (s i)) atTop p \u2264 blimsup (fun i => thickening (r i) (s i)) atTop p", "state_after": "case refine'_1.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\ni : \u2115\nhi : p i \u2192 0 < r i\npi : p i\n\u22a2 1 / 2 * r i < r i"}, {"tactic": "nlinarith [hi pi]", "state_before": "case refine'_1.h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\ni : \u2115\nhi : p i \u2192 0 < r i\npi : p i\n\u22a2 1 / 2 * r i < r i", "state_after": "no goals"}, {"tactic": "exact mono_blimsup fun i _ => thickening_subset_cthickening _ _", "state_before": "case refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\ninst\u271d\u2074 : SecondCountableTopology \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsUnifLocDoublingMeasure \u03bc\np : \u2115 \u2192 Prop\ns : \u2115 \u2192 Set \u03b1\nr : \u2115 \u2192 \u211d\nhr : Tendsto r atTop (\ud835\udcdd 0)\nhr' : \u2200\u1da0 (i : \u2115) in atTop, p i \u2192 0 < r i\n\u22a2 blimsup (fun i => thickening (r i) (s i)) atTop p \u2264 blimsup (fun i => cthickening (r i) (s i)) atTop p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.SimpleFunc.setToL1S_zero_left", "start": [699, 1], "end": [700, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Basic.lean", "full_name": "Submodule.map_comap_eq", "start": [1387, 1], "end": [1390, 51], "traced_tactics": [{"tactic": "rintro _ \u27e8\u27e8x, _, rfl\u27e9, hx\u27e9", "state_before": "R : Type u_1\nR\u2081 : Type ?u.1242999\nR\u2082 : Type u_2\nR\u2083 : Type ?u.1243005\nR\u2084 : Type ?u.1243008\nS : Type ?u.1243011\nK : Type ?u.1243014\nK\u2082 : Type ?u.1243017\nM : Type u_4\nM' : Type ?u.1243023\nM\u2081 : Type ?u.1243026\nM\u2082 : Type u_3\nM\u2083 : Type ?u.1243032\nM\u2084 : Type ?u.1243035\nN : Type ?u.1243038\nN\u2082 : Type ?u.1243041\n\u03b9 : Type ?u.1243044\nV : Type ?u.1243047\nV\u2082 : Type ?u.1243050\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid 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\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\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\nF : Type u_5\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nf : F\nq : Submodule R\u2082 M\u2082\n\u22a2 range f \u2293 q \u2264 map f (comap f q)", "state_after": "case intro.intro.refl\nR : Type u_1\nR\u2081 : Type ?u.1242999\nR\u2082 : Type u_2\nR\u2083 : Type ?u.1243005\nR\u2084 : Type ?u.1243008\nS : Type ?u.1243011\nK : Type ?u.1243014\nK\u2082 : Type ?u.1243017\nM : Type u_4\nM' : Type ?u.1243023\nM\u2081 : Type ?u.1243026\nM\u2082 : Type u_3\nM\u2083 : Type ?u.1243032\nM\u2084 : Type ?u.1243035\nN : Type ?u.1243038\nN\u2082 : Type ?u.1243041\n\u03b9 : Type ?u.1243044\nV : Type ?u.1243047\nV\u2082 : Type ?u.1243050\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid 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\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\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\nF : Type u_5\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nf : F\nq : Submodule R\u2082 M\u2082\nx : M\nhx : \u2191f x \u2208 \u2191q\n\u22a2 \u2191f x \u2208 map f (comap f q)"}, {"tactic": "exact \u27e8x, hx, rfl\u27e9", "state_before": "case intro.intro.refl\nR : Type u_1\nR\u2081 : Type ?u.1242999\nR\u2082 : Type u_2\nR\u2083 : Type ?u.1243005\nR\u2084 : Type ?u.1243008\nS : Type ?u.1243011\nK : Type ?u.1243014\nK\u2082 : Type ?u.1243017\nM : Type u_4\nM' : Type ?u.1243023\nM\u2081 : Type ?u.1243026\nM\u2082 : Type u_3\nM\u2083 : Type ?u.1243032\nM\u2084 : Type ?u.1243035\nN : Type ?u.1243038\nN\u2082 : Type ?u.1243041\n\u03b9 : Type ?u.1243044\nV : Type ?u.1243047\nV\u2082 : Type ?u.1243050\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid 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\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\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\nF : Type u_5\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nf : F\nq : Submodule R\u2082 M\u2082\nx : M\nhx : \u2191f x \u2208 \u2191q\n\u22a2 \u2191f x \u2208 map f (comap f q)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Constructions.lean", "full_name": "continuous_prod_mk", "start": [318, 9], "end": [322, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Function/Iterate.lean", "full_name": "Function.Commute.iterate_iterate", "start": [135, 1], "end": [136, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "full_name": "ContinuousLinearMap.map_sum", "start": [515, 11], "end": [517, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Dimension.lean", "full_name": "linearIndependent_le_span'", "start": [695, 1], "end": [700, 47], "traced_tactics": [{"tactic": "haveI : Fintype \u03b9 := linearIndependentFintypeOfLeSpanFintype v i w s", "state_before": "K : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.349139\nR : Type u\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : StrongRankCondition R\nM : Type v\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_1\nv : \u03b9 \u2192 M\ni : LinearIndependent R v\nw : Set M\ninst\u271d : Fintype \u2191w\ns : range v \u2264 \u2191(span R w)\n\u22a2 (#\u03b9) \u2264 \u2191(Fintype.card \u2191w)", "state_after": "K : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.349139\nR : Type u\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : StrongRankCondition R\nM : Type v\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_1\nv : \u03b9 \u2192 M\ni : LinearIndependent R v\nw : Set M\ninst\u271d : Fintype \u2191w\ns : range v \u2264 \u2191(span R w)\nthis : Fintype \u03b9\n\u22a2 (#\u03b9) \u2264 \u2191(Fintype.card \u2191w)"}, {"tactic": "rw [Cardinal.mk_fintype]", "state_before": "K : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.349139\nR : Type u\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : StrongRankCondition R\nM : Type v\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_1\nv : \u03b9 \u2192 M\ni : LinearIndependent R v\nw : Set M\ninst\u271d : Fintype \u2191w\ns : range v \u2264 \u2191(span R w)\nthis : Fintype \u03b9\n\u22a2 (#\u03b9) \u2264 \u2191(Fintype.card \u2191w)", "state_after": "K : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.349139\nR : Type u\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : StrongRankCondition R\nM : Type v\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_1\nv : \u03b9 \u2192 M\ni : LinearIndependent R v\nw : Set M\ninst\u271d : Fintype \u2191w\ns : range v \u2264 \u2191(span R w)\nthis : Fintype \u03b9\n\u22a2 \u2191(Fintype.card \u03b9) \u2264 \u2191(Fintype.card \u2191w)"}, {"tactic": "simp only [Cardinal.natCast_le]", "state_before": "K : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.349139\nR : Type u\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : StrongRankCondition R\nM : Type v\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_1\nv : \u03b9 \u2192 M\ni : LinearIndependent R v\nw : Set M\ninst\u271d : Fintype \u2191w\ns : range v \u2264 \u2191(span R w)\nthis : Fintype \u03b9\n\u22a2 \u2191(Fintype.card \u03b9) \u2264 \u2191(Fintype.card \u2191w)", "state_after": "K : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.349139\nR : Type u\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : StrongRankCondition R\nM : Type v\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_1\nv : \u03b9 \u2192 M\ni : LinearIndependent R v\nw : Set M\ninst\u271d : Fintype \u2191w\ns : range v \u2264 \u2191(span R w)\nthis : Fintype \u03b9\n\u22a2 Fintype.card \u03b9 \u2264 Fintype.card \u2191w"}, {"tactic": "exact linearIndependent_le_span_aux' v i w s", "state_before": "K : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.349139\nR : Type u\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : StrongRankCondition R\nM : Type v\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_1\nv : \u03b9 \u2192 M\ni : LinearIndependent R v\nw : Set M\ninst\u271d : Fintype \u2191w\ns : range v \u2264 \u2191(span R w)\nthis : Fintype \u03b9\n\u22a2 Fintype.card \u03b9 \u2264 Fintype.card \u2191w", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean", "full_name": "HasDerivAtFilter.sub_const", "start": [326, 1], "end": [328, 54], "traced_tactics": [{"tactic": "simpa only [sub_eq_add_neg] using hf.add_const (-c)", "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' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhf : HasDerivAtFilter f f' x L\nc : F\n\u22a2 HasDerivAtFilter (fun x => f x - c) f' x L", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Concrete.lean", "full_name": "List.cycleOf_formPerm", "start": [100, 1], "end": [105, 70], "traced_tactics": [{"tactic": "rwa [\u2190 length_attach] at hn", "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl l' : List \u03b1\nhl : Nodup l\nhn : 2 \u2264 length l\nx : { x // x \u2208 l }\n\u22a2 2 \u2264 length (attach l)", "state_after": "no goals"}, {"tactic": "rwa [\u2190 nodup_attach] at hl", "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl l' : List \u03b1\nhl : Nodup l\nhn\u271d : 2 \u2264 length l\nx : { x // x \u2208 l }\nhn : 2 \u2264 length (attach l)\n\u22a2 Nodup (attach l)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Submonoid/Operations.lean", "full_name": "Submonoid.comap_equiv_eq_map_symm", "start": [948, 1], "end": [950, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Module/Zlattice.lean", "full_name": "Zspan.fract_eq_fract", "start": [159, 1], "end": [164, 66], "traced_tactics": [{"tactic": "classical\nrw [eq_comm, Basis.ext_elem_iff b]\nsimp_rw [repr_fract_apply, Int.fract_eq_fract, eq_comm, Basis.mem_span_iff_repr_mem,\n  sub_eq_neg_add, map_add, LinearEquiv.map_neg, Finsupp.coe_add, Finsupp.coe_neg, Pi.add_apply,\n  Pi.neg_apply, \u2190 eq_intCast (algebraMap \u2124 K) _, Set.mem_range]", "state_before": "E : Type u_1\n\u03b9 : Type u_2\nK : Type u_3\ninst\u271d\u2074 : NormedLinearOrderedField K\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace K E\nb : Basis \u03b9 K E\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Fintype \u03b9\nm n : E\n\u22a2 fract b m = fract b n \u2194 -m + n \u2208 span \u2124 (Set.range \u2191b)", "state_after": "no goals"}, {"tactic": "rw [eq_comm, Basis.ext_elem_iff b]", "state_before": "E : Type u_1\n\u03b9 : Type u_2\nK : Type u_3\ninst\u271d\u2074 : NormedLinearOrderedField K\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace K E\nb : Basis \u03b9 K E\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Fintype \u03b9\nm n : E\n\u22a2 fract b m = fract b n \u2194 -m + n \u2208 span \u2124 (Set.range \u2191b)", "state_after": "E : Type u_1\n\u03b9 : Type u_2\nK : Type u_3\ninst\u271d\u2074 : NormedLinearOrderedField K\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace K E\nb : Basis \u03b9 K E\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Fintype \u03b9\nm n : E\n\u22a2 (\u2200 (i : \u03b9), \u2191(\u2191b.repr (fract b n)) i = \u2191(\u2191b.repr (fract b m)) i) \u2194 -m + n \u2208 span \u2124 (Set.range \u2191b)"}, {"tactic": "simp_rw [repr_fract_apply, Int.fract_eq_fract, eq_comm, Basis.mem_span_iff_repr_mem,\n  sub_eq_neg_add, map_add, LinearEquiv.map_neg, Finsupp.coe_add, Finsupp.coe_neg, Pi.add_apply,\n  Pi.neg_apply, \u2190 eq_intCast (algebraMap \u2124 K) _, Set.mem_range]", "state_before": "E : Type u_1\n\u03b9 : Type u_2\nK : Type u_3\ninst\u271d\u2074 : NormedLinearOrderedField K\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace K E\nb : Basis \u03b9 K E\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Fintype \u03b9\nm n : E\n\u22a2 (\u2200 (i : \u03b9), \u2191(\u2191b.repr (fract b n)) i = \u2191(\u2191b.repr (fract b m)) i) \u2194 -m + n \u2208 span \u2124 (Set.range \u2191b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/Equiv.lean", "full_name": "UniformEquiv.ext", "start": [102, 1], "end": [103, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "full_name": "FiniteDimensional.cardinal_mk_le_finrank_of_linearIndependent", "start": [277, 1], "end": [281, 49], "traced_tactics": [{"tactic": "rw [\u2190 lift_le.{_, max v w}]", "state_before": "K : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u03b9 : Type w\nb : \u03b9 \u2192 V\nh : LinearIndependent K b\n\u22a2 (#\u03b9) \u2264 \u2191(finrank K V)", "state_after": "K : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u03b9 : Type w\nb : \u03b9 \u2192 V\nh : LinearIndependent K b\n\u22a2 lift (#\u03b9) \u2264 lift \u2191(finrank K V)"}, {"tactic": "simpa [\u2190 finrank_eq_rank', -finrank_eq_rank] using\n  cardinal_lift_le_rank_of_linearIndependent h", "state_before": "K : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\n\u03b9 : Type w\nb : \u03b9 \u2192 V\nh : LinearIndependent K b\n\u22a2 lift (#\u03b9) \u2264 lift \u2191(finrank K V)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Sheaves/Functors.lean", "full_name": "TopCat.Sheaf.pushforward_sheaf_of_sheaf", "start": [85, 1], "end": [87, 75], "traced_tactics": [{"tactic": "rw [isSheaf_iff_isSheafPairwiseIntersections] at h \u22a2", "state_before": "C : Type u\ninst\u271d : Category C\nX Y : TopCat\nf : X \u27f6 Y\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191Y\nF : Presheaf C X\nh : Presheaf.IsSheaf F\n\u22a2 Presheaf.IsSheaf (f _* F)", "state_after": "C : Type u\ninst\u271d : Category C\nX Y : TopCat\nf : X \u27f6 Y\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\n\u22a2 IsSheafPairwiseIntersections (f _* F)"}, {"tactic": "exact SheafConditionPairwiseIntersections.pushforward_sheaf_of_sheaf f h", "state_before": "C : Type u\ninst\u271d : Category C\nX Y : TopCat\nf : X \u27f6 Y\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191Y\nF : Presheaf C X\nh : IsSheafPairwiseIntersections F\n\u22a2 IsSheafPairwiseIntersections (f _* F)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Basic.lean", "full_name": "SimpleGraph.Dart.edge_mk", "start": [730, 1], "end": [731, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Bounds/Basic.lean", "full_name": "Monotone.image_upperBounds_subset_upperBounds_image", "start": [1272, 1], "end": [1274, 36], "traced_tactics": [{"tactic": "rintro _ \u27e8a, ha, rfl\u27e9", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nHf : Monotone f\na : \u03b1\ns : Set \u03b1\n\u22a2 f '' upperBounds s \u2286 upperBounds (f '' s)", "state_after": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nHf : Monotone f\na\u271d : \u03b1\ns : Set \u03b1\na : \u03b1\nha : a \u2208 upperBounds s\n\u22a2 f a \u2208 upperBounds (f '' s)"}, {"tactic": "exact Hf.mem_upperBounds_image ha", "state_before": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\nHf : Monotone f\na\u271d : \u03b1\ns : Set \u03b1\na : \u03b1\nha : a \u2208 upperBounds s\n\u22a2 f a \u2208 upperBounds (f '' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "full_name": "Finset.hasSum_iff_compl", "start": [877, 11], "end": [879, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean", "full_name": "CategoryTheory.IsPullback.of_isLimit'", "start": [233, 1], "end": [235, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.support_add", "start": [542, 1], "end": [543, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Iterate.lean", "full_name": "Monotone.le_iterate_comp_of_le", "start": [96, 1], "end": [100, 23], "traced_tactics": [{"tactic": "apply hf.seq_le_seq n <;> intros <;>\n  simp [iterate_succ', -iterate_succ, comp_apply, id_eq, le_refl]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b1\nf : \u03b1 \u2192 \u03b1\nx\u271d y : \u2115 \u2192 \u03b1\ng : \u03b2 \u2192 \u03b2\nh : \u03b2 \u2192 \u03b1\nhf : Monotone f\nH : h \u2218 g \u2264 f \u2218 h\nn : \u2115\nx : \u03b2\n\u22a2 (h \u2218 g^[n]) x \u2264 (f^[n] \u2218 h) x", "state_after": "case hx\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b1\nf : \u03b1 \u2192 \u03b1\nx\u271d y : \u2115 \u2192 \u03b1\ng : \u03b2 \u2192 \u03b2\nh : \u03b2 \u2192 \u03b1\nhf : Monotone f\nH : h \u2218 g \u2264 f \u2218 h\nn : \u2115\nx : \u03b2\nk\u271d : \u2115\na\u271d : k\u271d < n\n\u22a2 h (g ((g^[k\u271d]) x)) \u2264 f (h ((g^[k\u271d]) x))"}, {"tactic": "case hx => exact H _", "state_before": "case hx\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b1\nf : \u03b1 \u2192 \u03b1\nx\u271d y : \u2115 \u2192 \u03b1\ng : \u03b2 \u2192 \u03b2\nh : \u03b2 \u2192 \u03b1\nhf : Monotone f\nH : h \u2218 g \u2264 f \u2218 h\nn : \u2115\nx : \u03b2\nk\u271d : \u2115\na\u271d : k\u271d < n\n\u22a2 h (g ((g^[k\u271d]) x)) \u2264 f (h ((g^[k\u271d]) x))", "state_after": "no goals"}, {"tactic": "exact H _", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b1\nf : \u03b1 \u2192 \u03b1\nx\u271d y : \u2115 \u2192 \u03b1\ng : \u03b2 \u2192 \u03b2\nh : \u03b2 \u2192 \u03b1\nhf : Monotone f\nH : h \u2218 g \u2264 f \u2218 h\nn : \u2115\nx : \u03b2\nk\u271d : \u2115\na\u271d : k\u271d < n\n\u22a2 h (g ((g^[k\u271d]) x)) \u2264 f (h ((g^[k\u271d]) x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Lipschitz.lean", "full_name": "LipschitzWith.const", "start": [199, 11], "end": [200, 34], "traced_tactics": [{"tactic": "simp only [edist_self, zero_le]", "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 : \u03b1 \u2192 \u03b2\nx\u271d y\u271d : \u03b1\nr : \u211d\u22650\u221e\nb : \u03b2\nx y : \u03b1\n\u22a2 edist ((fun x => b) x) ((fun x => b) y) \u2264 \u21910 * edist x y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/IntegralClosure.lean", "full_name": "IsIntegralClosure.isIntegral_algebra", "start": [847, 1], "end": [848, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/Exponential.lean", "full_name": "continuousOn_exp", "start": [251, 1], "end": [252, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Nat/Lemmas.lean", "full_name": "Nat.dvd_add_iff_left", "start": [676, 11], "end": [677, 51], "traced_tactics": [{"tactic": "rw [Nat.add_comm]", "state_before": "k m n : Nat\nh : k \u2223 n\n\u22a2 k \u2223 m \u2194 k \u2223 m + n", "state_after": "k m n : Nat\nh : k \u2223 n\n\u22a2 k \u2223 m \u2194 k \u2223 n + m"}, {"tactic": "exact Nat.dvd_add_iff_right h", "state_before": "k m n : Nat\nh : k \u2223 n\n\u22a2 k \u2223 m \u2194 k \u2223 n + m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.strongInductionOn_eq", "start": [821, 1], "end": [823, 25], "traced_tactics": [{"tactic": "rw [strongInductionOn]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.74726\n\u03b3 : Type ?u.74729\np : Multiset \u03b1 \u2192 Sort u_2\ns : Multiset \u03b1\nH : (s : Multiset \u03b1) \u2192 ((t : Multiset \u03b1) \u2192 t < s \u2192 p t) \u2192 p s\n\u22a2 strongInductionOn s H = H s fun t _h => strongInductionOn t H", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Int/DivMod.lean", "full_name": "Int.ofNat_fmod", "start": [239, 1], "end": [239, 83], "traced_tactics": [{"tactic": "cases m <;> simp [fmod]", "state_before": "m n : Nat\n\u22a2 \u2191(m % n) = fmod \u2191m \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Lie/Nilpotent.lean", "full_name": "LieModule.iterate_toEndomorphism_mem_lowerCentralSeries", "start": [151, 1], "end": [157, 67], "traced_tactics": [{"tactic": "induction' k with k ih", "state_before": "R : 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 : LieSubmodule R L M\nx : L\nm : M\nk : \u2115\n\u22a2 (\u2191(\u2191(toEndomorphism R L M) x)^[k]) m \u2208 lowerCentralSeries R L M k", "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 : LieSubmodule R L M\nx : L\nm : M\n\u22a2 (\u2191(\u2191(toEndomorphism R L M) x)^[Nat.zero]) m \u2208 lowerCentralSeries R L M Nat.zero\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 : LieSubmodule R L M\nx : L\nm : M\nk : \u2115\nih : (\u2191(\u2191(toEndomorphism R L M) x)^[k]) m \u2208 lowerCentralSeries R L M k\n\u22a2 (\u2191(\u2191(toEndomorphism R L M) x)^[Nat.succ k]) m \u2208 lowerCentralSeries R L M (Nat.succ k)"}, {"tactic": "simp only [Nat.zero_eq, Function.iterate_zero, lowerCentralSeries_zero, LieSubmodule.mem_top]", "state_before": "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 : LieSubmodule R L M\nx : L\nm : M\n\u22a2 (\u2191(\u2191(toEndomorphism R L M) x)^[Nat.zero]) m \u2208 lowerCentralSeries R L M Nat.zero", "state_after": "no goals"}, {"tactic": "simp only [lowerCentralSeries_succ, Function.comp_apply, Function.iterate_succ',\n  toEndomorphism_apply_apply]", "state_before": "case 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 : LieSubmodule R L M\nx : L\nm : M\nk : \u2115\nih : (\u2191(\u2191(toEndomorphism R L M) x)^[k]) m \u2208 lowerCentralSeries R L M k\n\u22a2 (\u2191(\u2191(toEndomorphism R L M) x)^[Nat.succ k]) m \u2208 lowerCentralSeries R L M (Nat.succ k)", "state_after": "case 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 : LieSubmodule R L M\nx : L\nm : M\nk : \u2115\nih : (\u2191(\u2191(toEndomorphism R L M) x)^[k]) m \u2208 lowerCentralSeries R L M k\n\u22a2 \u2045x, (\u2191(\u2191(toEndomorphism R L M) x)^[k]) m\u2046 \u2208 \u2045\u22a4, lowerCentralSeries R L M k\u2046"}, {"tactic": "exact LieSubmodule.lie_mem_lie _ _ (LieSubmodule.mem_top x) ih", "state_before": "case 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 : LieSubmodule R L M\nx : L\nm : M\nk : \u2115\nih : (\u2191(\u2191(toEndomorphism R L M) x)^[k]) m \u2208 lowerCentralSeries R L M k\n\u22a2 \u2045x, (\u2191(\u2191(toEndomorphism R L M) x)^[k]) m\u2046 \u2208 \u2045\u22a4, lowerCentralSeries R L M k\u2046", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/ContinuousOn.lean", "full_name": "continuousOn_iff'", "start": [617, 1], "end": [626, 78], "traced_tactics": [{"tactic": "rw [continuousOn_iff_continuous_restrict, continuous_def]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.313683\n\u03b4 : Type ?u.313686\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\nthis : \u2200 (t : Set \u03b2), IsOpen (restrict s f \u207b\u00b9' t) \u2194 \u2203 u, IsOpen u \u2227 f \u207b\u00b9' t \u2229 s = u \u2229 s\n\u22a2 ContinuousOn f s \u2194 \u2200 (t : Set \u03b2), IsOpen t \u2192 \u2203 u, IsOpen u \u2227 f \u207b\u00b9' t \u2229 s = u \u2229 s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.313683\n\u03b4 : Type ?u.313686\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\nthis : \u2200 (t : Set \u03b2), IsOpen (restrict s f \u207b\u00b9' t) \u2194 \u2203 u, IsOpen u \u2227 f \u207b\u00b9' t \u2229 s = u \u2229 s\n\u22a2 (\u2200 (s_1 : Set \u03b2), IsOpen s_1 \u2192 IsOpen (restrict s f \u207b\u00b9' s_1)) \u2194\n    \u2200 (t : Set 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?u.313686\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\n\u22a2 (\u2203 t_1, IsOpen t_1 \u2227 Subtype.val \u207b\u00b9' t_1 = Subtype.val \u207b\u00b9' (f \u207b\u00b9' t)) \u2194 \u2203 u, IsOpen u \u2227 f \u207b\u00b9' t \u2229 s = u \u2229 s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.313683\n\u03b4 : Type ?u.313686\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\n\u22a2 (\u2203 t_1, IsOpen t_1 \u2227 t_1 \u2229 s = f \u207b\u00b9' t \u2229 s) \u2194 \u2203 u, IsOpen u \u2227 f \u207b\u00b9' t \u2229 s = u \u2229 s"}, {"tactic": "rintro \u27e8u, ou, useq\u27e9", "state_before": "case mpr\n\u03b1 : 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Submodule R M\nN : Type ?u.17852\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\n\u22a2 IsSimpleModule R (M \u29f8 m) \u2194 IsSimpleOrder \u2191(Set.Ici m)", "state_after": "case f\nR : Type u_1\ninst\u271d\u2074 : Ring R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nm : Submodule R M\nN : Type ?u.17852\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\n\u22a2 Submodule R (M \u29f8 m) \u2243o \u2191(Set.Ici m)"}, {"tactic": "exact Submodule.comapMkQRelIso m", "state_before": "case f\nR : Type u_1\ninst\u271d\u2074 : Ring R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nm : Submodule R M\nN : Type ?u.17852\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\n\u22a2 Submodule R (M \u29f8 m) \u2243o \u2191(Set.Ici m)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Tactic/NormNum/Basic.lean", "full_name": "Mathlib.Meta.NormNum.isNat_natSucc", "start": [935, 1], "end": [937, 37], "traced_tactics": [{"tactic": "simp", "state_before": "n\u271d : \u2115\n\u22a2 Nat.succ \u2191n\u271d = \u2191(Nat.succ n\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/UpperLower/Basic.lean", "full_name": "isLowerSet_iff_Iic_subset", "start": [227, 1], "end": [228, 54], "traced_tactics": [{"tactic": "simp [IsLowerSet, subset_def, @forall_swap (_ \u2208 s)]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.12567\n\u03b3 : Type ?u.12570\n\u03b9 : Sort ?u.12573\n\u03ba : \u03b9 \u2192 Sort ?u.12578\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ns : Set \u03b1\np : \u03b1 \u2192 Prop\na : \u03b1\n\u22a2 IsLowerSet s \u2194 \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2192 Iic a \u2286 s", "state_after": "no 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"Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.lt_pred", "start": [211, 1], "end": [215, 61], "traced_tactics": [{"tactic": "let \u27e8c, e\u27e9 := h", "state_before": "\u03b1 : Type ?u.85911\n\u03b2 : Type ?u.85914\n\u03b3 : Type ?u.85917\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal\nh : \u2203 a, b = succ a\n\u22a2 a < pred b \u2194 succ a < b", "state_after": "\u03b1 : Type ?u.85911\n\u03b2 : Type ?u.85914\n\u03b3 : Type ?u.85917\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal\nh : \u2203 a, b = succ a\nc : Ordinal\ne : b = succ c\n\u22a2 a < pred b \u2194 succ a < b"}, {"tactic": "rw [e, pred_succ, succ_lt_succ_iff]", "state_before": "\u03b1 : Type ?u.85911\n\u03b2 : Type ?u.85914\n\u03b3 : Type ?u.85917\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal\nh : \u2203 a, b = succ a\nc : Ordinal\ne : b = succ c\n\u22a2 a < pred b \u2194 succ a < b", "state_after": "no goals"}, {"tactic": "simp only [pred, dif_neg h, succ_lt_of_not_succ h]", "state_before": "\u03b1 : Type ?u.85911\n\u03b2 : Type ?u.85914\n\u03b3 : Type ?u.85917\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal\nh : \u00ac\u2203 a, b = succ a\n\u22a2 a < pred b \u2194 succ a < b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Data/List/Control.lean", "full_name": "List.forIn'_eq_forIn", "start": [178, 9], "end": [185, 13], "traced_tactics": [{"tactic": "simp [forIn', forIn, List.forIn, List.forIn']", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nm : Type v \u2192 Type w\ninst\u271d : Monad m\nas : List \u03b1\ninit : \u03b2\nf : \u03b1 \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 (forIn' as init fun a x b => f a b) = forIn as init f", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nm : Type v \u2192 Type w\ninst\u271d : Monad m\nas : List \u03b1\ninit : \u03b2\nf : \u03b1 \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 forIn'.loop as (fun a x b => f a b) as init (_ : Exists fun bs => bs ++ as = as) = forIn.loop f as init"}, {"tactic": "have : \u2200 cs h, List.forIn'.loop cs (fun a _ b => f a b) as init h = List.forIn.loop f as init := by\n  intro cs h\n  induction as generalizing cs init with\n  | nil => intros; rfl\n  | cons a as ih => intros; simp [List.forIn.loop, List.forIn'.loop, ih]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nm : Type v \u2192 Type w\ninst\u271d : Monad m\nas : List \u03b1\ninit : \u03b2\nf : \u03b1 \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 forIn'.loop as (fun a x b => f a b) as init (_ : Exists fun bs => bs ++ as = as) = forIn.loop f as init", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nm : Type v \u2192 Type w\ninst\u271d : Monad m\nas : List \u03b1\ninit : \u03b2\nf : \u03b1 \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nthis :\n  \u2200 (cs : List \u03b1) (h : Exists fun bs => bs ++ as = cs),\n    forIn'.loop cs (fun a x b => f a b) as init h = forIn.loop f as init\n\u22a2 forIn'.loop as (fun a x b => f a b) as init (_ : Exists fun bs => bs ++ as = as) = forIn.loop f as init"}, {"tactic": "apply this", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nm : Type v \u2192 Type w\ninst\u271d : Monad m\nas : List \u03b1\ninit : \u03b2\nf : \u03b1 \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\nthis :\n  \u2200 (cs : List \u03b1) (h : Exists fun bs => bs ++ as = cs),\n    forIn'.loop cs (fun a x b => f a b) as init h = forIn.loop f as init\n\u22a2 forIn'.loop as (fun a x b => f a b) as init (_ : Exists fun bs => bs ++ as = as) = forIn.loop f as init", "state_after": "no goals"}, {"tactic": "intro cs h", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nm : Type v \u2192 Type w\ninst\u271d : Monad m\nas : List \u03b1\ninit : \u03b2\nf : \u03b1 \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\n\u22a2 \u2200 (cs : List \u03b1) (h : Exists fun bs => bs ++ as = cs),\n    forIn'.loop cs (fun a x b => f a b) as init h = forIn.loop f as init", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nm : Type v \u2192 Type w\ninst\u271d : Monad m\nas : List \u03b1\ninit : \u03b2\nf : \u03b1 \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\ncs : List \u03b1\nh : Exists fun bs => bs ++ as = cs\n\u22a2 forIn'.loop cs (fun a x b => f a b) as init h = forIn.loop f as init"}, {"tactic": "induction as generalizing cs init with\n| nil => intros; rfl\n| cons a as ih => intros; simp [List.forIn.loop, List.forIn'.loop, ih]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nm : Type v \u2192 Type w\ninst\u271d : Monad m\nas : List \u03b1\ninit : \u03b2\nf : \u03b1 \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\ncs : List \u03b1\nh : Exists fun bs => bs ++ as = cs\n\u22a2 forIn'.loop cs (fun a x b => f a b) as init h = forIn.loop f as init", "state_after": "no goals"}, {"tactic": "intros", "state_before": "case nil\n\u03b1 : Type u\n\u03b2 : Type v\nm : Type v \u2192 Type w\ninst\u271d : Monad m\nf : \u03b1 \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\ninit : \u03b2\ncs : List \u03b1\nh : Exists fun bs => bs ++ nil = cs\n\u22a2 forIn'.loop cs (fun a x b => f a b) nil init h = forIn.loop f nil init", "state_after": "case nil\n\u03b1 : Type u\n\u03b2 : Type v\nm : Type v \u2192 Type w\ninst\u271d : Monad m\nf : \u03b1 \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\ninit : \u03b2\ncs : List \u03b1\nh : Exists fun bs => bs ++ nil = cs\n\u22a2 forIn'.loop cs (fun a x b => f a b) nil init h = forIn.loop f nil init"}, {"tactic": "rfl", "state_before": "case nil\n\u03b1 : Type u\n\u03b2 : Type v\nm : Type v \u2192 Type w\ninst\u271d : Monad m\nf : \u03b1 \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\ninit : \u03b2\ncs : List \u03b1\nh : Exists fun bs => bs ++ nil = cs\n\u22a2 forIn'.loop cs (fun a x b => f a b) nil init h = forIn.loop f nil init", "state_after": "no goals"}, {"tactic": "intros", "state_before": "case cons\n\u03b1 : Type u\n\u03b2 : Type v\nm : Type v \u2192 Type w\ninst\u271d : Monad m\nf : \u03b1 \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\na : \u03b1\nas : List \u03b1\nih :\n  \u2200 (init : \u03b2) (cs : List \u03b1) (h : Exists fun bs => bs ++ as = cs),\n    forIn'.loop cs (fun a x b => f a b) as init h = forIn.loop f as init\ninit : \u03b2\ncs : List \u03b1\nh : Exists fun bs => bs ++ a :: as = cs\n\u22a2 forIn'.loop cs (fun a x b => f a b) (a :: as) init h = forIn.loop f (a :: as) init", "state_after": "case cons\n\u03b1 : Type u\n\u03b2 : Type v\nm : Type v \u2192 Type w\ninst\u271d : Monad m\nf : \u03b1 \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\na : \u03b1\nas : List \u03b1\nih :\n  \u2200 (init : \u03b2) (cs : List \u03b1) (h : Exists fun bs => bs ++ as = cs),\n    forIn'.loop cs (fun a x b => f a b) as init h = forIn.loop f as init\ninit : \u03b2\ncs : List \u03b1\nh : Exists fun bs => bs ++ a :: as = cs\n\u22a2 forIn'.loop cs (fun a x b => f a b) (a :: as) init h = forIn.loop f (a :: as) init"}, {"tactic": "simp [List.forIn.loop, List.forIn'.loop, ih]", "state_before": "case cons\n\u03b1 : Type u\n\u03b2 : Type v\nm : Type v \u2192 Type w\ninst\u271d : Monad m\nf : \u03b1 \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\na : \u03b1\nas : List \u03b1\nih :\n  \u2200 (init : \u03b2) (cs : List \u03b1) (h : Exists fun bs => bs ++ as = cs),\n    forIn'.loop cs (fun a x b => f a b) as init h = forIn.loop f as init\ninit : \u03b2\ncs : List \u03b1\nh : Exists fun bs => bs ++ a :: as = cs\n\u22a2 forIn'.loop cs (fun a x b => f a b) (a :: as) init h = forIn.loop f (a :: as) init", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Minimal.lean", "full_name": "IsGreatest.mem_maximals", "start": [221, 1], "end": [222, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fin/VecNotation.lean", "full_name": "Matrix.head_zero", "start": [545, 1], "end": [546, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Int.preimage_Icc", "start": [1287, 1], "end": [1289, 27], "traced_tactics": [{"tactic": "ext", "state_before": "F : Type ?u.243033\n\u03b1 : Type u_1\n\u03b2 : Type ?u.243039\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\n\u22a2 Int.cast \u207b\u00b9' Icc a b = Icc \u2308a\u2309 \u230ab\u230b", "state_after": "case h\nF : Type ?u.243033\n\u03b1 : Type u_1\n\u03b2 : Type ?u.243039\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\nx\u271d : \u2124\n\u22a2 x\u271d \u2208 Int.cast \u207b\u00b9' Icc a b \u2194 x\u271d \u2208 Icc \u2308a\u2309 \u230ab\u230b"}, {"tactic": "simp [ceil_le, le_floor]", "state_before": "case h\nF : Type ?u.243033\n\u03b1 : Type u_1\n\u03b2 : Type ?u.243039\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a b : \u03b1\nx\u271d : \u2124\n\u22a2 x\u271d \u2208 Int.cast \u207b\u00b9' Icc a b \u2194 x\u271d \u2208 Icc \u2308a\u2309 \u230ab\u230b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Action.lean", "full_name": "CategoryTheory.ActionCategory.\u03c0_map", "start": [67, 1], "end": [68, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Init/Data/Bool/Lemmas.lean", "full_name": "Bool.cond_self", "start": [45, 1], "end": [45, 93], "traced_tactics": [{"tactic": "cases b <;> rfl", "state_before": "\u03b1 : Type u\nb : Bool\na : \u03b1\n\u22a2 (bif b then a else a) = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Bool/Basic.lean", "full_name": "Bool.injective_iff", "start": [404, 1], "end": [407, 56], "traced_tactics": [{"tactic": "cases x <;> cases y", "state_before": "\u03b1 : Sort u_1\nf : Bool \u2192 \u03b1\nH : f false \u2260 f true\nx y : Bool\nhxy : f x = f y\n\u22a2 x = y", "state_after": "case false.false\n\u03b1 : Sort u_1\nf : Bool \u2192 \u03b1\nH : f false \u2260 f true\nhxy : f false = f false\n\u22a2 false = false\n\ncase false.true\n\u03b1 : Sort u_1\nf : Bool \u2192 \u03b1\nH : f false \u2260 f true\nhxy : f false = f true\n\u22a2 false = true\n\ncase true.false\n\u03b1 : Sort u_1\nf : Bool \u2192 \u03b1\nH : f false \u2260 f true\nhxy : f true = f false\n\u22a2 true = false\n\ncase true.true\n\u03b1 : Sort u_1\nf : Bool \u2192 \u03b1\nH : f false \u2260 f true\nhxy : f true = f true\n\u22a2 true = true"}, {"tactic": "exacts [rfl, (H hxy).elim, (H hxy.symm).elim, rfl]", "state_before": "case false.false\n\u03b1 : Sort u_1\nf : Bool \u2192 \u03b1\nH : f false \u2260 f true\nhxy : f false = f false\n\u22a2 false = false\n\ncase false.true\n\u03b1 : Sort u_1\nf : Bool \u2192 \u03b1\nH : f false \u2260 f true\nhxy : f false = f true\n\u22a2 false = true\n\ncase true.false\n\u03b1 : Sort u_1\nf : Bool \u2192 \u03b1\nH : f false \u2260 f true\nhxy : f true = f false\n\u22a2 true = false\n\ncase true.true\n\u03b1 : Sort u_1\nf : Bool \u2192 \u03b1\nH : f false \u2260 f true\nhxy : f true = f true\n\u22a2 true = true", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/WithBot.lean", "full_name": "WithBot.some_lt_some", "start": [260, 1], "end": [261, 15], "traced_tactics": [{"tactic": "simp [LT.lt]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.15780\n\u03b3 : Type ?u.15783\n\u03b4 : Type ?u.15786\na b : \u03b1\ninst\u271d : LT \u03b1\n\u22a2 Option.some a < Option.some b \u2194 a < b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/SymmDiff.lean", "full_name": "symmDiff_sdiff", "start": [169, 1], "end": [170, 71], "traced_tactics": [{"tactic": "rw [symmDiff, sup_sdiff_distrib, sdiff_sdiff_left, sdiff_sdiff_left]", "state_before": "\u03b9 : Type ?u.26724\n\u03b1 : Type u_1\n\u03b2 : Type ?u.26730\n\u03c0 : \u03b9 \u2192 Type ?u.26735\ninst\u271d : GeneralizedCoheytingAlgebra \u03b1\na b c d : \u03b1\n\u22a2 a \u2206 b \\ c = a \\ (b \u2294 c) \u2294 b \\ (a \u2294 c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Rdrop.lean", "full_name": "List.rtakeWhile_eq_nil_iff", "start": [230, 1], "end": [238, 20], "traced_tactics": [{"tactic": "induction' l using List.reverseRecOn with l a", "state_before": "\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nn : \u2115\n\u22a2 rtakeWhile p l = [] \u2194 \u2200 (hl : l \u2260 []), \u00acp (getLast l hl) = true", "state_after": "case H0\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nn : \u2115\n\u22a2 rtakeWhile p [] = [] \u2194 \u2200 (hl : [] \u2260 []), \u00acp (getLast [] hl) = true\n\ncase H1\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u271d : List \u03b1\nn : \u2115\nl : List \u03b1\na : \u03b1\na\u271d : rtakeWhile p l = [] \u2194 \u2200 (hl : l \u2260 []), \u00acp (getLast l hl) = true\n\u22a2 rtakeWhile p (l ++ [a]) = [] \u2194 \u2200 (hl : l ++ [a] \u2260 []), \u00acp (getLast (l ++ [a]) hl) = true"}, {"tactic": "simp only [rtakeWhile, takeWhile, reverse_nil, true_iff]", "state_before": "case H0\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nn : \u2115\n\u22a2 rtakeWhile p [] = [] \u2194 \u2200 (hl : [] \u2260 []), \u00acp (getLast [] hl) = true", "state_after": "case H0\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nn : \u2115\n\u22a2 \u2200 (hl : [] \u2260 []), \u00acp (getLast [] hl) = true"}, {"tactic": "intro f", "state_before": "case H0\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nn : \u2115\n\u22a2 \u2200 (hl : [] \u2260 []), \u00acp (getLast [] hl) = true", "state_after": "case H0\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nn : \u2115\nf : [] \u2260 []\n\u22a2 \u00acp (getLast [] f) = true"}, {"tactic": "contradiction", "state_before": "case H0\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nn : \u2115\nf : [] \u2260 []\n\u22a2 \u00acp (getLast [] f) = true", "state_after": "no goals"}, {"tactic": "simp only [rtakeWhile, reverse_append, takeWhile, reverse_eq_nil, getLast_append, ne_eq,\nappend_eq_nil, and_false, forall_true_left]", "state_before": "case H1\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u271d : List \u03b1\nn : \u2115\nl : List \u03b1\na : \u03b1\na\u271d : rtakeWhile p l = [] \u2194 \u2200 (hl : l \u2260 []), \u00acp (getLast l hl) = true\n\u22a2 rtakeWhile p (l ++ [a]) = [] \u2194 \u2200 (hl : l ++ [a] \u2260 []), \u00acp (getLast (l ++ [a]) hl) = true", "state_after": "case H1\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u271d : List \u03b1\nn : \u2115\nl : List \u03b1\na : \u03b1\na\u271d : rtakeWhile p l = [] \u2194 \u2200 (hl : l \u2260 []), \u00acp (getLast l hl) = true\n\u22a2 (match p a with\n      | true => a :: takeWhile p (List.append [] (reverse l))\n      | false => []) =\n      [] \u2194\n    \u00acp a = true"}, {"tactic": "refine' \u27e8fun h => _ , fun h => _\u27e9", "state_before": "case H1\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u271d : List \u03b1\nn : \u2115\nl : List \u03b1\na : \u03b1\na\u271d : rtakeWhile p l = [] \u2194 \u2200 (hl : l \u2260 []), \u00acp (getLast l hl) = true\n\u22a2 (match p a with\n      | true => a :: takeWhile p (List.append [] (reverse l))\n      | false => []) =\n      [] \u2194\n    \u00acp a = true", "state_after": "case H1.refine'_1\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u271d : List \u03b1\nn : \u2115\nl : List \u03b1\na : \u03b1\na\u271d : rtakeWhile p l = [] \u2194 \u2200 (hl : l \u2260 []), \u00acp (getLast l hl) = true\nh :\n  (match p a with\n    | true => a :: takeWhile p (List.append [] (reverse l))\n    | false => []) =\n    []\n\u22a2 \u00acp a = true\n\ncase H1.refine'_2\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u271d : List \u03b1\nn : \u2115\nl : List \u03b1\na : \u03b1\na\u271d : rtakeWhile p l = [] \u2194 \u2200 (hl : l \u2260 []), \u00acp (getLast l hl) = true\nh : \u00acp a = true\n\u22a2 (match p a with\n    | true => a :: takeWhile p (List.append [] (reverse l))\n    | false => []) =\n    []"}, {"tactic": "intro pa", "state_before": "case H1.refine'_1\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u271d : List \u03b1\nn : \u2115\nl : List \u03b1\na : \u03b1\na\u271d : rtakeWhile p l = [] \u2194 \u2200 (hl : l \u2260 []), \u00acp (getLast l hl) = true\nh :\n  (match p a with\n    | true => a :: takeWhile p (List.append [] (reverse l))\n    | false => []) =\n    []\n\u22a2 \u00acp a = true", "state_after": "case H1.refine'_1\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u271d : List \u03b1\nn : \u2115\nl : List \u03b1\na : \u03b1\na\u271d : rtakeWhile p l = [] \u2194 \u2200 (hl : l \u2260 []), \u00acp (getLast l hl) = true\nh :\n  (match p a with\n    | true => a :: takeWhile p (List.append [] (reverse l))\n    | false => []) =\n    []\npa : p a = true\n\u22a2 False"}, {"tactic": "simp only [pa] at h", "state_before": "case H1.refine'_1\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u271d : List \u03b1\nn : \u2115\nl : List \u03b1\na : \u03b1\na\u271d : rtakeWhile p l = [] \u2194 \u2200 (hl : l \u2260 []), \u00acp (getLast l hl) = true\nh :\n  (match p a with\n    | true => a :: takeWhile p (List.append [] (reverse l))\n    | false => []) =\n    []\npa : p a = true\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simp only [h]", "state_before": "case H1.refine'_2\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u271d : List \u03b1\nn : \u2115\nl : List \u03b1\na : \u03b1\na\u271d : rtakeWhile p l = [] \u2194 \u2200 (hl : l \u2260 []), \u00acp (getLast l hl) = true\nh : \u00acp a = true\n\u22a2 (match p a with\n    | true => a :: takeWhile p (List.append [] (reverse l))\n    | false => []) =\n    []", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean", "full_name": "HasFDerivWithinAt.csin", "start": [382, 1], "end": [384, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/PNat/Factors.lean", "full_name": "PNat.factorMultiset_one", "start": [295, 1], "end": [296, 78], "traced_tactics": [{"tactic": "simp [factorMultiset, PrimeMultiset.ofNatList, PrimeMultiset.ofNatMultiset]", "state_before": "\u22a2 factorMultiset 1 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Category/TopCat/Limits/Products.lean", "full_name": "TopCat.limit_topology", "start": [153, 1], "end": [155, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/FiberBundle/Trivialization.lean", "full_name": "Trivialization.symm_trans_source_eq", "start": [421, 1], "end": [423, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Schreier.lean", "full_name": "Subgroup.closure_mul_image_eq_top", "start": [88, 1], "end": [92, 77], "traced_tactics": [{"tactic": "rw [eq_top_iff, \u2190 map_subtype_le_map_subtype, MonoidHom.map_closure, Set.image_image]", "state_before": "G : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\n\u22a2 closure ((fun g => { val := g * (\u2191(toFun hR g))\u207b\u00b9, property := (_ : g * (\u2191(toFun hR g))\u207b\u00b9 \u2208 H) }) '' (R * S)) = \u22a4", "state_after": "G : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\n\u22a2 map (Subgroup.subtype H) \u22a4 \u2264\n    closure\n      ((fun x => \u2191(Subgroup.subtype H) { val := x * (\u2191(toFun hR x))\u207b\u00b9, property := (_ : x * (\u2191(toFun hR x))\u207b\u00b9 \u2208 H) }) ''\n        (R * S))"}, {"tactic": "exact (map_subtype_le \u22a4).trans (ge_of_eq (closure_mul_image_eq hR hR1 hS))", "state_before": "G : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nR S : Set G\nhR : R \u2208 rightTransversals \u2191H\nhR1 : 1 \u2208 R\nhS : closure S = \u22a4\n\u22a2 map (Subgroup.subtype H) \u22a4 \u2264\n    closure\n      ((fun x => \u2191(Subgroup.subtype H) { val := x * (\u2191(toFun hR x))\u207b\u00b9, property := (_ : x * (\u2191(toFun hR x))\u207b\u00b9 \u2208 H) }) ''\n        (R * S))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Order/T5.lean", "full_name": "Set.ordConnectedComponent_mem_nhds", "start": [30, 1], "end": [33, 65], "traced_tactics": [{"tactic": "refine' \u27e8fun h => mem_of_superset h ordConnectedComponent_subset, fun h => _\u27e9", "state_before": "X : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b c : X\ns t : Set X\n\u22a2 ordConnectedComponent s a \u2208 \ud835\udcdd a \u2194 s \u2208 \ud835\udcdd a", "state_after": "X : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b c : X\ns t : Set X\nh : s \u2208 \ud835\udcdd a\n\u22a2 ordConnectedComponent s a \u2208 \ud835\udcdd a"}, {"tactic": "rcases exists_Icc_mem_subset_of_mem_nhds h with \u27e8b, c, ha, ha', hs\u27e9", "state_before": "X : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b c : X\ns t : Set X\nh : s \u2208 \ud835\udcdd a\n\u22a2 ordConnectedComponent s a \u2208 \ud835\udcdd a", "state_after": "case intro.intro.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nh : s \u2208 \ud835\udcdd a\nb c : X\nha : a \u2208 Icc b c\nha' : Icc b c \u2208 \ud835\udcdd a\nhs : Icc b c \u2286 s\n\u22a2 ordConnectedComponent s a \u2208 \ud835\udcdd a"}, {"tactic": "exact mem_of_superset ha' (subset_ordConnectedComponent ha hs)", "state_before": "case intro.intro.intro.intro\nX : Type u_1\ninst\u271d\u00b2 : LinearOrder X\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : OrderTopology X\na b\u271d c\u271d : X\ns t : Set X\nh : s \u2208 \ud835\udcdd a\nb c : X\nha : a \u2208 Icc b c\nha' : Icc b c \u2208 \ud835\udcdd a\nhs : Icc b c \u2286 s\n\u22a2 ordConnectedComponent s a \u2208 \ud835\udcdd a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/AlgebraMap.lean", "full_name": "Polynomial.aeval_one", "start": [227, 1], "end": [228, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean", "full_name": "NonUnitalSubsemiring.coe_toSubsemigroup", "start": [241, 1], "end": [242, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "measurable_of_Iio", "start": [1041, 1], "end": [1044, 32], "traced_tactics": [{"tactic": "convert measurable_generateFrom (\u03b1 := \u03b4) _", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.1076994\n\u03b3 : Type ?u.1076997\n\u03b3\u2082 : Type ?u.1077000\n\u03b4 : Type u_1\n\u03b9 : Sort y\ns t u : 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 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nf : \u03b4 \u2192 \u03b1\nhf : \u2200 (x : \u03b1), MeasurableSet (f \u207b\u00b9' Iio x)\n\u22a2 Measurable f", "state_after": "case h.e'_4\n\u03b1 : Type u_2\n\u03b2 : Type ?u.1076994\n\u03b3 : Type ?u.1076997\n\u03b3\u2082 : Type ?u.1077000\n\u03b4 : Type u_1\n\u03b9 : Sort y\ns t u : 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 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nf : \u03b4 \u2192 \u03b1\nhf : \u2200 (x : \u03b1), MeasurableSet (f \u207b\u00b9' Iio x)\n\u22a2 inst\u271d\u00b9\u00b9 = MeasurableSpace.generateFrom ?convert_2\n\ncase convert_2\n\u03b1 : Type u_2\n\u03b2 : Type ?u.1076994\n\u03b3 : Type ?u.1076997\n\u03b3\u2082 : Type ?u.1077000\n\u03b4 : Type u_1\n\u03b9 : Sort y\ns t u : 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 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nf : \u03b4 \u2192 \u03b1\nhf : \u2200 (x : \u03b1), MeasurableSet (f \u207b\u00b9' Iio x)\n\u22a2 Set (Set \u03b1)\n\ncase convert_4\n\u03b1 : Type u_2\n\u03b2 : Type ?u.1076994\n\u03b3 : Type ?u.1076997\n\u03b3\u2082 : Type ?u.1077000\n\u03b4 : Type u_1\n\u03b9 : Sort y\ns t u : 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 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nf : \u03b4 \u2192 \u03b1\nhf : \u2200 (x : \u03b1), MeasurableSet (f \u207b\u00b9' Iio x)\n\u22a2 \u2200 (t : Set \u03b1), t \u2208 ?convert_2 \u2192 MeasurableSet (f \u207b\u00b9' t)"}, {"tactic": "exact BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Iio _)", "state_before": "case h.e'_4\n\u03b1 : Type u_2\n\u03b2 : Type ?u.1076994\n\u03b3 : Type ?u.1076997\n\u03b3\u2082 : Type ?u.1077000\n\u03b4 : Type u_1\n\u03b9 : Sort y\ns t u : 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 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nf : \u03b4 \u2192 \u03b1\nhf : \u2200 (x : \u03b1), MeasurableSet (f \u207b\u00b9' Iio x)\n\u22a2 inst\u271d\u00b9\u00b9 = MeasurableSpace.generateFrom ?convert_2\n\ncase convert_2\n\u03b1 : Type u_2\n\u03b2 : Type ?u.1076994\n\u03b3 : Type ?u.1076997\n\u03b3\u2082 : Type ?u.1077000\n\u03b4 : Type u_1\n\u03b9 : Sort y\ns t u : 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 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nf : \u03b4 \u2192 \u03b1\nhf : \u2200 (x : \u03b1), MeasurableSet (f \u207b\u00b9' Iio x)\n\u22a2 Set (Set \u03b1)\n\ncase convert_4\n\u03b1 : Type u_2\n\u03b2 : Type ?u.1076994\n\u03b3 : Type ?u.1076997\n\u03b3\u2082 : Type ?u.1077000\n\u03b4 : Type u_1\n\u03b9 : Sort y\ns t u : 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 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nf : \u03b4 \u2192 \u03b1\nhf : \u2200 (x : \u03b1), MeasurableSet (f \u207b\u00b9' Iio x)\n\u22a2 \u2200 (t : Set \u03b1), t \u2208 ?convert_2 \u2192 MeasurableSet (f \u207b\u00b9' t)", "state_after": "case convert_4\n\u03b1 : Type u_2\n\u03b2 : Type ?u.1076994\n\u03b3 : Type ?u.1076997\n\u03b3\u2082 : Type ?u.1077000\n\u03b4 : Type u_1\n\u03b9 : Sort y\ns t u : 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 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nf : \u03b4 \u2192 \u03b1\nhf : \u2200 (x : \u03b1), MeasurableSet (f \u207b\u00b9' Iio x)\n\u22a2 \u2200 (t : Set \u03b1), t \u2208 range Iio \u2192 MeasurableSet (f \u207b\u00b9' t)"}, {"tactic": "rintro _ \u27e8x, rfl\u27e9", "state_before": "case convert_4\n\u03b1 : Type u_2\n\u03b2 : Type ?u.1076994\n\u03b3 : Type ?u.1076997\n\u03b3\u2082 : Type ?u.1077000\n\u03b4 : Type u_1\n\u03b9 : Sort y\ns t u : 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 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nf : \u03b4 \u2192 \u03b1\nhf : \u2200 (x : \u03b1), MeasurableSet (f \u207b\u00b9' Iio x)\n\u22a2 \u2200 (t : Set \u03b1), t \u2208 range Iio \u2192 MeasurableSet (f \u207b\u00b9' t)", "state_after": "case convert_4.intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.1076994\n\u03b3 : Type ?u.1076997\n\u03b3\u2082 : Type ?u.1077000\n\u03b4 : Type u_1\n\u03b9 : Sort y\ns t u : 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 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nf : \u03b4 \u2192 \u03b1\nhf : \u2200 (x : \u03b1), MeasurableSet (f \u207b\u00b9' Iio x)\nx : \u03b1\n\u22a2 MeasurableSet (f \u207b\u00b9' Iio x)"}, {"tactic": "exact hf x", "state_before": "case convert_4.intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.1076994\n\u03b3 : Type ?u.1076997\n\u03b3\u2082 : Type ?u.1077000\n\u03b4 : Type u_1\n\u03b9 : Sort y\ns t u : 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 : LinearOrder \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : SecondCountableTopology \u03b1\nf : \u03b4 \u2192 \u03b1\nhf : \u2200 (x : \u03b1), MeasurableSet (f \u207b\u00b9' Iio x)\nx : \u03b1\n\u22a2 MeasurableSet (f \u207b\u00b9' Iio x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.image_surjective", "start": [1548, 1], "end": [1552, 16], "traced_tactics": [{"tactic": "refine' \u27e8fun h y => _, Surjective.image_surjective\u27e9", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\n\u22a2 Surjective (image f) \u2194 Surjective f", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (image f)\ny : \u03b2\n\u22a2 \u2203 a, f a = y"}, {"tactic": "cases' h {y} with s hs", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (image f)\ny : \u03b2\n\u22a2 \u2203 a, f a = y", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (image f)\ny : \u03b2\ns : Set \u03b1\nhs : f '' s = {y}\n\u22a2 \u2203 a, f a = y"}, {"tactic": "have := mem_singleton y", "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (image f)\ny : \u03b2\ns : Set \u03b1\nhs : f '' s = {y}\n\u22a2 \u2203 a, f a = y", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (image f)\ny : \u03b2\ns : Set \u03b1\nhs : f '' s = {y}\nthis : y \u2208 {y}\n\u22a2 \u2203 a, f a = y"}, {"tactic": "rw [\u2190 hs] at this", "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (image f)\ny : \u03b2\ns : Set \u03b1\nhs : f '' s = {y}\nthis : y \u2208 {y}\n\u22a2 \u2203 a, f a = y", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (image f)\ny : \u03b2\ns : Set \u03b1\nhs : f '' s = {y}\nthis : y \u2208 f '' s\n\u22a2 \u2203 a, f a = y"}, {"tactic": "rcases this with \u27e8x, _, hx\u27e9", "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (image f)\ny : \u03b2\ns : Set \u03b1\nhs : f '' s = {y}\nthis : y \u2208 f '' s\n\u22a2 \u2203 a, f a = y", "state_after": "case intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (image f)\ny : \u03b2\ns : Set \u03b1\nhs : f '' s = {y}\nx : \u03b1\nleft\u271d : x \u2208 s\nhx : f x = y\n\u22a2 \u2203 a, f a = y"}, {"tactic": "exact \u27e8x, hx\u27e9", "state_before": "case intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\nh : Surjective (image f)\ny : \u03b2\ns : Set \u03b1\nhs : f '' s = {y}\nx : \u03b1\nleft\u271d : x \u2208 s\nhx : f x = y\n\u22a2 \u2203 a, f a = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "iSup_le", "start": [816, 1], "end": [817, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.comp_rangeSplitting", "start": [1154, 1], "end": [1158, 29], "traced_tactics": [{"tactic": "ext", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.98077\n\u03b9 : Sort ?u.98080\n\u03b9' : Sort ?u.98083\nf\u271d : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 f \u2218 rangeSplitting f = Subtype.val", "state_after": "case h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.98077\n\u03b9 : Sort ?u.98080\n\u03b9' : Sort ?u.98083\nf\u271d : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nx\u271d : \u2191(range f)\n\u22a2 (f \u2218 rangeSplitting f) x\u271d = \u2191x\u271d"}, {"tactic": "simp only [Function.comp_apply]", "state_before": "case h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.98077\n\u03b9 : Sort ?u.98080\n\u03b9' : Sort ?u.98083\nf\u271d : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nx\u271d : \u2191(range f)\n\u22a2 (f \u2218 rangeSplitting f) x\u271d = \u2191x\u271d", "state_after": "case h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.98077\n\u03b9 : Sort ?u.98080\n\u03b9' : Sort ?u.98083\nf\u271d : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nx\u271d : \u2191(range f)\n\u22a2 f (rangeSplitting f x\u271d) = \u2191x\u271d"}, {"tactic": "apply apply_rangeSplitting", "state_before": "case h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.98077\n\u03b9 : Sort ?u.98080\n\u03b9' : Sort ?u.98083\nf\u271d : \u03b9 \u2192 \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nx\u271d : \u2191(range f)\n\u22a2 f (rangeSplitting f x\u271d) = \u2191x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Mul.lean", "full_name": "DifferentiableOn.pow", "start": [354, 1], "end": [355, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/Divisors.lean", "full_name": "Nat.image_fst_divisorsAntidiagonal", "start": [252, 1], "end": [254, 39], "traced_tactics": [{"tactic": "ext", "state_before": "n : \u2115\n\u22a2 image Prod.fst (divisorsAntidiagonal n) = divisors n", "state_after": "case a\nn a\u271d : \u2115\n\u22a2 a\u271d \u2208 image Prod.fst (divisorsAntidiagonal n) \u2194 a\u271d \u2208 divisors n"}, {"tactic": "simp [Dvd.dvd, @eq_comm _ n (_ * _)]", "state_before": "case a\nn a\u271d : \u2115\n\u22a2 a\u271d \u2208 image Prod.fst (divisorsAntidiagonal n) \u2194 a\u271d \u2208 divisors n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Logic.lean", "full_name": "not_and_self_iff", "start": [241, 1], "end": [241, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "IsClosed.leftCoset", "start": [98, 1], "end": [99, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Dimension.lean", "full_name": "LinearEquiv.rank_map_eq", "start": [210, 1], "end": [212, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/BoundedOrder.lean", "full_name": "Subtype.mk_top", "start": [767, 1], "end": [768, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Star.lean", "full_name": "StarConvex.affine_image", "start": [363, 1], "end": [367, 43], "traced_tactics": [{"tactic": "rintro y \u27e8y', \u27e8hy', hy'f\u27e9\u27e9 a b ha hb hab", "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y : E\ns\u271d t : Set E\nf : E \u2192\u1d43[\ud835\udd5c] F\ns : Set E\nhs : StarConvex \ud835\udd5c x s\n\u22a2 StarConvex \ud835\udd5c (\u2191f x) (\u2191f '' s)", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y\u271d : E\ns\u271d t : Set E\nf : E \u2192\u1d43[\ud835\udd5c] F\ns : Set E\nhs : StarConvex \ud835\udd5c x s\ny : F\ny' : E\nhy' : y' \u2208 s\nhy'f : \u2191f y' = y\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 \u2191f x + b \u2022 y \u2208 \u2191f '' s"}, {"tactic": "refine' \u27e8a \u2022 x + b \u2022 y', \u27e8hs hy' ha hb hab, _\u27e9\u27e9", "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y\u271d : E\ns\u271d t : Set E\nf : E \u2192\u1d43[\ud835\udd5c] F\ns : Set E\nhs : StarConvex \ud835\udd5c x s\ny : F\ny' : E\nhy' : y' \u2208 s\nhy'f : \u2191f y' = y\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 \u2191f x + b \u2022 y \u2208 \u2191f '' s", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y\u271d : E\ns\u271d t : Set E\nf : E \u2192\u1d43[\ud835\udd5c] F\ns : Set E\nhs : StarConvex \ud835\udd5c x s\ny : F\ny' : E\nhy' : y' \u2208 s\nhy'f : \u2191f y' = y\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 \u2191f (a \u2022 x + b \u2022 y') = a \u2022 \u2191f x + b \u2022 y"}, {"tactic": "rw [Convex.combo_affine_apply hab, hy'f]", "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y\u271d : E\ns\u271d t : Set E\nf : E \u2192\u1d43[\ud835\udd5c] F\ns : Set E\nhs : StarConvex \ud835\udd5c x s\ny : F\ny' : E\nhy' : y' \u2208 s\nhy'f : \u2191f y' = y\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 \u2191f (a \u2022 x + b \u2022 y') = a \u2022 \u2191f x + b \u2022 y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/NonZeroDivisors.lean", "full_name": "nonZeroDivisors_le_comap_nonZeroDivisors_of_injective", "start": [160, 1], "end": [162, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/MonoidLocalization.lean", "full_name": "Submonoid.LocalizationMap.map_spec", "start": [1169, 1], "end": [1170, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Basic.lean", "full_name": "Set.Icc_diff_both", "start": [795, 1], "end": [796, 61], "traced_tactics": [{"tactic": "rw [insert_eq, \u2190 diff_diff, Icc_diff_left, Ioc_diff_right]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.41354\ninst\u271d : PartialOrder \u03b1\na b c : \u03b1\n\u22a2 Icc a b \\ {a, b} = Ioo a b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Perfect.lean", "full_name": "Preperfect.perfect_closure", "start": [103, 1], "end": [111, 11], "traced_tactics": [{"tactic": "constructor", "state_before": "\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\n\u22a2 Perfect (closure C)", "state_after": "case closed\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\n\u22a2 IsClosed (closure C)\n\ncase acc\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\n\u22a2 Preperfect (closure C)"}, {"tactic": "intro x hx", "state_before": "case acc\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\n\u22a2 Preperfect (closure C)", "state_after": "case acc\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\nx : \u03b1\nhx : x \u2208 closure C\n\u22a2 AccPt x (\ud835\udcdf (closure C))"}, {"tactic": "by_cases h : x \u2208 C <;> apply AccPt.mono _ (principal_mono.mpr subset_closure)", "state_before": "case acc\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\nx : \u03b1\nhx : x \u2208 closure C\n\u22a2 AccPt x (\ud835\udcdf (closure C))", "state_after": "\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\nx : \u03b1\nhx : x \u2208 closure C\nh : x \u2208 C\n\u22a2 AccPt x (\ud835\udcdf C)\n\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\nx : \u03b1\nhx : x \u2208 closure C\nh : \u00acx \u2208 C\n\u22a2 AccPt x (\ud835\udcdf C)"}, {"tactic": "have : {x}\u1d9c \u2229 C = C := by simp [h]", "state_before": "\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\nx : \u03b1\nhx : x \u2208 closure C\nh : \u00acx \u2208 C\n\u22a2 AccPt x (\ud835\udcdf C)", "state_after": "\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\nx : \u03b1\nhx : x \u2208 closure C\nh : \u00acx \u2208 C\nthis : {x}\u1d9c \u2229 C = C\n\u22a2 AccPt x (\ud835\udcdf C)"}, {"tactic": "rw [AccPt, nhdsWithin, inf_assoc, inf_principal, this]", "state_before": "\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\nx : \u03b1\nhx : x \u2208 closure C\nh : \u00acx \u2208 C\nthis : {x}\u1d9c \u2229 C = C\n\u22a2 AccPt x (\ud835\udcdf C)", "state_after": "\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\nx : \u03b1\nhx : x \u2208 closure C\nh : \u00acx \u2208 C\nthis : {x}\u1d9c \u2229 C = C\n\u22a2 NeBot (\ud835\udcdd x \u2293 \ud835\udcdf C)"}, {"tactic": "rw [closure_eq_cluster_pts] at hx", "state_before": "\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\nx : \u03b1\nhx : x \u2208 closure C\nh : \u00acx \u2208 C\nthis : {x}\u1d9c \u2229 C = C\n\u22a2 NeBot (\ud835\udcdd x \u2293 \ud835\udcdf C)", "state_after": "\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\nx : \u03b1\nhx : x \u2208 {a | ClusterPt a (\ud835\udcdf C)}\nh : \u00acx \u2208 C\nthis : {x}\u1d9c \u2229 C = C\n\u22a2 NeBot (\ud835\udcdd x \u2293 \ud835\udcdf C)"}, {"tactic": "exact hx", "state_before": "\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\nx : \u03b1\nhx : x \u2208 {a | ClusterPt a (\ud835\udcdf C)}\nh : \u00acx \u2208 C\nthis : {x}\u1d9c \u2229 C = C\n\u22a2 NeBot (\ud835\udcdd x \u2293 \ud835\udcdf C)", "state_after": "no goals"}, {"tactic": "exact isClosed_closure", "state_before": "case closed\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\n\u22a2 IsClosed (closure C)", "state_after": "no goals"}, {"tactic": "exact hC _ h", "state_before": "\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\nx : \u03b1\nhx : x \u2208 closure C\nh : x \u2208 C\n\u22a2 AccPt x (\ud835\udcdf C)", "state_after": "no goals"}, {"tactic": "simp [h]", "state_before": "\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\nC : Set \u03b1\nhC : Preperfect C\nx : \u03b1\nhx : x \u2208 closure C\nh : \u00acx \u2208 C\n\u22a2 {x}\u1d9c \u2229 C = C", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Matrix/Basic.lean", "full_name": "Matrix.conjTranspose_col", "start": [2691, 1], "end": [2693, 6], "traced_tactics": [{"tactic": "ext", "state_before": "l : Type ?u.1214195\nm : Type u_1\nn : Type ?u.1214201\no : Type ?u.1214204\nm' : o \u2192 Type ?u.1214209\nn' : o \u2192 Type ?u.1214214\nR : Type ?u.1214217\nS : Type ?u.1214220\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type ?u.1214227\ninst\u271d : Star \u03b1\nv : m \u2192 \u03b1\n\u22a2 (col v)\u1d34 = row (star v)", "state_after": "case a.h\nl : Type ?u.1214195\nm : Type u_1\nn : Type ?u.1214201\no : Type ?u.1214204\nm' : o \u2192 Type ?u.1214209\nn' : o \u2192 Type ?u.1214214\nR : Type ?u.1214217\nS : Type ?u.1214220\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type ?u.1214227\ninst\u271d : Star \u03b1\nv : m \u2192 \u03b1\ni\u271d : Unit\nx\u271d : m\n\u22a2 (col v)\u1d34 i\u271d x\u271d = row (star v) i\u271d x\u271d"}, {"tactic": "rfl", "state_before": "case a.h\nl : Type ?u.1214195\nm : Type u_1\nn : Type ?u.1214201\no : Type ?u.1214204\nm' : o \u2192 Type ?u.1214209\nn' : o \u2192 Type ?u.1214214\nR : Type ?u.1214217\nS : Type ?u.1214220\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type ?u.1214227\ninst\u271d : Star \u03b1\nv : m \u2192 \u03b1\ni\u271d : Unit\nx\u271d : m\n\u22a2 (col v)\u1d34 i\u271d x\u271d = row (star v) i\u271d x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.mass_nonzero_iff", "start": [185, 1], "end": [187, 38], "traced_tactics": [{"tactic": "rw [not_iff_not]", "state_before": "\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 mass \u03bc \u2260 0 \u2194 \u03bc \u2260 0", "state_after": "\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 mass \u03bc = 0 \u2194 \u03bc = 0"}, {"tactic": "exact FiniteMeasure.mass_zero_iff \u03bc", "state_before": "\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 mass \u03bc = 0 \u2194 \u03bc = 0", "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.smul_measure", "start": [657, 1], "end": [659, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Constructions.lean", "full_name": "Units.inducing_embedProduct", "start": [104, 1], "end": [105, 8], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Pointwise.lean", "full_name": "Filter.pureOneHom_apply", "start": [166, 1], "end": [167, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "full_name": "zero_tsub", "start": [344, 1], "end": [345, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/CharP/MixedCharZero.lean", "full_name": "EqualCharZero.pnatCast_one", "start": [208, 1], "end": [213, 34], "traced_tactics": [{"tactic": "apply Units.ext", "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fact (\u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I))\n\u22a2 \u21911 = 1", "state_after": "case a\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fact (\u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I))\n\u22a2 \u2191\u21911 = \u21911"}, {"tactic": "rw [Units.val_one]", "state_before": "case a\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fact (\u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I))\n\u22a2 \u2191\u21911 = \u21911", "state_after": "case a\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fact (\u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I))\n\u22a2 \u2191\u21911 = 1"}, {"tactic": "change ((PNat.isUnit_natCast (R := R) 1).unit : R) = 1", "state_before": "case a\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fact (\u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I))\n\u22a2 \u2191\u21911 = 1", "state_after": "case a\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fact (\u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I))\n\u22a2 \u2191(IsUnit.unit (_ : IsUnit \u2191\u21911)) = 1"}, {"tactic": "rw [IsUnit.unit_spec (PNat.isUnit_natCast 1)]", "state_before": "case a\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fact (\u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I))\n\u22a2 \u2191(IsUnit.unit (_ : IsUnit \u2191\u21911)) = 1", "state_after": "case a\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fact (\u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I))\n\u22a2 \u2191\u21911 = 1"}, {"tactic": "rw [PNat.one_coe, Nat.cast_one]", "state_before": "case a\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fact (\u2200 (I : Ideal R), I \u2260 \u22a4 \u2192 CharZero (R \u29f8 I))\n\u22a2 \u2191\u21911 = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.lift_le_nat_iff", "start": [1297, 1], "end": [1298, 37], "traced_tactics": [{"tactic": "rw [\u2190 lift_natCast.{v,u}, lift_le]", "state_before": "\u03b1 \u03b2 : Type u\na : Cardinal\nn : \u2115\n\u22a2 lift a \u2264 \u2191n \u2194 a \u2264 \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Game/PGame.lean", "full_name": "PGame.fuzzy_congr_left", "start": [966, 1], "end": [967, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Constructions.lean", "full_name": "Continuous.Prod.mk", "start": [406, 1], "end": [407, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Interval.lean", "full_name": "Filter.tendstoIxxClass_of_subset", "start": [114, 1], "end": [116, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicTopology/SplitSimplicialObject.lean", "full_name": "SimplicialObject.Split.id_F", "start": [437, 1], "end": [438, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Int/GCD.lean", "full_name": "Int.gcd_dvd_gcd_mul_right_right", "start": [357, 1], "end": [358, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Ray.lean", "full_name": "sameRay_or_sameRay_neg_iff_not_linearIndependent", "start": [592, 1], "end": [632, 21], "traced_tactics": [{"tactic": "by_cases hx : x = 0", "state_before": "R : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\n\u22a2 SameRay R x y \u2228 SameRay R x (-y) \u2194 \u00acLinearIndependent R ![x, y]", "state_after": "case pos\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : x = 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y) \u2194 \u00acLinearIndependent R ![x, y]\n\ncase neg\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y) \u2194 \u00acLinearIndependent R ![x, y]"}, {"tactic": "by_cases hy : y = 0", "state_before": "case neg\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y) \u2194 \u00acLinearIndependent R ![x, y]", "state_after": "case pos\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y) \u2194 \u00acLinearIndependent R ![x, y]\n\ncase neg\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y) \u2194 \u00acLinearIndependent R ![x, y]"}, {"tactic": "simp_rw [Fintype.not_linearIndependent_iff]", "state_before": "case neg\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y) \u2194 \u00acLinearIndependent R ![x, y]", "state_after": "case neg\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y) \u2194\n    \u2203 g, \u2211 i : Fin (Nat.succ (Nat.succ 0)), g i \u2022 Matrix.vecCons x ![y] i = 0 \u2227 \u2203 i, g i \u2260 0"}, {"tactic": "refine' \u27e8fun h => _, fun h => _\u27e9", "state_before": "case neg\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y) \u2194\n    \u2203 g, \u2211 i : Fin (Nat.succ (Nat.succ 0)), g i \u2022 Matrix.vecCons x ![y] i = 0 \u2227 \u2203 i, g i \u2260 0", "state_after": "case neg.refine'_1\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nh : SameRay R x y \u2228 SameRay R x (-y)\n\u22a2 \u2203 g, \u2211 i : Fin (Nat.succ (Nat.succ 0)), g i \u2022 Matrix.vecCons x ![y] i = 0 \u2227 \u2203 i, g i \u2260 0\n\ncase neg.refine'_2\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nh : \u2203 g, \u2211 i : Fin (Nat.succ (Nat.succ 0)), g i \u2022 Matrix.vecCons x ![y] i = 0 \u2227 \u2203 i, g i \u2260 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)"}, {"tactic": "simpa [hx] using fun h : LinearIndependent R ![0, y] => h.ne_zero 0 rfl", "state_before": "case pos\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : x = 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y) \u2194 \u00acLinearIndependent R ![x, y]", "state_after": "no goals"}, {"tactic": "simpa [hy] using fun h : LinearIndependent R ![x, 0] => h.ne_zero 1 rfl", "state_before": "case pos\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y) \u2194 \u00acLinearIndependent R ![x, y]", "state_after": "no goals"}, {"tactic": "rcases h with ((hx0 | hy0 | \u27e8r\u2081, r\u2082, hr\u2081, _, h\u27e9) | (hx0 | hy0 | \u27e8r\u2081, r\u2082, hr\u2081, _, h\u27e9))", "state_before": "case neg.refine'_1\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nh : SameRay R x y \u2228 SameRay R x (-y)\n\u22a2 \u2203 g, \u2211 i : Fin (Nat.succ (Nat.succ 0)), g i \u2022 Matrix.vecCons x ![y] i = 0 \u2227 \u2203 i, g i \u2260 0", "state_after": "case neg.refine'_1.inl.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhx0 : x = 0\n\u22a2 \u2203 g, \u2211 i : Fin (Nat.succ (Nat.succ 0)), g i \u2022 Matrix.vecCons x ![y] i = 0 \u2227 \u2203 i, g i \u2260 0\n\ncase neg.refine'_1.inl.inr.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhy0 : y = 0\n\u22a2 \u2203 g, \u2211 i : Fin (Nat.succ (Nat.succ 0)), g i \u2022 Matrix.vecCons x ![y] i = 0 \u2227 \u2203 i, g i \u2260 0\n\ncase neg.refine'_1.inl.inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nleft\u271d : 0 < r\u2082\nh : r\u2081 \u2022 x = r\u2082 \u2022 y\n\u22a2 \u2203 g, \u2211 i : Fin (Nat.succ (Nat.succ 0)), g i \u2022 Matrix.vecCons x ![y] i = 0 \u2227 \u2203 i, g i \u2260 0\n\ncase neg.refine'_1.inr.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhx0 : x = 0\n\u22a2 \u2203 g, \u2211 i : Fin (Nat.succ (Nat.succ 0)), g i \u2022 Matrix.vecCons x ![y] i = 0 \u2227 \u2203 i, g i \u2260 0\n\ncase neg.refine'_1.inr.inr.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhy0 : -y = 0\n\u22a2 \u2203 g, \u2211 i : Fin (Nat.succ (Nat.succ 0)), g i \u2022 Matrix.vecCons x ![y] i = 0 \u2227 \u2203 i, g i \u2260 0\n\ncase neg.refine'_1.inr.inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nleft\u271d : 0 < r\u2082\nh : r\u2081 \u2022 x = r\u2082 \u2022 -y\n\u22a2 \u2203 g, \u2211 i : Fin (Nat.succ (Nat.succ 0)), g i \u2022 Matrix.vecCons x ![y] i = 0 \u2227 \u2203 i, g i \u2260 0"}, {"tactic": "exact False.elim (hx hx0)", "state_before": "case neg.refine'_1.inl.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhx0 : x = 0\n\u22a2 \u2203 g, \u2211 i : Fin (Nat.succ (Nat.succ 0)), g i \u2022 Matrix.vecCons x ![y] i = 0 \u2227 \u2203 i, g i \u2260 0", "state_after": "no goals"}, {"tactic": "exact False.elim (hy hy0)", "state_before": "case neg.refine'_1.inl.inr.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhy0 : y = 0\n\u22a2 \u2203 g, \u2211 i : Fin (Nat.succ (Nat.succ 0)), g i \u2022 Matrix.vecCons x ![y] i = 0 \u2227 \u2203 i, g i \u2260 0", "state_after": "no goals"}, {"tactic": "refine' \u27e8![r\u2081, -r\u2082], _\u27e9", "state_before": "case neg.refine'_1.inl.inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nleft\u271d : 0 < r\u2082\nh : r\u2081 \u2022 x = r\u2082 \u2022 y\n\u22a2 \u2203 g, \u2211 i : Fin (Nat.succ (Nat.succ 0)), g i \u2022 Matrix.vecCons x ![y] i = 0 \u2227 \u2203 i, g i \u2260 0", "state_after": "case neg.refine'_1.inl.inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nleft\u271d : 0 < r\u2082\nh : r\u2081 \u2022 x = r\u2082 \u2022 y\n\u22a2 \u2211 i : Fin (Nat.succ (Nat.succ 0)), Matrix.vecCons r\u2081 ![-r\u2082] i \u2022 Matrix.vecCons x ![y] i = 0 \u2227\n    \u2203 i, Matrix.vecCons r\u2081 ![-r\u2082] i \u2260 0"}, {"tactic": "rw [Fin.sum_univ_two, Fin.exists_fin_two]", "state_before": "case neg.refine'_1.inl.inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nleft\u271d : 0 < r\u2082\nh : r\u2081 \u2022 x = r\u2082 \u2022 y\n\u22a2 \u2211 i : Fin (Nat.succ (Nat.succ 0)), Matrix.vecCons r\u2081 ![-r\u2082] i \u2022 Matrix.vecCons x ![y] i = 0 \u2227\n    \u2203 i, Matrix.vecCons r\u2081 ![-r\u2082] i \u2260 0", "state_after": "case neg.refine'_1.inl.inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nleft\u271d : 0 < r\u2082\nh : r\u2081 \u2022 x = r\u2082 \u2022 y\n\u22a2 Matrix.vecCons r\u2081 ![-r\u2082] 0 \u2022 Matrix.vecCons x ![y] 0 + Matrix.vecCons r\u2081 ![-r\u2082] 1 \u2022 Matrix.vecCons x ![y] 1 = 0 \u2227\n    (Matrix.vecCons r\u2081 ![-r\u2082] 0 \u2260 0 \u2228 Matrix.vecCons r\u2081 ![-r\u2082] 1 \u2260 0)"}, {"tactic": "simp [h, hr\u2081.ne.symm]", "state_before": "case neg.refine'_1.inl.inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nleft\u271d : 0 < r\u2082\nh : r\u2081 \u2022 x = r\u2082 \u2022 y\n\u22a2 Matrix.vecCons r\u2081 ![-r\u2082] 0 \u2022 Matrix.vecCons x ![y] 0 + Matrix.vecCons r\u2081 ![-r\u2082] 1 \u2022 Matrix.vecCons x ![y] 1 = 0 \u2227\n    (Matrix.vecCons r\u2081 ![-r\u2082] 0 \u2260 0 \u2228 Matrix.vecCons r\u2081 ![-r\u2082] 1 \u2260 0)", "state_after": "no goals"}, {"tactic": "exact False.elim (hx hx0)", "state_before": "case neg.refine'_1.inr.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhx0 : x = 0\n\u22a2 \u2203 g, \u2211 i : Fin (Nat.succ (Nat.succ 0)), g i \u2022 Matrix.vecCons x ![y] i = 0 \u2227 \u2203 i, g i \u2260 0", "state_after": "no goals"}, {"tactic": "exact False.elim (hy (neg_eq_zero.1 hy0))", "state_before": "case neg.refine'_1.inr.inr.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nhy0 : -y = 0\n\u22a2 \u2203 g, \u2211 i : Fin (Nat.succ (Nat.succ 0)), g i \u2022 Matrix.vecCons x ![y] i = 0 \u2227 \u2203 i, g i \u2260 0", "state_after": "no goals"}, {"tactic": "refine' \u27e8![r\u2081, r\u2082], _\u27e9", "state_before": "case neg.refine'_1.inr.inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nleft\u271d : 0 < r\u2082\nh : r\u2081 \u2022 x = r\u2082 \u2022 -y\n\u22a2 \u2203 g, \u2211 i : Fin (Nat.succ (Nat.succ 0)), g i \u2022 Matrix.vecCons x ![y] i = 0 \u2227 \u2203 i, g i \u2260 0", "state_after": "case neg.refine'_1.inr.inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nleft\u271d : 0 < r\u2082\nh : r\u2081 \u2022 x = r\u2082 \u2022 -y\n\u22a2 \u2211 i : Fin (Nat.succ (Nat.succ 0)), Matrix.vecCons r\u2081 ![r\u2082] i \u2022 Matrix.vecCons x ![y] i = 0 \u2227\n    \u2203 i, Matrix.vecCons r\u2081 ![r\u2082] i \u2260 0"}, {"tactic": "rw [Fin.sum_univ_two, Fin.exists_fin_two]", "state_before": "case neg.refine'_1.inr.inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nleft\u271d : 0 < r\u2082\nh : r\u2081 \u2022 x = r\u2082 \u2022 -y\n\u22a2 \u2211 i : Fin (Nat.succ (Nat.succ 0)), Matrix.vecCons r\u2081 ![r\u2082] i \u2022 Matrix.vecCons x ![y] i = 0 \u2227\n    \u2203 i, Matrix.vecCons r\u2081 ![r\u2082] i \u2260 0", "state_after": "case neg.refine'_1.inr.inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nleft\u271d : 0 < r\u2082\nh : r\u2081 \u2022 x = r\u2082 \u2022 -y\n\u22a2 Matrix.vecCons r\u2081 ![r\u2082] 0 \u2022 Matrix.vecCons x ![y] 0 + Matrix.vecCons r\u2081 ![r\u2082] 1 \u2022 Matrix.vecCons x ![y] 1 = 0 \u2227\n    (Matrix.vecCons r\u2081 ![r\u2082] 0 \u2260 0 \u2228 Matrix.vecCons r\u2081 ![r\u2082] 1 \u2260 0)"}, {"tactic": "simp [h, hr\u2081.ne.symm]", "state_before": "case neg.refine'_1.inr.inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nleft\u271d : 0 < r\u2082\nh : r\u2081 \u2022 x = r\u2082 \u2022 -y\n\u22a2 Matrix.vecCons r\u2081 ![r\u2082] 0 \u2022 Matrix.vecCons x ![y] 0 + Matrix.vecCons r\u2081 ![r\u2082] 1 \u2022 Matrix.vecCons x ![y] 1 = 0 \u2227\n    (Matrix.vecCons r\u2081 ![r\u2082] 0 \u2260 0 \u2228 Matrix.vecCons r\u2081 ![r\u2082] 1 \u2260 0)", "state_after": "no goals"}, {"tactic": "rcases h with \u27e8m, hm, hmne\u27e9", "state_before": "case neg.refine'_2\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nh : \u2203 g, \u2211 i : Fin (Nat.succ (Nat.succ 0)), g i \u2022 Matrix.vecCons x ![y] i = 0 \u2227 \u2203 i, g i \u2260 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)", "state_after": "case neg.refine'_2.intro.intro\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : \u2211 i : Fin (Nat.succ (Nat.succ 0)), m i \u2022 Matrix.vecCons x ![y] i = 0\nhmne : \u2203 i, m i \u2260 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)"}, {"tactic": "rw [Fin.sum_univ_two, add_eq_zero_iff_eq_neg, Matrix.cons_val_zero,\n  Matrix.cons_val_one, Matrix.head_cons] at hm", "state_before": "case neg.refine'_2.intro.intro\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : \u2211 i : Fin (Nat.succ (Nat.succ 0)), m i \u2022 Matrix.vecCons x ![y] i = 0\nhmne : \u2203 i, m i \u2260 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)", "state_after": "case neg.refine'_2.intro.intro\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)"}, {"tactic": "rcases lt_trichotomy (m 0) 0 with (hm0 | hm0 | hm0) <;>\n  rcases lt_trichotomy (m 1) 0 with (hm1 | hm1 | hm1)", "state_before": "case neg.refine'_2.intro.intro\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)", "state_after": "case neg.refine'_2.intro.intro.inl.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : m 0 < 0\nhm1 : m 1 < 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)\n\ncase neg.refine'_2.intro.intro.inl.inr.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : m 0 < 0\nhm1 : m 1 = 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)\n\ncase neg.refine'_2.intro.intro.inl.inr.inr\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : m 0 < 0\nhm1 : 0 < m 1\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)\n\ncase neg.refine'_2.intro.intro.inr.inl.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : m 0 = 0\nhm1 : m 1 < 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)\n\ncase neg.refine'_2.intro.intro.inr.inl.inr.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : m 0 = 0\nhm1 : m 1 = 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)\n\ncase neg.refine'_2.intro.intro.inr.inl.inr.inr\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : m 0 = 0\nhm1 : 0 < m 1\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)\n\ncase neg.refine'_2.intro.intro.inr.inr.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : 0 < m 0\nhm1 : m 1 < 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)\n\ncase neg.refine'_2.intro.intro.inr.inr.inr.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : 0 < m 0\nhm1 : m 1 = 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)\n\ncase neg.refine'_2.intro.intro.inr.inr.inr.inr\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : 0 < m 0\nhm1 : 0 < m 1\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)"}, {"tactic": "refine'\n  Or.inr (Or.inr (Or.inr \u27e8-m 0, -m 1, Left.neg_pos_iff.2 hm0, Left.neg_pos_iff.2 hm1, _\u27e9))", "state_before": "case neg.refine'_2.intro.intro.inl.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : m 0 < 0\nhm1 : m 1 < 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)", "state_after": "case neg.refine'_2.intro.intro.inl.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : m 0 < 0\nhm1 : m 1 < 0\n\u22a2 -m 0 \u2022 x = -m 1 \u2022 -y"}, {"tactic": "rw [neg_smul_neg, neg_smul, hm, neg_neg]", "state_before": "case neg.refine'_2.intro.intro.inl.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : m 0 < 0\nhm1 : m 1 < 0\n\u22a2 -m 0 \u2022 x = -m 1 \u2022 -y", "state_after": "no goals"}, {"tactic": "exfalso", "state_before": "case neg.refine'_2.intro.intro.inl.inr.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : m 0 < 0\nhm1 : m 1 = 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)", "state_after": "case neg.refine'_2.intro.intro.inl.inr.inl.h\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : m 0 < 0\nhm1 : m 1 = 0\n\u22a2 False"}, {"tactic": "simp [hm1, hx, hm0.ne] at hm", "state_before": "case neg.refine'_2.intro.intro.inl.inr.inl.h\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : m 0 < 0\nhm1 : m 1 = 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "refine' Or.inl (Or.inr (Or.inr \u27e8-m 0, m 1, Left.neg_pos_iff.2 hm0, hm1, _\u27e9))", "state_before": "case neg.refine'_2.intro.intro.inl.inr.inr\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : m 0 < 0\nhm1 : 0 < m 1\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)", "state_after": "case neg.refine'_2.intro.intro.inl.inr.inr\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : m 0 < 0\nhm1 : 0 < m 1\n\u22a2 -m 0 \u2022 x = m 1 \u2022 y"}, {"tactic": "rw [neg_smul, hm, neg_neg]", "state_before": "case neg.refine'_2.intro.intro.inl.inr.inr\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : m 0 < 0\nhm1 : 0 < m 1\n\u22a2 -m 0 \u2022 x = m 1 \u2022 y", "state_after": "no goals"}, {"tactic": "exfalso", "state_before": "case neg.refine'_2.intro.intro.inr.inl.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : m 0 = 0\nhm1 : m 1 < 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)", "state_after": "case neg.refine'_2.intro.intro.inr.inl.inl.h\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : m 0 = 0\nhm1 : m 1 < 0\n\u22a2 False"}, {"tactic": "simp [hm0, hy, hm1.ne] at hm", "state_before": "case neg.refine'_2.intro.intro.inr.inl.inl.h\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : m 0 = 0\nhm1 : m 1 < 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rw [Fin.exists_fin_two] at hmne", "state_before": "case neg.refine'_2.intro.intro.inr.inl.inr.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : m 0 = 0\nhm1 : m 1 = 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)", "state_after": "case neg.refine'_2.intro.intro.inr.inl.inr.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : m 0 \u2260 0 \u2228 m 1 \u2260 0\nhm0 : m 0 = 0\nhm1 : m 1 = 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)"}, {"tactic": "exact False.elim (not_and_or.2 hmne \u27e8hm0, hm1\u27e9)", "state_before": "case neg.refine'_2.intro.intro.inr.inl.inr.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : m 0 \u2260 0 \u2228 m 1 \u2260 0\nhm0 : m 0 = 0\nhm1 : m 1 = 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)", "state_after": "no goals"}, {"tactic": "exfalso", "state_before": "case neg.refine'_2.intro.intro.inr.inl.inr.inr\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : m 0 = 0\nhm1 : 0 < m 1\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)", "state_after": "case neg.refine'_2.intro.intro.inr.inl.inr.inr.h\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : m 0 = 0\nhm1 : 0 < m 1\n\u22a2 False"}, {"tactic": "simp [hm0, hy, hm1.ne.symm] at hm", "state_before": "case neg.refine'_2.intro.intro.inr.inl.inr.inr.h\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : m 0 = 0\nhm1 : 0 < m 1\n\u22a2 False", "state_after": "no goals"}, {"tactic": "refine' Or.inl (Or.inr (Or.inr \u27e8m 0, -m 1, hm0, Left.neg_pos_iff.2 hm1, _\u27e9))", "state_before": "case neg.refine'_2.intro.intro.inr.inr.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : 0 < m 0\nhm1 : m 1 < 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)", "state_after": "case neg.refine'_2.intro.intro.inr.inr.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : 0 < m 0\nhm1 : m 1 < 0\n\u22a2 m 0 \u2022 x = -m 1 \u2022 y"}, {"tactic": "rwa [neg_smul]", "state_before": "case neg.refine'_2.intro.intro.inr.inr.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : 0 < m 0\nhm1 : m 1 < 0\n\u22a2 m 0 \u2022 x = -m 1 \u2022 y", "state_after": "no goals"}, {"tactic": "exfalso", "state_before": "case neg.refine'_2.intro.intro.inr.inr.inr.inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : 0 < m 0\nhm1 : m 1 = 0\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)", "state_after": "case neg.refine'_2.intro.intro.inr.inr.inr.inl.h\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : 0 < m 0\nhm1 : m 1 = 0\n\u22a2 False"}, {"tactic": "simp [hm1, hx, hm0.ne.symm] at hm", "state_before": "case neg.refine'_2.intro.intro.inr.inr.inr.inl.h\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : 0 < m 0\nhm1 : m 1 = 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "refine' Or.inr (Or.inr (Or.inr \u27e8m 0, m 1, hm0, hm1, _\u27e9))", "state_before": "case neg.refine'_2.intro.intro.inr.inr.inr.inr\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : 0 < m 0\nhm1 : 0 < m 1\n\u22a2 SameRay R x y \u2228 SameRay R x (-y)", "state_after": "case neg.refine'_2.intro.intro.inr.inr.inr.inr\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : 0 < m 0\nhm1 : 0 < m 1\n\u22a2 m 0 \u2022 x = m 1 \u2022 -y"}, {"tactic": "rwa [smul_neg]", "state_before": "case neg.refine'_2.intro.intro.inr.inr.inr.inr\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedCommRing R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nx y : M\nhx : \u00acx = 0\nhy : \u00acy = 0\nm : Fin (Nat.succ (Nat.succ 0)) \u2192 R\nhm : m 0 \u2022 x = -(m 1 \u2022 y)\nhmne : \u2203 i, m i \u2260 0\nhm0 : 0 < m 0\nhm1 : 0 < m 1\n\u22a2 m 0 \u2022 x = m 1 \u2022 -y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Basic.lean", "full_name": "Set.Ioc_diff_right", "start": [790, 1], "end": [791, 55], "traced_tactics": [{"tactic": "simp [and_assoc, \u2190 lt_iff_le_and_ne]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.39340\ninst\u271d : PartialOrder \u03b1\na b c x : \u03b1\n\u22a2 x \u2208 Ioc a b \\ {b} \u2194 x \u2208 Ioo a b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Coset.lean", "full_name": "QuotientGroup.eq'", "start": [524, 1], "end": [525, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Sites/SheafOfTypes.lean", "full_name": "CategoryTheory.Presieve.isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor", "start": [580, 1], "end": [593, 58], "traced_tactics": [{"tactic": "rw [IsSeparatedFor, \u2190 forall_and]", "state_before": "C : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\n\u22a2 (IsSeparatedFor P R \u2227\n      \u2200 (x : FamilyOfElements P R), FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t) \u2194\n    IsSheafFor P R", "state_after": "C : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\n\u22a2 (\u2200 (x : FamilyOfElements P R),\n      (\u2200 (t\u2081 t\u2082 : P.obj X.op), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n        (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t)) \u2194\n    IsSheafFor P R"}, {"tactic": "apply forall_congr'", "state_before": "C : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\n\u22a2 (\u2200 (x : FamilyOfElements P R),\n      (\u2200 (t\u2081 t\u2082 : P.obj X.op), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n        (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t)) \u2194\n    IsSheafFor P R", "state_after": "case h\nC : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\n\u22a2 \u2200 (a : FamilyOfElements P R),\n    (\u2200 (t\u2081 t\u2082 : P.obj X.op), FamilyOfElements.IsAmalgamation a t\u2081 \u2192 FamilyOfElements.IsAmalgamation a t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n        (FamilyOfElements.Compatible a \u2192 \u2203 t, FamilyOfElements.IsAmalgamation a t) \u2194\n      FamilyOfElements.Compatible a \u2192 \u2203! t, FamilyOfElements.IsAmalgamation a t"}, {"tactic": "intro x", "state_before": "case h\nC : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\n\u22a2 \u2200 (a : FamilyOfElements P R),\n    (\u2200 (t\u2081 t\u2082 : P.obj X.op), FamilyOfElements.IsAmalgamation a t\u2081 \u2192 FamilyOfElements.IsAmalgamation a t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n        (FamilyOfElements.Compatible a \u2192 \u2203 t, FamilyOfElements.IsAmalgamation a t) \u2194\n      FamilyOfElements.Compatible a \u2192 \u2203! t, FamilyOfElements.IsAmalgamation a t", "state_after": "case h\nC : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\n\u22a2 (\u2200 (t\u2081 t\u2082 : P.obj X.op), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n      (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t) \u2194\n    FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t"}, {"tactic": "constructor", "state_before": "case h\nC : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\n\u22a2 (\u2200 (t\u2081 t\u2082 : P.obj X.op), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n      (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t) \u2194\n    FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t", "state_after": "case h.mp\nC : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\n\u22a2 (\u2200 (t\u2081 t\u2082 : P.obj X.op), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n      (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t) \u2192\n    FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\n\ncase h.mpr\nC : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\n\u22a2 (FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t) \u2192\n    (\u2200 (t\u2081 t\u2082 : P.obj X.op), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n      (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t)"}, {"tactic": "intro z hx", "state_before": "case h.mp\nC : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\n\u22a2 (\u2200 (t\u2081 t\u2082 : P.obj X.op), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n      (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t) \u2192\n    FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t", "state_after": "case h.mp\nC : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nz :\n  (\u2200 (t\u2081 t\u2082 : P.obj X.op), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n    (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t)\nhx : FamilyOfElements.Compatible x\n\u22a2 \u2203! t, FamilyOfElements.IsAmalgamation x t"}, {"tactic": "exact exists_unique_of_exists_of_unique (z.2 hx) z.1", "state_before": "case h.mp\nC : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nz :\n  (\u2200 (t\u2081 t\u2082 : P.obj X.op), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n    (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t)\nhx : FamilyOfElements.Compatible x\n\u22a2 \u2203! t, FamilyOfElements.IsAmalgamation x t", "state_after": "no goals"}, {"tactic": "intro h", "state_before": "case h.mpr\nC : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\n\u22a2 (FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t) \u2192\n    (\u2200 (t\u2081 t\u2082 : P.obj X.op), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n      (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t)", "state_after": "case h.mpr\nC : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nh : FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\n\u22a2 (\u2200 (t\u2081 t\u2082 : P.obj X.op), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n    (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t)"}, {"tactic": "refine' \u27e8_, ExistsUnique.exists \u2218 h\u27e9", "state_before": "case h.mpr\nC : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nh : FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\n\u22a2 (\u2200 (t\u2081 t\u2082 : P.obj X.op), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082) \u2227\n    (FamilyOfElements.Compatible x \u2192 \u2203 t, FamilyOfElements.IsAmalgamation x t)", "state_after": "case h.mpr\nC : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nh : FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\n\u22a2 \u2200 (t\u2081 t\u2082 : P.obj X.op), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082"}, {"tactic": "intro t\u2081 t\u2082 ht\u2081 ht\u2082", "state_before": "case h.mpr\nC : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nh : FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\n\u22a2 \u2200 (t\u2081 t\u2082 : P.obj X.op), FamilyOfElements.IsAmalgamation x t\u2081 \u2192 FamilyOfElements.IsAmalgamation x t\u2082 \u2192 t\u2081 = t\u2082", "state_after": "case h.mpr\nC : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nh : FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\nt\u2081 t\u2082 : P.obj X.op\nht\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nht\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\n\u22a2 t\u2081 = t\u2082"}, {"tactic": "apply (h _).unique ht\u2081 ht\u2082", "state_before": "case h.mpr\nC : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nh : FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\nt\u2081 t\u2082 : P.obj X.op\nht\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nht\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\n\u22a2 t\u2081 = t\u2082", "state_after": "C : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nh : FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\nt\u2081 t\u2082 : P.obj X.op\nht\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nht\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\n\u22a2 FamilyOfElements.Compatible x"}, {"tactic": "exact is_compatible_of_exists_amalgamation x \u27e8_, ht\u2082\u27e9", "state_before": "C : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nh : FamilyOfElements.Compatible x \u2192 \u2203! t, FamilyOfElements.IsAmalgamation x t\nt\u2081 t\u2082 : P.obj X.op\nht\u2081 : FamilyOfElements.IsAmalgamation x t\u2081\nht\u2082 : FamilyOfElements.IsAmalgamation x t\u2082\n\u22a2 FamilyOfElements.Compatible x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/DFA.lean", "full_name": "DFA.eval_singleton", "start": [83, 1], "end": [84, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/InfiniteSum/Order.lean", "full_name": "tsum_nonneg", "start": [136, 1], "end": [139, 41], "traced_tactics": [{"tactic": "by_cases hg : Summable g", "state_before": "\u03b9 : Type u_2\n\u03ba : Type ?u.31235\n\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : \u2200 (i : \u03b9), 0 \u2264 g i\n\u22a2 0 \u2264 \u2211' (i : \u03b9), g i", "state_after": "case pos\n\u03b9 : Type u_2\n\u03ba : Type ?u.31235\n\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : \u2200 (i : \u03b9), 0 \u2264 g i\nhg : Summable g\n\u22a2 0 \u2264 \u2211' (i : \u03b9), g i\n\ncase neg\n\u03b9 : Type u_2\n\u03ba : Type ?u.31235\n\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : \u2200 (i : \u03b9), 0 \u2264 g i\nhg : \u00acSummable g\n\u22a2 0 \u2264 \u2211' (i : \u03b9), g i"}, {"tactic": "exact hg.hasSum.nonneg h", "state_before": "case pos\n\u03b9 : Type u_2\n\u03ba : Type ?u.31235\n\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : \u2200 (i : \u03b9), 0 \u2264 g i\nhg : Summable g\n\u22a2 0 \u2264 \u2211' (i : \u03b9), g i", "state_after": "no goals"}, {"tactic": "rw [tsum_eq_zero_of_not_summable hg]", "state_before": "case neg\n\u03b9 : Type u_2\n\u03ba : Type ?u.31235\n\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : \u2200 (i : \u03b9), 0 \u2264 g i\nhg : \u00acSummable g\n\u22a2 0 \u2264 \u2211' (i : \u03b9), g i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Ring.lean", "full_name": "Finset.prod_sum", "start": [96, 1], "end": [124, 63], "traced_tactics": [{"tactic": "induction' s using Finset.induction with a s ha ih", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\ns : Finset \u03b1\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\n\u22a2 \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))", "state_after": "case empty\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\n\u22a2 \u220f a in \u2205, \u2211 b in t a, f a b = \u2211 p in pi \u2205 t, \u220f x in attach \u2205, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 \u2205))\n\ncase insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\n\u22a2 \u220f a in insert a s, \u2211 b in t a, f a b =\n    \u2211 p in pi (insert a s) t, \u220f x in attach (insert a s), f (\u2191x) (p \u2191x (_ : \u2191x \u2208 insert a s))"}, {"tactic": "rw [pi_empty, sum_singleton]", "state_before": "case empty\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\n\u22a2 \u220f a in \u2205, \u2211 b in t a, f a b = \u2211 p in pi \u2205 t, \u220f x in attach \u2205, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 \u2205))", "state_after": "case empty\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\n\u22a2 \u220f a in \u2205, \u2211 b in t a, f a b = \u220f x in attach \u2205, f (\u2191x) (Pi.empty (fun a => \u03b4 a) \u2191x (_ : \u2191x \u2208 \u2205))"}, {"tactic": "rfl", "state_before": "case empty\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\n\u22a2 \u220f a in \u2205, \u2211 b in t a, f a b = \u220f x in attach \u2205, f (\u2191x) (Pi.empty (fun a => \u03b4 a) \u2191x (_ : \u2191x \u2208 \u2205))", "state_after": "no goals"}, {"tactic": "have h\u2081 : \u2200 x \u2208 t a, \u2200 y \u2208 t a, x \u2260 y \u2192\n  Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t)) := by\n  intro x _ y _ h\n  simp only [disjoint_iff_ne, mem_image]\n  rintro _ \u27e8p\u2082, _, eq\u2082\u27e9 _ \u27e8p\u2083, _, eq\u2083\u27e9 eq\n  have : Pi.cons s a x p\u2082 a (mem_insert_self _ _) = Pi.cons s a y p\u2083 a (mem_insert_self _ _) :=\n    by rw [eq\u2082, eq\u2083, eq]\n  rw [Pi.cons_same, Pi.cons_same] at this\n  exact h this", "state_before": "case insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\n\u22a2 \u220f a in insert a s, \u2211 b in t a, f a b =\n    \u2211 p in pi (insert a s) t, \u220f x in attach (insert a s), f (\u2191x) (p \u2191x (_ : \u2191x \u2208 insert a s))", "state_after": "case insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\n\u22a2 \u220f a in insert a s, \u2211 b in t a, f a b =\n    \u2211 p in pi (insert a s) t, \u220f x in attach (insert a s), f (\u2191x) (p \u2191x (_ : \u2191x \u2208 insert a s))"}, {"tactic": "rw [prod_insert ha, pi_insert ha, ih, sum_mul, sum_biUnion h\u2081]", "state_before": "case insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\n\u22a2 \u220f a in insert a s, \u2211 b in t a, f a b =\n    \u2211 p in pi (insert a s) t, \u220f x in attach (insert a s), f (\u2191x) (p \u2191x (_ : \u2191x \u2208 insert a s))", "state_after": "case insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\n\u22a2 \u2211 x in t a, f a x * \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s)) =\n    \u2211 x in t a, \u2211 i in image (Pi.cons s a x) (pi s t), \u220f x in attach (insert a s), f (\u2191x) (i \u2191x (_ : \u2191x \u2208 insert a s))"}, {"tactic": "refine' sum_congr rfl fun b _ => _", "state_before": "case insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\n\u22a2 \u2211 x in t a, f a x * \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s)) =\n    \u2211 x in t a, \u2211 i in image (Pi.cons s a x) (pi s t), \u220f x in attach (insert a s), f (\u2191x) (i \u2191x (_ : \u2191x \u2208 insert a s))", "state_after": "case insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d : b \u2208 t a\n\u22a2 f a b * \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s)) =\n    \u2211 i in image (Pi.cons s a b) (pi s t), \u220f x in attach (insert a s), f (\u2191x) (i \u2191x (_ : \u2191x \u2208 insert a s))"}, {"tactic": "have h\u2082 : \u2200 p\u2081 \u2208 pi s t, \u2200 p\u2082 \u2208 pi s t, Pi.cons s a b p\u2081 = Pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082 :=\n  fun p\u2081 _ p\u2082 _ eq => Pi.cons_injective ha eq", "state_before": "case insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d : b \u2208 t a\n\u22a2 f a b * \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s)) =\n    \u2211 i in image (Pi.cons s a b) (pi s t), \u220f x in attach (insert a s), f (\u2191x) (i \u2191x (_ : \u2191x \u2208 insert a s))", "state_after": "case insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d : b \u2208 t a\nh\u2082 :\n  \u2200 (p\u2081 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a),\n    p\u2081 \u2208 pi s t \u2192 \u2200 (p\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a), p\u2082 \u2208 pi s t \u2192 Pi.cons s a b p\u2081 = Pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082\n\u22a2 f a b * \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s)) =\n    \u2211 i in image (Pi.cons s a b) (pi s t), \u220f x in attach (insert a s), f (\u2191x) (i \u2191x (_ : \u2191x \u2208 insert a s))"}, {"tactic": "rw [sum_image h\u2082, mul_sum]", "state_before": "case insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d : b \u2208 t a\nh\u2082 :\n  \u2200 (p\u2081 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a),\n    p\u2081 \u2208 pi s t \u2192 \u2200 (p\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a), p\u2082 \u2208 pi s t \u2192 Pi.cons s a b p\u2081 = Pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082\n\u22a2 f a b * \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s)) =\n    \u2211 i in image (Pi.cons s a b) (pi s t), \u220f x in attach (insert a s), f (\u2191x) (i \u2191x (_ : \u2191x \u2208 insert a s))", "state_after": "case insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d : b \u2208 t a\nh\u2082 :\n  \u2200 (p\u2081 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a),\n    p\u2081 \u2208 pi s t \u2192 \u2200 (p\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a), p\u2082 \u2208 pi s t \u2192 Pi.cons s a b p\u2081 = Pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082\n\u22a2 \u2211 x in pi s t, f a b * \u220f x_1 in attach s, f (\u2191x_1) (x \u2191x_1 (_ : \u2191x_1 \u2208 s)) =\n    \u2211 x in pi s t, \u220f x_1 in attach (insert a s), f (\u2191x_1) (Pi.cons s a b x \u2191x_1 (_ : \u2191x_1 \u2208 insert a s))"}, {"tactic": "refine' sum_congr rfl fun g _ => _", "state_before": "case insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d : b \u2208 t a\nh\u2082 :\n  \u2200 (p\u2081 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a),\n    p\u2081 \u2208 pi s t \u2192 \u2200 (p\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a), p\u2082 \u2208 pi s t \u2192 Pi.cons s a b p\u2081 = Pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082\n\u22a2 \u2211 x in pi s t, f a b * \u220f x_1 in attach s, f (\u2191x_1) (x \u2191x_1 (_ : \u2191x_1 \u2208 s)) =\n    \u2211 x in pi s t, \u220f x_1 in attach (insert a s), f (\u2191x_1) (Pi.cons s a b x \u2191x_1 (_ : \u2191x_1 \u2208 insert a s))", "state_after": "case insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d\u00b9 : b \u2208 t a\nh\u2082 :\n  \u2200 (p\u2081 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a),\n    p\u2081 \u2208 pi s t \u2192 \u2200 (p\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a), p\u2082 \u2208 pi s t \u2192 Pi.cons s a b p\u2081 = Pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nx\u271d : g \u2208 pi s t\n\u22a2 f a b * \u220f x in attach s, f (\u2191x) (g \u2191x (_ : \u2191x \u2208 s)) =\n    \u220f x in attach (insert a s), f (\u2191x) (Pi.cons s a b g \u2191x (_ : \u2191x \u2208 insert a s))"}, {"tactic": "rw [attach_insert, prod_insert, prod_image]", "state_before": "case insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d\u00b9 : b \u2208 t a\nh\u2082 :\n  \u2200 (p\u2081 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a),\n    p\u2081 \u2208 pi s t \u2192 \u2200 (p\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a), p\u2082 \u2208 pi s t \u2192 Pi.cons s a b p\u2081 = Pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nx\u271d : g \u2208 pi s t\n\u22a2 f a b * \u220f x in attach s, f (\u2191x) (g \u2191x (_ : \u2191x \u2208 s)) =\n    \u220f x in attach (insert a s), f (\u2191x) (Pi.cons s a b g \u2191x (_ : \u2191x \u2208 insert a s))", "state_after": "case insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d\u00b9 : b \u2208 t a\nh\u2082 :\n  \u2200 (p\u2081 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a),\n    p\u2081 \u2208 pi s t \u2192 \u2200 (p\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a), p\u2082 \u2208 pi s t \u2192 Pi.cons s a b p\u2081 = Pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nx\u271d : g \u2208 pi s t\n\u22a2 f a b * \u220f x in attach s, f (\u2191x) (g \u2191x (_ : \u2191x \u2208 s)) =\n    f (\u2191{ val := a, property := (_ : a \u2208 insert a s) })\n        (Pi.cons s a b g \u2191{ val := a, property := (_ : a \u2208 insert a s) }\n          (_ : \u2191{ val := a, property := (_ : a \u2208 insert a s) } \u2208 insert a s)) *\n      \u220f x in attach s,\n        f (\u2191{ val := \u2191x, property := (_ : \u2191x \u2208 insert a s) })\n          (Pi.cons s a b g \u2191{ val := \u2191x, property := (_ : \u2191x \u2208 insert a s) }\n            (_ : \u2191{ val := \u2191x, property := (_ : \u2191x \u2208 insert a s) } \u2208 insert a s))\n\ncase insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d\u00b9 : b \u2208 t a\nh\u2082 :\n  \u2200 (p\u2081 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a),\n    p\u2081 \u2208 pi s t \u2192 \u2200 (p\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a), p\u2082 \u2208 pi s t \u2192 Pi.cons s a b p\u2081 = Pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nx\u271d : g \u2208 pi s t\n\u22a2 \u2200 (x : { x // x \u2208 s }),\n    x \u2208 attach s \u2192\n      \u2200 (y : { x // x \u2208 s }),\n        y \u2208 attach s \u2192\n          { val := \u2191x, property := (_ : \u2191x \u2208 insert a s) } = { val := \u2191y, property := (_ : \u2191y \u2208 insert a s) } \u2192 x = y\n\ncase insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d\u00b9 : b \u2208 t a\nh\u2082 :\n  \u2200 (p\u2081 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a),\n    p\u2081 \u2208 pi s t \u2192 \u2200 (p\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a), p\u2082 \u2208 pi s t \u2192 Pi.cons s a b p\u2081 = Pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nx\u271d : g \u2208 pi s t\n\u22a2 \u00ac{ val := a, property := (_ : a \u2208 insert a s) } \u2208\n      image (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 insert a s) }) (attach s)"}, {"tactic": "intro x _ y _ h", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\n\u22a2 \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d\u00b2 : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nx : \u03b4 a\na\u271d\u00b9 : x \u2208 t a\ny : \u03b4 a\na\u271d : y \u2208 t a\nh : x \u2260 y\n\u22a2 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))"}, {"tactic": "simp only [disjoint_iff_ne, mem_image]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d\u00b2 : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nx : \u03b4 a\na\u271d\u00b9 : x \u2208 t a\ny : \u03b4 a\na\u271d : y \u2208 t a\nh : x \u2260 y\n\u22a2 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d\u00b2 : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nx : \u03b4 a\na\u271d\u00b9 : x \u2208 t a\ny : \u03b4 a\na\u271d : y \u2208 t a\nh : x \u2260 y\n\u22a2 \u2200 (a_1 : (a' : \u03b1) \u2192 a' \u2208 insert a s \u2192 \u03b4 a'),\n    (\u2203 a_2, a_2 \u2208 pi s t \u2227 Pi.cons s a x a_2 = a_1) \u2192\n      \u2200 (b : (a' : \u03b1) \u2192 a' \u2208 insert a s \u2192 \u03b4 a'), (\u2203 a_3, a_3 \u2208 pi s t \u2227 Pi.cons s a y a_3 = b) \u2192 a_1 \u2260 b"}, {"tactic": "rintro _ \u27e8p\u2082, _, eq\u2082\u27e9 _ \u27e8p\u2083, _, eq\u2083\u27e9 eq", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d\u00b2 : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nx : \u03b4 a\na\u271d\u00b9 : x \u2208 t a\ny : \u03b4 a\na\u271d : y \u2208 t a\nh : x \u2260 y\n\u22a2 \u2200 (a_1 : (a' : \u03b1) \u2192 a' \u2208 insert a s \u2192 \u03b4 a'),\n    (\u2203 a_2, a_2 \u2208 pi s t \u2227 Pi.cons s a x a_2 = a_1) \u2192\n      \u2200 (b : (a' : \u03b1) \u2192 a' \u2208 insert a s \u2192 \u03b4 a'), (\u2203 a_3, a_3 \u2208 pi s t \u2227 Pi.cons s a y a_3 = b) \u2192 a_1 \u2260 b", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d\u00b3 : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nx : \u03b4 a\na\u271d\u00b2 : x \u2208 t a\ny : \u03b4 a\na\u271d\u00b9 : y \u2208 t a\nh : x \u2260 y\na\u271d : (a' : \u03b1) \u2192 a' \u2208 insert a s \u2192 \u03b4 a'\np\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nleft\u271d\u00b9 : p\u2082 \u2208 pi s t\neq\u2082 : Pi.cons s a x p\u2082 = a\u271d\nb\u271d : (a' : \u03b1) \u2192 a' \u2208 insert a s \u2192 \u03b4 a'\np\u2083 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nleft\u271d : p\u2083 \u2208 pi s t\neq\u2083 : Pi.cons s a y p\u2083 = b\u271d\neq : a\u271d = b\u271d\n\u22a2 False"}, {"tactic": "have : Pi.cons s a x p\u2082 a (mem_insert_self _ _) = Pi.cons s a y p\u2083 a (mem_insert_self _ _) :=\n  by rw [eq\u2082, eq\u2083, eq]", "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d\u00b3 : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nx : \u03b4 a\na\u271d\u00b2 : x \u2208 t a\ny : \u03b4 a\na\u271d\u00b9 : y \u2208 t a\nh : x \u2260 y\na\u271d : (a' : \u03b1) \u2192 a' \u2208 insert a s \u2192 \u03b4 a'\np\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nleft\u271d\u00b9 : p\u2082 \u2208 pi s t\neq\u2082 : Pi.cons s a x p\u2082 = a\u271d\nb\u271d : (a' : \u03b1) \u2192 a' \u2208 insert a s \u2192 \u03b4 a'\np\u2083 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nleft\u271d : p\u2083 \u2208 pi s t\neq\u2083 : Pi.cons s a y p\u2083 = b\u271d\neq : a\u271d = b\u271d\n\u22a2 False", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d\u00b3 : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nx : \u03b4 a\na\u271d\u00b2 : x \u2208 t a\ny : \u03b4 a\na\u271d\u00b9 : y \u2208 t a\nh : x \u2260 y\na\u271d : (a' : \u03b1) \u2192 a' \u2208 insert a s \u2192 \u03b4 a'\np\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nleft\u271d\u00b9 : p\u2082 \u2208 pi s t\neq\u2082 : Pi.cons s a x p\u2082 = a\u271d\nb\u271d : (a' : \u03b1) \u2192 a' \u2208 insert a s \u2192 \u03b4 a'\np\u2083 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nleft\u271d : p\u2083 \u2208 pi s t\neq\u2083 : Pi.cons s a y p\u2083 = b\u271d\neq : a\u271d = b\u271d\nthis : Pi.cons s a x p\u2082 a (_ : a \u2208 insert a s) = Pi.cons s a y p\u2083 a (_ : a \u2208 insert a s)\n\u22a2 False"}, {"tactic": "rw [Pi.cons_same, Pi.cons_same] at this", "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d\u00b3 : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nx : \u03b4 a\na\u271d\u00b2 : x \u2208 t a\ny : \u03b4 a\na\u271d\u00b9 : y \u2208 t a\nh : x \u2260 y\na\u271d : (a' : \u03b1) \u2192 a' \u2208 insert a s \u2192 \u03b4 a'\np\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nleft\u271d\u00b9 : p\u2082 \u2208 pi s t\neq\u2082 : Pi.cons s a x p\u2082 = a\u271d\nb\u271d : (a' : \u03b1) \u2192 a' \u2208 insert a s \u2192 \u03b4 a'\np\u2083 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nleft\u271d : p\u2083 \u2208 pi s t\neq\u2083 : Pi.cons s a y p\u2083 = b\u271d\neq : a\u271d = b\u271d\nthis : Pi.cons s a x p\u2082 a (_ : a \u2208 insert a s) = Pi.cons s a y p\u2083 a (_ : a \u2208 insert a s)\n\u22a2 False", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d\u00b3 : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nx : \u03b4 a\na\u271d\u00b2 : x \u2208 t a\ny : \u03b4 a\na\u271d\u00b9 : y \u2208 t a\nh : x \u2260 y\na\u271d : (a' : \u03b1) \u2192 a' \u2208 insert a s \u2192 \u03b4 a'\np\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nleft\u271d\u00b9 : p\u2082 \u2208 pi s t\neq\u2082 : Pi.cons s a x p\u2082 = a\u271d\nb\u271d : (a' : \u03b1) \u2192 a' \u2208 insert a s \u2192 \u03b4 a'\np\u2083 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nleft\u271d : p\u2083 \u2208 pi s t\neq\u2083 : Pi.cons s a y p\u2083 = b\u271d\neq : a\u271d = b\u271d\nthis : x = y\n\u22a2 False"}, {"tactic": "exact h this", "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d\u00b3 : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nx : \u03b4 a\na\u271d\u00b2 : x \u2208 t a\ny : \u03b4 a\na\u271d\u00b9 : y \u2208 t a\nh : x \u2260 y\na\u271d : (a' : \u03b1) \u2192 a' \u2208 insert a s \u2192 \u03b4 a'\np\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nleft\u271d\u00b9 : p\u2082 \u2208 pi s t\neq\u2082 : Pi.cons s a x p\u2082 = a\u271d\nb\u271d : (a' : \u03b1) \u2192 a' \u2208 insert a s \u2192 \u03b4 a'\np\u2083 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nleft\u271d : p\u2083 \u2208 pi s t\neq\u2083 : Pi.cons s a y p\u2083 = b\u271d\neq : a\u271d = b\u271d\nthis : x = y\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rw [eq\u2082, eq\u2083, eq]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d\u00b3 : \u03b1\nb : \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nx : \u03b4 a\na\u271d\u00b2 : x \u2208 t a\ny : \u03b4 a\na\u271d\u00b9 : y \u2208 t a\nh : x \u2260 y\na\u271d : (a' : \u03b1) \u2192 a' \u2208 insert a s \u2192 \u03b4 a'\np\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nleft\u271d\u00b9 : p\u2082 \u2208 pi s t\neq\u2082 : Pi.cons s a x p\u2082 = a\u271d\nb\u271d : (a' : \u03b1) \u2192 a' \u2208 insert a s \u2192 \u03b4 a'\np\u2083 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nleft\u271d : p\u2083 \u2208 pi s t\neq\u2083 : Pi.cons s a y p\u2083 = b\u271d\neq : a\u271d = b\u271d\n\u22a2 Pi.cons s a x p\u2082 a (_ : a \u2208 insert a s) = Pi.cons s a y p\u2083 a (_ : a \u2208 insert a s)", "state_after": "no goals"}, {"tactic": "simp only [Pi.cons_same]", "state_before": "case insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d\u00b9 : b \u2208 t a\nh\u2082 :\n  \u2200 (p\u2081 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a),\n    p\u2081 \u2208 pi s t \u2192 \u2200 (p\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a), p\u2082 \u2208 pi s t \u2192 Pi.cons s a b p\u2081 = Pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nx\u271d : g \u2208 pi s t\n\u22a2 f a b * \u220f x in attach s, f (\u2191x) (g \u2191x (_ : \u2191x \u2208 s)) =\n    f (\u2191{ val := a, property := (_ : a \u2208 insert a s) })\n        (Pi.cons s a b g \u2191{ val := a, property := (_ : a \u2208 insert a s) }\n          (_ : \u2191{ val := a, property := (_ : a \u2208 insert a s) } \u2208 insert a s)) *\n      \u220f x in attach s,\n        f (\u2191{ val := \u2191x, property := (_ : \u2191x \u2208 insert a s) })\n          (Pi.cons s a b g \u2191{ val := \u2191x, property := (_ : \u2191x \u2208 insert a s) }\n            (_ : \u2191{ val := \u2191x, property := (_ : \u2191x \u2208 insert a s) } \u2208 insert a s))", "state_after": "case insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d\u00b9 : b \u2208 t a\nh\u2082 :\n  \u2200 (p\u2081 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a),\n    p\u2081 \u2208 pi s t \u2192 \u2200 (p\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a), p\u2082 \u2208 pi s t \u2192 Pi.cons s a b p\u2081 = Pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nx\u271d : g \u2208 pi s t\n\u22a2 f a b * \u220f x in attach s, f (\u2191x) (g \u2191x (_ : \u2191x \u2208 s)) =\n    f a b *\n      \u220f x in attach s,\n        f (\u2191x)\n          (Pi.cons s a b g \u2191{ val := \u2191x, property := (_ : \u2191x \u2208 insert a s) }\n            (_ : \u2191{ val := \u2191x, property := (_ : \u2191x \u2208 insert a s) } \u2208 insert a s))"}, {"tactic": "congr with \u27e8v, hv\u27e9", "state_before": "case insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d\u00b9 : b \u2208 t a\nh\u2082 :\n  \u2200 (p\u2081 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a),\n    p\u2081 \u2208 pi s t \u2192 \u2200 (p\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a), p\u2082 \u2208 pi s t \u2192 Pi.cons s a b p\u2081 = Pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nx\u271d : g \u2208 pi s t\n\u22a2 f a b * \u220f x in attach s, f (\u2191x) (g \u2191x (_ : \u2191x \u2208 s)) =\n    f a b *\n      \u220f x in attach s,\n        f (\u2191x)\n          (Pi.cons s a b g \u2191{ val := \u2191x, property := (_ : \u2191x \u2208 insert a s) }\n            (_ : \u2191{ val := \u2191x, property := (_ : \u2191x \u2208 insert a s) } \u2208 insert a s))", "state_after": "case insert.e_a.e_f.h.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d\u00b9 : b \u2208 t a\nh\u2082 :\n  \u2200 (p\u2081 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a),\n    p\u2081 \u2208 pi s t \u2192 \u2200 (p\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a), p\u2082 \u2208 pi s t \u2192 Pi.cons s a b p\u2081 = Pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nx\u271d : g \u2208 pi s t\nv : \u03b1\nhv : v \u2208 s\n\u22a2 f (\u2191{ val := v, property := hv }) (g \u2191{ val := v, property := hv } (_ : \u2191{ val := v, property := hv } \u2208 s)) =\n    f (\u2191{ val := v, property := hv })\n      (Pi.cons s a b g\n        \u2191{ val := \u2191{ val := v, property := hv }, property := (_ : \u2191{ val := v, property := hv } \u2208 insert a s) }\n        (_ :\n          \u2191{ val := \u2191{ val := v, property := hv }, property := (_ : \u2191{ val := v, property := hv } \u2208 insert a s) } \u2208\n            insert a s))"}, {"tactic": "congr", "state_before": "case insert.e_a.e_f.h.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d\u00b9 : b \u2208 t a\nh\u2082 :\n  \u2200 (p\u2081 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a),\n    p\u2081 \u2208 pi s t \u2192 \u2200 (p\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a), p\u2082 \u2208 pi s t \u2192 Pi.cons s a b p\u2081 = Pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nx\u271d : g \u2208 pi s t\nv : \u03b1\nhv : v \u2208 s\n\u22a2 f (\u2191{ val := v, property := hv }) (g \u2191{ val := v, property := hv } (_ : \u2191{ val := v, property := hv } \u2208 s)) =\n    f (\u2191{ val := v, property := hv })\n      (Pi.cons s a b g\n        \u2191{ val := \u2191{ val := v, property := hv }, property := (_ : \u2191{ val := v, property := hv } \u2208 insert a s) }\n        (_ :\n          \u2191{ val := \u2191{ val := v, property := hv }, property := (_ : \u2191{ val := v, property := hv } \u2208 insert a s) } \u2208\n            insert a s))", "state_after": "case insert.e_a.e_f.h.mk.e_a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d\u00b9 : b \u2208 t a\nh\u2082 :\n  \u2200 (p\u2081 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a),\n    p\u2081 \u2208 pi s t \u2192 \u2200 (p\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a), p\u2082 \u2208 pi s t \u2192 Pi.cons s a b p\u2081 = Pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nx\u271d : g \u2208 pi s t\nv : \u03b1\nhv : v \u2208 s\n\u22a2 g \u2191{ val := v, property := hv } (_ : \u2191{ val := v, property := hv } \u2208 s) =\n    Pi.cons s a b g\n      \u2191{ val := \u2191{ val := v, property := hv }, property := (_ : \u2191{ val := v, property := hv } \u2208 insert a s) }\n      (_ :\n        \u2191{ val := \u2191{ val := v, property := hv }, property := (_ : \u2191{ val := v, property := hv } \u2208 insert a s) } \u2208\n          insert a s)"}, {"tactic": "exact (Pi.cons_ne (by rintro rfl; exact ha hv)).symm", "state_before": "case insert.e_a.e_f.h.mk.e_a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d\u00b9 : b \u2208 t a\nh\u2082 :\n  \u2200 (p\u2081 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a),\n    p\u2081 \u2208 pi s t \u2192 \u2200 (p\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a), p\u2082 \u2208 pi s t \u2192 Pi.cons s a b p\u2081 = Pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nx\u271d : g \u2208 pi s t\nv : \u03b1\nhv : v \u2208 s\n\u22a2 g \u2191{ val := v, property := hv } (_ : \u2191{ val := v, property := hv } \u2208 s) =\n    Pi.cons s a b g\n      \u2191{ val := \u2191{ val := v, property := hv }, property := (_ : \u2191{ val := v, property := hv } \u2208 insert a s) }\n      (_ :\n        \u2191{ val := \u2191{ val := v, property := hv }, property := (_ : \u2191{ val := v, property := hv } \u2208 insert a s) } \u2208\n          insert a s)", "state_after": "no goals"}, {"tactic": "rintro rfl", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d\u00b9 : b \u2208 t a\nh\u2082 :\n  \u2200 (p\u2081 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a),\n    p\u2081 \u2208 pi s t \u2192 \u2200 (p\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a), p\u2082 \u2208 pi s t \u2192 Pi.cons s a b p\u2081 = Pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nx\u271d : g \u2208 pi s t\nv : \u03b1\nhv : v \u2208 s\n\u22a2 a \u2260 \u2191{ val := v, property := hv }", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d\u00b9 : b \u2208 t a\nh\u2082 :\n  \u2200 (p\u2081 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a),\n    p\u2081 \u2208 pi s t \u2192 \u2200 (p\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a), p\u2082 \u2208 pi s t \u2192 Pi.cons s a b p\u2081 = Pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nx\u271d : g \u2208 pi s t\nhv : a \u2208 s\n\u22a2 False"}, {"tactic": "exact ha hv", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d\u00b9 : b \u2208 t a\nh\u2082 :\n  \u2200 (p\u2081 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a),\n    p\u2081 \u2208 pi s t \u2192 \u2200 (p\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a), p\u2082 \u2208 pi s t \u2192 Pi.cons s a b p\u2081 = Pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nx\u271d : g \u2208 pi s t\nhv : a \u2208 s\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exact fun _ _ _ _ => Subtype.eq \u2218 Subtype.mk.inj", "state_before": "case insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d\u00b9 : b \u2208 t a\nh\u2082 :\n  \u2200 (p\u2081 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a),\n    p\u2081 \u2208 pi s t \u2192 \u2200 (p\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a), p\u2082 \u2208 pi s t \u2192 Pi.cons s a b p\u2081 = Pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nx\u271d : g \u2208 pi s t\n\u22a2 \u2200 (x : { x // x \u2208 s }),\n    x \u2208 attach s \u2192\n      \u2200 (y : { x // x \u2208 s }),\n        y \u2208 attach s \u2192\n          { val := \u2191x, property := (_ : \u2191x \u2208 insert a s) } = { val := \u2191y, property := (_ : \u2191y \u2208 insert a s) } \u2192 x = y", "state_after": "no goals"}, {"tactic": "simpa only [mem_image, mem_attach, Subtype.mk.injEq, true_and,\n  Subtype.exists, exists_prop, exists_eq_right] using ha", "state_before": "case insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nb\u271d : \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b2 : CommSemiring \u03b2\n\u03b4 : \u03b1 \u2192 Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (\u03b4 a)\nt : (a : \u03b1) \u2192 Finset (\u03b4 a)\nf : (a : \u03b1) \u2192 \u03b4 a \u2192 \u03b2\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : \u220f a in s, \u2211 b in t a, f a b = \u2211 p in pi s t, \u220f x in attach s, f (\u2191x) (p \u2191x (_ : \u2191x \u2208 s))\nh\u2081 :\n  \u2200 (x : \u03b4 a),\n    x \u2208 t a \u2192 \u2200 (y : \u03b4 a), y \u2208 t a \u2192 x \u2260 y \u2192 Disjoint (image (Pi.cons s a x) (pi s t)) (image (Pi.cons s a y) (pi s t))\nb : \u03b4 a\nx\u271d\u00b9 : b \u2208 t a\nh\u2082 :\n  \u2200 (p\u2081 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a),\n    p\u2081 \u2208 pi s t \u2192 \u2200 (p\u2082 : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a), p\u2082 \u2208 pi s t \u2192 Pi.cons s a b p\u2081 = Pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b4 a\nx\u271d : g \u2208 pi s t\n\u22a2 \u00ac{ val := a, property := (_ : a \u2208 insert a s) } \u2208\n      image (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 insert a s) }) (attach s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subsemigroup/Membership.lean", "full_name": "Subsemigroup.mul_mem_sup", "start": [97, 1], "end": [98, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "full_name": "Subalgebra.mem_of_finset_sum_eq_one_of_pow_smul_mem", "start": [1429, 1], "end": [1457, 25], "traced_tactics": [{"tactic": "let _i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id _", "state_before": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\nH : \u2200 (i : \u03b9), \u2203 n, s i ^ n \u2022 x \u2208 S'\n\u22a2 x \u2208 S'", "state_after": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\nH : \u2200 (i : \u03b9), \u2203 n, s i ^ n \u2022 x \u2208 S'\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\n\u22a2 x \u2208 S'"}, {"tactic": "suffices x \u2208 Subalgebra.toSubmodule (Algebra.ofId S' S).range by\n  obtain \u27e8x, rfl\u27e9 := this\n  exact x.2", "state_before": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\nH : \u2200 (i : \u03b9), \u2203 n, s i ^ n \u2022 x \u2208 S'\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\n\u22a2 x \u2208 S'", "state_after": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\nH : \u2200 (i : \u03b9), \u2203 n, s i ^ n \u2022 x \u2208 S'\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\n\u22a2 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))"}, {"tactic": "choose n hn using H", "state_before": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\nH : \u2200 (i : \u03b9), \u2203 n, s i ^ n \u2022 x \u2208 S'\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\n\u22a2 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))", "state_after": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\n\u22a2 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))"}, {"tactic": "let s' : \u03b9 \u2192 S' := fun x => \u27e8s x, hs x\u27e9", "state_before": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\n\u22a2 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))", "state_after": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\n\u22a2 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))"}, {"tactic": "let l' : \u03b9 \u2192 S' := fun x => \u27e8l x, hl x\u27e9", "state_before": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\n\u22a2 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))", "state_after": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\n\u22a2 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))"}, {"tactic": "have e' : (\u2211 i in \u03b9', l' i * s' i) = 1 := by\n  ext\n  show S'.subtype (\u2211 i in \u03b9', l' i * s' i) = 1\n  simpa only [map_sum, map_mul] using e", "state_before": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\n\u22a2 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))", "state_after": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\n\u22a2 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))"}, {"tactic": "have : Ideal.span (s' '' \u03b9') = \u22a4 := by\n  rw [Ideal.eq_top_iff_one, \u2190 e']\n  apply sum_mem\n  intros i hi\n  exact Ideal.mul_mem_left _ _ <| Ideal.subset_span <| Set.mem_image_of_mem s' hi", "state_before": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\n\u22a2 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))", "state_after": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\n\u22a2 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))"}, {"tactic": "let N := \u03b9'.sup n", "state_before": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\n\u22a2 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))", "state_after": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := Finset.sup \u03b9' n\n\u22a2 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))"}, {"tactic": "have hN := Ideal.span_pow_eq_top _ this N", "state_before": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := Finset.sup \u03b9' n\n\u22a2 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))", "state_after": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := Finset.sup \u03b9' n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\n\u22a2 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))"}, {"tactic": "apply (Algebra.ofId S' S).range.toSubmodule.mem_of_span_top_of_smul_mem _ hN", "state_before": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := Finset.sup \u03b9' n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\n\u22a2 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))", "state_after": "case H\nR : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := Finset.sup \u03b9' n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\n\u22a2 \u2200 (r : \u2191((fun x => x ^ N) '' (s' '' \u2191\u03b9'))), \u2191r \u2022 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))"}, {"tactic": "rintro \u27e8_, _, \u27e8i, hi, rfl\u27e9, rfl\u27e9", "state_before": "case H\nR : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := Finset.sup \u03b9' n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\n\u22a2 \u2200 (r : \u2191((fun x => x ^ N) '' (s' '' \u2191\u03b9'))), \u2191r \u2022 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))", "state_after": "case H.mk.intro.intro.intro.intro\nR : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := Finset.sup \u03b9' n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\ni : \u03b9\nhi : i \u2208 \u2191\u03b9'\n\u22a2 \u2191{ val := (fun x => x ^ N) (s' i),\n          property := (_ : \u2203 a, a \u2208 s' '' \u2191\u03b9' \u2227 (fun x => x ^ N) a = (fun x => x ^ N) (s' i)) } \u2022\n      x \u2208\n    \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))"}, {"tactic": "change s' i ^ N \u2022 x \u2208 _", "state_before": "case H.mk.intro.intro.intro.intro\nR : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := Finset.sup \u03b9' n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\ni : \u03b9\nhi : i \u2208 \u2191\u03b9'\n\u22a2 \u2191{ val := (fun x => x ^ N) (s' i),\n          property := (_ : \u2203 a, a \u2208 s' '' \u2191\u03b9' \u2227 (fun x => x ^ N) a = (fun x => x ^ N) (s' i)) } \u2022\n      x \u2208\n    \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))", "state_after": "case H.mk.intro.intro.intro.intro\nR : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := Finset.sup \u03b9' n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\ni : \u03b9\nhi : i \u2208 \u2191\u03b9'\n\u22a2 s' i ^ N \u2022 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))"}, {"tactic": "rw [\u2190 tsub_add_cancel_of_le (show n i \u2264 N from Finset.le_sup hi), pow_add, mul_smul]", "state_before": "case H.mk.intro.intro.intro.intro\nR : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := Finset.sup \u03b9' n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\ni : \u03b9\nhi : i \u2208 \u2191\u03b9'\n\u22a2 s' i ^ N \u2022 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))", "state_after": "case H.mk.intro.intro.intro.intro\nR : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := Finset.sup \u03b9' n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\ni : \u03b9\nhi : i \u2208 \u2191\u03b9'\n\u22a2 s' i ^ (N - n i) \u2022 s' i ^ n i \u2022 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))"}, {"tactic": "refine' Submodule.smul_mem _ (\u27e8_, pow_mem (hs i) _\u27e9 : S') _", "state_before": "case H.mk.intro.intro.intro.intro\nR : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := Finset.sup \u03b9' n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\ni : \u03b9\nhi : i \u2208 \u2191\u03b9'\n\u22a2 s' i ^ (N - n i) \u2022 s' i ^ n i \u2022 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))", "state_after": "case H.mk.intro.intro.intro.intro\nR : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := Finset.sup \u03b9' n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\ni : \u03b9\nhi : i \u2208 \u2191\u03b9'\n\u22a2 s' i ^ n i \u2022 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))"}, {"tactic": "exact \u27e8\u27e8_, hn i\u27e9, rfl\u27e9", "state_before": "case H.mk.intro.intro.intro.intro\nR : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\nthis : Ideal.span (s' '' \u2191\u03b9') = \u22a4\nN : \u2115 := Finset.sup \u03b9' n\nhN : Ideal.span ((fun x => x ^ N) '' (s' '' \u2191\u03b9')) = \u22a4\ni : \u03b9\nhi : i \u2208 \u2191\u03b9'\n\u22a2 s' i ^ n i \u2022 x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))", "state_after": "no goals"}, {"tactic": "obtain \u27e8x, rfl\u27e9 := this", "state_before": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\nH : \u2200 (i : \u03b9), \u2203 n, s i ^ n \u2022 x \u2208 S'\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nthis : x \u2208 \u2191toSubmodule (AlgHom.range (ofId { x // x \u2208 S' } S))\n\u22a2 x \u2208 S'", "state_after": "case intro\nR : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nx : { x // x \u2208 S' }\nH : \u2200 (i : \u03b9), \u2203 n, s i ^ n \u2022 \u2191\u2191(ofId { x // x \u2208 S' } S) x \u2208 S'\n\u22a2 \u2191\u2191(ofId { x // x \u2208 S' } S) x \u2208 S'"}, {"tactic": "exact x.2", "state_before": "case intro\nR : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nx : { x // x \u2208 S' }\nH : \u2200 (i : \u03b9), \u2203 n, s i ^ n \u2022 \u2191\u2191(ofId { x // x \u2208 S' } S) x \u2208 S'\n\u22a2 \u2191\u2191(ofId { x // x \u2208 S' } S) x \u2208 S'", "state_after": "no goals"}, {"tactic": "ext", "state_before": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\n\u22a2 \u2211 i in \u03b9', l' i * s' i = 1", "state_after": "case a\nR : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\n\u22a2 \u2191(\u2211 i in \u03b9', l' i * s' i) = \u21911"}, {"tactic": "show S'.subtype (\u2211 i in \u03b9', l' i * s' i) = 1", "state_before": "case a\nR : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\n\u22a2 \u2191(\u2211 i in \u03b9', l' i * s' i) = \u21911", "state_after": "case a\nR : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\n\u22a2 \u2191(Subsemiring.subtype S'.toSubsemiring) (\u2211 i in \u03b9', l' i * s' i) = 1"}, {"tactic": "simpa only [map_sum, map_mul] using e", "state_before": "case a\nR : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\n\u22a2 \u2191(Subsemiring.subtype S'.toSubsemiring) (\u2211 i in \u03b9', l' i * s' i) = 1", "state_after": "no goals"}, {"tactic": "rw [Ideal.eq_top_iff_one, \u2190 e']", "state_before": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\n\u22a2 Ideal.span (s' '' \u2191\u03b9') = \u22a4", "state_after": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\n\u22a2 \u2211 i in \u03b9', l' i * s' i \u2208 Ideal.span (s' '' \u2191\u03b9')"}, {"tactic": "apply sum_mem", "state_before": "R : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\n\u22a2 \u2211 i in \u03b9', l' i * s' i \u2208 Ideal.span (s' '' \u2191\u03b9')", "state_after": "case h\nR : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\n\u22a2 \u2200 (c : \u03b9), c \u2208 \u03b9' \u2192 l' c * s' c \u2208 Ideal.span (s' '' \u2191\u03b9')"}, {"tactic": "intros i hi", "state_before": "case h\nR : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\n\u22a2 \u2200 (c : \u03b9), c \u2208 \u03b9' \u2192 l' c * s' c \u2208 Ideal.span (s' '' \u2191\u03b9')", "state_after": "case h\nR : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\ni : \u03b9\nhi : i \u2208 \u03b9'\n\u22a2 l' i * s' i \u2208 Ideal.span (s' '' \u2191\u03b9')"}, {"tactic": "exact Ideal.mul_mem_left _ _ <| Ideal.subset_span <| Set.mem_image_of_mem s' hi", "state_before": "case h\nR : Type u\nA : Type v\nB : 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\nS\u271d : Subalgebra R A\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nS' : Subalgebra R S\n\u03b9 : Type u_2\n\u03b9' : Finset \u03b9\ns l : \u03b9 \u2192 S\ne : \u2211 i in \u03b9', l i * s i = 1\nhs : \u2200 (i : \u03b9), s i \u2208 S'\nhl : \u2200 (i : \u03b9), l i \u2208 S'\nx : S\n_i : Algebra { x // x \u2208 S' } { x // x \u2208 S' } := Algebra.id { x // x \u2208 S' }\nn : \u03b9 \u2192 \u2115\nhn : \u2200 (i : \u03b9), s i ^ n i \u2022 x \u2208 S'\ns' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := s x, property := (_ : s x \u2208 S') }\nl' : \u03b9 \u2192 { x // x \u2208 S' } := fun x => { val := l x, property := (_ : l x \u2208 S') }\ne' : \u2211 i in \u03b9', l' i * s' i = 1\ni : \u03b9\nhi : i \u2208 \u03b9'\n\u22a2 l' i * s' i \u2208 Ideal.span (s' '' \u2191\u03b9')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fin/Interval.lean", "full_name": "Fin.card_fintypeIio", "start": [221, 1], "end": [222, 39], "traced_tactics": [{"tactic": "rw [Fintype.card_ofFinset, card_Iio]", "state_before": "n : \u2115\na b : Fin n\n\u22a2 Fintype.card \u2191(Set.Iio b) = \u2191b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "full_name": "LinearMap.coe_isometryOfInner", "start": [1293, 1], "end": [1294, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Polynomial/BigOperators.lean", "full_name": "Polynomial.natDegree_multiset_prod", "start": [323, 1], "end": [329, 16], "traced_tactics": [{"tactic": "nontriviality R", "state_before": "R : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u00ac0 \u2208 t\n\u22a2 natDegree (prod t) = Multiset.sum (Multiset.map natDegree t)", "state_after": "R : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u00ac0 \u2208 t\n\u271d : Nontrivial R\n\u22a2 natDegree (prod t) = Multiset.sum (Multiset.map natDegree t)"}, {"tactic": "rw [natDegree_multiset_prod']", "state_before": "R : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u00ac0 \u2208 t\n\u271d : Nontrivial R\n\u22a2 natDegree (prod t) = Multiset.sum (Multiset.map natDegree t)", "state_after": "case h\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u00ac0 \u2208 t\n\u271d : Nontrivial R\n\u22a2 prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0"}, {"tactic": "simp_rw [Ne.def, Multiset.prod_eq_zero_iff, Multiset.mem_map, leadingCoeff_eq_zero]", "state_before": "case h\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u00ac0 \u2208 t\n\u271d : Nontrivial R\n\u22a2 prod (Multiset.map (fun f => leadingCoeff f) t) \u2260 0", "state_after": "case h\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u00ac0 \u2208 t\n\u271d : Nontrivial R\n\u22a2 \u00ac\u2203 a, a \u2208 t \u2227 a = 0"}, {"tactic": "rintro \u27e8_, h, rfl\u27e9", "state_before": "case h\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh : \u00ac0 \u2208 t\n\u271d : Nontrivial R\n\u22a2 \u00ac\u2203 a, a \u2208 t \u2227 a = 0", "state_after": "case h.intro.intro\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh\u271d : \u00ac0 \u2208 t\n\u271d : Nontrivial R\nh : 0 \u2208 t\n\u22a2 False"}, {"tactic": "contradiction", "state_before": "case h.intro.intro\nR : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nh\u271d : \u00ac0 \u2208 t\n\u271d : Nontrivial R\nh : 0 \u2208 t\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/ModelTheory/Types.lean", "full_name": "FirstOrder.Language.Theory.CompleteType.mem_or_not_mem", "start": [89, 1], "end": [90, 31], "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_of_two_zsmul_eq", "start": [825, 1], "end": [827, 33], "traced_tactics": [{"tactic": "simp_rw [two_zsmul, \u2190 two_nsmul] at h", "state_before": "\u03b8 \u03c8 : Angle\nh : 2 \u2022 \u03b8 = 2 \u2022 \u03c8\n\u22a2 tan \u03b8 = tan \u03c8", "state_after": "\u03b8 \u03c8 : Angle\nh : 2 \u2022 \u03b8 = 2 \u2022 \u03c8\n\u22a2 tan \u03b8 = tan \u03c8"}, {"tactic": "exact tan_eq_of_two_nsmul_eq h", "state_before": "\u03b8 \u03c8 : Angle\nh : 2 \u2022 \u03b8 = 2 \u2022 \u03c8\n\u22a2 tan \u03b8 = tan \u03c8", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Prod.lean", "full_name": "Filter.eventually_swap_iff", "start": [298, 1], "end": [300, 22], "traced_tactics": [{"tactic": "rw [prod_comm]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.18541\n\u03b4 : Type ?u.18544\n\u03b9 : Sort ?u.18547\ns : Set \u03b1\nt : Set \u03b2\nf : Filter \u03b1\ng : Filter \u03b2\np : \u03b1 \u00d7 \u03b2 \u2192 Prop\n\u22a2 (\u2200\u1da0 (x : \u03b1 \u00d7 \u03b2) in f \u00d7\u02e2 g, p x) \u2194 \u2200\u1da0 (y : \u03b2 \u00d7 \u03b1) in g \u00d7\u02e2 f, p (Prod.swap y)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.18541\n\u03b4 : Type ?u.18544\n\u03b9 : Sort ?u.18547\ns : Set \u03b1\nt : Set \u03b2\nf : Filter \u03b1\ng : Filter \u03b2\np : \u03b1 \u00d7 \u03b2 \u2192 Prop\n\u22a2 (\u2200\u1da0 (x : \u03b1 \u00d7 \u03b2) in map (fun p => (p.snd, p.fst)) (g \u00d7\u02e2 f), p x) \u2194 \u2200\u1da0 (y : \u03b2 \u00d7 \u03b1) in g \u00d7\u02e2 f, p (Prod.swap y)"}, {"tactic": "rfl", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.18541\n\u03b4 : Type ?u.18544\n\u03b9 : Sort ?u.18547\ns : Set \u03b1\nt : Set \u03b2\nf : Filter \u03b1\ng : Filter \u03b2\np : \u03b1 \u00d7 \u03b2 \u2192 Prop\n\u22a2 (\u2200\u1da0 (x : \u03b1 \u00d7 \u03b2) in map (fun p => (p.snd, p.fst)) (g \u00d7\u02e2 f), p x) \u2194 \u2200\u1da0 (y : \u03b2 \u00d7 \u03b1) in g \u00d7\u02e2 f, p (Prod.swap y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Dimension.lean", "full_name": "le_rank_iff_exists_linearIndependent", "start": [1186, 1], "end": [1195, 51], "traced_tactics": [{"tactic": "constructor", "state_before": "K : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9 : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.1036363\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V'\ninst\u271d : Module K V'\nc : Cardinal\n\u22a2 c \u2264 Module.rank K V \u2194 \u2203 s, (#\u2191s) = c \u2227 LinearIndependent K Subtype.val", "state_after": "case mp\nK : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9 : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.1036363\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V'\ninst\u271d : Module K V'\nc : Cardinal\n\u22a2 c \u2264 Module.rank K V \u2192 \u2203 s, (#\u2191s) = c \u2227 LinearIndependent K Subtype.val\n\ncase mpr\nK : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9 : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.1036363\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V'\ninst\u271d : Module K V'\nc : Cardinal\n\u22a2 (\u2203 s, (#\u2191s) = c \u2227 LinearIndependent K Subtype.val) \u2192 c \u2264 Module.rank K V"}, {"tactic": "intro h", "state_before": "case mp\nK : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9 : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.1036363\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V'\ninst\u271d : Module K V'\nc : Cardinal\n\u22a2 c \u2264 Module.rank K V \u2192 \u2203 s, (#\u2191s) = c \u2227 LinearIndependent K Subtype.val", "state_after": "case mp\nK : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9 : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.1036363\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V'\ninst\u271d : Module K V'\nc : Cardinal\nh : c \u2264 Module.rank K V\n\u22a2 \u2203 s, (#\u2191s) = c \u2227 LinearIndependent K Subtype.val"}, {"tactic": "let t := Basis.ofVectorSpace K V", "state_before": "case mp\nK : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9 : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.1036363\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V'\ninst\u271d : Module K V'\nc : Cardinal\nh : c \u2264 Module.rank K V\n\u22a2 \u2203 s, (#\u2191s) = c \u2227 LinearIndependent K Subtype.val", "state_after": "case mp\nK : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9 : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.1036363\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V'\ninst\u271d : Module K V'\nc : Cardinal\nh : c \u2264 Module.rank K V\nt : Basis (\u2191(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\n\u22a2 \u2203 s, (#\u2191s) = c \u2227 LinearIndependent K Subtype.val"}, {"tactic": "rw [\u2190 t.mk_eq_rank'', Cardinal.le_mk_iff_exists_subset] at h", "state_before": "case mp\nK : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9 : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.1036363\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V'\ninst\u271d : Module K V'\nc : Cardinal\nh : c \u2264 Module.rank K V\nt : Basis (\u2191(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\n\u22a2 \u2203 s, (#\u2191s) = c \u2227 LinearIndependent K Subtype.val", "state_after": "case mp\nK : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9 : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.1036363\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V'\ninst\u271d : Module K V'\nc : Cardinal\nh : \u2203 p, p \u2286 ofVectorSpaceIndex K V \u2227 (#\u2191p) = c\nt : Basis (\u2191(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\n\u22a2 \u2203 s, (#\u2191s) = c \u2227 LinearIndependent K Subtype.val"}, {"tactic": "rcases h with \u27e8s, hst, hsc\u27e9", "state_before": "case mp\nK : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9 : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.1036363\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V'\ninst\u271d : Module K V'\nc : Cardinal\nh : \u2203 p, p \u2286 ofVectorSpaceIndex K V \u2227 (#\u2191p) = c\nt : Basis (\u2191(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\n\u22a2 \u2203 s, (#\u2191s) = c \u2227 LinearIndependent K Subtype.val", "state_after": "case mp.intro.intro\nK : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9 : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.1036363\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V'\ninst\u271d : Module K V'\nc : Cardinal\nt : Basis (\u2191(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\ns : Set V\nhst : s \u2286 ofVectorSpaceIndex K V\nhsc : (#\u2191s) = c\n\u22a2 \u2203 s, (#\u2191s) = c \u2227 LinearIndependent K Subtype.val"}, {"tactic": "exact \u27e8s, hsc, (ofVectorSpaceIndex.linearIndependent K V).mono hst\u27e9", "state_before": "case mp.intro.intro\nK : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9 : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.1036363\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V'\ninst\u271d : Module K V'\nc : Cardinal\nt : Basis (\u2191(ofVectorSpaceIndex K V)) K V := ofVectorSpace K V\ns : Set V\nhst : s \u2286 ofVectorSpaceIndex K V\nhsc : (#\u2191s) = c\n\u22a2 \u2203 s, (#\u2191s) = c \u2227 LinearIndependent K Subtype.val", "state_after": "no goals"}, {"tactic": "rintro \u27e8s, rfl, si\u27e9", "state_before": "case mpr\nK : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9 : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.1036363\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V'\ninst\u271d : Module K V'\nc : Cardinal\n\u22a2 (\u2203 s, (#\u2191s) = c \u2227 LinearIndependent K Subtype.val) \u2192 c \u2264 Module.rank K V", "state_after": "case mpr.intro.intro\nK : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9 : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.1036363\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V'\ninst\u271d : Module K V'\ns : Set V\nsi : LinearIndependent K Subtype.val\n\u22a2 (#\u2191s) \u2264 Module.rank K V"}, {"tactic": "exact cardinal_le_rank_of_linearIndependent si", "state_before": "case mpr.intro.intro\nK : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9 : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.1036363\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V'\ninst\u271d : Module K V'\ns : Set V\nsi : LinearIndependent K Subtype.val\n\u22a2 (#\u2191s) \u2264 Module.rank K V", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/Padics/PadicNumbers.lean", "full_name": "Padic.coe_neg", "start": [545, 1], "end": [546, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "full_name": "Orthonormal.codRestrict", "start": [1958, 1], "end": [1960, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Logic.lean", "full_name": "and_congr_right", "start": [161, 1], "end": [162, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Relation.lean", "full_name": "IsRefl.reflexive", "start": [56, 1], "end": [56, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Relation.lean", "full_name": "Relation.TransGen.closed", "start": [487, 1], "end": [489, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "full_name": "AffineSubspace.comap_mono", "start": [1627, 1], "end": [1628, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Eval.lean", "full_name": "Polynomial.IsRoot.dvd", "start": [511, 1], "end": [513, 68], "traced_tactics": [{"tactic": "rwa [IsRoot, eval, eval\u2082_eq_zero_of_dvd_of_eval\u2082_eq_zero _ _ hpq]", "state_before": "R\u271d : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\u271d\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\u271d\np\u271d q\u271d r : R\u271d[X]\nx\u271d : R\u271d\nR : Type u_1\ninst\u271d : CommSemiring R\np q : R[X]\nx : R\nh : IsRoot p x\nhpq : p \u2223 q\n\u22a2 IsRoot q x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Dfinsupp/Basic.lean", "full_name": "Dfinsupp.support_zipWith", "start": [1209, 1], "end": [1211, 21], "traced_tactics": [{"tactic": "simp [zipWith_def]", "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\u2075 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i\nhf : \u2200 (i : \u03b9), f i 0 0 = 0\ng\u2081 : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i\ng\u2082 : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i\n\u22a2 support (zipWith f hf g\u2081 g\u2082) \u2286 support g\u2081 \u222a support g\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/LocallyConvex/Basic.lean", "full_name": "absorbs_iUnion_finset", "start": [99, 1], "end": [112, 96], "traced_tactics": [{"tactic": "induction' t using Finset.induction_on with i t _ht hi", "state_before": "\ud835\udd5c : Type u_3\n\ud835\udd5d : Type ?u.3143\nE : Type u_2\n\u03b9\u271d : Sort ?u.3149\n\u03ba : \u03b9\u271d \u2192 Sort ?u.3154\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns t\u271d u v A B : Set E\n\u03b9 : Type u_1\nt : Finset \u03b9\nf : \u03b9 \u2192 Set E\n\u22a2 Absorbs \ud835\udd5c s (\u22c3 (i : \u03b9) (_ : i \u2208 t), f i) \u2194 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)", "state_after": "case empty\n\ud835\udd5c : Type u_3\n\ud835\udd5d : Type ?u.3143\nE : Type u_2\n\u03b9\u271d : Sort ?u.3149\n\u03ba : \u03b9\u271d \u2192 Sort ?u.3154\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns t u v A B : Set E\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Set E\n\u22a2 Absorbs \ud835\udd5c s (\u22c3 (i : \u03b9) (_ : i \u2208 \u2205), f i) \u2194 \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 Absorbs \ud835\udd5c s (f i)\n\ncase insert\n\ud835\udd5c : Type u_3\n\ud835\udd5d : Type ?u.3143\nE : Type u_2\n\u03b9\u271d : Sort ?u.3149\n\u03ba : \u03b9\u271d \u2192 Sort ?u.3154\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns t\u271d u v A B : Set E\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Set E\ni : \u03b9\nt : Finset \u03b9\n_ht : \u00aci \u2208 t\nhi : Absorbs \ud835\udd5c s (\u22c3 (i : \u03b9) (_ : i \u2208 t), f i) \u2194 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)\n\u22a2 Absorbs \ud835\udd5c s (\u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 insert i t), f i_1) \u2194 \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 Absorbs \ud835\udd5c s (f i_1)"}, {"tactic": "rw [Finset.set_biUnion_insert, absorbs_union, hi]", "state_before": "case insert\n\ud835\udd5c : Type u_3\n\ud835\udd5d : Type ?u.3143\nE : Type u_2\n\u03b9\u271d : Sort ?u.3149\n\u03ba : \u03b9\u271d \u2192 Sort ?u.3154\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns t\u271d u v A B : Set E\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Set E\ni : \u03b9\nt : Finset \u03b9\n_ht : \u00aci \u2208 t\nhi : Absorbs \ud835\udd5c s (\u22c3 (i : \u03b9) (_ : i \u2208 t), f i) \u2194 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)\n\u22a2 Absorbs \ud835\udd5c s (\u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 insert i t), f i_1) \u2194 \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 Absorbs \ud835\udd5c s (f i_1)", "state_after": "case insert\n\ud835\udd5c : Type u_3\n\ud835\udd5d : Type ?u.3143\nE : Type u_2\n\u03b9\u271d : Sort ?u.3149\n\u03ba : \u03b9\u271d \u2192 Sort ?u.3154\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns t\u271d u v A B : Set E\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Set E\ni : \u03b9\nt : Finset \u03b9\n_ht : \u00aci \u2208 t\nhi : Absorbs \ud835\udd5c s (\u22c3 (i : \u03b9) (_ : i \u2208 t), f i) \u2194 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)\n\u22a2 (Absorbs \ud835\udd5c s (f i) \u2227 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)) \u2194 \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 Absorbs \ud835\udd5c s (f i_1)"}, {"tactic": "constructor <;> intro h", "state_before": "case insert\n\ud835\udd5c : Type u_3\n\ud835\udd5d : Type ?u.3143\nE : Type u_2\n\u03b9\u271d : Sort ?u.3149\n\u03ba : \u03b9\u271d \u2192 Sort ?u.3154\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns t\u271d u v A B : Set E\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Set E\ni : \u03b9\nt : Finset \u03b9\n_ht : \u00aci \u2208 t\nhi : Absorbs \ud835\udd5c s (\u22c3 (i : \u03b9) (_ : i \u2208 t), f i) \u2194 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)\n\u22a2 (Absorbs \ud835\udd5c s (f i) \u2227 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)) \u2194 \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 Absorbs \ud835\udd5c s (f i_1)", "state_after": "case insert.mp\n\ud835\udd5c : Type u_3\n\ud835\udd5d : Type ?u.3143\nE : Type u_2\n\u03b9\u271d : Sort ?u.3149\n\u03ba : \u03b9\u271d \u2192 Sort ?u.3154\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns t\u271d u v A B : Set E\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Set E\ni : \u03b9\nt : Finset \u03b9\n_ht : \u00aci \u2208 t\nhi : Absorbs \ud835\udd5c s (\u22c3 (i : \u03b9) (_ : i \u2208 t), f i) \u2194 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)\nh : Absorbs \ud835\udd5c s (f i) \u2227 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)\n\u22a2 \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 Absorbs \ud835\udd5c s (f i_1)\n\ncase insert.mpr\n\ud835\udd5c : Type u_3\n\ud835\udd5d : Type ?u.3143\nE : Type u_2\n\u03b9\u271d : Sort ?u.3149\n\u03ba : \u03b9\u271d \u2192 Sort ?u.3154\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns t\u271d u v A B : Set E\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Set E\ni : \u03b9\nt : Finset \u03b9\n_ht : \u00aci \u2208 t\nhi : Absorbs \ud835\udd5c s (\u22c3 (i : \u03b9) (_ : i \u2208 t), f i) \u2194 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)\nh : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 Absorbs \ud835\udd5c s (f i_1)\n\u22a2 Absorbs \ud835\udd5c s (f i) \u2227 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)"}, {"tactic": "exact \u27e8h i (Finset.mem_insert_self i t), fun i' hi' => h i' (Finset.mem_insert_of_mem hi')\u27e9", "state_before": "case insert.mpr\n\ud835\udd5c : Type u_3\n\ud835\udd5d : Type ?u.3143\nE : Type u_2\n\u03b9\u271d : Sort ?u.3149\n\u03ba : \u03b9\u271d \u2192 Sort ?u.3154\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns t\u271d u v A B : Set E\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Set E\ni : \u03b9\nt : Finset \u03b9\n_ht : \u00aci \u2208 t\nhi : Absorbs \ud835\udd5c s (\u22c3 (i : \u03b9) (_ : i \u2208 t), f i) \u2194 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)\nh : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 Absorbs \ud835\udd5c s (f i_1)\n\u22a2 Absorbs \ud835\udd5c s (f i) \u2227 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)", "state_after": "no goals"}, {"tactic": "simp only [Finset.not_mem_empty, Set.iUnion_false, Set.iUnion_empty, absorbs_empty,\n  IsEmpty.forall_iff, imp_true_iff]", "state_before": "case empty\n\ud835\udd5c : Type u_3\n\ud835\udd5d : Type ?u.3143\nE : Type u_2\n\u03b9\u271d : Sort ?u.3149\n\u03ba : \u03b9\u271d \u2192 Sort ?u.3154\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns t u v A B : Set E\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Set E\n\u22a2 Absorbs \ud835\udd5c s (\u22c3 (i : \u03b9) (_ : i \u2208 \u2205), f i) \u2194 \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 Absorbs \ud835\udd5c s (f i)", "state_after": "no goals"}, {"tactic": "refine' fun _ hi' => (Finset.mem_insert.mp hi').elim _ (h.2 _)", "state_before": "case insert.mp\n\ud835\udd5c : Type u_3\n\ud835\udd5d : Type ?u.3143\nE : Type u_2\n\u03b9\u271d : Sort ?u.3149\n\u03ba : \u03b9\u271d \u2192 Sort ?u.3154\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns t\u271d u v A B : Set E\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Set E\ni : \u03b9\nt : Finset \u03b9\n_ht : \u00aci \u2208 t\nhi : Absorbs \ud835\udd5c s (\u22c3 (i : \u03b9) (_ : i \u2208 t), f i) \u2194 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)\nh : Absorbs \ud835\udd5c s (f i) \u2227 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)\n\u22a2 \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 Absorbs \ud835\udd5c s (f i_1)", "state_after": "case insert.mp\n\ud835\udd5c : Type u_3\n\ud835\udd5d : Type ?u.3143\nE : Type u_2\n\u03b9\u271d : Sort ?u.3149\n\u03ba : \u03b9\u271d \u2192 Sort ?u.3154\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns t\u271d u v A B : Set E\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Set E\ni : \u03b9\nt : Finset \u03b9\n_ht : \u00aci \u2208 t\nhi : Absorbs \ud835\udd5c s (\u22c3 (i : \u03b9) (_ : i \u2208 t), f i) \u2194 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)\nh : Absorbs \ud835\udd5c s (f i) \u2227 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)\nx\u271d : \u03b9\nhi' : x\u271d \u2208 insert i t\n\u22a2 x\u271d = i \u2192 Absorbs \ud835\udd5c s (f x\u271d)"}, {"tactic": "exact fun hi'' => by\n  rw [hi'']\n  exact h.1", "state_before": "case insert.mp\n\ud835\udd5c : Type u_3\n\ud835\udd5d : Type ?u.3143\nE : Type u_2\n\u03b9\u271d : Sort ?u.3149\n\u03ba : \u03b9\u271d \u2192 Sort ?u.3154\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns t\u271d u v A B : Set E\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Set E\ni : \u03b9\nt : Finset \u03b9\n_ht : \u00aci \u2208 t\nhi : Absorbs \ud835\udd5c s (\u22c3 (i : \u03b9) (_ : i \u2208 t), f i) \u2194 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)\nh : Absorbs \ud835\udd5c s (f i) \u2227 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)\nx\u271d : \u03b9\nhi' : x\u271d \u2208 insert i t\n\u22a2 x\u271d = i \u2192 Absorbs \ud835\udd5c s (f x\u271d)", "state_after": "no goals"}, {"tactic": "rw [hi'']", "state_before": "\ud835\udd5c : Type u_3\n\ud835\udd5d : Type ?u.3143\nE : Type u_2\n\u03b9\u271d : Sort ?u.3149\n\u03ba : \u03b9\u271d \u2192 Sort ?u.3154\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns t\u271d u v A B : Set E\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Set E\ni : \u03b9\nt : Finset \u03b9\n_ht : \u00aci \u2208 t\nhi : Absorbs \ud835\udd5c s (\u22c3 (i : \u03b9) (_ : i \u2208 t), f i) \u2194 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)\nh : Absorbs \ud835\udd5c s (f i) \u2227 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)\nx\u271d : \u03b9\nhi' : x\u271d \u2208 insert i t\nhi'' : x\u271d = i\n\u22a2 Absorbs \ud835\udd5c s (f x\u271d)", "state_after": "\ud835\udd5c : Type u_3\n\ud835\udd5d : Type ?u.3143\nE : Type u_2\n\u03b9\u271d : Sort ?u.3149\n\u03ba : \u03b9\u271d \u2192 Sort ?u.3154\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns t\u271d u v A B : Set E\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Set E\ni : \u03b9\nt : Finset \u03b9\n_ht : \u00aci \u2208 t\nhi : Absorbs \ud835\udd5c s (\u22c3 (i : \u03b9) (_ : i \u2208 t), f i) \u2194 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)\nh : Absorbs \ud835\udd5c s (f i) \u2227 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)\nx\u271d : \u03b9\nhi' : x\u271d \u2208 insert i t\nhi'' : x\u271d = i\n\u22a2 Absorbs \ud835\udd5c s (f i)"}, {"tactic": "exact h.1", "state_before": "\ud835\udd5c : Type u_3\n\ud835\udd5d : Type ?u.3143\nE : Type u_2\n\u03b9\u271d : Sort ?u.3149\n\u03ba : \u03b9\u271d \u2192 Sort ?u.3154\ninst\u271d\u00b9 : SeminormedRing \ud835\udd5c\ninst\u271d : SMul \ud835\udd5c E\ns t\u271d u v A B : Set E\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Set E\ni : \u03b9\nt : Finset \u03b9\n_ht : \u00aci \u2208 t\nhi : Absorbs \ud835\udd5c s (\u22c3 (i : \u03b9) (_ : i \u2208 t), f i) \u2194 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)\nh : Absorbs \ud835\udd5c s (f i) \u2227 \u2200 (i : \u03b9), i \u2208 t \u2192 Absorbs \ud835\udd5c s (f i)\nx\u271d : \u03b9\nhi' : x\u271d \u2208 insert i t\nhi'' : x\u271d = i\n\u22a2 Absorbs \ud835\udd5c s (f i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/Exponential.lean", "full_name": "exp_nsmul", "start": [512, 1], "end": [515, 86], "traced_tactics": [{"tactic": "induction' n with n ih", "state_before": "\ud835\udd42 : Type u_2\n\ud835\udd38 : Type u_1\n\ud835\udd39 : Type ?u.380774\ninst\u271d\u2075 : IsROrC \ud835\udd42\ninst\u271d\u2074 : NormedRing \ud835\udd38\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d\u00b2 : NormedRing \ud835\udd39\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd39\ninst\u271d : CompleteSpace \ud835\udd38\nn : \u2115\nx : \ud835\udd38\n\u22a2 exp \ud835\udd42 (n \u2022 x) = exp \ud835\udd42 x ^ n", "state_after": "case zero\n\ud835\udd42 : Type u_2\n\ud835\udd38 : Type u_1\n\ud835\udd39 : Type ?u.380774\ninst\u271d\u2075 : IsROrC \ud835\udd42\ninst\u271d\u2074 : NormedRing \ud835\udd38\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d\u00b2 : NormedRing \ud835\udd39\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd39\ninst\u271d : CompleteSpace \ud835\udd38\nx : \ud835\udd38\n\u22a2 exp \ud835\udd42 (Nat.zero \u2022 x) = exp \ud835\udd42 x ^ Nat.zero\n\ncase succ\n\ud835\udd42 : Type u_2\n\ud835\udd38 : Type u_1\n\ud835\udd39 : Type ?u.380774\ninst\u271d\u2075 : IsROrC \ud835\udd42\ninst\u271d\u2074 : NormedRing \ud835\udd38\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d\u00b2 : NormedRing \ud835\udd39\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd39\ninst\u271d : CompleteSpace \ud835\udd38\nx : \ud835\udd38\nn : \u2115\nih : exp \ud835\udd42 (n \u2022 x) = exp \ud835\udd42 x ^ n\n\u22a2 exp \ud835\udd42 (Nat.succ n \u2022 x) = exp \ud835\udd42 x ^ Nat.succ n"}, {"tactic": "rw [Nat.zero_eq, zero_smul, pow_zero, exp_zero]", "state_before": "case zero\n\ud835\udd42 : Type u_2\n\ud835\udd38 : Type u_1\n\ud835\udd39 : Type ?u.380774\ninst\u271d\u2075 : IsROrC \ud835\udd42\ninst\u271d\u2074 : NormedRing \ud835\udd38\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d\u00b2 : NormedRing \ud835\udd39\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd39\ninst\u271d : CompleteSpace \ud835\udd38\nx : \ud835\udd38\n\u22a2 exp \ud835\udd42 (Nat.zero \u2022 x) = exp \ud835\udd42 x ^ Nat.zero", "state_after": "no goals"}, {"tactic": "rw [succ_nsmul, pow_succ, exp_add_of_commute ((Commute.refl x).smul_right n), ih]", "state_before": "case succ\n\ud835\udd42 : Type u_2\n\ud835\udd38 : Type u_1\n\ud835\udd39 : Type ?u.380774\ninst\u271d\u2075 : IsROrC \ud835\udd42\ninst\u271d\u2074 : NormedRing \ud835\udd38\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d\u00b2 : NormedRing \ud835\udd39\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd39\ninst\u271d : CompleteSpace \ud835\udd38\nx : \ud835\udd38\nn : \u2115\nih : exp \ud835\udd42 (n \u2022 x) = exp \ud835\udd42 x ^ n\n\u22a2 exp \ud835\udd42 (Nat.succ n \u2022 x) = exp \ud835\udd42 x ^ Nat.succ n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GCDMonoid/Multiset.lean", "full_name": "Multiset.gcd_ndinsert", "start": [222, 1], "end": [224, 7], "traced_tactics": [{"tactic": "rw [\u2190 gcd_dedup, dedup_ext.2, gcd_dedup, gcd_cons]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\n\u22a2 gcd (ndinsert a s) = GCDMonoid.gcd a (gcd s)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \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 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : NormalizedGCDMonoid \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", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Directed.lean", "full_name": "directedOn_univ_iff", "start": [193, 1], "end": [198, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Subobject/Limits.lean", "full_name": "CategoryTheory.Limits.imageSubobjectCompIso_hom_arrow", "start": [415, 1], "end": [418, 34], "traced_tactics": [{"tactic": "simp [imageSubobjectCompIso]", "state_before": "C : Type u\ninst\u271d\u2074 : Category C\nX Y Z : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b3 : HasImage f\u271d\ninst\u271d\u00b2 : HasEqualizers C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasImage f\nY' : C\nh : Y \u27f6 Y'\ninst\u271d : IsIso h\n\u22a2 (imageSubobjectCompIso f h).hom \u226b arrow (imageSubobject f) = arrow (imageSubobject (f \u226b h)) \u226b inv h", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.disjoint_iff_inter_eq_empty", "start": [1319, 1], "end": [1320, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "Complex.exp_sub", "start": [553, 1], "end": [554, 58], "traced_tactics": [{"tactic": "simp [sub_eq_add_neg, exp_add, exp_neg, div_eq_mul_inv]", "state_before": "x y : \u2102\n\u22a2 exp (x - y) = exp x / exp y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/BilinearForm.lean", "full_name": "BilinForm.isAdjointPairLeftAdjointOfNondegenerate", "start": [1572, 1], "end": [1574, 59], "traced_tactics": []}, {"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": [711, 1], "end": [717, 68], "traced_tactics": [{"tactic": "by_contra h", "state_before": "\u03b1 : Type ?u.166364\n\u03b2 : Type ?u.166367\n\u03b3 : Type ?u.166370\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, x n < R", "state_after": "\u03b1 : Type ?u.166364\n\u03b2 : Type ?u.166367\n\u03b3 : Type ?u.166370\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, x n < R\n\u22a2 False"}, {"tactic": "simp_rw [not_exists, not_frequently, not_lt] at h", "state_before": "\u03b1 : Type ?u.166364\n\u03b2 : Type ?u.166367\n\u03b3 : Type ?u.166370\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, x n < R\n\u22a2 False", "state_after": "\u03b1 : Type ?u.166364\n\u03b2 : Type ?u.166367\n\u03b3 : Type ?u.166370\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_1 \u2264 x x_2\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.166364\n\u03b2 : Type ?u.166367\n\u03b3 : Type ?u.166370\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_1 \u2264 x x_2\n\u22a2 False", "state_after": "\u03b1 : Type ?u.166364\n\u03b2 : Type ?u.166367\n\u03b3 : Type ?u.166370\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_1 \u2264 x x_2\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.166364\n\u03b2 : Type ?u.166367\n\u03b3 : Type ?u.166370\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_1 \u2264 x x_2\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.166364\n\u03b2 : Type ?u.166367\n\u03b3 : Type ?u.166370\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_1 \u2264 x x_2\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 hi.trans (le_abs_self (x i))", "state_before": "\u03b1 : Type ?u.166364\n\u03b2 : Type ?u.166367\n\u03b3 : Type ?u.166370\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_1 \u2264 x x_2\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.166364\n\u03b2 : Type ?u.166367\n\u03b3 : Type ?u.166370\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_1 \u2264 x x_2\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/Algebra/Lie/UniversalEnveloping.lean", "full_name": "UniversalEnvelopingAlgebra.lift_\u03b9_apply", "start": [150, 1], "end": [151, 78], "traced_tactics": [{"tactic": "rw [\u2190 Function.comp_apply (f := lift R f) (g := \u03b9 R) (x := x), \u03b9_comp_lift]", "state_before": "R : Type u\u2081\nL : Type u\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\nA : Type u\u2083\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nf : L \u2192\u2097\u2045R\u2046 A\nx : L\n\u22a2 \u2191(\u2191(lift R) f) (\u2191(\u03b9 R) x) = \u2191f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/UpperLower.lean", "full_name": "IsLowerSet.mul_left", "start": [82, 1], "end": [83, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Hom/Monoid.lean", "full_name": "OrderMonoidWithZeroHom.toOrderMonoidHom_eq_coe", "start": [767, 1], "end": [768, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Factors.lean", "full_name": "Nat.factors_chain'", "start": [111, 1], "end": [112, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Heyting/Basic.lean", "full_name": "le_himp_comm", "start": [326, 1], "end": [326, 82], "traced_tactics": [{"tactic": "rw [le_himp_iff, le_himp_iff']", "state_before": "\u03b9 : Type ?u.26953\n\u03b1 : Type u_1\n\u03b2 : Type ?u.26959\ninst\u271d : GeneralizedHeytingAlgebra \u03b1\na b c d : \u03b1\n\u22a2 a \u2264 b \u21e8 c \u2194 b \u2264 a \u21e8 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/Deriv/Linear.lean", "full_name": "LinearMap.derivWithin", "start": [107, 11], "end": [109, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Normed.lean", "full_name": "convexOn_univ_norm", "start": [52, 1], "end": [53, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Antichain.lean", "full_name": "IsAntichain.isAntisymm", "start": [72, 11], "end": [73, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Polynomial/Chebyshev.lean", "full_name": "Polynomial.Chebyshev.map_U", "start": [173, 1], "end": [180, 13], "traced_tactics": [{"tactic": "simp only [U_zero, Polynomial.map_one]", "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\n\u22a2 map f (U R 0) = U S 0", "state_after": "no goals"}, {"tactic": "simp [U_one, map_X, Polynomial.map_mul, Polynomial.map_add, Polynomial.map_one]", "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\n\u22a2 map f (U R 1) = U S 1", "state_after": "no goals"}, {"tactic": "simp only [U_add_two, Polynomial.map_mul, Polynomial.map_sub, map_X, Polynomial.map_add,\n  Polynomial.map_one, map_U f (n + 1), map_U f n]", "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nn : \u2115\n\u22a2 map f (U R (n + 2)) = U S (n + 2)", "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nn : \u2115\n\u22a2 map f 2 * X * U S (n + 1) - U S n = 2 * X * U S (n + 1) - U S n"}, {"tactic": "norm_num", "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nn : \u2115\n\u22a2 map f 2 * X * U S (n + 1) - U S n = 2 * X * U S (n + 1) - U S n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Monotone/Monovary.lean", "full_name": "Antivary.comp_right", "start": [151, 1], "end": [152, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/IsFreeGroup.lean", "full_name": "IsFreeGroup.lift'_eq_freeGroup_lift", "start": [104, 1], "end": [105, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/IsEmpty.lean", "full_name": "isEmpty_sum", "start": [180, 1], "end": [181, 55], "traced_tactics": [{"tactic": "simp only [\u2190 not_nonempty_iff, nonempty_sum, not_or]", "state_before": "\u03b1\u271d : Sort ?u.3308\n\u03b2\u271d : Sort ?u.3311\n\u03b3 : Sort ?u.3314\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u22a2 IsEmpty (\u03b1 \u2295 \u03b2) \u2194 IsEmpty \u03b1 \u2227 IsEmpty \u03b2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean", "full_name": "Euclidean.nhds_basis_ball", "start": [120, 1], "end": [122, 39], "traced_tactics": [{"tactic": "rw [toEuclidean.toHomeomorph.nhds_eq_comap x]", "state_before": "E : Type u_1\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : TopologicalAddGroup E\ninst\u271d\u00b3 : T2Space E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ninst\u271d : FiniteDimensional \u211d E\nx : E\n\u22a2 Filter.HasBasis (\ud835\udcdd x) (fun r => 0 < r) (ball x)", "state_after": "E : Type u_1\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : TopologicalAddGroup E\ninst\u271d\u00b3 : T2Space E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ninst\u271d : FiniteDimensional \u211d E\nx : E\n\u22a2 Filter.HasBasis\n    (Filter.comap (\u2191(ContinuousLinearEquiv.toHomeomorph toEuclidean))\n      (\ud835\udcdd (\u2191(ContinuousLinearEquiv.toHomeomorph toEuclidean) x)))\n    (fun r => 0 < r) (ball x)"}, {"tactic": "exact Metric.nhds_basis_ball.comap _", "state_before": "E : Type u_1\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : TopologicalAddGroup E\ninst\u271d\u00b3 : T2Space E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ninst\u271d : FiniteDimensional \u211d E\nx : E\n\u22a2 Filter.HasBasis\n    (Filter.comap (\u2191(ContinuousLinearEquiv.toHomeomorph toEuclidean))\n      (\ud835\udcdd (\u2191(ContinuousLinearEquiv.toHomeomorph toEuclidean) x)))\n    (fun r => 0 < r) (ball x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/RootsOfUnity/Basic.lean", "full_name": "mem_rootsOfUnity_prime_pow_mul_iff'", "start": [281, 1], "end": [284, 40], "traced_tactics": [{"tactic": "rw [\u2190 PNat.mk_coe p hp.1.pos, \u2190 PNat.pow_coe, \u2190 PNat.mul_coe, \u2190 mem_rootsOfUnity,\n  mem_rootsOfUnity_prime_pow_mul_iff]", "state_before": "M : Type ?u.1342912\nN : Type ?u.1342915\nG : Type ?u.1342918\nR : Type u_1\nS : Type ?u.1342924\nF : Type ?u.1342927\ninst\u271d\u2075 : CommMonoid M\ninst\u271d\u2074 : CommMonoid N\ninst\u271d\u00b3 : DivisionCommMonoid G\nk\u271d l : \u2115+\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsReduced R\np k : \u2115\nm : \u2115+\nhp : Fact (Nat.Prime p)\ninst\u271d : CharP R p\n\u03b6 : R\u02e3\n\u22a2 \u03b6 ^ (p ^ k * \u2191m) = 1 \u2194 \u03b6 \u2208 rootsOfUnity m R", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Directed.lean", "full_name": "directedOn_pair", "start": [246, 1], "end": [247, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/NeZero.lean", "full_name": "not_neZero", "start": [44, 1], "end": [44, 93], "traced_tactics": [{"tactic": "simp [neZero_iff]", "state_before": "R : Type u_1\ninst\u271d : Zero R\nn : R\n\u22a2 \u00acNeZero n \u2194 n = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/FinitePresentation.lean", "full_name": "RingHom.FinitePresentation.comp_surjective", "start": [429, 1], "end": [438, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/TensorProduct.lean", "full_name": "Algebra.TensorProduct.mul_assoc'", "start": [393, 1], "end": [413, 58], "traced_tactics": [{"tactic": "intros x y z", "state_before": "R : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\n\u22a2 \u2200 (x y z : A \u2297[R] B), \u2191(\u2191mul (\u2191(\u2191mul x) y)) z = \u2191(\u2191mul x) (\u2191(\u2191mul y) z)", "state_after": "R : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2191(\u2191mul (\u2191(\u2191mul x) y)) z = \u2191(\u2191mul x) (\u2191(\u2191mul y) z)"}, {"tactic": "refine TensorProduct.induction_on x ?_ ?_ ?_", "state_before": "R : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2191(\u2191mul (\u2191(\u2191mul x) y)) z = \u2191(\u2191mul x) (\u2191(\u2191mul y) z)", "state_after": "case refine_1\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2191(\u2191mul (\u2191(\u2191mul 0) y)) z = \u2191(\u2191mul 0) (\u2191(\u2191mul y) z)\n\ncase refine_2\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2200 (x : A) (y_1 : B), \u2191(\u2191mul (\u2191(\u2191mul (x \u2297\u209c[R] y_1)) y)) z = \u2191(\u2191mul (x \u2297\u209c[R] y_1)) (\u2191(\u2191mul y) z)\n\ncase refine_3\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2200 (x y_1 : A \u2297[R] B),\n    \u2191(\u2191mul (\u2191(\u2191mul x) y)) z = \u2191(\u2191mul x) (\u2191(\u2191mul y) z) \u2192\n      \u2191(\u2191mul (\u2191(\u2191mul y_1) y)) z = \u2191(\u2191mul y_1) (\u2191(\u2191mul y) z) \u2192\n        \u2191(\u2191mul (\u2191(\u2191mul (x + y_1)) y)) z = \u2191(\u2191mul (x + y_1)) (\u2191(\u2191mul y) z)"}, {"tactic": "refine TensorProduct.induction_on y ?_ ?_ ?_", "state_before": "case refine_2\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2200 (x : A) (y_1 : B), \u2191(\u2191mul (\u2191(\u2191mul (x \u2297\u209c[R] y_1)) y)) z = \u2191(\u2191mul (x \u2297\u209c[R] y_1)) (\u2191(\u2191mul y) z)\n\ncase refine_3\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2200 (x y_1 : A \u2297[R] B),\n    \u2191(\u2191mul (\u2191(\u2191mul x) y)) z = \u2191(\u2191mul x) (\u2191(\u2191mul y) z) \u2192\n      \u2191(\u2191mul (\u2191(\u2191mul y_1) y)) z = \u2191(\u2191mul y_1) (\u2191(\u2191mul y) z) \u2192\n        \u2191(\u2191mul (\u2191(\u2191mul (x + y_1)) y)) z = \u2191(\u2191mul (x + y_1)) (\u2191(\u2191mul y) z)", "state_after": "case refine_2.refine_1\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2200 (x : A) (y : B), \u2191(\u2191mul (\u2191(\u2191mul (x \u2297\u209c[R] y)) 0)) z = \u2191(\u2191mul (x \u2297\u209c[R] y)) (\u2191(\u2191mul 0) z)\n\ncase refine_2.refine_2\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2200 (x : A) (y : B) (x_1 : A) (y_1 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (x \u2297\u209c[R] y))) z = \u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (\u2191(\u2191mul (x \u2297\u209c[R] y)) z)\n\ncase refine_2.refine_3\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2200 (x y : A \u2297[R] B),\n    (\u2200 (x_1 : A) (y : B), \u2191(\u2191mul (\u2191(\u2191mul (x_1 \u2297\u209c[R] y)) x)) z = \u2191(\u2191mul (x_1 \u2297\u209c[R] y)) (\u2191(\u2191mul x) z)) \u2192\n      (\u2200 (x : A) (y_1 : B), \u2191(\u2191mul (\u2191(\u2191mul (x \u2297\u209c[R] y_1)) y)) z = \u2191(\u2191mul (x \u2297\u209c[R] y_1)) (\u2191(\u2191mul y) z)) \u2192\n        \u2200 (x_1 : A) (y_1 : B), \u2191(\u2191mul (\u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (x + y))) z = \u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (\u2191(\u2191mul (x + y)) z)\n\ncase refine_3\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2200 (x y_1 : A \u2297[R] B),\n    \u2191(\u2191mul (\u2191(\u2191mul x) y)) z = \u2191(\u2191mul x) (\u2191(\u2191mul y) z) \u2192\n      \u2191(\u2191mul (\u2191(\u2191mul y_1) y)) z = \u2191(\u2191mul y_1) (\u2191(\u2191mul y) z) \u2192\n        \u2191(\u2191mul (\u2191(\u2191mul (x + y_1)) y)) z = \u2191(\u2191mul (x + y_1)) (\u2191(\u2191mul y) z)"}, {"tactic": "refine TensorProduct.induction_on z ?_ ?_ ?_", "state_before": "case refine_2.refine_2\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2200 (x : A) (y : B) (x_1 : A) (y_1 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (x \u2297\u209c[R] y))) z = \u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (\u2191(\u2191mul (x \u2297\u209c[R] y)) z)\n\ncase refine_2.refine_3\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2200 (x y : A \u2297[R] B),\n    (\u2200 (x_1 : A) (y : B), \u2191(\u2191mul (\u2191(\u2191mul (x_1 \u2297\u209c[R] y)) x)) z = \u2191(\u2191mul (x_1 \u2297\u209c[R] y)) (\u2191(\u2191mul x) z)) \u2192\n      (\u2200 (x : A) (y_1 : B), \u2191(\u2191mul (\u2191(\u2191mul (x \u2297\u209c[R] y_1)) y)) z = \u2191(\u2191mul (x \u2297\u209c[R] y_1)) (\u2191(\u2191mul y) z)) \u2192\n        \u2200 (x_1 : A) (y_1 : B), \u2191(\u2191mul (\u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (x + y))) z = \u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (\u2191(\u2191mul (x + y)) z)\n\ncase refine_3\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2200 (x y_1 : A \u2297[R] B),\n    \u2191(\u2191mul (\u2191(\u2191mul x) y)) z = \u2191(\u2191mul x) (\u2191(\u2191mul y) z) \u2192\n      \u2191(\u2191mul (\u2191(\u2191mul y_1) y)) z = \u2191(\u2191mul y_1) (\u2191(\u2191mul y) z) \u2192\n        \u2191(\u2191mul (\u2191(\u2191mul (x + y_1)) y)) z = \u2191(\u2191mul (x + y_1)) (\u2191(\u2191mul y) z)", "state_after": "case refine_2.refine_2.refine_1\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2200 (x : A) (y : B) (x_1 : A) (y_1 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (x \u2297\u209c[R] y))) 0 = \u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (\u2191(\u2191mul (x \u2297\u209c[R] y)) 0)\n\ncase refine_2.refine_2.refine_2\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2200 (x : A) (y : B) (x_1 : A) (y_1 : B) (x_2 : A) (y_2 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (x_2 \u2297\u209c[R] y_2)) (x_1 \u2297\u209c[R] y_1))) (x \u2297\u209c[R] y) =\n      \u2191(\u2191mul (x_2 \u2297\u209c[R] y_2)) (\u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (x \u2297\u209c[R] y))\n\ncase refine_2.refine_2.refine_3\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2200 (x y : A \u2297[R] B),\n    (\u2200 (x_1 : A) (y : B) (x_2 : A) (y_1 : B),\n        \u2191(\u2191mul (\u2191(\u2191mul (x_2 \u2297\u209c[R] y_1)) (x_1 \u2297\u209c[R] y))) x = \u2191(\u2191mul (x_2 \u2297\u209c[R] y_1)) (\u2191(\u2191mul (x_1 \u2297\u209c[R] y)) x)) \u2192\n      (\u2200 (x : A) (y_1 : B) (x_1 : A) (y_2 : B),\n          \u2191(\u2191mul (\u2191(\u2191mul (x_1 \u2297\u209c[R] y_2)) (x \u2297\u209c[R] y_1))) y = \u2191(\u2191mul (x_1 \u2297\u209c[R] y_2)) (\u2191(\u2191mul (x \u2297\u209c[R] y_1)) y)) \u2192\n        \u2200 (x_1 : A) (y_1 : B) (x_2 : A) (y_2 : B),\n          \u2191(\u2191mul (\u2191(\u2191mul (x_2 \u2297\u209c[R] y_2)) (x_1 \u2297\u209c[R] y_1))) (x + y) =\n            \u2191(\u2191mul (x_2 \u2297\u209c[R] y_2)) (\u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (x + y))\n\ncase refine_2.refine_3\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2200 (x y : A \u2297[R] B),\n    (\u2200 (x_1 : A) (y : B), \u2191(\u2191mul (\u2191(\u2191mul (x_1 \u2297\u209c[R] y)) x)) z = \u2191(\u2191mul (x_1 \u2297\u209c[R] y)) (\u2191(\u2191mul x) z)) \u2192\n      (\u2200 (x : A) (y_1 : B), \u2191(\u2191mul (\u2191(\u2191mul (x \u2297\u209c[R] y_1)) y)) z = \u2191(\u2191mul (x \u2297\u209c[R] y_1)) (\u2191(\u2191mul y) z)) \u2192\n        \u2200 (x_1 : A) (y_1 : B), \u2191(\u2191mul (\u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (x + y))) z = \u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (\u2191(\u2191mul (x + y)) z)\n\ncase refine_3\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2200 (x y_1 : A \u2297[R] B),\n    \u2191(\u2191mul (\u2191(\u2191mul x) y)) z = \u2191(\u2191mul x) (\u2191(\u2191mul y) z) \u2192\n      \u2191(\u2191mul (\u2191(\u2191mul y_1) y)) z = \u2191(\u2191mul y_1) (\u2191(\u2191mul y) z) \u2192\n        \u2191(\u2191mul (\u2191(\u2191mul (x + y_1)) y)) z = \u2191(\u2191mul (x + y_1)) (\u2191(\u2191mul y) z)"}, {"tactic": "simp only [LinearMap.map_zero, LinearMap.zero_apply]", "state_before": "case refine_1\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2191(\u2191mul (\u2191(\u2191mul 0) y)) z = \u2191(\u2191mul 0) (\u2191(\u2191mul y) z)", "state_after": "no goals"}, {"tactic": "simp only [LinearMap.map_zero, forall_const, LinearMap.zero_apply]", "state_before": "case refine_2.refine_1\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2200 (x : A) (y : B), \u2191(\u2191mul (\u2191(\u2191mul (x \u2297\u209c[R] y)) 0)) z = \u2191(\u2191mul (x \u2297\u209c[R] y)) (\u2191(\u2191mul 0) z)", "state_after": "no goals"}, {"tactic": "simp only [LinearMap.map_zero, forall_const]", "state_before": "case refine_2.refine_2.refine_1\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2200 (x : A) (y : B) (x_1 : A) (y_1 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (x \u2297\u209c[R] y))) 0 = \u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (\u2191(\u2191mul (x \u2297\u209c[R] y)) 0)", "state_after": "no goals"}, {"tactic": "intros", "state_before": "case refine_2.refine_2.refine_2\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2200 (x : A) (y : B) (x_1 : A) (y_1 : B) (x_2 : A) (y_2 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (x_2 \u2297\u209c[R] y_2)) (x_1 \u2297\u209c[R] y_1))) (x \u2297\u209c[R] y) =\n      \u2191(\u2191mul (x_2 \u2297\u209c[R] y_2)) (\u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (x \u2297\u209c[R] y))", "state_after": "case refine_2.refine_2.refine_2\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\nx\u271d\u00b2 : A\ny\u271d\u00b2 : B\nx\u271d\u00b9 : A\ny\u271d\u00b9 : B\nx\u271d : A\ny\u271d : B\n\u22a2 \u2191(\u2191mul (\u2191(\u2191mul (x\u271d \u2297\u209c[R] y\u271d)) (x\u271d\u00b9 \u2297\u209c[R] y\u271d\u00b9))) (x\u271d\u00b2 \u2297\u209c[R] y\u271d\u00b2) =\n    \u2191(\u2191mul (x\u271d \u2297\u209c[R] y\u271d)) (\u2191(\u2191mul (x\u271d\u00b9 \u2297\u209c[R] y\u271d\u00b9)) (x\u271d\u00b2 \u2297\u209c[R] y\u271d\u00b2))"}, {"tactic": "simp only [h]", "state_before": "case refine_2.refine_2.refine_2\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\nx\u271d\u00b2 : A\ny\u271d\u00b2 : B\nx\u271d\u00b9 : A\ny\u271d\u00b9 : B\nx\u271d : A\ny\u271d : B\n\u22a2 \u2191(\u2191mul (\u2191(\u2191mul (x\u271d \u2297\u209c[R] y\u271d)) (x\u271d\u00b9 \u2297\u209c[R] y\u271d\u00b9))) (x\u271d\u00b2 \u2297\u209c[R] y\u271d\u00b2) =\n    \u2191(\u2191mul (x\u271d \u2297\u209c[R] y\u271d)) (\u2191(\u2191mul (x\u271d\u00b9 \u2297\u209c[R] y\u271d\u00b9)) (x\u271d\u00b2 \u2297\u209c[R] y\u271d\u00b2))", "state_after": "no goals"}, {"tactic": "intros", "state_before": "case refine_2.refine_2.refine_3\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2200 (x y : A \u2297[R] B),\n    (\u2200 (x_1 : A) (y : B) (x_2 : A) (y_1 : B),\n        \u2191(\u2191mul (\u2191(\u2191mul (x_2 \u2297\u209c[R] y_1)) (x_1 \u2297\u209c[R] y))) x = \u2191(\u2191mul (x_2 \u2297\u209c[R] y_1)) (\u2191(\u2191mul (x_1 \u2297\u209c[R] y)) x)) \u2192\n      (\u2200 (x : A) (y_1 : B) (x_1 : A) (y_2 : B),\n          \u2191(\u2191mul (\u2191(\u2191mul (x_1 \u2297\u209c[R] y_2)) (x \u2297\u209c[R] y_1))) y = \u2191(\u2191mul (x_1 \u2297\u209c[R] y_2)) (\u2191(\u2191mul (x \u2297\u209c[R] y_1)) y)) \u2192\n        \u2200 (x_1 : A) (y_1 : B) (x_2 : A) (y_2 : B),\n          \u2191(\u2191mul (\u2191(\u2191mul (x_2 \u2297\u209c[R] y_2)) (x_1 \u2297\u209c[R] y_1))) (x + y) =\n            \u2191(\u2191mul (x_2 \u2297\u209c[R] y_2)) (\u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (x + y))", "state_after": "case refine_2.refine_2.refine_3\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z x\u271d\u00b2 y\u271d\u00b2 : A \u2297[R] B\na\u271d\u00b9 :\n  \u2200 (x : A) (y : B) (x_1 : A) (y_1 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (x \u2297\u209c[R] y))) x\u271d\u00b2 = \u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (\u2191(\u2191mul (x \u2297\u209c[R] y)) x\u271d\u00b2)\na\u271d :\n  \u2200 (x : A) (y : B) (x_1 : A) (y_1 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (x \u2297\u209c[R] y))) y\u271d\u00b2 = \u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (\u2191(\u2191mul (x \u2297\u209c[R] y)) y\u271d\u00b2)\nx\u271d\u00b9 : A\ny\u271d\u00b9 : B\nx\u271d : A\ny\u271d : B\n\u22a2 \u2191(\u2191mul (\u2191(\u2191mul (x\u271d \u2297\u209c[R] y\u271d)) (x\u271d\u00b9 \u2297\u209c[R] y\u271d\u00b9))) (x\u271d\u00b2 + y\u271d\u00b2) =\n    \u2191(\u2191mul (x\u271d \u2297\u209c[R] y\u271d)) (\u2191(\u2191mul (x\u271d\u00b9 \u2297\u209c[R] y\u271d\u00b9)) (x\u271d\u00b2 + y\u271d\u00b2))"}, {"tactic": "simp only [LinearMap.map_add, *]", "state_before": "case refine_2.refine_2.refine_3\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z x\u271d\u00b2 y\u271d\u00b2 : A \u2297[R] B\na\u271d\u00b9 :\n  \u2200 (x : A) (y : B) (x_1 : A) (y_1 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (x \u2297\u209c[R] y))) x\u271d\u00b2 = \u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (\u2191(\u2191mul (x \u2297\u209c[R] y)) x\u271d\u00b2)\na\u271d :\n  \u2200 (x : A) (y : B) (x_1 : A) (y_1 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (x \u2297\u209c[R] y))) y\u271d\u00b2 = \u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (\u2191(\u2191mul (x \u2297\u209c[R] y)) y\u271d\u00b2)\nx\u271d\u00b9 : A\ny\u271d\u00b9 : B\nx\u271d : A\ny\u271d : B\n\u22a2 \u2191(\u2191mul (\u2191(\u2191mul (x\u271d \u2297\u209c[R] y\u271d)) (x\u271d\u00b9 \u2297\u209c[R] y\u271d\u00b9))) (x\u271d\u00b2 + y\u271d\u00b2) =\n    \u2191(\u2191mul (x\u271d \u2297\u209c[R] y\u271d)) (\u2191(\u2191mul (x\u271d\u00b9 \u2297\u209c[R] y\u271d\u00b9)) (x\u271d\u00b2 + y\u271d\u00b2))", "state_after": "no goals"}, {"tactic": "intros", "state_before": "case refine_2.refine_3\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2200 (x y : A \u2297[R] B),\n    (\u2200 (x_1 : A) (y : B), \u2191(\u2191mul (\u2191(\u2191mul (x_1 \u2297\u209c[R] y)) x)) z = \u2191(\u2191mul (x_1 \u2297\u209c[R] y)) (\u2191(\u2191mul x) z)) \u2192\n      (\u2200 (x : A) (y_1 : B), \u2191(\u2191mul (\u2191(\u2191mul (x \u2297\u209c[R] y_1)) y)) z = \u2191(\u2191mul (x \u2297\u209c[R] y_1)) (\u2191(\u2191mul y) z)) \u2192\n        \u2200 (x_1 : A) (y_1 : B), \u2191(\u2191mul (\u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (x + y))) z = \u2191(\u2191mul (x_1 \u2297\u209c[R] y_1)) (\u2191(\u2191mul (x + y)) z)", "state_after": "case refine_2.refine_3\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z x\u271d\u00b9 y\u271d\u00b9 : A \u2297[R] B\na\u271d\u00b9 : \u2200 (x : A) (y : B), \u2191(\u2191mul (\u2191(\u2191mul (x \u2297\u209c[R] y)) x\u271d\u00b9)) z = \u2191(\u2191mul (x \u2297\u209c[R] y)) (\u2191(\u2191mul x\u271d\u00b9) z)\na\u271d : \u2200 (x : A) (y : B), \u2191(\u2191mul (\u2191(\u2191mul (x \u2297\u209c[R] y)) y\u271d\u00b9)) z = \u2191(\u2191mul (x \u2297\u209c[R] y)) (\u2191(\u2191mul y\u271d\u00b9) z)\nx\u271d : A\ny\u271d : B\n\u22a2 \u2191(\u2191mul (\u2191(\u2191mul (x\u271d \u2297\u209c[R] y\u271d)) (x\u271d\u00b9 + y\u271d\u00b9))) z = \u2191(\u2191mul (x\u271d \u2297\u209c[R] y\u271d)) (\u2191(\u2191mul (x\u271d\u00b9 + y\u271d\u00b9)) z)"}, {"tactic": "simp only [LinearMap.map_add, *, LinearMap.add_apply]", "state_before": "case refine_2.refine_3\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z x\u271d\u00b9 y\u271d\u00b9 : A \u2297[R] B\na\u271d\u00b9 : \u2200 (x : A) (y : B), \u2191(\u2191mul (\u2191(\u2191mul (x \u2297\u209c[R] y)) x\u271d\u00b9)) z = \u2191(\u2191mul (x \u2297\u209c[R] y)) (\u2191(\u2191mul x\u271d\u00b9) z)\na\u271d : \u2200 (x : A) (y : B), \u2191(\u2191mul (\u2191(\u2191mul (x \u2297\u209c[R] y)) y\u271d\u00b9)) z = \u2191(\u2191mul (x \u2297\u209c[R] y)) (\u2191(\u2191mul y\u271d\u00b9) z)\nx\u271d : A\ny\u271d : B\n\u22a2 \u2191(\u2191mul (\u2191(\u2191mul (x\u271d \u2297\u209c[R] y\u271d)) (x\u271d\u00b9 + y\u271d\u00b9))) z = \u2191(\u2191mul (x\u271d \u2297\u209c[R] y\u271d)) (\u2191(\u2191mul (x\u271d\u00b9 + y\u271d\u00b9)) z)", "state_after": "no goals"}, {"tactic": "intros", "state_before": "case refine_3\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z : A \u2297[R] B\n\u22a2 \u2200 (x y_1 : A \u2297[R] B),\n    \u2191(\u2191mul (\u2191(\u2191mul x) y)) z = \u2191(\u2191mul x) (\u2191(\u2191mul y) z) \u2192\n      \u2191(\u2191mul (\u2191(\u2191mul y_1) y)) z = \u2191(\u2191mul y_1) (\u2191(\u2191mul y) z) \u2192\n        \u2191(\u2191mul (\u2191(\u2191mul (x + y_1)) y)) z = \u2191(\u2191mul (x + y_1)) (\u2191(\u2191mul y) z)", "state_after": "case refine_3\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z x\u271d y\u271d : A \u2297[R] B\na\u271d\u00b9 : \u2191(\u2191mul (\u2191(\u2191mul x\u271d) y)) z = \u2191(\u2191mul x\u271d) (\u2191(\u2191mul y) z)\na\u271d : \u2191(\u2191mul (\u2191(\u2191mul y\u271d) y)) z = \u2191(\u2191mul y\u271d) (\u2191(\u2191mul y) z)\n\u22a2 \u2191(\u2191mul (\u2191(\u2191mul (x\u271d + y\u271d)) y)) z = \u2191(\u2191mul (x\u271d + y\u271d)) (\u2191(\u2191mul y) z)"}, {"tactic": "simp only [LinearMap.map_add, *, LinearMap.add_apply]", "state_before": "case refine_3\nR : Type u\ninst\u271d\u2074 : CommSemiring R\nA : Type v\u2081\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type v\u2082\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nmul : A \u2297[R] B \u2192\u2097[R] A \u2297[R] B \u2192\u2097[R] A \u2297[R] B\nh :\n  \u2200 (a\u2081 a\u2082 a\u2083 : A) (b\u2081 b\u2082 b\u2083 : B),\n    \u2191(\u2191mul (\u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (a\u2082 \u2297\u209c[R] b\u2082))) (a\u2083 \u2297\u209c[R] b\u2083) =\n      \u2191(\u2191mul (a\u2081 \u2297\u209c[R] b\u2081)) (\u2191(\u2191mul (a\u2082 \u2297\u209c[R] b\u2082)) (a\u2083 \u2297\u209c[R] b\u2083))\nx y z x\u271d y\u271d : A \u2297[R] B\na\u271d\u00b9 : \u2191(\u2191mul (\u2191(\u2191mul x\u271d) y)) z = \u2191(\u2191mul x\u271d) (\u2191(\u2191mul y) z)\na\u271d : \u2191(\u2191mul (\u2191(\u2191mul y\u271d) y)) z = \u2191(\u2191mul y\u271d) (\u2191(\u2191mul y) z)\n\u22a2 \u2191(\u2191mul (\u2191(\u2191mul (x\u271d + y\u271d)) y)) z = \u2191(\u2191mul (x\u271d + y\u271d)) (\u2191(\u2191mul y) z)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/NoncommPiCoprod.lean", "full_name": "MonoidHom.noncommPiCoprod_mulSingle", "start": [135, 1], "end": [147, 14], "traced_tactics": [{"tactic": "change Finset.univ.noncommProd (fun j => \u03d5 j (Pi.mulSingle i y j)) (fun _ _ _ _ h => hcomm h _ _)\n  = \u03d5 i y", "state_before": "M : Type u_1\ninst\u271d\u00b2 : Monoid M\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nN : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 Monoid (N i)\n\u03d5 : (i : \u03b9) \u2192 N i \u2192* M\nhcomm : Pairwise fun i j => \u2200 (x : N i) (y : N j), Commute (\u2191(\u03d5 i) x) (\u2191(\u03d5 j) y)\nf g : (i : \u03b9) \u2192 N i\ni : \u03b9\ny : N i\n\u22a2 \u2191(noncommPiCoprod \u03d5 hcomm) (Pi.mulSingle i y) = \u2191(\u03d5 i) y", "state_after": "M : Type u_1\ninst\u271d\u00b2 : Monoid M\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nN : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 Monoid (N i)\n\u03d5 : (i : \u03b9) \u2192 N i \u2192* M\nhcomm : Pairwise fun i j => \u2200 (x : N i) (y : N j), Commute (\u2191(\u03d5 i) x) (\u2191(\u03d5 j) y)\nf g : (i : \u03b9) \u2192 N i\ni : \u03b9\ny : N i\n\u22a2 Finset.noncommProd Finset.univ (fun j => \u2191(\u03d5 j) (Pi.mulSingle i y j))\n      (_ :\n        \u2200 (x : \u03b9),\n          x \u2208 \u2191Finset.univ \u2192\n            \u2200 (x_2 : \u03b9),\n              x_2 \u2208 \u2191Finset.univ \u2192 x \u2260 x_2 \u2192 Commute (\u2191(\u03d5 x) (Pi.mulSingle i y x)) (\u2191(\u03d5 x_2) (Pi.mulSingle i y x_2))) =\n    \u2191(\u03d5 i) y"}, {"tactic": "simp (config := { singlePass := true }) only [\u2190 Finset.insert_erase (Finset.mem_univ i)]", "state_before": "M : Type u_1\ninst\u271d\u00b2 : Monoid M\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nN : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 Monoid (N i)\n\u03d5 : (i : \u03b9) \u2192 N i \u2192* M\nhcomm : Pairwise fun i j => \u2200 (x : N i) (y : N j), Commute (\u2191(\u03d5 i) x) (\u2191(\u03d5 j) y)\nf g : (i : \u03b9) \u2192 N i\ni : \u03b9\ny : N i\n\u22a2 Finset.noncommProd Finset.univ (fun j => \u2191(\u03d5 j) (Pi.mulSingle i y j))\n      (_ :\n        \u2200 (x : \u03b9),\n          x \u2208 \u2191Finset.univ \u2192\n            \u2200 (x_2 : \u03b9),\n              x_2 \u2208 \u2191Finset.univ \u2192 x \u2260 x_2 \u2192 Commute (\u2191(\u03d5 x) (Pi.mulSingle i y x)) (\u2191(\u03d5 x_2) (Pi.mulSingle i y x_2))) =\n    \u2191(\u03d5 i) y", "state_after": "M : Type u_1\ninst\u271d\u00b2 : Monoid M\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nN : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 Monoid (N i)\n\u03d5 : (i : \u03b9) \u2192 N i \u2192* M\nhcomm : Pairwise fun i j => \u2200 (x : N i) (y : N j), Commute (\u2191(\u03d5 i) x) (\u2191(\u03d5 j) y)\nf g : (i : \u03b9) \u2192 N i\ni : \u03b9\ny : N i\n\u22a2 Finset.noncommProd (insert i (Finset.erase Finset.univ i)) (fun x => \u2191(\u03d5 x) (Pi.mulSingle i y x))\n      (_ :\n        \u2200 (x : \u03b9),\n          x \u2208 \u2191(insert i (Finset.erase Finset.univ i)) \u2192\n            \u2200 (y_1 : \u03b9),\n              y_1 \u2208 \u2191(insert i (Finset.erase Finset.univ i)) \u2192\n                x \u2260 y_1 \u2192 (fun a b => Commute (\u2191(\u03d5 a) (Pi.mulSingle i y a)) (\u2191(\u03d5 b) (Pi.mulSingle i y b))) x y_1) =\n    \u2191(\u03d5 i) y"}, {"tactic": "rw [Finset.noncommProd_insert_of_not_mem _ _ _ _ (Finset.not_mem_erase i _)]", "state_before": "M : Type u_1\ninst\u271d\u00b2 : Monoid M\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nN : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 Monoid (N i)\n\u03d5 : (i : \u03b9) \u2192 N i \u2192* M\nhcomm : Pairwise fun i j => \u2200 (x : N i) (y : N j), Commute (\u2191(\u03d5 i) x) (\u2191(\u03d5 j) y)\nf g : (i : \u03b9) \u2192 N i\ni : \u03b9\ny : N i\n\u22a2 Finset.noncommProd (insert i (Finset.erase Finset.univ i)) (fun x => \u2191(\u03d5 x) (Pi.mulSingle i y x))\n      (_ :\n        \u2200 (x : \u03b9),\n          x \u2208 \u2191(insert i (Finset.erase Finset.univ i)) \u2192\n            \u2200 (y_1 : \u03b9),\n              y_1 \u2208 \u2191(insert i (Finset.erase Finset.univ i)) \u2192\n                x \u2260 y_1 \u2192 (fun a b => Commute (\u2191(\u03d5 a) (Pi.mulSingle i y a)) (\u2191(\u03d5 b) (Pi.mulSingle i y b))) x y_1) =\n    \u2191(\u03d5 i) y", "state_after": "M : Type u_1\ninst\u271d\u00b2 : Monoid M\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nN : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 Monoid (N i)\n\u03d5 : (i : \u03b9) \u2192 N i \u2192* M\nhcomm : Pairwise fun i j => \u2200 (x : N i) (y : N j), Commute (\u2191(\u03d5 i) x) (\u2191(\u03d5 j) y)\nf g : (i : \u03b9) \u2192 N i\ni : \u03b9\ny : N i\n\u22a2 \u2191(\u03d5 i) (Pi.mulSingle i y i) *\n      Finset.noncommProd (Finset.erase Finset.univ i) (fun x => \u2191(\u03d5 x) (Pi.mulSingle i y x))\n        (_ :\n          Set.Pairwise \u2191(Finset.erase Finset.univ i) fun a b =>\n            Commute (\u2191(\u03d5 a) (Pi.mulSingle i y a)) (\u2191(\u03d5 b) (Pi.mulSingle i y b))) =\n    \u2191(\u03d5 i) y"}, {"tactic": "rw [Pi.mulSingle_eq_same]", "state_before": "M : Type u_1\ninst\u271d\u00b2 : Monoid M\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nN : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 Monoid (N i)\n\u03d5 : (i : \u03b9) \u2192 N i \u2192* M\nhcomm : Pairwise fun i j => \u2200 (x : N i) (y : N j), Commute (\u2191(\u03d5 i) x) (\u2191(\u03d5 j) y)\nf g : (i : \u03b9) \u2192 N i\ni : \u03b9\ny : N i\n\u22a2 \u2191(\u03d5 i) (Pi.mulSingle i y i) *\n      Finset.noncommProd (Finset.erase Finset.univ i) (fun x => \u2191(\u03d5 x) (Pi.mulSingle i y x))\n        (_ :\n          Set.Pairwise \u2191(Finset.erase Finset.univ i) fun a b =>\n            Commute (\u2191(\u03d5 a) (Pi.mulSingle i y a)) (\u2191(\u03d5 b) (Pi.mulSingle i y b))) =\n    \u2191(\u03d5 i) y", "state_after": "M : Type u_1\ninst\u271d\u00b2 : Monoid M\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nN : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 Monoid (N i)\n\u03d5 : (i : \u03b9) \u2192 N i \u2192* M\nhcomm : Pairwise fun i j => \u2200 (x : N i) (y : N j), Commute (\u2191(\u03d5 i) x) (\u2191(\u03d5 j) y)\nf g : (i : \u03b9) \u2192 N i\ni : \u03b9\ny : N i\n\u22a2 \u2191(\u03d5 i) y *\n      Finset.noncommProd (Finset.erase Finset.univ i) (fun x => \u2191(\u03d5 x) (Pi.mulSingle i y x))\n        (_ :\n          Set.Pairwise \u2191(Finset.erase Finset.univ i) fun a b =>\n            Commute (\u2191(\u03d5 a) (Pi.mulSingle i y a)) (\u2191(\u03d5 b) (Pi.mulSingle i y b))) =\n    \u2191(\u03d5 i) y"}, {"tactic": "rw [Finset.noncommProd_eq_pow_card]", "state_before": "M : Type u_1\ninst\u271d\u00b2 : Monoid M\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nN : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 Monoid (N i)\n\u03d5 : (i : \u03b9) \u2192 N i \u2192* M\nhcomm : Pairwise fun i j => \u2200 (x : N i) (y : N j), Commute (\u2191(\u03d5 i) x) (\u2191(\u03d5 j) y)\nf g : (i : \u03b9) \u2192 N i\ni : \u03b9\ny : N i\n\u22a2 \u2191(\u03d5 i) y *\n      Finset.noncommProd (Finset.erase Finset.univ i) (fun x => \u2191(\u03d5 x) (Pi.mulSingle i y x))\n        (_ :\n          Set.Pairwise \u2191(Finset.erase Finset.univ i) fun a b =>\n            Commute (\u2191(\u03d5 a) (Pi.mulSingle i y a)) (\u2191(\u03d5 b) (Pi.mulSingle i y b))) =\n    \u2191(\u03d5 i) y", "state_after": "M : Type u_1\ninst\u271d\u00b2 : Monoid M\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nN : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 Monoid (N i)\n\u03d5 : (i : \u03b9) \u2192 N i \u2192* M\nhcomm : Pairwise fun i j => \u2200 (x : N i) (y : N j), Commute (\u2191(\u03d5 i) x) (\u2191(\u03d5 j) y)\nf g : (i : \u03b9) \u2192 N i\ni : \u03b9\ny : N i\n\u22a2 \u2191(\u03d5 i) y * ?m ^ Finset.card (Finset.erase Finset.univ i) = \u2191(\u03d5 i) y\n\ncase m\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nN : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 Monoid (N i)\n\u03d5 : (i : \u03b9) \u2192 N i \u2192* M\nhcomm : Pairwise fun i j => \u2200 (x : N i) (y : N j), Commute (\u2191(\u03d5 i) x) (\u2191(\u03d5 j) y)\nf g : (i : \u03b9) \u2192 N i\ni : \u03b9\ny : N i\n\u22a2 M\n\ncase h\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nN : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 Monoid (N i)\n\u03d5 : (i : \u03b9) \u2192 N i \u2192* M\nhcomm : Pairwise fun i j => \u2200 (x : N i) (y : N j), Commute (\u2191(\u03d5 i) x) (\u2191(\u03d5 j) y)\nf g : (i : \u03b9) \u2192 N i\ni : \u03b9\ny : N i\n\u22a2 \u2200 (x : \u03b9), x \u2208 Finset.erase Finset.univ i \u2192 \u2191(\u03d5 x) (Pi.mulSingle i y x) = ?m"}, {"tactic": "rw [one_pow]", "state_before": "M : Type u_1\ninst\u271d\u00b2 : Monoid M\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nN : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 Monoid (N i)\n\u03d5 : (i : \u03b9) \u2192 N i \u2192* M\nhcomm : Pairwise fun i j => \u2200 (x : N i) (y : N j), Commute (\u2191(\u03d5 i) x) (\u2191(\u03d5 j) y)\nf g : (i : \u03b9) \u2192 N i\ni : \u03b9\ny : N i\n\u22a2 \u2191(\u03d5 i) y * ?m ^ Finset.card (Finset.erase Finset.univ i) = \u2191(\u03d5 i) y", "state_after": "M : Type u_1\ninst\u271d\u00b2 : Monoid M\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nN : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 Monoid (N i)\n\u03d5 : (i : \u03b9) \u2192 N i \u2192* M\nhcomm : Pairwise fun i j => \u2200 (x : N i) (y : N j), Commute (\u2191(\u03d5 i) x) (\u2191(\u03d5 j) y)\nf g : (i : \u03b9) \u2192 N i\ni : \u03b9\ny : N i\n\u22a2 \u2191(\u03d5 i) y * 1 = \u2191(\u03d5 i) y"}, {"tactic": "exact mul_one _", "state_before": "M : Type u_1\ninst\u271d\u00b2 : Monoid M\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nN : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 Monoid (N i)\n\u03d5 : (i : \u03b9) \u2192 N i \u2192* M\nhcomm : Pairwise fun i j => \u2200 (x : N i) (y : N j), Commute (\u2191(\u03d5 i) x) (\u2191(\u03d5 j) y)\nf g : (i : \u03b9) \u2192 N i\ni : \u03b9\ny : N i\n\u22a2 \u2191(\u03d5 i) y * 1 = \u2191(\u03d5 i) y", "state_after": "no goals"}, {"tactic": "intro j hj", "state_before": "case h\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nN : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 Monoid (N i)\n\u03d5 : (i : \u03b9) \u2192 N i \u2192* M\nhcomm : Pairwise fun i j => \u2200 (x : N i) (y : N j), Commute (\u2191(\u03d5 i) x) (\u2191(\u03d5 j) y)\nf g : (i : \u03b9) \u2192 N i\ni : \u03b9\ny : N i\n\u22a2 \u2200 (x : \u03b9), x \u2208 Finset.erase Finset.univ i \u2192 \u2191(\u03d5 x) (Pi.mulSingle i y x) = 1", "state_after": "case h\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nN : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 Monoid (N i)\n\u03d5 : (i : \u03b9) \u2192 N i \u2192* M\nhcomm : Pairwise fun i j => \u2200 (x : N i) (y : N j), Commute (\u2191(\u03d5 i) x) (\u2191(\u03d5 j) y)\nf g : (i : \u03b9) \u2192 N i\ni : \u03b9\ny : N i\nj : \u03b9\nhj : j \u2208 Finset.erase Finset.univ i\n\u22a2 \u2191(\u03d5 j) (Pi.mulSingle i y j) = 1"}, {"tactic": "simp only [Finset.mem_erase] at hj", "state_before": "case h\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nN : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 Monoid (N i)\n\u03d5 : (i : \u03b9) \u2192 N i \u2192* M\nhcomm : Pairwise fun i j => \u2200 (x : N i) (y : N j), Commute (\u2191(\u03d5 i) x) (\u2191(\u03d5 j) y)\nf g : (i : \u03b9) \u2192 N i\ni : \u03b9\ny : N i\nj : \u03b9\nhj : j \u2208 Finset.erase Finset.univ i\n\u22a2 \u2191(\u03d5 j) (Pi.mulSingle i y j) = 1", "state_after": "case h\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nN : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 Monoid (N i)\n\u03d5 : (i : \u03b9) \u2192 N i \u2192* M\nhcomm : Pairwise fun i j => \u2200 (x : N i) (y : N j), Commute (\u2191(\u03d5 i) x) (\u2191(\u03d5 j) y)\nf g : (i : \u03b9) \u2192 N i\ni : \u03b9\ny : N i\nj : \u03b9\nhj : j \u2260 i \u2227 j \u2208 Finset.univ\n\u22a2 \u2191(\u03d5 j) (Pi.mulSingle i y j) = 1"}, {"tactic": "simp [hj]", "state_before": "case h\nM : Type u_1\ninst\u271d\u00b2 : Monoid M\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nN : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 Monoid (N i)\n\u03d5 : (i : \u03b9) \u2192 N i \u2192* M\nhcomm : Pairwise fun i j => \u2200 (x : N i) (y : N j), Commute (\u2191(\u03d5 i) x) (\u2191(\u03d5 j) y)\nf g : (i : \u03b9) \u2192 N i\ni : \u03b9\ny : N i\nj : \u03b9\nhj : j \u2260 i \u2227 j \u2208 Finset.univ\n\u22a2 \u2191(\u03d5 j) (Pi.mulSingle i y j) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Logic.lean", "full_name": "Not.intro", "start": [14, 1], "end": [14, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Nat/Gcd.lean", "full_name": "Nat.coprime.gcd_right", "start": [343, 1], "end": [344, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "full_name": "Ordinal.lift_card", "start": [792, 1], "end": [793, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.AEStronglyMeasurable.ennnorm", "start": [1474, 11], "end": [1476, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "full_name": "ciSup_le_iff", "start": [512, 1], "end": [514, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Module/Star.lean", "full_name": "continuous_skewAdjointPart", "start": [58, 1], "end": [60, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/Hyperreal.lean", "full_name": "Hyperreal.infinitePos_mul_of_infiniteNeg_not_infinitesimal_neg", "start": [829, 1], "end": [832, 61], "traced_tactics": [{"tactic": "rw [\u2190 infinitePos_neg, \u2190 neg_pos, \u2190 neg_mul_neg, \u2190 infinitesimal_neg]", "state_before": "x y : \u211d*\n\u22a2 InfiniteNeg x \u2192 \u00acInfinitesimal y \u2192 y < 0 \u2192 InfinitePos (x * y)", "state_after": "x y : \u211d*\n\u22a2 InfinitePos (-x) \u2192 \u00acInfinitesimal (-y) \u2192 0 < -y \u2192 InfinitePos (-x * -y)"}, {"tactic": "exact infinitePos_mul_of_infinitePos_not_infinitesimal_pos", "state_before": "x y : \u211d*\n\u22a2 InfinitePos (-x) \u2192 \u00acInfinitesimal (-y) \u2192 0 < -y \u2192 InfinitePos (-x * -y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Ultrafilter.lean", "full_name": "Ultrafilter.finite_sUnion_mem_iff", "start": [202, 1], "end": [204, 55], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type ?u.12991\nf g : Ultrafilter \u03b1\ns\u271d t : Set \u03b1\np q : \u03b1 \u2192 Prop\ns : Set (Set \u03b1)\nhs : Set.Finite s\n\u22a2 \u22c3\u2080 \u2205 \u2208 f \u2194 \u2203 t, t \u2208 \u2205 \u2227 t \u2208 f", "state_after": "no goals"}, {"tactic": "simp [union_mem_iff, his, or_and_right, exists_or]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type ?u.12991\nf g : Ultrafilter \u03b1\ns\u271d\u00b9 t : Set \u03b1\np q : \u03b1 \u2192 Prop\ns : Set (Set \u03b1)\nhs : Set.Finite s\na\u271d : Set \u03b1\ns\u271d : Set (Set \u03b1)\nx\u271d\u00b9 : \u00aca\u271d \u2208 s\u271d\nx\u271d : Set.Finite s\u271d\nhis : \u22c3\u2080 s\u271d \u2208 f \u2194 \u2203 t, t \u2208 s\u271d \u2227 t \u2208 f\n\u22a2 \u22c3\u2080 insert a\u271d s\u271d \u2208 f \u2194 \u2203 t, t \u2208 insert a\u271d s\u271d \u2227 t \u2208 f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/DotProduct.lean", "full_name": "Matrix.dotProduct_eq", "start": [59, 1], "end": [61, 81], "traced_tactics": [{"tactic": "funext x", "state_before": "R : Type v\nn : Type w\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Fintype n\nv w : n \u2192 R\nh : \u2200 (u : n \u2192 R), v \u2b1d\u1d65 u = w \u2b1d\u1d65 u\n\u22a2 v = w", "state_after": "case h\nR : Type v\nn : Type w\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Fintype n\nv w : n \u2192 R\nh : \u2200 (u : n \u2192 R), v \u2b1d\u1d65 u = w \u2b1d\u1d65 u\nx : n\n\u22a2 v x = w x"}, {"tactic": "classical rw [\u2190 dotProduct_stdBasis_one v x, \u2190 dotProduct_stdBasis_one w x, h]", "state_before": "case h\nR : Type v\nn : Type w\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Fintype n\nv w : n \u2192 R\nh : \u2200 (u : n \u2192 R), v \u2b1d\u1d65 u = w \u2b1d\u1d65 u\nx : n\n\u22a2 v x = w x", "state_after": "no goals"}, {"tactic": "rw [\u2190 dotProduct_stdBasis_one v x, \u2190 dotProduct_stdBasis_one w x, h]", "state_before": "case h\nR : Type v\nn : Type w\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Fintype n\nv w : n \u2192 R\nh : \u2200 (u : n \u2192 R), v \u2b1d\u1d65 u = w \u2b1d\u1d65 u\nx : n\n\u22a2 v x = w x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/Hyperreal.lean", "full_name": "Hyperreal.infiniteNeg_add_not_infinite", "start": [577, 1], "end": [579, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Nonneg/Ring.lean", "full_name": "Nonneg.nsmul_mk", "start": [149, 1], "end": [151, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Hom/Centroid.lean", "full_name": "CentroidHom.toAddMonoidHom_id", "start": [182, 1], "end": [183, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/ModelTheory/Substructures.lean", "full_name": "FirstOrder.Language.Substructure.le_comap_of_map_le", "start": [482, 1], "end": [483, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Sheaves/SheafCondition/Sites.lean", "full_name": "TopCat.Presheaf.coveringOfPresieve.iSup_eq_of_mem_grothendieck", "start": [61, 1], "end": [70, 39], "traced_tactics": [{"tactic": "apply le_antisymm", "state_before": "X : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\n\u22a2 iSup (coveringOfPresieve U R) = U", "state_after": "case a\nX : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\n\u22a2 iSup (coveringOfPresieve U R) \u2264 U\n\ncase a\nX : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\n\u22a2 U \u2264 iSup (coveringOfPresieve U R)"}, {"tactic": "intro x hxU", "state_before": "case a\nX : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\n\u22a2 U \u2264 iSup (coveringOfPresieve U R)", "state_after": "case a\nX : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\nx : \u2191X\nhxU : x \u2208 \u2191U\n\u22a2 x \u2208 \u2191(iSup (coveringOfPresieve U R))"}, {"tactic": "rw [Opens.coe_iSup, Set.mem_iUnion]", "state_before": "case a\nX : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\nx : \u2191X\nhxU : x \u2208 \u2191U\n\u22a2 x \u2208 \u2191(iSup (coveringOfPresieve U R))", "state_after": "case a\nX : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\nx : \u2191X\nhxU : x \u2208 \u2191U\n\u22a2 \u2203 i, x \u2208 \u2191(coveringOfPresieve U R i)"}, {"tactic": "obtain \u27e8V, iVU, \u27e8W, iVW, iWU, hiWU, -\u27e9, hxV\u27e9 := hR x hxU", "state_before": "case a\nX : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\nx : \u2191X\nhxU : x \u2208 \u2191U\n\u22a2 \u2203 i, x \u2208 \u2191(coveringOfPresieve U R i)", "state_after": "case a.intro.intro.intro.intro.intro.intro.intro\nX : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\nx : \u2191X\nhxU : x \u2208 \u2191U\nV : Opens \u2191X\niVU : V \u27f6 U\nhxV : x \u2208 V\nW : Opens \u2191X\niVW : V \u27f6 W\niWU : W \u27f6 U\nhiWU : R iWU\n\u22a2 \u2203 i, x \u2208 \u2191(coveringOfPresieve U R i)"}, {"tactic": "exact \u27e8\u27e8W, \u27e8iWU, hiWU\u27e9\u27e9, iVW.le hxV\u27e9", "state_before": "case a.intro.intro.intro.intro.intro.intro.intro\nX : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\nx : \u2191X\nhxU : x \u2208 \u2191U\nV : Opens \u2191X\niVU : V \u27f6 U\nhxV : x \u2208 V\nW : Opens \u2191X\niVW : V \u27f6 W\niWU : W \u27f6 U\nhiWU : R iWU\n\u22a2 \u2203 i, x \u2208 \u2191(coveringOfPresieve U R i)", "state_after": "no goals"}, {"tactic": "refine' iSup_le _", "state_before": "case a\nX : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\n\u22a2 iSup (coveringOfPresieve U R) \u2264 U", "state_after": "case a\nX : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\n\u22a2 \u2200 (i : (V : Opens \u2191X) \u00d7 { f // R f }), coveringOfPresieve U R i \u2264 U"}, {"tactic": "intro f", "state_before": "case a\nX : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\n\u22a2 \u2200 (i : (V : Opens \u2191X) \u00d7 { f // R f }), coveringOfPresieve U R i \u2264 U", "state_after": "case a\nX : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\nf : (V : Opens \u2191X) \u00d7 { f // R f }\n\u22a2 coveringOfPresieve U R f \u2264 U"}, {"tactic": "exact f.2.1.le", "state_before": "case a\nX : TopCat\nU : Opens \u2191X\nR : Presieve U\nhR : Sieve.generate R \u2208 GrothendieckTopology.sieves (Opens.grothendieckTopology \u2191X) U\nf : (V : Opens \u2191X) \u00d7 { f // R f }\n\u22a2 coveringOfPresieve U R f \u2264 U", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/DoubleCounting.lean", "full_name": "Finset.card_le_card_of_forall_subsingleton", "start": [114, 1], "end": [124, 23], "traced_tactics": [{"tactic": "classical\n  rw [\u2190 mul_one s.card, \u2190 mul_one t.card]\n  exact card_mul_le_card_mul r\n    (fun a h \u21a6 card_pos.2 (by\n      rw [\u2190 coe_nonempty, coe_bipartiteAbove]\n      exact hs _ h : (t.bipartiteAbove r a).Nonempty))\n    (fun b h \u21a6 card_le_one.2 (by\n      simp_rw [mem_bipartiteBelow]\n      exact ht _ h))", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Finset \u03b1\nt : Finset \u03b2\na a' : \u03b1\nb b' : \u03b2\ninst\u271d\u00b9 : DecidablePred (r a)\ninst\u271d : (a : \u03b1) \u2192 Decidable (r a b)\nm n : \u2115\nhs : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\nht : \u2200 (b : \u03b2), b \u2208 t \u2192 Set.Subsingleton {a | a \u2208 s \u2227 r a b}\n\u22a2 card s \u2264 card t", "state_after": "no goals"}, {"tactic": "rw [\u2190 mul_one s.card, \u2190 mul_one t.card]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Finset \u03b1\nt : Finset \u03b2\na a' : \u03b1\nb b' : \u03b2\ninst\u271d\u00b9 : DecidablePred (r a)\ninst\u271d : (a : \u03b1) \u2192 Decidable (r a b)\nm n : \u2115\nhs : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\nht : \u2200 (b : \u03b2), b \u2208 t \u2192 Set.Subsingleton {a | a \u2208 s \u2227 r a b}\n\u22a2 card s \u2264 card t", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Finset \u03b1\nt : Finset \u03b2\na a' : \u03b1\nb b' : \u03b2\ninst\u271d\u00b9 : DecidablePred (r a)\ninst\u271d : (a : \u03b1) \u2192 Decidable (r a b)\nm n : \u2115\nhs : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\nht : \u2200 (b : \u03b2), b \u2208 t \u2192 Set.Subsingleton {a | a \u2208 s \u2227 r a b}\n\u22a2 card s * 1 \u2264 card t * 1"}, {"tactic": "exact card_mul_le_card_mul r\n  (fun a h \u21a6 card_pos.2 (by\n    rw [\u2190 coe_nonempty, coe_bipartiteAbove]\n    exact hs _ h : (t.bipartiteAbove r a).Nonempty))\n  (fun b h \u21a6 card_le_one.2 (by\n    simp_rw [mem_bipartiteBelow]\n    exact ht _ h))", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Finset \u03b1\nt : Finset \u03b2\na a' : \u03b1\nb b' : \u03b2\ninst\u271d\u00b9 : DecidablePred (r a)\ninst\u271d : (a : \u03b1) \u2192 Decidable (r a b)\nm n : \u2115\nhs : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\nht : \u2200 (b : \u03b2), b \u2208 t \u2192 Set.Subsingleton {a | a \u2208 s \u2227 r a b}\n\u22a2 card s * 1 \u2264 card t * 1", "state_after": "no goals"}, {"tactic": "rw [\u2190 coe_nonempty, coe_bipartiteAbove]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Finset \u03b1\nt : Finset \u03b2\na\u271d a' : \u03b1\nb b' : \u03b2\ninst\u271d\u00b9 : DecidablePred (r a\u271d)\ninst\u271d : (a : \u03b1) \u2192 Decidable (r a b)\nm n : \u2115\nhs : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\nht : \u2200 (b : \u03b2), b \u2208 t \u2192 Set.Subsingleton {a | a \u2208 s \u2227 r a b}\na : \u03b1\nh : a \u2208 s\n\u22a2 Finset.Nonempty (bipartiteAbove r t a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Finset \u03b1\nt : Finset \u03b2\na\u271d a' : \u03b1\nb b' : \u03b2\ninst\u271d\u00b9 : DecidablePred (r a\u271d)\ninst\u271d : (a : \u03b1) \u2192 Decidable (r a b)\nm n : \u2115\nhs : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\nht : \u2200 (b : \u03b2), b \u2208 t \u2192 Set.Subsingleton {a | a \u2208 s \u2227 r a b}\na : \u03b1\nh : a \u2208 s\n\u22a2 Set.Nonempty {b | b \u2208 t \u2227 r a b}"}, {"tactic": "exact hs _ h", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Finset \u03b1\nt : Finset \u03b2\na\u271d a' : \u03b1\nb b' : \u03b2\ninst\u271d\u00b9 : DecidablePred (r a\u271d)\ninst\u271d : (a : \u03b1) \u2192 Decidable (r a b)\nm n : \u2115\nhs : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\nht : \u2200 (b : \u03b2), b \u2208 t \u2192 Set.Subsingleton {a | a \u2208 s \u2227 r a b}\na : \u03b1\nh : a \u2208 s\n\u22a2 Set.Nonempty {b | b \u2208 t \u2227 r a b}", "state_after": "no goals"}, {"tactic": "simp_rw [mem_bipartiteBelow]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Finset \u03b1\nt : Finset \u03b2\na a' : \u03b1\nb\u271d b' : \u03b2\ninst\u271d\u00b9 : DecidablePred (r a)\ninst\u271d : (a : \u03b1) \u2192 Decidable (r a b\u271d)\nm n : \u2115\nhs : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\nht : \u2200 (b : \u03b2), b \u2208 t \u2192 Set.Subsingleton {a | a \u2208 s \u2227 r a b}\nb : \u03b2\nh : b \u2208 t\n\u22a2 \u2200 (a : \u03b1), a \u2208 bipartiteBelow r s b \u2192 \u2200 (b_1 : \u03b1), b_1 \u2208 bipartiteBelow r s b \u2192 a = b_1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Finset \u03b1\nt : Finset \u03b2\na a' : \u03b1\nb\u271d b' : \u03b2\ninst\u271d\u00b9 : DecidablePred (r a)\ninst\u271d : (a : \u03b1) \u2192 Decidable (r a b\u271d)\nm n : \u2115\nhs : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\nht : \u2200 (b : \u03b2), b \u2208 t \u2192 Set.Subsingleton {a | a \u2208 s \u2227 r a b}\nb : \u03b2\nh : b \u2208 t\n\u22a2 \u2200 (a : \u03b1), a \u2208 s \u2227 r a b \u2192 \u2200 (b_1 : \u03b1), b_1 \u2208 s \u2227 r b_1 b \u2192 a = b_1"}, {"tactic": "exact ht _ h", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Finset \u03b1\nt : Finset \u03b2\na a' : \u03b1\nb\u271d b' : \u03b2\ninst\u271d\u00b9 : DecidablePred (r a)\ninst\u271d : (a : \u03b1) \u2192 Decidable (r a b\u271d)\nm n : \u2115\nhs : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\nht : \u2200 (b : \u03b2), b \u2208 t \u2192 Set.Subsingleton {a | a \u2208 s \u2227 r a b}\nb : \u03b2\nh : b \u2208 t\n\u22a2 \u2200 (a : \u03b1), a \u2208 s \u2227 r a b \u2192 \u2200 (b_1 : \u03b1), b_1 \u2208 s \u2227 r b_1 b \u2192 a = b_1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Dynamics/OmegaLimit.lean", "full_name": "omegaLimit_iInter", "start": [173, 1], "end": [174, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Interval.lean", "full_name": "Nat.Ico_image_const_sub_eq_Ico", "start": [222, 1], "end": [247, 78], "traced_tactics": [{"tactic": "ext x", "state_before": "a b c : \u2115\nhac : a \u2264 c\n\u22a2 image (fun x => c - x) (Ico a b) = Ico (c + 1 - b) (c + 1 - a)", "state_after": "case a\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\n\u22a2 x \u2208 image (fun x => c - x) (Ico a b) \u2194 x \u2208 Ico (c + 1 - b) (c + 1 - a)"}, {"tactic": "rw [mem_image, mem_Ico]", "state_before": "case a\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\n\u22a2 x \u2208 image (fun x => c - x) (Ico a b) \u2194 x \u2208 Ico (c + 1 - b) (c + 1 - a)", "state_after": "case a\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\n\u22a2 (\u2203 a_1, a_1 \u2208 Ico a b \u2227 c - a_1 = x) \u2194 c + 1 - b \u2264 x \u2227 x < c + 1 - a"}, {"tactic": "constructor", "state_before": "case a\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\n\u22a2 (\u2203 a_1, a_1 \u2208 Ico a b \u2227 c - a_1 = x) \u2194 c + 1 - b \u2264 x \u2227 x < c + 1 - a", "state_after": "case a.mp\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\n\u22a2 (\u2203 a_1, a_1 \u2208 Ico a b \u2227 c - a_1 = x) \u2192 c + 1 - b \u2264 x \u2227 x < c + 1 - a\n\ncase a.mpr\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\n\u22a2 c + 1 - b \u2264 x \u2227 x < c + 1 - a \u2192 \u2203 a_2, a_2 \u2208 Ico a b \u2227 c - a_2 = x"}, {"tactic": "rintro \u27e8x, hx, rfl\u27e9", "state_before": "case a.mp\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\n\u22a2 (\u2203 a_1, a_1 \u2208 Ico a b \u2227 c - a_1 = x) \u2192 c + 1 - b \u2264 x \u2227 x < c + 1 - a", "state_after": "case a.mp.intro.intro\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\nhx : x \u2208 Ico a b\n\u22a2 c + 1 - b \u2264 c - x \u2227 c - x < c + 1 - a"}, {"tactic": "rw [mem_Ico] at hx", "state_before": "case a.mp.intro.intro\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\nhx : x \u2208 Ico a b\n\u22a2 c + 1 - b \u2264 c - x \u2227 c - x < c + 1 - a", "state_after": "case a.mp.intro.intro\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\nhx : a \u2264 x \u2227 x < b\n\u22a2 c + 1 - b \u2264 c - x \u2227 c - x < c + 1 - a"}, {"tactic": "refine'\n  \u27e8_,\n    ((tsub_le_tsub_iff_left hac).2 hx.1).trans_lt\n      ((tsub_lt_tsub_iff_right hac).2 (Nat.lt_succ_self _))\u27e9", "state_before": "case a.mp.intro.intro\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\nhx : a \u2264 x \u2227 x < b\n\u22a2 c + 1 - b \u2264 c - x \u2227 c - x < c + 1 - a", "state_after": "case a.mp.intro.intro\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\nhx : a \u2264 x \u2227 x < b\n\u22a2 c + 1 - b \u2264 c - x"}, {"tactic": "cases lt_or_le c b with\n| inl h =>\n  rw [tsub_eq_zero_iff_le.mpr (succ_le_of_lt h)]\n  exact zero_le _\n| inr h =>\n  rw [\u2190 succ_sub_succ c]\n  exact (tsub_le_tsub_iff_left (succ_le_succ <| hx.2.le.trans h)).2 hx.2", "state_before": "case a.mp.intro.intro\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\nhx : a \u2264 x \u2227 x < b\n\u22a2 c + 1 - b \u2264 c - x", "state_after": "no goals"}, {"tactic": "rw [tsub_eq_zero_iff_le.mpr (succ_le_of_lt h)]", "state_before": "case a.mp.intro.intro.inl\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\nhx : a \u2264 x \u2227 x < b\nh : c < b\n\u22a2 c + 1 - b \u2264 c - x", "state_after": "case a.mp.intro.intro.inl\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\nhx : a \u2264 x \u2227 x < b\nh : c < b\n\u22a2 0 \u2264 c - x"}, {"tactic": "exact zero_le _", "state_before": "case a.mp.intro.intro.inl\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\nhx : a \u2264 x \u2227 x < b\nh : c < b\n\u22a2 0 \u2264 c - x", "state_after": "no goals"}, {"tactic": "rw [\u2190 succ_sub_succ c]", "state_before": "case a.mp.intro.intro.inr\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\nhx : a \u2264 x \u2227 x < b\nh : b \u2264 c\n\u22a2 c + 1 - b \u2264 c - x", "state_after": "case a.mp.intro.intro.inr\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\nhx : a \u2264 x \u2227 x < b\nh : b \u2264 c\n\u22a2 c + 1 - b \u2264 succ c - succ x"}, {"tactic": "exact (tsub_le_tsub_iff_left (succ_le_succ <| hx.2.le.trans h)).2 hx.2", "state_before": "case a.mp.intro.intro.inr\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\nhx : a \u2264 x \u2227 x < b\nh : b \u2264 c\n\u22a2 c + 1 - b \u2264 succ c - succ x", "state_after": "no goals"}, {"tactic": "rintro \u27e8hb, ha\u27e9", "state_before": "case a.mpr\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\n\u22a2 c + 1 - b \u2264 x \u2227 x < c + 1 - a \u2192 \u2203 a_2, a_2 \u2208 Ico a b \u2227 c - a_2 = x", "state_after": "case a.mpr.intro\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\nhb : c + 1 - b \u2264 x\nha : x < c + 1 - a\n\u22a2 \u2203 a_1, a_1 \u2208 Ico a b \u2227 c - a_1 = x"}, {"tactic": "rw [lt_tsub_iff_left, lt_succ_iff] at ha", "state_before": "case a.mpr.intro\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\nhb : c + 1 - b \u2264 x\nha : x < c + 1 - a\n\u22a2 \u2203 a_1, a_1 \u2208 Ico a b \u2227 c - a_1 = x", "state_after": "case a.mpr.intro\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\nhb : c + 1 - b \u2264 x\nha\u271d : x < c + 1 - a\nha : a + x \u2264 c\n\u22a2 \u2203 a_1, a_1 \u2208 Ico a b \u2227 c - a_1 = x"}, {"tactic": "have hx : x \u2264 c := (Nat.le_add_left _ _).trans ha", "state_before": "case a.mpr.intro\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\nhb : c + 1 - b \u2264 x\nha\u271d : x < c + 1 - a\nha : a + x \u2264 c\n\u22a2 \u2203 a_1, a_1 \u2208 Ico a b \u2227 c - a_1 = x", "state_after": "case a.mpr.intro\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\nhb : c + 1 - b \u2264 x\nha\u271d : x < c + 1 - a\nha : a + x \u2264 c\nhx : x \u2264 c\n\u22a2 \u2203 a_1, a_1 \u2208 Ico a b \u2227 c - a_1 = x"}, {"tactic": "refine' \u27e8c - x, _, tsub_tsub_cancel_of_le hx\u27e9", "state_before": "case a.mpr.intro\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\nhb : c + 1 - b \u2264 x\nha\u271d : x < c + 1 - a\nha : a + x \u2264 c\nhx : x \u2264 c\n\u22a2 \u2203 a_1, a_1 \u2208 Ico a b \u2227 c - a_1 = x", "state_after": "case a.mpr.intro\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\nhb : c + 1 - b \u2264 x\nha\u271d : x < c + 1 - a\nha : a + x \u2264 c\nhx : x \u2264 c\n\u22a2 c - x \u2208 Ico a b"}, {"tactic": "rw [mem_Ico]", "state_before": "case a.mpr.intro\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\nhb : c + 1 - b \u2264 x\nha\u271d : x < c + 1 - a\nha : a + x \u2264 c\nhx : x \u2264 c\n\u22a2 c - x \u2208 Ico a b", "state_after": "case a.mpr.intro\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\nhb : c + 1 - b \u2264 x\nha\u271d : x < c + 1 - a\nha : a + x \u2264 c\nhx : x \u2264 c\n\u22a2 a \u2264 c - x \u2227 c - x < b"}, {"tactic": "exact\n  \u27e8le_tsub_of_add_le_right ha,\n    (tsub_lt_iff_left hx).2 <| succ_le_iff.1 <| tsub_le_iff_right.1 hb\u27e9", "state_before": "case a.mpr.intro\na b c : \u2115\nhac : a \u2264 c\nx : \u2115\nhb : c + 1 - b \u2264 x\nha\u271d : x < c + 1 - a\nha : a + x \u2264 c\nhx : x \u2264 c\n\u22a2 a \u2264 c - x \u2227 c - x < b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/List/Lemmas.lean", "full_name": "List.Sublist.filterMap", "start": [1226, 1], "end": [1227, 59], "traced_tactics": [{"tactic": "induction s <;> simp <;> split <;> simp [*, cons, cons\u2082]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u2081 l\u2082 : List \u03b1\nf : \u03b1 \u2192 Option \u03b2\ns : l\u2081 <+ l\u2082\n\u22a2 List.filterMap f l\u2081 <+ List.filterMap f l\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "full_name": "tsum_subtype_add_tsum_subtype_compl", "start": [1232, 1], "end": [1234, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "full_name": "piChartedSpace_chartAt", "start": [751, 1], "end": [755, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Data/Nat/Basic.lean", "full_name": "Nat.succ_sub_succ", "start": [247, 1], "end": [248, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Hom/Open.lean", "full_name": "ContinuousOpenMap.comp_assoc", "start": [142, 1], "end": [144, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "full_name": "SimpleGraph.Walk.reverse_nil", "start": [321, 1], "end": [321, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Ultrafilter.lean", "full_name": "Filter.bot_ne_hyperfilter", "start": [487, 1], "end": [488, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.sublist_replicate_iff", "start": [1087, 1], "end": [1092, 72], "traced_tactics": [{"tactic": "rintro \u27e8k, h, rfl\u27e9", "state_before": "\u03b9 : Type ?u.72813\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 l : List \u03b1\na : \u03b1\nn : \u2115\n\u22a2 (\u2203 k, k \u2264 n \u2227 l = replicate k a) \u2192 l <+ replicate n a", "state_after": "case intro.intro\n\u03b9 : Type ?u.72813\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\na : \u03b1\nn k : \u2115\nh : k \u2264 n\n\u22a2 replicate k a <+ replicate n a"}, {"tactic": "exact (replicate_sublist_replicate _).mpr h", "state_before": "case intro.intro\n\u03b9 : Type ?u.72813\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\nl\u2081 l\u2082 : List \u03b1\na : \u03b1\nn k : \u2115\nh : k \u2264 n\n\u22a2 replicate k a <+ replicate n a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Functor/LeftDerived.lean", "full_name": "CategoryTheory.NatTrans.leftDerived_eq", "start": [156, 1], "end": [173, 29], "traced_tactics": [{"tactic": "symm", "state_before": "C : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (leftDerived \u03b1 n).app X =\n    (Functor.leftDerivedObjIso F n P).hom \u226b\n      (homologyFunctor D (ComplexShape.down \u2115) n).map ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) \u226b\n        (Functor.leftDerivedObjIso G n P).inv", "state_after": "C : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (Functor.leftDerivedObjIso F n P).hom \u226b\n      (homologyFunctor D (ComplexShape.down \u2115) n).map ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) \u226b\n        (Functor.leftDerivedObjIso G n P).inv =\n    (leftDerived \u03b1 n).app X"}, {"tactic": "dsimp [NatTrans.leftDerived, Functor.leftDerivedObjIso]", "state_before": "C : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (Functor.leftDerivedObjIso F n P).hom \u226b\n      (homologyFunctor D (ComplexShape.down \u2115) n).map ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) \u226b\n        (Functor.leftDerivedObjIso G n P).inv =\n    (leftDerived \u03b1 n).app X", "state_after": "C : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 ((HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom)) \u226b\n        \ud835\udfd9\n          ((HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).obj\n            ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).obj\n              ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex)))) \u226b\n      homology.map\n          (_ :\n            HomologicalComplex.dTo ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n \u226b\n                HomologicalComplex.dFrom ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n =\n              0)\n          (_ :\n            HomologicalComplex.dTo ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n \u226b\n                HomologicalComplex.dFrom ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n =\n              0)\n          (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n)\n          (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n)\n          (_ :\n            (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n).right =\n              (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n).right) \u226b\n        \ud835\udfd9 (HomologicalComplex.homology ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n) \u226b\n          (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n            ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n              ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n                (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as))"}, {"tactic": "simp only [Category.comp_id, Category.id_comp]", "state_before": "C : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 ((HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom)) \u226b\n        \ud835\udfd9\n          ((HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).obj\n            ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).obj\n              ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex)))) \u226b\n      homology.map\n          (_ :\n            HomologicalComplex.dTo ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n \u226b\n                HomologicalComplex.dFrom ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n =\n              0)\n          (_ :\n            HomologicalComplex.dTo ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n \u226b\n                HomologicalComplex.dFrom ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n =\n              0)\n          (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n)\n          (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n)\n          (_ :\n            (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n).right =\n              (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n).right) \u226b\n        \ud835\udfd9 (HomologicalComplex.homology ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n) \u226b\n          (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n            ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n              ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n                (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as))", "state_after": "C : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n        ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n          ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n            (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom)) \u226b\n      homology.map\n          (_ :\n            HomologicalComplex.dTo ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n \u226b\n                HomologicalComplex.dFrom ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n =\n              0)\n          (_ :\n            HomologicalComplex.dTo ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n \u226b\n                HomologicalComplex.dFrom ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n =\n              0)\n          (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n)\n          (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n)\n          (_ :\n            (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n).right =\n              (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n).right) \u226b\n        (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as))"}, {"tactic": "rw [\u2190 homologyFunctor_map, HomotopyCategory.homologyFunctor_map_factors]", "state_before": "C : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n        ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n          ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n            (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom)) \u226b\n      homology.map\n          (_ :\n            HomologicalComplex.dTo ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n \u226b\n                HomologicalComplex.dFrom ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n =\n              0)\n          (_ :\n            HomologicalComplex.dTo ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n \u226b\n                HomologicalComplex.dFrom ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n =\n              0)\n          (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n)\n          (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n)\n          (_ :\n            (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n).right =\n              (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n).right) \u226b\n        (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as))", "state_after": "C : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n        ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n          ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n            (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom)) \u226b\n      (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex)) \u226b\n        (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as))"}, {"tactic": "simp only [\u2190 Functor.map_comp]", "state_before": "C : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n        ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n          ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n            (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom)) \u226b\n      (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex)) \u226b\n        (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as))", "state_after": "C : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n            (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n          (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex \u226b\n            (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as))"}, {"tactic": "congr 1", "state_before": "C : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n            (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n          (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex \u226b\n            (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as))", "state_after": "case e_a\nC : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n      ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n          (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n        (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex \u226b\n          (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n            (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv) =\n    (HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n      ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as)"}, {"tactic": "apply HomotopyCategory.eq_of_homotopy", "state_before": "case e_a\nC : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n      ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n          (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n        (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex \u226b\n          (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n            (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv) =\n    (HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n      ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as)", "state_after": "case e_a.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 Homotopy\n    ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n        (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n      (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex \u226b\n        (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n          (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)\n    ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as)"}, {"tactic": "simp only [NatTrans.mapHomologicalComplex_naturality_assoc, \u2190 Functor.map_comp]", "state_before": "case e_a.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 Homotopy\n    ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n        (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n      (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex \u226b\n        (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n          (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)\n    ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as)", "state_after": "case e_a.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 Homotopy\n    ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as \u226b\n      (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n        ((ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n          (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv))\n    ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as)"}, {"tactic": "apply Homotopy.compLeftId", "state_before": "case e_a.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 Homotopy\n    ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as \u226b\n      (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n        ((ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n          (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv))\n    ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as)", "state_after": "case e_a.h.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 Homotopy\n    ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n      ((ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n        (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv))\n    (\ud835\udfd9 ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj ((projectiveResolutions C).obj X).as))"}, {"tactic": "refine' (Functor.mapHomotopy _ (HomotopyEquiv.homotopyHomInvId _) ).trans _", "state_before": "case e_a.h.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 Homotopy\n    ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n      ((ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n        (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv))\n    (\ud835\udfd9 ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj ((projectiveResolutions C).obj X).as))", "state_after": "case e_a.h.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 Homotopy ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map (\ud835\udfd9 ((projectiveResolutions C).obj X).as))\n    (\ud835\udfd9 ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj ((projectiveResolutions C).obj X).as))"}, {"tactic": "apply Homotopy.ofEq", "state_before": "case e_a.h.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 Homotopy ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map (\ud835\udfd9 ((projectiveResolutions C).obj X).as))\n    (\ud835\udfd9 ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj ((projectiveResolutions C).obj X).as))", "state_after": "case e_a.h.h.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map (\ud835\udfd9 ((projectiveResolutions C).obj X).as) =\n    \ud835\udfd9 ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj ((projectiveResolutions C).obj X).as)"}, {"tactic": "simp only [Functor.map_id]", "state_before": "case e_a.h.h.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category C\nD : Type u_2\ninst\u271d\u00b9\u00b2 : Category D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : 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"https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Type.lean", "full_name": "Equiv.Perm.cycleType_def", "start": [54, 1], "end": [56, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "full_name": "ciInf_le'", "start": [1068, 1], "end": [1069, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "dense_closure", "start": [621, 1], "end": [622, 37], "traced_tactics": [{"tactic": "rw [Dense, Dense, closure_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 Dense (closure s) \u2194 Dense s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/Basic.lean", "full_name": "Ideal.mem_inf", "start": [429, 1], "end": [430, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/ModEq.lean", "full_name": "AddCommGroup.ModEq.add_right_cancel'", "start": [250, 11], "end": [251, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.setToL1_nonneg", "start": [1187, 1], "end": [1200, 83], "traced_tactics": [{"tactic": "suffices : \u2200 f : { g : \u03b1 \u2192\u2081[\u03bc] G' // 0 \u2264 g }, 0 \u2264 setToL1 hT f", "state_before": "\u03b1 : Type u_1\nE : Type ?u.1222330\nF : Type ?u.1222333\nF' : Type ?u.1222336\nG : Type ?u.1222339\n\ud835\udd5c : Type ?u.1222342\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G'), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 Lp G' 1 }\nhf : 0 \u2264 f\n\u22a2 0 \u2264 \u2191(setToL1 hT) f", "state_after": "\u03b1 : Type u_1\nE : Type ?u.1222330\nF : Type ?u.1222333\nF' : Type ?u.1222336\nG : Type ?u.1222339\n\ud835\udd5c : Type ?u.1222342\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G'), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 Lp G' 1 }\nhf : 0 \u2264 f\nthis : \u2200 (f : { g // 0 \u2264 g }), 0 \u2264 \u2191(setToL1 hT) \u2191f\n\u22a2 0 \u2264 \u2191(setToL1 hT) f\n\ncase this\n\u03b1 : Type u_1\nE : Type ?u.1222330\nF : Type ?u.1222333\nF' : Type ?u.1222336\nG : Type ?u.1222339\n\ud835\udd5c : Type ?u.1222342\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G'), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 Lp G' 1 }\nhf : 0 \u2264 f\n\u22a2 \u2200 (f : { g // 0 \u2264 g }), 0 \u2264 \u2191(setToL1 hT) \u2191f"}, {"tactic": "exact this (\u27e8f, hf\u27e9 : { g : \u03b1 \u2192\u2081[\u03bc] G' // 0 \u2264 g })", "state_before": "\u03b1 : Type u_1\nE : Type ?u.1222330\nF : Type ?u.1222333\nF' : Type ?u.1222336\nG : Type ?u.1222339\n\ud835\udd5c : Type ?u.1222342\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G'), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 Lp G' 1 }\nhf : 0 \u2264 f\nthis : \u2200 (f : { g // 0 \u2264 g }), 0 \u2264 \u2191(setToL1 hT) \u2191f\n\u22a2 0 \u2264 \u2191(setToL1 hT) f\n\ncase this\n\u03b1 : Type u_1\nE : Type ?u.1222330\nF : Type ?u.1222333\nF' : Type ?u.1222336\nG : Type ?u.1222339\n\ud835\udd5c : Type ?u.1222342\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G'), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 Lp G' 1 }\nhf : 0 \u2264 f\n\u22a2 \u2200 (f : { g // 0 \u2264 g }), 0 \u2264 \u2191(setToL1 hT) \u2191f", "state_after": "case this\n\u03b1 : Type u_1\nE : Type ?u.1222330\nF : Type ?u.1222333\nF' : Type ?u.1222336\nG : Type ?u.1222339\n\ud835\udd5c : Type ?u.1222342\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G'), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 Lp G' 1 }\nhf : 0 \u2264 f\n\u22a2 \u2200 (f : { g // 0 \u2264 g }), 0 \u2264 \u2191(setToL1 hT) \u2191f"}, {"tactic": "refine' fun g =>\n  @isClosed_property { g : \u03b1 \u2192\u2081\u209b[\u03bc] G' // 0 \u2264 g } { g : \u03b1 \u2192\u2081[\u03bc] G' // 0 \u2264 g } _ _\n    (fun g => 0 \u2264 setToL1 hT g)\n    (denseRange_coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' one_ne_top) _ _ g", "state_before": "case this\n\u03b1 : Type u_1\nE : Type ?u.1222330\nF : Type ?u.1222333\nF' : Type ?u.1222336\nG : Type ?u.1222339\n\ud835\udd5c : Type ?u.1222342\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G'), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 Lp G' 1 }\nhf : 0 \u2264 f\n\u22a2 \u2200 (f : { g // 0 \u2264 g }), 0 \u2264 \u2191(setToL1 hT) \u2191f", "state_after": "case this.refine'_1\n\u03b1 : Type u_1\nE : Type ?u.1222330\nF : Type ?u.1222333\nF' : Type ?u.1222336\nG : Type ?u.1222339\n\ud835\udd5c : Type ?u.1222342\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G'), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 Lp G' 1 }\nhf : 0 \u2264 f\ng : { g // 0 \u2264 g }\n\u22a2 IsClosed {x | (fun g => 0 \u2264 \u2191(setToL1 hT) \u2191g) x}\n\ncase this.refine'_2\n\u03b1 : Type u_1\nE : Type ?u.1222330\nF : Type ?u.1222333\nF' : Type ?u.1222336\nG : Type ?u.1222339\n\ud835\udd5c : Type ?u.1222342\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G'), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 Lp G' 1 }\nhf : 0 \u2264 f\ng : { g // 0 \u2264 g }\n\u22a2 \u2200 (a : { g // 0 \u2264 g }), (fun g => 0 \u2264 \u2191(setToL1 hT) \u2191g) (coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' a)"}, {"tactic": "exact isClosed_le continuous_zero ((setToL1 hT).continuous.comp continuous_induced_dom)", "state_before": "case this.refine'_1\n\u03b1 : Type u_1\nE : Type ?u.1222330\nF : Type ?u.1222333\nF' : Type ?u.1222336\nG : Type ?u.1222339\n\ud835\udd5c : Type ?u.1222342\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G'), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 Lp G' 1 }\nhf : 0 \u2264 f\ng : { g // 0 \u2264 g }\n\u22a2 IsClosed {x | (fun g => 0 \u2264 \u2191(setToL1 hT) \u2191g) x}", "state_after": "no goals"}, {"tactic": "intro g", "state_before": "case this.refine'_2\n\u03b1 : Type u_1\nE : Type ?u.1222330\nF : Type ?u.1222333\nF' : Type ?u.1222336\nG : Type ?u.1222339\n\ud835\udd5c : Type ?u.1222342\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G'), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 Lp G' 1 }\nhf : 0 \u2264 f\ng : { g // 0 \u2264 g }\n\u22a2 \u2200 (a : { g // 0 \u2264 g }), (fun g => 0 \u2264 \u2191(setToL1 hT) \u2191g) (coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' a)", "state_after": "case this.refine'_2\n\u03b1 : Type u_1\nE : Type ?u.1222330\nF : Type ?u.1222333\nF' : Type ?u.1222336\nG : Type ?u.1222339\n\ud835\udd5c : Type ?u.1222342\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G'), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 Lp G' 1 }\nhf : 0 \u2264 f\ng\u271d : { g // 0 \u2264 g }\ng : { g // 0 \u2264 g }\n\u22a2 0 \u2264 \u2191(setToL1 hT) \u2191(coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g)"}, {"tactic": "have : (coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g : \u03b1 \u2192\u2081[\u03bc] G') = (g : \u03b1 \u2192\u2081\u209b[\u03bc] G') := rfl", "state_before": "case this.refine'_2\n\u03b1 : Type u_1\nE : Type ?u.1222330\nF : Type ?u.1222333\nF' : Type ?u.1222336\nG : Type ?u.1222339\n\ud835\udd5c : Type ?u.1222342\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G'), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 Lp G' 1 }\nhf : 0 \u2264 f\ng\u271d : { g // 0 \u2264 g }\ng : { g // 0 \u2264 g }\n\u22a2 0 \u2264 \u2191(setToL1 hT) \u2191(coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g)", "state_after": "case this.refine'_2\n\u03b1 : Type u_1\nE : Type ?u.1222330\nF : Type ?u.1222333\nF' : Type ?u.1222336\nG : Type ?u.1222339\n\ud835\udd5c : Type ?u.1222342\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G'), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 Lp G' 1 }\nhf : 0 \u2264 f\ng\u271d : { g // 0 \u2264 g }\ng : { g // 0 \u2264 g }\nthis : \u2191(coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g) = \u2191\u2191g\n\u22a2 0 \u2264 \u2191(setToL1 hT) \u2191(coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g)"}, {"tactic": "rw [this, setToL1_eq_setToL1SCLM]", "state_before": "case this.refine'_2\n\u03b1 : Type u_1\nE : Type ?u.1222330\nF : Type ?u.1222333\nF' : Type ?u.1222336\nG : Type ?u.1222339\n\ud835\udd5c : Type ?u.1222342\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G'), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 Lp G' 1 }\nhf : 0 \u2264 f\ng\u271d : { g // 0 \u2264 g }\ng : { g // 0 \u2264 g }\nthis : \u2191(coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g) = \u2191\u2191g\n\u22a2 0 \u2264 \u2191(setToL1 hT) \u2191(coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g)", "state_after": "case this.refine'_2\n\u03b1 : Type u_1\nE : Type ?u.1222330\nF : Type ?u.1222333\nF' : Type ?u.1222336\nG : Type ?u.1222339\n\ud835\udd5c : Type ?u.1222342\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G'), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 Lp G' 1 }\nhf : 0 \u2264 f\ng\u271d : { g // 0 \u2264 g }\ng : { g // 0 \u2264 g }\nthis : \u2191(coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g) = \u2191\u2191g\n\u22a2 0 \u2264 \u2191(setToL1SCLM \u03b1 G' \u03bc hT) \u2191g"}, {"tactic": "exact setToL1S_nonneg (fun s => hT.eq_zero_of_measure_zero) hT.1 hT_nonneg g.2", "state_before": "case this.refine'_2\n\u03b1 : Type u_1\nE : Type ?u.1222330\nF : Type ?u.1222333\nF' : Type ?u.1222336\nG : Type ?u.1222339\n\ud835\udd5c : Type ?u.1222342\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F'\ninst\u271d\u2079 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\ninst\u271d\u2075 : CompleteSpace F\nT\u271d T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C' C'' : \u211d\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b3 : NormedSpace \u211d G''\ninst\u271d\u00b2 : CompleteSpace G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nC : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT_nonneg : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 (x : G'), 0 \u2264 x \u2192 0 \u2264 \u2191(T s) x\nf : { x // x \u2208 Lp G' 1 }\nhf : 0 \u2264 f\ng\u271d : { g // 0 \u2264 g }\ng : { g // 0 \u2264 g }\nthis : \u2191(coeSimpleFuncNonnegToLpNonneg 1 \u03bc G' g) = \u2191\u2191g\n\u22a2 0 \u2264 \u2191(setToL1SCLM \u03b1 G' \u03bc hT) \u2191g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/Encoding.lean", "full_name": "Computability.leftInverse_section_inclusion", "start": [90, 1], "end": [91, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.count_pos", "start": [2407, 1], "end": [2407, 98], "traced_tactics": [{"tactic": "simp [count, countp_pos]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.357322\n\u03b3 : Type ?u.357325\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\n\u22a2 0 < count a s \u2194 a \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Tropical/Basic.lean", "full_name": "Tropical.untrop_zero", "start": [234, 1], "end": [235, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Homology/HomologicalComplex.lean", "full_name": "HomologicalComplex.dFrom_eq_zero", "start": [426, 1], "end": [427, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/Finite/Polynomial.lean", "full_name": "MvPolynomial.finrank_R", "start": [236, 1], "end": [237, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/GroupWithZero.lean", "full_name": "Filter.Tendsto.zpow\u2080", "start": [329, 1], "end": [331, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Filtration.lean", "full_name": "Ideal.Filtration.Stable.exists_forall_le", "start": [241, 1], "end": [250, 9], "traced_tactics": [{"tactic": "obtain \u27e8n\u2080, hF\u27e9 := h", "state_before": "R M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh\u271d h : Stable F\ne : N F 0 \u2264 N F' 0\n\u22a2 \u2203 n\u2080, \u2200 (n : \u2115), N F (n + n\u2080) \u2264 N F' n", "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\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\n\u22a2 \u2203 n\u2080, \u2200 (n : \u2115), N F (n + n\u2080) \u2264 N F' n"}, {"tactic": "use n\u2080", "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\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\n\u22a2 \u2203 n\u2080, \u2200 (n : \u2115), N F (n + n\u2080) \u2264 N F' n", "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\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\n\u22a2 \u2200 (n : \u2115), N F (n + n\u2080) \u2264 N F' n"}, {"tactic": "intro n", "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\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\n\u22a2 \u2200 (n : \u2115), N F (n + n\u2080) \u2264 N F' n", "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\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nn : \u2115\n\u22a2 N F (n + n\u2080) \u2264 N F' n"}, {"tactic": "induction' n with n hn", "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\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nn : \u2115\n\u22a2 N F (n + n\u2080) \u2264 N F' n", "state_after": "case intro.zero\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\n\u22a2 N F (Nat.zero + n\u2080) \u2264 N F' Nat.zero\n\ncase intro.succ\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nn : \u2115\nhn : N F (n + n\u2080) \u2264 N F' n\n\u22a2 N F (Nat.succ n + n\u2080) \u2264 N F' (Nat.succ n)"}, {"tactic": "refine' (F.antitone _).trans e", "state_before": "case intro.zero\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\n\u22a2 N F (Nat.zero + n\u2080) \u2264 N F' Nat.zero", "state_after": "case intro.zero\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\n\u22a2 0 \u2264 Nat.zero + n\u2080"}, {"tactic": "simp", "state_before": "case intro.zero\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\n\u22a2 0 \u2264 Nat.zero + n\u2080", "state_after": "no goals"}, {"tactic": "rw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, \u2190 hF]", "state_before": "case intro.succ\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nn : \u2115\nhn : N F (n + n\u2080) \u2264 N F' n\n\u22a2 N F (Nat.succ n + n\u2080) \u2264 N F' (Nat.succ n)", "state_after": "case intro.succ\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nn : \u2115\nhn : N F (n + n\u2080) \u2264 N F' n\n\u22a2 I \u2022 N F (n + n\u2080) \u2264 N F' (n + 1)\n\ncase intro.succ.a\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nn : \u2115\nhn : N F (n + n\u2080) \u2264 N F' n\n\u22a2 n + n\u2080 \u2265 n\u2080"}, {"tactic": "exact (Submodule.smul_mono_right hn).trans (F'.smul_le _)", "state_before": "case intro.succ\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nn : \u2115\nhn : N F (n + n\u2080) \u2264 N F' n\n\u22a2 I \u2022 N F (n + n\u2080) \u2264 N F' (n + 1)\n\ncase intro.succ.a\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nn : \u2115\nhn : N F (n + n\u2080) \u2264 N F' n\n\u22a2 n + n\u2080 \u2265 n\u2080", "state_after": "case intro.succ.a\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nn : \u2115\nhn : N F (n + n\u2080) \u2264 N F' n\n\u22a2 n + n\u2080 \u2265 n\u2080"}, {"tactic": "simp", "state_before": "case intro.succ.a\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nn : \u2115\nhn : N F (n + n\u2080) \u2264 N F' n\n\u22a2 n + n\u2080 \u2265 n\u2080", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean", "full_name": "CategoryTheory.Limits.zero_of_source_iso_zero", "start": [390, 1], "end": [392, 16], "traced_tactics": [{"tactic": "have h : f = i.hom \u226b \ud835\udfd9 0 \u226b i.inv \u226b f := by simp only [Iso.hom_inv_id_assoc, id_comp, comp_id]", "state_before": "C : Type u\ninst\u271d\u00b3 : Category C\nD : Type u'\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ni : X \u2245 0\n\u22a2 f = 0", "state_after": "C : Type u\ninst\u271d\u00b3 : Category C\nD : Type u'\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ni : X \u2245 0\nh : f = i.hom \u226b \ud835\udfd9 0 \u226b i.inv \u226b f\n\u22a2 f = 0"}, {"tactic": "simpa using h", "state_before": "C : Type u\ninst\u271d\u00b3 : Category C\nD : Type u'\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ni : X \u2245 0\nh : f = i.hom \u226b \ud835\udfd9 0 \u226b i.inv \u226b f\n\u22a2 f = 0", "state_after": "no goals"}, {"tactic": "simp only [Iso.hom_inv_id_assoc, id_comp, comp_id]", "state_before": "C : Type u\ninst\u271d\u00b3 : Category C\nD : Type u'\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nX Y : C\nf : X \u27f6 Y\ni : X \u2245 0\n\u22a2 f = i.hom \u226b \ud835\udfd9 0 \u226b i.inv \u226b f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Antichain.lean", "full_name": "isAntichain_insert", "start": [103, 1], "end": [105, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.mem_of_mem_erase", "start": [1086, 1], "end": [1087, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Covering/VitaliFamily.lean", "full_name": "VitaliFamily.FineSubfamilyOn.covering_mem_family", "start": [150, 1], "end": [151, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Connected.lean", "full_name": "connectedComponentIn_eq", "start": [681, 1], "end": [686, 72], "traced_tactics": [{"tactic": "have hx : x \u2208 F := connectedComponentIn_nonempty_iff.mp \u27e8y, h\u27e9", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.56608\n\u03c0 : \u03b9 \u2192 Type ?u.56613\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx y : \u03b1\nF : Set \u03b1\nh : y \u2208 connectedComponentIn F x\n\u22a2 connectedComponentIn F x = connectedComponentIn F y", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.56608\n\u03c0 : \u03b9 \u2192 Type ?u.56613\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx y : \u03b1\nF : Set \u03b1\nh : y \u2208 connectedComponentIn F x\nhx : x \u2208 F\n\u22a2 connectedComponentIn F x = connectedComponentIn F y"}, {"tactic": "simp_rw [connectedComponentIn_eq_image hx] at h\u22a2", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.56608\n\u03c0 : \u03b9 \u2192 Type ?u.56613\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx y : \u03b1\nF : Set \u03b1\nh : y \u2208 connectedComponentIn F x\nhx : x \u2208 F\n\u22a2 connectedComponentIn F x = connectedComponentIn F y", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.56608\n\u03c0 : \u03b9 \u2192 Type ?u.56613\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx y : \u03b1\nF : Set \u03b1\nhx : x \u2208 F\nh : y \u2208 Subtype.val '' connectedComponent { val := x, property := hx }\n\u22a2 Subtype.val '' connectedComponent { val := x, property := hx } = connectedComponentIn F y"}, {"tactic": "obtain \u27e8\u27e8y, hy\u27e9, h2y, rfl\u27e9 := h", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.56608\n\u03c0 : \u03b9 \u2192 Type ?u.56613\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx y : \u03b1\nF : Set \u03b1\nhx : x \u2208 F\nh : y \u2208 Subtype.val '' connectedComponent { val := x, property := hx }\n\u22a2 Subtype.val '' connectedComponent { val := x, property := hx } = connectedComponentIn F y", "state_after": "case intro.mk.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.56608\n\u03c0 : \u03b9 \u2192 Type ?u.56613\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\nhx : x \u2208 F\ny : \u03b1\nhy : y \u2208 F\nh2y : { val := y, property := hy } \u2208 connectedComponent { val := x, property := hx }\n\u22a2 Subtype.val '' connectedComponent { val := x, property := hx } = connectedComponentIn F \u2191{ val := y, property := hy }"}, {"tactic": "simp_rw [connectedComponentIn_eq_image hy, connectedComponent_eq h2y]", "state_before": "case intro.mk.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.56608\n\u03c0 : \u03b9 \u2192 Type ?u.56613\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\nhx : x \u2208 F\ny : \u03b1\nhy : y \u2208 F\nh2y : { val := y, property := hy } \u2208 connectedComponent { val := x, property := hx }\n\u22a2 Subtype.val '' connectedComponent { val := x, property := hx } = connectedComponentIn F \u2191{ val := y, property := hy }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "iInf_ne_top_subtype", "start": [1608, 1], "end": [1609, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/BilinearForm.lean", "full_name": "BilinForm.isOrtho_def", "start": [760, 1], "end": [761, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Matrix/Basic.lean", "full_name": "Matrix.vecMul_smul", "start": [1772, 1], "end": [1776, 81], "traced_tactics": [{"tactic": "ext i", "state_before": "l : Type ?u.858279\nm : Type u_4\nn : Type u_1\no : Type ?u.858288\nm' : o \u2192 Type ?u.858293\nn' : o \u2192 Type ?u.858298\nR : Type u_2\nS : Type u_3\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type ?u.858311\ninst\u271d\u2075 : NonUnitalNonAssocSemiring \u03b1\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Monoid R\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring S\ninst\u271d\u00b9 : DistribMulAction R S\ninst\u271d : IsScalarTower R S S\nM : Matrix n m S\nb : R\nv : n \u2192 S\n\u22a2 vecMul (b \u2022 v) M = b \u2022 vecMul v M", "state_after": "case h\nl : Type ?u.858279\nm : Type u_4\nn : Type u_1\no : Type ?u.858288\nm' : o \u2192 Type ?u.858293\nn' : o \u2192 Type ?u.858298\nR : Type u_2\nS : Type u_3\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type ?u.858311\ninst\u271d\u2075 : NonUnitalNonAssocSemiring \u03b1\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Monoid R\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring S\ninst\u271d\u00b9 : DistribMulAction R S\ninst\u271d : IsScalarTower R S S\nM : Matrix n m S\nb : R\nv : n \u2192 S\ni : m\n\u22a2 vecMul (b \u2022 v) M i = (b \u2022 vecMul v M) i"}, {"tactic": "simp only [vecMul, dotProduct, Finset.smul_sum, Pi.smul_apply, smul_mul_assoc]", "state_before": "case h\nl : Type ?u.858279\nm : Type u_4\nn : Type u_1\no : Type ?u.858288\nm' : o \u2192 Type ?u.858293\nn' : o \u2192 Type ?u.858298\nR : Type u_2\nS : Type u_3\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type ?u.858311\ninst\u271d\u2075 : NonUnitalNonAssocSemiring \u03b1\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : 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Semiring R\ninst\u271d\u00b9\u2078 : AddCommMonoid M\ninst\u271d\u00b9\u2077 : Module R M\nR\u2081 : Type ?u.3009745\nM\u2081 : Type ?u.3009748\ninst\u271d\u00b9\u2076 : Ring R\u2081\ninst\u271d\u00b9\u2075 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2074 : Module R\u2081 M\u2081\nR\u2082 : Type ?u.3010357\nM\u2082 : Type ?u.3010360\ninst\u271d\u00b9\u00b3 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b9 : Module R\u2082 M\u2082\nR\u2083 : Type ?u.3010547\nM\u2083 : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\u2083\ninst\u271d\u2079 : AddCommGroup M\u2083\ninst\u271d\u2078 : Module R\u2083 M\u2083\nV : Type ?u.3011138\nK : Type ?u.3011141\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nA : Type u_1\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : IsDomain A\ninst\u271d\u00b2 : Module A M\u2083\nB\u2083 : BilinForm A M\u2083\n\u03b9 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nB : BilinForm A M\u2083\nb : Basis \u03b9 A M\u2083\n\u22a2 Nondegenerate B \u2194 det (\u2191(toMatrix b) B) \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "full_name": "PowerSeries.rescale_zero", "start": [1769, 1], "end": [1773, 32], "traced_tactics": [{"tactic": "ext x n", "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\n\u22a2 rescale 0 = RingHom.comp (C R) (constantCoeff R)", "state_after": "case a.h\nR : Type u_1\ninst\u271d : CommSemiring R\nx : PowerSeries R\nn : \u2115\n\u22a2 \u2191(coeff R n) (\u2191(rescale 0) x) = \u2191(coeff R n) (\u2191(RingHom.comp (C R) (constantCoeff R)) x)"}, {"tactic": "simp only [Function.comp_apply, RingHom.coe_comp, rescale, RingHom.coe_mk,\n  PowerSeries.coeff_mk _ _, coeff_C]", "state_before": "case a.h\nR : Type u_1\ninst\u271d : CommSemiring R\nx : PowerSeries R\nn : \u2115\n\u22a2 \u2191(coeff R n) (\u2191(rescale 0) x) = \u2191(coeff R n) (\u2191(RingHom.comp (C R) (constantCoeff R)) x)", "state_after": "case a.h\nR : Type u_1\ninst\u271d : CommSemiring R\nx : PowerSeries R\nn : \u2115\n\u22a2 \u2191(coeff R n)\n      (\u2191{\n            toOneHom :=\n              { toFun := fun f => mk fun n => 0 ^ n * \u2191(coeff R n) f,\n                map_one' := (_ : (fun f => mk fun n => 0 ^ n * \u2191(coeff R n) f) 1 = 1) },\n            map_mul' :=\n              (_ :\n                \u2200 (f g : PowerSeries R),\n                  OneHom.toFun\n                      { toFun := fun f => mk fun n => 0 ^ n * \u2191(coeff R n) f,\n                        map_one' := (_ : (fun f => mk fun n => 0 ^ n * \u2191(coeff R n) f) 1 = 1) }\n                      (f * g) =\n                    OneHom.toFun\n                        { toFun := fun f => mk fun n => 0 ^ n * \u2191(coeff R n) f,\n                          map_one' := (_ : (fun f => mk fun n => 0 ^ n * \u2191(coeff R n) f) 1 = 1) }\n                        f *\n                      OneHom.toFun\n                        { toFun := fun f => mk fun n => 0 ^ n * \u2191(coeff R n) f,\n                          map_one' := (_ : (fun f => mk fun n => 0 ^ n * \u2191(coeff R n) f) 1 = 1) }\n                        g) }\n        x) =\n    if n = 0 then \u2191(constantCoeff R) x else 0"}, {"tactic": "split_ifs with h <;> simp [h]", "state_before": "case a.h\nR : Type u_1\ninst\u271d : CommSemiring R\nx : PowerSeries R\nn : \u2115\n\u22a2 \u2191(coeff R n)\n      (\u2191{\n            toOneHom :=\n              { toFun := fun f => mk fun n => 0 ^ n * \u2191(coeff R n) f,\n                map_one' := (_ : (fun f => mk fun n => 0 ^ n * \u2191(coeff R n) f) 1 = 1) },\n            map_mul' :=\n              (_ :\n                \u2200 (f g : PowerSeries R),\n                  OneHom.toFun\n                      { toFun := fun f => mk fun n => 0 ^ n * \u2191(coeff R n) f,\n                        map_one' := (_ : (fun f => mk fun n => 0 ^ n * \u2191(coeff R n) f) 1 = 1) }\n                      (f * g) =\n                    OneHom.toFun\n                        { toFun := fun f => mk fun n => 0 ^ n * \u2191(coeff R n) f,\n                          map_one' := (_ : (fun f => mk fun n => 0 ^ n * \u2191(coeff R n) f) 1 = 1) }\n                        f *\n                      OneHom.toFun\n                        { toFun := fun f => mk fun n => 0 ^ n * \u2191(coeff R n) f,\n                          map_one' := (_ : (fun f => mk fun n => 0 ^ n * \u2191(coeff R n) f) 1 = 1) }\n                        g) }\n        x) =\n    if n = 0 then \u2191(constantCoeff R) x else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Ring/Idempotents.lean", "full_name": "IsIdempotentElem.pow", "start": [77, 1], "end": [81, 45], "traced_tactics": [{"tactic": "conv_rhs => rw [\u2190 h.eq]", "state_before": "M : Type ?u.3776\nN : Type u_1\nS : Type ?u.3782\nM\u2080 : Type ?u.3785\nM\u2081 : Type ?u.3788\nR : Type ?u.3791\nG : Type ?u.3794\nG\u2080 : Type ?u.3797\ninst\u271d\u2077 : Mul M\ninst\u271d\u2076 : Monoid N\ninst\u271d\u2075 : Semigroup S\ninst\u271d\u2074 : MulZeroClass M\u2080\ninst\u271d\u00b3 : MulOneClass M\u2081\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : Group G\ninst\u271d : CancelMonoidWithZero G\u2080\np : N\nn\u271d : \u2115\nh : IsIdempotentElem p\nn : \u2115\nx\u271d : IsIdempotentElem (p ^ n)\n\u22a2 p ^ Nat.succ n * p ^ Nat.succ n = p ^ Nat.succ n", "state_after": "M : Type ?u.3776\nN : Type u_1\nS : Type ?u.3782\nM\u2080 : Type ?u.3785\nM\u2081 : Type ?u.3788\nR : Type ?u.3791\nG : Type ?u.3794\nG\u2080 : Type ?u.3797\ninst\u271d\u2077 : Mul M\ninst\u271d\u2076 : Monoid N\ninst\u271d\u2075 : Semigroup S\ninst\u271d\u2074 : MulZeroClass M\u2080\ninst\u271d\u00b3 : MulOneClass M\u2081\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : Group G\ninst\u271d : CancelMonoidWithZero G\u2080\np : N\nn\u271d : \u2115\nh : IsIdempotentElem p\nn : \u2115\nx\u271d : IsIdempotentElem (p ^ n)\n\u22a2 p ^ Nat.succ n * p ^ Nat.succ n = (p * p) ^ Nat.succ n"}, {"tactic": "rw [\u2190 sq, \u2190 sq, \u2190 pow_mul, \u2190 pow_mul']", "state_before": "M : Type ?u.3776\nN : Type u_1\nS : Type ?u.3782\nM\u2080 : Type ?u.3785\nM\u2081 : Type ?u.3788\nR : Type ?u.3791\nG : Type ?u.3794\nG\u2080 : Type ?u.3797\ninst\u271d\u2077 : Mul M\ninst\u271d\u2076 : Monoid N\ninst\u271d\u2075 : Semigroup S\ninst\u271d\u2074 : MulZeroClass M\u2080\ninst\u271d\u00b3 : MulOneClass M\u2081\ninst\u271d\u00b2 : NonAssocRing R\ninst\u271d\u00b9 : Group G\ninst\u271d : CancelMonoidWithZero G\u2080\np : N\nn\u271d : \u2115\nh : IsIdempotentElem p\nn : \u2115\nx\u271d : IsIdempotentElem (p ^ n)\n\u22a2 p ^ Nat.succ n * p ^ Nat.succ n = (p * p) ^ Nat.succ n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Nodup.lean", "full_name": "List.nodup_append_comm", "start": [230, 1], "end": [231, 57], "traced_tactics": [{"tactic": "simp only [nodup_append, and_left_comm, disjoint_comm]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nl l\u2081\u271d l\u2082\u271d : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 Nodup (l\u2081 ++ l\u2082) \u2194 Nodup (l\u2082 ++ l\u2081)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Ultrafilter.lean", "full_name": "Ultrafilter.comap_inf_principal_neBot_of_image_mem", "start": [519, 1], "end": [520, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/FilterBasis.lean", "full_name": "ModuleFilterBasis.smul", "start": [357, 1], "end": [358, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Interval.lean", "full_name": "Interval.mem_pure_self", "start": [522, 1], "end": [523, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/MinMax.lean", "full_name": "List.minimum_not_lt_of_mem", "start": [327, 1], "end": [328, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SetFamily/Compression/UV.lean", "full_name": "UV.card_shadow_compression_le", "start": [432, 1], "end": [436, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Pointwise.lean", "full_name": "Filter.inv_le_inv_iff", "start": [256, 11], "end": [257, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/MvPolynomial/Symmetric.lean", "full_name": "MvPolynomial.support_esymm'", "start": [245, 1], "end": [251, 53], "traced_tactics": [{"tactic": "rw [support_esymm'']", "state_before": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type ?u.190525\nS : Type ?u.190528\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\n\u22a2 support (esymm \u03c3 R n) = Finset.biUnion (powersetLen n univ) fun t => {\u2211 i in t, Finsupp.single i 1}", "state_after": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type ?u.190525\nS : Type ?u.190528\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\n\u22a2 (Finset.biUnion (powersetLen n univ) fun t => (Finsupp.single (\u2211 i in t, Finsupp.single i 1) 1).support) =\n    Finset.biUnion (powersetLen n univ) fun t => {\u2211 i in t, Finsupp.single i 1}"}, {"tactic": "congr", "state_before": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type ?u.190525\nS : Type ?u.190528\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\n\u22a2 (Finset.biUnion (powersetLen n univ) fun t => (Finsupp.single (\u2211 i in t, Finsupp.single i 1) 1).support) =\n    Finset.biUnion (powersetLen n univ) fun t => {\u2211 i in t, Finsupp.single i 1}", "state_after": "case e_t\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type ?u.190525\nS : Type ?u.190528\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\n\u22a2 (fun t => (Finsupp.single (\u2211 i in t, Finsupp.single i 1) 1).support) = fun t => {\u2211 i in t, Finsupp.single i 1}"}, {"tactic": "funext", "state_before": "case e_t\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type ?u.190525\nS : Type ?u.190528\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\n\u22a2 (fun t => (Finsupp.single (\u2211 i in t, Finsupp.single i 1) 1).support) = fun t => {\u2211 i in t, Finsupp.single i 1}", "state_after": "case e_t.h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type ?u.190525\nS : Type ?u.190528\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\nx\u271d : Finset \u03c3\n\u22a2 (Finsupp.single (\u2211 i in x\u271d, Finsupp.single i 1) 1).support = {\u2211 i in x\u271d, Finsupp.single i 1}"}, {"tactic": "exact Finsupp.support_single_ne_zero _ one_ne_zero", "state_before": "case e_t.h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type ?u.190525\nS : Type ?u.190528\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\nx\u271d : Finset \u03c3\n\u22a2 (Finsupp.single (\u2211 i in x\u271d, Finsupp.single i 1) 1).support = {\u2211 i in x\u271d, Finsupp.single i 1}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.preimage_id_eq", "start": [120, 1], "end": [121, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Relation.lean", "full_name": "Relation.comp_eq", "start": [139, 1], "end": [141, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/CofilteredSystem.lean", "full_name": "CategoryTheory.Functor.eventualRange_mapsTo", "start": [204, 1], "end": [211, 17], "traced_tactics": [{"tactic": "rw [mem_eventualRange_iff] at hx\u22a2", "state_before": "J : Type u\ninst\u271d\u00b9 : Category J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : j \u27f6 i\nx : F.obj j\nhx : x \u2208 eventualRange F j\n\u22a2 F.map f x \u2208 eventualRange F i", "state_after": "J : Type u\ninst\u271d\u00b9 : Category J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : j \u27f6 i\nx : F.obj j\nhx : \u2200 \u2983i : J\u2984 (f : i \u27f6 j), x \u2208 range (F.map f)\n\u22a2 \u2200 \u2983i_1 : J\u2984 (f_1 : i_1 \u27f6 i), F.map f x \u2208 range (F.map f_1)"}, {"tactic": "intro k f'", "state_before": "J : Type u\ninst\u271d\u00b9 : Category J\nF : J \u2964 Type v\ni j k : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : j \u27f6 i\nx : F.obj j\nhx : \u2200 \u2983i : J\u2984 (f : i \u27f6 j), x \u2208 range (F.map f)\n\u22a2 \u2200 \u2983i_1 : J\u2984 (f_1 : i_1 \u27f6 i), F.map f x \u2208 range (F.map f_1)", "state_after": "J : Type u\ninst\u271d\u00b9 : Category J\nF : J \u2964 Type v\ni j k\u271d : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : j \u27f6 i\nx : F.obj j\nhx : \u2200 \u2983i : J\u2984 (f : i \u27f6 j), x \u2208 range (F.map f)\nk : J\nf' : k \u27f6 i\n\u22a2 F.map f x \u2208 range (F.map f')"}, {"tactic": "obtain \u27e8l, g, g', he\u27e9 := cospan f f'", "state_before": "J : Type u\ninst\u271d\u00b9 : Category J\nF : J \u2964 Type v\ni j k\u271d : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : j \u27f6 i\nx : F.obj j\nhx : \u2200 \u2983i : J\u2984 (f : i \u27f6 j), x \u2208 range (F.map f)\nk : J\nf' : k \u27f6 i\n\u22a2 F.map f x \u2208 range (F.map f')", "state_after": "case intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category J\nF : J \u2964 Type v\ni j k\u271d : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : j \u27f6 i\nx : F.obj j\nhx : \u2200 \u2983i : J\u2984 (f : i \u27f6 j), x \u2208 range (F.map f)\nk : J\nf' : k \u27f6 i\nl : J\ng : l \u27f6 j\ng' : l \u27f6 k\nhe : g \u226b f = g' \u226b f'\n\u22a2 F.map f x \u2208 range (F.map f')"}, {"tactic": "obtain \u27e8x, rfl\u27e9 := hx g", "state_before": "case intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category J\nF : J \u2964 Type v\ni j k\u271d : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : j \u27f6 i\nx : F.obj j\nhx : \u2200 \u2983i : J\u2984 (f : i \u27f6 j), x \u2208 range (F.map f)\nk : J\nf' : k \u27f6 i\nl : J\ng : l \u27f6 j\ng' : l \u27f6 k\nhe : g \u226b f = g' \u226b f'\n\u22a2 F.map f x \u2208 range (F.map f')", "state_after": "case intro.intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category J\nF : J \u2964 Type v\ni j k\u271d : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : j \u27f6 i\nk : J\nf' : k \u27f6 i\nl : J\ng : l \u27f6 j\ng' : l \u27f6 k\nhe : g \u226b f = g' \u226b f'\nx : F.obj l\nhx : \u2200 \u2983i : J\u2984 (f : i \u27f6 j), F.map g x \u2208 range (F.map f)\n\u22a2 F.map f (F.map g x) \u2208 range (F.map f')"}, {"tactic": "rw [\u2190 map_comp_apply, he, F.map_comp]", "state_before": "case intro.intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category J\nF : J \u2964 Type v\ni j k\u271d : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : j \u27f6 i\nk : J\nf' : k \u27f6 i\nl : J\ng : l \u27f6 j\ng' : l \u27f6 k\nhe : g \u226b f = g' \u226b f'\nx : F.obj l\nhx : \u2200 \u2983i : J\u2984 (f : i \u27f6 j), F.map g x \u2208 range (F.map f)\n\u22a2 F.map f (F.map g x) \u2208 range (F.map f')", "state_after": "case intro.intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category J\nF : J \u2964 Type v\ni j k\u271d : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : j \u27f6 i\nk : J\nf' : k \u27f6 i\nl : J\ng : l \u27f6 j\ng' : l \u27f6 k\nhe : g \u226b f = g' \u226b f'\nx : F.obj l\nhx : \u2200 \u2983i : J\u2984 (f : i \u27f6 j), F.map g x \u2208 range (F.map f)\n\u22a2 (F.map g' \u226b F.map f') x \u2208 range (F.map f')"}, {"tactic": "exact \u27e8_, rfl\u27e9", "state_before": "case intro.intro.intro.intro\nJ : Type u\ninst\u271d\u00b9 : Category J\nF : J \u2964 Type v\ni j k\u271d : J\ns : Set (F.obj i)\ninst\u271d : IsCofilteredOrEmpty J\nf : j \u27f6 i\nk : J\nf' : k \u27f6 i\nl : J\ng : l \u27f6 j\ng' : l \u27f6 k\nhe : g \u226b f = g' \u226b f'\nx : F.obj l\nhx : \u2200 \u2983i : J\u2984 (f : i \u27f6 j), F.map g x \u2208 range (F.map f)\n\u22a2 (F.map g' \u226b F.map f') x \u2208 range (F.map f')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Extr.lean", "full_name": "isMaxOn_const", "start": [196, 1], "end": [197, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Subadditive.lean", "full_name": "Subadditive.apply_mul_add_le", "start": [54, 1], "end": [62, 51], "traced_tactics": [{"tactic": "induction k with\n| zero => simp only [Nat.zero_eq, Nat.cast_zero, zero_mul, zero_add]; rfl\n| succ k IH =>\n  calc\n    u ((k + 1) * n + r) = u (n + (k * n + r)) := by congr 1; ring\n    _ \u2264 u n + u (k * n + r) := h _ _\n    _ \u2264 u n + (k * u n + u r) := add_le_add_left IH _\n    _ = (k + 1 : \u2115) * u n + u r := by simp; ring", "state_before": "u : \u2115 \u2192 \u211d\nh : Subadditive u\nk n r : \u2115\n\u22a2 u (k * n + r) \u2264 \u2191k * u n + u r", "state_after": "no goals"}, {"tactic": "simp only [Nat.zero_eq, Nat.cast_zero, zero_mul, zero_add]", "state_before": "case zero\nu : \u2115 \u2192 \u211d\nh : Subadditive u\nn r : \u2115\n\u22a2 u (Nat.zero * n + r) \u2264 \u2191Nat.zero * u n + u r", "state_after": "case zero\nu : \u2115 \u2192 \u211d\nh : Subadditive u\nn r : \u2115\n\u22a2 u r \u2264 u r"}, {"tactic": "rfl", "state_before": "case zero\nu : \u2115 \u2192 \u211d\nh : Subadditive u\nn r : \u2115\n\u22a2 u r \u2264 u r", "state_after": "no goals"}, {"tactic": "calc\n  u ((k + 1) * n + r) = u (n + (k * n + r)) := by congr 1; ring\n  _ \u2264 u n + u (k * n + r) := h _ _\n  _ \u2264 u n + (k * u n + u r) := add_le_add_left IH _\n  _ = (k + 1 : \u2115) * u n + u r := by simp; ring", "state_before": "case succ\nu : \u2115 \u2192 \u211d\nh : Subadditive u\nn r k : \u2115\nIH : u (k * n + r) \u2264 \u2191k * u n + u r\n\u22a2 u (Nat.succ k * n + r) \u2264 \u2191(Nat.succ k) * u n + u r", "state_after": "no goals"}, {"tactic": "congr 1", "state_before": "u : \u2115 \u2192 \u211d\nh : Subadditive u\nn r k : \u2115\nIH : u (k * n + r) \u2264 \u2191k * u n + u r\n\u22a2 u ((k + 1) * n + r) = u (n + (k * n + r))", "state_after": "case e_a\nu : \u2115 \u2192 \u211d\nh : Subadditive u\nn r k : \u2115\nIH : u (k * n + r) \u2264 \u2191k * u n + u r\n\u22a2 (k + 1) * n + r = n + (k * n + r)"}, {"tactic": "ring", "state_before": "case e_a\nu : \u2115 \u2192 \u211d\nh : Subadditive u\nn r k : \u2115\nIH : u (k * n + r) \u2264 \u2191k * u n + u r\n\u22a2 (k + 1) * n + r = n + (k * n + r)", "state_after": "no goals"}, {"tactic": "simp", "state_before": "u : \u2115 \u2192 \u211d\nh : Subadditive u\nn r k : \u2115\nIH : u (k * n + r) \u2264 \u2191k * u n + u r\n\u22a2 u n + (\u2191k * u n + u r) = \u2191(k + 1) * u n + u r", "state_after": "u : \u2115 \u2192 \u211d\nh : Subadditive u\nn r k : \u2115\nIH : u (k * n + r) \u2264 \u2191k * u n + u r\n\u22a2 u n + (\u2191k * u n + u r) = (\u2191k + 1) * u n + u r"}, {"tactic": "ring", "state_before": "u : \u2115 \u2192 \u211d\nh : Subadditive u\nn r k : \u2115\nIH : u (k * n + r) \u2264 \u2191k * u n + u r\n\u22a2 u n + (\u2191k * u n + u r) = (\u2191k + 1) * u n + u r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Side.lean", "full_name": "Wbtw.wSameSide\u2082\u2081", "start": [390, 1], "end": [392, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finsupp/Defs.lean", "full_name": "Finsupp.single_apply_mem", "start": [369, 1], "end": [370, 74], "traced_tactics": [{"tactic": "rcases em (a = x) with (rfl | hx) <;> [simp; simp [single_eq_of_ne hx]]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.47865\n\u03b3 : Type ?u.47868\n\u03b9 : Type ?u.47871\nM : Type u_1\nM' : Type ?u.47877\nN : Type ?u.47880\nP : Type ?u.47883\nG : Type ?u.47886\nH : Type ?u.47889\nR : Type ?u.47892\nS : Type ?u.47895\ninst\u271d : Zero M\na a' : \u03b1\nb : M\nx : \u03b1\n\u22a2 \u2191(single a b) x \u2208 {0, b}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean", "full_name": "GromovHausdorff.HD_below_aux1", "start": [296, 1], "end": [299, 97], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Sups.lean", "full_name": "Finset.infs_sups_subset_right", "start": [413, 1], "end": [414, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Instances.lean", "full_name": "Set.Ioc.coe_pos", "start": [276, 1], "end": [277, 8], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Nodup.lean", "full_name": "Multiset.Nodup.pairwise", "start": [108, 11], "end": [109, 94], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean", 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"https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RepresentationTheory/Action.lean", "full_name": "Action.rightDual_v", "start": [735, 1], "end": [736, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Game/PGame.lean", "full_name": "PGame.zero_lf_le", "start": [655, 1], "end": [657, 7], "traced_tactics": [{"tactic": "rw [lf_iff_exists_le]", "state_before": "x : PGame\n\u22a2 0 \u29cf x \u2194 \u2203 i, 0 \u2264 moveLeft x i", "state_after": "x : PGame\n\u22a2 ((\u2203 i, 0 \u2264 moveLeft x i) \u2228 \u2203 j, moveRight 0 j \u2264 x) \u2194 \u2203 i, 0 \u2264 moveLeft x i"}, {"tactic": "simp", "state_before": "x : PGame\n\u22a2 ((\u2203 i, 0 \u2264 moveLeft x i) \u2228 \u2203 j, moveRight 0 j \u2264 x) \u2194 \u2203 i, 0 \u2264 moveLeft x i", "state_after": "no goals"}]}, {"url": 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"traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Sites/InducedTopology.lean", "full_name": "CategoryTheory.CoverDense.locallyCoverDense", "start": [115, 1], "end": [122, 7], "traced_tactics": [{"tactic": "intro X T", "state_before": "C : Type u_3\ninst\u271d\u00b3 : Category C\nD : Type u_4\ninst\u271d\u00b2 : Category D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b9 : Category A\ninst\u271d : Full G\nH : CoverDense K G\n\u22a2 LocallyCoverDense K G", "state_after": "C : Type u_3\ninst\u271d\u00b3 : Category C\nD : Type u_4\ninst\u271d\u00b2 : Category D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b9 : Category A\ninst\u271d : Full G\nH : CoverDense K G\nX : C\nT : \u2191(GrothendieckTopology.sieves K (G.obj X))\n\u22a2 Sieve.functorPushforward G 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Sieve.coverByImage G Y) \u2264 Sieve.functorPushforward G (Sieve.functorPullback G \u2191T)"}, {"tactic": "rintro Y _ \u27e8Z, _, f, hf, \u27e8W, g, f', rfl : _ = _\u27e9, rfl\u27e9", "state_before": "C : Type u_3\ninst\u271d\u00b3 : Category C\nD : Type u_4\ninst\u271d\u00b2 : Category D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b9 : Category A\ninst\u271d : Full G\nH : CoverDense K G\nX : C\nT : \u2191(GrothendieckTopology.sieves K (G.obj X))\n\u22a2 (Sieve.bind (\u2191T).arrows fun Y f x => Sieve.coverByImage G Y) \u2264 Sieve.functorPushforward G (Sieve.functorPullback G \u2191T)", "state_after": "case intro.intro.intro.intro.intro.intro.mk\nC : Type u_3\ninst\u271d\u00b3 : Category C\nD : Type u_4\ninst\u271d\u00b2 : Category D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b9 : Category A\ninst\u271d : Full G\nH : CoverDense K G\nX : C\nT : 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v\ninst\u271d\u00b9 : Category A\ninst\u271d : Full G\nH : CoverDense K G\nX : C\nT : \u2191(GrothendieckTopology.sieves K (G.obj X))\nY Z : D\nf : Z \u27f6 G.obj X\nhf : (\u2191T).arrows f\nW : C\ng : Y \u27f6 G.obj W\nf' : G.obj W \u27f6 Z\n\u22a2 \u2203 h, (Sieve.functorPullback G \u2191T).arrows (G.preimage (f' \u226b f)) \u2227 (g \u226b f') \u226b f = h \u226b G.map (G.preimage (f' \u226b f))", "state_after": "case intro.intro.intro.intro.intro.intro.mk\nC : Type u_3\ninst\u271d\u00b3 : Category C\nD : Type u_4\ninst\u271d\u00b2 : Category D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b9 : Category A\ninst\u271d : Full G\nH : CoverDense K G\nX : C\nT : \u2191(GrothendieckTopology.sieves K (G.obj X))\nY Z : D\nf : Z \u27f6 G.obj X\nhf : (\u2191T).arrows f\nW : C\ng : Y \u27f6 G.obj W\nf' : G.obj W \u27f6 Z\n\u22a2 (Sieve.functorPullback G \u2191T).arrows (G.preimage (f' \u226b f)) \u2227 (g \u226b f') \u226b f = g \u226b G.map (G.preimage (f' \u226b f))"}, {"tactic": "constructor", "state_before": "case intro.intro.intro.intro.intro.intro.mk\nC : Type u_3\ninst\u271d\u00b3 : Category C\nD : Type u_4\ninst\u271d\u00b2 : Category D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b9 : Category A\ninst\u271d : Full G\nH : CoverDense K G\nX : C\nT : \u2191(GrothendieckTopology.sieves K (G.obj X))\nY Z : D\nf : Z \u27f6 G.obj X\nhf : (\u2191T).arrows f\nW : C\ng : Y \u27f6 G.obj W\nf' : G.obj W \u27f6 Z\n\u22a2 (Sieve.functorPullback G \u2191T).arrows (G.preimage (f' \u226b f)) \u2227 (g \u226b f') \u226b f = g \u226b G.map (G.preimage (f' \u226b f))", "state_after": "case intro.intro.intro.intro.intro.intro.mk.left\nC : Type u_3\ninst\u271d\u00b3 : Category C\nD : Type u_4\ninst\u271d\u00b2 : Category D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b9 : Category A\ninst\u271d : Full G\nH : CoverDense K G\nX : C\nT : \u2191(GrothendieckTopology.sieves K (G.obj X))\nY Z : D\nf : Z \u27f6 G.obj X\nhf : (\u2191T).arrows f\nW : C\ng : Y \u27f6 G.obj W\nf' : G.obj W \u27f6 Z\n\u22a2 (Sieve.functorPullback G \u2191T).arrows (G.preimage (f' \u226b f))\n\ncase intro.intro.intro.intro.intro.intro.mk.right\nC : Type u_3\ninst\u271d\u00b3 : Category C\nD : Type u_4\ninst\u271d\u00b2 : Category D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b9 : Category A\ninst\u271d : Full G\nH : CoverDense K G\nX : C\nT : \u2191(GrothendieckTopology.sieves K (G.obj X))\nY Z : D\nf : Z \u27f6 G.obj X\nhf : (\u2191T).arrows f\nW : C\ng : Y \u27f6 G.obj W\nf' : G.obj W \u27f6 Z\n\u22a2 (g \u226b f') \u226b f = g \u226b G.map (G.preimage (f' \u226b f))"}, {"tactic": "simpa using T.val.downward_closed hf f'", "state_before": "case intro.intro.intro.intro.intro.intro.mk.left\nC : Type u_3\ninst\u271d\u00b3 : Category C\nD : Type u_4\ninst\u271d\u00b2 : Category D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b9 : Category A\ninst\u271d : Full G\nH : CoverDense K G\nX : C\nT : \u2191(GrothendieckTopology.sieves K (G.obj X))\nY Z : D\nf : Z \u27f6 G.obj X\nhf : (\u2191T).arrows f\nW : C\ng : Y \u27f6 G.obj W\nf' : G.obj W \u27f6 Z\n\u22a2 (Sieve.functorPullback G \u2191T).arrows (G.preimage (f' \u226b f))\n\ncase intro.intro.intro.intro.intro.intro.mk.right\nC : Type u_3\ninst\u271d\u00b3 : Category C\nD : Type u_4\ninst\u271d\u00b2 : Category D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b9 : Category A\ninst\u271d : Full G\nH : CoverDense K G\nX : C\nT : \u2191(GrothendieckTopology.sieves K (G.obj X))\nY Z : D\nf : Z \u27f6 G.obj X\nhf : (\u2191T).arrows f\nW : C\ng : Y \u27f6 G.obj W\nf' : G.obj W \u27f6 Z\n\u22a2 (g \u226b f') \u226b f = g \u226b G.map (G.preimage (f' \u226b f))", "state_after": "case 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\u27f6 G.obj W\nf' : G.obj W \u27f6 Z\n\u22a2 (g \u226b f') \u226b f = g \u226b G.map (G.preimage (f' \u226b f))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Logic.lean", "full_name": "forall_imp", "start": [353, 1], "end": [354, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/StructuredArrow.lean", "full_name": "CategoryTheory.StructuredArrow.map_id", "start": [209, 1], "end": [211, 7], "traced_tactics": [{"tactic": "rw [eq_mk f]", "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category C\nD : Type u\u2082\ninst\u271d : Category D\nS S' S'' : D\nY Y' : C\nT : C \u2964 D\nf : StructuredArrow S T\n\u22a2 (map (\ud835\udfd9 S)).obj f = f", "state_after": "C : Type u\u2081\ninst\u271d\u00b9 : Category C\nD : Type u\u2082\ninst\u271d : Category D\nS S' S'' : D\nY Y' : C\nT : C \u2964 D\nf : StructuredArrow S T\n\u22a2 (map (\ud835\udfd9 S)).obj (mk f.hom) = mk f.hom"}, {"tactic": "simp", "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category C\nD : Type u\u2082\ninst\u271d : Category D\nS S' S'' : D\nY Y' : C\nT : C \u2964 D\nf : StructuredArrow S T\n\u22a2 (map (\ud835\udfd9 S)).obj (mk f.hom) = mk f.hom", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean", "full_name": "Complex.measurable_exp", "start": [92, 1], "end": [93, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Basic.lean", "full_name": "LinearEquiv.ofTop_symm_apply", "start": [2111, 1], "end": [2112, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Nat/Lemmas.lean", "full_name": "Nat.sub_right_comm", "start": [414, 11], "end": [415, 46], "traced_tactics": [{"tactic": "rw [Nat.sub_sub, Nat.sub_sub, Nat.add_comm]", "state_before": "m n k : Nat\n\u22a2 m - n - k = m - k - n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "full_name": "LinearMap.toMatrix_mul", "start": [653, 1], "end": [655, 67], "traced_tactics": [{"tactic": "rw [LinearMap.mul_eq_comp, LinearMap.toMatrix_comp v\u2081 v\u2081 v\u2081 f g]", "state_before": "R : Type u_1\ninst\u271d\u2079 : CommSemiring R\nl : Type ?u.1900809\nm : Type ?u.1900812\nn : Type u_3\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : DecidableEq n\nM\u2081 : Type u_2\nM\u2082 : Type ?u.1900836\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type ?u.1901338\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : Module R M\u2083\nv\u2083 : Basis l R M\u2083\nf g : M\u2081 \u2192\u2097[R] M\u2081\n\u22a2 \u2191(toMatrix v\u2081 v\u2081) (f * g) = \u2191(toMatrix v\u2081 v\u2081) f \u2b1d \u2191(toMatrix v\u2081 v\u2081) g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.aestronglyMeasurable_zero_measure", "start": [1175, 1], "end": [1179, 60], "traced_tactics": [{"tactic": "nontriviality \u03b1", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.285120\n\u03b9 : Type ?u.285123\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 AEStronglyMeasurable f 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.285120\n\u03b9 : Type ?u.285123\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u271d : Nontrivial \u03b1\n\u22a2 AEStronglyMeasurable f 0"}, {"tactic": "inhabit \u03b1", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.285120\n\u03b9 : Type ?u.285123\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u271d : Nontrivial \u03b1\n\u22a2 AEStronglyMeasurable f 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.285120\n\u03b9 : Type ?u.285123\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u271d : Nontrivial \u03b1\ninhabited_h : Inhabited \u03b1\n\u22a2 AEStronglyMeasurable f 0"}, {"tactic": "exact \u27e8fun _ => f default, stronglyMeasurable_const, rfl\u27e9", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.285120\n\u03b9 : Type ?u.285123\ninst\u271d\u00b2 : Countable \u03b9\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u271d : Nontrivial \u03b1\ninhabited_h : Inhabited \u03b1\n\u22a2 AEStronglyMeasurable f 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/Adjugate.lean", "full_name": "Matrix.adjugate_transpose", "start": [222, 1], "end": [249, 38], "traced_tactics": [{"tactic": "ext (i j)", "state_before": "m : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\n\u22a2 (adjugate A)\u1d40 = adjugate A\u1d40", "state_after": "case a.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u22a2 (adjugate A)\u1d40 i j = adjugate A\u1d40 i j"}, {"tactic": "rw [transpose_apply, adjugate_apply, adjugate_apply, updateRow_transpose, det_transpose]", "state_before": "case a.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u22a2 (adjugate A)\u1d40 i j = adjugate A\u1d40 i j", "state_after": "case a.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u22a2 det (updateRow A i (Pi.single j 1)) = det (updateColumn A j (Pi.single i 1))"}, {"tactic": "rw [det_apply', det_apply']", "state_before": "case a.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u22a2 det (updateRow A i (Pi.single j 1)) = det (updateColumn A j (Pi.single i 1))", "state_after": "case a.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u22a2 \u2211 \u03c3 : Perm n, \u2191\u2191(\u2191sign \u03c3) * \u220f i_1 : n, updateRow A i (Pi.single j 1) (\u2191\u03c3 i_1) i_1 =\n    \u2211 \u03c3 : Perm n, \u2191\u2191(\u2191sign \u03c3) * \u220f i_1 : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 i_1) i_1"}, {"tactic": "apply Finset.sum_congr rfl", "state_before": "case a.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u22a2 \u2211 \u03c3 : Perm n, \u2191\u2191(\u2191sign \u03c3) * \u220f i_1 : n, updateRow A i (Pi.single j 1) (\u2191\u03c3 i_1) i_1 =\n    \u2211 \u03c3 : Perm n, \u2191\u2191(\u2191sign \u03c3) * \u220f i_1 : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 i_1) i_1", "state_after": "case a.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u22a2 \u2200 (x : Perm n),\n    x \u2208 univ \u2192\n      \u2191\u2191(\u2191sign x) * \u220f i_1 : n, updateRow A i (Pi.single j 1) (\u2191x i_1) i_1 =\n        \u2191\u2191(\u2191sign x) * \u220f i_1 : n, updateColumn A j (Pi.single i 1) (\u2191x i_1) i_1"}, {"tactic": "intro \u03c3 _", "state_before": "case a.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u22a2 \u2200 (x : Perm n),\n    x \u2208 univ \u2192\n      \u2191\u2191(\u2191sign x) * \u220f i_1 : n, updateRow A i (Pi.single j 1) (\u2191x i_1) i_1 =\n        \u2191\u2191(\u2191sign x) * \u220f i_1 : n, updateColumn A j (Pi.single i 1) (\u2191x i_1) i_1", "state_after": "case a.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f i_1 : n, updateRow A i (Pi.single j 1) (\u2191\u03c3 i_1) i_1 =\n    \u2191\u2191(\u2191sign \u03c3) * \u220f i_1 : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 i_1) i_1"}, {"tactic": "congr 1", "state_before": "case a.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f i_1 : n, updateRow A i (Pi.single j 1) (\u2191\u03c3 i_1) i_1 =\n    \u2191\u2191(\u2191sign \u03c3) * \u220f i_1 : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 i_1) i_1", "state_after": "case a.h.e_a\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\n\u22a2 \u220f i_1 : n, updateRow A i (Pi.single j 1) (\u2191\u03c3 i_1) i_1 = \u220f i_1 : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 i_1) i_1"}, {"tactic": "by_cases i = \u03c3 j", "state_before": "case a.h.e_a\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\n\u22a2 \u220f i_1 : n, updateRow A i (Pi.single j 1) (\u2191\u03c3 i_1) i_1 = \u220f i_1 : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 i_1) i_1", "state_after": "case pos\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : i = \u2191\u03c3 j\n\u22a2 \u220f i_1 : n, updateRow A i (Pi.single j 1) (\u2191\u03c3 i_1) i_1 = \u220f i_1 : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 i_1) i_1\n\ncase neg\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : \u00aci = \u2191\u03c3 j\n\u22a2 \u220f i_1 : n, updateRow A i (Pi.single j 1) (\u2191\u03c3 i_1) i_1 = \u220f i_1 : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 i_1) i_1"}, {"tactic": "congr", "state_before": "case pos\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : i = \u2191\u03c3 j\n\u22a2 \u220f i_1 : n, updateRow A i (Pi.single j 1) (\u2191\u03c3 i_1) i_1 = \u220f i_1 : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 i_1) i_1", "state_after": "case pos.e_f\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : i = \u2191\u03c3 j\n\u22a2 (fun i_1 => updateRow A i (Pi.single j 1) (\u2191\u03c3 i_1) i_1) = fun i_1 => updateColumn A j (Pi.single i 1) (\u2191\u03c3 i_1) i_1"}, {"tactic": "ext j'", "state_before": "case pos.e_f\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : i = \u2191\u03c3 j\n\u22a2 (fun i_1 => updateRow A i (Pi.single j 1) (\u2191\u03c3 i_1) i_1) = fun i_1 => updateColumn A j (Pi.single i 1) (\u2191\u03c3 i_1) i_1", "state_after": "case pos.e_f.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : i = \u2191\u03c3 j\nj' : n\n\u22a2 updateRow A i (Pi.single j 1) (\u2191\u03c3 j') j' = updateColumn A j (Pi.single i 1) (\u2191\u03c3 j') j'"}, {"tactic": "subst h", "state_before": "case pos.e_f.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : i = \u2191\u03c3 j\nj' : n\n\u22a2 updateRow A i (Pi.single j 1) (\u2191\u03c3 j') j' = updateColumn A j (Pi.single i 1) (\u2191\u03c3 j') j'", "state_after": "case pos.e_f.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nj : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nj' : n\n\u22a2 updateRow A (\u2191\u03c3 j) (Pi.single j 1) (\u2191\u03c3 j') j' = updateColumn A j (Pi.single (\u2191\u03c3 j) 1) (\u2191\u03c3 j') j'"}, {"tactic": "have : \u03c3 j' = \u03c3 j \u2194 j' = j := \u03c3.injective.eq_iff", "state_before": "case pos.e_f.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nj : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nj' : n\n\u22a2 updateRow A (\u2191\u03c3 j) (Pi.single j 1) (\u2191\u03c3 j') j' = updateColumn A j (Pi.single (\u2191\u03c3 j) 1) (\u2191\u03c3 j') j'", "state_after": "case pos.e_f.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nj : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nj' : n\nthis : \u2191\u03c3 j' = \u2191\u03c3 j \u2194 j' = j\n\u22a2 updateRow A (\u2191\u03c3 j) (Pi.single j 1) (\u2191\u03c3 j') j' = updateColumn A j (Pi.single (\u2191\u03c3 j) 1) (\u2191\u03c3 j') j'"}, {"tactic": "rw [updateRow_apply, updateColumn_apply]", "state_before": "case pos.e_f.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nj : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nj' : n\nthis : \u2191\u03c3 j' = \u2191\u03c3 j \u2194 j' = j\n\u22a2 updateRow A (\u2191\u03c3 j) (Pi.single j 1) (\u2191\u03c3 j') j' = updateColumn A j (Pi.single (\u2191\u03c3 j) 1) (\u2191\u03c3 j') j'", "state_after": "case pos.e_f.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nj : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nj' : n\nthis : \u2191\u03c3 j' = \u2191\u03c3 j \u2194 j' = j\n\u22a2 (if \u2191\u03c3 j' = \u2191\u03c3 j then Pi.single j 1 j' else A (\u2191\u03c3 j') j') =\n    if j' = j then Pi.single (\u2191\u03c3 j) 1 (\u2191\u03c3 j') else A (\u2191\u03c3 j') j'"}, {"tactic": "simp_rw [this]", "state_before": "case pos.e_f.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nj : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nj' : n\nthis : \u2191\u03c3 j' = \u2191\u03c3 j \u2194 j' = j\n\u22a2 (if \u2191\u03c3 j' = \u2191\u03c3 j then Pi.single j 1 j' else A (\u2191\u03c3 j') j') =\n    if j' = j then Pi.single (\u2191\u03c3 j) 1 (\u2191\u03c3 j') else A (\u2191\u03c3 j') j'", "state_after": "case pos.e_f.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nj : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nj' : n\nthis : \u2191\u03c3 j' = \u2191\u03c3 j \u2194 j' = j\n\u22a2 (if j' = j then Pi.single j 1 j' else A (\u2191\u03c3 j') j') = if j' = j then Pi.single (\u2191\u03c3 j) 1 (\u2191\u03c3 j') else A (\u2191\u03c3 j') j'"}, {"tactic": "rw [\u2190 dite_eq_ite, \u2190 dite_eq_ite]", "state_before": "case pos.e_f.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nj : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nj' : n\nthis : \u2191\u03c3 j' = \u2191\u03c3 j \u2194 j' = j\n\u22a2 (if j' = j then Pi.single j 1 j' else A (\u2191\u03c3 j') j') = if j' = j then Pi.single (\u2191\u03c3 j) 1 (\u2191\u03c3 j') else A (\u2191\u03c3 j') j'", "state_after": "case pos.e_f.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nj : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nj' : n\nthis : \u2191\u03c3 j' = \u2191\u03c3 j \u2194 j' = j\n\u22a2 (if x : j' = j then Pi.single j 1 j' else A (\u2191\u03c3 j') j') =\n    if x : j' = j then Pi.single (\u2191\u03c3 j) 1 (\u2191\u03c3 j') else A (\u2191\u03c3 j') j'"}, {"tactic": "congr 1 with rfl", "state_before": "case pos.e_f.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nj : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nj' : n\nthis : \u2191\u03c3 j' = \u2191\u03c3 j \u2194 j' = j\n\u22a2 (if x : j' = j then Pi.single j 1 j' else A (\u2191\u03c3 j') j') =\n    if x : j' = j then Pi.single (\u2191\u03c3 j) 1 (\u2191\u03c3 j') else A (\u2191\u03c3 j') j'", "state_after": "case pos.e_f.h.e_t.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nj' : n\nthis : \u2191\u03c3 j' = \u2191\u03c3 j' \u2194 j' = j'\n\u22a2 Pi.single j' 1 j' = Pi.single (\u2191\u03c3 j') 1 (\u2191\u03c3 j')"}, {"tactic": "rw [Pi.single_eq_same, Pi.single_eq_same]", "state_before": "case pos.e_f.h.e_t.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nj' : n\nthis : \u2191\u03c3 j' = \u2191\u03c3 j' \u2194 j' = j'\n\u22a2 Pi.single j' 1 j' = Pi.single (\u2191\u03c3 j') 1 (\u2191\u03c3 j')", "state_after": "no goals"}, {"tactic": "have : (\u220f j' : n, updateColumn A j (Pi.single i 1) (\u03c3 j') j') = 0 := by\n  apply prod_eq_zero (mem_univ j)\n  rw [updateColumn_self, Pi.single_eq_of_ne' h]", "state_before": "case neg\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : \u00aci = \u2191\u03c3 j\n\u22a2 \u220f i_1 : n, updateRow A i (Pi.single j 1) (\u2191\u03c3 i_1) i_1 = \u220f i_1 : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 i_1) i_1", "state_after": "case neg\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : \u00aci = \u2191\u03c3 j\nthis : \u220f j' : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 j') j' = 0\n\u22a2 \u220f i_1 : n, updateRow A i (Pi.single j 1) (\u2191\u03c3 i_1) i_1 = \u220f i_1 : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 i_1) i_1"}, {"tactic": "rw [this]", "state_before": "case neg\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : \u00aci = \u2191\u03c3 j\nthis : \u220f j' : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 j') j' = 0\n\u22a2 \u220f i_1 : n, updateRow A i (Pi.single j 1) (\u2191\u03c3 i_1) i_1 = \u220f i_1 : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 i_1) i_1", "state_after": "case neg\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : \u00aci = \u2191\u03c3 j\nthis : \u220f j' : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 j') j' = 0\n\u22a2 \u220f i_1 : n, updateRow A i (Pi.single j 1) (\u2191\u03c3 i_1) i_1 = 0"}, {"tactic": "apply prod_eq_zero (mem_univ (\u03c3\u207b\u00b9 i))", "state_before": "case neg\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : \u00aci = \u2191\u03c3 j\nthis : \u220f j' : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 j') j' = 0\n\u22a2 \u220f i_1 : n, updateRow A i (Pi.single j 1) (\u2191\u03c3 i_1) i_1 = 0", "state_after": "case neg\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : \u00aci = \u2191\u03c3 j\nthis : \u220f j' : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 j') j' = 0\n\u22a2 updateRow A i (Pi.single j 1) (\u2191\u03c3 (\u2191\u03c3\u207b\u00b9 i)) (\u2191\u03c3\u207b\u00b9 i) = 0"}, {"tactic": "erw [apply_symm_apply \u03c3 i, updateRow_self]", "state_before": "case neg\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : \u00aci = \u2191\u03c3 j\nthis : \u220f j' : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 j') j' = 0\n\u22a2 updateRow A i (Pi.single j 1) (\u2191\u03c3 (\u2191\u03c3\u207b\u00b9 i)) (\u2191\u03c3\u207b\u00b9 i) = 0", "state_after": "case neg\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : \u00aci = \u2191\u03c3 j\nthis : \u220f j' : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 j') j' = 0\n\u22a2 Pi.single j 1 (\u2191\u03c3\u207b\u00b9 i) = 0"}, {"tactic": "apply Pi.single_eq_of_ne", "state_before": "case neg\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : \u00aci = \u2191\u03c3 j\nthis : \u220f j' : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 j') j' = 0\n\u22a2 Pi.single j 1 (\u2191\u03c3\u207b\u00b9 i) = 0", "state_after": "case neg.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : \u00aci = \u2191\u03c3 j\nthis : \u220f j' : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 j') j' = 0\n\u22a2 \u2191\u03c3\u207b\u00b9 i \u2260 j"}, {"tactic": "intro h'", "state_before": "case neg.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : \u00aci = \u2191\u03c3 j\nthis : \u220f j' : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 j') j' = 0\n\u22a2 \u2191\u03c3\u207b\u00b9 i \u2260 j", "state_after": "case neg.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : \u00aci = \u2191\u03c3 j\nthis : \u220f j' : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 j') j' = 0\nh' : \u2191\u03c3\u207b\u00b9 i = j\n\u22a2 False"}, {"tactic": "exact h ((symm_apply_eq \u03c3).mp h')", "state_before": "case neg.h\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : \u00aci = \u2191\u03c3 j\nthis : \u220f j' : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 j') j' = 0\nh' : \u2191\u03c3\u207b\u00b9 i = j\n\u22a2 False", "state_after": "no goals"}, {"tactic": "apply prod_eq_zero (mem_univ j)", "state_before": "m : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : \u00aci = \u2191\u03c3 j\n\u22a2 \u220f j' : n, updateColumn A j (Pi.single i 1) (\u2191\u03c3 j') j' = 0", "state_after": "m : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : \u00aci = \u2191\u03c3 j\n\u22a2 updateColumn A j (Pi.single i 1) (\u2191\u03c3 j) j = 0"}, {"tactic": "rw [updateColumn_self, Pi.single_eq_of_ne' h]", "state_before": "m : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni j : n\n\u03c3 : Perm n\na\u271d : \u03c3 \u2208 univ\nh : \u00aci = \u2191\u03c3 j\n\u22a2 updateColumn A j (Pi.single i 1) (\u2191\u03c3 j) j = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Hom/Equiv/Basic.lean", "full_name": "MulEquiv.apply_symm_apply", "start": [347, 1], "end": [348, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "full_name": "Pmf.coe_le_one", "start": [135, 1], "end": [137, 55], "traced_tactics": [{"tactic": "refine' hasSum_le (fun b => _) (hasSum_ite_eq a (p a)) (hasSum_coe_one p)", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.16612\n\u03b3 : Type ?u.16615\np : Pmf \u03b1\na : \u03b1\n\u22a2 \u2191p a \u2264 1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.16612\n\u03b3 : Type ?u.16615\np : Pmf \u03b1\na b : \u03b1\n\u22a2 (if b = a then \u2191p a else 0) \u2264 \u2191p b"}, {"tactic": "split_ifs with h <;> simp only [h, zero_le', le_rfl]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.16612\n\u03b3 : Type ?u.16615\np : Pmf \u03b1\na b : \u03b1\n\u22a2 (if b = a then \u2191p a else 0) \u2264 \u2191p b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Submonoid/Pointwise.lean", "full_name": "Submonoid.mem_inv_pointwise_smul_iff", "start": [288, 1], "end": [289, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Derivative.lean", "full_name": "Polynomial.derivative_X_sub_C", "start": [606, 1], "end": [607, 60], "traced_tactics": [{"tactic": "rw [derivative_sub, derivative_X, derivative_C, sub_zero]", "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\nc : R\n\u22a2 \u2191derivative (X - \u2191C c) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "full_name": "hasGroupoid_of_pregroupoid", "start": [880, 1], "end": [883, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "full_name": "edist_le_of_edist_le_geometric_of_tendsto\u2080", "start": [330, 1], "end": [332, 100], "traced_tactics": [{"tactic": "simpa only [_root_.pow_zero, mul_one] using edist_le_of_edist_le_geometric_of_tendsto r C hu ha 0", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.436933\n\u03b9 : Type ?u.436936\ninst\u271d : PseudoEMetricSpace \u03b1\nr C : \u211d\u22650\u221e\nhr : r < 1\nhC : C \u2260 \u22a4\nf : \u2115 \u2192 \u03b1\nhu : \u2200 (n : \u2115), edist (f n) (f (n + 1)) \u2264 C * r ^ n\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\n\u22a2 edist (f 0) a \u2264 C / (1 - r)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM1.stmts_supportsStmt", "start": [1377, 1], "end": [1381, 60], "traced_tactics": [{"tactic": "simp only [stmts, Finset.mem_insertNone, Finset.mem_biUnion, Option.mem_def, Option.some.injEq,\n  forall_eq', exists_imp, and_imp]", "state_before": "\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nq : Stmt\u2081\nss : Supports M S\n\u22a2 some q \u2208 stmts M S \u2192 SupportsStmt S q", "state_after": "\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nq : Stmt\u2081\nss : Supports M S\n\u22a2 \u2200 (x : \u039b), x \u2208 S \u2192 q \u2208 stmts\u2081 (M x) \u2192 SupportsStmt S q"}, {"tactic": "exact fun l ls h \u21a6 stmts\u2081_supportsStmt_mono h (ss.2 _ ls)", "state_before": "\u0393 : Type u_1\ninst\u271d\u00b9 : Inhabited \u0393\n\u039b : Type u_2\n\u03c3 : Type u_3\ninst\u271d : Inhabited \u039b\nM : \u039b \u2192 Stmt\u2081\nS : Finset \u039b\nq : Stmt\u2081\nss : Supports M S\n\u22a2 \u2200 (x : \u039b), x \u2208 S \u2192 q \u2208 stmts\u2081 (M x) \u2192 SupportsStmt S q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.union_left_idem", "start": [1721, 1], "end": [1722, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Sigma.lean", "full_name": "Set.image_sigmaMk_subset_sigma_left", "start": [228, 1], "end": [230, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Antichain.lean", "full_name": "IsAntichain.swap", "start": [88, 1], "end": [89, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.takeI_left", "start": [2341, 1], "end": [2342, 89], "traced_tactics": [{"tactic": "simp only [length_append, Nat.le_add_right]", "state_before": "\u03b9 : Type ?u.226237\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 : Inhabited \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 length l\u2081 \u2264 length (l\u2081 ++ l\u2082)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/BigOperators/Basic.lean", "full_name": "List.prod_eq_pow_card", "start": [89, 1], "end": [90, 58], "traced_tactics": [{"tactic": "rw [\u2190 prod_replicate, \u2190 List.eq_replicate.mpr \u27e8rfl, h\u27e9]", "state_before": "\u03b9 : Type ?u.16618\n\u03b1 : Type ?u.16621\nM : Type u_1\nN : Type ?u.16627\nP : Type ?u.16630\nM\u2080 : Type ?u.16633\nG : Type ?u.16636\nR : Type ?u.16639\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\nl : List M\nm : M\nh : \u2200 (x : M), x \u2208 l \u2192 x = m\n\u22a2 prod l = m ^ length l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "full_name": "MvPolynomial.finSuccEquiv_X_zero", "start": [357, 1], "end": [357, 79], "traced_tactics": [{"tactic": "simp", "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type ?u.978708\na a' a\u2081 a\u2082 : R\ne : \u2115\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d : CommSemiring R\nn : \u2115\n\u22a2 \u2191(finSuccEquiv R n) (X 0) = Polynomial.X", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Defs.lean", "full_name": "pow_succ", "start": [635, 1], "end": [636, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Constructions.lean", "full_name": "Filter.Tendsto.apply", "start": [1212, 1], "end": [1214, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Basic.lean", "full_name": "exists\u2082_comm", "start": [828, 1], "end": [830, 51], "traced_tactics": [{"tactic": "simp only [@exists_comm (\u03ba\u2081 _), @exists_comm \u03b9\u2081]", "state_before": "\u03b9 : Sort ?u.20987\n\u03b1 : Sort ?u.20992\n\u03ba : \u03b9 \u2192 Sort ?u.20989\np\u271d q : \u03b1 \u2192 Prop\n\u03b9\u2081 : Sort u_1\n\u03b9\u2082 : Sort u_2\n\u03ba\u2081 : \u03b9\u2081 \u2192 Sort u_3\n\u03ba\u2082 : \u03b9\u2082 \u2192 Sort u_4\np : (i\u2081 : \u03b9\u2081) \u2192 \u03ba\u2081 i\u2081 \u2192 (i\u2082 : \u03b9\u2082) \u2192 \u03ba\u2082 i\u2082 \u2192 Prop\n\u22a2 (\u2203 i\u2081 j\u2081 i\u2082 j\u2082, p i\u2081 j\u2081 i\u2082 j\u2082) \u2194 \u2203 i\u2082 j\u2082 i\u2081 j\u2081, p i\u2081 j\u2081 i\u2082 j\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "full_name": "GroupWithZero.mul_right_injective", "start": [273, 1], "end": [275, 91], "traced_tactics": [{"tactic": "simpa only [mul_assoc, mul_inv_cancel _ h, mul_one] using congr_arg (fun y => y * x\u207b\u00b9) w", "state_before": "\u03b1 : Type ?u.18629\nM\u2080 : Type ?u.18632\nG\u2080 : Type u_1\nM\u2080' : Type ?u.18638\nG\u2080' : Type ?u.18641\nF : Type ?u.18644\nF' : Type ?u.18647\ninst\u271d : GroupWithZero G\u2080\na b c g h\u271d x : G\u2080\nh : x \u2260 0\ny y' : G\u2080\nw : (fun y => y * x) y = (fun y => y * x) y'\n\u22a2 y = y'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/Ordered.lean", "full_name": "lineMap_strict_mono_right", "start": [69, 1], "end": [71, 54], "traced_tactics": [{"tactic": "simp only [lineMap_apply_module]", "state_before": "k : Type u_2\nE : Type u_1\nPE : Type ?u.22259\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\na a' b b' : E\nr r' : k\nhb : b < b'\nhr : 0 < r\n\u22a2 \u2191(lineMap a b) r < \u2191(lineMap a b') r", "state_after": "k : Type u_2\nE : Type u_1\nPE : Type ?u.22259\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\na a' b b' : E\nr r' : k\nhb : b < b'\nhr : 0 < r\n\u22a2 (1 - r) \u2022 a + r \u2022 b < (1 - r) \u2022 a + r \u2022 b'"}, {"tactic": "exact add_lt_add_left (smul_lt_smul_of_pos hb hr) _", "state_before": "k : Type u_2\nE : Type u_1\nPE : Type ?u.22259\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : OrderedSMul k E\na a' b b' : E\nr r' : k\nhb : b < b'\nhr : 0 < r\n\u22a2 (1 - r) \u2022 a + r \u2022 b < (1 - r) \u2022 a + r \u2022 b'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GroupWithZero/Power.lean", "full_name": "inv_pow_sub\u2080", "start": [44, 1], "end": [45, 64], "traced_tactics": [{"tactic": "rw [pow_sub\u2080 _ (inv_ne_zero ha) h, inv_pow, inv_pow, inv_inv]", "state_before": "G\u2080 : Type u_1\ninst\u271d : GroupWithZero G\u2080\na : G\u2080\nm n : \u2115\nha : a \u2260 0\nh : n \u2264 m\n\u22a2 a\u207b\u00b9 ^ (m - n) = (a ^ m)\u207b\u00b9 * a ^ n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Fold.lean", "full_name": "Finset.fold_max_lt", "start": [254, 1], "end": [259, 19], "traced_tactics": [{"tactic": "show _ > _ \u2194 _", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.70885\nop : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nhc : IsCommutative \u03b2 op\nha : IsAssociative \u03b2 op\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns : Finset \u03b1\na : \u03b1\ninst\u271d : LinearOrder \u03b2\nc : \u03b2\n\u22a2 fold max b f s < c \u2194 b < c \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 f x < c", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.70885\nop : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nhc : IsCommutative \u03b2 op\nha : IsAssociative \u03b2 op\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns : Finset \u03b1\na : \u03b1\ninst\u271d : LinearOrder \u03b2\nc : \u03b2\n\u22a2 c > fold max b f s \u2194 b < c \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 f x < c"}, {"tactic": "apply fold_op_rel_iff_and", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.70885\nop : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nhc : IsCommutative \u03b2 op\nha : IsAssociative \u03b2 op\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns : Finset \u03b1\na : \u03b1\ninst\u271d : LinearOrder \u03b2\nc : \u03b2\n\u22a2 c > fold max b f s \u2194 b < c \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 f x < c", "state_after": "case hr\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.70885\nop : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nhc : IsCommutative \u03b2 op\nha : IsAssociative \u03b2 op\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns : Finset \u03b1\na : \u03b1\ninst\u271d : LinearOrder \u03b2\nc : \u03b2\n\u22a2 \u2200 {x y z : \u03b2}, x > max y z \u2194 x > y \u2227 x > z"}, {"tactic": "intro x y z", "state_before": "case hr\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.70885\nop : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nhc : IsCommutative \u03b2 op\nha : IsAssociative \u03b2 op\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns : Finset \u03b1\na : \u03b1\ninst\u271d : LinearOrder \u03b2\nc : \u03b2\n\u22a2 \u2200 {x y z : \u03b2}, x > max y z \u2194 x > y \u2227 x > z", "state_after": "case hr\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.70885\nop : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nhc : IsCommutative \u03b2 op\nha : IsAssociative \u03b2 op\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns : Finset \u03b1\na : \u03b1\ninst\u271d : LinearOrder \u03b2\nc x y z : \u03b2\n\u22a2 x > max y z \u2194 x > y \u2227 x > z"}, {"tactic": "show _ < _ \u2194 _", "state_before": "case hr\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.70885\nop : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nhc : IsCommutative \u03b2 op\nha : IsAssociative \u03b2 op\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns : Finset \u03b1\na : \u03b1\ninst\u271d : LinearOrder \u03b2\nc x y z : \u03b2\n\u22a2 x > max y z \u2194 x > y \u2227 x > z", "state_after": "case hr\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.70885\nop : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nhc : IsCommutative \u03b2 op\nha : IsAssociative \u03b2 op\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns : Finset \u03b1\na : \u03b1\ninst\u271d : LinearOrder \u03b2\nc x y z : \u03b2\n\u22a2 max y z < x \u2194 x > y \u2227 x > z"}, {"tactic": "exact max_lt_iff", "state_before": "case hr\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.70885\nop : \u03b2 \u2192 \u03b2 \u2192 \u03b2\nhc : IsCommutative \u03b2 op\nha : IsAssociative \u03b2 op\nf : \u03b1 \u2192 \u03b2\nb : \u03b2\ns : Finset \u03b1\na : \u03b1\ninst\u271d : LinearOrder \u03b2\nc x y z : \u03b2\n\u22a2 max y z < x \u2194 x > y \u2227 x > z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Exponent.lean", "full_name": "Monoid.exponent_eq_iSup_orderOf'", "start": [331, 1], "end": [337, 64], "traced_tactics": [{"tactic": "split_ifs with h", "state_before": "G : Type u\ninst\u271d : CommMonoid G\n\u22a2 exponent G = if \u2203 g, orderOf g = 0 then 0 else \u2a06 (g : G), orderOf g", "state_after": "case inl\nG : Type u\ninst\u271d : CommMonoid G\nh : \u2203 g, orderOf g = 0\n\u22a2 exponent G = 0\n\ncase inr\nG : Type u\ninst\u271d : CommMonoid G\nh : \u00ac\u2203 g, orderOf g = 0\n\u22a2 exponent G = \u2a06 (g : G), orderOf g"}, {"tactic": "obtain \u27e8g, hg\u27e9 := h", "state_before": "case inl\nG : Type u\ninst\u271d : CommMonoid G\nh : \u2203 g, orderOf g = 0\n\u22a2 exponent G = 0", "state_after": "case inl.intro\nG : Type u\ninst\u271d : CommMonoid G\ng : G\nhg : orderOf g = 0\n\u22a2 exponent G = 0"}, {"tactic": "exact exponent_eq_zero_of_order_zero hg", "state_before": "case inl.intro\nG : Type u\ninst\u271d : CommMonoid G\ng : G\nhg : orderOf g = 0\n\u22a2 exponent G = 0", "state_after": "no goals"}, {"tactic": "have := not_exists.mp h", "state_before": "case inr\nG : Type u\ninst\u271d : CommMonoid G\nh : \u00ac\u2203 g, orderOf g = 0\n\u22a2 exponent G = \u2a06 (g : G), orderOf g", "state_after": "case inr\nG : Type u\ninst\u271d : CommMonoid G\nh : \u00ac\u2203 g, orderOf g = 0\nthis : \u2200 (x : G), \u00acorderOf x = 0\n\u22a2 exponent G = \u2a06 (g : G), orderOf g"}, {"tactic": "exact exponent_eq_iSup_orderOf fun g => Ne.bot_lt <| this g", "state_before": "case inr\nG : Type u\ninst\u271d : CommMonoid G\nh : \u00ac\u2203 g, orderOf g = 0\nthis : \u2200 (x : G), \u00acorderOf x = 0\n\u22a2 exponent G = \u2a06 (g : G), orderOf g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/ToIntervalMod.lean", "full_name": "toIcoDiv_sub_zsmul'", "start": [284, 1], "end": [285, 74], "traced_tactics": [{"tactic": "rw [sub_eq_add_neg, \u2190 neg_smul, toIcoDiv_add_zsmul', sub_neg_eq_add]", "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\nm : \u2124\n\u22a2 toIcoDiv hp (a - m \u2022 p) b = toIcoDiv hp a b + m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "interior_frontier", "start": [774, 1], "end": [780, 50], "traced_tactics": [{"tactic": "have A : frontier s = s \\ interior s := h.frontier_eq", "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\nh : IsClosed s\n\u22a2 interior (frontier s) = \u2205", "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\nh : IsClosed s\nA : frontier s = s \\ interior s\n\u22a2 interior (frontier s) = \u2205"}, {"tactic": "have B : interior (frontier s) \u2286 interior s := by rw [A]; exact interior_mono (diff_subset _ _)", "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\nh : IsClosed s\nA : frontier s = s \\ interior s\n\u22a2 interior (frontier s) = \u2205", "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\nh : IsClosed s\nA : frontier s = s \\ interior s\nB : interior (frontier s) \u2286 interior s\n\u22a2 interior (frontier s) = \u2205"}, {"tactic": "have C : interior (frontier s) \u2286 frontier s := interior_subset", "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\nh : IsClosed s\nA : frontier s = s \\ interior s\nB : interior (frontier s) \u2286 interior s\n\u22a2 interior (frontier s) = \u2205", "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\nh : IsClosed s\nA : frontier s = s \\ interior s\nB : interior (frontier s) \u2286 interior s\nC : interior (frontier s) \u2286 frontier s\n\u22a2 interior (frontier s) = \u2205"}, {"tactic": "have : interior (frontier s) \u2286 interior s \u2229 (s \\ interior s) :=\n  subset_inter B (by simpa [A] using C)", "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\nh : IsClosed s\nA : frontier s = s \\ interior s\nB : interior (frontier s) \u2286 interior s\nC : interior (frontier s) \u2286 frontier s\n\u22a2 interior (frontier s) = \u2205", "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\nh : IsClosed s\nA : frontier s = s \\ interior s\nB : interior (frontier s) \u2286 interior s\nC : interior (frontier s) \u2286 frontier s\nthis : interior (frontier s) \u2286 interior s \u2229 (s \\ interior s)\n\u22a2 interior (frontier s) = \u2205"}, {"tactic": "rwa [inter_diff_self, subset_empty_iff] at this", "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\nh : IsClosed s\nA : frontier s = s \\ interior s\nB : interior (frontier s) \u2286 interior s\nC : interior (frontier s) \u2286 frontier s\nthis : interior (frontier s) \u2286 interior s \u2229 (s \\ interior s)\n\u22a2 interior (frontier s) = \u2205", "state_after": "no goals"}, {"tactic": "rw [A]", "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\nh : IsClosed s\nA : frontier s = s \\ interior s\n\u22a2 interior (frontier s) \u2286 interior 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\nh : IsClosed s\nA : frontier s = s \\ interior s\n\u22a2 interior (s \\ interior s) \u2286 interior s"}, {"tactic": "exact interior_mono (diff_subset _ _)", "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\nh : IsClosed s\nA : frontier s = s \\ interior s\n\u22a2 interior (s \\ interior s) \u2286 interior s", "state_after": "no goals"}, {"tactic": "simpa [A] using C", "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\nh : IsClosed s\nA : frontier s = s \\ interior s\nB : interior (frontier s) \u2286 interior s\nC : interior (frontier s) \u2286 frontier s\n\u22a2 interior (frontier s) \u2286 s \\ interior s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Hom/Lattice.lean", "full_name": "SupHom.sup_apply", "start": [499, 1], "end": [500, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Coprime/Basic.lean", "full_name": "IsCoprime.ne_zero", "start": [77, 1], "end": [79, 34], "traced_tactics": [{"tactic": "rintro rfl", "state_before": "R : Type u\ninst\u271d\u00b9 : CommSemiring R\nx y z : R\ninst\u271d : Nontrivial R\np : Fin 2 \u2192 R\nh : IsCoprime (p 0) (p 1)\n\u22a2 p \u2260 0", "state_after": "R : Type u\ninst\u271d\u00b9 : CommSemiring R\nx y z : R\ninst\u271d : Nontrivial R\nh : IsCoprime (OfNat.ofNat 0 0) (OfNat.ofNat 0 1)\n\u22a2 False"}, {"tactic": "exact not_isCoprime_zero_zero h", "state_before": "R : Type u\ninst\u271d\u00b9 : CommSemiring R\nx y z : R\ninst\u271d : Nontrivial R\nh : IsCoprime (OfNat.ofNat 0 0) (OfNat.ofNat 0 1)\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "isOpen_iff_mem_nhds", "start": [1236, 1], "end": [1237, 92], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Hom/Iterate.lean", "full_name": "hom_coe_pow", "start": [37, 1], "end": [42, 78], "traced_tactics": [{"tactic": "rw [pow_zero, h1]", "state_before": "M : Type u_2\nN : Type ?u.17\nG : Type ?u.20\nH : Type ?u.23\nF : Type u_1\ninst\u271d : Monoid F\nc : F \u2192 M \u2192 M\nh1 : c 1 = id\nhmul : \u2200 (f g : F), c (f * g) = c f \u2218 c g\nf : F\n\u22a2 c (f ^ 0) = c f^[0]", "state_after": "M : Type u_2\nN : Type ?u.17\nG : Type ?u.20\nH : Type ?u.23\nF : Type u_1\ninst\u271d : Monoid F\nc : F \u2192 M \u2192 M\nh1 : c 1 = id\nhmul : \u2200 (f g : F), c (f * g) = c f \u2218 c g\nf : F\n\u22a2 id = c f^[0]"}, {"tactic": "rfl", "state_before": "M : Type u_2\nN : Type ?u.17\nG : Type ?u.20\nH : Type ?u.23\nF : Type u_1\ninst\u271d : Monoid F\nc : F \u2192 M \u2192 M\nh1 : c 1 = id\nhmul : \u2200 (f g : F), c (f * g) = c f \u2218 c g\nf : F\n\u22a2 id = c f^[0]", "state_after": "no goals"}, {"tactic": "rw [pow_succ, iterate_succ', hmul, hom_coe_pow c h1 hmul f n]", "state_before": "M : Type u_2\nN : Type ?u.17\nG : Type ?u.20\nH : Type ?u.23\nF : Type u_1\ninst\u271d : Monoid F\nc : F \u2192 M \u2192 M\nh1 : c 1 = id\nhmul : \u2200 (f g : F), c (f * g) = c f \u2218 c g\nf : F\nn : \u2115\n\u22a2 c (f ^ (n + 1)) = c f^[n + 1]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/Basic.lean", "full_name": "CauSeq.Completion.ofRat_rat", "start": [48, 1], "end": [50, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Choose/Multinomial.lean", "full_name": "Nat.binomial_one", "start": [122, 1], "end": [124, 74], "traced_tactics": [{"tactic": "simp [multinomial_insert_one {b} f (Finset.not_mem_singleton.mpr h) h\u2081]", "state_before": "\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na b : \u03b1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\nh : a \u2260 b\nh\u2081 : f a = 1\n\u22a2 multinomial {a, b} f = succ (f b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Sylow.lean", "full_name": "Sylow.conj_eq_normalizer_conj_of_mem_centralizer", "start": [375, 1], "end": [389, 101], "traced_tactics": [{"tactic": "have h1 : \u2191P \u2264 (zpowers x).centralizer := by rwa [le_centralizer_iff, zpowers_le]", "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nx g : G\nhx : x \u2208 centralizer \u2191P\nhy : g\u207b\u00b9 * x * g \u2208 centralizer \u2191P\n\u22a2 \u2203 n, n \u2208 normalizer \u2191P \u2227 g\u207b\u00b9 * x * g = n\u207b\u00b9 * x * n", "state_after": "p : \u2115\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nx g : G\nhx : x \u2208 centralizer \u2191P\nhy : g\u207b\u00b9 * x * g \u2208 centralizer \u2191P\nh1 : \u2191P \u2264 centralizer (zpowers x)\n\u22a2 \u2203 n, n \u2208 normalizer \u2191P \u2227 g\u207b\u00b9 * x * g = n\u207b\u00b9 * x * n"}, {"tactic": "have h2 : \u2191(g \u2022 P) \u2264 (zpowers x).centralizer := by\n  rw [le_centralizer_iff, zpowers_le]\n  rintro - \u27e8z, hz, rfl\u27e9\n  specialize hy z hz\n  rwa [\u2190 mul_assoc, \u2190 eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, \u2190 mul_assoc,\n    eq_inv_mul_iff_mul_eq, \u2190 mul_assoc, \u2190 mul_assoc] at hy", "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nx g : G\nhx : x \u2208 centralizer \u2191P\nhy : g\u207b\u00b9 * x * g \u2208 centralizer \u2191P\nh1 : \u2191P \u2264 centralizer (zpowers x)\n\u22a2 \u2203 n, n \u2208 normalizer \u2191P \u2227 g\u207b\u00b9 * x * g = n\u207b\u00b9 * x * n", "state_after": "p : \u2115\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nx g : G\nhx : x \u2208 centralizer \u2191P\nhy : g\u207b\u00b9 * x * g \u2208 centralizer \u2191P\nh1 : \u2191P \u2264 centralizer (zpowers x)\nh2 : \u2191(g \u2022 P) \u2264 centralizer (zpowers x)\n\u22a2 \u2203 n, n \u2208 normalizer \u2191P \u2227 g\u207b\u00b9 * x * g = n\u207b\u00b9 * x * n"}, {"tactic": "obtain \u27e8h, hh\u27e9 := exists_smul_eq (zpowers x).centralizer ((g \u2022 P).subtype h2) (P.subtype h1)", "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nx g : G\nhx : x \u2208 centralizer \u2191P\nhy : g\u207b\u00b9 * x * g \u2208 centralizer \u2191P\nh1 : \u2191P \u2264 centralizer (zpowers x)\nh2 : \u2191(g \u2022 P) \u2264 centralizer (zpowers x)\n\u22a2 \u2203 n, n \u2208 normalizer \u2191P \u2227 g\u207b\u00b9 * x * g = n\u207b\u00b9 * x * n", "state_after": "case intro\np : \u2115\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nx g : G\nhx : x \u2208 centralizer \u2191P\nhy : g\u207b\u00b9 * x * g \u2208 centralizer \u2191P\nh1 : \u2191P \u2264 centralizer (zpowers x)\nh2 : \u2191(g \u2022 P) \u2264 centralizer (zpowers x)\nh : { x_1 // x_1 \u2208 centralizer (zpowers x) }\nhh : h \u2022 Sylow.subtype (g \u2022 P) h2 = Sylow.subtype P h1\n\u22a2 \u2203 n, n \u2208 normalizer \u2191P \u2227 g\u207b\u00b9 * x * g = n\u207b\u00b9 * x * n"}, {"tactic": "simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh", "state_before": "case intro\np : \u2115\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nx g : G\nhx : x \u2208 centralizer \u2191P\nhy : g\u207b\u00b9 * x * g \u2208 centralizer \u2191P\nh1 : \u2191P \u2264 centralizer (zpowers x)\nh2 : \u2191(g \u2022 P) \u2264 centralizer (zpowers x)\nh : { x_1 // x_1 \u2208 centralizer (zpowers x) }\nhh : h \u2022 Sylow.subtype (g \u2022 P) h2 = Sylow.subtype P h1\n\u22a2 \u2203 n, n \u2208 normalizer \u2191P \u2227 g\u207b\u00b9 * x * g = n\u207b\u00b9 * x * n", "state_after": "case intro\np : \u2115\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nx g : G\nhx : x \u2208 centralizer \u2191P\nhy : g\u207b\u00b9 * x * g \u2208 centralizer \u2191P\nh1 : \u2191P \u2264 centralizer (zpowers x)\nh2 : \u2191(g \u2022 P) \u2264 centralizer (zpowers x)\nh : { x_1 // x_1 \u2208 centralizer (zpowers x) }\nhh : Sylow.subtype ((\u2191h * g) \u2022 P) (_ : \u2191((\u2191h * g) \u2022 P) \u2264 centralizer (zpowers x)) = Sylow.subtype P h1\n\u22a2 \u2203 n, n \u2208 normalizer \u2191P \u2227 g\u207b\u00b9 * x * g = n\u207b\u00b9 * x * n"}, {"tactic": "refine' \u27e8h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _\u27e9", "state_before": "case intro\np : \u2115\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nx g : G\nhx : x \u2208 centralizer \u2191P\nhy : g\u207b\u00b9 * x * g \u2208 centralizer \u2191P\nh1 : \u2191P \u2264 centralizer (zpowers x)\nh2 : \u2191(g \u2022 P) \u2264 centralizer (zpowers x)\nh : { x_1 // x_1 \u2208 centralizer (zpowers x) }\nhh : Sylow.subtype ((\u2191h * g) \u2022 P) (_ : \u2191((\u2191h * g) \u2022 P) \u2264 centralizer (zpowers x)) = Sylow.subtype P h1\n\u22a2 \u2203 n, n \u2208 normalizer \u2191P \u2227 g\u207b\u00b9 * x * g = n\u207b\u00b9 * x * n", "state_after": "case intro\np : \u2115\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nx g : G\nhx : x \u2208 centralizer \u2191P\nhy : g\u207b\u00b9 * x * g \u2208 centralizer \u2191P\nh1 : \u2191P \u2264 centralizer (zpowers x)\nh2 : \u2191(g \u2022 P) \u2264 centralizer (zpowers x)\nh : { x_1 // x_1 \u2208 centralizer (zpowers x) }\nhh : Sylow.subtype ((\u2191h * g) \u2022 P) (_ : \u2191((\u2191h * g) \u2022 P) \u2264 centralizer (zpowers x)) = Sylow.subtype P h1\n\u22a2 g\u207b\u00b9 * x * g = (\u2191h * g)\u207b\u00b9 * x * (\u2191h * g)"}, {"tactic": "rw [\u2190 mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]", "state_before": "case intro\np : \u2115\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nx g : G\nhx : x \u2208 centralizer \u2191P\nhy : g\u207b\u00b9 * x * g \u2208 centralizer \u2191P\nh1 : \u2191P \u2264 centralizer (zpowers x)\nh2 : \u2191(g \u2022 P) \u2264 centralizer (zpowers x)\nh : { x_1 // x_1 \u2208 centralizer (zpowers x) }\nhh : Sylow.subtype ((\u2191h * g) \u2022 P) (_ : \u2191((\u2191h * g) \u2022 P) \u2264 centralizer (zpowers x)) = Sylow.subtype P h1\n\u22a2 g\u207b\u00b9 * x * g = (\u2191h * g)\u207b\u00b9 * x * (\u2191h * g)", "state_after": "no goals"}, {"tactic": "rwa [le_centralizer_iff, zpowers_le]", "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nx g : G\nhx : x \u2208 centralizer \u2191P\nhy : g\u207b\u00b9 * x * g \u2208 centralizer \u2191P\n\u22a2 \u2191P \u2264 centralizer (zpowers x)", "state_after": "no goals"}, {"tactic": "rw [le_centralizer_iff, zpowers_le]", "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nx g : G\nhx : x \u2208 centralizer \u2191P\nhy : g\u207b\u00b9 * x * g \u2208 centralizer \u2191P\nh1 : \u2191P \u2264 centralizer (zpowers x)\n\u22a2 \u2191(g \u2022 P) \u2264 centralizer (zpowers x)", "state_after": "p : \u2115\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nx g : G\nhx : x \u2208 centralizer \u2191P\nhy : g\u207b\u00b9 * x * g \u2208 centralizer \u2191P\nh1 : \u2191P \u2264 centralizer (zpowers x)\n\u22a2 x \u2208 centralizer \u2191(g \u2022 P)"}, {"tactic": "rintro - \u27e8z, hz, rfl\u27e9", "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nx g : G\nhx : x \u2208 centralizer \u2191P\nhy : g\u207b\u00b9 * x * g \u2208 centralizer \u2191P\nh1 : \u2191P \u2264 centralizer (zpowers x)\n\u22a2 x \u2208 centralizer \u2191(g \u2022 P)", "state_after": "case intro.intro\np : \u2115\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nx g : G\nhx : x \u2208 centralizer \u2191P\nhy : g\u207b\u00b9 * x * g \u2208 centralizer \u2191P\nh1 : \u2191P \u2264 centralizer (zpowers x)\nz : G\nhz : z \u2208 \u2191\u2191P\n\u22a2 \u2191(\u2191(MulDistribMulAction.toMonoidEnd (MulAut G) G) (\u2191MulAut.conj g)) z * x =\n    x * \u2191(\u2191(MulDistribMulAction.toMonoidEnd (MulAut G) G) (\u2191MulAut.conj g)) z"}, {"tactic": "specialize hy z hz", "state_before": "case intro.intro\np : \u2115\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nx g : G\nhx : x \u2208 centralizer \u2191P\nhy : g\u207b\u00b9 * x * g \u2208 centralizer \u2191P\nh1 : \u2191P \u2264 centralizer (zpowers x)\nz : G\nhz : z \u2208 \u2191\u2191P\n\u22a2 \u2191(\u2191(MulDistribMulAction.toMonoidEnd (MulAut G) G) (\u2191MulAut.conj g)) z * x =\n    x * \u2191(\u2191(MulDistribMulAction.toMonoidEnd (MulAut G) G) (\u2191MulAut.conj g)) z", "state_after": "case intro.intro\np : \u2115\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nx g : G\nhx : x \u2208 centralizer \u2191P\nh1 : \u2191P \u2264 centralizer (zpowers x)\nz : G\nhz : z \u2208 \u2191\u2191P\nhy : z * (g\u207b\u00b9 * x * g) = g\u207b\u00b9 * x * g * z\n\u22a2 \u2191(\u2191(MulDistribMulAction.toMonoidEnd (MulAut G) G) (\u2191MulAut.conj g)) z * x =\n    x * \u2191(\u2191(MulDistribMulAction.toMonoidEnd (MulAut G) G) (\u2191MulAut.conj g)) z"}, {"tactic": "rwa [\u2190 mul_assoc, \u2190 eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, \u2190 mul_assoc,\n  eq_inv_mul_iff_mul_eq, \u2190 mul_assoc, \u2190 mul_assoc] at hy", "state_before": "case intro.intro\np : \u2115\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nx g : G\nhx : x \u2208 centralizer \u2191P\nh1 : \u2191P \u2264 centralizer (zpowers x)\nz : G\nhz : z \u2208 \u2191\u2191P\nhy : z * (g\u207b\u00b9 * x * g) = g\u207b\u00b9 * x * g * z\n\u22a2 \u2191(\u2191(MulDistribMulAction.toMonoidEnd (MulAut G) G) (\u2191MulAut.conj g)) z * x =\n    x * \u2191(\u2191(MulDistribMulAction.toMonoidEnd (MulAut G) G) (\u2191MulAut.conj g)) z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "full_name": "ModelWithCorners.continuous", "start": [222, 11], "end": [223, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/List/Lemmas.lean", "full_name": "List.length_removeNth", "start": [855, 1], "end": [861, 93], "traced_tactics": [{"tactic": "simp [removeNth]", "state_before": "\u03b1 : Type u_1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nx\u271d : 0 < length (head\u271d :: tail\u271d)\n\u22a2 length (removeNth (head\u271d :: tail\u271d) 0) = length (head\u271d :: tail\u271d) - 1", "state_after": "no goals"}, {"tactic": "have : i < length xs := Nat.lt_of_succ_lt_succ h", "state_before": "\u03b1 : Type u_1\nx : \u03b1\nxs : List \u03b1\ni : Nat\nh : i + 1 < length (x :: xs)\n\u22a2 length (removeNth (x :: xs) (i + 1)) = length (x :: xs) - 1", "state_after": "\u03b1 : Type u_1\nx : \u03b1\nxs : List \u03b1\ni : Nat\nh : i + 1 < length (x :: xs)\nthis : i < length xs\n\u22a2 length (removeNth (x :: xs) (i + 1)) = length (x :: xs) - 1"}, {"tactic": "simp [removeNth, \u2190 Nat.add_one]", "state_before": "\u03b1 : Type u_1\nx : \u03b1\nxs : List \u03b1\ni : Nat\nh : i + 1 < length (x :: xs)\nthis : i < length xs\n\u22a2 length (removeNth (x :: xs) (i + 1)) = length (x :: xs) - 1", "state_after": "\u03b1 : Type u_1\nx : \u03b1\nxs : List \u03b1\ni : Nat\nh : i + 1 < length (x :: xs)\nthis : i < length xs\n\u22a2 length (removeNth xs i) + 1 = length xs"}, {"tactic": "rw [length_removeNth this, Nat.sub_add_cancel (Nat.lt_of_le_of_lt (Nat.zero_le _) this)]", "state_before": "\u03b1 : Type u_1\nx : \u03b1\nxs : List \u03b1\ni : Nat\nh : i + 1 < length (x :: xs)\nthis : i < length xs\n\u22a2 length (removeNth xs i) + 1 = length xs", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean", "full_name": "PrimeSpectrum.basicOpen_mul_le_left", "start": [806, 1], "end": [808, 20], "traced_tactics": [{"tactic": "rw [basicOpen_mul f g]", "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf g : R\n\u22a2 basicOpen (f * g) \u2264 basicOpen f", "state_after": "R : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf g : R\n\u22a2 basicOpen f \u2293 basicOpen g \u2264 basicOpen f"}, {"tactic": "exact inf_le_left", "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf g : R\n\u22a2 basicOpen f \u2293 basicOpen g \u2264 basicOpen f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "aestronglyMeasurable_of_aestronglyMeasurable_trim", "start": [1540, 1], "end": [1543, 98], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Laurent.lean", "full_name": "LaurentPolynomial.induction_on_mul_T", "start": [409, 1], "end": [414, 11], "traced_tactics": [{"tactic": "rcases f.exists_T_pow with \u27e8n, f', hf\u27e9", "state_before": "R : Type u_1\ninst\u271d : Semiring R\nQ : R[T;T\u207b\u00b9] \u2192 Prop\nf : R[T;T\u207b\u00b9]\nQf : \u2200 {f : R[X]} {n : \u2115}, Q (\u2191toLaurent f * T (-\u2191n))\n\u22a2 Q f", "state_after": "case intro.intro\nR : Type u_1\ninst\u271d : Semiring R\nQ : R[T;T\u207b\u00b9] \u2192 Prop\nf : R[T;T\u207b\u00b9]\nQf : \u2200 {f : R[X]} {n : \u2115}, Q (\u2191toLaurent f * T (-\u2191n))\nn : \u2115\nf' : R[X]\nhf : \u2191toLaurent f' = f * T \u2191n\n\u22a2 Q f"}, {"tactic": "rw [\u2190 mul_one f, \u2190 T_zero, \u2190 Nat.cast_zero, \u2190 Nat.sub_self n, Nat.cast_sub rfl.le, T_sub,\n  \u2190 mul_assoc, \u2190 hf]", "state_before": "case intro.intro\nR : Type u_1\ninst\u271d : Semiring R\nQ : R[T;T\u207b\u00b9] \u2192 Prop\nf : R[T;T\u207b\u00b9]\nQf : \u2200 {f : R[X]} {n : \u2115}, Q (\u2191toLaurent f * T (-\u2191n))\nn : \u2115\nf' : R[X]\nhf : \u2191toLaurent f' = f * T \u2191n\n\u22a2 Q f", "state_after": "case intro.intro\nR : Type u_1\ninst\u271d : Semiring R\nQ : R[T;T\u207b\u00b9] \u2192 Prop\nf : R[T;T\u207b\u00b9]\nQf : \u2200 {f : R[X]} {n : \u2115}, Q (\u2191toLaurent f * T (-\u2191n))\nn : \u2115\nf' : R[X]\nhf : \u2191toLaurent f' = f * T \u2191n\n\u22a2 Q (\u2191toLaurent f' * T (-\u2191n))"}, {"tactic": "exact Qf", "state_before": "case intro.intro\nR : Type u_1\ninst\u271d : Semiring R\nQ : R[T;T\u207b\u00b9] \u2192 Prop\nf : R[T;T\u207b\u00b9]\nQf : \u2200 {f : R[X]} {n : \u2115}, Q (\u2191toLaurent f * T (-\u2191n))\nn : \u2115\nf' : R[X]\nhf : \u2191toLaurent f' = f * T \u2191n\n\u22a2 Q (\u2191toLaurent f' * T (-\u2191n))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Abelian/Homology.lean", "full_name": "homology.condition_\u03c0'", "start": [154, 1], "end": [156, 7], "traced_tactics": [{"tactic": "dsimp [\u03c0']", "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\n\u22a2 kernel.lift g f w \u226b \u03c0' f g w = 0", "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\n\u22a2 kernel.lift g f w \u226b cokernel.\u03c0 (kernel.lift g f w) \u226b (homologyIsoCokernelLift f g w).inv = 0"}, {"tactic": "simp", "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\n\u22a2 kernel.lift g f w \u226b cokernel.\u03c0 (kernel.lift g f w) \u226b (homologyIsoCokernelLift f g w).inv = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Pairwise.lean", "full_name": "Pairwise.mono", "start": [41, 1], "end": [42, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Sym/Sym2.lean", "full_name": "Sym2.fromRel_bot", "start": [503, 1], "end": [506, 46], "traced_tactics": [{"tactic": "apply Set.eq_empty_of_forall_not_mem fun e => _", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.42295\n\u03b3 : Type ?u.42298\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 fromRel (_ : \u2200 (x y : \u03b1), \u22a5 x y \u2192 \u22a5 x y) = \u2205", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.42295\n\u03b3 : Type ?u.42298\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u2200 (e : Sym2 \u03b1), \u00ace \u2208 fromRel (_ : \u2200 (x y : \u03b1), \u22a5 x y \u2192 \u22a5 x y)"}, {"tactic": "apply Sym2.ind", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.42295\n\u03b3 : Type ?u.42298\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u2200 (e : Sym2 \u03b1), \u00ace \u2208 fromRel (_ : \u2200 (x y : \u03b1), \u22a5 x y \u2192 \u22a5 x y)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.42295\n\u03b3 : Type ?u.42298\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u2200 (x y : \u03b1), \u00acQuotient.mk (Rel.setoid \u03b1) (x, y) \u2208 fromRel (_ : \u2200 (x y : \u03b1), \u22a5 x y \u2192 \u22a5 x y)"}, {"tactic": "simp [-Set.bot_eq_empty, Prop.bot_eq_false]", "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.42295\n\u03b3 : Type ?u.42298\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u2200 (x y : \u03b1), \u00acQuotient.mk (Rel.setoid \u03b1) (x, y) \u2208 fromRel (_ : \u2200 (x y : \u03b1), \u22a5 x y \u2192 \u22a5 x y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Logic.lean", "full_name": "Decidable.not_imp_symm", "start": [534, 1], "end": [535, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/Positive.lean", "full_name": "ContinuousLinearMap.IsPositive.inner_nonneg_left", "start": [68, 1], "end": [70, 9], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/IntegralClosure.lean", "full_name": "isIntegral_of_submodule_noetherian", "start": [104, 1], "end": [114, 45], "traced_tactics": [{"tactic": "suffices IsIntegral R (show S from \u27e8x, hx\u27e9) by\n  rcases this with \u27e8p, hpm, hpx\u27e9\n  replace hpx := congr_arg S.val hpx\n  refine' \u27e8p, hpm, Eq.trans _ hpx\u27e9\n  simp only [aeval_def, eval\u2082, sum_def]\n  rw [S.val.map_sum]\n  refine' Finset.sum_congr rfl fun n _hn => _\n  rw [S.val.map_mul, S.val.map_pow, S.val.commutes, S.val_apply, Subtype.coe_mk]", "state_before": "R : Type u_1\nS\u271d : Type ?u.86903\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\n\u22a2 IsIntegral R x", "state_after": "R : Type u_1\nS\u271d : Type ?u.86903\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\n\u22a2 IsIntegral R\n    (let_fun this := { val := x, property := hx };\n    this)"}, {"tactic": "refine' isIntegral_of_noetherian H \u27e8x, hx\u27e9", "state_before": "R : Type u_1\nS\u271d : Type ?u.86903\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\n\u22a2 IsIntegral R\n    (let_fun this := { val := x, property := hx };\n    this)", "state_after": "no goals"}, {"tactic": "rcases this with \u27e8p, hpm, hpx\u27e9", "state_before": "R : Type u_1\nS\u271d : Type ?u.86903\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\nthis :\n  IsIntegral R\n    (let_fun this := { val := x, property := hx };\n    this)\n\u22a2 IsIntegral R x", "state_after": "case intro.intro\nR : Type u_1\nS\u271d : Type ?u.86903\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\np : R[X]\nhpm : Monic p\nhpx :\n  eval\u2082 (algebraMap R { x // x \u2208 S })\n      (let_fun this := { val := x, property := hx };\n      this)\n      p =\n    0\n\u22a2 IsIntegral R x"}, {"tactic": "replace hpx := congr_arg S.val hpx", "state_before": "case intro.intro\nR : Type u_1\nS\u271d : Type ?u.86903\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\np : R[X]\nhpm : Monic p\nhpx :\n  eval\u2082 (algebraMap R { x // x \u2208 S })\n      (let_fun this := { val := x, property := hx };\n      this)\n      p =\n    0\n\u22a2 IsIntegral R x", "state_after": "case intro.intro\nR : Type u_1\nS\u271d : Type ?u.86903\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\np : R[X]\nhpm : Monic p\nhpx :\n  \u2191(Subalgebra.val S)\n      (eval\u2082 (algebraMap R { x // x \u2208 S })\n        (let_fun this := { val := x, property := hx };\n        this)\n        p) =\n    \u2191(Subalgebra.val S) 0\n\u22a2 IsIntegral R x"}, {"tactic": "refine' \u27e8p, hpm, Eq.trans _ hpx\u27e9", "state_before": "case intro.intro\nR : Type u_1\nS\u271d : Type ?u.86903\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\np : R[X]\nhpm : Monic p\nhpx :\n  \u2191(Subalgebra.val S)\n      (eval\u2082 (algebraMap R { x // x \u2208 S })\n        (let_fun this := { val := x, property := hx };\n        this)\n        p) =\n    \u2191(Subalgebra.val S) 0\n\u22a2 IsIntegral R x", "state_after": "case intro.intro\nR : Type u_1\nS\u271d : Type ?u.86903\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\np : R[X]\nhpm : Monic p\nhpx :\n  \u2191(Subalgebra.val S)\n      (eval\u2082 (algebraMap R { x // x \u2208 S })\n        (let_fun this := { val := x, property := hx };\n        this)\n        p) =\n    \u2191(Subalgebra.val S) 0\n\u22a2 eval\u2082 (algebraMap R A) x p =\n    \u2191(Subalgebra.val S)\n      (eval\u2082 (algebraMap R { x // x \u2208 S })\n        (let_fun this := { val := x, property := hx };\n        this)\n        p)"}, {"tactic": "simp only [aeval_def, eval\u2082, sum_def]", "state_before": "case intro.intro\nR : Type u_1\nS\u271d : Type ?u.86903\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\np : R[X]\nhpm : Monic p\nhpx :\n  \u2191(Subalgebra.val S)\n      (eval\u2082 (algebraMap R { x // x \u2208 S })\n        (let_fun this := { val := x, property := hx };\n        this)\n        p) =\n    \u2191(Subalgebra.val S) 0\n\u22a2 eval\u2082 (algebraMap R A) x p =\n    \u2191(Subalgebra.val S)\n      (eval\u2082 (algebraMap R { x // x \u2208 S })\n        (let_fun this := { val := x, property := hx };\n        this)\n        p)", "state_after": "case intro.intro\nR : Type u_1\nS\u271d : Type ?u.86903\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\np : R[X]\nhpm : Monic p\nhpx :\n  \u2191(Subalgebra.val S)\n      (eval\u2082 (algebraMap R { x // x \u2208 S })\n        (let_fun this := { val := x, property := hx };\n        this)\n        p) =\n    \u2191(Subalgebra.val S) 0\n\u22a2 \u2211 n in support p, \u2191(algebraMap R A) (coeff p n) * x ^ n =\n    \u2191(Subalgebra.val S)\n      (\u2211 n in support p, \u2191(algebraMap R { x // x \u2208 S }) (coeff p n) * { val := x, property := hx } ^ n)"}, {"tactic": "rw [S.val.map_sum]", "state_before": "case intro.intro\nR : Type u_1\nS\u271d : Type ?u.86903\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\np : R[X]\nhpm : Monic p\nhpx :\n  \u2191(Subalgebra.val S)\n      (eval\u2082 (algebraMap R { x // x \u2208 S })\n        (let_fun this := { val := x, property := hx };\n        this)\n        p) =\n    \u2191(Subalgebra.val S) 0\n\u22a2 \u2211 n in support p, \u2191(algebraMap R A) (coeff p n) * x ^ n =\n    \u2191(Subalgebra.val S)\n      (\u2211 n in support p, \u2191(algebraMap R { x // x \u2208 S }) (coeff p n) * { val := x, property := hx } ^ n)", "state_after": "case intro.intro\nR : Type u_1\nS\u271d : Type ?u.86903\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\np : R[X]\nhpm : Monic p\nhpx :\n  \u2191(Subalgebra.val S)\n      (eval\u2082 (algebraMap R { x // x \u2208 S })\n        (let_fun this := { val := x, property := hx };\n        this)\n        p) =\n    \u2191(Subalgebra.val S) 0\n\u22a2 \u2211 n in support p, \u2191(algebraMap R A) (coeff p n) * x ^ n =\n    \u2211 x_1 in support p,\n      \u2191(Subalgebra.val S) (\u2191(algebraMap R { x // x \u2208 S }) (coeff p x_1) * { val := x, property := hx } ^ x_1)"}, {"tactic": "refine' Finset.sum_congr rfl fun n _hn => _", "state_before": "case intro.intro\nR : Type u_1\nS\u271d : Type ?u.86903\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\np : R[X]\nhpm : Monic p\nhpx :\n  \u2191(Subalgebra.val S)\n      (eval\u2082 (algebraMap R { x // x \u2208 S })\n        (let_fun this := { val := x, property := hx };\n        this)\n        p) =\n    \u2191(Subalgebra.val S) 0\n\u22a2 \u2211 n in support p, \u2191(algebraMap R A) (coeff p n) * x ^ n =\n    \u2211 x_1 in support p,\n      \u2191(Subalgebra.val S) (\u2191(algebraMap R { x // x \u2208 S }) (coeff p x_1) * { val := x, property := hx } ^ x_1)", "state_after": "case intro.intro\nR : Type u_1\nS\u271d : Type ?u.86903\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\np : R[X]\nhpm : Monic p\nhpx :\n  \u2191(Subalgebra.val S)\n      (eval\u2082 (algebraMap R { x // x \u2208 S })\n        (let_fun this := { val := x, property := hx };\n        this)\n        p) =\n    \u2191(Subalgebra.val S) 0\nn : \u2115\n_hn : n \u2208 support p\n\u22a2 \u2191(algebraMap R A) (coeff p n) * x ^ n =\n    \u2191(Subalgebra.val S) (\u2191(algebraMap R { x // x \u2208 S }) (coeff p n) * { val := x, property := hx } ^ n)"}, {"tactic": "rw [S.val.map_mul, S.val.map_pow, S.val.commutes, S.val_apply, Subtype.coe_mk]", "state_before": "case intro.intro\nR : Type u_1\nS\u271d : Type ?u.86903\nA : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring S\u271d\nf : R \u2192+* S\u271d\ninst\u271d : Algebra R A\nS : Subalgebra R A\nH : IsNoetherian R { x // x \u2208 \u2191Subalgebra.toSubmodule S }\nx : A\nhx : x \u2208 S\np : R[X]\nhpm : Monic p\nhpx :\n  \u2191(Subalgebra.val S)\n      (eval\u2082 (algebraMap R { x // x \u2208 S })\n        (let_fun this := { val := x, property := hx };\n        this)\n        p) =\n    \u2191(Subalgebra.val S) 0\nn : \u2115\n_hn : n \u2208 support p\n\u22a2 \u2191(algebraMap R A) (coeff p n) * x ^ n =\n    \u2191(Subalgebra.val S) (\u2191(algebraMap R { x // x \u2208 S }) (coeff p n) * { val := x, property := hx } ^ n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/OrderIsoNat.lean", "full_name": "Nat.coe_orderEmbeddingOfSet", "start": [133, 1], "end": [135, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Finiteness.lean", "full_name": "AlgHom.Finite.comp", "start": [738, 1], "end": [739, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/EReal.lean", "full_name": "EReal.add_lt_top", "start": [721, 1], "end": [723, 45], "traced_tactics": [{"tactic": "rw [\u2190 EReal.top_add_top]", "state_before": "x y : EReal\nhx : x \u2260 \u22a4\nhy : y \u2260 \u22a4\n\u22a2 x + y < \u22a4", "state_after": "x y : EReal\nhx : x \u2260 \u22a4\nhy : y \u2260 \u22a4\n\u22a2 x + y < \u22a4 + \u22a4"}, {"tactic": "exact EReal.add_lt_add hx.lt_top hy.lt_top", "state_before": "x y : EReal\nhx : x \u2260 \u22a4\nhy : y \u2260 \u22a4\n\u22a2 x + y < \u22a4 + \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "full_name": "MeasureTheory.AEEqFun.comp\u2082_eq_pair", "start": [308, 1], "end": [310, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/Projection.lean", "full_name": "orthogonalProjection_minimal", "start": [502, 1], "end": [505, 45], "traced_tactics": [{"tactic": "rw [norm_eq_iInf_iff_inner_eq_zero _ (Submodule.coe_mem _)]", "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.503637\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b9 : CompleteSpace { x // x \u2208 K }\nU : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace { x // x \u2208 U }\ny : E\n\u22a2 \u2016y - \u2191(\u2191(orthogonalProjection U) y)\u2016 = \u2a05 (x : { x // x \u2208 U }), \u2016y - \u2191x\u2016", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.503637\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b9 : CompleteSpace { x // x \u2208 K }\nU : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace { x // x \u2208 U }\ny : E\n\u22a2 \u2200 (w : E), w \u2208 U \u2192 inner (y - \u2191(\u2191(orthogonalProjection U) y)) w = 0"}, {"tactic": "exact orthogonalProjection_inner_eq_zero _", "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.503637\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b9 : CompleteSpace { x // x \u2208 K }\nU : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace { x // x \u2208 U }\ny : E\n\u22a2 \u2200 (w : E), w \u2208 U \u2192 inner (y - \u2191(\u2191(orthogonalProjection U) y)) w = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Compactification/OnePoint.lean", "full_name": "OnePoint.isOpen_iff_of_not_mem", "start": [224, 1], "end": [225, 23], "traced_tactics": [{"tactic": "simp [isOpen_def, h]", "state_before": "X : Type u_1\ninst\u271d : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nh : \u00ac\u221e \u2208 s\n\u22a2 IsOpen s \u2194 IsOpen (some \u207b\u00b9' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Sober.lean", "full_name": "IsGenericPoint.image", "start": [103, 11], "end": [105, 80], "traced_tactics": [{"tactic": "rw [isGenericPoint_def, \u2190 h.def, \u2190 image_singleton, closure_image_closure hf]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nx y : \u03b1\nS U Z : Set \u03b1\nh : IsGenericPoint x S\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\n\u22a2 IsGenericPoint (f x) (closure (f '' S))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Seminorm.lean", "full_name": "Seminorm.comp_apply", "start": [323, 1], "end": [324, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/GaloisConnection.lean", "full_name": "sInf_image2_eq_sSup_sSup", "start": [419, 1], "end": [422, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean", "full_name": "Matrix.IsAdjMatrix.compl", "start": [133, 1], "end": [134, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Abelian/Basic.lean", "full_name": "CategoryTheory.Abelian.epi_of_cokernel_\u03c0_eq_zero", "start": [335, 1], "end": [338, 81], "traced_tactics": [{"tactic": "apply NormalMonoCategory.epi_of_zero_cokernel _ (cokernel f)", "state_before": "C : Type u\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nP Q : C\nf : P \u27f6 Q\nh : cokernel.\u03c0 f = 0\n\u22a2 Epi f", "state_after": "C : Type u\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nP Q : C\nf : P \u27f6 Q\nh : cokernel.\u03c0 f = 0\n\u22a2 IsColimit (CokernelCofork.of\u03c0 0 (_ : f \u226b 0 = 0))"}, {"tactic": "simp_rw [\u2190 h]", "state_before": "C : Type u\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nP Q : C\nf : P \u27f6 Q\nh : cokernel.\u03c0 f = 0\n\u22a2 IsColimit (CokernelCofork.of\u03c0 0 (_ : f \u226b 0 = 0))", "state_after": "C : Type u\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nP Q : C\nf : P \u27f6 Q\nh : cokernel.\u03c0 f = 0\n\u22a2 IsColimit (CokernelCofork.of\u03c0 (cokernel.\u03c0 f) (_ : f \u226b cokernel.\u03c0 f = 0))"}, {"tactic": "exact IsColimit.ofIsoColimit (colimit.isColimit (parallelPair f 0)) (isoOf\u03c0 _)", "state_before": "C : Type u\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nP Q : C\nf : P \u27f6 Q\nh : cokernel.\u03c0 f = 0\n\u22a2 IsColimit (CokernelCofork.of\u03c0 (cokernel.\u03c0 f) (_ : f \u226b cokernel.\u03c0 f = 0))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Powerset.lean", "full_name": "Finset.empty_mem_powerset", "start": [51, 1], "end": [52, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.aevalTower_id", "start": [1633, 1], "end": [1636, 36], "traced_tactics": [{"tactic": "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\u2077 : CommSemiring R\ninst\u271d\u2076 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nS : Type u_2\nA : Type ?u.5316365\nB : Type ?u.5316368\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : CommSemiring A\ninst\u271d\u00b3 : CommSemiring B\ninst\u271d\u00b2 : Algebra S R\ninst\u271d\u00b9 : Algebra S A\ninst\u271d : Algebra S B\ng : R \u2192\u2090[S] A\ny : \u03c3 \u2192 A\n\u22a2 aevalTower (AlgHom.id S S) = aeval", "state_after": "case h.hf\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\u2077 : CommSemiring R\ninst\u271d\u2076 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nS : Type u_2\nA : Type ?u.5316365\nB : Type ?u.5316368\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : CommSemiring A\ninst\u271d\u00b3 : CommSemiring B\ninst\u271d\u00b2 : Algebra S R\ninst\u271d\u00b9 : Algebra S A\ninst\u271d : Algebra S B\ng : R \u2192\u2090[S] A\ny : \u03c3 \u2192 A\nx\u271d : \u03c3 \u2192 S\ni\u271d : \u03c3\n\u22a2 \u2191(aevalTower (AlgHom.id S S) x\u271d) (X i\u271d) = \u2191(aeval x\u271d) (X i\u271d)"}, {"tactic": "simp only [aevalTower_X, aeval_X]", "state_before": "case h.hf\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\u2077 : CommSemiring R\ninst\u271d\u2076 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\nS : Type u_2\nA : Type ?u.5316365\nB : Type ?u.5316368\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : CommSemiring A\ninst\u271d\u00b3 : CommSemiring B\ninst\u271d\u00b2 : Algebra S R\ninst\u271d\u00b9 : Algebra S A\ninst\u271d : Algebra S B\ng : R \u2192\u2090[S] A\ny : \u03c3 \u2192 A\nx\u271d : \u03c3 \u2192 S\ni\u271d : \u03c3\n\u22a2 \u2191(aevalTower (AlgHom.id S S) x\u271d) (X i\u271d) = \u2191(aeval x\u271d) (X i\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/AffineIsometry.lean", "full_name": "AffineIsometryEquiv.symm_symm", "start": [533, 1], "end": [534, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean", "full_name": "Matrix.Represents.add", "start": [136, 1], "end": [138, 65], "traced_tactics": [{"tactic": "delta Matrix.Represents at h h'\u22a2", "state_before": "\u03b9 : Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nb : \u03b9 \u2192 M\nhb : Submodule.span R (Set.range b) = \u22a4\ninst\u271d : DecidableEq \u03b9\nA A' : Matrix \u03b9 \u03b9 R\nf f' : Module.End R M\nh : Represents b A f\nh' : Represents b A' f'\n\u22a2 Represents b (A + A') (f + f')", "state_after": "\u03b9 : Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nb : \u03b9 \u2192 M\nhb : Submodule.span R (Set.range b) = \u22a4\ninst\u271d : DecidableEq \u03b9\nA A' : Matrix \u03b9 \u03b9 R\nf f' : Module.End R M\nh : \u2191(PiToModule.fromMatrix R b) A = \u2191(PiToModule.fromEnd R b) f\nh' : \u2191(PiToModule.fromMatrix R b) A' = \u2191(PiToModule.fromEnd R b) f'\n\u22a2 \u2191(PiToModule.fromMatrix R b) (A + A') = \u2191(PiToModule.fromEnd R b) (f + f')"}, {"tactic": "rw [map_add, map_add, h, h']", "state_before": "\u03b9 : Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nb : \u03b9 \u2192 M\nhb : Submodule.span R (Set.range b) = \u22a4\ninst\u271d : DecidableEq \u03b9\nA A' : Matrix \u03b9 \u03b9 R\nf f' : Module.End R M\nh : \u2191(PiToModule.fromMatrix R b) A = \u2191(PiToModule.fromEnd R b) f\nh' : \u2191(PiToModule.fromMatrix R b) A' = \u2191(PiToModule.fromEnd R b) f'\n\u22a2 \u2191(PiToModule.fromMatrix R b) (A + A') = \u2191(PiToModule.fromEnd R b) (f + f')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/DiscreteQuotient.lean", "full_name": "DiscreteQuotient.eq_of_forall_proj_eq", "start": [367, 1], "end": [372, 53], "traced_tactics": [{"tactic": "rw [\u2190 mem_singleton_iff, \u2190 connectedComponent_eq_singleton, connectedComponent_eq_iInter_clopen,\n  mem_iInter]", "state_before": "\u03b1 : Type ?u.36994\nX : Type u_1\nY : Type ?u.37000\nZ : Type ?u.37003\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : TopologicalSpace Z\nS : DiscreteQuotient X\ninst\u271d\u00b9 : T2Space X\ninst\u271d : CompactSpace X\ndisc : TotallyDisconnectedSpace X\nx y : X\nh : \u2200 (Q : DiscreteQuotient X), proj Q x = proj Q y\n\u22a2 x = y", "state_after": "\u03b1 : Type ?u.36994\nX : Type u_1\nY : Type ?u.37000\nZ : Type ?u.37003\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : TopologicalSpace Z\nS : DiscreteQuotient X\ninst\u271d\u00b9 : T2Space X\ninst\u271d : CompactSpace X\ndisc : TotallyDisconnectedSpace X\nx y : X\nh : \u2200 (Q : DiscreteQuotient X), proj Q x = proj Q y\n\u22a2 \u2200 (i : { Z // IsClopen Z \u2227 y \u2208 Z }), x \u2208 \u2191i"}, {"tactic": "rintro \u27e8U, hU1, hU2\u27e9", "state_before": "\u03b1 : Type ?u.36994\nX : Type u_1\nY : Type ?u.37000\nZ : Type ?u.37003\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : TopologicalSpace Z\nS : DiscreteQuotient X\ninst\u271d\u00b9 : T2Space X\ninst\u271d : CompactSpace X\ndisc : TotallyDisconnectedSpace X\nx y : X\nh : \u2200 (Q : DiscreteQuotient X), proj Q x = proj Q y\n\u22a2 \u2200 (i : { Z // IsClopen Z \u2227 y \u2208 Z }), x \u2208 \u2191i", "state_after": "case mk.intro\n\u03b1 : Type ?u.36994\nX : Type u_1\nY : Type ?u.37000\nZ : Type ?u.37003\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : TopologicalSpace Z\nS : DiscreteQuotient X\ninst\u271d\u00b9 : T2Space X\ninst\u271d : CompactSpace X\ndisc : TotallyDisconnectedSpace X\nx y : X\nh : \u2200 (Q : DiscreteQuotient X), proj Q x = proj Q y\nU : Set X\nhU1 : IsClopen U\nhU2 : y \u2208 U\n\u22a2 x \u2208 \u2191{ val := U, property := (_ : IsClopen U \u2227 y \u2208 U) }"}, {"tactic": "exact (Quotient.exact' (h (ofClopen hU1))).mpr hU2", "state_before": "case mk.intro\n\u03b1 : Type ?u.36994\nX : Type u_1\nY : Type ?u.37000\nZ : Type ?u.37003\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : TopologicalSpace Z\nS : DiscreteQuotient X\ninst\u271d\u00b9 : T2Space X\ninst\u271d : CompactSpace X\ndisc : TotallyDisconnectedSpace X\nx y : X\nh : \u2200 (Q : DiscreteQuotient X), proj Q x = proj Q y\nU : Set X\nhU1 : IsClopen U\nhU2 : y \u2208 U\n\u22a2 x \u2208 \u2191{ val := U, property := (_ : IsClopen U \u2227 y \u2208 U) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/StrictConvexSpace.lean", "full_name": "Isometry.affineIsometryOfStrictConvexSpace_apply", "start": [303, 1], "end": [305, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/ContinuousFunction/Bounded.lean", "full_name": "BoundedContinuousFunction.dist_set_exists", "start": [164, 1], "end": [168, 29], "traced_tactics": [{"tactic": "rcases f.bounded_range.union g.bounded_range with \u27e8C, hC\u27e9", "state_before": "F : Type ?u.274313\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : PseudoMetricSpace \u03b3\nf g : \u03b1 \u2192\u1d47 \u03b2\nx : \u03b1\nC : \u211d\n\u22a2 \u2203 C, 0 \u2264 C \u2227 \u2200 (x : \u03b1), dist (\u2191f x) (\u2191g x) \u2264 C", "state_after": "case intro\nF : Type ?u.274313\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : PseudoMetricSpace \u03b3\nf g : \u03b1 \u2192\u1d47 \u03b2\nx : \u03b1\nC\u271d C : \u211d\nhC : \u2200 (x : \u03b2), x \u2208 range \u2191f \u222a range \u2191g \u2192 \u2200 (y : \u03b2), y \u2208 range \u2191f \u222a range \u2191g \u2192 dist x y \u2264 C\n\u22a2 \u2203 C, 0 \u2264 C \u2227 \u2200 (x : \u03b1), dist (\u2191f x) (\u2191g x) \u2264 C"}, {"tactic": "refine' \u27e8max 0 C, le_max_left _ _, fun x => (hC _ _ _ _).trans (le_max_right _ _)\u27e9\n  <;> [left; right]\n  <;> apply mem_range_self", "state_before": "case intro\nF : Type ?u.274313\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : PseudoMetricSpace \u03b2\ninst\u271d : PseudoMetricSpace \u03b3\nf g : \u03b1 \u2192\u1d47 \u03b2\nx : \u03b1\nC\u271d C : \u211d\nhC : \u2200 (x : \u03b2), x \u2208 range \u2191f \u222a range \u2191g \u2192 \u2200 (y : \u03b2), y \u2208 range \u2191f \u222a range \u2191g \u2192 dist x y \u2264 C\n\u22a2 \u2203 C, 0 \u2264 C \u2227 \u2200 (x : \u03b1), dist (\u2191f x) (\u2191g x) \u2264 C", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Dfinsupp/Lex.lean", "full_name": "Dfinsupp.lex_def", "start": [45, 1], "end": [47, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/ENNReal.lean", "full_name": "Set.OrdConnected.image_coe_nnreal_ennreal", "start": [2512, 1], "end": [2515, 96], "traced_tactics": [{"tactic": "refine' \u27e8ball_image_iff.2 fun x hx => ball_image_iff.2 fun y hy z hz => _\u27e9", "state_before": "\u03b1 : Type ?u.871449\n\u03b2 : Type ?u.871452\ns : Set \u211d\nt : Set \u211d\u22650\nu : Set \u211d\u22650\u221e\nh : OrdConnected t\n\u22a2 OrdConnected (ENNReal.some '' t)", "state_after": "\u03b1 : Type ?u.871449\n\u03b2 : Type ?u.871452\ns : Set \u211d\nt : Set \u211d\u22650\nu : Set \u211d\u22650\u221e\nh : OrdConnected t\nx : \u211d\u22650\nhx : x \u2208 t\ny : \u211d\u22650\nhy : y \u2208 t\nz : \u211d\u22650\u221e\nhz : z \u2208 Icc \u2191x \u2191y\n\u22a2 z \u2208 ENNReal.some '' t"}, {"tactic": "rcases ENNReal.le_coe_iff.1 hz.2 with \u27e8z, rfl, -\u27e9", "state_before": "\u03b1 : Type ?u.871449\n\u03b2 : Type ?u.871452\ns : Set \u211d\nt : Set \u211d\u22650\nu : Set \u211d\u22650\u221e\nh : OrdConnected t\nx : \u211d\u22650\nhx : x \u2208 t\ny : \u211d\u22650\nhy : y \u2208 t\nz : \u211d\u22650\u221e\nhz : z \u2208 Icc \u2191x \u2191y\n\u22a2 z \u2208 ENNReal.some '' t", "state_after": "case intro.intro\n\u03b1 : Type ?u.871449\n\u03b2 : Type ?u.871452\ns : Set \u211d\nt : Set \u211d\u22650\nu : Set \u211d\u22650\u221e\nh : OrdConnected t\nx : \u211d\u22650\nhx : x \u2208 t\ny : \u211d\u22650\nhy : y \u2208 t\nz : \u211d\u22650\nhz : \u2191z \u2208 Icc \u2191x \u2191y\n\u22a2 \u2191z \u2208 ENNReal.some '' t"}, {"tactic": "exact mem_image_of_mem _ (h.out hx hy \u27e8ENNReal.coe_le_coe.1 hz.1, ENNReal.coe_le_coe.1 hz.2\u27e9)", "state_before": "case intro.intro\n\u03b1 : Type ?u.871449\n\u03b2 : Type ?u.871452\ns : Set \u211d\nt : Set \u211d\u22650\nu : Set \u211d\u22650\u221e\nh : OrdConnected t\nx : \u211d\u22650\nhx : x \u2208 t\ny : \u211d\u22650\nhy : y \u2208 t\nz : \u211d\u22650\nhz : \u2191z \u2208 Icc \u2191x \u2191y\n\u22a2 \u2191z \u2208 ENNReal.some '' t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Density.lean", "full_name": "Rel.mk_mem_interedges_iff", "start": [64, 1], "end": [65, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "full_name": "LocalHomeomorph.isBigOWith_congr", "start": [2179, 1], "end": [2188, 73], "traced_tactics": [{"tactic": "have := e.continuousAt (e.map_target hb)", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nE : Type u_3\ninst\u271d\u00b9 : Norm E\nF : Type u_4\ninst\u271d : Norm F\ne : LocalHomeomorph \u03b1 \u03b2\nb : \u03b2\nhb : b \u2208 e.target\nf : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nC : \u211d\nh : IsBigOWith C (\ud835\udcdd b) f g\n\u22a2 Tendsto (\u2191e) (\ud835\udcdd (\u2191(LocalHomeomorph.symm e) b)) (\ud835\udcdd b)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nE : Type u_3\ninst\u271d\u00b9 : Norm E\nF : Type u_4\ninst\u271d : Norm F\ne : LocalHomeomorph \u03b1 \u03b2\nb : \u03b2\nhb : b \u2208 e.target\nf : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nC : \u211d\nh : IsBigOWith C (\ud835\udcdd b) f g\nthis : ContinuousAt (\u2191e) (\u2191(LocalHomeomorph.symm e) b)\n\u22a2 Tendsto (\u2191e) (\ud835\udcdd (\u2191(LocalHomeomorph.symm e) b)) (\ud835\udcdd b)"}, {"tactic": "rwa [ContinuousAt, e.rightInvOn hb] at this", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\nE : Type u_3\ninst\u271d\u00b9 : Norm E\nF : Type u_4\ninst\u271d : Norm F\ne : LocalHomeomorph \u03b1 \u03b2\nb : \u03b2\nhb : b \u2208 e.target\nf : \u03b2 \u2192 E\ng : \u03b2 \u2192 F\nC : \u211d\nh : IsBigOWith C (\ud835\udcdd b) f g\nthis : ContinuousAt (\u2191e) (\u2191(LocalHomeomorph.symm e) b)\n\u22a2 Tendsto (\u2191e) (\ud835\udcdd (\u2191(LocalHomeomorph.symm e) b)) (\ud835\udcdd b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/Basic.lean", "full_name": "Ideal.IsMaximal.isPrime", "start": [529, 1], "end": [541, 36], "traced_tactics": [{"tactic": "let J : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\n\u22a2 y \u2208 I", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\n\u22a2 y \u2208 I"}, {"tactic": "have IJ : I \u2264 J := Set.Subset.trans (subset_insert _ _) subset_span", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\n\u22a2 y \u2208 I", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\n\u22a2 y \u2208 I"}, {"tactic": "have xJ : x \u2208 J := Ideal.subset_span (Set.mem_insert x I)", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\n\u22a2 y \u2208 I", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\nxJ : x \u2208 J\n\u22a2 y \u2208 I"}, {"tactic": "cases' isMaximal_iff.1 H with _ oJ", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\nxJ : x \u2208 J\n\u22a2 y \u2208 I", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\nxJ : x \u2208 J\nleft\u271d : \u00ac1 \u2208 I\noJ : \u2200 (J : Ideal \u03b1) (x : \u03b1), I \u2264 J \u2192 \u00acx \u2208 I \u2192 x \u2208 J \u2192 1 \u2208 J\n\u22a2 y \u2208 I"}, {"tactic": "specialize oJ J x IJ hx xJ", "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\nxJ : x \u2208 J\nleft\u271d : \u00ac1 \u2208 I\noJ : \u2200 (J : Ideal \u03b1) (x : \u03b1), I \u2264 J \u2192 \u00acx \u2208 I \u2192 x \u2208 J \u2192 1 \u2208 J\n\u22a2 y \u2208 I", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\nxJ : x \u2208 J\nleft\u271d : \u00ac1 \u2208 I\noJ : 1 \u2208 J\n\u22a2 y \u2208 I"}, {"tactic": "rcases Submodule.mem_span_insert.mp oJ with \u27e8a, b, h, oe\u27e9", "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\na b : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\nxJ : x \u2208 J\nleft\u271d : \u00ac1 \u2208 I\noJ : 1 \u2208 J\n\u22a2 y \u2208 I", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b\u271d : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\nxJ : x \u2208 J\nleft\u271d : \u00ac1 \u2208 I\noJ : 1 \u2208 J\na b : \u03b1\nh : b \u2208 Submodule.span \u03b1 \u2191I\noe : 1 = a \u2022 x + b\n\u22a2 y \u2208 I"}, {"tactic": "obtain F : y * 1 = y * (a \u2022 x + b) := congr_arg (fun g : \u03b1 => y * g) oe", "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b\u271d : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\nxJ : x \u2208 J\nleft\u271d : \u00ac1 \u2208 I\noJ : 1 \u2208 J\na b : \u03b1\nh : b \u2208 Submodule.span \u03b1 \u2191I\noe : 1 = a \u2022 x + b\n\u22a2 y \u2208 I", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b\u271d : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\nxJ : x \u2208 J\nleft\u271d : \u00ac1 \u2208 I\noJ : 1 \u2208 J\na b : \u03b1\nh : b \u2208 Submodule.span \u03b1 \u2191I\noe : 1 = a \u2022 x + b\nF : y * 1 = y * (a \u2022 x + b)\n\u22a2 y \u2208 I"}, {"tactic": "rw [\u2190 mul_one y, F, mul_add, mul_comm, smul_eq_mul, mul_assoc]", "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b\u271d : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\nxJ : x \u2208 J\nleft\u271d : \u00ac1 \u2208 I\noJ : 1 \u2208 J\na b : \u03b1\nh : b \u2208 Submodule.span \u03b1 \u2191I\noe : 1 = a \u2022 x + b\nF : y * 1 = y * (a \u2022 x + b)\n\u22a2 y \u2208 I", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b\u271d : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\nxJ : x \u2208 J\nleft\u271d : \u00ac1 \u2208 I\noJ : 1 \u2208 J\na b : \u03b1\nh : b \u2208 Submodule.span \u03b1 \u2191I\noe : 1 = a \u2022 x + b\nF : y * 1 = y * (a \u2022 x + b)\n\u22a2 a * (x * y) + y * b \u2208 I"}, {"tactic": "refine' Submodule.add_mem I (I.mul_mem_left a hxy) (Submodule.smul_mem I y _)", "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b\u271d : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\nxJ : x \u2208 J\nleft\u271d : \u00ac1 \u2208 I\noJ : 1 \u2208 J\na b : \u03b1\nh : b \u2208 Submodule.span \u03b1 \u2191I\noe : 1 = a \u2022 x + b\nF : y * 1 = y * (a \u2022 x + b)\n\u22a2 a * (x * y) + y * b \u2208 I", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b\u271d : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\nxJ : x \u2208 J\nleft\u271d : \u00ac1 \u2208 I\noJ : 1 \u2208 J\na b : \u03b1\nh : b \u2208 Submodule.span \u03b1 \u2191I\noe : 1 = a \u2022 x + b\nF : y * 1 = y * (a \u2022 x + b)\n\u22a2 b \u2208 I"}, {"tactic": "rwa [Submodule.span_eq] at h", "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\na\u271d b\u271d : \u03b1\ninst\u271d : CommSemiring \u03b1\nI\u271d I : Ideal \u03b1\nH : IsMaximal I\nx y : \u03b1\nhxy : x * y \u2208 I\nhx : \u00acx \u2208 I\nJ : Ideal \u03b1 := Submodule.span \u03b1 (insert x \u2191I)\nIJ : I \u2264 J\nxJ : x \u2208 J\nleft\u271d : \u00ac1 \u2208 I\noJ : 1 \u2208 J\na b : \u03b1\nh : b \u2208 Submodule.span \u03b1 \u2191I\noe : 1 = a \u2022 x + b\nF : y * 1 = y * (a \u2022 x + b)\n\u22a2 b \u2208 I", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/ContDiff.lean", "full_name": "norm_iteratedFDerivWithin_mul_le", "start": [2506, 1], "end": [2513, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/MeasurableSpace.lean", "full_name": "measurable_to_countable", "start": [398, 1], "end": [405, 71], "traced_tactics": [{"tactic": "rw [\u2190 biUnion_preimage_singleton]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.48063\n\u03b4 : Type ?u.48066\n\u03b4' : Type ?u.48069\n\u03b9 : Sort u\u03b9\ns\u271d t u : Set \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (y : \u03b2), MeasurableSet (f \u207b\u00b9' {f y})\ns : Set \u03b1\nx\u271d : MeasurableSet s\n\u22a2 MeasurableSet (f \u207b\u00b9' s)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.48063\n\u03b4 : Type ?u.48066\n\u03b4' : Type ?u.48069\n\u03b9 : Sort u\u03b9\ns\u271d t u : Set \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (y : \u03b2), MeasurableSet (f \u207b\u00b9' {f y})\ns : Set \u03b1\nx\u271d : MeasurableSet s\n\u22a2 MeasurableSet (\u22c3 (y : \u03b1) (_ : y \u2208 s), f \u207b\u00b9' {y})"}, {"tactic": "refine' MeasurableSet.iUnion fun y => MeasurableSet.iUnion fun hy => _", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.48063\n\u03b4 : Type ?u.48066\n\u03b4' : Type ?u.48069\n\u03b9 : Sort u\u03b9\ns\u271d t u : Set \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (y : \u03b2), MeasurableSet (f \u207b\u00b9' {f y})\ns : Set \u03b1\nx\u271d : MeasurableSet s\n\u22a2 MeasurableSet (\u22c3 (y : \u03b1) (_ : y \u2208 s), f \u207b\u00b9' {y})", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.48063\n\u03b4 : Type ?u.48066\n\u03b4' : Type ?u.48069\n\u03b9 : Sort u\u03b9\ns\u271d t u : Set \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (y : \u03b2), MeasurableSet (f \u207b\u00b9' {f y})\ns : Set \u03b1\nx\u271d : MeasurableSet s\ny : \u03b1\nhy : y \u2208 s\n\u22a2 MeasurableSet (f \u207b\u00b9' {y})"}, {"tactic": "by_cases hyf : y \u2208 range f", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.48063\n\u03b4 : Type ?u.48066\n\u03b4' : Type ?u.48069\n\u03b9 : Sort u\u03b9\ns\u271d t u : Set \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (y : \u03b2), MeasurableSet (f \u207b\u00b9' {f y})\ns : Set \u03b1\nx\u271d : MeasurableSet s\ny : \u03b1\nhy : y \u2208 s\n\u22a2 MeasurableSet (f \u207b\u00b9' {y})", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.48063\n\u03b4 : Type ?u.48066\n\u03b4' : Type ?u.48069\n\u03b9 : Sort u\u03b9\ns\u271d t u : Set \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (y : \u03b2), MeasurableSet (f \u207b\u00b9' {f y})\ns : Set \u03b1\nx\u271d : MeasurableSet s\ny : \u03b1\nhy : y \u2208 s\nhyf : y \u2208 range f\n\u22a2 MeasurableSet (f \u207b\u00b9' {y})\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.48063\n\u03b4 : Type ?u.48066\n\u03b4' : Type ?u.48069\n\u03b9 : Sort u\u03b9\ns\u271d t u : Set \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (y : \u03b2), MeasurableSet (f \u207b\u00b9' {f y})\ns : Set \u03b1\nx\u271d : MeasurableSet s\ny : \u03b1\nhy : y \u2208 s\nhyf : \u00acy \u2208 range f\n\u22a2 MeasurableSet (f \u207b\u00b9' {y})"}, {"tactic": "rcases hyf with \u27e8y, rfl\u27e9", "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.48063\n\u03b4 : Type ?u.48066\n\u03b4' : Type ?u.48069\n\u03b9 : Sort u\u03b9\ns\u271d t u : Set \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (y : \u03b2), MeasurableSet (f \u207b\u00b9' {f y})\ns : Set \u03b1\nx\u271d : MeasurableSet s\ny : \u03b1\nhy : y \u2208 s\nhyf : y \u2208 range f\n\u22a2 MeasurableSet (f \u207b\u00b9' {y})", "state_after": "case pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.48063\n\u03b4 : Type ?u.48066\n\u03b4' : Type ?u.48069\n\u03b9 : Sort u\u03b9\ns\u271d t u : Set \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (y : \u03b2), MeasurableSet (f \u207b\u00b9' {f y})\ns : Set \u03b1\nx\u271d : MeasurableSet s\ny : \u03b2\nhy : f y \u2208 s\n\u22a2 MeasurableSet (f \u207b\u00b9' {f y})"}, {"tactic": "apply h", "state_before": "case pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.48063\n\u03b4 : Type ?u.48066\n\u03b4' : Type ?u.48069\n\u03b9 : Sort u\u03b9\ns\u271d t u : Set \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (y : \u03b2), MeasurableSet (f \u207b\u00b9' {f y})\ns : Set \u03b1\nx\u271d : MeasurableSet s\ny : \u03b2\nhy : f y \u2208 s\n\u22a2 MeasurableSet (f \u207b\u00b9' {f y})", "state_after": "no goals"}, {"tactic": "simp only [preimage_singleton_eq_empty.2 hyf, MeasurableSet.empty]", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.48063\n\u03b4 : Type ?u.48066\n\u03b4' : Type ?u.48069\n\u03b9 : Sort u\u03b9\ns\u271d t u : Set \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : Countable \u03b1\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nh : \u2200 (y : \u03b2), MeasurableSet (f \u207b\u00b9' {f y})\ns : Set \u03b1\nx\u271d : MeasurableSet s\ny : \u03b1\nhy : y \u2208 s\nhyf : \u00acy \u2208 range f\n\u22a2 MeasurableSet (f \u207b\u00b9' {y})", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Sites/SheafOfTypes.lean", "full_name": "CategoryTheory.Presieve.FamilyOfElements.Compatible.compPresheafMap", "start": [364, 1], "end": [368, 68], "traced_tactics": [{"tactic": "intro Z\u2081 Z\u2082 W g\u2081 g\u2082 f\u2081 f\u2082 h\u2081 h\u2082 eq", "state_before": "C : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nf : P \u27f6 Q\nx : FamilyOfElements P R\nh : Compatible x\n\u22a2 Compatible (FamilyOfElements.compPresheafMap f x)", "state_after": "C : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nf : P \u27f6 Q\nx : FamilyOfElements P R\nh : Compatible x\nZ\u2081 Z\u2082 W : C\ng\u2081 : W \u27f6 Z\u2081\ng\u2082 : W \u27f6 Z\u2082\nf\u2081 : Z\u2081 \u27f6 X\nf\u2082 : Z\u2082 \u27f6 X\nh\u2081 : R f\u2081\nh\u2082 : R f\u2082\neq : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\n\u22a2 Q.map g\u2081.op (FamilyOfElements.compPresheafMap f x f\u2081 h\u2081) = Q.map g\u2082.op (FamilyOfElements.compPresheafMap f x f\u2082 h\u2082)"}, {"tactic": "unfold FamilyOfElements.compPresheafMap", "state_before": "C : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nf : P \u27f6 Q\nx : FamilyOfElements P R\nh : Compatible x\nZ\u2081 Z\u2082 W : C\ng\u2081 : W \u27f6 Z\u2081\ng\u2082 : W \u27f6 Z\u2082\nf\u2081 : Z\u2081 \u27f6 X\nf\u2082 : Z\u2082 \u27f6 X\nh\u2081 : R f\u2081\nh\u2082 : R f\u2082\neq : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\n\u22a2 Q.map g\u2081.op (FamilyOfElements.compPresheafMap f x f\u2081 h\u2081) = Q.map g\u2082.op (FamilyOfElements.compPresheafMap f x f\u2082 h\u2082)", "state_after": "C : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nf : P \u27f6 Q\nx : FamilyOfElements P R\nh : Compatible x\nZ\u2081 Z\u2082 W : C\ng\u2081 : W \u27f6 Z\u2081\ng\u2082 : W \u27f6 Z\u2082\nf\u2081 : Z\u2081 \u27f6 X\nf\u2082 : Z\u2082 \u27f6 X\nh\u2081 : R f\u2081\nh\u2082 : R f\u2082\neq : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\n\u22a2 Q.map g\u2081.op (f.app Z\u2081.op (x f\u2081 h\u2081)) = Q.map g\u2082.op (f.app Z\u2082.op (x f\u2082 h\u2082))"}, {"tactic": "rwa [\u2190 FunctorToTypes.naturality, \u2190 FunctorToTypes.naturality, h]", "state_before": "C : Type u\u2081\ninst\u271d : Category C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nf : P \u27f6 Q\nx : FamilyOfElements P R\nh : Compatible x\nZ\u2081 Z\u2082 W : C\ng\u2081 : W \u27f6 Z\u2081\ng\u2082 : W \u27f6 Z\u2082\nf\u2081 : Z\u2081 \u27f6 X\nf\u2082 : Z\u2082 \u27f6 X\nh\u2081 : R f\u2081\nh\u2082 : R f\u2082\neq : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\n\u22a2 Q.map g\u2081.op (f.app Z\u2081.op (x f\u2081 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Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(withDensity \u03bc 0) s = \u2191\u21910 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Perm/List.lean", "full_name": "List.zipWith_swap_prod_support'", "start": [168, 1], "end": [190, 49], "traced_tactics": [{"tactic": "simp only [Set.sup_eq_union, Set.le_eq_subset]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl l' : List \u03b1\n\u22a2 {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2264 \u2191(toFinset l \u2294 toFinset l')", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl l' : List \u03b1\n\u22a2 {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')"}, {"tactic": "induction' l with y l hl generalizing l'", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl l' : List \u03b1\n\u22a2 {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')", "state_after": "case nil\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nl'\u271d l' : List \u03b1\n\u22a2 {x | \u2191(prod (zipWith swap [] l')) x \u2260 x} \u2286 \u2191(toFinset [] \u2294 toFinset l')\n\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nhl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')\nl' : List \u03b1\n\u22a2 {x | \u2191(prod (zipWith swap (y :: l) l')) x \u2260 x} \u2286 \u2191(toFinset (y :: l) \u2294 toFinset l')"}, {"tactic": "simp", "state_before": "case nil\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nx : \u03b1\nl'\u271d l' : List \u03b1\n\u22a2 {x | \u2191(prod (zipWith swap [] l')) x \u2260 x} \u2286 \u2191(toFinset [] \u2294 toFinset l')", "state_after": "no goals"}, {"tactic": "cases' l' with z l'", "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nhl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')\nl' : List \u03b1\n\u22a2 {x | \u2191(prod (zipWith swap (y :: l) l')) x \u2260 x} \u2286 \u2191(toFinset (y :: l) \u2294 toFinset l')", "state_after": "case cons.nil\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl' : List \u03b1\ny : \u03b1\nl : List \u03b1\nhl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')\n\u22a2 {x | \u2191(prod (zipWith swap (y :: l) [])) x \u2260 x} \u2286 \u2191(toFinset (y :: l) \u2294 toFinset [])\n\ncase cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nhl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')\nz : \u03b1\nl' : List \u03b1\n\u22a2 {x | \u2191(prod (zipWith swap (y :: l) (z :: l'))) x \u2260 x} \u2286 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l'))"}, {"tactic": "simp", "state_before": "case cons.nil\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl' : List \u03b1\ny : \u03b1\nl : List \u03b1\nhl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')\n\u22a2 {x | \u2191(prod (zipWith swap (y :: l) [])) x \u2260 x} \u2286 \u2191(toFinset (y :: l) \u2294 toFinset [])", "state_after": "no goals"}, {"tactic": "intro x", "state_before": "case cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nhl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')\nz : \u03b1\nl' : List \u03b1\n\u22a2 {x | \u2191(prod (zipWith swap (y :: l) (z :: l'))) x \u2260 x} \u2286 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l'))", "state_after": "case cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nhl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')\nz : \u03b1\nl' : List \u03b1\nx : \u03b1\n\u22a2 x \u2208 {x | \u2191(prod (zipWith swap (y :: l) (z :: l'))) x \u2260 x} \u2192 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l'))"}, {"tactic": "simp only [Set.union_subset_iff, mem_cons, zipWith_cons_cons, foldr, prod_cons,\n  mul_apply]", "state_before": "case cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nhl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')\nz : \u03b1\nl' : List \u03b1\nx : \u03b1\n\u22a2 x \u2208 {x | \u2191(prod (zipWith swap (y :: l) (z :: l'))) x \u2260 x} \u2192 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l'))", "state_after": "case cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nhl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')\nz : \u03b1\nl' : List \u03b1\nx : \u03b1\n\u22a2 x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x} \u2192 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l'))"}, {"tactic": "intro hx", "state_before": "case cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nhl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')\nz : \u03b1\nl' : List \u03b1\nx : \u03b1\n\u22a2 x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x} \u2192 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l'))", "state_after": "case cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nhl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')\nz : \u03b1\nl' : List \u03b1\nx : \u03b1\nhx : x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x}\n\u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l'))"}, {"tactic": "by_cases h : x \u2208 { x | (zipWith swap l l').prod x \u2260 x }", "state_before": "case cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nhl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')\nz : \u03b1\nl' : List \u03b1\nx : \u03b1\nhx : x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x}\n\u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l'))", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nhl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')\nz : \u03b1\nl' : List \u03b1\nx : \u03b1\nhx : x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x}\nh : x \u2208 {x | \u2191(prod (zipWith swap l l')) x \u2260 x}\n\u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l'))\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nhl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')\nz : \u03b1\nl' : List \u03b1\nx : \u03b1\nhx : x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x}\nh : \u00acx \u2208 {x | \u2191(prod (zipWith swap l l')) x \u2260 x}\n\u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l'))"}, {"tactic": "specialize hl l' h", "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nhl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')\nz : \u03b1\nl' : List \u03b1\nx : \u03b1\nhx : x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x}\nh : x \u2208 {x | \u2191(prod (zipWith swap l l')) x \u2260 x}\n\u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l'))", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nz : \u03b1\nl' : List \u03b1\nx : \u03b1\nhx : x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x}\nh : x \u2208 {x | \u2191(prod (zipWith swap l l')) x \u2260 x}\nhl : x \u2208 \u2191(toFinset l \u2294 toFinset l')\n\u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l'))"}, {"tactic": "simp only [ge_iff_le, Finset.le_eq_subset, Finset.sup_eq_union, Finset.coe_union,\n  coe_toFinset, Set.mem_union, Set.mem_setOf_eq] at hl", "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nz : \u03b1\nl' : List \u03b1\nx : \u03b1\nhx : x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x}\nh : x \u2208 {x | \u2191(prod (zipWith swap l l')) x \u2260 x}\nhl : x \u2208 \u2191(toFinset l \u2294 toFinset l')\n\u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l'))", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nz : \u03b1\nl' : List \u03b1\nx : \u03b1\nhx : x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x}\nh : x \u2208 {x | \u2191(prod (zipWith swap l l')) x \u2260 x}\nhl : x \u2208 l \u2228 x \u2208 l'\n\u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l'))"}, {"tactic": "simp only [Finset.coe_insert, Set.mem_insert_iff, Finset.mem_coe, toFinset_cons,\n  mem_toFinset] at hm\u22a2", "state_before": "case pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nz : \u03b1\nl' : List \u03b1\nx : \u03b1\nhx : x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x}\nh : x \u2208 {x | \u2191(prod (zipWith swap l l')) x \u2260 x}\nhl : x \u2208 l \u2228 x \u2208 l'\nhm : x \u2208 l'\n\u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l'))", "state_after": "case pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nz : \u03b1\nl' : List \u03b1\nx : \u03b1\nhx : x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x}\nh : x \u2208 {x | \u2191(prod (zipWith swap l l')) x \u2260 x}\nhl : x \u2208 l \u2228 x \u2208 l'\nhm : x \u2208 l'\n\u22a2 x \u2208 insert y (toFinset l) \u2294 insert z (toFinset l')"}, {"tactic": "simp [hm]", "state_before": "case pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nz : \u03b1\nl' : List \u03b1\nx : \u03b1\nhx : x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x}\nh : x \u2208 {x | \u2191(prod (zipWith swap l l')) x \u2260 x}\nhl : x \u2208 l \u2228 x \u2208 l'\nhm : x \u2208 l'\n\u22a2 x \u2208 insert y (toFinset l) \u2294 insert z (toFinset l')", "state_after": "no goals"}, {"tactic": "simp only [not_not, Set.mem_setOf_eq] at h", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nhl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')\nz : \u03b1\nl' : List \u03b1\nx : \u03b1\nhx : x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x}\nh : \u00acx \u2208 {x | \u2191(prod (zipWith swap l l')) x \u2260 x}\n\u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l'))", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nhl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')\nz : \u03b1\nl' : List \u03b1\nx : \u03b1\nhx : x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x}\nh : \u2191(prod (zipWith swap l l')) x = x\n\u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l'))"}, {"tactic": "simp only [h, Set.mem_setOf_eq] at hx", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nhl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')\nz : \u03b1\nl' : List \u03b1\nx : \u03b1\nhx : x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x}\nh : \u2191(prod (zipWith swap l l')) x = x\n\u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l'))", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nhl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')\nz : \u03b1\nl' : List \u03b1\nx : \u03b1\nh : \u2191(prod (zipWith swap l l')) x = x\nhx : \u2191(swap y z) x \u2260 x\n\u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l'))"}, {"tactic": "rw [swap_apply_ne_self_iff] at hx", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nhl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')\nz : \u03b1\nl' : List \u03b1\nx : \u03b1\nh : \u2191(prod (zipWith swap l l')) x = x\nhx : \u2191(swap y z) x \u2260 x\n\u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l'))", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nhl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')\nz : \u03b1\nl' : List \u03b1\nx : \u03b1\nh : \u2191(prod (zipWith swap l l')) x = x\nhx : y \u2260 z \u2227 (x = y \u2228 x = z)\n\u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l'))"}, {"tactic": "rcases hx with \u27e8hyz, rfl | rfl\u27e9 <;> simp", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.671533\ninst\u271d : DecidableEq \u03b1\nl\u271d : List \u03b1\nx\u271d : \u03b1\nl'\u271d : List \u03b1\ny : \u03b1\nl : List \u03b1\nhl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l')\nz : \u03b1\nl' : List \u03b1\nx : \u03b1\nh : \u2191(prod (zipWith swap l l')) x = x\nhx : y \u2260 z \u2227 (x = y \u2228 x = z)\n\u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l'))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/LazyList/Basic.lean", "full_name": "LazyList.forall_mem_cons", "start": [261, 1], "end": [263, 74], "traced_tactics": [{"tactic": "simp only [Membership.mem, LazyList.Mem, or_imp, forall_and, forall_eq]", "state_before": "\u03b1 : Type u_1\np : \u03b1 \u2192 Prop\na : \u03b1\nl : Thunk (LazyList \u03b1)\n\u22a2 (\u2200 (x : \u03b1), x \u2208 cons a l \u2192 p x) \u2194 p a \u2227 \u2200 (x : \u03b1), x \u2208 Thunk.get l \u2192 p x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Sym/Card.lean", "full_name": "Sym.card_sym_eq_multichoose", "start": [113, 1], "end": [116, 45], "traced_tactics": [{"tactic": "rw [\u2190 card_sym_fin_eq_multichoose]", "state_before": "\u03b1\u271d : Type ?u.22617\n\u03b2 : Type ?u.22620\nn : \u2115\n\u03b1 : Type u_1\nk : \u2115\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype (Sym \u03b1 k)\n\u22a2 Fintype.card (Sym \u03b1 k) = multichoose (Fintype.card \u03b1) k", "state_after": "\u03b1\u271d : Type ?u.22617\n\u03b2 : Type ?u.22620\nn : \u2115\n\u03b1 : Type u_1\nk : \u2115\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype (Sym \u03b1 k)\n\u22a2 Fintype.card (Sym \u03b1 k) = Fintype.card (Sym (Fin (Fintype.card \u03b1)) k)"}, {"tactic": "exact card_congr (equivCongr (equivFin \u03b1))", "state_before": "\u03b1\u271d : Type ?u.22617\n\u03b2 : Type ?u.22620\nn : \u2115\n\u03b1 : Type 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Type u_2\nV' : Type ?u.205544\nP : Type u_3\nP' : Type ?u.205550\ninst\u271d\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx y z : P\n\u22a2 Wbtw R z y x \u2227 Wbtw R y z x \u2194 z = y"}, {"tactic": "exact wbtw_swap_left_iff R x", "state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.205544\nP : Type u_3\nP' : Type ?u.205550\ninst\u271d\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nx y z : P\n\u22a2 Wbtw R z y x \u2227 Wbtw R y z x \u2194 z = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": 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{"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.mem_ae_map_iff", "start": [2697, 1], "end": [2699, 80], "traced_tactics": [{"tactic": "simp only [mem_ae_iff, map_apply_of_aemeasurable hf hs.compl, preimage_compl]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.549927\n\u03b4 : Type ?u.549930\n\u03b9 : Type ?u.549933\nR : Type ?u.549936\nR' : Type ?u.549939\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 : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 s \u2208 ae (Measure.map f \u03bc) \u2194 f \u207b\u00b9' s \u2208 ae \u03bc", "state_after": "no goals"}]}, {"url": 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u_1\nM\u2083 : Type u_3\ninst\u271d\u2076 : TopologicalSpace M\u2081\ninst\u271d\u2075 : TopologicalSpace M\u2082\ninst\u271d\u2074 : TopologicalSpace M\u2083\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : Module R\u2081 M\u2081\ninst\u271d\u00b9 : AddCommMonoid M\u2082\ninst\u271d : Module R\u2082 M\u2082\nf : M\u2082 \u2192 M\u2083\nhf : IsCompactOperator f\ng : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\n\u22a2 IsCompactOperator (f \u2218 \u2191g)", "state_after": "R\u2081 : Type u_4\nR\u2082 : Type u_2\nR\u2083 : Type ?u.166093\ninst\u271d\u2079 : Semiring R\u2081\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\nM\u2081 : Type u_5\nM\u2082 : Type u_1\nM\u2083 : Type u_3\ninst\u271d\u2076 : TopologicalSpace M\u2081\ninst\u271d\u2075 : TopologicalSpace M\u2082\ninst\u271d\u2074 : TopologicalSpace M\u2083\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : Module R\u2081 M\u2081\ninst\u271d\u00b9 : AddCommMonoid M\u2082\ninst\u271d : Module R\u2082 M\u2082\nf : M\u2082 \u2192 M\u2083\nhf : IsCompactOperator f\ng : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nthis : Tendsto (\u2191g) (\ud835\udcdd 0) (\ud835\udcdd (\u2191g 0))\n\u22a2 IsCompactOperator (f \u2218 \u2191g)"}, {"tactic": "rw [map_zero] at this", "state_before": "R\u2081 : Type u_4\nR\u2082 : Type u_2\nR\u2083 : Type ?u.166093\ninst\u271d\u2079 : Semiring R\u2081\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\nM\u2081 : Type u_5\nM\u2082 : Type u_1\nM\u2083 : Type u_3\ninst\u271d\u2076 : TopologicalSpace M\u2081\ninst\u271d\u2075 : TopologicalSpace M\u2082\ninst\u271d\u2074 : TopologicalSpace M\u2083\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : Module R\u2081 M\u2081\ninst\u271d\u00b9 : AddCommMonoid M\u2082\ninst\u271d : Module R\u2082 M\u2082\nf : M\u2082 \u2192 M\u2083\nhf : IsCompactOperator f\ng : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nthis : Tendsto (\u2191g) (\ud835\udcdd 0) (\ud835\udcdd (\u2191g 0))\n\u22a2 IsCompactOperator (f \u2218 \u2191g)", "state_after": "R\u2081 : Type u_4\nR\u2082 : Type u_2\nR\u2083 : Type ?u.166093\ninst\u271d\u2079 : Semiring R\u2081\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\nM\u2081 : Type u_5\nM\u2082 : Type u_1\nM\u2083 : Type u_3\ninst\u271d\u2076 : TopologicalSpace M\u2081\ninst\u271d\u2075 : TopologicalSpace M\u2082\ninst\u271d\u2074 : TopologicalSpace M\u2083\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : Module R\u2081 M\u2081\ninst\u271d\u00b9 : AddCommMonoid M\u2082\ninst\u271d : Module R\u2082 M\u2082\nf : M\u2082 \u2192 M\u2083\nhf : IsCompactOperator f\ng : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nthis : Tendsto (\u2191g) (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u22a2 IsCompactOperator (f \u2218 \u2191g)"}, {"tactic": "rcases hf with \u27e8K, hK, hKf\u27e9", "state_before": "R\u2081 : Type u_4\nR\u2082 : Type u_2\nR\u2083 : Type ?u.166093\ninst\u271d\u2079 : Semiring R\u2081\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\nM\u2081 : Type u_5\nM\u2082 : Type u_1\nM\u2083 : Type u_3\ninst\u271d\u2076 : TopologicalSpace M\u2081\ninst\u271d\u2075 : TopologicalSpace M\u2082\ninst\u271d\u2074 : TopologicalSpace M\u2083\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : Module R\u2081 M\u2081\ninst\u271d\u00b9 : AddCommMonoid M\u2082\ninst\u271d : Module R\u2082 M\u2082\nf : M\u2082 \u2192 M\u2083\nhf : IsCompactOperator f\ng : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nthis : Tendsto (\u2191g) (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u22a2 IsCompactOperator (f \u2218 \u2191g)", "state_after": "case intro.intro\nR\u2081 : Type u_4\nR\u2082 : Type u_2\nR\u2083 : Type ?u.166093\ninst\u271d\u2079 : Semiring R\u2081\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\nM\u2081 : Type u_5\nM\u2082 : Type u_1\nM\u2083 : Type u_3\ninst\u271d\u2076 : TopologicalSpace M\u2081\ninst\u271d\u2075 : TopologicalSpace M\u2082\ninst\u271d\u2074 : TopologicalSpace M\u2083\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : Module R\u2081 M\u2081\ninst\u271d\u00b9 : AddCommMonoid M\u2082\ninst\u271d : Module R\u2082 M\u2082\nf : M\u2082 \u2192 M\u2083\ng : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nthis : Tendsto (\u2191g) (\ud835\udcdd 0) (\ud835\udcdd 0)\nK : Set M\u2083\nhK : IsCompact K\nhKf : f \u207b\u00b9' K \u2208 \ud835\udcdd 0\n\u22a2 IsCompactOperator (f \u2218 \u2191g)"}, {"tactic": "exact \u27e8K, hK, this hKf\u27e9", "state_before": "case intro.intro\nR\u2081 : Type u_4\nR\u2082 : Type u_2\nR\u2083 : Type ?u.166093\ninst\u271d\u2079 : Semiring R\u2081\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\nM\u2081 : Type u_5\nM\u2082 : Type u_1\nM\u2083 : Type u_3\ninst\u271d\u2076 : TopologicalSpace M\u2081\ninst\u271d\u2075 : TopologicalSpace M\u2082\ninst\u271d\u2074 : TopologicalSpace M\u2083\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : Module R\u2081 M\u2081\ninst\u271d\u00b9 : AddCommMonoid M\u2082\ninst\u271d : Module R\u2082 M\u2082\nf : M\u2082 \u2192 M\u2083\ng : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nthis : Tendsto (\u2191g) (\ud835\udcdd 0) (\ud835\udcdd 0)\nK : Set M\u2083\nhK : IsCompact K\nhKf : f \u207b\u00b9' K \u2208 \ud835\udcdd 0\n\u22a2 IsCompactOperator (f \u2218 \u2191g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Lattice.lean", "full_name": "left_lt_sup", "start": [208, 1], "end": [209, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/LocalExtr.lean", "full_name": "IsLocalMinOn.congr", "start": [573, 8], "end": [575, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "AEMeasurable.ennnorm", "start": [2074, 1], "end": [2076, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Int/Lemmas.lean", "full_name": "Int.add_def", "start": [49, 9], "end": [49, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean", "full_name": "CategoryTheory.IsPullback.of_vert_isIso", "start": [656, 1], "end": [657, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Basic.lean", "full_name": "Metric.mem_closedBall_comm", "start": [576, 1], "end": [577, 39], "traced_tactics": [{"tactic": "rw [mem_closedBall', mem_closedBall]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.38777\n\u03b9 : Type ?u.38780\ninst\u271d : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns : Set \u03b1\n\u22a2 x \u2208 closedBall y \u03b5 \u2194 y \u2208 closedBall x \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Separation.lean", "full_name": "discrete_of_t1_of_finite", "start": [785, 1], "end": [790, 34], "traced_tactics": [{"tactic": "apply singletons_open_iff_discrete.mp", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : T1Space X\ninst\u271d : Finite X\n\u22a2 DiscreteTopology X", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : T1Space X\ninst\u271d : Finite X\n\u22a2 \u2200 (a : X), IsOpen {a}"}, {"tactic": "intro x", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : T1Space X\ninst\u271d : Finite X\n\u22a2 \u2200 (a : X), IsOpen {a}", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : T1Space X\ninst\u271d : Finite X\nx : X\n\u22a2 IsOpen {x}"}, {"tactic": "rw [\u2190 isClosed_compl_iff]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : T1Space X\ninst\u271d : Finite X\nx : X\n\u22a2 IsOpen {x}", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : T1Space X\ninst\u271d : Finite X\nx : X\n\u22a2 IsClosed ({x}\u1d9c)"}, {"tactic": "exact (Set.toFinite _).isClosed", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : T1Space X\ninst\u271d : Finite X\nx : X\n\u22a2 IsClosed ({x}\u1d9c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Int/CharZero.lean", "full_name": "Int.cast_eq_zero", "start": [26, 1], "end": [33, 35], "traced_tactics": [{"tactic": "cases n", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne \u03b1\ninst\u271d : CharZero \u03b1\nn : \u2124\nh : \u2191n = 0\n\u22a2 n = 0", "state_after": "case ofNat\n\u03b1 : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne \u03b1\ninst\u271d : CharZero \u03b1\na\u271d : \u2115\nh : \u2191(ofNat a\u271d) = 0\n\u22a2 ofNat a\u271d = 0\n\ncase negSucc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne \u03b1\ninst\u271d : CharZero \u03b1\na\u271d : \u2115\nh : \u2191-[a\u271d+1] = 0\n\u22a2 -[a\u271d+1] = 0"}, {"tactic": "erw [Int.cast_ofNat] at h", "state_before": "case ofNat\n\u03b1 : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne \u03b1\ninst\u271d : CharZero \u03b1\na\u271d : \u2115\nh : \u2191(ofNat a\u271d) = 0\n\u22a2 ofNat a\u271d = 0", "state_after": "case ofNat\n\u03b1 : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne \u03b1\ninst\u271d : CharZero \u03b1\na\u271d : \u2115\nh : \u2191a\u271d = 0\n\u22a2 ofNat a\u271d = 0"}, {"tactic": "exact congr_arg _ (Nat.cast_eq_zero.1 h)", "state_before": "case ofNat\n\u03b1 : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne \u03b1\ninst\u271d : CharZero \u03b1\na\u271d : \u2115\nh : \u2191a\u271d = 0\n\u22a2 ofNat a\u271d = 0", "state_after": "no goals"}, {"tactic": "rw [cast_negSucc, neg_eq_zero, Nat.cast_eq_zero] at h", "state_before": "case negSucc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne \u03b1\ninst\u271d : CharZero \u03b1\na\u271d : \u2115\nh : \u2191-[a\u271d+1] = 0\n\u22a2 -[a\u271d+1] = 0", "state_after": "case negSucc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne \u03b1\ninst\u271d : CharZero \u03b1\na\u271d : \u2115\nh : a\u271d + 1 = 0\n\u22a2 -[a\u271d+1] = 0"}, {"tactic": "contradiction", "state_before": "case negSucc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne \u03b1\ninst\u271d : CharZero \u03b1\na\u271d : \u2115\nh : a\u271d + 1 = 0\n\u22a2 -[a\u271d+1] = 0", "state_after": "no goals"}, {"tactic": "rw [h, cast_zero]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne \u03b1\ninst\u271d : CharZero \u03b1\nn : \u2124\nh : n = 0\n\u22a2 \u2191n = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.mk_preimage_prod_left_eq_if", "start": [248, 1], "end": [249, 94], "traced_tactics": [{"tactic": "split_ifs with h <;> simp [h]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.27643\n\u03b4 : Type ?u.27646\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d : DecidablePred fun x => x \u2208 t\n\u22a2 (fun a => (a, b)) \u207b\u00b9' s \u00d7\u02e2 t = if b \u2208 t then s else \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Option/NAry.lean", "full_name": "Option.map\u2082_map_right", "start": [101, 1], "end": [102, 79], "traced_tactics": [{"tactic": "cases b <;> rfl", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_4\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : Option \u03b1\nb : Option \u03b2\nc : Option \u03b3\n\u03b4 : Type u_1\nf : \u03b1 \u2192 \u03b3 \u2192 \u03b4\ng : \u03b2 \u2192 \u03b3\n\u22a2 map\u2082 f a (Option.map g b) = map\u2082 (fun a b => f a (g b)) a b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Monotone/Basic.lean", "full_name": "StrictMono.imp", "start": [371, 1], "end": [372, 7], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/FiberBundle/Trivialization.lean", "full_name": "Pretrivialization.coe_symm_of_not_mem", "start": [269, 1], "end": [271, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/FiniteDimension.lean", "full_name": "exists_seq_norm_le_one_le_norm_sub", "start": [437, 1], "end": [442, 45], "traced_tactics": [{"tactic": "obtain \u27e8c, hc\u27e9 : \u2203 c : \ud835\udd5c, 1 < \u2016c\u2016 := NormedField.exists_one_lt_norm \ud835\udd5c", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nh : \u00acFiniteDimensional \ud835\udd5c E\n\u22a2 \u2203 R f, 1 < R \u2227 (\u2200 (n : \u2115), \u2016f n\u2016 \u2264 R) \u2227 \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016", "state_after": "case intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nh : \u00acFiniteDimensional \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\n\u22a2 \u2203 R f, 1 < R \u2227 (\u2200 (n : \u2115), \u2016f n\u2016 \u2264 R) \u2227 \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016"}, {"tactic": "have A : \u2016c\u2016 < \u2016c\u2016 + 1 := by linarith", "state_before": "case intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nh : \u00acFiniteDimensional \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\n\u22a2 \u2203 R f, 1 < R \u2227 (\u2200 (n : \u2115), \u2016f n\u2016 \u2264 R) \u2227 \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016", "state_after": "case intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nh : \u00acFiniteDimensional \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nA : \u2016c\u2016 < \u2016c\u2016 + 1\n\u22a2 \u2203 R f, 1 < R \u2227 (\u2200 (n : \u2115), \u2016f n\u2016 \u2264 R) \u2227 \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016"}, {"tactic": "rcases exists_seq_norm_le_one_le_norm_sub' hc A h with \u27e8f, hf\u27e9", "state_before": "case intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nh : \u00acFiniteDimensional \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nA : \u2016c\u2016 < \u2016c\u2016 + 1\n\u22a2 \u2203 R f, 1 < R \u2227 (\u2200 (n : \u2115), \u2016f n\u2016 \u2264 R) \u2227 \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016", "state_after": "case intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nh : \u00acFiniteDimensional \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nA : \u2016c\u2016 < \u2016c\u2016 + 1\nf : \u2115 \u2192 E\nhf : (\u2200 (n : \u2115), \u2016f n\u2016 \u2264 \u2016c\u2016 + 1) \u2227 \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\n\u22a2 \u2203 R f, 1 < R \u2227 (\u2200 (n : \u2115), \u2016f n\u2016 \u2264 R) \u2227 \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016"}, {"tactic": "exact \u27e8\u2016c\u2016 + 1, f, hc.trans A, hf.1, hf.2\u27e9", "state_before": "case intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nh : \u00acFiniteDimensional \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nA : \u2016c\u2016 < \u2016c\u2016 + 1\nf : \u2115 \u2192 E\nhf : (\u2200 (n : \u2115), \u2016f n\u2016 \u2264 \u2016c\u2016 + 1) \u2227 \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\n\u22a2 \u2203 R f, 1 < R \u2227 (\u2200 (n : \u2115), \u2016f n\u2016 \u2264 R) \u2227 \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016", "state_after": "no goals"}, {"tactic": "linarith", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nh : \u00acFiniteDimensional \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\n\u22a2 \u2016c\u2016 < \u2016c\u2016 + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/ContinuousFunction/Bounded.lean", "full_name": "BoundedContinuousFunction.continuous_comp", "start": [440, 1], "end": [442, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Init/Algebra/Order.lean", "full_name": "le_trans", "start": [60, 1], "end": [61, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.vsub_iInter_subset", "start": [738, 1], "end": [739, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean", "full_name": "PrimeSpectrum.gc_set", "start": [199, 1], "end": [203, 70], "traced_tactics": [{"tactic": "have ideal_gc : GaloisConnection Ideal.span _ := (Submodule.gi R R).gc", "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 GaloisConnection (fun s => zeroLocus s) fun t => \u2191(vanishingIdeal t)", "state_after": "R : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nideal_gc : GaloisConnection Ideal.span SetLike.coe\n\u22a2 GaloisConnection (fun s => zeroLocus s) fun t => \u2191(vanishingIdeal t)"}, {"tactic": "simpa [zeroLocus_span, Function.comp] using ideal_gc.compose (gc R)", "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nideal_gc : GaloisConnection Ideal.span SetLike.coe\n\u22a2 GaloisConnection (fun s => zeroLocus s) fun t => \u2191(vanishingIdeal t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Digits.lean", "full_name": "Nat.nine_dvd_iff", "start": [599, 1], "end": [600, 46], "traced_tactics": [{"tactic": "norm_num", "state_before": "n\u271d n : \u2115\n\u22a2 10 % 9 = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Ncard.lean", "full_name": "Set.exists_ne_map_eq_of_ncard_lt_of_maps_to", "start": [396, 1], "end": [401, 55], "traced_tactics": [{"tactic": "by_contra h'", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\ns t\u271d : Set \u03b1\na b x y : \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nt : Set \u03b2\nhc : ncard t < ncard s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nht : autoParam (Set.Finite t) _auto\u271d\n\u22a2 \u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y \u2227 f x = f y", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_1\ns t\u271d : Set \u03b1\na b x y : \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nt : Set \u03b2\nhc : ncard t < ncard s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nht : autoParam (Set.Finite t) _auto\u271d\nh' : \u00ac\u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y \u2227 f x = f y\n\u22a2 False"}, {"tactic": "simp only [Ne.def, exists_prop, not_exists, not_and, not_imp_not] at h'", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\ns t\u271d : Set \u03b1\na b x y : \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nt : Set \u03b2\nhc : ncard t < ncard s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nht : autoParam (Set.Finite t) _auto\u271d\nh' : \u00ac\u2203 x, x \u2208 s \u2227 \u2203 y, y \u2208 s \u2227 x \u2260 y \u2227 f x = f y\n\u22a2 False", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_1\ns t\u271d : Set \u03b1\na b x y : \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nt : Set \u03b2\nhc : ncard t < ncard s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nht : autoParam (Set.Finite t) _auto\u271d\nh' : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (x_1 : \u03b1), x_1 \u2208 s \u2192 f x = f x_1 \u2192 x = x_1\n\u22a2 False"}, {"tactic": "exact (ncard_le_ncard_of_injOn f hf h' ht).not_lt hc", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\ns t\u271d : Set \u03b1\na b x y : \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nt : Set \u03b2\nhc : ncard t < ncard s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nht : autoParam (Set.Finite t) _auto\u271d\nh' : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (x_1 : \u03b1), x_1 \u2208 s \u2192 f x = f x_1 \u2192 x = x_1\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Associated.lean", "full_name": "Associates.units_eq_one", "start": [873, 1], "end": [874, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Function/Conjugate.lean", "full_name": "Function.Commute.refl", "start": [105, 1], "end": [106, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Extr.lean", "full_name": "IsMaxOn.sub", "start": [518, 1], "end": [519, 53], "traced_tactics": [{"tactic": "simpa only [sub_eq_add_neg] using hf.add hg.neg", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d : OrderedAddCommGroup \u03b2\nf g : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Set \u03b1\nl : Filter \u03b1\nhf : IsMaxOn f s a\nhg : IsMinOn g s a\n\u22a2 IsMaxOn (fun x => f x - g x) s a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Finprod.lean", "full_name": "finprod_eq_of_bijective", "start": [976, 1], "end": [979, 85], "traced_tactics": [{"tactic": "rw [\u2190 finprod_mem_univ f, \u2190 finprod_mem_univ g]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.356791\nG : Type ?u.356794\nM : Type u_3\nN : Type ?u.356800\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf\u271d g\u271d : \u03b1 \u2192 M\na b : \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 M\ng : \u03b2 \u2192 M\ne : \u03b1 \u2192 \u03b2\nhe\u2080 : Bijective e\nhe\u2081 : \u2200 (x : \u03b1), f x = g (e x)\n\u22a2 (\u220f\u1da0 (i : \u03b1), f i) = \u220f\u1da0 (j : \u03b2), g j", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.356791\nG : Type ?u.356794\nM : Type u_3\nN : Type ?u.356800\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf\u271d g\u271d : \u03b1 \u2192 M\na b : \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 M\ng : \u03b2 \u2192 M\ne : \u03b1 \u2192 \u03b2\nhe\u2080 : Bijective e\nhe\u2081 : \u2200 (x : \u03b1), f x = g (e x)\n\u22a2 (\u220f\u1da0 (i : \u03b1) (_ : i \u2208 univ), f i) = \u220f\u1da0 (i : \u03b2) (_ : i \u2208 univ), g i"}, {"tactic": "exact finprod_mem_eq_of_bijOn _ (bijective_iff_bijOn_univ.mp he\u2080) fun x _ => he\u2081 x", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.356791\nG : Type ?u.356794\nM : Type u_3\nN : Type ?u.356800\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf\u271d g\u271d : \u03b1 \u2192 M\na b : \u03b1\ns t : Set \u03b1\nf : \u03b1 \u2192 M\ng : \u03b2 \u2192 M\ne : \u03b1 \u2192 \u03b2\nhe\u2080 : Bijective e\nhe\u2081 : \u2200 (x : \u03b1), f x = g (e x)\n\u22a2 (\u220f\u1da0 (i : \u03b1) (_ : i \u2208 univ), f i) = \u220f\u1da0 (i : \u03b2) (_ : i \u2208 univ), g i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "full_name": "IsCompact.exists_bound_of_continuous_on'", "start": [856, 1], "end": [859, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.ListBlank.head_map", "start": [398, 1], "end": [401, 44], "traced_tactics": [{"tactic": "conv => lhs; rw [\u2190 ListBlank.cons_head_tail l]", "state_before": "\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : ListBlank \u0393\n\u22a2 head (map f l) = PointedMap.f f (head l)", "state_after": "\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : ListBlank \u0393\n\u22a2 head (map f (cons (head l) (tail l))) = PointedMap.f f (head l)"}, {"tactic": "exact Quotient.inductionOn' l fun a \u21a6 rfl", "state_before": "\u0393 : Type u_1\n\u0393' : Type u_2\ninst\u271d\u00b9 : Inhabited \u0393\ninst\u271d : Inhabited \u0393'\nf : PointedMap \u0393 \u0393'\nl : ListBlank \u0393\n\u22a2 head (map f (cons (head l) (tail l))) = PointedMap.f f (head l)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/ENNReal.lean", "full_name": "ENNReal.coe_nnreal_eq", "start": [200, 1], "end": [201, 41], "traced_tactics": [{"tactic": "rw [ENNReal.ofReal, Real.toNNReal_coe]", "state_before": "\u03b1 : Type ?u.18923\n\u03b2 : Type ?u.18926\na b c d : \u211d\u22650\u221e\nr\u271d p q r : \u211d\u22650\n\u22a2 \u2191r = ENNReal.ofReal \u2191r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/ModelTheory/LanguageMap.lean", "full_name": "FirstOrder.Language.LHom.map_onFunction", "start": [255, 1], "end": [257, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/BilinearForm.lean", "full_name": "Matrix.toBilin'_toMatrix'", "start": [205, 1], "end": [207, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/List/Init/Lemmas.lean", "full_name": "List.foldr_self", "start": [205, 1], "end": [205, 65], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b1 : Type u_1\nl : List \u03b1\n\u22a2 foldr cons [] l = l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "full_name": "Summable.sub", "start": [825, 1], "end": [826, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Basic.lean", "full_name": "AffineSubspace.convex", "start": [490, 1], "end": [493, 38], "traced_tactics": [{"tactic": "intro x hx y hy a b _ _ hab", "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.233124\n\u03b2 : Type ?u.233127\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns t : Set E\nQ : AffineSubspace \ud835\udd5c E\n\u22a2 Convex \ud835\udd5c \u2191Q", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.233124\n\u03b2 : Type ?u.233127\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns t : Set E\nQ : AffineSubspace \ud835\udd5c E\nx : E\nhx : x \u2208 \u2191Q\ny : E\nhy : y \u2208 \u2191Q\na b : \ud835\udd5c\na\u271d\u00b9 : 0 \u2264 a\na\u271d : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 x + b \u2022 y \u2208 \u2191Q"}, {"tactic": "rw [eq_sub_of_add_eq hab, \u2190 AffineMap.lineMap_apply_module]", "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.233124\n\u03b2 : Type ?u.233127\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns t : Set E\nQ : AffineSubspace \ud835\udd5c E\nx : E\nhx : x \u2208 \u2191Q\ny : E\nhy : y \u2208 \u2191Q\na b : \ud835\udd5c\na\u271d\u00b9 : 0 \u2264 a\na\u271d : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 x + b \u2022 y \u2208 \u2191Q", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.233124\n\u03b2 : Type ?u.233127\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns t : Set E\nQ : AffineSubspace \ud835\udd5c E\nx : E\nhx : x \u2208 \u2191Q\ny : E\nhy : y \u2208 \u2191Q\na b : \ud835\udd5c\na\u271d\u00b9 : 0 \u2264 a\na\u271d : 0 \u2264 b\nhab : a + b = 1\n\u22a2 \u2191(AffineMap.lineMap x y) b \u2208 \u2191Q"}, {"tactic": "exact AffineMap.lineMap_mem b hx hy", "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.233124\n\u03b2 : Type ?u.233127\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns t : Set E\nQ : AffineSubspace \ud835\udd5c E\nx : E\nhx : x \u2208 \u2191Q\ny : E\nhy : y \u2208 \u2191Q\na b : \ud835\udd5c\na\u271d\u00b9 : 0 \u2264 a\na\u271d : 0 \u2264 b\nhab : a + b = 1\n\u22a2 \u2191(AffineMap.lineMap x y) b \u2208 \u2191Q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/FiniteType.lean", "full_name": "Algebra.FiniteType.trans", "start": [120, 1], "end": [122, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Subobject/Lattice.lean", "full_name": "CategoryTheory.Subobject.sInf_le", "start": [640, 1], "end": [651, 14], "traced_tactics": [{"tactic": "fapply le_of_comm", "state_before": "C : Type u\u2081\ninst\u271d\u00b3 : Category C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : WellPowered C\ninst\u271d : HasWidePullbacks C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f \u2208 s\n\u22a2 sInf s \u2264 f", "state_after": "case f\nC : Type u\u2081\ninst\u271d\u00b3 : Category C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : WellPowered C\ninst\u271d : HasWidePullbacks C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f \u2208 s\n\u22a2 underlying.obj (sInf s) \u27f6 underlying.obj f\n\ncase w\nC : Type u\u2081\ninst\u271d\u00b3 : Category C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : WellPowered C\ninst\u271d : HasWidePullbacks C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f \u2208 s\n\u22a2 ?f \u226b arrow f = arrow (sInf s)"}, {"tactic": ". exact (underlyingIso _).hom \u226b\n    Limits.limit.\u03c0 (wideCospan s)\n      (some \u27e8equivShrink (Subobject A) f,\n        Set.mem_image_of_mem (equivShrink (Subobject A)) hf\u27e9) \u226b\n    eqToHom (congr_arg (fun X : Subobject A => (X : C)) (Equiv.symm_apply_apply _ _))", "state_before": "case f\nC : Type u\u2081\ninst\u271d\u00b3 : Category C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : WellPowered C\ninst\u271d : HasWidePullbacks C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f \u2208 s\n\u22a2 underlying.obj (sInf s) \u27f6 underlying.obj f\n\ncase w\nC : Type u\u2081\ninst\u271d\u00b3 : Category C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : WellPowered C\ninst\u271d : HasWidePullbacks C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f \u2208 s\n\u22a2 ?f \u226b arrow f = arrow (sInf s)", "state_after": "case w\nC : Type u\u2081\ninst\u271d\u00b3 : Category C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : WellPowered C\ninst\u271d : HasWidePullbacks C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f \u2208 s\n\u22a2 ((underlyingIso (widePullback\u03b9 s)).hom \u226b\n        limit.\u03c0 (wideCospan s)\n            (some\n              { val := \u2191(equivShrink (Subobject A)) f,\n                property := (_ : \u2191(equivShrink (Subobject A)) f \u2208 \u2191(equivShrink (Subobject A)) '' s) }) \u226b\n          eqToHom\n            (_ :\n              underlying.obj\n                  (\u2191(equivShrink (Subobject A)).symm\n                    \u2191{ val := \u2191(equivShrink (Subobject A)) f,\n                        property := (_ : \u2191(equivShrink (Subobject A)) f \u2208 \u2191(equivShrink (Subobject A)) '' s) }) =\n                underlying.obj f)) \u226b\n      arrow f =\n    arrow (sInf s)"}, {"tactic": "exact (underlyingIso _).hom \u226b\n  Limits.limit.\u03c0 (wideCospan s)\n    (some \u27e8equivShrink (Subobject A) f,\n      Set.mem_image_of_mem (equivShrink (Subobject A)) hf\u27e9) \u226b\n  eqToHom (congr_arg (fun X : Subobject A => (X : C)) (Equiv.symm_apply_apply _ _))", "state_before": "case f\nC : Type u\u2081\ninst\u271d\u00b3 : Category C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : WellPowered C\ninst\u271d : HasWidePullbacks C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f \u2208 s\n\u22a2 underlying.obj (sInf s) \u27f6 underlying.obj f", "state_after": "no goals"}, {"tactic": "dsimp [sInf]", "state_before": "case w\nC : Type u\u2081\ninst\u271d\u00b3 : Category C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : WellPowered C\ninst\u271d : HasWidePullbacks C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f \u2208 s\n\u22a2 ((underlyingIso (widePullback\u03b9 s)).hom \u226b\n        limit.\u03c0 (wideCospan s)\n            (some\n              { val := \u2191(equivShrink (Subobject A)) f,\n                property := (_ : \u2191(equivShrink (Subobject A)) f \u2208 \u2191(equivShrink (Subobject A)) '' s) }) \u226b\n          eqToHom\n            (_ :\n              underlying.obj\n                  (\u2191(equivShrink (Subobject A)).symm\n                    \u2191{ val := \u2191(equivShrink (Subobject A)) f,\n                        property := (_ : \u2191(equivShrink (Subobject A)) f \u2208 \u2191(equivShrink (Subobject A)) '' s) }) =\n                underlying.obj f)) \u226b\n      arrow f =\n    arrow (sInf s)", "state_after": "case w\nC : Type u\u2081\ninst\u271d\u00b3 : Category C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : WellPowered C\ninst\u271d : HasWidePullbacks C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f \u2208 s\n\u22a2 ((underlyingIso (widePullback\u03b9 s)).hom \u226b\n        limit.\u03c0 (wideCospan s)\n            (some\n              { val := \u2191(equivShrink (Subobject A)) f,\n                property := (_ : \u2191(equivShrink (Subobject A)) f \u2208 \u2191(equivShrink (Subobject A)) '' s) }) \u226b\n          eqToHom\n            (_ :\n              underlying.obj (\u2191(equivShrink (Subobject A)).symm (\u2191(equivShrink (Subobject A)) f)) = underlying.obj f)) \u226b\n      arrow f =\n    arrow (mk (widePullback\u03b9 s))"}, {"tactic": "simp only [Category.comp_id, Category.assoc, \u2190 underlyingIso_hom_comp_eq_mk,\n  Subobject.arrow_congr, congrArg_mpr_hom_left, Iso.cancel_iso_hom_left]", "state_before": "case w\nC : Type u\u2081\ninst\u271d\u00b3 : Category C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : WellPowered C\ninst\u271d : HasWidePullbacks C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f \u2208 s\n\u22a2 ((underlyingIso (widePullback\u03b9 s)).hom \u226b\n        limit.\u03c0 (wideCospan s)\n            (some\n              { val := \u2191(equivShrink (Subobject A)) f,\n                property := (_ : \u2191(equivShrink (Subobject A)) f \u2208 \u2191(equivShrink (Subobject A)) '' s) }) \u226b\n          eqToHom\n            (_ :\n              underlying.obj (\u2191(equivShrink (Subobject A)).symm (\u2191(equivShrink (Subobject A)) f)) = underlying.obj f)) \u226b\n      arrow f =\n    arrow (mk (widePullback\u03b9 s))", "state_after": "case w\nC : Type u\u2081\ninst\u271d\u00b3 : Category C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : WellPowered C\ninst\u271d : HasWidePullbacks C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f \u2208 s\n\u22a2 limit.\u03c0 (wideCospan s)\n        (some\n          { val := \u2191(equivShrink (Subobject A)) f,\n            property := (_ : \u2191(equivShrink (Subobject A)) f \u2208 \u2191(equivShrink (Subobject A)) '' s) }) \u226b\n      eqToHom\n          (_ : underlying.obj (\u2191(equivShrink (Subobject A)).symm (\u2191(equivShrink (Subobject A)) f)) = underlying.obj f) \u226b\n        arrow f =\n    widePullback\u03b9 s"}, {"tactic": "convert limit.w (wideCospan s) (WidePullbackShape.Hom.term _)", "state_before": "case w\nC : Type u\u2081\ninst\u271d\u00b3 : Category C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : WellPowered C\ninst\u271d : HasWidePullbacks C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f \u2208 s\n\u22a2 limit.\u03c0 (wideCospan s)\n        (some\n          { val := \u2191(equivShrink (Subobject A)) f,\n            property := (_ : \u2191(equivShrink (Subobject A)) f \u2208 \u2191(equivShrink (Subobject A)) '' s) }) \u226b\n      eqToHom\n          (_ : underlying.obj (\u2191(equivShrink (Subobject A)).symm (\u2191(equivShrink (Subobject A)) f)) = underlying.obj f) \u226b\n        arrow f =\n    widePullback\u03b9 s", "state_after": "case h.e'_2.h.h.e'_7.h\nC : Type u\u2081\ninst\u271d\u00b3 : Category C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : WellPowered C\ninst\u271d : HasWidePullbacks C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f \u2208 s\ne_1\u271d : (widePullback s \u27f6 A) = (limit (wideCospan s) \u27f6 (wideCospan s).obj none)\ne_4\u271d :\n  (wideCospan s).obj\n      (some\n        { val := \u2191(equivShrink (Subobject A)) f,\n          property := (_ : \u2191(equivShrink (Subobject A)) f \u2208 \u2191(equivShrink (Subobject A)) '' s) }) =\n    (wideCospan s).obj\n      (some\n        { val := \u2191(equivShrink (Subobject A)) f,\n          property := (_ : \u2191(equivShrink (Subobject A)) f \u2208 \u2191(equivShrink (Subobject A)) '' s) })\ne_5\u271d : A = (wideCospan s).obj none\n\u22a2 eqToHom (_ : underlying.obj (\u2191(equivShrink (Subobject A)).symm (\u2191(equivShrink (Subobject A)) f)) = underlying.obj f) \u226b\n      arrow f =\n    (wideCospan s).map\n      (WidePullbackShape.Hom.term\n        { val := \u2191(equivShrink (Subobject A)) f,\n          property := (_ : \u2191(equivShrink (Subobject A)) f \u2208 \u2191(equivShrink (Subobject A)) '' s) })"}, {"tactic": "aesop_cat", "state_before": "case h.e'_2.h.h.e'_7.h\nC : Type u\u2081\ninst\u271d\u00b3 : Category C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b2 : Category D\ninst\u271d\u00b9 : WellPowered C\ninst\u271d : HasWidePullbacks C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nhf : f \u2208 s\ne_1\u271d : (widePullback s \u27f6 A) = (limit (wideCospan s) \u27f6 (wideCospan s).obj none)\ne_4\u271d :\n  (wideCospan s).obj\n      (some\n        { val := \u2191(equivShrink (Subobject A)) f,\n          property := (_ : \u2191(equivShrink (Subobject A)) f \u2208 \u2191(equivShrink (Subobject A)) '' s) }) =\n    (wideCospan s).obj\n      (some\n        { val := \u2191(equivShrink (Subobject A)) f,\n          property := (_ : \u2191(equivShrink (Subobject A)) f \u2208 \u2191(equivShrink (Subobject A)) '' s) })\ne_5\u271d : A = (wideCospan s).obj none\n\u22a2 eqToHom (_ : underlying.obj (\u2191(equivShrink (Subobject A)).symm (\u2191(equivShrink (Subobject A)) f)) = underlying.obj f) \u226b\n      arrow f =\n    (wideCospan s).map\n      (WidePullbackShape.Hom.term\n        { val := \u2191(equivShrink (Subobject A)) f,\n          property := (_ : \u2191(equivShrink (Subobject A)) f \u2208 \u2191(equivShrink (Subobject A)) '' s) })", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Dimension.lean", "full_name": "rank_fin_fun", "start": [1031, 1], "end": [1031, 84], "traced_tactics": [{"tactic": "simp [rank_fun']", "state_before": "K : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9 : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type ?u.546106\ninst\u271d\u00b9\u2074 : Ring K\ninst\u271d\u00b9\u00b3 : StrongRankCondition K\ninst\u271d\u00b9\u00b2 : AddCommGroup V\ninst\u271d\u00b9\u00b9 : Module K V\ninst\u271d\u00b9\u2070 : Module.Free K V\ninst\u271d\u2079 : AddCommGroup V'\ninst\u271d\u2078 : Module K V'\ninst\u271d\u2077 : Module.Free K V'\ninst\u271d\u2076 : AddCommGroup V\u2081\ninst\u271d\u2075 : Module K V\u2081\ninst\u271d\u2074 : Module.Free K V\u2081\ninst\u271d\u00b3 : (i : \u03b7) \u2192 AddCommGroup (\u03c6 i)\ninst\u271d\u00b2 : (i : \u03b7) \u2192 Module K (\u03c6 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b7), Module.Free K (\u03c6 i)\ninst\u271d : Fintype \u03b7\nn : \u2115\n\u22a2 Module.rank K (Fin n \u2192 K) = \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Deprecated/Subgroup.lean", "full_name": "IsAddSubgroup.of_sub", "start": [101, 1], "end": [104, 52], "traced_tactics": [{"tactic": "simpa only [sub_eq_add_neg] using sub_mem hx hy", "state_before": "G : Type ?u.9247\nH : Type ?u.9250\nA : Type u_1\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set A\nzero_mem : 0 \u2208 s\nsub_mem : \u2200 {a b : A}, a \u2208 s \u2192 b \u2208 s \u2192 a - b \u2208 s\nx y : A\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 x + -y \u2208 s", "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.isHenstock_single", "start": [329, 1], "end": [330, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Chain.lean", "full_name": "List.Chain.sublist", "start": [132, 11], "end": [135, 35], "traced_tactics": [{"tactic": "rw [chain_iff_pairwise] at hl\u22a2", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\ninst\u271d : IsTrans \u03b1 R\nhl : Chain R a l\u2082\nh : l\u2081 <+ l\u2082\n\u22a2 Chain R a l\u2081", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\ninst\u271d : IsTrans \u03b1 R\nhl : Pairwise R (a :: l\u2082)\nh : l\u2081 <+ l\u2082\n\u22a2 Pairwise R (a :: l\u2081)"}, {"tactic": "exact hl.sublist (h.cons_cons a)", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nR r : \u03b1 \u2192 \u03b1 \u2192 Prop\nl l\u2081 l\u2082 : List \u03b1\na b : \u03b1\ninst\u271d : IsTrans \u03b1 R\nhl : Pairwise R (a :: l\u2082)\nh : l\u2081 <+ l\u2082\n\u22a2 Pairwise R (a :: l\u2081)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/Projection.lean", "full_name": "reflection_reflection", "start": [671, 1], "end": [672, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/ContDiff.lean", "full_name": "ContDiff.comp_contDiff_on\u2082", "start": [868, 1], "end": [871, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.ext_of_generate_finite", "start": [4060, 1], "end": [4063, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean", "full_name": "Matrix.SpecialLinearGroup.SL2_inv_expl", "start": [298, 1], "end": [303, 6], "traced_tactics": [{"tactic": "ext", "state_before": "n : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type ?u.739651\ninst\u271d : CommRing S\nA : SL(2, R)\n\u22a2 A\u207b\u00b9 =\n    { val := ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]],\n      property := (_ : det ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]] = 1) }", "state_after": "case a\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type ?u.739651\ninst\u271d : CommRing S\nA : SL(2, R)\ni\u271d j\u271d : Fin 2\n\u22a2 \u2191A\u207b\u00b9 i\u271d j\u271d =\n    \u2191{ val := ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]],\n          property := (_ : det ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]] = 1) }\n      i\u271d j\u271d"}, {"tactic": "have := Matrix.adjugate_fin_two A.1", "state_before": "case a\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type ?u.739651\ninst\u271d : CommRing S\nA : SL(2, R)\ni\u271d j\u271d : Fin 2\n\u22a2 \u2191A\u207b\u00b9 i\u271d j\u271d =\n    \u2191{ val := ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]],\n          property := (_ : det ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]] = 1) }\n      i\u271d j\u271d", "state_after": "case a\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type ?u.739651\ninst\u271d : CommRing S\nA : SL(2, R)\ni\u271d j\u271d : Fin 2\nthis : adjugate \u2191A = \u2191of ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]]\n\u22a2 \u2191A\u207b\u00b9 i\u271d j\u271d =\n    \u2191{ val := ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]],\n          property := (_ : det ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]] = 1) }\n      i\u271d j\u271d"}, {"tactic": "rw [coe_inv, this]", "state_before": "case a\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type ?u.739651\ninst\u271d : CommRing S\nA : SL(2, R)\ni\u271d j\u271d : Fin 2\nthis : adjugate \u2191A = \u2191of ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]]\n\u22a2 \u2191A\u207b\u00b9 i\u271d j\u271d =\n    \u2191{ val := ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]],\n          property := (_ : det ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]] = 1) }\n      i\u271d j\u271d", "state_after": "case a\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type ?u.739651\ninst\u271d : CommRing S\nA : SL(2, R)\ni\u271d j\u271d : Fin 2\nthis : adjugate \u2191A = \u2191of ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]]\n\u22a2 \u2191of ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]] i\u271d j\u271d =\n    \u2191{ val := ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]],\n          property := (_ : det ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]] = 1) }\n      i\u271d j\u271d"}, {"tactic": "rfl", "state_before": "case a\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type ?u.739651\ninst\u271d : CommRing S\nA : SL(2, R)\ni\u271d j\u271d : Fin 2\nthis : adjugate \u2191A = \u2191of ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]]\n\u22a2 \u2191of ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]] i\u271d j\u271d =\n    \u2191{ val := ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]],\n          property := (_ : det ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]] = 1) }\n      i\u271d j\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.ext_iff", "start": [169, 1], "end": [170, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.Lp.simpleFunc.norm_toLp", "start": [584, 8], "end": [586, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean", "full_name": "CategoryTheory.Limits.Multicofork.condition", "start": [580, 1], "end": [581, 44], "traced_tactics": [{"tactic": "rw [\u2190 K.snd_app_right, \u2190 K.fst_app_right]", "state_before": "C : Type u\ninst\u271d : Category C\nI : MultispanIndex C\nK : Multicofork I\na : I.L\n\u22a2 MultispanIndex.fst I a \u226b \u03c0 K (MultispanIndex.fstFrom I a) = MultispanIndex.snd I a \u226b \u03c0 K (MultispanIndex.sndFrom I a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Pointwise.lean", "full_name": "Filter.smul_le_smul", "start": [1036, 1], "end": [1037, 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"SemilatticeInf.ext_inf", "start": [563, 1], "end": [567, 92], "traced_tactics": [{"tactic": "simp only [le_inf_iff]", "state_before": "\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeInf \u03b1\u271d\na b c\u271d d : \u03b1\u271d\n\u03b1 : Type u_1\nA B : SemilatticeInf \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\nx y c : \u03b1\n\u22a2 c \u2264 x \u2293 y \u2194 c \u2264 x \u2293 y", "state_after": "\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeInf \u03b1\u271d\na b c\u271d d : \u03b1\u271d\n\u03b1 : Type u_1\nA B : SemilatticeInf \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\nx y c : \u03b1\n\u22a2 c \u2264 x \u2293 y \u2194 c \u2264 x \u2227 c \u2264 y"}, {"tactic": "rw [\u2190 H, @le_inf_iff \u03b1 A, H, H]", "state_before": "\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeInf \u03b1\u271d\na b c\u271d d : \u03b1\u271d\n\u03b1 : Type u_1\nA B : SemilatticeInf \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\nx y c : \u03b1\n\u22a2 c \u2264 x \u2293 y \u2194 c \u2264 x \u2227 c \u2264 y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Monoidal/Free/Basic.lean", "full_name": "CategoryTheory.FreeMonoidalCategory.unit_eq_unit", "start": [240, 1], "end": [241, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "Prod.fst_sInf", "start": [1831, 1], "end": [1832, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Hom/Lattice.lean", "full_name": "Codisjoint.map", "start": [273, 1], "end": [274, 52], "traced_tactics": [{"tactic": "rw [codisjoint_iff, \u2190 map_sup, h.eq_top, map_top]", "state_before": "F : Type u_3\n\u03b9 : Type ?u.29958\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.29967\n\u03b4 : Type ?u.29970\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : Lattice \u03b2\ninst\u271d\u00b9 : BoundedOrder \u03b2\ninst\u271d : BoundedLatticeHomClass F \u03b1 \u03b2\nf : F\na b : \u03b1\nh : Codisjoint a b\n\u22a2 Codisjoint (\u2191f a) (\u2191f b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Periodic.lean", "full_name": "Function.Periodic.int_mul_eq", "start": [281, 1], "end": [282, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean", "full_name": "ContinuousLinearMap.compFormalMultilinearSeries_apply", "start": [194, 1], "end": [196, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "dense_compl_singleton_iff_not_open", "start": [674, 1], "end": [682, 16], "traced_tactics": [{"tactic": "constructor", "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 : TopologicalSpace \u03b1\nx : \u03b1\n\u22a2 Dense ({x}\u1d9c) \u2194 \u00acIsOpen {x}", "state_after": "case mp\n\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 : TopologicalSpace \u03b1\nx : \u03b1\n\u22a2 Dense ({x}\u1d9c) \u2192 \u00acIsOpen {x}\n\ncase mpr\n\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 : TopologicalSpace \u03b1\nx : \u03b1\n\u22a2 \u00acIsOpen {x} \u2192 Dense ({x}\u1d9c)"}, {"tactic": "intro hd ho", "state_before": "case mp\n\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 : TopologicalSpace \u03b1\nx : \u03b1\n\u22a2 Dense ({x}\u1d9c) \u2192 \u00acIsOpen {x}", "state_after": "case mp\n\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 : TopologicalSpace \u03b1\nx : \u03b1\nhd : Dense ({x}\u1d9c)\nho : IsOpen {x}\n\u22a2 False"}, {"tactic": "exact (hd.inter_open_nonempty _ ho (singleton_nonempty _)).ne_empty (inter_compl_self _)", "state_before": "case mp\n\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 : TopologicalSpace \u03b1\nx : \u03b1\nhd : Dense ({x}\u1d9c)\nho : IsOpen {x}\n\u22a2 False", "state_after": "no goals"}, {"tactic": "refine' fun ho => dense_iff_inter_open.2 fun U hU hne => inter_compl_nonempty_iff.2 fun hUx => _", "state_before": "case mpr\n\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 : TopologicalSpace \u03b1\nx : \u03b1\n\u22a2 \u00acIsOpen {x} \u2192 Dense ({x}\u1d9c)", "state_after": "case mpr\n\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 : TopologicalSpace \u03b1\nx : \u03b1\nho : \u00acIsOpen {x}\nU : Set \u03b1\nhU : IsOpen U\nhne : Set.Nonempty U\nhUx : U \u2286 {x}\n\u22a2 False"}, {"tactic": "obtain rfl : U = {x}", "state_before": "case mpr\n\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 : TopologicalSpace \u03b1\nx : \u03b1\nho : \u00acIsOpen {x}\nU : Set \u03b1\nhU : IsOpen U\nhne : Set.Nonempty U\nhUx : U \u2286 {x}\n\u22a2 False", "state_after": "\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 : TopologicalSpace \u03b1\nx : \u03b1\nho : \u00acIsOpen {x}\nU : Set \u03b1\nhU : IsOpen U\nhne : Set.Nonempty U\nhUx : U \u2286 {x}\n\u22a2 U = {x}\n\ncase mpr\n\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 : TopologicalSpace \u03b1\nx : \u03b1\nho : \u00acIsOpen {x}\nhU : IsOpen {x}\nhne : Set.Nonempty {x}\nhUx : {x} \u2286 {x}\n\u22a2 False"}, {"tactic": "exact eq_singleton_iff_nonempty_unique_mem.2 \u27e8hne, hUx\u27e9", "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 : TopologicalSpace \u03b1\nx : \u03b1\nho : \u00acIsOpen {x}\nU : Set \u03b1\nhU : IsOpen U\nhne : Set.Nonempty U\nhUx : U \u2286 {x}\n\u22a2 U = {x}\n\ncase mpr\n\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 : TopologicalSpace \u03b1\nx : \u03b1\nho : \u00acIsOpen {x}\nhU : IsOpen {x}\nhne : Set.Nonempty {x}\nhUx : {x} \u2286 {x}\n\u22a2 False", "state_after": "case mpr\n\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 : TopologicalSpace \u03b1\nx : \u03b1\nho : \u00acIsOpen {x}\nhU : IsOpen {x}\nhne : Set.Nonempty {x}\nhUx : {x} \u2286 {x}\n\u22a2 False"}, {"tactic": "exact ho hU", "state_before": "case mpr\n\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 : TopologicalSpace \u03b1\nx : \u03b1\nho : \u00acIsOpen {x}\nhU : IsOpen {x}\nhne : Set.Nonempty {x}\nhUx : {x} \u2286 {x}\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Basic.lean", "full_name": "Set.right_mem_Icc", "start": [218, 1], "end": [218, 65], "traced_tactics": [{"tactic": "simp [le_refl]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.5251\ninst\u271d : Preorder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 b \u2208 Icc a b \u2194 a \u2264 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf_cons", "start": [333, 1], "end": [334, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "full_name": "Set.preimage_mul_const_Iio", "start": [511, 1], "end": [513, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Quandle.lean", "full_name": "Rack.invAct_apply", "start": [214, 1], "end": [215, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Int/Order/Basic.lean", "full_name": "Int.le_induction_down", "start": [179, 11], "end": [188, 28], "traced_tactics": [{"tactic": "refine Int.inductionOn' n m ?_ ?_ ?_", "state_before": "P : \u2124 \u2192 Prop\nm : \u2124\nh0 : P m\nh1 : \u2200 (n : \u2124), n \u2264 m \u2192 P n \u2192 P (n - 1)\nn : \u2124\n\u22a2 n \u2264 m \u2192 P n", "state_after": "case refine_1\nP : \u2124 \u2192 Prop\nm : \u2124\nh0 : P m\nh1 : \u2200 (n : \u2124), n \u2264 m \u2192 P n \u2192 P (n - 1)\nn : \u2124\n\u22a2 m \u2264 m \u2192 P m\n\ncase refine_2\nP : \u2124 \u2192 Prop\nm : \u2124\nh0 : P m\nh1 : \u2200 (n : \u2124), n \u2264 m \u2192 P n \u2192 P (n - 1)\nn : \u2124\n\u22a2 \u2200 (k : \u2124), m \u2264 k \u2192 (k \u2264 m \u2192 P k) \u2192 k + 1 \u2264 m \u2192 P (k + 1)\n\ncase refine_3\nP : \u2124 \u2192 Prop\nm : \u2124\nh0 : P m\nh1 : \u2200 (n : \u2124), n \u2264 m \u2192 P n \u2192 P (n - 1)\nn : \u2124\n\u22a2 \u2200 (k : \u2124), k \u2264 m \u2192 (k \u2264 m \u2192 P k) \u2192 k - 1 \u2264 m \u2192 P (k - 1)"}, {"tactic": "intro", "state_before": "case refine_1\nP : \u2124 \u2192 Prop\nm : \u2124\nh0 : P m\nh1 : \u2200 (n : \u2124), n \u2264 m \u2192 P n \u2192 P (n - 1)\nn : \u2124\n\u22a2 m \u2264 m \u2192 P m", "state_after": "case refine_1\nP : \u2124 \u2192 Prop\nm : \u2124\nh0 : P m\nh1 : \u2200 (n : \u2124), n \u2264 m \u2192 P n \u2192 P (n - 1)\nn : \u2124\na\u271d : m \u2264 m\n\u22a2 P m"}, {"tactic": "exact h0", "state_before": "case refine_1\nP : \u2124 \u2192 Prop\nm : \u2124\nh0 : P m\nh1 : \u2200 (n : \u2124), n \u2264 m \u2192 P n \u2192 P (n - 1)\nn : \u2124\na\u271d : m \u2264 m\n\u22a2 P m", "state_after": "no goals"}, {"tactic": "intro k hle _ hle'", "state_before": "case refine_2\nP : \u2124 \u2192 Prop\nm : \u2124\nh0 : P m\nh1 : \u2200 (n : \u2124), n \u2264 m \u2192 P n \u2192 P (n - 1)\nn : \u2124\n\u22a2 \u2200 (k : \u2124), m \u2264 k \u2192 (k \u2264 m \u2192 P k) \u2192 k + 1 \u2264 m \u2192 P (k + 1)", "state_after": "case refine_2\nP : \u2124 \u2192 Prop\nm : \u2124\nh0 : P m\nh1 : \u2200 (n : \u2124), n \u2264 m \u2192 P n \u2192 P (n - 1)\nn k : \u2124\nhle : m \u2264 k\na\u271d : k \u2264 m \u2192 P k\nhle' : k + 1 \u2264 m\n\u22a2 P (k + 1)"}, {"tactic": "exfalso", "state_before": "case refine_2\nP : \u2124 \u2192 Prop\nm : \u2124\nh0 : P m\nh1 : \u2200 (n : \u2124), n \u2264 m \u2192 P n \u2192 P (n - 1)\nn k : \u2124\nhle : m \u2264 k\na\u271d : k \u2264 m \u2192 P k\nhle' : k + 1 \u2264 m\n\u22a2 P (k + 1)", "state_after": "case refine_2.h\nP : \u2124 \u2192 Prop\nm : \u2124\nh0 : P m\nh1 : \u2200 (n : \u2124), n \u2264 m \u2192 P n \u2192 P (n - 1)\nn k : \u2124\nhle : m \u2264 k\na\u271d : k \u2264 m \u2192 P k\nhle' : k + 1 \u2264 m\n\u22a2 False"}, {"tactic": "exact lt_irrefl k (add_one_le_iff.mp (hle'.trans hle))", "state_before": "case refine_2.h\nP : \u2124 \u2192 Prop\nm : \u2124\nh0 : P m\nh1 : \u2200 (n : \u2124), n \u2264 m \u2192 P n \u2192 P (n - 1)\nn k : \u2124\nhle : m \u2264 k\na\u271d : k \u2264 m \u2192 P k\nhle' : k + 1 \u2264 m\n\u22a2 False", "state_after": "no goals"}, {"tactic": "intro k hle hi _", "state_before": "case refine_3\nP : \u2124 \u2192 Prop\nm : \u2124\nh0 : P m\nh1 : \u2200 (n : \u2124), n \u2264 m \u2192 P n \u2192 P (n - 1)\nn : \u2124\n\u22a2 \u2200 (k : \u2124), k \u2264 m \u2192 (k \u2264 m \u2192 P k) \u2192 k - 1 \u2264 m \u2192 P (k - 1)", "state_after": "case refine_3\nP : \u2124 \u2192 Prop\nm : \u2124\nh0 : P m\nh1 : \u2200 (n : \u2124), n \u2264 m \u2192 P n \u2192 P (n - 1)\nn k : \u2124\nhle : k \u2264 m\nhi : k \u2264 m \u2192 P k\na\u271d : k - 1 \u2264 m\n\u22a2 P (k - 1)"}, {"tactic": "exact h1 k hle (hi hle)", "state_before": "case refine_3\nP : \u2124 \u2192 Prop\nm : \u2124\nh0 : P m\nh1 : \u2200 (n : \u2124), n \u2264 m \u2192 P n \u2192 P (n - 1)\nn k : \u2124\nhle : k \u2264 m\nhi : k \u2264 m \u2192 P k\na\u271d : k - 1 \u2264 m\n\u22a2 P (k - 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Basic.lean", "full_name": "totallyBounded_Ico", "start": [1423, 1], "end": [1424, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/LinearIndependent.lean", "full_name": "LinearIndependent.total_comp_repr", "start": [785, 1], "end": [787, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Basic.lean", "full_name": "Set.Ioo_self", "start": [415, 1], "end": [416, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "Continuous.closure_preimage_subset", "start": [1736, 1], "end": [1739, 52], "traced_tactics": [{"tactic": "rw [\u2190 (isClosed_closure.preimage hf).closure_eq]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.171989\n\u03b4 : Type ?u.171992\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\nt : Set \u03b2\n\u22a2 closure (f \u207b\u00b9' t) \u2286 f \u207b\u00b9' closure t", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.171989\n\u03b4 : Type ?u.171992\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\nt : Set \u03b2\n\u22a2 closure (f \u207b\u00b9' t) \u2286 closure (f \u207b\u00b9' closure t)"}, {"tactic": "exact closure_mono (preimage_mono subset_closure)", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.171989\n\u03b4 : Type ?u.171992\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\nt : Set \u03b2\n\u22a2 closure (f \u207b\u00b9' t) \u2286 closure (f \u207b\u00b9' closure t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Monotone/Monovary.lean", "full_name": "AntivaryOn.comp_right", "start": [159, 1], "end": [160, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Basic.lean", "full_name": "LinearMap.eqLocus_same", "start": [1278, 1], "end": [1280, 37], "traced_tactics": [{"tactic": "simp only [mem_eqLocus, mem_top]", "state_before": "R : Type u_1\nR\u2081 : Type ?u.1134796\nR\u2082 : Type u_2\nR\u2083 : Type ?u.1134802\nR\u2084 : Type ?u.1134805\nS : Type ?u.1134808\nK : Type ?u.1134811\nK\u2082 : Type ?u.1134814\nM : Type u_3\nM' : Type ?u.1134820\nM\u2081 : Type ?u.1134823\nM\u2082 : Type u_4\nM\u2083 : Type ?u.1134829\nM\u2084 : Type ?u.1134832\nN : Type ?u.1134835\nN\u2082 : Type ?u.1134838\n\u03b9 : Type ?u.1134841\nV : Type ?u.1134844\nV\u2082 : Type ?u.1134847\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid 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\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\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 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type ?u.1135257\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nx\u271d : M\n\u22a2 x\u271d \u2208 eqLocus f f \u2194 x\u271d \u2208 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Uniform.lean", "full_name": "Pmf.toOuterMeasure_uniformOfFinset_apply", "start": [83, 1], "end": [97, 67], "traced_tactics": [{"tactic": "simp_rw [uniformOfFinset_apply, and_comm, \u2190 ite_and, and_comm]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.9058\n\u03b3 : Type ?u.9061\ns : Finset \u03b1\nhs : Finset.Nonempty s\na : \u03b1\nt : Set \u03b1\nx : \u03b1\n\u22a2 (if x \u2208 t then \u2191(uniformOfFinset s hs) x else 0) = if x \u2208 s \u2227 x \u2208 t then (\u2191(Finset.card s))\u207b\u00b9 else 0", "state_after": "no goals"}, {"tactic": "let this : x \u2208 s \u2227 x \u2208 t := by simpa using hx", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.9058\n\u03b3 : Type ?u.9061\ns : Finset \u03b1\nhs : Finset.Nonempty s\na : \u03b1\nt : Set \u03b1\nx : \u03b1\nhx : x \u2208 Finset.filter (fun x => x \u2208 t) s\n\u22a2 (if x \u2208 s \u2227 x \u2208 t then (\u2191(Finset.card s))\u207b\u00b9 else 0) = (\u2191(Finset.card s))\u207b\u00b9", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.9058\n\u03b3 : Type ?u.9061\ns : Finset \u03b1\nhs : Finset.Nonempty s\na : \u03b1\nt : Set \u03b1\nx : \u03b1\nhx : x \u2208 Finset.filter (fun x => x \u2208 t) s\nthis : x \u2208 s \u2227 x \u2208 t := Eq.mp Mathlib.Data.Finset.Basic._auxLemma.130 hx\n\u22a2 (if x \u2208 s \u2227 x \u2208 t then (\u2191(Finset.card s))\u207b\u00b9 else 0) = (\u2191(Finset.card s))\u207b\u00b9"}, {"tactic": "simp only [this, and_self_iff, if_true]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.9058\n\u03b3 : Type ?u.9061\ns : Finset \u03b1\nhs : Finset.Nonempty s\na : \u03b1\nt : Set \u03b1\nx : \u03b1\nhx : x \u2208 Finset.filter (fun x => x \u2208 t) s\nthis : x \u2208 s \u2227 x \u2208 t := Eq.mp Mathlib.Data.Finset.Basic._auxLemma.130 hx\n\u22a2 (if x \u2208 s \u2227 x \u2208 t then (\u2191(Finset.card s))\u207b\u00b9 else 0) = (\u2191(Finset.card s))\u207b\u00b9", "state_after": "no goals"}, {"tactic": "simpa using hx", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.9058\n\u03b3 : Type ?u.9061\ns : Finset \u03b1\nhs : Finset.Nonempty s\na : \u03b1\nt : Set \u03b1\nx : \u03b1\nhx : x \u2208 Finset.filter (fun x => x \u2208 t) s\n\u22a2 x \u2208 s \u2227 x \u2208 t", "state_after": "no goals"}, {"tactic": "simp only [div_eq_mul_inv, Finset.sum_const, nsmul_eq_mul]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.9058\n\u03b3 : Type ?u.9061\ns : Finset \u03b1\nhs : Finset.Nonempty s\na : \u03b1\nt : Set \u03b1\n\u22a2 \u2211 _x in Finset.filter (fun x => x \u2208 t) s, (\u2191(Finset.card s))\u207b\u00b9 =\n    \u2191(Finset.card (Finset.filter (fun x => x \u2208 t) s)) / \u2191(Finset.card s)", "state_after": "no goals"}]}, {"url": 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"5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Sigma.lean", "full_name": "Set.sigma_univ_range_eq", "start": [200, 1], "end": [202, 25], "traced_tactics": [{"tactic": "simp [range]", "state_before": "\u03b9 : Type u_1\n\u03b9' : Type ?u.14429\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_2\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nu : Set ((i : \u03b9) \u00d7 \u03b1 i)\nx : (i : \u03b9) \u00d7 \u03b1 i\ni j : \u03b9\na : \u03b1 i\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\n\u22a2 \u2200 (x : (i : \u03b9) \u00d7 \u03b2 i),\n    (x \u2208 Set.Sigma univ fun i => range (f i)) \u2194 x \u2208 range fun x => { fst := x.fst, snd := f x.fst x.snd }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "full_name": "Finsupp.coe_finset_sum", "start": [318, 1], "end": [320, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/DirectSum/Ring.lean", "full_name": "DirectSum.of_zero_smul", "start": [438, 1], "end": [439, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/Option.lean", "full_name": "Equiv.optionSubtype_apply_apply", "start": [217, 1], "end": [221, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Tropical/Basic.lean", "full_name": "Tropical.untrop_trop", "start": [101, 1], "end": [102, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Associated.lean", "full_name": "Irreducible.isUnit_or_isUnit", "start": [179, 1], "end": [181, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Ordmap/Ordset.lean", "full_name": "Ordnode.Valid'.valid", "start": [1055, 1], "end": [1056, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean", "full_name": "AffineMap.coe_homothetyAffine", "start": [856, 1], "end": [857, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/ModelTheory/Basic.lean", "full_name": "FirstOrder.Language.Embedding.coe_toHom", "start": [642, 1], "end": [643, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Nat.preimage_Ico", "start": [389, 1], "end": [391, 26], "traced_tactics": [{"tactic": "ext", "state_before": "F : Type ?u.63320\n\u03b1 : Type u_1\n\u03b2 : Type ?u.63326\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na b : \u03b1\n\u22a2 Nat.cast \u207b\u00b9' Ico a b = Ico \u2308a\u2309\u208a \u2308b\u2309\u208a", "state_after": "case h\nF : Type ?u.63320\n\u03b1 : Type u_1\n\u03b2 : Type ?u.63326\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na b : \u03b1\nx\u271d : \u2115\n\u22a2 x\u271d \u2208 Nat.cast \u207b\u00b9' Ico a b \u2194 x\u271d \u2208 Ico \u2308a\u2309\u208a \u2308b\u2309\u208a"}, {"tactic": "simp [ceil_le, lt_ceil]", "state_before": "case h\nF : Type ?u.63320\n\u03b1 : Type u_1\n\u03b2 : Type ?u.63326\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na\u271d : \u03b1\nn : \u2115\na b : \u03b1\nx\u271d : \u2115\n\u22a2 x\u271d \u2208 Nat.cast \u207b\u00b9' Ico a b \u2194 x\u271d \u2208 Ico \u2308a\u2309\u208a \u2308b\u2309\u208a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Algebra.lean", "full_name": "continuousSMul_of_algebraMap", "start": [59, 1], "end": [60, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.val_eq_zero", "start": [839, 1], "end": [843, 18], "traced_tactics": [{"tactic": "rw [Fin.ext_iff]", "state_before": "n : \u2115\na : ZMod (n + 1)\n\u22a2 val a = 0 \u2194 a = 0", "state_after": "n : \u2115\na : ZMod (n + 1)\n\u22a2 val a = 0 \u2194 \u2191a = \u21910"}, {"tactic": "exact Iff.rfl", "state_before": "n : \u2115\na : ZMod (n + 1)\n\u22a2 val a = 0 \u2194 \u2191a = \u21910", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Extr.lean", "full_name": "Filter.EventuallyLE.isMaxFilter", "start": [637, 1], "end": [642, 24], "traced_tactics": [{"tactic": "refine' hle.mp (h.mono fun x hf hgf => _)", "state_before": "\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03b4 : Type x\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\na : \u03b1\nl : Filter \u03b1\nhle : g \u2264\u1da0[l] f\nhfga : f a = g a\nh : IsMaxFilter f l a\n\u22a2 IsMaxFilter g l a", "state_after": "\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03b4 : Type x\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder 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"exact le_trans hgf hf", "state_before": "\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03b4 : Type x\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\na : \u03b1\nl : Filter \u03b1\nhle : g \u2264\u1da0[l] f\nhfga : f a = g a\nh : IsMaxFilter f l a\nx : \u03b1\nhf : f x \u2264 f a\nhgf : g x \u2264 f x\n\u22a2 g x \u2264 f a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "full_name": "CategoryTheory.Limits.walkingParallelPairOp_zero", "start": [142, 1], "end": [142, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Complex/LocallyUniformLimit.lean", "full_name": "TendstoUniformlyOn.cderiv", "start": [99, 1], "end": [115, 90], "traced_tactics": [{"tactic": "by_cases \u03c6 = \u22a5", "state_before": "E : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : TendstoUniformlyOn F f \u03c6 (cthickening \u03b4 K)\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\n\u22a2 TendstoUniformlyOn (Complex.cderiv \u03b4 \u2218 F) (Complex.cderiv \u03b4 f) \u03c6 K", "state_after": "case pos\nE : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : TendstoUniformlyOn F f \u03c6 (cthickening \u03b4 K)\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\nh : \u03c6 = \u22a5\n\u22a2 TendstoUniformlyOn (Complex.cderiv \u03b4 \u2218 F) (Complex.cderiv \u03b4 f) \u03c6 K\n\ncase neg\nE : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : TendstoUniformlyOn F f \u03c6 (cthickening \u03b4 K)\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\nh : \u00ac\u03c6 = \u22a5\n\u22a2 TendstoUniformlyOn (Complex.cderiv \u03b4 \u2218 F) (Complex.cderiv \u03b4 f) \u03c6 K"}, {"tactic": "haveI : \u03c6.NeBot := neBot_iff.2 h", "state_before": "case neg\nE : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : TendstoUniformlyOn F f \u03c6 (cthickening \u03b4 K)\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\nh : \u00ac\u03c6 = \u22a5\n\u22a2 TendstoUniformlyOn (Complex.cderiv \u03b4 \u2218 F) (Complex.cderiv \u03b4 f) \u03c6 K", "state_after": "case neg\nE : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : TendstoUniformlyOn F f \u03c6 (cthickening \u03b4 K)\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\nh : \u00ac\u03c6 = \u22a5\nthis : NeBot \u03c6\n\u22a2 TendstoUniformlyOn (Complex.cderiv \u03b4 \u2218 F) (Complex.cderiv \u03b4 f) \u03c6 K"}, {"tactic": "have e1 : ContinuousOn f (cthickening \u03b4 K) := TendstoUniformlyOn.continuousOn hF hFn", "state_before": "case neg\nE : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : TendstoUniformlyOn F f \u03c6 (cthickening \u03b4 K)\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\nh : \u00ac\u03c6 = \u22a5\nthis : NeBot \u03c6\n\u22a2 TendstoUniformlyOn (Complex.cderiv \u03b4 \u2218 F) (Complex.cderiv \u03b4 f) \u03c6 K", "state_after": "case neg\nE : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : TendstoUniformlyOn F f \u03c6 (cthickening \u03b4 K)\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\nh : \u00ac\u03c6 = \u22a5\nthis : NeBot \u03c6\ne1 : ContinuousOn f (cthickening \u03b4 K)\n\u22a2 TendstoUniformlyOn (Complex.cderiv \u03b4 \u2218 F) (Complex.cderiv \u03b4 f) \u03c6 K"}, {"tactic": "rw [tendstoUniformlyOn_iff] at hF \u22a2", "state_before": "case neg\nE : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : TendstoUniformlyOn F f \u03c6 (cthickening \u03b4 K)\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\nh : \u00ac\u03c6 = \u22a5\nthis : NeBot \u03c6\ne1 : ContinuousOn f (cthickening \u03b4 K)\n\u22a2 TendstoUniformlyOn (Complex.cderiv \u03b4 \u2218 F) (Complex.cderiv \u03b4 f) \u03c6 K", "state_after": "case neg\nE : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9) in \u03c6, \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 dist (f x) (F n x) < \u03b5\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\nh : \u00ac\u03c6 = \u22a5\nthis : NeBot \u03c6\ne1 : ContinuousOn f (cthickening \u03b4 K)\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9) in \u03c6, \u2200 (x : \u2102), x \u2208 K \u2192 dist (Complex.cderiv \u03b4 f x) ((Complex.cderiv \u03b4 \u2218 F) n x) < \u03b5"}, {"tactic": "rintro \u03b5 h\u03b5", "state_before": "case neg\nE : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9) in \u03c6, \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 dist (f x) (F n x) < \u03b5\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\nh : \u00ac\u03c6 = \u22a5\nthis : NeBot \u03c6\ne1 : ContinuousOn f (cthickening \u03b4 K)\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9) in \u03c6, \u2200 (x : \u2102), x \u2208 K \u2192 dist (Complex.cderiv \u03b4 f x) ((Complex.cderiv \u03b4 \u2218 F) n x) < \u03b5", "state_after": "case neg\nE : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9) in \u03c6, \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 dist (f x) (F n x) < \u03b5\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\nh : \u00ac\u03c6 = \u22a5\nthis : NeBot \u03c6\ne1 : ContinuousOn f (cthickening \u03b4 K)\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2200\u1da0 (n : \u03b9) in \u03c6, \u2200 (x : \u2102), x \u2208 K \u2192 dist (Complex.cderiv \u03b4 f x) ((Complex.cderiv \u03b4 \u2218 F) n x) < \u03b5"}, {"tactic": "filter_upwards [hF (\u03b5 * \u03b4) (mul_pos h\u03b5 h\u03b4), hFn] with n h h' z hz", "state_before": "case neg\nE : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9) in \u03c6, \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 dist (f x) (F n x) < \u03b5\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\nh : \u00ac\u03c6 = \u22a5\nthis : NeBot \u03c6\ne1 : ContinuousOn f (cthickening \u03b4 K)\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2200\u1da0 (n : \u03b9) in \u03c6, \u2200 (x : \u2102), x \u2208 K \u2192 dist (Complex.cderiv \u03b4 f x) ((Complex.cderiv \u03b4 \u2218 F) n x) < \u03b5", "state_after": "case h\nE : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz\u271d : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9) in \u03c6, \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 dist (f x) (F n x) < \u03b5\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\nh\u271d : \u00ac\u03c6 = \u22a5\nthis : NeBot \u03c6\ne1 : ContinuousOn f (cthickening \u03b4 K)\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nn : \u03b9\nh : \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 dist (f x) (F n x) < \u03b5 * \u03b4\nh' : ContinuousOn (F n) (cthickening \u03b4 K)\nz : \u2102\nhz : z \u2208 K\n\u22a2 dist (Complex.cderiv \u03b4 f z) ((Complex.cderiv \u03b4 \u2218 F) n z) < \u03b5"}, {"tactic": "simp_rw [dist_eq_norm] at h \u22a2", "state_before": "case h\nE : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz\u271d : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9) in \u03c6, \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 dist (f x) (F n x) < \u03b5\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\nh\u271d : \u00ac\u03c6 = \u22a5\nthis : NeBot \u03c6\ne1 : ContinuousOn f (cthickening \u03b4 K)\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nn : \u03b9\nh : \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 dist (f x) (F n x) < \u03b5 * \u03b4\nh' : ContinuousOn (F n) (cthickening \u03b4 K)\nz : \u2102\nhz : z \u2208 K\n\u22a2 dist (Complex.cderiv \u03b4 f z) ((Complex.cderiv \u03b4 \u2218 F) n z) < \u03b5", "state_after": "case h\nE : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz\u271d : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9) in \u03c6, \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 dist (f x) (F n x) < \u03b5\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\nh\u271d : \u00ac\u03c6 = \u22a5\nthis : NeBot \u03c6\ne1 : ContinuousOn f (cthickening \u03b4 K)\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nn : \u03b9\nh' : ContinuousOn (F n) (cthickening \u03b4 K)\nz : \u2102\nhz : z \u2208 K\nh : \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 \u2016f x - F n x\u2016 < \u03b5 * \u03b4\n\u22a2 \u2016Complex.cderiv \u03b4 f z - (Complex.cderiv \u03b4 \u2218 F) n z\u2016 < \u03b5"}, {"tactic": "have e2 : \u2200 w \u2208 sphere z \u03b4, \u2016f w - F n w\u2016 < \u03b5 * \u03b4 := fun w hw1 =>\n  h w (closedBall_subset_cthickening hz \u03b4 (sphere_subset_closedBall hw1))", "state_before": "case h\nE : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz\u271d : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9) in \u03c6, \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 dist (f x) (F n x) < \u03b5\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\nh\u271d : \u00ac\u03c6 = \u22a5\nthis : NeBot \u03c6\ne1 : ContinuousOn f (cthickening \u03b4 K)\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nn : \u03b9\nh' : ContinuousOn (F n) (cthickening \u03b4 K)\nz : \u2102\nhz : z \u2208 K\nh : \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 \u2016f x - F n x\u2016 < \u03b5 * \u03b4\n\u22a2 \u2016Complex.cderiv \u03b4 f z - (Complex.cderiv \u03b4 \u2218 F) n z\u2016 < \u03b5", "state_after": "case h\nE : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz\u271d : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9) in \u03c6, \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 dist (f x) (F n x) < \u03b5\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\nh\u271d : \u00ac\u03c6 = \u22a5\nthis : NeBot \u03c6\ne1 : ContinuousOn f (cthickening \u03b4 K)\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nn : \u03b9\nh' : ContinuousOn (F n) (cthickening \u03b4 K)\nz : \u2102\nhz : z \u2208 K\nh : \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 \u2016f x - F n x\u2016 < \u03b5 * \u03b4\ne2 : \u2200 (w : \u2102), w \u2208 sphere z \u03b4 \u2192 \u2016f w - F n w\u2016 < \u03b5 * \u03b4\n\u22a2 \u2016Complex.cderiv \u03b4 f z - (Complex.cderiv \u03b4 \u2218 F) n z\u2016 < \u03b5"}, {"tactic": "have e3 := sphere_subset_closedBall.trans (closedBall_subset_cthickening hz \u03b4)", "state_before": "case h\nE : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz\u271d : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9) in \u03c6, \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 dist (f x) (F n x) < \u03b5\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\nh\u271d : \u00ac\u03c6 = \u22a5\nthis : NeBot \u03c6\ne1 : ContinuousOn f (cthickening \u03b4 K)\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nn : \u03b9\nh' : ContinuousOn (F n) (cthickening \u03b4 K)\nz : \u2102\nhz : z \u2208 K\nh : \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 \u2016f x - F n x\u2016 < \u03b5 * \u03b4\ne2 : \u2200 (w : \u2102), w \u2208 sphere z \u03b4 \u2192 \u2016f w - F n w\u2016 < \u03b5 * \u03b4\n\u22a2 \u2016Complex.cderiv \u03b4 f z - (Complex.cderiv \u03b4 \u2218 F) n z\u2016 < \u03b5", "state_after": "case h\nE : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz\u271d : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9) in \u03c6, \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 dist (f x) (F n x) < \u03b5\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\nh\u271d : \u00ac\u03c6 = \u22a5\nthis : NeBot \u03c6\ne1 : ContinuousOn f (cthickening \u03b4 K)\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nn : \u03b9\nh' : ContinuousOn (F n) (cthickening \u03b4 K)\nz : \u2102\nhz : z \u2208 K\nh : \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 \u2016f x - F n x\u2016 < \u03b5 * \u03b4\ne2 : \u2200 (w : \u2102), w \u2208 sphere z \u03b4 \u2192 \u2016f w - F n w\u2016 < \u03b5 * \u03b4\ne3 : sphere z \u03b4 \u2286 cthickening \u03b4 K\n\u22a2 \u2016Complex.cderiv \u03b4 f z - (Complex.cderiv \u03b4 \u2218 F) n z\u2016 < \u03b5"}, {"tactic": "have hf : ContinuousOn f (sphere z \u03b4) :=\n  e1.mono (sphere_subset_closedBall.trans (closedBall_subset_cthickening hz \u03b4))", "state_before": "case h\nE : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz\u271d : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9) in \u03c6, \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 dist (f x) (F n x) < \u03b5\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\nh\u271d : \u00ac\u03c6 = \u22a5\nthis : NeBot \u03c6\ne1 : ContinuousOn f (cthickening \u03b4 K)\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nn : \u03b9\nh' : ContinuousOn (F n) (cthickening \u03b4 K)\nz : \u2102\nhz : z \u2208 K\nh : \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 \u2016f x - F n x\u2016 < \u03b5 * \u03b4\ne2 : \u2200 (w : \u2102), w \u2208 sphere z \u03b4 \u2192 \u2016f w - F n w\u2016 < \u03b5 * \u03b4\ne3 : sphere z \u03b4 \u2286 cthickening \u03b4 K\n\u22a2 \u2016Complex.cderiv \u03b4 f z - (Complex.cderiv \u03b4 \u2218 F) n z\u2016 < \u03b5", "state_after": "case h\nE : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz\u271d : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9) in \u03c6, \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 dist (f x) (F n x) < \u03b5\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\nh\u271d : \u00ac\u03c6 = \u22a5\nthis : NeBot \u03c6\ne1 : ContinuousOn f (cthickening \u03b4 K)\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nn : \u03b9\nh' : ContinuousOn (F n) (cthickening \u03b4 K)\nz : \u2102\nhz : z \u2208 K\nh : \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 \u2016f x - F n x\u2016 < \u03b5 * \u03b4\ne2 : \u2200 (w : \u2102), w \u2208 sphere z \u03b4 \u2192 \u2016f w - F n w\u2016 < \u03b5 * \u03b4\ne3 : sphere z \u03b4 \u2286 cthickening \u03b4 K\nhf : ContinuousOn f (sphere z \u03b4)\n\u22a2 \u2016Complex.cderiv \u03b4 f z - (Complex.cderiv \u03b4 \u2218 F) n z\u2016 < \u03b5"}, {"tactic": "simpa only [mul_div_cancel _ h\u03b4.ne.symm] using norm_cderiv_sub_lt h\u03b4 e2 hf (h'.mono e3)", "state_before": "case h\nE : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz\u271d : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u03b9) in \u03c6, \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 dist (f x) (F n x) < \u03b5\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\nh\u271d : \u00ac\u03c6 = \u22a5\nthis : NeBot \u03c6\ne1 : ContinuousOn f (cthickening \u03b4 K)\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nn : \u03b9\nh' : ContinuousOn (F n) (cthickening \u03b4 K)\nz : \u2102\nhz : z \u2208 K\nh : \u2200 (x : \u2102), x \u2208 cthickening \u03b4 K \u2192 \u2016f x - F n x\u2016 < \u03b5 * \u03b4\ne2 : \u2200 (w : \u2102), w \u2208 sphere z \u03b4 \u2192 \u2016f w - F n w\u2016 < \u03b5 * \u03b4\ne3 : sphere z \u03b4 \u2286 cthickening \u03b4 K\nhf : ContinuousOn f (sphere z \u03b4)\n\u22a2 \u2016Complex.cderiv \u03b4 f z - (Complex.cderiv \u03b4 \u2218 F) n z\u2016 < \u03b5", "state_after": "no goals"}, {"tactic": "simp only [h, TendstoUniformlyOn, eventually_bot, imp_true_iff]", "state_before": "case pos\nE : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nU K : Set \u2102\nz : \u2102\nM r \u03b4 : \u211d\n\u03c6 : Filter \u03b9\nF : \u03b9 \u2192 \u2102 \u2192 E\nf g : \u2102 \u2192 E\nhF : TendstoUniformlyOn F f \u03c6 (cthickening \u03b4 K)\nh\u03b4 : 0 < \u03b4\nhFn : \u2200\u1da0 (n : \u03b9) in \u03c6, ContinuousOn (F n) (cthickening \u03b4 K)\nh : \u03c6 = \u22a5\n\u22a2 TendstoUniformlyOn (Complex.cderiv \u03b4 \u2218 F) (Complex.cderiv \u03b4 f) \u03c6 K", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.eq_zero_iff", "start": [807, 1], "end": [809, 25], "traced_tactics": [{"tactic": "rw [ext_iff]", "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\u271d q p : MvPolynomial \u03c3 R\n\u22a2 p = 0 \u2194 \u2200 (d : \u03c3 \u2192\u2080 \u2115), coeff d p = 0", "state_after": "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\u271d q p : MvPolynomial \u03c3 R\n\u22a2 (\u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m p = coeff m 0) \u2194 \u2200 (d : \u03c3 \u2192\u2080 \u2115), coeff d p = 0"}, {"tactic": "simp only [coeff_zero]", "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\u271d q p : MvPolynomial \u03c3 R\n\u22a2 (\u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m p = coeff m 0) \u2194 \u2200 (d : \u03c3 \u2192\u2080 \u2115), coeff d p = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean", "full_name": "CategoryTheory.InjectiveResolution.desc_commutes", "start": [103, 1], "end": [106, 35], "traced_tactics": [{"tactic": "ext", "state_before": "C : Type u\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nY Z : C\nf : Z \u27f6 Y\nI : InjectiveResolution Y\nJ : InjectiveResolution Z\n\u22a2 J.\u03b9 \u226b desc f I J = (CochainComplex.single\u2080 C).map f \u226b I.\u03b9", "state_after": "case h\nC : Type u\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nY Z : C\nf : Z \u27f6 Y\nI : InjectiveResolution Y\nJ : InjectiveResolution Z\n\u22a2 HomologicalComplex.Hom.f (J.\u03b9 \u226b desc f I J) 0 = HomologicalComplex.Hom.f ((CochainComplex.single\u2080 C).map f \u226b I.\u03b9) 0"}, {"tactic": "simp [desc, descFOne, descFZero]", "state_before": "case h\nC : Type u\ninst\u271d\u00b9 : Category C\ninst\u271d : Abelian C\nY Z : C\nf : Z \u27f6 Y\nI : InjectiveResolution Y\nJ : InjectiveResolution Z\n\u22a2 HomologicalComplex.Hom.f (J.\u03b9 \u226b desc f I J) 0 = HomologicalComplex.Hom.f ((CochainComplex.single\u2080 C).map f \u226b I.\u03b9) 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.mul_subset_iff_right", "start": [1855, 1], "end": [1856, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Perm.lean", "full_name": "List.Subperm.refl", "start": [427, 1], "end": [428, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/ModelTheory/Graph.lean", "full_name": "SimpleGraph.simpleGraphOfStructure", "start": [111, 1], "end": [114, 6], "traced_tactics": [{"tactic": "ext", "state_before": "L : Language\n\u03b1 : Type w\nV : Type w'\nn : \u2115\nG : SimpleGraph V\n\u22a2 FirstOrder.Language.simpleGraphOfStructure V = G", "state_after": "case Adj.h.h.a\nL : Language\n\u03b1 : Type w\nV : Type w'\nn : \u2115\nG : SimpleGraph V\nx\u271d\u00b9 x\u271d : V\n\u22a2 SimpleGraph.Adj (FirstOrder.Language.simpleGraphOfStructure V) x\u271d\u00b9 x\u271d \u2194 SimpleGraph.Adj G x\u271d\u00b9 x\u271d"}, {"tactic": "rfl", "state_before": "case Adj.h.h.a\nL : Language\n\u03b1 : Type w\nV : Type w'\nn : \u2115\nG : SimpleGraph V\nx\u271d\u00b9 x\u271d : V\n\u22a2 SimpleGraph.Adj (FirstOrder.Language.simpleGraphOfStructure V) x\u271d\u00b9 x\u271d \u2194 SimpleGraph.Adj G x\u271d\u00b9 x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/LocallyConvex/Basic.lean", "full_name": "balanced_iff_smul_mem", "start": [167, 1], "end": [168, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/SymmDiff.lean", "full_name": "Disjoint.le_symmDiff_sup_symmDiff_left", "start": [794, 1], "end": [798, 47], "traced_tactics": [{"tactic": "trans c \\ (a \u2293 b)", "state_before": "\u03b9 : Type ?u.93896\n\u03b1 : Type u_1\n\u03b2 : Type ?u.93902\n\u03c0 : \u03b9 \u2192 Type ?u.93907\ninst\u271d : BooleanAlgebra \u03b1\na b c d : \u03b1\nh : Disjoint a b\n\u22a2 c \u2264 a \u2206 c \u2294 b \u2206 c", "state_after": "\u03b9 : Type ?u.93896\n\u03b1 : Type u_1\n\u03b2 : Type ?u.93902\n\u03c0 : \u03b9 \u2192 Type ?u.93907\ninst\u271d : BooleanAlgebra \u03b1\na b c d : \u03b1\nh : Disjoint a b\n\u22a2 c \u2264 c \\ (a \u2293 b)\n\n\u03b9 : Type ?u.93896\n\u03b1 : Type u_1\n\u03b2 : Type ?u.93902\n\u03c0 : \u03b9 \u2192 Type ?u.93907\ninst\u271d : BooleanAlgebra \u03b1\na b c d : \u03b1\nh : Disjoint a b\n\u22a2 c \\ (a \u2293 b) \u2264 a \u2206 c \u2294 b \u2206 c"}, {"tactic": "rw [h.eq_bot, sdiff_bot]", "state_before": "\u03b9 : Type ?u.93896\n\u03b1 : Type u_1\n\u03b2 : Type ?u.93902\n\u03c0 : \u03b9 \u2192 Type ?u.93907\ninst\u271d : BooleanAlgebra \u03b1\na b c d : \u03b1\nh : Disjoint a b\n\u22a2 c \u2264 c \\ (a \u2293 b)", "state_after": "no goals"}, {"tactic": "rw [sdiff_inf]", "state_before": "\u03b9 : Type ?u.93896\n\u03b1 : Type u_1\n\u03b2 : Type ?u.93902\n\u03c0 : \u03b9 \u2192 Type ?u.93907\ninst\u271d : BooleanAlgebra \u03b1\na b c d : \u03b1\nh : Disjoint a b\n\u22a2 c \\ (a \u2293 b) \u2264 a \u2206 c \u2294 b \u2206 c", "state_after": "\u03b9 : Type ?u.93896\n\u03b1 : Type u_1\n\u03b2 : Type ?u.93902\n\u03c0 : \u03b9 \u2192 Type ?u.93907\ninst\u271d : BooleanAlgebra \u03b1\na b c d : \u03b1\nh : Disjoint a b\n\u22a2 c \\ a \u2294 c \\ b \u2264 a \u2206 c \u2294 b \u2206 c"}, {"tactic": "exact sup_le_sup le_sup_right le_sup_right", "state_before": "\u03b9 : Type ?u.93896\n\u03b1 : Type u_1\n\u03b2 : Type ?u.93902\n\u03c0 : \u03b9 \u2192 Type ?u.93907\ninst\u271d : BooleanAlgebra \u03b1\na b c d : \u03b1\nh : Disjoint a b\n\u22a2 c \\ a \u2294 c \\ b \u2264 a \u2206 c \u2294 b \u2206 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Instances/EReal.lean", "full_name": "EReal.continuousAt_add_coe_coe", "start": [178, 1], "end": [181, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subgroup/Pointwise.lean", "full_name": "AddSubgroup.pointwise_smul_le_iff", "start": [512, 1], "end": [513, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Closeds.lean", "full_name": "Metric.NonemptyCompacts.dist_eq", "start": [418, 1], "end": [420, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.Finite.toFinset_image", "start": [291, 11], "end": [294, 7], "traced_tactics": [{"tactic": "ext", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns t : Set \u03b1\na : \u03b1\nhs\u271d : Set.Finite s\nht : Set.Finite t\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\nhs : Set.Finite s\nh : Set.Finite (f '' s)\n\u22a2 Finite.toFinset h = Finset.image f (Finite.toFinset hs)", "state_after": "case a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns t : Set \u03b1\na : \u03b1\nhs\u271d : Set.Finite s\nht : Set.Finite t\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\nhs : Set.Finite s\nh : Set.Finite (f '' s)\na\u271d : \u03b2\n\u22a2 a\u271d \u2208 Finite.toFinset h \u2194 a\u271d \u2208 Finset.image f (Finite.toFinset hs)"}, {"tactic": "simp", "state_before": "case a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3 : Type x\ns t : Set \u03b1\na : \u03b1\nhs\u271d : Set.Finite s\nht : Set.Finite t\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\nhs : Set.Finite s\nh : Set.Finite (f '' s)\na\u271d : \u03b2\n\u22a2 a\u271d \u2208 Finite.toFinset h \u2194 a\u271d \u2208 Finset.image f (Finite.toFinset hs)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean", "full_name": "HasFDerivAt.const_rpow", "start": [491, 1], "end": [493, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "full_name": "EMetric.infEdist_zero_of_mem", 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"state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Complex/ReImTopology.lean", "full_name": "Complex.frontier_setOf_le_re", "start": [147, 1], "end": [148, 63], "traced_tactics": [{"tactic": "simpa only [frontier_Ici] using frontier_preimage_re (Ici a)", "state_before": "a : \u211d\n\u22a2 frontier {z | a \u2264 z.re} = {z | z.re = a}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Interval.lean", "full_name": "Nat.Icc_eq_range'", "start": [67, 1], "end": [68, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/PairingHeap.lean", "full_name": "Std.PairingHeapImp.Heap.size_merge_node", "start": [120, 1], "end": [122, 51], "traced_tactics": [{"tactic": "unfold merge", "state_before": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\na\u2081 : \u03b1\nc\u2081 s\u2081 : Heap \u03b1\na\u2082 : \u03b1\nc\u2082 s\u2082 : Heap \u03b1\n\u22a2 size (merge le (node a\u2081 c\u2081 s\u2081) (node a\u2082 c\u2082 s\u2082)) = size c\u2081 + size c\u2082 + 2", "state_after": "\u03b1 : Type u_1\nle : \u03b1 \u2192 \u03b1 \u2192 Bool\na\u2081 : \u03b1\nc\u2081 s\u2081 : Heap \u03b1\na\u2082 : \u03b1\nc\u2082 s\u2082 : Heap \u03b1\n\u22a2 size\n      (match node a\u2081 c\u2081 s\u2081, node a\u2082 c\u2082 s\u2082 with\n      | nil, nil => nil\n      | nil, node a\u2082 c\u2082 sibling => node a\u2082 c\u2082 nil\n      | node a\u2081 c\u2081 sibling, nil => node a\u2081 c\u2081 nil\n      | node a\u2081 c\u2081 sibling, node a\u2082 c\u2082 sibling_1 =>\n        if le a\u2081 a\u2082 = true then node a\u2081 (node a\u2082 c\u2082 c\u2081) nil else node a\u2082 (node a\u2081 c\u2081 c\u2082) nil) =\n    size c\u2081 + size 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Module R' M'\ninst\u271d\u00b2 : Algebra R R'\ninst\u271d\u00b9 : Module R M'\ninst\u271d : IsScalarTower R R' M'\np : R[X]\nq : PolynomialModule R M\nr : R\n\u22a2 \u2191(LinearMap.comp (eval r) (comp p)) 0 = \u2191(eval (Polynomial.eval r p)) 0\n\ncase hadd\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\nI : Ideal R\nS : Type ?u.687011\ninst\u271d\u2079 : CommSemiring S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : Module S M\ninst\u271d\u2076 : IsScalarTower S R M\nR' : Type ?u.689663\nM' : Type ?u.689666\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : Module R' M'\ninst\u271d\u00b2 : Algebra R R'\ninst\u271d\u00b9 : Module R M'\ninst\u271d : IsScalarTower R R' M'\np : R[X]\nq : PolynomialModule R M\nr : R\n\u22a2 \u2200 (f g : PolynomialModule R M),\n    \u2191(LinearMap.comp (eval r) (comp p)) f = \u2191(eval (Polynomial.eval r p)) f \u2192\n      \u2191(LinearMap.comp (eval r) (comp p)) g = \u2191(eval (Polynomial.eval r p)) g \u2192\n        \u2191(LinearMap.comp (eval r) (comp p)) (f + g) = \u2191(eval (Polynomial.eval r p)) (f + g)\n\ncase hsingle\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\nI : Ideal R\nS : Type ?u.687011\ninst\u271d\u2079 : CommSemiring S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : Module S M\ninst\u271d\u2076 : IsScalarTower S R M\nR' : Type ?u.689663\nM' : Type ?u.689666\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : Module R' M'\ninst\u271d\u00b2 : Algebra R R'\ninst\u271d\u00b9 : Module R M'\ninst\u271d : IsScalarTower R R' M'\np : R[X]\nq : PolynomialModule R M\nr : R\n\u22a2 \u2200 (a : \u2115) (b : M),\n    \u2191(LinearMap.comp (eval r) (comp p)) (\u2191(single R a) b) = \u2191(eval (Polynomial.eval r p)) (\u2191(single R a) b)"}, {"tactic": "simp_rw [map_zero]", "state_before": "case h0\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\nI : Ideal R\nS : Type ?u.687011\ninst\u271d\u2079 : CommSemiring S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : Module S M\ninst\u271d\u2076 : IsScalarTower S R M\nR' : Type ?u.689663\nM' : Type ?u.689666\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : Module R' M'\ninst\u271d\u00b2 : Algebra R R'\ninst\u271d\u00b9 : Module R M'\ninst\u271d : IsScalarTower R R' M'\np : R[X]\nq : PolynomialModule R M\nr : R\n\u22a2 \u2191(LinearMap.comp (eval r) (comp p)) 0 = \u2191(eval (Polynomial.eval r p)) 0", "state_after": "no goals"}, {"tactic": "intro _ _ e\u2081 e\u2082", "state_before": "case hadd\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\nI : Ideal R\nS : Type ?u.687011\ninst\u271d\u2079 : CommSemiring S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : Module S M\ninst\u271d\u2076 : IsScalarTower S R M\nR' : Type ?u.689663\nM' : Type ?u.689666\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : Module R' M'\ninst\u271d\u00b2 : Algebra R R'\ninst\u271d\u00b9 : Module R M'\ninst\u271d : IsScalarTower R R' M'\np : R[X]\nq : PolynomialModule R M\nr : R\n\u22a2 \u2200 (f g : PolynomialModule R M),\n    \u2191(LinearMap.comp (eval r) (comp p)) f = \u2191(eval (Polynomial.eval r p)) f \u2192\n      \u2191(LinearMap.comp (eval r) (comp p)) g = \u2191(eval (Polynomial.eval r p)) g \u2192\n        \u2191(LinearMap.comp (eval r) (comp p)) (f + g) = \u2191(eval (Polynomial.eval r p)) (f + g)", "state_after": "case hadd\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\nI : Ideal R\nS : Type ?u.687011\ninst\u271d\u2079 : CommSemiring S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : Module S M\ninst\u271d\u2076 : IsScalarTower S R M\nR' : Type ?u.689663\nM' : Type ?u.689666\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : Module R' M'\ninst\u271d\u00b2 : Algebra R R'\ninst\u271d\u00b9 : Module R M'\ninst\u271d : IsScalarTower R R' M'\np : R[X]\nq : PolynomialModule R M\nr : R\nf\u271d g\u271d : PolynomialModule R M\ne\u2081 : \u2191(LinearMap.comp (eval r) (comp p)) f\u271d = \u2191(eval (Polynomial.eval r p)) f\u271d\ne\u2082 : \u2191(LinearMap.comp (eval r) (comp p)) g\u271d = \u2191(eval (Polynomial.eval r p)) g\u271d\n\u22a2 \u2191(LinearMap.comp (eval r) (comp p)) (f\u271d + g\u271d) = \u2191(eval (Polynomial.eval r p)) (f\u271d + g\u271d)"}, {"tactic": "simp_rw [map_add, e\u2081, e\u2082]", "state_before": "case hadd\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\nI : Ideal R\nS : Type ?u.687011\ninst\u271d\u2079 : CommSemiring S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : Module S M\ninst\u271d\u2076 : IsScalarTower S R M\nR' : Type ?u.689663\nM' : Type ?u.689666\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : Module R' M'\ninst\u271d\u00b2 : Algebra R R'\ninst\u271d\u00b9 : Module R M'\ninst\u271d : IsScalarTower R R' M'\np : R[X]\nq : PolynomialModule R M\nr : R\nf\u271d g\u271d : PolynomialModule R M\ne\u2081 : \u2191(LinearMap.comp (eval r) (comp p)) f\u271d = \u2191(eval (Polynomial.eval r p)) f\u271d\ne\u2082 : \u2191(LinearMap.comp (eval r) (comp p)) g\u271d = \u2191(eval (Polynomial.eval r p)) g\u271d\n\u22a2 \u2191(LinearMap.comp (eval r) (comp p)) (f\u271d + g\u271d) = \u2191(eval (Polynomial.eval r p)) (f\u271d + g\u271d)", "state_after": "no goals"}, {"tactic": "intro i m", "state_before": "case hsingle\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\nI : Ideal R\nS : Type ?u.687011\ninst\u271d\u2079 : CommSemiring S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : Module S M\ninst\u271d\u2076 : IsScalarTower S R M\nR' : Type ?u.689663\nM' : Type ?u.689666\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : Module R' M'\ninst\u271d\u00b2 : Algebra R R'\ninst\u271d\u00b9 : Module R M'\ninst\u271d : IsScalarTower R R' M'\np : R[X]\nq : PolynomialModule R M\nr : R\n\u22a2 \u2200 (a : \u2115) (b : M),\n    \u2191(LinearMap.comp (eval r) (comp p)) (\u2191(single R a) b) = \u2191(eval (Polynomial.eval r p)) (\u2191(single R a) b)", "state_after": "case hsingle\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\nI : Ideal R\nS : Type ?u.687011\ninst\u271d\u2079 : CommSemiring S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : Module S M\ninst\u271d\u2076 : IsScalarTower S R M\nR' : Type ?u.689663\nM' : Type ?u.689666\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : Module R' M'\ninst\u271d\u00b2 : Algebra R R'\ninst\u271d\u00b9 : Module R M'\ninst\u271d : IsScalarTower R R' M'\np : R[X]\nq : PolynomialModule R M\nr : R\ni : \u2115\nm : M\n\u22a2 \u2191(LinearMap.comp (eval r) (comp p)) (\u2191(single R i) m) = \u2191(eval (Polynomial.eval r p)) (\u2191(single R i) m)"}, {"tactic": "rw [LinearMap.comp_apply, comp_single, eval_single, eval_smul, eval_single, pow_zero, one_smul,\n  Polynomial.eval_pow]", "state_before": "case hsingle\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\nI : Ideal R\nS : Type ?u.687011\ninst\u271d\u2079 : CommSemiring S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : Module S M\ninst\u271d\u2076 : IsScalarTower S R M\nR' : Type ?u.689663\nM' : Type ?u.689666\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : Module R' M'\ninst\u271d\u00b2 : Algebra R R'\ninst\u271d\u00b9 : Module R M'\ninst\u271d : IsScalarTower R R' M'\np : R[X]\nq : PolynomialModule R M\nr : R\ni : \u2115\nm : M\n\u22a2 \u2191(LinearMap.comp (eval r) (comp p)) (\u2191(single R i) m) = \u2191(eval (Polynomial.eval r p)) (\u2191(single R i) m)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Monic.lean", "full_name": "Polynomial.Monic.eq_one_of_map_eq_one", "start": [220, 1], "end": [230, 70], "traced_tactics": [{"tactic": "nontriviality R", "state_before": "R : Type u\nS\u271d : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Nontrivial S\nf : R \u2192+* S\nhp : Monic p\nmap_eq : Polynomial.map f p = 1\n\u22a2 p = 1", "state_after": "R : Type u\nS\u271d : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Nontrivial S\nf : R \u2192+* S\nhp : Monic p\nmap_eq : Polynomial.map f p = 1\n\u271d : Nontrivial R\n\u22a2 p = 1"}, {"tactic": "have hndeg : p.natDegree = 0 :=\n  WithBot.coe_eq_coe.mp ((degree_eq_natDegree hp.ne_zero).symm.trans hdeg)", "state_before": "R : Type u\nS\u271d : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Nontrivial S\nf : R \u2192+* S\nhp : Monic p\nmap_eq : Polynomial.map f p = 1\n\u271d : Nontrivial R\nhdeg : degree p = 0\n\u22a2 p = 1", "state_after": "R : Type u\nS\u271d : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Nontrivial S\nf : R \u2192+* S\nhp : Monic p\nmap_eq : Polynomial.map f p = 1\n\u271d : Nontrivial R\nhdeg : degree p = 0\nhndeg : natDegree p = 0\n\u22a2 p = 1"}, {"tactic": "convert eq_C_of_degree_eq_zero hdeg", "state_before": "R : Type u\nS\u271d : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Nontrivial S\nf : R \u2192+* S\nhp : Monic p\nmap_eq : Polynomial.map f p = 1\n\u271d : Nontrivial R\nhdeg : degree p = 0\nhndeg : natDegree p = 0\n\u22a2 p = 1", "state_after": "case h.e'_3\nR : Type u\nS\u271d : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Nontrivial S\nf : R \u2192+* S\nhp : Monic p\nmap_eq : Polynomial.map f p = 1\n\u271d : Nontrivial R\nhdeg : degree p = 0\nhndeg : natDegree p = 0\n\u22a2 1 = \u2191C (coeff p 0)"}, {"tactic": "rw [\u2190 hndeg, \u2190 Polynomial.leadingCoeff, hp.leadingCoeff, C.map_one]", "state_before": "case h.e'_3\nR : Type u\nS\u271d : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Nontrivial S\nf : R \u2192+* S\nhp : Monic p\nmap_eq : Polynomial.map f p = 1\n\u271d : Nontrivial R\nhdeg : degree p = 0\nhndeg : natDegree p = 0\n\u22a2 1 = \u2191C (coeff p 0)", "state_after": "no goals"}, {"tactic": "rw [\u2190 degree_map_eq_of_leadingCoeff_ne_zero f _, map_eq, degree_one]", "state_before": "R : Type u\nS\u271d : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Nontrivial S\nf : R \u2192+* S\nhp : Monic p\nmap_eq : Polynomial.map f p = 1\n\u271d : Nontrivial R\n\u22a2 degree p = 0", "state_after": "R : Type u\nS\u271d : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Nontrivial S\nf : R \u2192+* S\nhp : Monic p\nmap_eq : Polynomial.map f p = 1\n\u271d : Nontrivial R\n\u22a2 \u2191f (Polynomial.leadingCoeff p) \u2260 0"}, {"tactic": "rw [hp.leadingCoeff, f.map_one]", "state_before": "R : Type u\nS\u271d : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Nontrivial S\nf : R \u2192+* S\nhp : Monic p\nmap_eq : Polynomial.map f p = 1\n\u271d : Nontrivial R\n\u22a2 \u2191f (Polynomial.leadingCoeff p) \u2260 0", "state_after": "R : Type u\nS\u271d : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Nontrivial S\nf : R \u2192+* S\nhp : Monic p\nmap_eq : Polynomial.map f p = 1\n\u271d : Nontrivial R\n\u22a2 1 \u2260 0"}, {"tactic": "exact one_ne_zero", "state_before": "R : Type u\nS\u271d : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d\u00b2 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Nontrivial S\nf : R \u2192+* S\nhp : Monic p\nmap_eq : Polynomial.map f p = 1\n\u271d : Nontrivial R\n\u22a2 1 \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Preadditive/Basic.lean", "full_name": "CategoryTheory.Preadditive.sub_comp", "start": [143, 1], "end": [144, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Additive/Etransform.lean", "full_name": "Finset.mulEtransformRight_inv", "start": [198, 1], "end": [199, 73], "traced_tactics": [{"tactic": "simp [-op_inv, op_smul_eq_smul, mulEtransformLeft, mulEtransformRight]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CommGroup \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 mulEtransformRight e\u207b\u00b9 x = Prod.swap (mulEtransformLeft e (Prod.swap x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Stream/Init.lean", "full_name": "Stream'.take_succ_cons", "start": [581, 9], "end": [581, 100], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.div_singleton", "start": [618, 1], "end": [619, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Lie/Subalgebra.lean", "full_name": "LieSubalgebra.span_iUnion", "start": [749, 1], "end": [750, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/ODE/PicardLindelof.lean", "full_name": "IsPicardLindelof.exists_forall_hasDerivWithinAt_Icc_eq", "start": [386, 1], "end": [394, 80], "traced_tactics": [{"tactic": "lift C to \u211d\u22650 using (norm_nonneg _).trans hpl.norm_le\u2080", "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nv : \u211d \u2192 E \u2192 E\ntMin t\u2080 tMax : \u211d\nx\u2080 : E\nC R : \u211d\nL : \u211d\u22650\nhpl : IsPicardLindelof v tMin t\u2080 tMax x\u2080 L R C\n\u22a2 \u2203 f, f t\u2080 = x\u2080 \u2227 \u2200 (t : \u211d), t \u2208 Icc tMin tMax \u2192 HasDerivWithinAt f (v t (f t)) (Icc tMin tMax) t", "state_after": "case intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nv : \u211d \u2192 E \u2192 E\ntMin t\u2080 tMax : \u211d\nx\u2080 : E\nR : \u211d\nL C : \u211d\u22650\nhpl : IsPicardLindelof v tMin t\u2080 tMax x\u2080 L R \u2191C\n\u22a2 \u2203 f, f t\u2080 = x\u2080 \u2227 \u2200 (t : \u211d), t \u2208 Icc tMin tMax \u2192 HasDerivWithinAt f (v t (f t)) (Icc tMin tMax) t"}, {"tactic": "lift t\u2080 to Icc tMin tMax using hpl.ht\u2080", "state_before": "case intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nv : \u211d \u2192 E \u2192 E\ntMin t\u2080 tMax : \u211d\nx\u2080 : E\nR : \u211d\nL C : \u211d\u22650\nhpl : IsPicardLindelof v tMin t\u2080 tMax x\u2080 L R \u2191C\n\u22a2 \u2203 f, f t\u2080 = x\u2080 \u2227 \u2200 (t : \u211d), t \u2208 Icc tMin tMax \u2192 HasDerivWithinAt f (v t (f t)) (Icc tMin tMax) t", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nv : \u211d \u2192 E \u2192 E\ntMin tMax : \u211d\nx\u2080 : E\nR : \u211d\nL C : \u211d\u22650\nt\u2080 : { x // x \u2208 Icc tMin tMax }\nhpl : IsPicardLindelof v tMin (\u2191t\u2080) tMax x\u2080 L R \u2191C\n\u22a2 \u2203 f, f \u2191t\u2080 = x\u2080 \u2227 \u2200 (t : \u211d), t \u2208 Icc tMin tMax \u2192 HasDerivWithinAt f (v t (f t)) (Icc tMin tMax) t"}, {"tactic": "exact PicardLindelof.exists_solution\n  \u27e8v, tMin, tMax, t\u2080, x\u2080, C, \u27e8R, hpl.hR\u27e9, L, { hpl with ht\u2080 := t\u2080.property }\u27e9", "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nv : \u211d \u2192 E \u2192 E\ntMin tMax : \u211d\nx\u2080 : E\nR : \u211d\nL C : \u211d\u22650\nt\u2080 : { x // x \u2208 Icc tMin tMax }\nhpl : IsPicardLindelof v tMin (\u2191t\u2080) tMax x\u2080 L R \u2191C\n\u22a2 \u2203 f, f \u2191t\u2080 = x\u2080 \u2227 \u2200 (t : \u211d), t \u2208 Icc tMin tMax \u2192 HasDerivWithinAt f (v t (f t)) (Icc tMin tMax) t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Basic.lean", "full_name": "forall_or_of_or_forall", "start": [837, 1], "end": [837, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "full_name": "mul_le_mul_of_nonneg_right", "start": [159, 1], "end": [160, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.lift_comp_coe", "start": [1216, 1], "end": [1217, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Deprecated/Group.lean", "full_name": "Additive.isAddMonoidHom", "start": [455, 1], "end": [457, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Operations.lean", "full_name": "Submodule.mem_span_singleton_mul", "start": [373, 1], "end": [376, 6], "traced_tactics": [{"tactic": "rfl", "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 x y : A\n\u22a2 x \u2208 map (\u2191(LinearMap.mul R A) y) P \u2194 \u2203 z, z \u2208 P \u2227 Mul.mul y z = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/ContDiffDef.lean", "full_name": "hasFTaylorSeriesUpToOn_univ_iff", "start": [1226, 1], "end": [1246, 24], "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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 HasFTaylorSeriesUpToOn n f p univ \u2194 HasFTaylorSeriesUpTo n f p", "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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 HasFTaylorSeriesUpToOn n f p univ \u2192 HasFTaylorSeriesUpTo n f p\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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 HasFTaylorSeriesUpTo n f p \u2192 HasFTaylorSeriesUpToOn n f p univ"}, {"tactic": "intro H", "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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 HasFTaylorSeriesUpToOn n f p univ \u2192 HasFTaylorSeriesUpTo n f p", "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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\n\u22a2 HasFTaylorSeriesUpTo n f p"}, {"tactic": "constructor", "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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\n\u22a2 HasFTaylorSeriesUpTo n f p", "state_after": "case mp.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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\n\u22a2 \u2200 (x : E), ContinuousMultilinearMap.uncurry0 (p x 0) = f x\n\ncase mp.fderiv\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 : HasFTaylorSeriesUpToOn n f p univ\n\u22a2 \u2200 (m : \u2115), \u2191m < n \u2192 \u2200 (x : E), HasFDerivAt (fun y => p y m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) x\n\ncase mp.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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 n \u2192 Continuous fun x => p x m"}, {"tactic": "exact fun x => H.zero_eq x (mem_univ x)", "state_before": "case mp.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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\n\u22a2 \u2200 (x : E), ContinuousMultilinearMap.uncurry0 (p x 0) = f x", "state_after": "no goals"}, {"tactic": "intro m hm x", "state_before": "case mp.fderiv\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 : HasFTaylorSeriesUpToOn n f p univ\n\u22a2 \u2200 (m : \u2115), \u2191m < n \u2192 \u2200 (x : E), HasFDerivAt (fun y => p y m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) x", "state_after": "case mp.fderiv\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\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\nm : \u2115\nhm : \u2191m < n\nx : E\n\u22a2 HasFDerivAt (fun y => p y m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) x"}, {"tactic": "rw [\u2190 hasFDerivWithinAt_univ]", "state_before": "case mp.fderiv\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\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\nm : \u2115\nhm : \u2191m < n\nx : E\n\u22a2 HasFDerivAt (fun y => p y m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) x", "state_after": "case mp.fderiv\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\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\nm : \u2115\nhm : \u2191m < n\nx : E\n\u22a2 HasFDerivWithinAt (fun y => p y m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) univ x"}, {"tactic": "exact H.fderivWithin m hm x (mem_univ x)", "state_before": "case mp.fderiv\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\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\nm : \u2115\nhm : \u2191m < n\nx : E\n\u22a2 HasFDerivWithinAt (fun y => p y m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) univ x", "state_after": "no goals"}, {"tactic": "intro m hm", "state_before": "case mp.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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 n \u2192 Continuous fun x => p x m", "state_after": "case mp.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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 Continuous fun x => p x m"}, {"tactic": "rw [continuous_iff_continuousOn_univ]", "state_before": "case mp.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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 Continuous fun x => p x m", "state_after": "case mp.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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 ContinuousOn (fun x => p x m) univ"}, {"tactic": "exact H.cont m hm", "state_before": "case mp.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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 ContinuousOn (fun x => p x m) univ", "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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 HasFTaylorSeriesUpTo n f p \u2192 HasFTaylorSeriesUpToOn n f p univ", "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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\n\u22a2 HasFTaylorSeriesUpToOn n f p univ"}, {"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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\n\u22a2 HasFTaylorSeriesUpToOn n f p univ", "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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\n\u22a2 \u2200 (x : E), x \u2208 univ \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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\n\u22a2 \u2200 (m : \u2115),\n    \u2191m < n \u2192\n      \u2200 (x : E),\n        x \u2208 univ \u2192 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) univ 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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => p x m) univ"}, {"tactic": "exact fun x _ => H.zero_eq x", "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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\n\u22a2 \u2200 (x : E), x \u2208 univ \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x", "state_after": "no goals"}, {"tactic": "intro m hm x _", "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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\n\u22a2 \u2200 (m : \u2115),\n    \u2191m < n \u2192\n      \u2200 (x : E),\n        x \u2208 univ \u2192 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) univ 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\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\nm : \u2115\nhm : \u2191m < n\nx : E\na\u271d : x \u2208 univ\n\u22a2 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) univ x"}, {"tactic": "rw [hasFDerivWithinAt_univ]", "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\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\nm : \u2115\nhm : \u2191m < n\nx : E\na\u271d : x \u2208 univ\n\u22a2 HasFDerivWithinAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) univ 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\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\nm : \u2115\nhm : \u2191m < n\nx : E\na\u271d : x \u2208 univ\n\u22a2 HasFDerivAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) x"}, {"tactic": "exact H.fderiv m hm x", "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\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\nm : \u2115\nhm : \u2191m < n\nx : E\na\u271d : x \u2208 univ\n\u22a2 HasFDerivAt (fun x => p x m) (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) 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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => p x m) univ", "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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 ContinuousOn (fun x => p x m) univ"}, {"tactic": "rw [\u2190 continuous_iff_continuousOn_univ]", "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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 ContinuousOn (fun x => p x m) univ", "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 : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 Continuous fun x => p x m"}, {"tactic": "exact H.cont 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\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 Continuous fun x => p x m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean", "full_name": "midpoint_vsub", "start": [127, 1], "end": [131, 44], "traced_tactics": [{"tactic": "rw [\u2190 vsub_sub_vsub_cancel_right p\u2081 p p\u2082, smul_sub, sub_eq_add_neg, \u2190 smul_neg,\n  neg_vsub_eq_vsub_rev, add_assoc, invOf_two_smul_add_invOf_two_smul, \u2190 vadd_vsub_assoc,\n  midpoint_comm, midpoint, lineMap_apply]", "state_before": "R : Type u_3\nV : Type u_1\nV' : Type ?u.57244\nP : Type u_2\nP' : Type ?u.57250\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 y z p\u2081 p\u2082 p : P\n\u22a2 midpoint R p\u2081 p\u2082 -\u1d65 p = \u215f2 \u2022 (p\u2081 -\u1d65 p) + \u215f2 \u2022 (p\u2082 -\u1d65 p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/ConjugateExponents.lean", "full_name": "Real.IsConjugateExponent.div_conj_eq_sub_one", "start": [99, 1], "end": [101, 26], "traced_tactics": [{"tactic": "field_simp [h.symm.ne_zero]", "state_before": "p q : \u211d\nh : IsConjugateExponent p q\n\u22a2 p / q = p - 1", "state_after": "p q : \u211d\nh : IsConjugateExponent p q\n\u22a2 p = (p - 1) * q"}, {"tactic": "rw [h.sub_one_mul_conj]", "state_before": "p q : \u211d\nh : IsConjugateExponent p q\n\u22a2 p = (p - 1) * q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/MulAction.lean", "full_name": "AddTorsor.connectedSpace", "start": [222, 11], "end": [228, 34], "traced_tactics": [{"tactic": "convert\n  isPreconnected_univ.image (Equiv.vaddConst (Classical.arbitrary P) : G \u2192 P)\n    (continuous_id.vadd continuous_const).continuousOn", "state_before": "G : Type ?u.30653\nP : Type u_1\ninst\u271d\u2075 : AddGroup G\ninst\u271d\u2074 : AddTorsor G P\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : PreconnectedSpace G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : ContinuousVAdd G P\n\u22a2 IsPreconnected Set.univ", "state_after": "case h.e'_3\nG : Type ?u.30653\nP : Type u_1\ninst\u271d\u2075 : AddGroup G\ninst\u271d\u2074 : AddTorsor G P\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : PreconnectedSpace G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : ContinuousVAdd G P\n\u22a2 Set.univ = \u2191(Equiv.vaddConst (Classical.arbitrary P)) '' Set.univ"}, {"tactic": "rw [Set.image_univ, Equiv.range_eq_univ]", "state_before": "case h.e'_3\nG : Type ?u.30653\nP : Type u_1\ninst\u271d\u2075 : AddGroup G\ninst\u271d\u2074 : AddTorsor G P\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : PreconnectedSpace G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : ContinuousVAdd G P\n\u22a2 Set.univ = \u2191(Equiv.vaddConst (Classical.arbitrary P)) '' Set.univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.Finite.seq'", "start": [971, 1], "end": [973, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Hom/Lattice.lean", "full_name": "InfTopHom.symm_dual_id", "start": [1502, 1], "end": [1503, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/Adjoin.lean", "full_name": "IntermediateField.adjoin_adjoin_comm", "start": [407, 1], "end": [409, 62], "traced_tactics": [{"tactic": "rw [adjoin_adjoin_left, adjoin_adjoin_left, Set.union_comm]", "state_before": "F : Type u_2\ninst\u271d\u00b2 : Field F\nE : Type u_1\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nS T : Set E\n\u22a2 restrictScalars F (adjoin { x // x \u2208 adjoin F S } T) = restrictScalars F (adjoin { x // x \u2208 adjoin F T } S)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.blsub_zero", "start": [1890, 1], "end": [1890, 99], "traced_tactics": [{"tactic": "rw [blsub_eq_zero_iff]", "state_before": "\u03b1 : Type ?u.358145\n\u03b2 : Type ?u.358148\n\u03b3 : Type ?u.358151\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\nf : (a : Ordinal) \u2192 a < 0 \u2192 Ordinal\n\u22a2 blsub 0 f = 0", "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.snorm_lt_top", "start": [138, 1], "end": [139, 8], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/ContinuousFunction/CocompactMap.lean", "full_name": "CocompactMap.copy_eq", "start": [115, 1], "end": [116, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.setToL1_congr_left'", "start": [1071, 1], "end": [1080, 50], "traced_tactics": [{"tactic": "suffices setToL1 hT = setToL1 hT' by rw [this]", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1059932\nG : Type ?u.1059935\n\ud835\udd5c : Type ?u.1059938\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf : { x // x \u2208 Lp E 1 }\n\u22a2 \u2191(setToL1 hT) f = \u2191(setToL1 hT') f", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1059932\nG : Type ?u.1059935\n\ud835\udd5c : Type ?u.1059938\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf : { x // x \u2208 Lp E 1 }\n\u22a2 setToL1 hT = setToL1 hT'"}, {"tactic": "refine' ContinuousLinearMap.extend_unique (setToL1SCLM \u03b1 E \u03bc hT) _ _ _ _ _", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1059932\nG : Type ?u.1059935\n\ud835\udd5c : Type ?u.1059938\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf : { x // x \u2208 Lp E 1 }\n\u22a2 setToL1 hT = setToL1 hT'", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1059932\nG : Type ?u.1059935\n\ud835\udd5c : Type ?u.1059938\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d) = setToL1SCLM \u03b1 E \u03bc hT"}, {"tactic": "ext1 f", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1059932\nG : Type ?u.1059935\n\ud835\udd5c : Type ?u.1059938\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf : { x // x \u2208 Lp E 1 }\n\u22a2 ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d) = setToL1SCLM \u03b1 E \u03bc hT", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1059932\nG : Type ?u.1059935\n\ud835\udd5c : Type ?u.1059938\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f"}, {"tactic": "suffices setToL1 hT' f = setToL1SCLM \u03b1 E \u03bc hT f by rw [\u2190 this]; rfl", "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1059932\nG : Type ?u.1059935\n\ud835\udd5c : Type ?u.1059938\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1059932\nG : Type ?u.1059935\n\ud835\udd5c : Type ?u.1059938\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f"}, {"tactic": "rw [setToL1_eq_setToL1SCLM]", "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1059932\nG : Type ?u.1059935\n\ud835\udd5c : Type ?u.1059938\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1059932\nG : Type ?u.1059935\n\ud835\udd5c : Type ?u.1059938\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(setToL1SCLM \u03b1 E \u03bc hT') f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f"}, {"tactic": "exact (setToL1SCLM_congr_left' hT hT' h f).symm", "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1059932\nG : Type ?u.1059935\n\ud835\udd5c : Type ?u.1059938\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\n\u22a2 \u2191(setToL1SCLM \u03b1 E \u03bc hT') f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f", "state_after": "no goals"}, {"tactic": "rw [this]", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1059932\nG : Type ?u.1059935\n\ud835\udd5c : Type ?u.1059938\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf : { x // x \u2208 Lp E 1 }\nthis : setToL1 hT = setToL1 hT'\n\u22a2 \u2191(setToL1 hT) f = \u2191(setToL1 hT') f", "state_after": "no goals"}, {"tactic": "rw [\u2190 this]", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1059932\nG : Type ?u.1059935\n\ud835\udd5c : Type ?u.1059938\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\nthis : \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f\n\u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1059932\nG : Type ?u.1059935\n\ud835\udd5c : Type ?u.1059938\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\nthis : \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f\n\u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1 hT') \u2191f"}, {"tactic": "rfl", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1059932\nG : Type ?u.1059935\n\ud835\udd5c : Type ?u.1059938\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : { x // x \u2208 Lp E 1 }\nf : { x // x \u2208 simpleFunc E 1 \u03bc }\nthis : \u2191(setToL1 hT') \u2191f = \u2191(setToL1SCLM \u03b1 E \u03bc hT) f\n\u22a2 \u2191(ContinuousLinearMap.comp (setToL1 hT') (coeToLp \u03b1 E \u211d)) f = \u2191(setToL1 hT') \u2191f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/Separation.lean", "full_name": "UniformSpace.SeparationQuotient.map_unique", "start": [404, 1], "end": [410, 34], "traced_tactics": [{"tactic": "ext \u27e8a\u27e9", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\nhf : UniformContinuous f\ng : SeparationQuotient \u03b1 \u2192 SeparationQuotient \u03b2\ncomm : Quotient.mk (separationSetoid \u03b2) \u2218 f = g \u2218 Quotient.mk (separationSetoid \u03b1)\n\u22a2 map f = g", "state_after": "case h.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2\nhf : UniformContinuous f\ng : SeparationQuotient \u03b1 \u2192 SeparationQuotient \u03b2\ncomm : Quotient.mk (separationSetoid \u03b2) \u2218 f = g \u2218 Quotient.mk (separationSetoid \u03b1)\nx\u271d : SeparationQuotient \u03b1\na : \u03b1\n\u22a2 map f (Quot.mk Setoid.r a) = g (Quot.mk Setoid.r a)"}, {"tactic": "calc\n  map f \u27e6a\u27e7 = \u27e6f a\u27e7 := map_mk hf a\n  _ = g \u27e6a\u27e7 := congr_fun comm 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Algebra R\u2081 K\nfrac : IsFractionRing R\u2081 K\nI J : FractionalIdeal R\u2081\u2070 K\nh : I * J = 1\nhI : I = 0\n\u22a2 0 = I * J"}, {"tactic": "simp [hI]", "state_before": "case h.e'_2\nR : Type ?u.1132184\ninst\u271d\u2075 : CommRing R\nS : Submonoid R\nP : Type ?u.1132391\ninst\u271d\u2074 : CommRing P\ninst\u271d\u00b3 : Algebra R P\nloc : IsLocalization S P\nR\u2081 : Type u_1\ninst\u271d\u00b2 : CommRing R\u2081\nK : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra R\u2081 K\nfrac : IsFractionRing R\u2081 K\nI J : FractionalIdeal R\u2081\u2070 K\nh : I * J = 1\nhI : I = 0\n\u22a2 0 = I * J", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "full_name": "Asymptotics.isBigOWith_refl", "start": [563, 1], "end": [564, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.LeftInvOn.injOn", "start": [1041, 1], "end": [1045, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.mk_Prop", "start": [570, 1], "end": [570, 41], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b1 \u03b2 : Type u\n\u22a2 (#Prop) = 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.range_eq_iff", "start": [729, 1], "end": [732, 24], "traced_tactics": [{"tactic": "rw [\u2190 range_subset_iff]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.71655\n\u03b9 : Sort ?u.71658\n\u03b9' : Sort ?u.71661\nf\u271d : \u03b9 \u2192 \u03b1\ns\u271d t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u22a2 range f = s \u2194 (\u2200 (a : \u03b1), f a \u2208 s) \u2227 \u2200 (b : \u03b2), b \u2208 s \u2192 \u2203 a, f a = b", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.71655\n\u03b9 : Sort ?u.71658\n\u03b9' : Sort ?u.71661\nf\u271d : \u03b9 \u2192 \u03b1\ns\u271d t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u22a2 range f = s \u2194 (range fun a => f a) \u2286 s \u2227 \u2200 (b : \u03b2), b \u2208 s \u2192 \u2203 a, f a = b"}, {"tactic": "exact le_antisymm_iff", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.71655\n\u03b9 : Sort ?u.71658\n\u03b9' : Sort ?u.71661\nf\u271d : \u03b9 \u2192 \u03b1\ns\u271d t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b2\n\u22a2 range f = s \u2194 (range fun a => f a) \u2286 s \u2227 \u2200 (b : \u03b2), b \u2208 s \u2192 \u2203 a, f a = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "full_name": "Matrix.det_eq_zero_of_row_eq_zero", "start": [366, 1], "end": [367, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "full_name": "UniqueFactorizationMonoid.dvd_of_dvd_mul_right_of_no_prime_factors", "start": [888, 1], "end": [890, 84], "traced_tactics": [{"tactic": "simpa [mul_comm b c] using dvd_of_dvd_mul_left_of_no_prime_factors ha @no_factors", "state_before": "\u03b1 : Type ?u.944204\nR : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero R\ninst\u271d : UniqueFactorizationMonoid R\na b c : R\nha : a \u2260 0\nno_factors : \u2200 {d : R}, d \u2223 a \u2192 d \u2223 b \u2192 \u00acPrime d\n\u22a2 a \u2223 b * c \u2192 a \u2223 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Coloring.lean", "full_name": "SimpleGraph.Colorable.of_embedding", "start": [221, 1], "end": [223, 36], "traced_tactics": [{"tactic": "simp", "state_before": "V : Type u\nG : SimpleGraph V\n\u03b1 : Type v\nC : Coloring G \u03b1\nV' : Type u_1\nG' : SimpleGraph V'\nf : G \u21aag G'\nn : \u2115\nh : Colorable G' n\n\u22a2 n \u2264 Fintype.card (Fin n)", "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.sign_coe_pi", "start": [862, 1], "end": [862, 89], "traced_tactics": [{"tactic": "rw [sign, sin_coe_pi, _root_.sign_zero]", "state_before": "\u22a2 sign \u2191\u03c0 = 0", "state_after": "no goals"}]}, {"url": 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\u03b1\nf : Filter \u03b1\ns : Set \u03b1\n\u22a2 f \u2293 \ud835\udcdf s = \u22a5 \u2194 s\u1d9c \u2208 f", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.119534\n\u03b9 : Sort x\nf\u271d g : Filter \u03b1\ns\u271d t : Set \u03b1\nf : Filter \u03b1\ns : Set \u03b1\n\u22a2 {x | x \u2208 s \u2192 x \u2208 \u2205} \u2208 f \u2194 s\u1d9c \u2208 f"}, {"tactic": "rfl", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type ?u.119534\n\u03b9 : Sort x\nf\u271d g : Filter \u03b1\ns\u271d t : Set \u03b1\nf : Filter \u03b1\ns : Set \u03b1\n\u22a2 {x | x \u2208 s \u2192 x \u2208 \u2205} \u2208 f \u2194 s\u1d9c \u2208 f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Basic.lean", "full_name": "SimpleGraph.degree_lt_card_verts", "start": [1579, 1], "end": [1583, 43], "traced_tactics": [{"tactic": "classical\napply Finset.card_lt_card\nrw [Finset.ssubset_iff]\nexact \u27e8v, by simp, Finset.subset_univ _\u27e9", "state_before": "\u03b9 : Sort ?u.295859\n\ud835\udd5c : Type ?u.295862\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v\u271d w : V\ne : Sym2 V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\n\u22a2 degree G v < Fintype.card V", "state_after": "no goals"}, {"tactic": "apply Finset.card_lt_card", "state_before": "\u03b9 : Sort ?u.295859\n\ud835\udd5c : Type ?u.295862\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v\u271d w : V\ne : Sym2 V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\n\u22a2 degree G v < Fintype.card V", "state_after": "case h\n\u03b9 : Sort ?u.295859\n\ud835\udd5c : Type ?u.295862\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v\u271d w : V\ne : Sym2 V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\n\u22a2 neighborFinset G v \u2282 univ"}, {"tactic": "rw [Finset.ssubset_iff]", "state_before": "case h\n\u03b9 : Sort ?u.295859\n\ud835\udd5c : Type ?u.295862\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v\u271d w : V\ne : Sym2 V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\n\u22a2 neighborFinset G v \u2282 univ", "state_after": "case h\n\u03b9 : Sort ?u.295859\n\ud835\udd5c : Type ?u.295862\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v\u271d w : V\ne : Sym2 V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\n\u22a2 \u2203 a x, insert a (neighborFinset G v) \u2286 univ"}, {"tactic": "exact \u27e8v, by simp, Finset.subset_univ _\u27e9", "state_before": "case h\n\u03b9 : Sort ?u.295859\n\ud835\udd5c : Type ?u.295862\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v\u271d w : V\ne : Sym2 V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\n\u22a2 \u2203 a x, insert a (neighborFinset G v) \u2286 univ", "state_after": "no goals"}, {"tactic": "simp", "state_before": "\u03b9 : Sort ?u.295859\n\ud835\udd5c : Type ?u.295862\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v\u271d w : V\ne : Sym2 V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\n\u22a2 \u00acv \u2208 neighborFinset G v", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/SuccPred/Basic.lean", "full_name": "Order.Ioc_pred_right_of_not_isMin", "start": [687, 1], "end": [688, 64], "traced_tactics": [{"tactic": "rw [\u2190 Ioi_inter_Iic, Iic_pred_of_not_isMin ha, Ioi_inter_Iio]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nha : \u00acIsMin b\n\u22a2 Ioc a (pred b) = Ioo a b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Lagrange.lean", "full_name": "Lagrange.natDegree_basisDivisor_of_ne", "start": [165, 1], "end": [166, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/GroupAction/Prod.lean", "full_name": "Prod.smul_def", "start": [66, 1], "end": [67, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Coprime/Basic.lean", "full_name": "IsCoprime.add_mul_left_left", "start": [284, 1], "end": [285, 94], "traced_tactics": [{"tactic": "simpa only [mul_neg, add_neg_cancel_right] using h", "state_before": "R : Type u\ninst\u271d : CommRing R\nx y : R\nh : IsCoprime x y\nz : R\n\u22a2 IsCoprime (x + y * z + y * -z) y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/IsDiag.lean", "full_name": "Matrix.isDiag_one", "start": [75, 1], "end": [76, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "full_name": "MeasureTheory.IntegrableAtFilter.eventually", "start": [401, 11], "end": [403, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.iInf_neBot_iff_of_directed'", "start": [937, 1], "end": [939, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "TopologicalGroup.of_nhds_one'", "start": [920, 1], "end": [932, 30], "traced_tactics": [{"tactic": "rw [show (fun x => x\u2080 * x * x\u2080\u207b\u00b9) = (fun x => x * x\u2080\u207b\u00b9) \u2218 fun x => x\u2080 * x from rfl, \u2190\n  map_map, \u2190 hleft, hright, map_map]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nG\u271d : Type w\nH : Type x\ninst\u271d\u2075 : TopologicalSpace G\u271d\ninst\u271d\u2074 : Group G\u271d\ninst\u271d\u00b3 : TopologicalGroup G\u271d\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf : \u03b1 \u2192 G\u271d\ns : Set \u03b1\nx : \u03b1\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalSpace G\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhinv : Tendsto (fun x => x\u207b\u00b9) (\ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : G), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhright : \u2200 (x\u2080 : G), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1)\nx\u2080 : G\n\u22a2 map (fun x => x\u2080 * x * x\u2080\u207b\u00b9) (\ud835\udcdd 1) = \ud835\udcdd 1", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nG\u271d : Type w\nH : Type x\ninst\u271d\u2075 : TopologicalSpace G\u271d\ninst\u271d\u2074 : Group G\u271d\ninst\u271d\u00b3 : TopologicalGroup G\u271d\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf : \u03b1 \u2192 G\u271d\ns : Set \u03b1\nx : \u03b1\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalSpace G\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhinv : Tendsto (fun x => x\u207b\u00b9) (\ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : G), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhright : \u2200 (x\u2080 : G), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1)\nx\u2080 : G\n\u22a2 map ((fun x => x * x\u2080\u207b\u00b9) \u2218 fun x => x * x\u2080) (\ud835\udcdd 1) = \ud835\udcdd 1"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean", "full_name": "IsBoundedLinearMap.smul", "start": [129, 1], "end": [135, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Rat/Cast.lean", "full_name": "Rat.cast_mono", "start": [310, 1], "end": [311, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Preadditive/ProjectiveResolution.lean", "full_name": "CategoryTheory.ProjectiveResolution.complex_d_succ_comp", "start": [130, 1], "end": [132, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "full_name": "MeasureTheory.SignedMeasure.null_of_totalVariation_zero", "start": [511, 1], "end": [518, 21], "traced_tactics": [{"tactic": "rw [totalVariation, Measure.coe_add, Pi.add_apply, add_eq_zero_iff] at hs", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.88005\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(totalVariation s) i = 0\n\u22a2 \u2191s i = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.88005\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\n\u22a2 \u2191s i = 0"}, {"tactic": "rw [\u2190 toSignedMeasure_toJordanDecomposition s, toSignedMeasure, VectorMeasure.coe_sub,\n  Pi.sub_apply, Measure.toSignedMeasure_apply, Measure.toSignedMeasure_apply]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.88005\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\n\u22a2 \u2191s i = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.88005\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\n\u22a2 ((if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) else 0) -\n      if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) else 0) =\n    0"}, {"tactic": "by_cases hi : MeasurableSet i", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.88005\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\n\u22a2 ((if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) else 0) -\n      if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) else 0) =\n    0", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.88005\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\nhi : MeasurableSet i\n\u22a2 ((if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) else 0) -\n      if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) else 0) =\n    0\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.88005\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\nhi : \u00acMeasurableSet i\n\u22a2 ((if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) else 0) -\n      if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) else 0) =\n    0"}, {"tactic": "rw [if_pos hi, if_pos hi]", "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.88005\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\nhi : MeasurableSet i\n\u22a2 ((if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) else 0) -\n      if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) else 0) =\n    0", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.88005\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\nhi : MeasurableSet i\n\u22a2 ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) - ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) = 0"}, {"tactic": "simp [hs.1, hs.2]", "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.88005\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\nhi : MeasurableSet i\n\u22a2 ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) - ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) = 0", "state_after": "no goals"}, {"tactic": "simp [if_neg hi]", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.88005\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\ni : Set \u03b1\nhs : \u2191\u2191(toJordanDecomposition s).posPart i = 0 \u2227 \u2191\u2191(toJordanDecomposition s).negPart i = 0\nhi : \u00acMeasurableSet i\n\u22a2 ((if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).posPart i) else 0) -\n      if MeasurableSet i then ENNReal.toReal (\u2191\u2191(toJordanDecomposition s).negPart i) else 0) =\n    0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Init/Logic.lean", "full_name": "heq_of_eq_rec_left", "start": [70, 1], "end": [72, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean", "full_name": "CategoryTheory.Limits.Multicoequalizer.hom_ext", "start": [888, 1], "end": [894, 17], "traced_tactics": [{"tactic": "rintro (a | b)", "state_before": "C : Type u\ninst\u271d\u00b9 : Category C\nI : MultispanIndex C\ninst\u271d : HasMulticoequalizer I\nW : C\ni j : multicoequalizer I \u27f6 W\nh : \u2200 (b : I.R), \u03c0 I b \u226b i = \u03c0 I b \u226b j\n\u22a2 \u2200 (j_1 : WalkingMultispan I.fstFrom I.sndFrom),\n    colimit.\u03b9 (MultispanIndex.multispan I) j_1 \u226b i = colimit.\u03b9 (MultispanIndex.multispan I) j_1 \u226b j", "state_after": "case left\nC : Type u\ninst\u271d\u00b9 : Category C\nI : MultispanIndex C\ninst\u271d : HasMulticoequalizer I\nW : C\ni j : multicoequalizer I \u27f6 W\nh : \u2200 (b : I.R), \u03c0 I b \u226b i = \u03c0 I b \u226b j\na : I.L\n\u22a2 colimit.\u03b9 (MultispanIndex.multispan I) (WalkingMultispan.left a) \u226b i =\n    colimit.\u03b9 (MultispanIndex.multispan I) (WalkingMultispan.left a) \u226b j\n\ncase right\nC : Type u\ninst\u271d\u00b9 : Category C\nI : MultispanIndex C\ninst\u271d : HasMulticoequalizer I\nW : C\ni j : multicoequalizer I \u27f6 W\nh : \u2200 (b : I.R), \u03c0 I b \u226b i = \u03c0 I b \u226b j\nb : I.R\n\u22a2 colimit.\u03b9 (MultispanIndex.multispan I) (WalkingMultispan.right b) \u226b i =\n    colimit.\u03b9 (MultispanIndex.multispan I) (WalkingMultispan.right b) \u226b j"}, {"tactic": "simp_rw [\u2190 colimit.w I.multispan (WalkingMultispan.Hom.fst a), Category.assoc, h]", "state_before": "case left\nC : Type u\ninst\u271d\u00b9 : Category C\nI : MultispanIndex C\ninst\u271d : HasMulticoequalizer I\nW : C\ni j : multicoequalizer I \u27f6 W\nh : \u2200 (b : I.R), \u03c0 I b \u226b i = \u03c0 I b \u226b j\na : I.L\n\u22a2 colimit.\u03b9 (MultispanIndex.multispan I) (WalkingMultispan.left a) \u226b i =\n    colimit.\u03b9 (MultispanIndex.multispan I) (WalkingMultispan.left a) \u226b j", "state_after": "no goals"}, {"tactic": "apply h", "state_before": "case right\nC : Type u\ninst\u271d\u00b9 : Category C\nI : MultispanIndex C\ninst\u271d : HasMulticoequalizer I\nW : C\ni j : multicoequalizer I \u27f6 W\nh : \u2200 (b : I.R), \u03c0 I b \u226b i = \u03c0 I b \u226b j\nb : I.R\n\u22a2 colimit.\u03b9 (MultispanIndex.multispan I) (WalkingMultispan.right b) \u226b i =\n    colimit.\u03b9 (MultispanIndex.multispan I) (WalkingMultispan.right b) \u226b j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finsupp/Basic.lean", "full_name": "Finsupp.prod_mapDomain_index_inj", "start": [608, 1], "end": [610, 84], "traced_tactics": [{"tactic": "rw [\u2190 Function.Embedding.coeFn_mk f hf, \u2190 embDomain_eq_mapDomain, prod_embDomain]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_4\n\u03b3 : Type ?u.249071\n\u03b9 : Type ?u.249074\nM : Type u_3\nM' : Type ?u.249080\nN : Type u_1\nP : Type ?u.249086\nG : Type ?u.249089\nH : Type ?u.249092\nR : Type ?u.249095\nS : Type ?u.249098\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : CommMonoid N\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nh : \u03b2 \u2192 M \u2192 N\nhf : Injective f\n\u22a2 prod (mapDomain f s) h = prod s fun a b => h (f a) b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.mk_subtype_le", "start": [289, 1], "end": [290, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean", "full_name": "CategoryTheory.Limits.zeroProdIso_hom", "start": [56, 1], "end": [57, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.tendsto_atBot_add_nonpos_left'", "start": [614, 1], "end": [616, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UrysohnsLemma.lean", "full_name": "Urysohns.CU.approx_of_nmem_U", "start": [149, 1], "end": [155, 52], "traced_tactics": [{"tactic": "induction' n with n ihn generalizing c", "state_before": "X : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc : CU X\nn : \u2115\nx : X\nhx : \u00acx \u2208 c.U\n\u22a2 approx n c x = 1", "state_after": "case zero\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : \u00acx \u2208 c\u271d.U\nc : CU X\nhx : \u00acx \u2208 c.U\n\u22a2 approx Nat.zero c x = 1\n\ncase succ\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : \u00acx \u2208 c\u271d.U\nn : \u2115\nihn : \u2200 (c : CU X), \u00acx \u2208 c.U \u2192 approx n c x = 1\nc : CU X\nhx : \u00acx \u2208 c.U\n\u22a2 approx (Nat.succ n) c x = 1"}, {"tactic": "rw [\u2190 mem_compl_iff] at hx", "state_before": "case zero\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : \u00acx \u2208 c\u271d.U\nc : CU X\nhx : \u00acx \u2208 c.U\n\u22a2 approx Nat.zero c x = 1", "state_after": "case zero\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : \u00acx \u2208 c\u271d.U\nc : CU X\nhx : x \u2208 c.U\u1d9c\n\u22a2 approx Nat.zero c x = 1"}, {"tactic": "exact indicator_of_mem hx _", "state_before": "case zero\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : \u00acx \u2208 c\u271d.U\nc : CU X\nhx : x \u2208 c.U\u1d9c\n\u22a2 approx Nat.zero c x = 1", "state_after": "no goals"}, {"tactic": "simp only [approx]", "state_before": "case succ\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : \u00acx \u2208 c\u271d.U\nn : \u2115\nihn : \u2200 (c : CU X), \u00acx \u2208 c.U \u2192 approx n c x = 1\nc : CU X\nhx : \u00acx \u2208 c.U\n\u22a2 approx (Nat.succ n) c x = 1", "state_after": "case succ\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : \u00acx \u2208 c\u271d.U\nn : \u2115\nihn : \u2200 (c : CU X), \u00acx \u2208 c.U \u2192 approx n c x = 1\nc : CU X\nhx : \u00acx \u2208 c.U\n\u22a2 midpoint \u211d (approx n (left c) x) (approx n (right c) x) = 1"}, {"tactic": "rw [ihn, ihn, midpoint_self]", "state_before": "case succ\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : \u00acx \u2208 c\u271d.U\nn : \u2115\nihn : \u2200 (c : CU X), \u00acx \u2208 c.U \u2192 approx n c x = 1\nc : CU X\nhx : \u00acx \u2208 c.U\n\u22a2 midpoint \u211d (approx n (left c) x) (approx n (right c) x) = 1", "state_after": "case succ.hx\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : \u00acx \u2208 c\u271d.U\nn : \u2115\nihn : \u2200 (c : CU X), \u00acx \u2208 c.U \u2192 approx n c x = 1\nc : CU X\nhx : \u00acx \u2208 c.U\n\u22a2 \u00acx \u2208 (right c).U\n\ncase succ.hx\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : \u00acx \u2208 c\u271d.U\nn : \u2115\nihn : \u2200 (c : CU X), \u00acx \u2208 c.U \u2192 approx n c x = 1\nc : CU X\nhx : \u00acx \u2208 c.U\n\u22a2 \u00acx \u2208 (left c).U"}, {"tactic": "exacts [hx, fun hU => hx <| c.left_U_subset hU]", "state_before": "case succ.hx\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : \u00acx \u2208 c\u271d.U\nn : \u2115\nihn : \u2200 (c : CU X), \u00acx \u2208 c.U \u2192 approx n c x = 1\nc : CU X\nhx : \u00acx \u2208 c.U\n\u22a2 \u00acx \u2208 (right c).U\n\ncase succ.hx\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\nc\u271d : CU X\nx : X\nhx\u271d : \u00acx \u2208 c\u271d.U\nn : \u2115\nihn : \u2200 (c : CU X), \u00acx \u2208 c.U \u2192 approx n c x = 1\nc : CU X\nhx : \u00acx \u2208 c.U\n\u22a2 \u00acx \u2208 (left c).U", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec\u2082.option_some_iff", "start": [452, 1], "end": [453, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finite/Defs.lean", "full_name": "Equiv.infinite_iff", "start": [113, 1], "end": [114, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "full_name": "Real.lt_rpow_of_log_lt", "start": [606, 1], "end": [610, 34], "traced_tactics": [{"tactic": "obtain hx | rfl := hx.lt_or_eq", "state_before": "x y z : \u211d\nhx : 0 \u2264 x\nhy : 0 < y\nh : log x < z * log y\n\u22a2 x < y ^ z", "state_after": "case inl\nx y z : \u211d\nhx\u271d : 0 \u2264 x\nhy : 0 < y\nh : log x < z * log y\nhx : 0 < x\n\u22a2 x < y ^ z\n\ncase inr\ny z : \u211d\nhy : 0 < y\nhx : 0 \u2264 0\nh : log 0 < z * log y\n\u22a2 0 < y ^ z"}, {"tactic": "exact Real.rpow_pos_of_pos hy z", "state_before": "case inr\ny z : \u211d\nhy : 0 < y\nhx : 0 \u2264 0\nh : log 0 < z * log y\n\u22a2 0 < y ^ z", "state_after": "no goals"}, {"tactic": "exact (lt_rpow_iff_log_lt hx hy).2 h", "state_before": "case inl\nx y z : \u211d\nhx\u271d : 0 \u2264 x\nhy : 0 < y\nh : log x < z * log y\nhx : 0 < x\n\u22a2 x < y ^ z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Lattice.lean", "full_name": "inf_left_idem", "start": [514, 1], "end": [515, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/LinearIndependent.lean", "full_name": "exists_linearIndependent", "start": [1290, 1], "end": [1294, 76], "traced_tactics": [{"tactic": "obtain \u27e8b, hb\u2081, -, hb\u2082, hb\u2083\u27e9 :=\n  exists_linearIndependent_extension (linearIndependent_empty K V) (Set.empty_subset t)", "state_before": "\u03b9 : Type u'\n\u03b9' : Type ?u.1261935\nR : Type ?u.1261938\nK : Type u_1\nM : Type ?u.1261944\nM' : Type ?u.1261947\nM'' : Type ?u.1261950\nV : Type u\nV' : Type ?u.1261955\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 y z : V\n\u22a2 \u2203 b x, span K b = span K t \u2227 LinearIndependent K Subtype.val", "state_after": "case intro.intro.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type ?u.1261935\nR : Type ?u.1261938\nK : Type u_1\nM : Type ?u.1261944\nM' : Type ?u.1261947\nM'' : Type ?u.1261950\nV : Type u\nV' : Type ?u.1261955\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 y z : V\nb : Set V\nhb\u2081 : b \u2286 t\nhb\u2082 : t \u2286 \u2191(span K b)\nhb\u2083 : LinearIndependent K Subtype.val\n\u22a2 \u2203 b x, span K b = span K t \u2227 LinearIndependent K Subtype.val"}, {"tactic": "exact \u27e8b, hb\u2081, (span_eq_of_le _ hb\u2082 (Submodule.span_mono hb\u2081)).symm, hb\u2083\u27e9", "state_before": "case intro.intro.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type ?u.1261935\nR : Type ?u.1261938\nK : Type u_1\nM : Type ?u.1261944\nM' : Type ?u.1261947\nM'' : Type ?u.1261950\nV : Type u\nV' : Type ?u.1261955\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 y z : V\nb : Set V\nhb\u2081 : b \u2286 t\nhb\u2082 : t \u2286 \u2191(span K b)\nhb\u2083 : LinearIndependent K Subtype.val\n\u22a2 \u2203 b x, span K b = span K t \u2227 LinearIndependent K Subtype.val", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/ODE/PicardLindelof.lean", "full_name": "exists_forall_deriv_at_Ioo_eq_of_contDiffOn_nhds", "start": [440, 1], "end": [462, 50], "traced_tactics": [{"tactic": "obtain \u27e8\u03b5, h\u03b5, L, R, C, hpl\u27e9 := exists_isPicardLindelof_const_of_contDiffOn_nhds t\u2080 x\u2080 hv hs", "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 f, f t\u2080 = x\u2080 \u2227 \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 f t \u2208 s \u2227 HasDerivAt f (v (f t)) t", "state_after": "case intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 f, f t\u2080 = x\u2080 \u2227 \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 f t \u2208 s \u2227 HasDerivAt f (v (f t)) t"}, {"tactic": "obtain \u27e8f, hf1, hf2\u27e9 := hpl.exists_forall_hasDerivWithinAt_Icc_eq x\u2080", "state_before": "case intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 f, f t\u2080 = x\u2080 \u2227 \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 f t \u2208 s \u2227 HasDerivAt f (v (f t)) t", "state_after": "case intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 f, f t\u2080 = x\u2080 \u2227 \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 f t \u2208 s \u2227 HasDerivAt f (v (f t)) t"}, {"tactic": "have hf2' : \u2200 t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5), HasDerivAt f (v (f t)) t := fun t ht =>\n  (hf2 t (Ioo_subset_Icc_self ht)).hasDerivAt (Icc_mem_nhds ht.1 ht.2)", "state_before": "case intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 f, f t\u2080 = x\u2080 \u2227 \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 f t \u2208 s \u2227 HasDerivAt f (v (f t)) t", "state_after": "case intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 f, f t\u2080 = x\u2080 \u2227 \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 f t \u2208 s \u2227 HasDerivAt f (v (f t)) t"}, {"tactic": "have h : f \u207b\u00b9' s \u2208 \ud835\udcdd t\u2080 := by\n  have := hf2' t\u2080 (mem_Ioo.mpr \u27e8sub_lt_self _ h\u03b5, lt_add_of_pos_right _ h\u03b5\u27e9)\n  apply ContinuousAt.preimage_mem_nhds this.continuousAt\n  rw [hf1]\n  exact hs", "state_before": "case intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 f, f t\u2080 = x\u2080 \u2227 \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 f t \u2208 s \u2227 HasDerivAt f (v (f t)) t", "state_after": "case intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\nh : f \u207b\u00b9' s \u2208 \ud835\udcdd t\u2080\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 f, f t\u2080 = x\u2080 \u2227 \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 f t \u2208 s \u2227 HasDerivAt f (v (f t)) t"}, {"tactic": "rw [Metric.mem_nhds_iff] at h", "state_before": "case intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\nh : f \u207b\u00b9' s \u2208 \ud835\udcdd t\u2080\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 f, f t\u2080 = x\u2080 \u2227 \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 f t \u2208 s \u2227 HasDerivAt f (v (f t)) t", "state_after": "case intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\nh : \u2203 \u03b5, \u03b5 > 0 \u2227 ball t\u2080 \u03b5 \u2286 f \u207b\u00b9' s\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 f, f t\u2080 = x\u2080 \u2227 \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 f t \u2208 s \u2227 HasDerivAt f (v (f t)) t"}, {"tactic": "obtain \u27e8r, hr1, hr2\u27e9 := h", "state_before": "case intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\nh : \u2203 \u03b5, \u03b5 > 0 \u2227 ball t\u2080 \u03b5 \u2286 f \u207b\u00b9' s\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 f, f t\u2080 = x\u2080 \u2227 \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 f t \u2208 s \u2227 HasDerivAt f (v (f t)) t", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\nr : \u211d\nhr1 : r > 0\nhr2 : ball t\u2080 r \u2286 f \u207b\u00b9' s\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 f, f t\u2080 = x\u2080 \u2227 \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 f t \u2208 s \u2227 HasDerivAt f (v (f t)) t"}, {"tactic": "refine \u27e8min r \u03b5, lt_min hr1 h\u03b5, f, hf1, fun t ht => \u27e8?_,\n  hf2' t (mem_of_mem_of_subset ht (Ioo_subset_Ioo (sub_le_sub_left (min_le_right _ _) _)\n    (add_le_add_left (min_le_right _ _) _)))\u27e9\u27e9", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\nr : \u211d\nhr1 : r > 0\nhr2 : ball t\u2080 r \u2286 f \u207b\u00b9' s\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 f, f t\u2080 = x\u2080 \u2227 \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 f t \u2208 s \u2227 HasDerivAt f (v (f t)) t", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\nr : \u211d\nhr1 : r > 0\nhr2 : ball t\u2080 r \u2286 f \u207b\u00b9' s\nt : \u211d\nht : t \u2208 Ioo (t\u2080 - min r \u03b5) (t\u2080 + min r \u03b5)\n\u22a2 f t \u2208 s"}, {"tactic": "rw [\u2190 Set.mem_preimage]", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\nr : \u211d\nhr1 : r > 0\nhr2 : ball t\u2080 r \u2286 f \u207b\u00b9' s\nt : \u211d\nht : t \u2208 Ioo (t\u2080 - min r \u03b5) (t\u2080 + min r \u03b5)\n\u22a2 f t \u2208 s", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\nr : \u211d\nhr1 : r > 0\nhr2 : ball t\u2080 r \u2286 f \u207b\u00b9' s\nt : \u211d\nht : t \u2208 Ioo (t\u2080 - min r \u03b5) (t\u2080 + min r \u03b5)\n\u22a2 t \u2208 f \u207b\u00b9' s"}, {"tactic": "apply Set.mem_of_mem_of_subset _ hr2", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\nr : \u211d\nhr1 : r > 0\nhr2 : ball t\u2080 r \u2286 f \u207b\u00b9' s\nt : \u211d\nht : t \u2208 Ioo (t\u2080 - min r \u03b5) (t\u2080 + min r \u03b5)\n\u22a2 t \u2208 f \u207b\u00b9' s", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\nr : \u211d\nhr1 : r > 0\nhr2 : ball t\u2080 r \u2286 f \u207b\u00b9' s\nt : \u211d\nht : t \u2208 Ioo (t\u2080 - min r \u03b5) (t\u2080 + min r \u03b5)\n\u22a2 t \u2208 ball t\u2080 r"}, {"tactic": "apply Set.mem_of_mem_of_subset ht", "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\nr : \u211d\nhr1 : r > 0\nhr2 : ball t\u2080 r \u2286 f \u207b\u00b9' s\nt : \u211d\nht : t \u2208 Ioo (t\u2080 - min r \u03b5) (t\u2080 + min r \u03b5)\n\u22a2 t \u2208 ball t\u2080 r", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\nr : \u211d\nhr1 : r > 0\nhr2 : ball t\u2080 r \u2286 f \u207b\u00b9' s\nt : \u211d\nht : t \u2208 Ioo (t\u2080 - min r \u03b5) (t\u2080 + min r \u03b5)\n\u22a2 Ioo (t\u2080 - min r \u03b5) (t\u2080 + min r \u03b5) \u2286 ball t\u2080 r"}, {"tactic": "rw [\u2190 Real.ball_eq_Ioo]", "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\nr : \u211d\nhr1 : r > 0\nhr2 : ball t\u2080 r \u2286 f \u207b\u00b9' s\nt : \u211d\nht : t \u2208 Ioo (t\u2080 - min r \u03b5) (t\u2080 + min r \u03b5)\n\u22a2 Ioo (t\u2080 - min r \u03b5) (t\u2080 + min r \u03b5) \u2286 ball t\u2080 r", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\nr : \u211d\nhr1 : r > 0\nhr2 : ball t\u2080 r \u2286 f \u207b\u00b9' s\nt : \u211d\nht : t \u2208 Ioo (t\u2080 - min r \u03b5) (t\u2080 + min r \u03b5)\n\u22a2 ball t\u2080 (min r \u03b5) \u2286 ball t\u2080 r"}, {"tactic": "exact Metric.ball_subset_ball (min_le_left _ _)", "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\nr : \u211d\nhr1 : r > 0\nhr2 : ball t\u2080 r \u2286 f \u207b\u00b9' s\nt : \u211d\nht : t \u2208 Ioo (t\u2080 - min r \u03b5) (t\u2080 + min r \u03b5)\n\u22a2 ball t\u2080 (min r \u03b5) \u2286 ball t\u2080 r", "state_after": "no goals"}, {"tactic": "have := hf2' t\u2080 (mem_Ioo.mpr \u27e8sub_lt_self _ h\u03b5, lt_add_of_pos_right _ h\u03b5\u27e9)", "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\n\u22a2 f \u207b\u00b9' s \u2208 \ud835\udcdd t\u2080", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\nthis : HasDerivAt f (v (f t\u2080)) t\u2080\n\u22a2 f \u207b\u00b9' s \u2208 \ud835\udcdd t\u2080"}, {"tactic": "apply ContinuousAt.preimage_mem_nhds this.continuousAt", "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\nthis : HasDerivAt f (v (f t\u2080)) t\u2080\n\u22a2 f \u207b\u00b9' s \u2208 \ud835\udcdd t\u2080", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\nthis : HasDerivAt f (v (f t\u2080)) t\u2080\n\u22a2 s \u2208 \ud835\udcdd (f t\u2080)"}, {"tactic": "rw [hf1]", "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\nthis : HasDerivAt f (v (f t\u2080)) t\u2080\n\u22a2 s \u2208 \ud835\udcdd (f t\u2080)", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\nthis : HasDerivAt f (v (f t\u2080)) t\u2080\n\u22a2 s \u2208 \ud835\udcdd x\u2080"}, {"tactic": "exact hs", "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : ProperSpace E\nv : E \u2192 E\nt\u2080 : \u211d\nx\u2080 : E\ns : Set E\nhv : ContDiffOn \u211d 1 v s\nhs : s \u2208 \ud835\udcdd x\u2080\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nL : \u211d\u22650\nR C : \u211d\nhpl : IsPicardLindelof (fun x => v) (t\u2080 - \u03b5) t\u2080 (t\u2080 + \u03b5) x\u2080 L R C\nf : \u211d \u2192 E\nhf1 : f t\u2080 = x\u2080\nhf2 : \u2200 (t : \u211d), t \u2208 Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivWithinAt f (v (f t)) (Icc (t\u2080 - \u03b5) (t\u2080 + \u03b5)) t\nhf2' : \u2200 (t : \u211d), t \u2208 Ioo (t\u2080 - \u03b5) (t\u2080 + \u03b5) \u2192 HasDerivAt f (v (f t)) t\nthis : HasDerivAt f (v (f t\u2080)) t\u2080\n\u22a2 s \u2208 \ud835\udcdd x\u2080", "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.biproduct.hom_ext", "start": [464, 1], "end": [466, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Countable.lean", "full_name": "Set.countable_insert", "start": [227, 1], "end": [228, 76], "traced_tactics": [{"tactic": "simp only [insert_eq, countable_union, countable_singleton, true_and_iff]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ns : Set \u03b1\na : \u03b1\n\u22a2 Set.Countable (insert a s) \u2194 Set.Countable s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "full_name": "SimpleGraph.Path.loop_eq", "start": [1324, 1], "end": [1327, 14], "traced_tactics": [{"tactic": "obtain \u27e8_ | _, h\u27e9 := p", "state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv : V\np : Path G v v\n\u22a2 p = Path.nil", "state_after": "case mk.nil\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv : V\nh : Walk.IsPath Walk.nil\n\u22a2 { val := Walk.nil, property := h } = Path.nil\n\ncase mk.cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv v\u271d : V\nh\u271d : Adj G v v\u271d\np\u271d : Walk G v\u271d v\nh : Walk.IsPath (Walk.cons h\u271d p\u271d)\n\u22a2 { val := Walk.cons h\u271d p\u271d, property := h } = Path.nil"}, {"tactic": "rfl", "state_before": "case mk.nil\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv : V\nh : Walk.IsPath Walk.nil\n\u22a2 { val := Walk.nil, property := h } = Path.nil", "state_after": "no goals"}, {"tactic": "simp at h", "state_before": "case mk.cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv v\u271d : V\nh\u271d : Adj G v v\u271d\np\u271d : Walk G v\u271d v\nh : Walk.IsPath (Walk.cons h\u271d p\u271d)\n\u22a2 { val := Walk.cons h\u271d p\u271d, property := h } = Path.nil", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Arrow.lean", "full_name": "CategoryTheory.Arrow.mk_inj", "start": [100, 1], "end": [101, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.mem_compl_image", "start": [368, 1], "end": [370, 41], "traced_tactics": [{"tactic": "simp [\u2190 preimage_compl_eq_image_compl]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.38945\n\u03b3 : Type ?u.38948\n\u03b9 : Sort ?u.38951\n\u03b9' : Sort ?u.38954\nf : \u03b1 \u2192 \u03b2\ns t\u271d : Set \u03b1\ninst\u271d : BooleanAlgebra \u03b1\nt : \u03b1\nS : Set \u03b1\n\u22a2 t \u2208 compl '' S \u2194 t\u1d9c \u2208 S", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/PFun.lean", "full_name": "PFun.image_mono", "start": [412, 1], "end": [413, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Basic.lean", "full_name": "div_mul_eq_div_div_swap", "start": [472, 1], "end": [473, 56], "traced_tactics": [{"tactic": "simp only [mul_assoc, mul_inv_rev, div_eq_mul_inv]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.29041\nG : Type ?u.29044\ninst\u271d : DivisionMonoid \u03b1\na b c : \u03b1\n\u22a2 a / (b * c) = a / c / b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Composition.lean", "full_name": "List.length_splitWrtCompositionAux", "start": [677, 1], "end": [681, 15], "traced_tactics": [{"tactic": "induction ns generalizing l", "state_before": "n : \u2115\n\u03b1 : Type u_1\nl : List \u03b1\nns : List \u2115\n\u22a2 length (splitWrtCompositionAux l ns) = length ns", "state_after": "case nil\nn : \u2115\n\u03b1 : Type u_1\nl : List \u03b1\n\u22a2 length (splitWrtCompositionAux l []) = length []\n\ncase cons\nn : \u2115\n\u03b1 : Type u_1\nhead\u271d : \u2115\ntail\u271d : List \u2115\ntail_ih\u271d : \u2200 (l : List \u03b1), length (splitWrtCompositionAux l tail\u271d) = length tail\u271d\nl : List \u03b1\n\u22a2 length (splitWrtCompositionAux l (head\u271d :: tail\u271d)) = length (head\u271d :: tail\u271d)"}, {"tactic": ". simp [splitWrtCompositionAux, *]", "state_before": "case nil\nn : \u2115\n\u03b1 : Type u_1\nl : List \u03b1\n\u22a2 length (splitWrtCompositionAux l []) = length []\n\ncase cons\nn : \u2115\n\u03b1 : Type u_1\nhead\u271d : \u2115\ntail\u271d : List \u2115\ntail_ih\u271d : \u2200 (l : List \u03b1), length (splitWrtCompositionAux l tail\u271d) = length tail\u271d\nl : List \u03b1\n\u22a2 length (splitWrtCompositionAux l (head\u271d :: tail\u271d)) = length (head\u271d :: tail\u271d)", "state_after": "case cons\nn : \u2115\n\u03b1 : Type u_1\nhead\u271d : \u2115\ntail\u271d : List \u2115\ntail_ih\u271d : \u2200 (l : List \u03b1), length (splitWrtCompositionAux l tail\u271d) = length tail\u271d\nl : List \u03b1\n\u22a2 length (splitWrtCompositionAux l (head\u271d :: tail\u271d)) = length (head\u271d :: tail\u271d)"}, {"tactic": ". simp [*]", "state_before": "case cons\nn : \u2115\n\u03b1 : Type u_1\nhead\u271d : \u2115\ntail\u271d : List \u2115\ntail_ih\u271d : \u2200 (l : List \u03b1), length (splitWrtCompositionAux l tail\u271d) = length tail\u271d\nl : List \u03b1\n\u22a2 length (splitWrtCompositionAux l (head\u271d :: tail\u271d)) = length (head\u271d :: tail\u271d)", "state_after": "no goals"}, {"tactic": "simp [splitWrtCompositionAux, *]", "state_before": "case nil\nn : \u2115\n\u03b1 : Type u_1\nl : List \u03b1\n\u22a2 length (splitWrtCompositionAux l []) = length []", "state_after": "no goals"}, {"tactic": "simp [*]", "state_before": "case cons\nn : \u2115\n\u03b1 : Type u_1\nhead\u271d : \u2115\ntail\u271d : List \u2115\ntail_ih\u271d : \u2200 (l : List \u03b1), length (splitWrtCompositionAux l tail\u271d) = length tail\u271d\nl : List \u03b1\n\u22a2 length (splitWrtCompositionAux l (head\u271d :: tail\u271d)) = length (head\u271d :: tail\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "denseRange_iff_closure_range", "start": [1806, 1], "end": [1807, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf_le_of_le", "start": [410, 1], "end": [410, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Covering.lean", "full_name": "IsEvenlyCovered.to_isEvenlyCovered_preimage", "start": [72, 1], "end": [77, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Valuation/Basic.lean", "full_name": "AddValuation.ext", "start": [720, 1], "end": [721, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "iSup_option", "start": [1575, 1], "end": [1576, 85], "traced_tactics": [{"tactic": "simp only [iSup_le_iff, sup_le_iff, Option.forall]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b2\u2082 : Type ?u.169814\n\u03b3 : Type ?u.169817\n\u03b9 : Sort ?u.169820\n\u03b9' : Sort ?u.169823\n\u03ba : \u03b9 \u2192 Sort ?u.169828\n\u03ba' : \u03b9' \u2192 Sort ?u.169833\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : Option \u03b2 \u2192 \u03b1\nc : \u03b1\n\u22a2 (\u2a06 (o : Option \u03b2), f o) \u2264 c \u2194 (f none \u2294 \u2a06 (b : \u03b2), f (some b)) \u2264 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Coeff.lean", "full_name": "Polynomial.coeff_X_add_one_pow", "start": [322, 1], "end": [323, 99], "traced_tactics": [{"tactic": "rw [\u2190 C_1, coeff_X_add_C_pow, one_pow, one_mul]", "state_before": "R\u271d : Type u\nS : Type v\na b : R\u271d\nn\u271d m : \u2115\ninst\u271d\u00b9 : Semiring R\u271d\np q r : R\u271d[X]\nR : Type u_1\ninst\u271d : Semiring R\nn k : \u2115\n\u22a2 coeff ((X + 1) ^ n) k = \u2191(Nat.choose n k)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/PartENat.lean", "full_name": "PartENat.find_le", "start": [812, 1], "end": [814, 45], "traced_tactics": [{"tactic": "rw [le_coe_iff]", "state_before": "P : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : P n\n\u22a2 find P \u2264 \u2191n", "state_after": "P : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : P n\n\u22a2 \u2203 h, Part.get (find P) h \u2264 n"}, {"tactic": "refine' \u27e8\u27e8_, h\u27e9, @Nat.find_min' P _ _ _ h\u27e9", "state_before": "P : \u2115 \u2192 Prop\ninst\u271d : DecidablePred P\nn : \u2115\nh : P n\n\u22a2 \u2203 h, Part.get (find P) h \u2264 n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "full_name": "Algebra.inf_toSubmodule", "start": [844, 1], "end": [845, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Basic.lean", "full_name": "Function.const_le_const", "start": [923, 1], "end": [923, 78], "traced_tactics": [{"tactic": "simp [Pi.le_def]", "state_before": "\u03b9 : Type ?u.73033\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type ?u.73044\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Nonempty \u03b2\na b : \u03b1\n\u22a2 const \u03b2 a \u2264 const \u03b2 b \u2194 a \u2264 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/Fin.lean", "full_name": "Fin.preimage_apply_01_prod", "start": [60, 1], "end": [65, 34], "traced_tactics": [{"tactic": "ext f", "state_before": "m n : \u2115\n\u03b1 : Fin 2 \u2192 Type u\ns : Set (\u03b1 0)\nt : Set (\u03b1 1)\n\u22a2 (fun f => (f 0, f 1)) \u207b\u00b9' s \u00d7\u02e2 t = Set.pi Set.univ (cons s (cons t finZeroElim))", "state_after": "case h\nm n : \u2115\n\u03b1 : Fin 2 \u2192 Type u\ns : Set (\u03b1 0)\nt : Set (\u03b1 1)\nf : (i : Fin 2) \u2192 \u03b1 i\n\u22a2 f \u2208 (fun f => (f 0, f 1)) \u207b\u00b9' s \u00d7\u02e2 t \u2194 f \u2208 Set.pi Set.univ (cons s (cons t finZeroElim))"}, {"tactic": "have : (Fin.cons s (Fin.cons t finZeroElim) : \u2200 i, Set (\u03b1 i)) 1 = t := rfl", "state_before": "case h\nm n : \u2115\n\u03b1 : Fin 2 \u2192 Type u\ns : Set (\u03b1 0)\nt : Set (\u03b1 1)\nf : (i : Fin 2) \u2192 \u03b1 i\n\u22a2 f \u2208 (fun f => (f 0, f 1)) \u207b\u00b9' s \u00d7\u02e2 t \u2194 f \u2208 Set.pi Set.univ (cons s (cons t finZeroElim))", "state_after": "case h\nm n : \u2115\n\u03b1 : Fin 2 \u2192 Type u\ns : Set (\u03b1 0)\nt : Set (\u03b1 1)\nf : (i : Fin 2) \u2192 \u03b1 i\nthis : cons s (cons t finZeroElim) 1 = t\n\u22a2 f \u2208 (fun f => (f 0, f 1)) \u207b\u00b9' s \u00d7\u02e2 t \u2194 f \u2208 Set.pi Set.univ (cons s (cons t finZeroElim))"}, {"tactic": "simp [Fin.forall_fin_two, this]", "state_before": "case h\nm n : \u2115\n\u03b1 : Fin 2 \u2192 Type u\ns : Set (\u03b1 0)\nt : Set (\u03b1 1)\nf : (i : Fin 2) \u2192 \u03b1 i\nthis : cons s (cons t finZeroElim) 1 = t\n\u22a2 f \u2208 (fun f => (f 0, f 1)) \u207b\u00b9' s \u00d7\u02e2 t \u2194 f \u2208 Set.pi Set.univ (cons s (cons t finZeroElim))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "full_name": "Cardinal.mk_compl_eq_mk_compl_finite_lift", "start": [1166, 1], "end": [1178, 42], "traced_tactics": [{"tactic": "cases nonempty_fintype \u03b1", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Finite \u03b1\ns : Set \u03b1\nt : Set \u03b2\nh1 : lift (#\u03b1) = lift (#\u03b2)\nh2 : lift (#\u2191s) = lift (#\u2191t)\n\u22a2 lift (#\u2191(s\u1d9c)) = lift (#\u2191(t\u1d9c))", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Finite \u03b1\ns : Set \u03b1\nt : Set \u03b2\nh1 : lift (#\u03b1) = lift (#\u03b2)\nh2 : lift (#\u2191s) = lift (#\u2191t)\nval\u271d : Fintype \u03b1\n\u22a2 lift (#\u2191(s\u1d9c)) = lift (#\u2191(t\u1d9c))"}, {"tactic": "rcases lift_mk_eq.{u, v, w}.1 h1 with \u27e8e\u27e9", "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Finite \u03b1\ns : Set \u03b1\nt : Set \u03b2\nh1 : lift (#\u03b1) = lift (#\u03b2)\nh2 : lift (#\u2191s) = lift (#\u2191t)\nval\u271d : Fintype \u03b1\n\u22a2 lift (#\u2191(s\u1d9c)) = lift (#\u2191(t\u1d9c))", "state_after": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Finite \u03b1\ns : Set \u03b1\nt : Set \u03b2\nh1 : lift (#\u03b1) = lift (#\u03b2)\nh2 : lift (#\u2191s) = lift (#\u2191t)\nval\u271d : Fintype \u03b1\ne : \u03b1 \u2243 \u03b2\n\u22a2 lift (#\u2191(s\u1d9c)) = lift (#\u2191(t\u1d9c))"}, {"tactic": "letI : Fintype \u03b2 := Fintype.ofEquiv \u03b1 e", "state_before": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Finite \u03b1\ns : Set \u03b1\nt : Set \u03b2\nh1 : lift (#\u03b1) = lift (#\u03b2)\nh2 : lift (#\u2191s) = lift (#\u2191t)\nval\u271d : Fintype \u03b1\ne : \u03b1 \u2243 \u03b2\n\u22a2 lift (#\u2191(s\u1d9c)) = lift (#\u2191(t\u1d9c))", "state_after": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Finite \u03b1\ns : Set \u03b1\nt : Set \u03b2\nh1 : lift (#\u03b1) = lift (#\u03b2)\nh2 : lift (#\u2191s) = lift (#\u2191t)\nval\u271d : Fintype \u03b1\ne : \u03b1 \u2243 \u03b2\nthis : Fintype \u03b2 := Fintype.ofEquiv \u03b1 e\n\u22a2 lift (#\u2191(s\u1d9c)) = lift (#\u2191(t\u1d9c))"}, {"tactic": "replace h1 : Fintype.card \u03b1 = Fintype.card \u03b2 := (Fintype.ofEquiv_card _).symm", "state_before": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Finite \u03b1\ns : Set \u03b1\nt : Set \u03b2\nh1 : lift (#\u03b1) = lift (#\u03b2)\nh2 : lift (#\u2191s) = lift (#\u2191t)\nval\u271d : Fintype \u03b1\ne : \u03b1 \u2243 \u03b2\nthis : Fintype \u03b2 := Fintype.ofEquiv \u03b1 e\n\u22a2 lift (#\u2191(s\u1d9c)) = lift (#\u2191(t\u1d9c))", "state_after": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Finite \u03b1\ns : Set \u03b1\nt : Set \u03b2\nh2 : lift (#\u2191s) = lift (#\u2191t)\nval\u271d : Fintype \u03b1\ne : \u03b1 \u2243 \u03b2\nthis : Fintype \u03b2 := Fintype.ofEquiv \u03b1 e\nh1 : Fintype.card \u03b1 = Fintype.card \u03b2\n\u22a2 lift (#\u2191(s\u1d9c)) = lift (#\u2191(t\u1d9c))"}, {"tactic": "classical\n  lift s to Finset \u03b1 using s.toFinite\n  lift t to Finset \u03b2 using t.toFinite\n  simp only [Finset.coe_sort_coe, mk_fintype, Fintype.card_coe, lift_natCast, Nat.cast_inj] at h2\n  simp only [\u2190 Finset.coe_compl, Finset.coe_sort_coe, mk_coe_finset, Finset.card_compl,\n    lift_natCast, Nat.cast_inj, h1, h2]", "state_before": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Finite \u03b1\ns : Set \u03b1\nt : Set \u03b2\nh2 : lift (#\u2191s) = lift (#\u2191t)\nval\u271d : Fintype \u03b1\ne : \u03b1 \u2243 \u03b2\nthis : Fintype \u03b2 := Fintype.ofEquiv \u03b1 e\nh1 : Fintype.card \u03b1 = Fintype.card \u03b2\n\u22a2 lift (#\u2191(s\u1d9c)) = lift (#\u2191(t\u1d9c))", "state_after": "no goals"}, {"tactic": "lift s to Finset \u03b1 using s.toFinite", "state_before": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Finite \u03b1\ns : Set \u03b1\nt : Set \u03b2\nh2 : lift (#\u2191s) = lift (#\u2191t)\nval\u271d : Fintype \u03b1\ne : \u03b1 \u2243 \u03b2\nthis : Fintype \u03b2 := Fintype.ofEquiv \u03b1 e\nh1 : Fintype.card \u03b1 = Fintype.card \u03b2\n\u22a2 lift (#\u2191(s\u1d9c)) = lift (#\u2191(t\u1d9c))", "state_after": "case intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Finite \u03b1\nt : Set \u03b2\nval\u271d : Fintype \u03b1\ne : \u03b1 \u2243 \u03b2\nthis : Fintype \u03b2 := Fintype.ofEquiv \u03b1 e\nh1 : Fintype.card \u03b1 = Fintype.card \u03b2\ns : Finset \u03b1\nh2 : lift (#\u2191\u2191s) = lift (#\u2191t)\n\u22a2 lift (#\u2191(\u2191s\u1d9c)) = lift (#\u2191(t\u1d9c))"}, {"tactic": "lift t to Finset \u03b2 using t.toFinite", "state_before": "case intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Finite \u03b1\nt : Set \u03b2\nval\u271d : Fintype \u03b1\ne : \u03b1 \u2243 \u03b2\nthis : Fintype \u03b2 := Fintype.ofEquiv \u03b1 e\nh1 : Fintype.card \u03b1 = Fintype.card \u03b2\ns : Finset \u03b1\nh2 : lift (#\u2191\u2191s) = lift (#\u2191t)\n\u22a2 lift (#\u2191(\u2191s\u1d9c)) = lift (#\u2191(t\u1d9c))", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Finite \u03b1\nval\u271d : Fintype \u03b1\ne : \u03b1 \u2243 \u03b2\nthis : Fintype \u03b2 := Fintype.ofEquiv \u03b1 e\nh1 : Fintype.card \u03b1 = Fintype.card \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nh2 : lift (#\u2191\u2191s) = lift (#\u2191\u2191t)\n\u22a2 lift (#\u2191(\u2191s\u1d9c)) = lift (#\u2191(\u2191t\u1d9c))"}, {"tactic": "simp only [Finset.coe_sort_coe, mk_fintype, Fintype.card_coe, lift_natCast, Nat.cast_inj] at h2", "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Finite \u03b1\nval\u271d : Fintype \u03b1\ne : \u03b1 \u2243 \u03b2\nthis : Fintype \u03b2 := Fintype.ofEquiv \u03b1 e\nh1 : Fintype.card \u03b1 = Fintype.card \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nh2 : lift (#\u2191\u2191s) = lift (#\u2191\u2191t)\n\u22a2 lift (#\u2191(\u2191s\u1d9c)) = lift (#\u2191(\u2191t\u1d9c))", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Finite \u03b1\nval\u271d : Fintype \u03b1\ne : \u03b1 \u2243 \u03b2\nthis : Fintype \u03b2 := Fintype.ofEquiv \u03b1 e\nh1 : Fintype.card \u03b1 = Fintype.card \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nh2 : Finset.card s = Finset.card t\n\u22a2 lift (#\u2191(\u2191s\u1d9c)) = lift (#\u2191(\u2191t\u1d9c))"}, {"tactic": "simp only [\u2190 Finset.coe_compl, Finset.coe_sort_coe, mk_coe_finset, Finset.card_compl,\n  lift_natCast, Nat.cast_inj, h1, h2]", "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Finite \u03b1\nval\u271d : Fintype \u03b1\ne : \u03b1 \u2243 \u03b2\nthis : Fintype \u03b2 := Fintype.ofEquiv \u03b1 e\nh1 : Fintype.card \u03b1 = Fintype.card \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nh2 : Finset.card s = Finset.card t\n\u22a2 lift (#\u2191(\u2191s\u1d9c)) = lift (#\u2191(\u2191t\u1d9c))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean", "full_name": "InnerProductGeometry.norm_add_sq_eq_norm_sq_add_norm_sq_iff_angle_eq_pi_div_two", "start": [50, 1], "end": [53, 50], "traced_tactics": [{"tactic": "rw [norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero]", "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\n\u22a2 \u2016x + y\u2016 * \u2016x + y\u2016 = \u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 \u2194 angle x y = \u03c0 / 2", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\n\u22a2 inner x y = 0 \u2194 angle x y = \u03c0 / 2"}, {"tactic": "exact inner_eq_zero_iff_angle_eq_pi_div_two x y", "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\n\u22a2 inner x y = 0 \u2194 angle x y = \u03c0 / 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.erase_comm", "start": [1090, 1], "end": [1091, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Int/Parity.lean", "full_name": "Int.odd_pow", "start": [168, 1], "end": [169, 91], "traced_tactics": [{"tactic": "rw [\u2190 not_iff_not, \u2190 Int.even_iff_not_odd, not_or, \u2190 Int.even_iff_not_odd, Int.even_pow]", "state_before": "m n\u271d : \u2124\nn : \u2115\n\u22a2 Odd (m ^ n) \u2194 Odd m \u2228 n = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.inter_nonempty_iff_exists_right", "start": [527, 1], "end": [528, 37], "traced_tactics": [{"tactic": "simp_rw [inter_nonempty, and_comm]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\na b : \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 u : Set \u03b1\n\u22a2 Set.Nonempty (s \u2229 t) \u2194 \u2203 x, x \u2208 t \u2227 x \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "full_name": "MeasureTheory.AnalyticSet.image_of_continuous", "start": [141, 1], "end": [143, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/EReal.lean", "full_name": "EReal.coe_ennreal_pow", "start": [1203, 1], "end": [1204, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Localization/NumDen.lean", "full_name": "IsFractionRing.mk'_num_den'", "start": [75, 1], "end": [77, 20], "traced_tactics": [{"tactic": "rw [\u2190 mk'_eq_div]", "state_before": "R : Type ?u.14437\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type ?u.14643\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type ?u.14897\ninst\u271d\u2076 : CommRing P\nA : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_1\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\n\u22a2 \u2191(algebraMap A K) (num A x) / \u2191(algebraMap A K) \u2191(den A x) = x", "state_after": "R : Type ?u.14437\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type ?u.14643\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type ?u.14897\ninst\u271d\u2076 : CommRing P\nA : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_1\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\n\u22a2 mk' K (num A x) (den A x) = x"}, {"tactic": "apply mk'_num_den", "state_before": "R : Type ?u.14437\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type ?u.14643\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type ?u.14897\ninst\u271d\u2076 : CommRing P\nA : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_1\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\n\u22a2 mk' K (num A x) (den A 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.choose_spec", "start": [3707, 1], "end": [3708, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Ordmap/Ordset.lean", "full_name": "Ordnode.rotateL_nil", "start": [264, 1], "end": [265, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Dfinsupp/Basic.lean", "full_name": "Dfinsupp.erase_add", "start": [888, 1], "end": [889, 38], "traced_tactics": [{"tactic": "simp [ite_zero_add]", "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 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i)\ni : \u03b9\nf\u2081 f\u2082 : \u03a0\u2080 (i : \u03b9), \u03b2 i\nx\u271d : \u03b9\n\u22a2 \u2191(erase i (f\u2081 + f\u2082)) x\u271d = \u2191(erase i f\u2081 + erase i f\u2082) x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/MonoidAlgebra/Basic.lean", "full_name": "AddMonoidAlgebra.opRingEquiv_single", "start": [1870, 1], "end": [1871, 80], "traced_tactics": [{"tactic": "simp", "state_before": "k : Type u\u2081\nG : Type u\u2082\nR : Type ?u.2383656\ninst\u271d\u00b9 : Semiring k\ninst\u271d : AddCommMonoid G\nr : k\nx : G\n\u22a2 \u2191AddMonoidAlgebra.opRingEquiv (op (single x r)) = single x (op r)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.Lp.simpleFunc.toLp_zero", "start": [558, 1], "end": [559, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Nat/Gcd.lean", "full_name": "Nat.coprime_one_right_eq_true", "start": [363, 9], "end": [363, 100], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/StarSubalgebra.lean", "full_name": "StarAlgHom.ext_topologicalClosure", "start": [129, 1], "end": [146, 41], "traced_tactics": [{"tactic": "rw [FunLike.ext'_iff]", "state_before": "R : Type u_2\nA : Type u_3\nB : Type u_1\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n  StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n    StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\n\u22a2 \u03c6 = \u03c8", "state_after": "R : Type u_2\nA : Type u_3\nB : Type u_1\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n  StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n    StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\n\u22a2 \u2191\u03c6 = \u2191\u03c8"}, {"tactic": "have : Dense (Set.range <| inclusion (le_topologicalClosure S)) := by\n  refine' embedding_subtype_val.toInducing.dense_iff.2 fun x => _\n  convert show \u2191x \u2208 closure (S : Set A) from x.prop\n  rw [\u2190 Set.range_comp]\n  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": "R : Type u_2\nA : Type u_3\nB : Type u_1\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n  StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n    StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\n\u22a2 \u2191\u03c6 = \u2191\u03c8", "state_after": "R : Type u_2\nA : Type u_3\nB : Type u_1\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n  StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n    StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nthis : Dense (range \u2191(inclusion (_ : S \u2264 topologicalClosure S)))\n\u22a2 \u2191\u03c6 = \u2191\u03c8"}, {"tactic": "refine' Continuous.ext_on this h\u03c6 h\u03c8 _", "state_before": "R : Type u_2\nA : Type u_3\nB : Type u_1\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n  StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n    StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nthis : Dense (range \u2191(inclusion (_ : S \u2264 topologicalClosure S)))\n\u22a2 \u2191\u03c6 = \u2191\u03c8", "state_after": "R : Type u_2\nA : Type u_3\nB : Type u_1\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n  StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n    StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nthis : Dense (range \u2191(inclusion (_ : S \u2264 topologicalClosure S)))\n\u22a2 EqOn (\u2191\u03c6) (\u2191\u03c8) (range \u2191(inclusion (_ : S \u2264 topologicalClosure S)))"}, {"tactic": "rintro _ \u27e8x, rfl\u27e9", "state_before": "R : Type u_2\nA : Type u_3\nB : Type u_1\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n  StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n    StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nthis : Dense (range \u2191(inclusion (_ : S \u2264 topologicalClosure S)))\n\u22a2 EqOn (\u2191\u03c6) (\u2191\u03c8) (range \u2191(inclusion (_ : S \u2264 topologicalClosure S)))", "state_after": "case intro\nR : Type u_2\nA : Type u_3\nB : Type u_1\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n  StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n    StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nthis : Dense (range \u2191(inclusion (_ : S \u2264 topologicalClosure S)))\nx : { x // x \u2208 S }\n\u22a2 \u2191\u03c6 (\u2191(inclusion (_ : S \u2264 topologicalClosure S)) x) = \u2191\u03c8 (\u2191(inclusion (_ : S \u2264 topologicalClosure S)) x)"}, {"tactic": "simpa only using FunLike.congr_fun h x", "state_before": "case intro\nR : Type u_2\nA : Type u_3\nB : Type u_1\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n  StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n    StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nthis : Dense (range \u2191(inclusion (_ : S \u2264 topologicalClosure S)))\nx : { x // x \u2208 S }\n\u22a2 \u2191\u03c6 (\u2191(inclusion (_ : S \u2264 topologicalClosure S)) x) = \u2191\u03c8 (\u2191(inclusion (_ : S \u2264 topologicalClosure S)) x)", "state_after": "no goals"}, {"tactic": "refine' embedding_subtype_val.toInducing.dense_iff.2 fun x => _", "state_before": "R : Type u_2\nA : Type u_3\nB : Type u_1\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n  StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n    StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\n\u22a2 Dense (range \u2191(inclusion (_ : S \u2264 topologicalClosure S)))", "state_after": "R : Type u_2\nA : Type u_3\nB : Type u_1\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n  StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n    StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nx : { x // x \u2208 topologicalClosure S }\n\u22a2 \u2191x \u2208 closure (Subtype.val '' range \u2191(inclusion (_ : S \u2264 topologicalClosure S)))"}, {"tactic": "convert show \u2191x \u2208 closure (S : Set A) from x.prop", "state_before": "R : Type u_2\nA : Type u_3\nB : Type u_1\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n  StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n    StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nx : { x // x \u2208 topologicalClosure S }\n\u22a2 \u2191x \u2208 closure (Subtype.val '' range \u2191(inclusion (_ : S \u2264 topologicalClosure S)))", "state_after": "case h.e'_5.h.e'_3\nR : Type u_2\nA : Type u_3\nB : Type u_1\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n  StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n    StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nx : { x // x \u2208 topologicalClosure S }\n\u22a2 Subtype.val '' range \u2191(inclusion (_ : S \u2264 topologicalClosure S)) = \u2191S"}, {"tactic": "rw [\u2190 Set.range_comp]", "state_before": "case h.e'_5.h.e'_3\nR : Type u_2\nA : Type u_3\nB : Type u_1\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n  StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n    StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nx : { x // x \u2208 topologicalClosure S }\n\u22a2 Subtype.val '' range \u2191(inclusion (_ : S \u2264 topologicalClosure S)) = \u2191S", "state_after": "case h.e'_5.h.e'_3\nR : Type u_2\nA : Type u_3\nB : Type u_1\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n  StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n    StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nx : { x // x \u2208 topologicalClosure S }\n\u22a2 range (Subtype.val \u2218 \u2191(inclusion (_ : S \u2264 topologicalClosure S))) = \u2191S"}, {"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'_5.h.e'_3\nR : Type u_2\nA : Type u_3\nB : Type u_1\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n  StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n    StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nx : { x // x \u2208 topologicalClosure S }\n\u22a2 range (Subtype.val \u2218 \u2191(inclusion (_ : S \u2264 topologicalClosure S))) = \u2191S", "state_after": "no goals"}, {"tactic": "rintro \u27e8y, rfl\u27e9", "state_before": "R : Type u_2\nA : Type u_3\nB : Type u_1\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n  StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n    StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nx : { x // x \u2208 topologicalClosure S }\ny : A\n\u22a2 y \u2208 range (Subtype.val \u2218 \u2191(inclusion (_ : S \u2264 topologicalClosure S))) \u2192 y \u2208 \u2191S", "state_after": "case intro\nR : Type u_2\nA : Type u_3\nB : Type u_1\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n  StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n    StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nx : { x // x \u2208 topologicalClosure S }\ny : { x // x \u2208 S }\n\u22a2 (Subtype.val \u2218 \u2191(inclusion (_ : S \u2264 topologicalClosure S))) y \u2208 \u2191S"}, {"tactic": "exact y.prop", "state_before": "case intro\nR : Type u_2\nA : Type u_3\nB : Type u_1\ninst\u271d\u00b9\u00b3 : CommSemiring R\ninst\u271d\u00b9\u00b2 : StarRing R\ninst\u271d\u00b9\u00b9 : TopologicalSpace A\ninst\u271d\u00b9\u2070 : Semiring A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarRing A\ninst\u271d\u2077 : StarModule R A\ninst\u271d\u2076 : TopologicalSemiring A\ninst\u271d\u2075 : ContinuousStar A\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : Semiring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : StarRing B\ninst\u271d : T2Space B\nS : StarSubalgebra R A\n\u03c6 \u03c8 : { x // x \u2208 topologicalClosure S } \u2192\u22c6\u2090[R] B\nh\u03c6 : Continuous \u2191\u03c6\nh\u03c8 : Continuous \u2191\u03c8\nh :\n  StarAlgHom.comp \u03c6 (inclusion (_ : S \u2264 topologicalClosure S)) =\n    StarAlgHom.comp \u03c8 (inclusion (_ : S \u2264 topologicalClosure S))\nx : { x // x \u2208 topologicalClosure S }\ny : { x // x \u2208 S }\n\u22a2 (Subtype.val \u2218 \u2191(inclusion (_ : S \u2264 topologicalClosure S))) y \u2208 \u2191S", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Synonym.lean", "full_name": "OrderDual.ofDual_toDual", "start": [76, 1], "end": [77, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subsemigroup/Operations.lean", "full_name": "MulMemClass.mk_mul_mk", "start": [540, 1], "end": [542, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Extr.lean", "full_name": "IsMaxOn.add", "start": [469, 1], "end": [470, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.mem_range_le", "start": [3033, 1], "end": [3034, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Acc.rank_eq", "start": [2538, 1], "end": [2541, 6], "traced_tactics": [{"tactic": "change (Acc.intro a fun _ => h.inv).rank = _", "state_before": "\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nh : Acc r a\n\u22a2 rank h = sup fun b => succ (rank (_ : Acc r \u2191b))", "state_after": "\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nh : Acc r a\n\u22a2 rank (_ : Acc (fun x => r x) a) = sup fun b => succ (rank (_ : Acc r \u2191b))"}, {"tactic": "rfl", "state_before": "\u03b1 : Type u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nh : Acc r a\n\u22a2 rank (_ : Acc (fun x => r x) a) = sup fun b => succ (rank (_ : Acc r \u2191b))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/LucasLehmer.lean", "full_name": "LucasLehmer.\u03c9_pow_eq_one", "start": [449, 1], "end": [455, 21], "traced_tactics": [{"tactic": "rw [\u2190 pow_mul, \u2190 Nat.pow_succ]", "state_before": "p' : \u2115\nh : lucasLehmerResidue (p' + 2) = 0\n\u22a2 \u03c9 ^ 2 ^ (p' + 2) = (\u03c9 ^ 2 ^ (p' + 1)) ^ 2", "state_after": "no goals"}, {"tactic": "rw [\u03c9_pow_eq_neg_one p' h]", "state_before": "p' : \u2115\nh : lucasLehmerResidue (p' + 2) = 0\n\u22a2 (\u03c9 ^ 2 ^ (p' + 1)) ^ 2 = (-1) ^ 2", "state_after": "no goals"}, {"tactic": "simp", "state_before": "p' : \u2115\nh : lucasLehmerResidue (p' + 2) = 0\n\u22a2 (-1) ^ 2 = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.ae_mono", "start": [2693, 1], "end": [2694, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean", "full_name": "ContinuousAffineMap.coe_contLinear_eq_linear", "start": [74, 1], "end": [75, 71], "traced_tactics": [{"tactic": "ext", "state_before": "\ud835\udd5c : Type ?u.13843\nR : Type u_1\nV : Type u_2\nW : Type u_3\nW\u2082 : Type ?u.13855\nP : Type u_4\nQ : Type u_5\nQ\u2082 : Type ?u.13864\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\n\u22a2 \u2191(contLinear f) = f.linear", "state_after": "case h\n\ud835\udd5c : Type ?u.13843\nR : Type u_1\nV : Type u_2\nW : Type u_3\nW\u2082 : Type ?u.13855\nP : Type u_4\nQ : Type u_5\nQ\u2082 : Type ?u.13864\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\nx\u271d : V\n\u22a2 \u2191\u2191(contLinear f) x\u271d = \u2191f.linear x\u271d"}, {"tactic": "rfl", "state_before": "case h\n\ud835\udd5c : Type ?u.13843\nR : Type u_1\nV : Type u_2\nW : Type u_3\nW\u2082 : Type ?u.13855\nP : Type u_4\nQ : Type u_5\nQ\u2082 : Type ?u.13864\ninst\u271d\u00b9\u2076 : NormedAddCommGroup V\ninst\u271d\u00b9\u2075 : MetricSpace P\ninst\u271d\u00b9\u2074 : NormedAddTorsor V P\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup W\ninst\u271d\u00b9\u00b2 : MetricSpace Q\ninst\u271d\u00b9\u00b9 : NormedAddTorsor W Q\ninst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082\ninst\u271d\u2079 : MetricSpace Q\u2082\ninst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082\ninst\u271d\u2077 : NormedField R\ninst\u271d\u2076 : NormedSpace R V\ninst\u271d\u2075 : NormedSpace R W\ninst\u271d\u2074 : NormedSpace R W\u2082\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : NormedSpace \ud835\udd5c W\u2082\nf : P \u2192A[R] Q\nx\u271d : V\n\u22a2 \u2191\u2191(contLinear f) x\u271d = \u2191f.linear x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Pairing.lean", "full_name": "Nat.unpair_pair", "start": [68, 1], "end": [77, 50], "traced_tactics": [{"tactic": "dsimp only [pair]", "state_before": "a b : \u2115\n\u22a2 unpair (pair a b) = (a, b)", "state_after": "a b : \u2115\n\u22a2 unpair (if a < b then b * b + a else a * a + a + b) = (a, b)"}, {"tactic": "split_ifs with h", "state_before": "a b : \u2115\n\u22a2 unpair (if a < b then b * b + a else a * a + a + b) = (a, b)", "state_after": "case inl\na b : \u2115\nh : a < b\n\u22a2 unpair (b * b + a) = (a, b)\n\ncase inr\na b : \u2115\nh : \u00aca < b\n\u22a2 unpair (a * a + a + b) = (a, b)"}, {"tactic": "show unpair (b * b + a) = (a, b)", "state_before": "case inl\na b : \u2115\nh : a < b\n\u22a2 unpair (b * b + a) = (a, b)", "state_after": "case inl\na b : \u2115\nh : a < b\n\u22a2 unpair (b * b + a) = (a, b)"}, {"tactic": "have be : sqrt (b * b + a) = b := sqrt_add_eq _ (le_trans (le_of_lt h) (Nat.le_add_left _ _))", "state_before": "case inl\na b : \u2115\nh : a < b\n\u22a2 unpair (b * b + a) = (a, b)", "state_after": "case inl\na b : \u2115\nh : a < b\nbe : sqrt (b * b + a) = b\n\u22a2 unpair (b * b + a) = (a, b)"}, {"tactic": "simp [unpair, be, add_tsub_cancel_right, h]", "state_before": "case inl\na b : \u2115\nh : a < b\nbe : sqrt (b * b + a) = b\n\u22a2 unpair (b * b + a) = (a, b)", "state_after": "no goals"}, {"tactic": "show unpair (a * a + a + b) = (a, b)", "state_before": "case inr\na b : \u2115\nh : \u00aca < b\n\u22a2 unpair (a * a + a + b) = (a, b)", "state_after": "case inr\na b : \u2115\nh : \u00aca < b\n\u22a2 unpair (a * a + a + b) = (a, b)"}, {"tactic": "have ae : sqrt (a * a + (a + b)) = a := by\n  rw [sqrt_add_eq]\n  exact add_le_add_left (le_of_not_gt h) _", "state_before": "case inr\na b : \u2115\nh : \u00aca < b\n\u22a2 unpair (a * a + a + b) = (a, b)", "state_after": "case inr\na b : \u2115\nh : \u00aca < b\nae : sqrt (a * a + (a + b)) = a\n\u22a2 unpair (a * a + a + b) = (a, b)"}, {"tactic": "simp [unpair, ae, Nat.not_lt_zero, add_assoc]", "state_before": "case inr\na b : \u2115\nh : \u00aca < b\nae : sqrt (a * a + (a + b)) = a\n\u22a2 unpair (a * a + a + b) = (a, b)", "state_after": "no goals"}, {"tactic": "rw [sqrt_add_eq]", "state_before": "a b : \u2115\nh : \u00aca < b\n\u22a2 sqrt (a * a + (a + b)) = a", "state_after": "case h\na b : \u2115\nh : \u00aca < b\n\u22a2 a + b \u2264 a + a"}, {"tactic": "exact add_le_add_left (le_of_not_gt h) _", "state_before": "case h\na b : \u2115\nh : \u00aca < b\n\u22a2 a + b \u2264 a + a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/OrderOfElement.lean", "full_name": "orderOf_dvd_nat_card", "start": [927, 1], "end": [930, 51], "traced_tactics": [{"tactic": "cases' fintypeOrInfinite G with h h", "state_before": "G\u271d : Type u\nA : Type v\nx\u271d y : G\u271d\na b : A\nn m : \u2115\ninst\u271d\u00b3 : Group G\u271d\ninst\u271d\u00b2 : AddGroup A\ninst\u271d\u00b9 : Fintype G\u271d\nG : Type u_1\ninst\u271d : Group G\nx : G\n\u22a2 orderOf x \u2223 Nat.card G", "state_after": "case inl\nG\u271d : Type u\nA : Type v\nx\u271d y : G\u271d\na b : A\nn m : \u2115\ninst\u271d\u00b3 : Group G\u271d\ninst\u271d\u00b2 : AddGroup A\ninst\u271d\u00b9 : Fintype G\u271d\nG : Type u_1\ninst\u271d : Group G\nx : G\nh : Fintype G\n\u22a2 orderOf x \u2223 Nat.card G\n\ncase inr\nG\u271d : Type u\nA : Type v\nx\u271d y : G\u271d\na b : A\nn m : \u2115\ninst\u271d\u00b3 : Group G\u271d\ninst\u271d\u00b2 : AddGroup A\ninst\u271d\u00b9 : Fintype G\u271d\nG : Type u_1\ninst\u271d : Group G\nx : G\nh : Infinite G\n\u22a2 orderOf x \u2223 Nat.card G"}, {"tactic": "simp only [Nat.card_eq_fintype_card, orderOf_dvd_card_univ]", "state_before": "case inl\nG\u271d : Type u\nA : Type v\nx\u271d y : G\u271d\na b : A\nn m : \u2115\ninst\u271d\u00b3 : Group G\u271d\ninst\u271d\u00b2 : AddGroup A\ninst\u271d\u00b9 : Fintype G\u271d\nG : Type u_1\ninst\u271d : Group G\nx : G\nh : Fintype G\n\u22a2 orderOf x \u2223 Nat.card G", "state_after": "no goals"}, {"tactic": "simp only [card_eq_zero_of_infinite, dvd_zero]", "state_before": "case inr\nG\u271d : Type u\nA : Type v\nx\u271d y : G\u271d\na b : A\nn m : \u2115\ninst\u271d\u00b3 : Group G\u271d\ninst\u271d\u00b2 : AddGroup A\ninst\u271d\u00b9 : Fintype G\u271d\nG : Type u_1\ninst\u271d : Group G\nx : G\nh : Infinite G\n\u22a2 orderOf x \u2223 Nat.card G", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/CharP/Basic.lean", "full_name": "RingHom.iterate_map_frobenius", "start": [390, 1], "end": [392, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Basic.lean", "full_name": "Set.Ioi_subset_Ioc_union_Ici", "start": [1324, 1], "end": [1325, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Basic.lean", "full_name": "not_imp", "start": [424, 1], "end": [424, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/SpecificGroups/Dihedral.lean", "full_name": "DihedralGroup.orderOf_r_one", "start": [165, 1], "end": [179, 44], "traced_tactics": [{"tactic": "rcases eq_zero_or_neZero n with (rfl | hn)", "state_before": "n : \u2115\n\u22a2 orderOf (r 1) = n", "state_after": "case inl\n\n\u22a2 orderOf (r 1) = 0\n\ncase inr\nn : \u2115\nhn : NeZero n\n\u22a2 orderOf (r 1) = n"}, {"tactic": "rw [orderOf_eq_zero_iff']", "state_before": "case inl\n\n\u22a2 orderOf (r 1) = 0", "state_after": "case inl\n\n\u22a2 \u2200 (n : \u2115), 0 < n \u2192 r 1 ^ n \u2260 1"}, {"tactic": "intro n hn", "state_before": "case inl\n\n\u22a2 \u2200 (n : \u2115), 0 < n \u2192 r 1 ^ n \u2260 1", "state_after": "case inl\nn : \u2115\nhn : 0 < n\n\u22a2 r 1 ^ n \u2260 1"}, {"tactic": "rw [r_one_pow, one_def]", "state_before": "case inl\nn : \u2115\nhn : 0 < n\n\u22a2 r 1 ^ n \u2260 1", "state_after": "case inl\nn : \u2115\nhn : 0 < n\n\u22a2 r \u2191n \u2260 r 0"}, {"tactic": "apply mt r.inj", "state_before": "case inl\nn : \u2115\nhn : 0 < n\n\u22a2 r \u2191n \u2260 r 0", "state_after": "case inl\nn : \u2115\nhn : 0 < n\n\u22a2 \u00ac\u2191n = 0"}, {"tactic": "simpa using hn.ne'", "state_before": "case inl\nn : \u2115\nhn : 0 < n\n\u22a2 \u00ac\u2191n = 0", "state_after": "no goals"}, {"tactic": "apply (Nat.le_of_dvd (NeZero.pos n) <|\n  orderOf_dvd_of_pow_eq_one <| @r_one_pow_n n).lt_or_eq.resolve_left", "state_before": "case inr\nn : \u2115\nhn : NeZero n\n\u22a2 orderOf (r 1) = n", "state_after": "case inr\nn : \u2115\nhn : NeZero n\n\u22a2 \u00acorderOf (r 1) < n"}, {"tactic": "intro h", "state_before": "case inr\nn : \u2115\nhn : NeZero n\n\u22a2 \u00acorderOf (r 1) < n", "state_after": "case inr\nn : \u2115\nhn : NeZero n\nh : orderOf (r 1) < n\n\u22a2 False"}, {"tactic": "have h1 : (r 1 : DihedralGroup n) ^ orderOf (r 1) = 1 := pow_orderOf_eq_one _", "state_before": "case inr\nn : \u2115\nhn : NeZero n\nh : orderOf (r 1) < n\n\u22a2 False", "state_after": "case inr\nn : \u2115\nhn : NeZero n\nh : orderOf (r 1) < n\nh1 : r 1 ^ orderOf (r 1) = 1\n\u22a2 False"}, {"tactic": "rw [r_one_pow] at h1", "state_before": "case inr\nn : \u2115\nhn : NeZero n\nh : orderOf (r 1) < n\nh1 : r 1 ^ orderOf (r 1) = 1\n\u22a2 False", "state_after": "case inr\nn : \u2115\nhn : NeZero n\nh : orderOf (r 1) < n\nh1 : r \u2191(orderOf (r 1)) = 1\n\u22a2 False"}, {"tactic": "injection h1 with h2", "state_before": "case inr\nn : \u2115\nhn : NeZero n\nh : orderOf (r 1) < n\nh1 : r \u2191(orderOf (r 1)) = 1\n\u22a2 False", "state_after": "case inr\nn : \u2115\nhn : NeZero n\nh : orderOf (r 1) < n\nh2 : \u2191(orderOf (r 1)) = 0\n\u22a2 False"}, {"tactic": "rw [\u2190 ZMod.val_eq_zero, ZMod.val_nat_cast, Nat.mod_eq_of_lt h] at h2", "state_before": "case inr\nn : \u2115\nhn : NeZero n\nh : orderOf (r 1) < n\nh2 : \u2191(orderOf (r 1)) = 0\n\u22a2 False", "state_after": "case inr\nn : \u2115\nhn : NeZero n\nh : orderOf (r 1) < n\nh2 : orderOf (r 1) = 0\n\u22a2 False"}, {"tactic": "exact absurd h2.symm (orderOf_pos _).ne", "state_before": "case inr\nn : \u2115\nhn : NeZero n\nh : orderOf (r 1) < n\nh2 : orderOf (r 1) = 0\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Sups.lean", "full_name": "Finset.disjSups_inter_subset_right", "start": [513, 1], "end": [514, 92], "traced_tactics": [{"tactic": "simpa only [disjSups, product_inter, filter_inter_distrib] using image_inter_subset _ _ _", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : DecidableRel Disjoint\ns s\u2081 s\u2082 t t\u2081 t\u2082 u : Finset \u03b1\na b c : \u03b1\n\u22a2 s \u25cb (t\u2081 \u2229 t\u2082) \u2286 s \u25cb t\u2081 \u2229 s \u25cb t\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/NatAntidiagonal.lean", "full_name": "List.Nat.antidiagonal_succ_succ'", "start": [88, 1], "end": [94, 7], "traced_tactics": [{"tactic": "rw [antidiagonal_succ']", "state_before": "n : \u2115\n\u22a2 antidiagonal (n + 2) = (0, n + 2) :: map (Prod.map succ succ) (antidiagonal n) ++ [(n + 2, 0)]", "state_after": "n : \u2115\n\u22a2 map (Prod.map id succ) (antidiagonal (n + 1)) ++ [(n + 1 + 1, 0)] =\n    (0, n + 2) :: map (Prod.map succ succ) (antidiagonal n) ++ [(n + 2, 0)]"}, {"tactic": "simp", "state_before": "n : \u2115\n\u22a2 map (Prod.map id succ) (antidiagonal (n + 1)) ++ [(n + 1 + 1, 0)] =\n    (0, n + 2) :: map (Prod.map succ succ) (antidiagonal n) ++ [(n + 2, 0)]", "state_after": "n : \u2115\n\u22a2 map (Prod.map id succ \u2218 Prod.map succ id) (antidiagonal n) = map (Prod.map succ succ) (antidiagonal n)"}, {"tactic": "ext", "state_before": "n : \u2115\n\u22a2 map (Prod.map id succ \u2218 Prod.map succ id) (antidiagonal n) = map (Prod.map succ succ) (antidiagonal n)", "state_after": "case a.a\nn n\u271d : \u2115\na\u271d : \u2115 \u00d7 \u2115\n\u22a2 a\u271d \u2208 get? (map (Prod.map id succ \u2218 Prod.map succ id) (antidiagonal n)) n\u271d \u2194\n    a\u271d \u2208 get? (map (Prod.map succ succ) (antidiagonal n)) n\u271d"}, {"tactic": "simp", "state_before": "case a.a\nn n\u271d : \u2115\na\u271d : \u2115 \u00d7 \u2115\n\u22a2 a\u271d \u2208 get? (map (Prod.map id succ \u2218 Prod.map succ id) (antidiagonal n)) n\u271d \u2194\n    a\u271d \u2208 get? (map (Prod.map succ succ) (antidiagonal n)) n\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Seminorm.lean", "full_name": "Seminorm.balanced_ball_zero", "start": [839, 1], "end": [844, 42], "traced_tactics": [{"tactic": "rintro a ha x \u27e8y, hy, hx\u27e9", "state_before": "R : Type ?u.1051890\nR' : Type ?u.1051893\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type ?u.1051899\n\ud835\udd5c\u2083 : Type ?u.1051902\n\ud835\udd5d : Type ?u.1051905\nE : Type u_2\nE\u2082 : Type ?u.1051911\nE\u2083 : Type ?u.1051914\nF : Type ?u.1051917\nG : Type ?u.1051920\n\u03b9 : Type ?u.1051923\ninst\u271d\u2076 : SeminormedRing \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b2 : AddCommGroup E\u2082\ninst\u271d\u00b9 : Module \ud835\udd5c\u2082 E\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\np : Seminorm \ud835\udd5c E\nr : \u211d\n\u22a2 Balanced \ud835\udd5c (ball p 0 r)", "state_after": "case intro.intro\nR : Type ?u.1051890\nR' : Type ?u.1051893\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type ?u.1051899\n\ud835\udd5c\u2083 : Type ?u.1051902\n\ud835\udd5d : Type ?u.1051905\nE : Type u_2\nE\u2082 : Type ?u.1051911\nE\u2083 : Type ?u.1051914\nF : Type ?u.1051917\nG : Type ?u.1051920\n\u03b9 : Type ?u.1051923\ninst\u271d\u2076 : SeminormedRing \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b2 : AddCommGroup E\u2082\ninst\u271d\u00b9 : Module \ud835\udd5c\u2082 E\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\np : Seminorm \ud835\udd5c E\nr : \u211d\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\nx y : E\nhy : y \u2208 ball p 0 r\nhx : (fun x => a \u2022 x) y = x\n\u22a2 x \u2208 ball p 0 r"}, {"tactic": "rw [mem_ball_zero, \u2190 hx, map_smul_eq_mul]", "state_before": "case intro.intro\nR : Type ?u.1051890\nR' : Type ?u.1051893\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type ?u.1051899\n\ud835\udd5c\u2083 : Type ?u.1051902\n\ud835\udd5d : Type ?u.1051905\nE : Type u_2\nE\u2082 : Type ?u.1051911\nE\u2083 : Type ?u.1051914\nF : Type ?u.1051917\nG : Type ?u.1051920\n\u03b9 : Type ?u.1051923\ninst\u271d\u2076 : SeminormedRing \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b2 : AddCommGroup E\u2082\ninst\u271d\u00b9 : Module \ud835\udd5c\u2082 E\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\np : Seminorm \ud835\udd5c E\nr : \u211d\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\nx y : E\nhy : y \u2208 ball p 0 r\nhx : (fun x => a \u2022 x) y = x\n\u22a2 x \u2208 ball p 0 r", "state_after": "case intro.intro\nR : Type ?u.1051890\nR' : Type ?u.1051893\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type ?u.1051899\n\ud835\udd5c\u2083 : Type ?u.1051902\n\ud835\udd5d : Type ?u.1051905\nE : Type u_2\nE\u2082 : Type ?u.1051911\nE\u2083 : Type ?u.1051914\nF : Type ?u.1051917\nG : Type ?u.1051920\n\u03b9 : Type ?u.1051923\ninst\u271d\u2076 : SeminormedRing \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b2 : AddCommGroup E\u2082\ninst\u271d\u00b9 : Module \ud835\udd5c\u2082 E\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\np : Seminorm \ud835\udd5c E\nr : \u211d\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\nx y : E\nhy : y \u2208 ball p 0 r\nhx : (fun x => a \u2022 x) y = x\n\u22a2 \u2016a\u2016 * \u2191p y < r"}, {"tactic": "calc\n  _ \u2264 p y := mul_le_of_le_one_left (map_nonneg p _) ha\n  _ < r := by rwa [mem_ball_zero] at hy", "state_before": "case intro.intro\nR : Type ?u.1051890\nR' : Type ?u.1051893\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type ?u.1051899\n\ud835\udd5c\u2083 : Type ?u.1051902\n\ud835\udd5d : Type ?u.1051905\nE : Type u_2\nE\u2082 : Type ?u.1051911\nE\u2083 : Type ?u.1051914\nF : Type ?u.1051917\nG : Type ?u.1051920\n\u03b9 : Type ?u.1051923\ninst\u271d\u2076 : SeminormedRing \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b2 : AddCommGroup E\u2082\ninst\u271d\u00b9 : Module \ud835\udd5c\u2082 E\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\np : Seminorm \ud835\udd5c E\nr : \u211d\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\nx y : E\nhy : y \u2208 ball p 0 r\nhx : (fun x => a \u2022 x) y = x\n\u22a2 \u2016a\u2016 * \u2191p y < r", "state_after": "no goals"}, {"tactic": "rwa [mem_ball_zero] at hy", "state_before": "R : Type ?u.1051890\nR' : Type ?u.1051893\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type ?u.1051899\n\ud835\udd5c\u2083 : Type ?u.1051902\n\ud835\udd5d : Type ?u.1051905\nE : Type u_2\nE\u2082 : Type ?u.1051911\nE\u2083 : Type ?u.1051914\nF : Type ?u.1051917\nG : Type ?u.1051920\n\u03b9 : Type ?u.1051923\ninst\u271d\u2076 : SeminormedRing \ud835\udd5c\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module \ud835\udd5c E\ninst\u271d\u00b3 : SeminormedRing \ud835\udd5c\u2082\ninst\u271d\u00b2 : AddCommGroup E\u2082\ninst\u271d\u00b9 : Module \ud835\udd5c\u2082 E\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\np : Seminorm \ud835\udd5c E\nr : \u211d\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\nx y : E\nhy : y \u2208 ball p 0 r\nhx : (fun x => a \u2022 x) y = x\n\u22a2 \u2191p y < r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/ConstMulAction.lean", "full_name": "IsUnit.continuousWithinAt_const_smul_iff", "start": [409, 8], "end": [412, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/SmallSets.lean", "full_name": "Filter.comap_smallSets", "start": [118, 1], "end": [120, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finite/Defs.lean", "full_name": "not_finite_iff_infinite", "start": [104, 1], "end": [105, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Instances/Int.lean", "full_name": "Int.preimage_closedBall", "start": [59, 1], "end": [59, 99], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Basic.lean", "full_name": "Polynomial.C_inj", "start": [758, 1], "end": [759, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Subring/Basic.lean", "full_name": "RingHom.eqLocus_same", "start": [1230, 1], "end": [1231, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "full_name": "gcd_mul_left'", "start": [449, 1], "end": [460, 101], "traced_tactics": [{"tactic": "obtain rfl | ha := eq_or_ne a 0", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\n\u22a2 Associated (gcd (a * b) (a * c)) (a * gcd b c)", "state_after": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nb c : \u03b1\n\u22a2 Associated (gcd (0 * b) (0 * c)) (0 * gcd b c)\n\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nha : a \u2260 0\n\u22a2 Associated (gcd (a * b) (a * c)) (a * gcd b c)"}, {"tactic": "obtain \u27e8d, eq\u27e9 := dvd_gcd (dvd_mul_right a b) (dvd_mul_right a c)", "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nha : a \u2260 0\n\u22a2 Associated (gcd (a * b) (a * c)) (a * gcd b c)", "state_after": "case inr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nha : a \u2260 0\nd : \u03b1\neq : gcd (a * b) (a * c) = a * d\n\u22a2 Associated (gcd (a * b) (a * c)) (a * gcd b c)"}, {"tactic": "apply associated_of_dvd_dvd", "state_before": "case inr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nha : a \u2260 0\nd : \u03b1\neq : gcd (a * b) (a * c) = a * d\n\u22a2 Associated (gcd (a * b) (a * c)) (a * gcd b c)", "state_after": "case inr.intro.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nha : a \u2260 0\nd : \u03b1\neq : gcd (a * b) (a * c) = a * d\n\u22a2 gcd (a * b) (a * c) \u2223 a * gcd b c\n\ncase inr.intro.hba\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nha : a \u2260 0\nd : \u03b1\neq : gcd (a * b) (a * c) = a * d\n\u22a2 a * gcd b c \u2223 gcd (a * b) (a * c)"}, {"tactic": "simp only [zero_mul, gcd_zero_left']", "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\nb c : \u03b1\n\u22a2 Associated (gcd (0 * b) (0 * c)) (0 * gcd b c)", "state_after": "no goals"}, {"tactic": "rw [eq]", "state_before": "case inr.intro.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nha : a \u2260 0\nd : \u03b1\neq : gcd (a * b) (a * c) = a * d\n\u22a2 gcd (a * b) (a * c) \u2223 a * gcd b c", "state_after": "case inr.intro.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nha : a \u2260 0\nd : \u03b1\neq : gcd (a * b) (a * c) = a * d\n\u22a2 a * d \u2223 a * gcd b c"}, {"tactic": "apply mul_dvd_mul_left", "state_before": "case inr.intro.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nha : a \u2260 0\nd : \u03b1\neq : gcd (a * b) (a * c) = a * d\n\u22a2 a * d \u2223 a * gcd b c", "state_after": "case inr.intro.hab.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nha : a \u2260 0\nd : \u03b1\neq : gcd (a * b) (a * c) = a * d\n\u22a2 d \u2223 gcd b c"}, {"tactic": "exact\n  dvd_gcd ((mul_dvd_mul_iff_left ha).1 <| eq \u25b8 gcd_dvd_left _ _)\n    ((mul_dvd_mul_iff_left ha).1 <| eq \u25b8 gcd_dvd_right _ _)", "state_before": "case inr.intro.hab.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nha : a \u2260 0\nd : \u03b1\neq : gcd (a * b) (a * c) = a * d\n\u22a2 d \u2223 gcd b c", "state_after": "no goals"}, {"tactic": "exact dvd_gcd (mul_dvd_mul_left a <| gcd_dvd_left _ _) (mul_dvd_mul_left a <| gcd_dvd_right _ _)", "state_before": "case inr.intro.hba\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : GCDMonoid \u03b1\na b c : \u03b1\nha : a \u2260 0\nd : \u03b1\neq : gcd (a * b) (a * c) = a * d\n\u22a2 a * gcd b c \u2223 gcd (a * b) (a * c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Quotient.lean", "full_name": "LinearMap.ker_le_range_iff", "start": [590, 1], "end": [592, 70], "traced_tactics": [{"tactic": "rw [\u2190 range_le_ker_iff, Submodule.ker_mkQ, Submodule.range_subtype]", "state_before": "R : Type u_1\nM : Type u_3\nR\u2082 : Type u_2\nM\u2082 : Type u_4\nR\u2083 : Type u_5\nM\u2083 : Type u_6\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\ng : M\u2082 \u2192\u209b\u2097[\u03c4\u2082\u2083] M\u2083\n\u22a2 ker g \u2264 range f \u2194 comp (mkQ (range f)) (Submodule.subtype (ker g)) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "full_name": "LinearMap.toMatrix_comp", "start": [646, 1], "end": [650, 30], "traced_tactics": [{"tactic": "simp_rw [LinearMap.toMatrix, LinearEquiv.trans_apply, LinearEquiv.arrowCongr_comp _ v\u2082.equivFun,\n  LinearMap.toMatrix'_comp]", "state_before": "R : Type u_3\ninst\u271d\u00b9\u00b9 : CommSemiring R\nl : Type u_1\nm : Type u_2\nn : Type u_7\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : Fintype m\ninst\u271d\u2078 : DecidableEq n\nM\u2081 : Type u_6\nM\u2082 : Type u_4\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R M\u2081\ninst\u271d\u2074 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_5\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\u2083\nv\u2083 : Basis l R M\u2083\ninst\u271d\u00b9 : Fintype l\ninst\u271d : DecidableEq m\nf : M\u2082 \u2192\u2097[R] M\u2083\ng : M\u2081 \u2192\u2097[R] M\u2082\n\u22a2 \u2191(toMatrix v\u2081 v\u2083) (comp f g) = \u2191(toMatrix v\u2082 v\u2083) f \u2b1d \u2191(toMatrix v\u2081 v\u2082) g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Monotone/Extension.lean", "full_name": "AntitoneOn.exists_antitone_extension", "start": [56, 1], "end": [58, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/MvPolynomial/Rename.lean", "full_name": "MvPolynomial.rename_monomial", "start": [101, 1], "end": [107, 39], "traced_tactics": [{"tactic": "rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),\n  Finsupp.prod_mapDomain_index]", "state_before": "\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03b1 : Type ?u.227167\nR : Type u_3\nS : Type ?u.227173\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : \u03c3 \u2192 \u03c4\nd : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 \u2191(rename f) (\u2191(monomial d) r) = \u2191(monomial (Finsupp.mapDomain f d)) r", "state_after": "\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03b1 : Type ?u.227167\nR : Type u_3\nS : Type ?u.227173\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : \u03c3 \u2192 \u03c4\nd : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 (\u2191(algebraMap R (MvPolynomial \u03c4 R)) r * Finsupp.prod d fun i k => (X \u2218 f) i ^ k) =\n    \u2191C r * Finsupp.prod d fun a m => X (f a) ^ m\n\ncase h_zero\n\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03b1 : Type ?u.227167\nR : Type u_3\nS : Type ?u.227173\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : \u03c3 \u2192 \u03c4\nd : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 \u2200 (b : \u03c4), X b ^ 0 = 1\n\ncase h_add\n\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03b1 : Type ?u.227167\nR : Type u_3\nS : Type ?u.227173\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : \u03c3 \u2192 \u03c4\nd : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 \u2200 (b : \u03c4) (m\u2081 m\u2082 : \u2115), X b ^ (m\u2081 + m\u2082) = X b ^ m\u2081 * X b ^ m\u2082"}, {"tactic": "rfl", "state_before": "\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03b1 : Type ?u.227167\nR : Type u_3\nS : Type ?u.227173\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : \u03c3 \u2192 \u03c4\nd : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 (\u2191(algebraMap R (MvPolynomial \u03c4 R)) r * Finsupp.prod d fun i k => (X \u2218 f) i ^ k) =\n    \u2191C r * Finsupp.prod d fun a m => X (f a) ^ m", "state_after": "no goals"}, {"tactic": "exact fun n => pow_zero _", "state_before": "case h_zero\n\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03b1 : Type ?u.227167\nR : Type u_3\nS : Type ?u.227173\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : \u03c3 \u2192 \u03c4\nd : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 \u2200 (b : \u03c4), X b ^ 0 = 1", "state_after": "no goals"}, {"tactic": "exact fun n i\u2081 i\u2082 => pow_add _ _ _", "state_before": "case h_add\n\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03b1 : Type ?u.227167\nR : Type u_3\nS : Type ?u.227173\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : \u03c3 \u2192 \u03c4\nd : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 \u2200 (b : \u03c4) (m\u2081 m\u2082 : \u2115), X b ^ (m\u2081 + m\u2082) = X b ^ m\u2081 * X b ^ m\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Hom/Units.lean", "full_name": "IsUnit.div_eq_one_iff_eq", "start": [435, 11], "end": [436, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Sylow.lean", "full_name": "QuotientGroup.card_preimage_mk", "start": [499, 1], "end": [501, 84], "traced_tactics": [{"tactic": "rw [\u2190 Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]", "state_before": "G : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\ns : Subgroup G\nt : Set (G \u29f8 s)\n\u22a2 Fintype.card \u2191(mk \u207b\u00b9' t) = Fintype.card { x // x \u2208 s } * Fintype.card \u2191t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Basic.lean", "full_name": "SimpleGraph.incidence_other_neighbor_edge", "start": [1075, 1], "end": [1077, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "measurableSet_closedBall", "start": [1528, 1], "end": [1529, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.eqOn_piecewise", "start": [1452, 1], "end": [1455, 64], "traced_tactics": [{"tactic": "simp only [EqOn, \u2190 forall_and]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.85260\n\u03b9 : Sort ?u.85263\n\u03c0 : \u03b1 \u2192 Type ?u.85268\n\u03b4 : \u03b1 \u2192 Sort ?u.85273\ns : Set \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\nf f' g : \u03b1 \u2192 \u03b2\nt : Set \u03b1\n\u22a2 EqOn (piecewise s f f') g t \u2194 EqOn f g (t \u2229 s) \u2227 EqOn f' g (t \u2229 s\u1d9c)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.85260\n\u03b9 : Sort ?u.85263\n\u03c0 : \u03b1 \u2192 Type ?u.85268\n\u03b4 : \u03b1 \u2192 Sort ?u.85273\ns : Set \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\nf f' g : \u03b1 \u2192 \u03b2\nt : Set \u03b1\n\u22a2 (\u2200 \u2983x : \u03b1\u2984, x \u2208 t \u2192 piecewise s f f' x = g x) \u2194 \u2200 (x : \u03b1), (x \u2208 t \u2229 s \u2192 f x = g x) \u2227 (x \u2208 t \u2229 s\u1d9c \u2192 f' x = g x)"}, {"tactic": "refine' forall_congr' fun a => _", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.85260\n\u03b9 : Sort ?u.85263\n\u03c0 : \u03b1 \u2192 Type ?u.85268\n\u03b4 : \u03b1 \u2192 Sort ?u.85273\ns : Set \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\nf f' g : \u03b1 \u2192 \u03b2\nt : Set \u03b1\n\u22a2 (\u2200 \u2983x : \u03b1\u2984, x \u2208 t \u2192 piecewise s f f' x = g x) \u2194 \u2200 (x : \u03b1), (x \u2208 t \u2229 s \u2192 f x = g x) \u2227 (x \u2208 t \u2229 s\u1d9c \u2192 f' x = g x)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.85260\n\u03b9 : Sort ?u.85263\n\u03c0 : \u03b1 \u2192 Type ?u.85268\n\u03b4 : \u03b1 \u2192 Sort ?u.85273\ns : Set \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\nf f' g : \u03b1 \u2192 \u03b2\nt : Set \u03b1\na : \u03b1\n\u22a2 a \u2208 t \u2192 piecewise s f f' a = g a \u2194 (a \u2208 t \u2229 s \u2192 f a = g a) \u2227 (a \u2208 t \u2229 s\u1d9c \u2192 f' a = g a)"}, {"tactic": "by_cases a \u2208 s <;> simp [*]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.85260\n\u03b9 : Sort ?u.85263\n\u03c0 : \u03b1 \u2192 Type ?u.85268\n\u03b4 : \u03b1 \u2192 Sort ?u.85273\ns : Set \u03b1\nf\u271d g\u271d : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s)\nf f' g : \u03b1 \u2192 \u03b2\nt : Set \u03b1\na : \u03b1\n\u22a2 a \u2208 t \u2192 piecewise s f f' a = g a \u2194 (a \u2208 t \u2229 s \u2192 f a = g a) \u2227 (a \u2208 t \u2229 s\u1d9c \u2192 f' a = g a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/PiLp.lean", "full_name": "PiLp.nnnorm_equiv_symm_one", "start": [930, 1], "end": [933, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "full_name": "Ordinal.nat_cast_succ", "start": [1105, 1], "end": [1106, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/IsROrC/Basic.lean", "full_name": "IsROrC.conj_eq_iff_real", "start": [424, 1], "end": [425, 61], "traced_tactics": [{"tactic": "simp only [eq_comm]", "state_before": "K : Type u_1\nE : Type ?u.3717020\ninst\u271d : IsROrC K\nz : K\n\u22a2 (\u2203 r, \u2191r = z) \u2194 \u2203 r, z = \u2191r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/FreeAlgebra.lean", "full_name": "FreeAlgebra.quot_mk_eq_\u03b9", "start": [233, 1], "end": [233, 88], 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"commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Basic.lean", "full_name": "Set.Icc_def", "start": [99, 1], "end": [100, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/BilinearMap.lean", "full_name": "LinearMap.llcomp_apply", "start": [318, 1], "end": [319, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/ModEq.lean", "full_name": "AddCommGroup.modEq_zero", "start": [104, 1], "end": [104, 85], "traced_tactics": [{"tactic": "simp [ModEq, sub_eq_zero, eq_comm]", "state_before": "\u03b1 : Type u_1\ninst\u271d : AddCommGroup \u03b1\np a a\u2081 a\u2082 b b\u2081 b\u2082 c : \u03b1\nn : \u2115\nz : \u2124\n\u22a2 a \u2261 b [PMOD 0] \u2194 a = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/RatFunc.lean", "full_name": "RatFunc.liftOn'_mk", "start": [274, 1], "end": [278, 50], "traced_tactics": [{"tactic": "rw [RatFunc.liftOn', RatFunc.liftOn_mk _ _ _ f0]", "state_before": "K : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nP : Sort v\np q : K[X]\nf : K[X] \u2192 K[X] \u2192 P\nf0 : \u2200 (p : K[X]), f p 0 = f 0 1\nH : \u2200 {p q a : K[X]}, q \u2260 0 \u2192 a \u2260 0 \u2192 f (a * p) (a * q) = f p q\n\u22a2 RatFunc.liftOn' (RatFunc.mk p q) f H = f p q", "state_after": "K : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nP : Sort v\np q : K[X]\nf : K[X] \u2192 K[X] \u2192 P\nf0 : \u2200 (p : K[X]), f p 0 = f 0 1\nH : \u2200 {p q a : K[X]}, q \u2260 0 \u2192 a \u2260 0 \u2192 f (a * p) (a * q) = f p q\n\u22a2 \u2200 {p q p' q' : K[X]}, q \u2260 0 \u2192 q' \u2260 0 \u2192 q' * p = q * p' \u2192 f p q = f p' q'"}, {"tactic": "apply lift_on_condition_of_lift_on'_condition H", "state_before": "K : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nP : Sort v\np q : K[X]\nf : K[X] \u2192 K[X] \u2192 P\nf0 : \u2200 (p : K[X]), f p 0 = f 0 1\nH : \u2200 {p q a : K[X]}, q \u2260 0 \u2192 a \u2260 0 \u2192 f (a * p) (a * q) = f p q\n\u22a2 \u2200 {p q p' q' : K[X]}, q \u2260 0 \u2192 q' \u2260 0 \u2192 q' * p = q * p' \u2192 f p q = f p' q'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.eq_of_mem_singleton", "start": [685, 1], "end": [686, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/LiminfLimsup.lean", "full_name": "OrderIso.isBoundedUnder_le_comp", "start": [251, 1], "end": [254, 69], "traced_tactics": [{"tactic": "simp only [eventually_map, e.le_iff_le]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type ?u.36752\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\nl : Filter \u03b3\nu : \u03b3 \u2192 \u03b1\na : \u03b1\n\u22a2 (\u2200\u1da0 (x : \u03b2) in map (fun x => \u2191e (u x)) l, (fun x x_1 => x \u2264 x_1) x (\u2191e a)) \u2194\n    \u2200\u1da0 (x : \u03b1) in map u l, (fun x x_1 => x \u2264 x_1) x a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Powerset.lean", "full_name": "Multiset.nodup_powerset", "start": [318, 1], "end": [324, 88], "traced_tactics": [{"tactic": "simp only [quot_mk_to_coe, powerset_coe', coe_nodup]", "state_before": "\u03b1 : Type u_1\ns : Multiset \u03b1\nl : List \u03b1\nh : Nodup (Quotient.mk (isSetoid \u03b1) l)\n\u22a2 Nodup (powerset (Quotient.mk (isSetoid \u03b1) l))", "state_after": "\u03b1 : Type u_1\ns : Multiset \u03b1\nl : List \u03b1\nh : Nodup (Quotient.mk (isSetoid \u03b1) l)\n\u22a2 List.Nodup (List.map ofList (sublists' l))"}, {"tactic": "refine' (nodup_sublists'.2 h).map_on _", "state_before": "\u03b1 : Type u_1\ns : Multiset \u03b1\nl : List \u03b1\nh : Nodup (Quotient.mk (isSetoid \u03b1) l)\n\u22a2 List.Nodup (List.map ofList (sublists' l))", "state_after": "\u03b1 : Type u_1\ns : Multiset \u03b1\nl : List \u03b1\nh : Nodup (Quotient.mk (isSetoid \u03b1) l)\n\u22a2 \u2200 (x : List \u03b1), x \u2208 sublists' l \u2192 \u2200 (y : List \u03b1), y \u2208 sublists' l \u2192 \u2191x = \u2191y \u2192 x = y"}, {"tactic": "exact fun x sx y sy e =>\n  (h.sublist_ext (mem_sublists'.1 sx) (mem_sublists'.1 sy)).1 (Quotient.exact e)", "state_before": "\u03b1 : Type u_1\ns : Multiset \u03b1\nl : List \u03b1\nh : Nodup (Quotient.mk (isSetoid \u03b1) l)\n\u22a2 \u2200 (x : List \u03b1), x \u2208 sublists' l \u2192 \u2200 (y : List \u03b1), y \u2208 sublists' l \u2192 \u2191x = \u2191y \u2192 x = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Basic.lean", "full_name": "convex_empty", "start": [91, 1], "end": [91, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/GroupWithZero.lean", "full_name": "Homeomorph.mulRight\u2080_symm_apply", "start": [274, 1], "end": [276, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Set.Infinite.not_bddAbove", "start": [497, 1], "end": [498, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm.lean", "full_name": "ContinuousLinearMap.isBigOWith_id", "start": [661, 1], "end": [662, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/VectorBundle/Basic.lean", "full_name": "Trivialization.linearMapAt_symm\u2097", "start": [267, 1], "end": [269, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/WithZeroTopology.lean", "full_name": "WithZeroTopology.isOpen_iff", "start": [139, 1], "end": [142, 32], "traced_tactics": [{"tactic": "rw [isOpen_iff_mem_nhds, \u2190 and_forall_ne (0 : \u0393\u2080)]", "state_before": "\u03b1 : Type ?u.86003\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\ns : Set \u0393\u2080\n\u22a2 IsOpen s \u2194 \u00ac0 \u2208 s \u2228 \u2203 \u03b3, \u03b3 \u2260 0 \u2227 Iio \u03b3 \u2286 s", "state_after": "\u03b1 : Type ?u.86003\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\ns : Set \u0393\u2080\n\u22a2 ((0 \u2208 s \u2192 s \u2208 \ud835\udcdd 0) \u2227 \u2200 (b : \u0393\u2080), b \u2260 0 \u2192 b \u2208 s \u2192 s \u2208 \ud835\udcdd b) \u2194 \u00ac0 \u2208 s \u2228 \u2203 \u03b3, \u03b3 \u2260 0 \u2227 Iio \u03b3 \u2286 s"}, {"tactic": "simp (config := { contextual := true }) [nhds_of_ne_zero, imp_iff_not_or,\n  hasBasis_nhds_zero.mem_iff]", "state_before": "\u03b1 : Type ?u.86003\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\ns : Set \u0393\u2080\n\u22a2 ((0 \u2208 s \u2192 s \u2208 \ud835\udcdd 0) \u2227 \u2200 (b : \u0393\u2080), b \u2260 0 \u2192 b \u2208 s \u2192 s \u2208 \ud835\udcdd b) \u2194 \u00ac0 \u2208 s \u2228 \u2203 \u03b3, \u03b3 \u2260 0 \u2227 Iio \u03b3 \u2286 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/LinearRecurrence.lean", "full_name": "LinearRecurrence.mkSol_eq_init", "start": [95, 1], "end": [98, 52], "traced_tactics": [{"tactic": "intro n", "state_before": "\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\ninit : Fin E.order \u2192 \u03b1\n\u22a2 \u2200 (n : Fin E.order), mkSol E init \u2191n = init n", "state_after": "\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\ninit : Fin E.order \u2192 \u03b1\nn : Fin E.order\n\u22a2 mkSol E init \u2191n = init n"}, {"tactic": "rw [mkSol]", "state_before": "\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\ninit : Fin E.order \u2192 \u03b1\nn : Fin E.order\n\u22a2 mkSol E init \u2191n = init n", "state_after": "\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\ninit : Fin E.order \u2192 \u03b1\nn : Fin E.order\n\u22a2 (if h : \u2191n < E.order then init { val := \u2191n, isLt := h }\n    else\n      \u2211 k : Fin E.order,\n        let_fun x := (_ : \u2191n - E.order + \u2191k < \u2191n);\n        coeffs E k * mkSol E init (\u2191n - E.order + \u2191k)) =\n    init n"}, {"tactic": "simp only [n.is_lt, dif_pos, Fin.mk_val, Fin.eta]", "state_before": "\u03b1 : Type u_1\ninst\u271d : CommSemiring \u03b1\nE : LinearRecurrence \u03b1\ninit : Fin E.order \u2192 \u03b1\nn : Fin E.order\n\u22a2 (if h : \u2191n < E.order then init { val := \u2191n, isLt := h }\n    else\n      \u2211 k : Fin E.order,\n        let_fun x := (_ : \u2191n - E.order + \u2191k < \u2191n);\n        coeffs E k * mkSol E init (\u2191n - E.order + \u2191k)) =\n    init n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/NAry.lean", "full_name": "Filter.map\u2082_map_left", "start": [269, 1], "end": [271, 100], "traced_tactics": [{"tactic": "rw [\u2190 map_prod_eq_map\u2082, \u2190 map_prod_eq_map\u2082, \u2190 @map_id _ g, prod_map_map_eq, map_map, map_id]", "state_before": "\u03b1 : Type u_4\n\u03b1' : Type ?u.36407\n\u03b2 : Type u_3\n\u03b2' : Type ?u.36413\n\u03b3 : Type u_2\n\u03b3' : Type ?u.36419\n\u03b4 : Type u_1\n\u03b4' : Type ?u.36425\n\u03b5 : Type ?u.36428\n\u03b5' : Type ?u.36431\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b3 \u2192 \u03b2 \u2192 \u03b4\nn : \u03b1 \u2192 \u03b3\n\u22a2 map\u2082 m (map n f) g = map\u2082 (fun a b => m (n a) b) f g", "state_after": "\u03b1 : Type u_4\n\u03b1' : Type ?u.36407\n\u03b2 : Type u_3\n\u03b2' : Type ?u.36413\n\u03b3 : Type u_2\n\u03b3' : Type ?u.36419\n\u03b4 : Type u_1\n\u03b4' : Type ?u.36425\n\u03b5 : Type ?u.36428\n\u03b5' : Type ?u.36431\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b3 \u2192 \u03b2 \u2192 \u03b4\nn : \u03b1 \u2192 \u03b3\n\u22a2 map ((fun p => m p.fst p.snd) \u2218 fun p => (n p.fst, id p.snd)) (f \u00d7\u02e2 g) = map (fun p => m (n p.fst) p.snd) (f \u00d7\u02e2 g)"}, {"tactic": "rfl", "state_before": "\u03b1 : Type u_4\n\u03b1' : Type ?u.36407\n\u03b2 : Type u_3\n\u03b2' : Type ?u.36413\n\u03b3 : Type u_2\n\u03b3' : Type ?u.36419\n\u03b4 : Type u_1\n\u03b4' : Type ?u.36425\n\u03b5 : Type ?u.36428\n\u03b5' : Type ?u.36431\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b3 \u2192 \u03b2 \u2192 \u03b4\nn : \u03b1 \u2192 \u03b3\n\u22a2 map ((fun p => m p.fst p.snd) \u2218 fun p => (n p.fst, id p.snd)) (f \u00d7\u02e2 g) = map (fun p => m (n p.fst) p.snd) (f \u00d7\u02e2 g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Span.lean", "full_name": 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_, fun h x => _\u27e9", "state_before": "R : Type ?u.756570\nM : Type u_1\ninst\u271d : AddCommGroup M\n\u22a2 IsTorsion M \u2194 Module.IsTorsion \u2124 M", "state_after": "case refine'_1\nR : Type ?u.756570\nM : Type u_1\ninst\u271d : AddCommGroup M\nh : IsTorsion M\nx : M\n\u22a2 \u2203 a, a \u2022 x = 0\n\ncase refine'_2\nR : Type ?u.756570\nM : Type u_1\ninst\u271d : AddCommGroup M\nh : Module.IsTorsion \u2124 M\nx : M\n\u22a2 IsOfFinAddOrder x"}, {"tactic": "obtain \u27e8n, h0, hn\u27e9 := (isOfFinAddOrder_iff_nsmul_eq_zero x).mp (h x)", "state_before": "case refine'_1\nR : Type ?u.756570\nM : Type u_1\ninst\u271d : AddCommGroup M\nh : IsTorsion M\nx : M\n\u22a2 \u2203 a, a \u2022 x = 0", "state_after": "case refine'_1.intro.intro\nR : Type ?u.756570\nM : Type u_1\ninst\u271d : AddCommGroup M\nh : IsTorsion M\nx : M\nn : \u2115\nh0 : 0 < n\nhn : n \u2022 x = 0\n\u22a2 \u2203 a, a \u2022 x = 0"}, {"tactic": "exact\n  \u27e8\u27e8n, mem_nonZeroDivisors_of_ne_zero <| ne_of_gt <| Int.coe_nat_pos.mpr h0\u27e9,\n    (coe_nat_zsmul _ _).trans hn\u27e9", "state_before": "case refine'_1.intro.intro\nR : Type ?u.756570\nM : Type u_1\ninst\u271d : AddCommGroup M\nh : IsTorsion M\nx : M\nn : \u2115\nh0 : 0 < n\nhn : n \u2022 x = 0\n\u22a2 \u2203 a, a \u2022 x = 0", "state_after": "no goals"}, {"tactic": "rw [isOfFinAddOrder_iff_nsmul_eq_zero]", "state_before": "case refine'_2\nR : Type ?u.756570\nM : Type u_1\ninst\u271d : AddCommGroup M\nh : Module.IsTorsion \u2124 M\nx : M\n\u22a2 IsOfFinAddOrder x", "state_after": "case refine'_2\nR : Type ?u.756570\nM : Type u_1\ninst\u271d : AddCommGroup M\nh : Module.IsTorsion \u2124 M\nx : M\n\u22a2 \u2203 n, 0 < n \u2227 n \u2022 x = 0"}, {"tactic": "obtain \u27e8n, hn\u27e9 := @h x", "state_before": "case refine'_2\nR : Type ?u.756570\nM : Type u_1\ninst\u271d : AddCommGroup M\nh : Module.IsTorsion \u2124 M\nx : M\n\u22a2 \u2203 n, 0 < n \u2227 n \u2022 x = 0", "state_after": "case refine'_2.intro\nR : Type ?u.756570\nM : Type u_1\ninst\u271d : AddCommGroup M\nh : Module.IsTorsion \u2124 M\nx : M\nn : { x // x \u2208 \u2124\u2070 }\nhn : n \u2022 x = 0\n\u22a2 \u2203 n, 0 < n \u2227 n \u2022 x = 0"}, {"tactic": "exact exists_nsmul_eq_zero_of_zsmul_eq_zero (nonZeroDivisors.coe_ne_zero n) hn", "state_before": "case refine'_2.intro\nR : Type ?u.756570\nM : Type u_1\ninst\u271d : AddCommGroup M\nh : Module.IsTorsion \u2124 M\nx : M\nn : { x // x \u2208 \u2124\u2070 }\nhn : n \u2022 x = 0\n\u22a2 \u2203 n, 0 < n \u2227 n \u2022 x = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/l2Space.lean", "full_name": "lp.inner_single_right", "start": [184, 1], "end": [185, 86], "traced_tactics": [{"tactic": "simpa [inner_conj_symm] using congr_arg conj (@inner_single_left _ \ud835\udd5c _ _ _ _ i a f)", "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type ?u.335242\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\ni : \u03b9\na : G i\nf : { x // x \u2208 lp G 2 }\n\u22a2 inner f (lp.single 2 i a) = inner (\u2191f i) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean", "full_name": "HasDerivAtFilter.add", "start": [55, 8], "end": [57, 43], "traced_tactics": [{"tactic": "simpa using (hf.add hg).hasDerivAtFilter", "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' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL : Filter \ud835\udd5c\nhf : HasDerivAtFilter f f' x L\nhg : HasDerivAtFilter g g' x L\n\u22a2 HasDerivAtFilter (fun y => f y + g y) (f' + g') x L", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fintype/Card.lean", "full_name": "Fintype.card_eq_zero", "start": [521, 1], "end": [522, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Bitvec/Lemmas.lean", "full_name": "Bitvec.toFin_le_toFin_of_le", "start": [155, 1], "end": [158, 12], "traced_tactics": [{"tactic": "rw [toFin_val, toFin_val]", "state_before": "n : \u2115\nv\u2080 v\u2081 : Bitvec n\nh : v\u2080 \u2264 v\u2081\n\u22a2 \u2191(toFin v\u2080) \u2264 \u2191(toFin v\u2081)", "state_after": "n : \u2115\nv\u2080 v\u2081 : Bitvec n\nh : v\u2080 \u2264 v\u2081\n\u22a2 Bitvec.toNat v\u2080 \u2264 Bitvec.toNat v\u2081"}, {"tactic": "exact h", "state_before": "n : \u2115\nv\u2080 v\u2081 : Bitvec n\nh : v\u2080 \u2264 v\u2081\n\u22a2 Bitvec.toNat v\u2080 \u2264 Bitvec.toNat v\u2081", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Seq/WSeq.lean", "full_name": "Stream'.WSeq.LiftRel.swap", "start": [562, 1], "end": [563, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Concrete.lean", "full_name": "Equiv.Perm.toList_formPerm_singleton", "start": [349, 1], "end": [349, 86], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx\u271d x y : \u03b1\n\u22a2 toList (formPerm [x]) y = []", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/ContinuousMapDense.lean", "full_name": "MeasureTheory.Lp.boundedContinuousFunction_dense", "start": [332, 1], "end": [353, 65], "traced_tactics": [{"tactic": "rw [AddSubgroup.eq_top_iff']", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\n\u22a2 AddSubgroup.topologicalClosure (boundedContinuousFunction E p \u03bc) = \u22a4", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\n\u22a2 \u2200 (x : { x // x \u2208 Lp E p }), x \u2208 AddSubgroup.topologicalClosure (boundedContinuousFunction E p \u03bc)"}, {"tactic": "intro f", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\n\u22a2 \u2200 (x : { x // x \u2208 Lp E p }), x \u2208 AddSubgroup.topologicalClosure (boundedContinuousFunction E p \u03bc)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u22a2 f \u2208 AddSubgroup.topologicalClosure (boundedContinuousFunction E p \u03bc)"}, {"tactic": "refine' mem_closure_iff_frequently.mpr _", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u22a2 f \u2208 AddSubgroup.topologicalClosure (boundedContinuousFunction E p \u03bc)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2203\u1da0 (x : { x // x \u2208 Lp E p }) in \ud835\udcdd f, x \u2208 \u2191(boundedContinuousFunction E p \u03bc)"}, {"tactic": "rw [Metric.nhds_basis_closedBall.frequently_iff]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2203\u1da0 (x : { x // x \u2208 Lp E p }) in \ud835\udcdd f, x \u2208 \u2191(boundedContinuousFunction E p \u03bc)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2200 (i : \u211d), 0 < i \u2192 \u2203 x, x \u2208 Metric.closedBall f i \u2227 x \u2208 \u2191(boundedContinuousFunction E p \u03bc)"}, {"tactic": "intro \u03b5 h\u03b5", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2200 (i : \u211d), 0 < i \u2192 \u2203 x, x \u2208 Metric.closedBall f i \u2227 x \u2208 \u2191(boundedContinuousFunction E p \u03bc)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 x, x \u2208 Metric.closedBall f \u03b5 \u2227 x \u2208 \u2191(boundedContinuousFunction E p \u03bc)"}, {"tactic": "have A : ENNReal.ofReal \u03b5 \u2260 0 := by simp only [Ne.def, ENNReal.ofReal_eq_zero, not_le, h\u03b5]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 x, x \u2208 Metric.closedBall f \u03b5 \u2227 x \u2208 \u2191(boundedContinuousFunction E p \u03bc)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\n\u22a2 \u2203 x, x \u2208 Metric.closedBall f \u03b5 \u2227 x \u2208 \u2191(boundedContinuousFunction E p \u03bc)"}, {"tactic": "obtain \u27e8g, hg, g_mem\u27e9 :\n    \u2203 g : \u03b1 \u2192\u1d47 E, snorm ((f : \u03b1 \u2192 E) - (g : \u03b1 \u2192 E)) p \u03bc \u2264 ENNReal.ofReal \u03b5 \u2227 Mem\u2112p g p \u03bc :=\n  (Lp.mem\u2112p f).exists_boundedContinuous_snorm_sub_le hp A", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\n\u22a2 \u2203 x, x \u2208 Metric.closedBall f \u03b5 \u2227 x \u2208 \u2191(boundedContinuousFunction E p \u03bc)", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 \u2203 x, x \u2208 Metric.closedBall f \u03b5 \u2227 x \u2208 \u2191(boundedContinuousFunction E p \u03bc)"}, {"tactic": "refine' \u27e8g_mem.toLp _, _, \u27e8g, rfl\u27e9\u27e9", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 \u2203 x, x \u2208 Metric.closedBall f \u03b5 \u2227 x \u2208 \u2191(boundedContinuousFunction E p \u03bc)", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 Mem\u2112p.toLp (\u2191g) g_mem \u2208 Metric.closedBall f \u03b5"}, {"tactic": "simp only [dist_eq_norm, Metric.mem_closedBall']", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 Mem\u2112p.toLp (\u2191g) g_mem \u2208 Metric.closedBall f \u03b5", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 \u2016f - Mem\u2112p.toLp (\u2191g) g_mem\u2016 \u2264 \u03b5"}, {"tactic": "rw [Lp.norm_def]", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 \u2016f - Mem\u2112p.toLp (\u2191g) g_mem\u2016 \u2264 \u03b5", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 ENNReal.toReal (snorm (\u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem)) p \u03bc) \u2264 \u03b5"}, {"tactic": "have key : snorm ((f : \u03b1 \u2192 E) - (g : \u03b1 \u2192 E)) p \u03bc = snorm (f - Mem\u2112p.toLp (\u2191g) g_mem) p \u03bc := by\n  apply snorm_congr_ae\n  filter_upwards [coeFn_sub f (g_mem.toLp g), g_mem.coeFn_toLp] with x hx h'x\n  simp only [hx, Pi.sub_apply, sub_right_inj, h'x]", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 ENNReal.toReal (snorm (\u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem)) p \u03bc) \u2264 \u03b5", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\nkey : snorm (\u2191\u2191f - \u2191g) p \u03bc = snorm (\u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem)) p \u03bc\n\u22a2 ENNReal.toReal (snorm (\u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem)) p \u03bc) \u2264 \u03b5"}, {"tactic": "simpa only [key] using ENNReal.toReal_le_of_le_ofReal h\u03b5.le hg", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\nkey : snorm (\u2191\u2191f - \u2191g) p \u03bc = snorm (\u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem)) p \u03bc\n\u22a2 ENNReal.toReal (snorm (\u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem)) p \u03bc) \u2264 \u03b5", "state_after": "no goals"}, {"tactic": "simp only [Ne.def, ENNReal.ofReal_eq_zero, not_le, h\u03b5]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 ENNReal.ofReal \u03b5 \u2260 0", "state_after": "no goals"}, {"tactic": "apply snorm_congr_ae", "state_before": "\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 snorm (\u2191\u2191f - \u2191g) p \u03bc = snorm (\u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem)) p \u03bc", "state_after": "case hfg\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 \u2191\u2191f - \u2191g =\u1d50[\u03bc] \u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem)"}, {"tactic": "filter_upwards [coeFn_sub f (g_mem.toLp g), g_mem.coeFn_toLp] with x hx h'x", "state_before": "case hfg\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 \u2191\u2191f - \u2191g =\u1d50[\u03bc] \u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\nx : \u03b1\nhx : \u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem) x = (\u2191\u2191f - \u2191\u2191(Mem\u2112p.toLp (\u2191g) g_mem)) x\nh'x : \u2191\u2191(Mem\u2112p.toLp (\u2191g) g_mem) x = \u2191g x\n\u22a2 (\u2191\u2191f - \u2191g) x = \u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem) x"}, {"tactic": "simp only [hx, Pi.sub_apply, sub_right_inj, h'x]", "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : NormalSpace \u03b1\ninst\u271d\u2074 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : SecondCountableTopologyEither \u03b1 E\n_i : Fact (1 \u2264 p)\nhp : p \u2260 \u22a4\ninst\u271d : Measure.WeaklyRegular \u03bc\nf : { x // x \u2208 Lp E p }\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nA : ENNReal.ofReal \u03b5 \u2260 0\ng : \u03b1 \u2192\u1d47 E\nhg : snorm (\u2191\u2191f - \u2191g) p \u03bc \u2264 ENNReal.ofReal \u03b5\ng_mem : Mem\u2112p (\u2191g) p\nx : \u03b1\nhx : \u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem) x = (\u2191\u2191f - \u2191\u2191(Mem\u2112p.toLp (\u2191g) g_mem)) x\nh'x : \u2191\u2191(Mem\u2112p.toLp (\u2191g) g_mem) x = \u2191g x\n\u22a2 (\u2191\u2191f - \u2191g) x = \u2191\u2191(f - Mem\u2112p.toLp (\u2191g) g_mem) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/DedekindDomain/Ideal.lean", "full_name": "IsDedekindDomainInv.isNoetherianRing", "start": [291, 1], "end": [296, 83], "traced_tactics": [{"tactic": "refine' isNoetherianRing_iff.mpr \u27e8fun I : Ideal A => _\u27e9", "state_before": "R : Type ?u.127506\nA : Type u_1\nK : Type ?u.127512\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nh : IsDedekindDomainInv A\n\u22a2 IsNoetherianRing A", "state_after": "R : Type ?u.127506\nA : Type u_1\nK : Type ?u.127512\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nh : IsDedekindDomainInv A\nI : Ideal A\n\u22a2 Submodule.FG I"}, {"tactic": "by_cases hI : I = \u22a5", "state_before": "R : Type ?u.127506\nA : Type u_1\nK : Type ?u.127512\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nh : IsDedekindDomainInv A\nI : Ideal A\n\u22a2 Submodule.FG I", "state_after": "case pos\nR : Type ?u.127506\nA : Type u_1\nK : Type ?u.127512\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nh : IsDedekindDomainInv A\nI : Ideal A\nhI : I = \u22a5\n\u22a2 Submodule.FG I\n\ncase neg\nR : Type ?u.127506\nA : Type u_1\nK : Type ?u.127512\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nh : IsDedekindDomainInv A\nI : Ideal A\nhI : \u00acI = \u22a5\n\u22a2 Submodule.FG I"}, {"tactic": "have hI : (I : FractionalIdeal A\u2070 (FractionRing A)) \u2260 0 := coeIdeal_ne_zero.mpr hI", "state_before": "case neg\nR : Type ?u.127506\nA : Type u_1\nK : Type ?u.127512\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nh : IsDedekindDomainInv A\nI : Ideal A\nhI : \u00acI = \u22a5\n\u22a2 Submodule.FG I", "state_after": "case neg\nR : Type ?u.127506\nA : Type u_1\nK : Type ?u.127512\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nh : IsDedekindDomainInv A\nI : Ideal A\nhI\u271d : \u00acI = \u22a5\nhI : \u2191I \u2260 0\n\u22a2 Submodule.FG I"}, {"tactic": "exact I.fg_of_isUnit (IsFractionRing.injective A (FractionRing A)) (h.isUnit hI)", "state_before": "case neg\nR : Type ?u.127506\nA : Type u_1\nK : Type ?u.127512\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nh : IsDedekindDomainInv A\nI : Ideal A\nhI\u271d : \u00acI = \u22a5\nhI : \u2191I \u2260 0\n\u22a2 Submodule.FG I", "state_after": "no goals"}, {"tactic": "rw [hI]", "state_before": "case pos\nR : Type ?u.127506\nA : Type u_1\nK : Type ?u.127512\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nh : IsDedekindDomainInv A\nI : Ideal A\nhI : I = \u22a5\n\u22a2 Submodule.FG I", "state_after": "case pos\nR : Type ?u.127506\nA : Type u_1\nK : Type ?u.127512\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nh : IsDedekindDomainInv A\nI : Ideal A\nhI : I = \u22a5\n\u22a2 Submodule.FG \u22a5"}, {"tactic": "apply Submodule.fg_bot", "state_before": "case pos\nR : Type ?u.127506\nA : Type u_1\nK : Type ?u.127512\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nh : IsDedekindDomainInv A\nI : Ideal A\nhI : I = \u22a5\n\u22a2 Submodule.FG \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Int/DivMod.lean", "full_name": "Int.div_eq_ediv", "start": [59, 1], "end": [61, 32], "traced_tactics": [{"tactic": "simp", "state_before": "x\u271d\u00b2 : Int\nx\u271d\u00b9 : 0 \u2264 x\u271d\u00b2\nx\u271d : 0 \u2264 0\n\u22a2 div x\u271d\u00b2 0 = x\u271d\u00b2 / 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Tactic/Ring/Basic.lean", "full_name": "Mathlib.Tactic.Ring.pow_add", "start": [798, 1], "end": [799, 66], "traced_tactics": [{"tactic": "subst_vars", "state_before": "u : Lean.Level\nR : Type u_1\n\u03b1 : Q(Type u)\ns\u03b1 : Q(CommSemiring \u00ab$\u03b1\u00bb)\ninst\u271d : CommSemiring R\na : R\nb\u2081 : \u2115\nc\u2081 : R\nb\u2082 : \u2115\nc\u2082 d : R\nx\u271d\u00b2 : a ^ b\u2081 = c\u2081\nx\u271d\u00b9 : a ^ b\u2082 = c\u2082\nx\u271d : c\u2081 * c\u2082 = d\n\u22a2 a ^ (b\u2081 + b\u2082) = d", "state_after": "u : Lean.Level\nR : Type u_1\n\u03b1 : Q(Type u)\ns\u03b1 : Q(CommSemiring \u00ab$\u03b1\u00bb)\ninst\u271d : CommSemiring R\na : R\nb\u2081 b\u2082 : \u2115\n\u22a2 a ^ (b\u2081 + b\u2082) = a ^ b\u2081 * a ^ b\u2082"}, {"tactic": "simp [_root_.pow_add]", "state_before": "u : Lean.Level\nR : Type u_1\n\u03b1 : Q(Type u)\ns\u03b1 : Q(CommSemiring \u00ab$\u03b1\u00bb)\ninst\u271d : CommSemiring R\na : R\nb\u2081 b\u2082 : \u2115\n\u22a2 a ^ (b\u2081 + b\u2082) = a ^ b\u2081 * a ^ b\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "StrictAntiOn.comp", "start": [1564, 1], "end": [1566, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/FieldDivision.lean", "full_name": "Polynomial.isUnit_map", "start": [297, 1], "end": [298, 50], "traced_tactics": [{"tactic": "simp_rw [isUnit_iff_degree_eq_zero, degree_map]", "state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\ninst\u271d : Field k\nf : R \u2192+* k\n\u22a2 IsUnit (map f p) \u2194 IsUnit p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean", "full_name": "nndist_lineMap_right", "start": [126, 1], "end": [128, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Interval.lean", "full_name": "NonemptyInterval.toDualProd_apply", "start": [74, 1], "end": [75, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/LinearIndependent.lean", "full_name": "linearIndependent_equiv'", "start": [260, 1], "end": [262, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/EuclideanDomain/Basic.lean", "full_name": "EuclideanDomain.lcm_zero_right", "start": [305, 1], "end": [305, 80], "traced_tactics": [{"tactic": "rw [lcm, mul_zero, zero_div]", "state_before": "R : Type u\ninst\u271d\u00b9 : EuclideanDomain R\ninst\u271d : DecidableEq R\nx : R\n\u22a2 lcm x 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/List/Lemmas.lean", "full_name": "List.getLast?_eq_get?", "start": [631, 1], "end": [633, 81], "traced_tactics": [{"tactic": "rw [getLast?_eq_getLast (a::l) fun., getLast_eq_get, get?_eq_get]", "state_before": "\u03b1 : Type u_1\na : \u03b1\nl : List \u03b1\n\u22a2 getLast? (a :: l) = get? (a :: l) (length (a :: l) - 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/TangentCone.lean", "full_name": "tangentCone_mono", "start": [105, 1], "end": [107, 64], "traced_tactics": [{"tactic": "rintro y \u27e8c, d, ds, ctop, clim\u27e9", "state_before": "\ud835\udd5c : Type u_2\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type ?u.19394\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type ?u.19484\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\nh : s \u2286 t\n\u22a2 tangentConeAt \ud835\udd5c s x \u2286 tangentConeAt \ud835\udd5c t x", "state_after": "case intro.intro.intro.intro\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type ?u.19394\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type ?u.19484\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t : Set E\nh : s \u2286 t\ny : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nds : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nctop : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nclim : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\n\u22a2 y \u2208 tangentConeAt \ud835\udd5c t x"}, {"tactic": "exact \u27e8c, d, mem_of_superset ds fun n hn => h hn, ctop, clim\u27e9", "state_before": "case intro.intro.intro.intro\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type ?u.19394\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type ?u.19484\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t : Set E\nh : s \u2286 t\ny : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nds : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nctop : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nclim : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\n\u22a2 y \u2208 tangentConeAt \ud835\udd5c t x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/StructuredArrow.lean", "full_name": "CategoryTheory.StructuredArrow.left_eq_id", "start": [97, 1], "end": [98, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/FiberBundle/Trivialization.lean", "full_name": "Trivialization.symm_trans_target_eq", "start": [426, 1], "end": [428, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Atoms.lean", "full_name": "isCoatomic_of_isAtomic_of_complementedLattice_of_isModular", "start": [860, 1], "end": [871, 20], "traced_tactics": [{"tactic": "rcases exists_isCompl x with \u27e8y, xy\u27e9", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.78195\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\ninst\u271d\u00b9 : ComplementedLattice \u03b1\ninst\u271d : IsAtomic \u03b1\nx : \u03b1\n\u22a2 x = \u22a4 \u2228 \u2203 a, IsCoatom a \u2227 x \u2264 a", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.78195\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\ninst\u271d\u00b9 : ComplementedLattice \u03b1\ninst\u271d : IsAtomic \u03b1\nx y : \u03b1\nxy : IsCompl x y\n\u22a2 x = \u22a4 \u2228 \u2203 a, IsCoatom a \u2227 x \u2264 a"}, {"tactic": "apply (eq_bot_or_exists_atom_le y).imp _ _", "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.78195\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\ninst\u271d\u00b9 : ComplementedLattice \u03b1\ninst\u271d : IsAtomic \u03b1\nx y : \u03b1\nxy : IsCompl x y\n\u22a2 x = \u22a4 \u2228 \u2203 a, IsCoatom a \u2227 x \u2264 a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.78195\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\ninst\u271d\u00b9 : ComplementedLattice \u03b1\ninst\u271d : IsAtomic \u03b1\nx y : \u03b1\nxy : IsCompl x y\n\u22a2 y = \u22a5 \u2192 x = \u22a4\n\n\u03b1 : Type u_1\n\u03b2 : Type ?u.78195\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\ninst\u271d\u00b9 : ComplementedLattice \u03b1\ninst\u271d : IsAtomic \u03b1\nx y : \u03b1\nxy : IsCompl x y\n\u22a2 (\u2203 a, IsAtom a \u2227 a \u2264 y) \u2192 \u2203 a, IsCoatom a \u2227 x \u2264 a"}, {"tactic": "rintro rfl", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.78195\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\ninst\u271d\u00b9 : ComplementedLattice \u03b1\ninst\u271d : IsAtomic \u03b1\nx y : \u03b1\nxy : IsCompl x y\n\u22a2 y = \u22a5 \u2192 x = \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.78195\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\ninst\u271d\u00b9 : ComplementedLattice \u03b1\ninst\u271d : IsAtomic \u03b1\nx : \u03b1\nxy : IsCompl x \u22a5\n\u22a2 x = \u22a4"}, {"tactic": "exact eq_top_of_isCompl_bot xy", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.78195\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\ninst\u271d\u00b9 : ComplementedLattice \u03b1\ninst\u271d : IsAtomic \u03b1\nx : \u03b1\nxy : IsCompl x \u22a5\n\u22a2 x = \u22a4", "state_after": "no goals"}, {"tactic": "rintro \u27e8a, ha, ay\u27e9", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.78195\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\ninst\u271d\u00b9 : ComplementedLattice \u03b1\ninst\u271d : IsAtomic \u03b1\nx y : \u03b1\nxy : IsCompl x y\n\u22a2 (\u2203 a, IsAtom a \u2227 a \u2264 y) \u2192 \u2203 a, IsCoatom a \u2227 x \u2264 a", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.78195\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\ninst\u271d\u00b9 : ComplementedLattice \u03b1\ninst\u271d : IsAtomic \u03b1\nx y : \u03b1\nxy : IsCompl x y\na : \u03b1\nha : IsAtom a\nay : a \u2264 y\n\u22a2 \u2203 a, IsCoatom a \u2227 x \u2264 a"}, {"tactic": "rcases exists_isCompl (xy.symm.IicOrderIsoIci \u27e8a, ay\u27e9) with \u27e8\u27e8b, xb\u27e9, hb\u27e9", "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.78195\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\ninst\u271d\u00b9 : ComplementedLattice \u03b1\ninst\u271d : IsAtomic \u03b1\nx y : \u03b1\nxy : IsCompl x y\na : \u03b1\nha : IsAtom a\nay : a \u2264 y\n\u22a2 \u2203 a, IsCoatom a \u2227 x \u2264 a", "state_after": "case intro.intro.intro.mk\n\u03b1 : Type u_1\n\u03b2 : Type ?u.78195\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\ninst\u271d\u00b9 : ComplementedLattice \u03b1\ninst\u271d : IsAtomic \u03b1\nx y : \u03b1\nxy : IsCompl x y\na : \u03b1\nha : IsAtom a\nay : a \u2264 y\nb : \u03b1\nxb : b \u2208 Set.Ici x\nhb : IsCompl (\u2191(IsCompl.IicOrderIsoIci (_ : IsCompl y x)) { val := a, property := ay }) { val := b, property := xb }\n\u22a2 \u2203 a, IsCoatom a \u2227 x \u2264 a"}, {"tactic": "refine' \u27e8\u2191(\u27e8b, xb\u27e9 : Set.Ici x), IsCoatom.of_isCoatom_coe_Ici _, xb\u27e9", "state_before": "case intro.intro.intro.mk\n\u03b1 : Type u_1\n\u03b2 : Type ?u.78195\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\ninst\u271d\u00b9 : ComplementedLattice \u03b1\ninst\u271d : IsAtomic \u03b1\nx y : \u03b1\nxy : IsCompl x y\na : \u03b1\nha : IsAtom a\nay : a \u2264 y\nb : \u03b1\nxb : b \u2208 Set.Ici x\nhb : IsCompl (\u2191(IsCompl.IicOrderIsoIci (_ : IsCompl y x)) { val := a, property := ay }) { val := b, property := xb }\n\u22a2 \u2203 a, IsCoatom a \u2227 x \u2264 a", "state_after": "case intro.intro.intro.mk\n\u03b1 : Type u_1\n\u03b2 : Type ?u.78195\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\ninst\u271d\u00b9 : ComplementedLattice \u03b1\ninst\u271d : IsAtomic \u03b1\nx y : \u03b1\nxy : IsCompl x y\na : \u03b1\nha : IsAtom a\nay : a \u2264 y\nb : \u03b1\nxb : b \u2208 Set.Ici x\nhb : IsCompl (\u2191(IsCompl.IicOrderIsoIci (_ : IsCompl y x)) { val := a, property := ay }) { val := b, property := xb }\n\u22a2 IsCoatom { val := b, property := xb }"}, {"tactic": "rw [\u2190 hb.isAtom_iff_isCoatom, OrderIso.isAtom_iff]", "state_before": "case intro.intro.intro.mk\n\u03b1 : Type u_1\n\u03b2 : Type ?u.78195\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\ninst\u271d\u00b9 : ComplementedLattice \u03b1\ninst\u271d : IsAtomic \u03b1\nx y : \u03b1\nxy : IsCompl x y\na : \u03b1\nha : IsAtom a\nay : a \u2264 y\nb : \u03b1\nxb : b \u2208 Set.Ici x\nhb : IsCompl (\u2191(IsCompl.IicOrderIsoIci (_ : IsCompl y x)) { val := a, property := ay }) { val := b, property := xb }\n\u22a2 IsCoatom { val := b, property := xb }", "state_after": "case intro.intro.intro.mk\n\u03b1 : Type u_1\n\u03b2 : Type ?u.78195\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\ninst\u271d\u00b9 : ComplementedLattice \u03b1\ninst\u271d : IsAtomic \u03b1\nx y : \u03b1\nxy : IsCompl x y\na : \u03b1\nha : IsAtom a\nay : a \u2264 y\nb : \u03b1\nxb : b \u2208 Set.Ici x\nhb : IsCompl (\u2191(IsCompl.IicOrderIsoIci (_ : IsCompl y x)) { val := a, property := ay }) { val := b, property := xb }\n\u22a2 IsAtom { val := a, property := ay }"}, {"tactic": "apply ha.Iic", "state_before": "case intro.intro.intro.mk\n\u03b1 : Type u_1\n\u03b2 : Type ?u.78195\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : IsModularLattice \u03b1\ninst\u271d\u00b9 : ComplementedLattice \u03b1\ninst\u271d : IsAtomic \u03b1\nx y : \u03b1\nxy : IsCompl x y\na : \u03b1\nha : IsAtom a\nay : a \u2264 y\nb : \u03b1\nxb : b \u2208 Set.Ici x\nhb : IsCompl (\u2191(IsCompl.IicOrderIsoIci (_ : IsCompl y x)) { val := a, property := ay }) { val := b, property := xb }\n\u22a2 IsAtom { val := a, property := ay }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Star/Subalgebra.lean", "full_name": "StarSubalgebra.mem_inf", "start": [658, 1], "end": [659, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Bounds/Basic.lean", "full_name": "isLeast_bot_iff", "start": [109, 1], "end": [110, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/String/Lemmas.lean", "full_name": "String.Iterator.ValidFor.hasNext", "start": [569, 1], "end": [570, 67], "traced_tactics": [{"tactic": "simp [Iterator.hasNext, \u2190 h.atEnd, Iterator.atEnd]", "state_before": "l r : List Char\nit : Iterator\nh : ValidFor l r it\n\u22a2 Iterator.hasNext it = true \u2194 r \u2260 []", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Int/Cast/Basic.lean", "full_name": "Int.cast_negSucc", "start": [55, 1], "end": [56, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Polynomial/Content.lean", "full_name": "Polynomial.primPart_ne_zero", "start": [279, 1], "end": [280, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "full_name": "Complex.sin_pi_sub", "start": [1160, 1], "end": [1161, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean", "full_name": "AffineEquiv.coe_trans", "start": [339, 1], "end": [340, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Dfinsupp/NeLocus.lean", "full_name": "Dfinsupp.neLocus_self_sub_left", "start": [178, 1], "end": [179, 44], "traced_tactics": [{"tactic": "rw [neLocus_comm, neLocus_self_sub_right]", "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) f = support g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.forall_mem_empty_iff", "start": [3076, 1], "end": [3077, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Sum/Basic.lean", "full_name": "Sum.swap_inr", "start": [345, 1], "end": [346, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/WithBot.lean", "full_name": "WithBot.bot_lt_iff_ne_bot", "start": [295, 11], "end": [297, 36], "traced_tactics": [{"tactic": "simpa using not_lt_none \u22a5", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.21635\n\u03b3 : Type ?u.21638\n\u03b4 : Type ?u.21641\na b : \u03b1\ninst\u271d : LT \u03b1\n\u22a2 \u22a5 < \u22a5 \u2194 \u22a5 \u2260 \u22a5", "state_after": "no goals"}, {"tactic": "simp [bot_lt_coe]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.21635\n\u03b3 : Type ?u.21638\n\u03b4 : Type ?u.21641\na b : \u03b1\ninst\u271d : LT \u03b1\nx : \u03b1\n\u22a2 \u22a5 < \u2191x \u2194 \u2191x \u2260 \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/NormalMono/Equalizers.lean", "full_name": "CategoryTheory.NormalEpiCategory.mono_of_cancel_zero", "start": [346, 1], "end": [348, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Order.lean", "full_name": "Continuous.coinduced_le", "start": [405, 1], "end": [406, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Sym/Basic.lean", "full_name": "Sym.mem_attach", "start": [446, 1], "end": [447, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/Basic.lean", "full_name": "abs_dist", "start": [292, 9], "end": [292, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Basic.lean", "full_name": "Polynomial.C_add", "start": [510, 1], "end": [511, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Lie/Submodule.lean", "full_name": "LieSubmodule.subsingleton_iff", "start": [528, 1], "end": [532, 42], "traced_tactics": [{"tactic": "rw [\u2190 subsingleton_iff_bot_eq_top, \u2190 subsingleton_iff_bot_eq_top, \u2190 coe_toSubmodule_eq_iff,\n  top_coeSubmodule, bot_coeSubmodule]", "state_before": "R : 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\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\n\u22a2 Subsingleton (LieSubmodule R L M) \u2194 Subsingleton (Submodule R M)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Extr.lean", "full_name": "IsMaxOn.comp_mono", "start": [346, 1], "end": [348, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean", "full_name": "Matrix.inv_inv_of_invertible", "start": [320, 1], "end": [321, 50], "traced_tactics": [{"tactic": "simp only [\u2190 invOf_eq_nonsing_inv, invOf_invOf]", "state_before": "l : Type ?u.198656\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA B : Matrix n n \u03b1\ninst\u271d : Invertible A\n\u22a2 A\u207b\u00b9\u207b\u00b9 = A", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Hom/Equiv/Basic.lean", "full_name": "MulEquiv.coe_toEquiv_symm", "start": [278, 1], "end": [278, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/ContDiff.lean", "full_name": "ContinuousLinearMap.norm_iteratedFDeriv_le_of_bilinear", "start": [2440, 1], "end": [2447, 20], "traced_tactics": [{"tactic": "simp_rw [\u2190 iteratedFDerivWithin_univ]", "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.4049704\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns 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\nB : E \u2192L[\ud835\udd5c] F \u2192L[\ud835\udd5c] G\nf : D \u2192 E\ng : D \u2192 F\nN : \u2115\u221e\nhf : ContDiff \ud835\udd5c N f\nhg : ContDiff \ud835\udd5c N g\nx : D\nn : \u2115\nhn : \u2191n \u2264 N\n\u22a2 \u2016iteratedFDeriv \ud835\udd5c n (fun y => \u2191(\u2191B (f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 * \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \ud835\udd5c i f x\u2016 * \u2016iteratedFDeriv \ud835\udd5c (n - i) g 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.4049704\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns 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\nB : E \u2192L[\ud835\udd5c] F \u2192L[\ud835\udd5c] G\nf : D \u2192 E\ng : D \u2192 F\nN : \u2115\u221e\nhf : ContDiff \ud835\udd5c N f\nhg : ContDiff \ud835\udd5c N g\nx : D\nn : \u2115\nhn : \u2191n \u2264 N\n\u22a2 \u2016iteratedFDerivWithin \ud835\udd5c n (fun y => \u2191(\u2191B (f y)) (g y)) univ x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 x_1 in Finset.range (n + 1),\n        \u2191(Nat.choose n x_1) * \u2016iteratedFDerivWithin \ud835\udd5c x_1 f univ x\u2016 * \u2016iteratedFDerivWithin \ud835\udd5c (n - x_1) g univ x\u2016"}, {"tactic": "exact B.norm_iteratedFDerivWithin_le_of_bilinear hf.contDiffOn hg.contDiffOn uniqueDiffOn_univ\n  (mem_univ x) hn", "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.4049704\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns 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\nB : E \u2192L[\ud835\udd5c] F \u2192L[\ud835\udd5c] G\nf : D \u2192 E\ng : D \u2192 F\nN : \u2115\u221e\nhf : ContDiff \ud835\udd5c N f\nhg : ContDiff \ud835\udd5c N g\nx : D\nn : \u2115\nhn : \u2191n \u2264 N\n\u22a2 \u2016iteratedFDerivWithin \ud835\udd5c n (fun y => \u2191(\u2191B (f y)) (g y)) univ x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 x_1 in Finset.range (n + 1),\n        \u2191(Nat.choose n x_1) * \u2016iteratedFDerivWithin \ud835\udd5c x_1 f univ x\u2016 * \u2016iteratedFDerivWithin \ud835\udd5c (n - x_1) g univ x\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Control/Lawful.lean", "full_name": "ReaderT.ext", "start": [180, 1], "end": [182, 17], "traced_tactics": [{"tactic": "simp [run] at h", "state_before": "\u03c1 : Type u_1\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_1\nx y : ReaderT \u03c1 m \u03b1\nh : \u2200 (ctx : \u03c1), run x ctx = run y ctx\n\u22a2 x = y", "state_after": "\u03c1 : Type u_1\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_1\nx y : ReaderT \u03c1 m \u03b1\nh : \u2200 (ctx : \u03c1), x ctx = y ctx\n\u22a2 x = y"}, {"tactic": "exact funext h", "state_before": "\u03c1 : Type u_1\nm : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_1\nx y : ReaderT \u03c1 m \u03b1\nh : \u2200 (ctx : \u03c1), x ctx = y ctx\n\u22a2 x = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Data/Nat/Basic.lean", "full_name": "Nat.pred_le", "start": [223, 1], "end": [225, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Int/Order/Units.lean", "full_name": "Int.isUnit_iff_abs_eq", "start": [22, 1], "end": [23, 74], "traced_tactics": [{"tactic": "rw [isUnit_iff_natAbs_eq, abs_eq_natAbs, \u2190 Int.ofNat_one, coe_nat_inj']", "state_before": "x : \u2124\n\u22a2 IsUnit x \u2194 abs x = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Seq/Seq.lean", "full_name": "Stream'.Seq1.map_join", "start": [1009, 1], "end": [1010, 62], "traced_tactics": [{"tactic": "apply recOn s <;> intros <;> simp [map]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Seq \u03b1\nS : Seq (Seq1 \u03b1)\n\u22a2 map f (join ((a, s), S)) = join (map (map f) ((a, s), S))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Euclidean/Sphere/Basic.lean", "full_name": "EuclideanGeometry.eq_of_mem_sphere_of_mem_sphere_of_mem_of_finrank_eq_two", "start": [311, 1], "end": [317, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Sqrt.lean", "full_name": "Nat.sqrt_pos", "start": [138, 1], "end": [139, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.setToFun_congr_ae", "start": [1424, 1], "end": [1430, 54], "traced_tactics": [{"tactic": "by_cases hfi : Integrable f \u03bc", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1485634\nG : Type ?u.1485637\n\ud835\udd5c : Type ?u.1485640\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nh : f =\u1d50[\u03bc] g\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1485634\nG : Type ?u.1485637\n\ud835\udd5c : Type ?u.1485640\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nh : f =\u1d50[\u03bc] g\nhfi : Integrable f\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1485634\nG : Type ?u.1485637\n\ud835\udd5c : Type ?u.1485640\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nh : f =\u1d50[\u03bc] g\nhfi : \u00acIntegrable f\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g"}, {"tactic": "have hgi : Integrable g \u03bc := hfi.congr h", "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1485634\nG : Type ?u.1485637\n\ud835\udd5c : Type ?u.1485640\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nh : f =\u1d50[\u03bc] g\nhfi : Integrable f\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1485634\nG : Type ?u.1485637\n\ud835\udd5c : Type ?u.1485640\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nh : f =\u1d50[\u03bc] g\nhfi : Integrable f\nhgi : Integrable g\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g"}, {"tactic": "rw [setToFun_eq hT hfi, setToFun_eq hT hgi, (Integrable.toL1_eq_toL1_iff f g hfi hgi).2 h]", "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1485634\nG : Type ?u.1485637\n\ud835\udd5c : Type ?u.1485640\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nh : f =\u1d50[\u03bc] g\nhfi : Integrable f\nhgi : Integrable g\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g", "state_after": "no goals"}, {"tactic": "have hgi : \u00acIntegrable g \u03bc := by rw [integrable_congr h] at hfi ; exact hfi", "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1485634\nG : Type ?u.1485637\n\ud835\udd5c : Type ?u.1485640\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nh : f =\u1d50[\u03bc] g\nhfi : \u00acIntegrable f\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1485634\nG : Type ?u.1485637\n\ud835\udd5c : Type ?u.1485640\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nh : f =\u1d50[\u03bc] g\nhfi : \u00acIntegrable f\nhgi : \u00acIntegrable g\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g"}, {"tactic": "rw [setToFun_undef hT hfi, setToFun_undef hT hgi]", "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1485634\nG : Type ?u.1485637\n\ud835\udd5c : Type ?u.1485640\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nh : f =\u1d50[\u03bc] g\nhfi : \u00acIntegrable f\nhgi : \u00acIntegrable g\n\u22a2 setToFun \u03bc T hT f = setToFun \u03bc T hT g", "state_after": "no goals"}, {"tactic": "rw [integrable_congr h] at hfi", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1485634\nG : Type ?u.1485637\n\ud835\udd5c : Type ?u.1485640\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nh : f =\u1d50[\u03bc] g\nhfi : \u00acIntegrable f\n\u22a2 \u00acIntegrable g", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1485634\nG : Type ?u.1485637\n\ud835\udd5c : Type ?u.1485640\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nh : f =\u1d50[\u03bc] g\nhfi : \u00acIntegrable g\n\u22a2 \u00acIntegrable g"}, {"tactic": "exact hfi", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1485634\nG : Type ?u.1485637\n\ud835\udd5c : Type ?u.1485640\np : \u211d\u22650\u221e\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : CompleteSpace F\nT T' T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' C'' : \u211d\nf g : \u03b1 \u2192 E\nhT : DominatedFinMeasAdditive \u03bc T C\nh : f =\u1d50[\u03bc] g\nhfi : \u00acIntegrable g\n\u22a2 \u00acIntegrable g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "exists_seq_tendsto_sSup", "start": [2152, 1], "end": [2156, 26], "traced_tactics": [{"tactic": "rcases(isLUB_csSup hS hS').exists_seq_monotone_tendsto hS with \u27e8u, hu\u27e9", "state_before": "\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2079 : TopologicalSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : LinearOrder \u03b1\u271d\ninst\u271d\u2076 : LinearOrder \u03b2\ninst\u271d\u2075 : OrderTopology \u03b1\u271d\ninst\u271d\u2074 : OrderTopology \u03b2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : FirstCountableTopology \u03b1\nS : Set \u03b1\nhS : Set.Nonempty S\nhS' : BddAbove S\n\u22a2 \u2203 u, Monotone u \u2227 Tendsto u atTop (\ud835\udcdd (sSup S)) \u2227 \u2200 (n : \u2115), u n \u2208 S", "state_after": "case intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2079 : TopologicalSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : LinearOrder \u03b1\u271d\ninst\u271d\u2076 : LinearOrder \u03b2\ninst\u271d\u2075 : OrderTopology \u03b1\u271d\ninst\u271d\u2074 : OrderTopology \u03b2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : FirstCountableTopology \u03b1\nS : Set \u03b1\nhS : Set.Nonempty S\nhS' : BddAbove S\nu : \u2115 \u2192 \u03b1\nhu : Monotone u \u2227 (\u2200 (n : \u2115), u n \u2264 sSup S) \u2227 Tendsto u atTop (\ud835\udcdd (sSup S)) \u2227 \u2200 (n : \u2115), u n \u2208 S\n\u22a2 \u2203 u, Monotone u \u2227 Tendsto u atTop (\ud835\udcdd (sSup S)) \u2227 \u2200 (n : \u2115), u n \u2208 S"}, {"tactic": "exact \u27e8u, hu.1, hu.2.2\u27e9", "state_before": "case intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2079 : TopologicalSpace \u03b1\u271d\ninst\u271d\u2078 : TopologicalSpace \u03b2\ninst\u271d\u2077 : LinearOrder \u03b1\u271d\ninst\u271d\u2076 : LinearOrder \u03b2\ninst\u271d\u2075 : OrderTopology \u03b1\u271d\ninst\u271d\u2074 : OrderTopology \u03b2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderTopology \u03b1\ninst\u271d : FirstCountableTopology \u03b1\nS : Set \u03b1\nhS : Set.Nonempty S\nhS' : BddAbove S\nu : \u2115 \u2192 \u03b1\nhu : Monotone u \u2227 (\u2200 (n : \u2115), u n \u2264 sSup S) \u2227 Tendsto u atTop (\ud835\udcdd (sSup S)) \u2227 \u2200 (n : \u2115), u n \u2208 S\n\u22a2 \u2203 u, Monotone u \u2227 Tendsto u atTop (\ud835\udcdd (sSup S)) \u2227 \u2200 (n : \u2115), u n \u2208 S", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Lie/Submodule.lean", "full_name": "LieSubmodule.range_incl", "start": [1245, 1], "end": [1245, 89], "traced_tactics": [{"tactic": "simp [\u2190 LieSubmodule.coe_toSubmodule_eq_iff]", "state_before": "R : 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\nN : LieSubmodule R L M\n\u22a2 LieModuleHom.range (incl N) = N", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Matrix/Kronecker.lean", "full_name": "Matrix.kroneckerTMul_add", "start": [482, 1], "end": [484, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/ConjugateExponents.lean", "full_name": "Real.IsConjugateExponent.sub_one_ne_zero", "start": [62, 1], "end": [62, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Preadditive/ProjectiveResolution.lean", "full_name": "CategoryTheory.ProjectiveResolution.liftOne_zero_comm", "start": [178, 1], "end": [182, 7], "traced_tactics": [{"tactic": "dsimp [liftZero, liftOne]", "state_before": "C : Type u\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : HasZeroObject C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasEqualizers C\ninst\u271d : HasImages C\nY Z : C\nf : Y \u27f6 Z\nP : ProjectiveResolution Y\nQ : ProjectiveResolution Z\n\u22a2 liftOne f P Q \u226b HomologicalComplex.d Q.complex 1 0 = HomologicalComplex.d P.complex 1 0 \u226b liftZero f P Q", "state_after": "C : Type u\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : HasZeroObject C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasEqualizers C\ninst\u271d : HasImages C\nY Z : C\nf : Y \u27f6 Z\nP : ProjectiveResolution Y\nQ : ProjectiveResolution Z\n\u22a2 Exact.lift\n        (HomologicalComplex.d P.complex 1 0 \u226b\n          factorThru (HomologicalComplex.Hom.f P.\u03c0 0 \u226b f) (HomologicalComplex.Hom.f Q.\u03c0 0))\n        (HomologicalComplex.d Q.complex 1 0) (HomologicalComplex.Hom.f Q.\u03c0 0)\n        (_ : Exact (HomologicalComplex.d Q.complex 1 0) (HomologicalComplex.Hom.f Q.\u03c0 0))\n        (_ :\n          (HomologicalComplex.d P.complex 1 0 \u226b\n                factorThru (HomologicalComplex.Hom.f P.\u03c0 0 \u226b f) (HomologicalComplex.Hom.f Q.\u03c0 0)) \u226b\n              HomologicalComplex.Hom.f Q.\u03c0 0 =\n            0) \u226b\n      HomologicalComplex.d Q.complex 1 0 =\n    HomologicalComplex.d P.complex 1 0 \u226b\n      factorThru (HomologicalComplex.Hom.f P.\u03c0 0 \u226b f) (HomologicalComplex.Hom.f Q.\u03c0 0)"}, {"tactic": "simp", "state_before": "C : Type u\ninst\u271d\u2074 : Category C\ninst\u271d\u00b3 : HasZeroObject C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasEqualizers C\ninst\u271d : HasImages C\nY Z : C\nf : Y \u27f6 Z\nP : ProjectiveResolution Y\nQ : ProjectiveResolution Z\n\u22a2 Exact.lift\n        (HomologicalComplex.d P.complex 1 0 \u226b\n          factorThru (HomologicalComplex.Hom.f P.\u03c0 0 \u226b f) (HomologicalComplex.Hom.f Q.\u03c0 0))\n        (HomologicalComplex.d Q.complex 1 0) (HomologicalComplex.Hom.f Q.\u03c0 0)\n        (_ : Exact (HomologicalComplex.d Q.complex 1 0) (HomologicalComplex.Hom.f Q.\u03c0 0))\n        (_ :\n          (HomologicalComplex.d P.complex 1 0 \u226b\n                factorThru (HomologicalComplex.Hom.f P.\u03c0 0 \u226b f) (HomologicalComplex.Hom.f Q.\u03c0 0)) \u226b\n              HomologicalComplex.Hom.f Q.\u03c0 0 =\n            0) \u226b\n      HomologicalComplex.d Q.complex 1 0 =\n    HomologicalComplex.d P.complex 1 0 \u226b\n      factorThru (HomologicalComplex.Hom.f P.\u03c0 0 \u226b f) (HomologicalComplex.Hom.f Q.\u03c0 0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Connected.lean", "full_name": "locallyConnectedSpace_iff_connected_subsets", "start": [1183, 1], "end": [1194, 96], "traced_tactics": [{"tactic": "constructor", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.115745\n\u03c0 : \u03b9 \u2192 Type ?u.115750\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 LocallyConnectedSpace \u03b1 \u2194 \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U", "state_after": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.115745\n\u03c0 : \u03b9 \u2192 Type ?u.115750\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 LocallyConnectedSpace \u03b1 \u2192 \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U\n\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.115745\n\u03c0 : \u03b9 \u2192 Type ?u.115750\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 (\u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U) \u2192 LocallyConnectedSpace \u03b1"}, {"tactic": "rw [locallyConnectedSpace_iff_open_connected_subsets]", "state_before": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.115745\n\u03c0 : \u03b9 \u2192 Type ?u.115750\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 LocallyConnectedSpace \u03b1 \u2192 \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U", "state_after": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.115745\n\u03c0 : \u03b9 \u2192 Type ?u.115750\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 (\u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x \u2208 V \u2227 IsConnected V) \u2192\n    \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U"}, {"tactic": "intro h x U hxU", "state_before": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.115745\n\u03c0 : \u03b9 \u2192 Type ?u.115750\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 (\u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x \u2208 V \u2227 IsConnected V) \u2192\n    \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U", "state_after": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.115745\n\u03c0 : \u03b9 \u2192 Type ?u.115750\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x \u2208 V \u2227 IsConnected V\nx : \u03b1\nU : Set \u03b1\nhxU : U \u2208 \ud835\udcdd x\n\u22a2 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U"}, {"tactic": "rcases h x U hxU with \u27e8V, hVU, hV\u2081, hxV, hV\u2082\u27e9", "state_before": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.115745\n\u03c0 : \u03b9 \u2192 Type ?u.115750\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x \u2208 V \u2227 IsConnected V\nx : \u03b1\nU : Set \u03b1\nhxU : U \u2208 \ud835\udcdd x\n\u22a2 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U", "state_after": "case mp.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.115745\n\u03c0 : \u03b9 \u2192 Type ?u.115750\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x \u2208 V \u2227 IsConnected V\nx : \u03b1\nU : Set \u03b1\nhxU : U \u2208 \ud835\udcdd x\nV : Set \u03b1\nhVU : V \u2286 U\nhV\u2081 : IsOpen V\nhxV : x \u2208 V\nhV\u2082 : IsConnected V\n\u22a2 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U"}, {"tactic": "exact \u27e8V, hV\u2081.mem_nhds hxV, hV\u2082.isPreconnected, hVU\u27e9", "state_before": "case mp.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.115745\n\u03c0 : \u03b9 \u2192 Type ?u.115750\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x \u2208 V \u2227 IsConnected V\nx : \u03b1\nU : Set \u03b1\nhxU : U \u2208 \ud835\udcdd x\nV : Set \u03b1\nhVU : V \u2286 U\nhV\u2081 : IsOpen V\nhxV : x \u2208 V\nhV\u2082 : IsConnected V\n\u22a2 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U", "state_after": "no goals"}, {"tactic": "rw [locallyConnectedSpace_iff_connectedComponentIn_open]", "state_before": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.115745\n\u03c0 : \u03b9 \u2192 Type ?u.115750\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 (\u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U) \u2192 LocallyConnectedSpace \u03b1", "state_after": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.115745\n\u03c0 : \u03b9 \u2192 Type ?u.115750\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 (\u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U) \u2192\n    \u2200 (F : Set \u03b1), IsOpen F \u2192 \u2200 (x : \u03b1), x \u2208 F \u2192 IsOpen (connectedComponentIn F x)"}, {"tactic": "refine' fun h U hU x _ => isOpen_iff_mem_nhds.mpr fun y hy => _", "state_before": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.115745\n\u03c0 : \u03b9 \u2192 Type ?u.115750\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 (\u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U) \u2192\n    \u2200 (F : Set \u03b1), IsOpen F \u2192 \u2200 (x : \u03b1), x \u2208 F \u2192 IsOpen (connectedComponentIn F x)", "state_after": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.115745\n\u03c0 : \u03b9 \u2192 Type ?u.115750\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U\nU : Set \u03b1\nhU : IsOpen U\nx : \u03b1\nx\u271d : x \u2208 U\ny : \u03b1\nhy : y \u2208 connectedComponentIn U x\n\u22a2 connectedComponentIn U x \u2208 \ud835\udcdd y"}, {"tactic": "rw [connectedComponentIn_eq hy]", "state_before": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.115745\n\u03c0 : \u03b9 \u2192 Type ?u.115750\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U\nU : Set \u03b1\nhU : IsOpen U\nx : \u03b1\nx\u271d : x \u2208 U\ny : \u03b1\nhy : y \u2208 connectedComponentIn U x\n\u22a2 connectedComponentIn U x \u2208 \ud835\udcdd y", "state_after": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.115745\n\u03c0 : \u03b9 \u2192 Type ?u.115750\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U\nU : Set \u03b1\nhU : IsOpen U\nx : \u03b1\nx\u271d : x \u2208 U\ny : \u03b1\nhy : y \u2208 connectedComponentIn U x\n\u22a2 connectedComponentIn U y \u2208 \ud835\udcdd y"}, {"tactic": "rcases h y U (hU.mem_nhds <| (connectedComponentIn_subset _ _) hy) with \u27e8V, hVy, hV, hVU\u27e9", "state_before": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.115745\n\u03c0 : \u03b9 \u2192 Type ?u.115750\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U\nU : Set \u03b1\nhU : IsOpen U\nx : \u03b1\nx\u271d : x \u2208 U\ny : \u03b1\nhy : y \u2208 connectedComponentIn U x\n\u22a2 connectedComponentIn U y \u2208 \ud835\udcdd y", "state_after": "case mpr.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.115745\n\u03c0 : \u03b9 \u2192 Type ?u.115750\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U\nU : Set \u03b1\nhU : IsOpen U\nx : \u03b1\nx\u271d : x \u2208 U\ny : \u03b1\nhy : y \u2208 connectedComponentIn U x\nV : Set \u03b1\nhVy : V \u2208 \ud835\udcdd y\nhV : IsPreconnected V\nhVU : V \u2286 U\n\u22a2 connectedComponentIn U y \u2208 \ud835\udcdd y"}, {"tactic": "exact Filter.mem_of_superset hVy (hV.subset_connectedComponentIn (mem_of_mem_nhds hVy) hVU)", "state_before": "case mpr.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type ?u.115745\n\u03c0 : \u03b9 \u2192 Type ?u.115750\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U\nU : Set \u03b1\nhU : IsOpen U\nx : \u03b1\nx\u271d : x \u2208 U\ny : \u03b1\nhy : y \u2208 connectedComponentIn U x\nV : Set \u03b1\nhVy : V \u2208 \ud835\udcdd y\nhV : IsPreconnected V\nhVU : V \u2286 U\n\u22a2 connectedComponentIn U y \u2208 \ud835\udcdd y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Matrix/Block.lean", "full_name": "Matrix.toBlock_mul_eq_add", "start": [930, 1], "end": [936, 82], "traced_tactics": [{"tactic": "classical\n  ext (i k)\n  simp only [toBlock_apply, mul_apply, Pi.add_apply]\n  exact (Fintype.sum_subtype_add_sum_subtype q fun x => A (\u2191i) x * B x \u2191k).symm", "state_before": "l : Type ?u.270844\nm\u271d : Type ?u.270847\nn\u271d : Type ?u.270850\no : Type ?u.270853\np\u271d : Type ?u.270856\nq\u271d : Type ?u.270859\nm' : o \u2192 Type ?u.270864\nn' : o \u2192 Type ?u.270869\np' : o \u2192 Type ?u.270874\nR : Type u_4\nS : Type ?u.270880\n\u03b1 : Type ?u.270883\n\u03b2 : Type ?u.270886\ninst\u271d\u00b2 : CommRing R\nm : Type u_1\nn : Type u_2\nk : Type u_3\ninst\u271d\u00b9 : Fintype n\np : m \u2192 Prop\nq : n \u2192 Prop\ninst\u271d : DecidablePred q\nr : k \u2192 Prop\nA : Matrix m n R\nB : Matrix n k R\n\u22a2 toBlock (A \u2b1d B) p r = toBlock A p q \u2b1d toBlock B q r + (toBlock A p fun i => \u00acq i) \u2b1d toBlock B (fun i => \u00acq i) r", "state_after": "no goals"}, {"tactic": "ext (i k)", "state_before": "l : Type ?u.270844\nm\u271d : Type ?u.270847\nn\u271d : Type ?u.270850\no : Type ?u.270853\np\u271d : Type ?u.270856\nq\u271d : Type ?u.270859\nm' : o \u2192 Type ?u.270864\nn' : o \u2192 Type ?u.270869\np' : o \u2192 Type ?u.270874\nR : Type u_4\nS : Type ?u.270880\n\u03b1 : Type ?u.270883\n\u03b2 : Type ?u.270886\ninst\u271d\u00b2 : CommRing R\nm : Type u_1\nn : Type u_2\nk : Type u_3\ninst\u271d\u00b9 : Fintype n\np : m \u2192 Prop\nq : n \u2192 Prop\ninst\u271d : DecidablePred q\nr : k \u2192 Prop\nA : Matrix m n R\nB : Matrix n k R\n\u22a2 toBlock (A \u2b1d B) p r = toBlock A p q \u2b1d toBlock B q r + (toBlock A p fun i => \u00acq i) \u2b1d toBlock B (fun i => \u00acq i) r", "state_after": "case a.h\nl : Type ?u.270844\nm\u271d : Type ?u.270847\nn\u271d : Type ?u.270850\no : Type ?u.270853\np\u271d : Type ?u.270856\nq\u271d : Type ?u.270859\nm' : o \u2192 Type ?u.270864\nn' : o \u2192 Type ?u.270869\np' : o \u2192 Type ?u.270874\nR : Type u_4\nS : Type ?u.270880\n\u03b1 : Type ?u.270883\n\u03b2 : Type ?u.270886\ninst\u271d\u00b2 : CommRing R\nm : Type u_1\nn : Type u_2\nk\u271d : Type u_3\ninst\u271d\u00b9 : Fintype n\np : m \u2192 Prop\nq : n \u2192 Prop\ninst\u271d : DecidablePred q\nr : k\u271d \u2192 Prop\nA : Matrix m n R\nB : Matrix n k\u271d R\ni : { a // p a }\nk : { a // r a }\n\u22a2 toBlock (A \u2b1d B) p r i k =\n    (toBlock A p q \u2b1d toBlock B q r + (toBlock A p fun i => \u00acq i) \u2b1d toBlock B (fun i => \u00acq i) r) i k"}, {"tactic": "simp only [toBlock_apply, mul_apply, Pi.add_apply]", "state_before": "case a.h\nl : Type ?u.270844\nm\u271d : Type ?u.270847\nn\u271d : Type ?u.270850\no : Type ?u.270853\np\u271d : Type ?u.270856\nq\u271d : Type ?u.270859\nm' : o \u2192 Type ?u.270864\nn' : o \u2192 Type ?u.270869\np' : o \u2192 Type ?u.270874\nR : Type u_4\nS : Type ?u.270880\n\u03b1 : Type ?u.270883\n\u03b2 : Type ?u.270886\ninst\u271d\u00b2 : CommRing R\nm : Type u_1\nn : Type u_2\nk\u271d : Type u_3\ninst\u271d\u00b9 : Fintype n\np : m \u2192 Prop\nq : n \u2192 Prop\ninst\u271d : DecidablePred q\nr : k\u271d \u2192 Prop\nA : Matrix m n R\nB : Matrix n k\u271d R\ni : { a // p a }\nk : { a // r a }\n\u22a2 toBlock (A \u2b1d B) p r i k =\n    (toBlock A p q \u2b1d toBlock B q r + (toBlock A p fun i => \u00acq i) \u2b1d toBlock B (fun i => \u00acq i) r) i k", "state_after": "case a.h\nl : Type ?u.270844\nm\u271d : Type ?u.270847\nn\u271d : Type ?u.270850\no : Type ?u.270853\np\u271d : Type ?u.270856\nq\u271d : Type ?u.270859\nm' : o \u2192 Type ?u.270864\nn' : o \u2192 Type ?u.270869\np' : o \u2192 Type ?u.270874\nR : Type u_4\nS : Type ?u.270880\n\u03b1 : Type ?u.270883\n\u03b2 : Type ?u.270886\ninst\u271d\u00b2 : CommRing R\nm : Type u_1\nn : Type u_2\nk\u271d : Type u_3\ninst\u271d\u00b9 : Fintype n\np : m \u2192 Prop\nq : n \u2192 Prop\ninst\u271d : DecidablePred q\nr : k\u271d \u2192 Prop\nA : Matrix m n R\nB : Matrix n k\u271d R\ni : { a // p a }\nk : { a // r a }\n\u22a2 \u2211 j : n, A (\u2191i) j * B j \u2191k =\n    (toBlock A p q \u2b1d toBlock B q r + (toBlock A p fun i => \u00acq i) \u2b1d toBlock B (fun i => \u00acq i) r) i k"}, {"tactic": "exact (Fintype.sum_subtype_add_sum_subtype q fun x => A (\u2191i) x * B x \u2191k).symm", "state_before": "case a.h\nl : Type ?u.270844\nm\u271d : Type ?u.270847\nn\u271d : Type ?u.270850\no : Type ?u.270853\np\u271d : Type ?u.270856\nq\u271d : Type ?u.270859\nm' : o \u2192 Type ?u.270864\nn' : o \u2192 Type ?u.270869\np' : o \u2192 Type ?u.270874\nR : Type u_4\nS : Type ?u.270880\n\u03b1 : Type ?u.270883\n\u03b2 : Type ?u.270886\ninst\u271d\u00b2 : CommRing R\nm : Type u_1\nn : Type u_2\nk\u271d : Type u_3\ninst\u271d\u00b9 : Fintype n\np : m \u2192 Prop\nq : n \u2192 Prop\ninst\u271d : DecidablePred q\nr : k\u271d \u2192 Prop\nA : Matrix m n R\nB : Matrix n k\u271d R\ni : { a // p a }\nk : { a // r a }\n\u22a2 \u2211 j : n, A (\u2191i) j * B j \u2191k =\n    (toBlock A p q \u2b1d toBlock B q r + (toBlock A p fun i => \u00acq i) \u2b1d toBlock B (fun i => \u00acq i) r) i k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "full_name": "LocalHomeomorph.extend_coe", "start": [786, 1], "end": [787, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm.lean", "full_name": "ContinuousLinearMap.op_norm_eq_of_bounds", "start": [179, 1], "end": [184, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "full_name": "zpowersMulHom_symm_apply", "start": [991, 1], "end": [993, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.seq_of_forall_finite_exists", "start": [1174, 1], "end": [1183, 30], "traced_tactics": [{"tactic": "haveI : Nonempty \u03b3 := (h \u2205 finite_empty).nonempty", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3\u271d : Type x\n\u03b3 : Type u_1\nP : \u03b3 \u2192 Set \u03b3 \u2192 Prop\nh : \u2200 (t : Set \u03b3), Set.Finite t \u2192 \u2203 c, P c t\n\u22a2 \u2203 u, \u2200 (n : \u2115), P (u n) (u '' Iio n)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3\u271d : Type x\n\u03b3 : Type u_1\nP : \u03b3 \u2192 Set \u03b3 \u2192 Prop\nh : \u2200 (t : Set \u03b3), Set.Finite t \u2192 \u2203 c, P c t\nthis : Nonempty \u03b3\n\u22a2 \u2203 u, \u2200 (n : \u2115), P (u n) (u '' Iio n)"}, {"tactic": "choose! c hc using h", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3\u271d : Type x\n\u03b3 : Type u_1\nP : \u03b3 \u2192 Set \u03b3 \u2192 Prop\nh : \u2200 (t : Set \u03b3), Set.Finite t \u2192 \u2203 c, P c t\nthis : Nonempty \u03b3\n\u22a2 \u2203 u, \u2200 (n : \u2115), P (u n) (u '' Iio n)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3\u271d : Type x\n\u03b3 : Type u_1\nP : \u03b3 \u2192 Set \u03b3 \u2192 Prop\nthis : Nonempty \u03b3\nc : Set \u03b3 \u2192 \u03b3\nhc : \u2200 (t : Set \u03b3), Set.Finite t \u2192 P (c t) t\n\u22a2 \u2203 u, \u2200 (n : \u2115), P (u n) (u '' Iio n)"}, {"tactic": "set f : (n : \u2115) \u2192 (g : (m : \u2115) \u2192 m < n \u2192 \u03b3) \u2192 \u03b3 := fun n g => c (range fun k : Iio n => g k.1 k.2)", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3\u271d : Type x\n\u03b3 : Type u_1\nP : \u03b3 \u2192 Set \u03b3 \u2192 Prop\nthis : Nonempty \u03b3\nc : Set \u03b3 \u2192 \u03b3\nhc : \u2200 (t : Set \u03b3), Set.Finite t \u2192 P (c t) t\n\u22a2 \u2203 u, \u2200 (n : \u2115), P (u n) (u '' Iio n)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3\u271d : Type x\n\u03b3 : Type u_1\nP : \u03b3 \u2192 Set \u03b3 \u2192 Prop\nthis : Nonempty \u03b3\nc : Set \u03b3 \u2192 \u03b3\nhc : \u2200 (t : Set \u03b3), Set.Finite t \u2192 P (c t) t\nf : (n : \u2115) \u2192 ((m : \u2115) \u2192 m < n \u2192 \u03b3) \u2192 \u03b3 := fun n g => c (range fun k => g \u2191k (_ : \u2191k \u2208 Iio n))\n\u22a2 \u2203 u, \u2200 (n : \u2115), P (u n) (u '' Iio n)"}, {"tactic": "set u : \u2115 \u2192 \u03b3 := fun n => Nat.strongRecOn' n f", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3\u271d : Type x\n\u03b3 : Type u_1\nP : \u03b3 \u2192 Set \u03b3 \u2192 Prop\nthis : Nonempty \u03b3\nc : Set \u03b3 \u2192 \u03b3\nhc : \u2200 (t : Set \u03b3), Set.Finite t \u2192 P (c t) t\nf : (n : \u2115) \u2192 ((m : \u2115) \u2192 m < n \u2192 \u03b3) \u2192 \u03b3 := fun n g => c (range fun k => g \u2191k (_ : \u2191k \u2208 Iio n))\n\u22a2 \u2203 u, \u2200 (n : \u2115), P (u n) (u '' Iio n)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3\u271d : Type x\n\u03b3 : Type u_1\nP : \u03b3 \u2192 Set \u03b3 \u2192 Prop\nthis : Nonempty \u03b3\nc : Set \u03b3 \u2192 \u03b3\nhc : \u2200 (t : Set \u03b3), Set.Finite t \u2192 P (c t) t\nf : (n : \u2115) \u2192 ((m : \u2115) \u2192 m < n \u2192 \u03b3) \u2192 \u03b3 := fun n g => c (range fun k => g \u2191k (_ : \u2191k \u2208 Iio n))\nu : \u2115 \u2192 \u03b3 := fun n => Nat.strongRecOn' n f\n\u22a2 \u2203 u, \u2200 (n : \u2115), P (u n) (u '' Iio n)"}, {"tactic": "refine' \u27e8u, fun n => _\u27e9", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3\u271d : Type x\n\u03b3 : Type u_1\nP : \u03b3 \u2192 Set \u03b3 \u2192 Prop\nthis : Nonempty \u03b3\nc : Set \u03b3 \u2192 \u03b3\nhc : \u2200 (t : Set \u03b3), Set.Finite t \u2192 P (c t) t\nf : (n : \u2115) \u2192 ((m : \u2115) \u2192 m < n \u2192 \u03b3) \u2192 \u03b3 := fun n g => c (range fun k => g \u2191k (_ : \u2191k \u2208 Iio n))\nu : \u2115 \u2192 \u03b3 := fun n => Nat.strongRecOn' n f\n\u22a2 \u2203 u, \u2200 (n : \u2115), P (u n) (u '' Iio n)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3\u271d : Type x\n\u03b3 : Type u_1\nP : \u03b3 \u2192 Set \u03b3 \u2192 Prop\nthis : Nonempty \u03b3\nc : Set \u03b3 \u2192 \u03b3\nhc : \u2200 (t : Set \u03b3), Set.Finite t \u2192 P (c t) t\nf : (n : \u2115) \u2192 ((m : \u2115) \u2192 m < n \u2192 \u03b3) \u2192 \u03b3 := fun n g => c (range fun k => g \u2191k (_ : \u2191k \u2208 Iio n))\nu : \u2115 \u2192 \u03b3 := fun n => Nat.strongRecOn' n f\nn : \u2115\n\u22a2 P (u n) (u '' Iio n)"}, {"tactic": "convert hc (u '' Iio n) ((finite_lt_nat _).image _)", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3\u271d : Type x\n\u03b3 : Type u_1\nP : \u03b3 \u2192 Set \u03b3 \u2192 Prop\nthis : Nonempty \u03b3\nc : Set \u03b3 \u2192 \u03b3\nhc : \u2200 (t : Set \u03b3), Set.Finite t \u2192 P (c t) t\nf : (n : \u2115) \u2192 ((m : \u2115) \u2192 m < n \u2192 \u03b3) \u2192 \u03b3 := fun n g => c (range fun k => g \u2191k (_ : \u2191k \u2208 Iio n))\nu : \u2115 \u2192 \u03b3 := fun n => Nat.strongRecOn' n f\nn : \u2115\n\u22a2 P (u n) (u '' Iio n)", "state_after": "case h.e'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3\u271d : Type x\n\u03b3 : Type u_1\nP : \u03b3 \u2192 Set \u03b3 \u2192 Prop\nthis : Nonempty \u03b3\nc : Set \u03b3 \u2192 \u03b3\nhc : \u2200 (t : Set \u03b3), Set.Finite t \u2192 P (c t) t\nf : (n : \u2115) \u2192 ((m : \u2115) \u2192 m < n \u2192 \u03b3) \u2192 \u03b3 := fun n g => c (range fun k => g \u2191k (_ : \u2191k \u2208 Iio n))\nu : \u2115 \u2192 \u03b3 := fun n => Nat.strongRecOn' n f\nn : \u2115\n\u22a2 u n = c (u '' Iio n)"}, {"tactic": "rw [image_eq_range]", "state_before": "case h.e'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3\u271d : Type x\n\u03b3 : Type u_1\nP : \u03b3 \u2192 Set \u03b3 \u2192 Prop\nthis : Nonempty \u03b3\nc : Set \u03b3 \u2192 \u03b3\nhc : \u2200 (t : Set \u03b3), Set.Finite t \u2192 P (c t) t\nf : (n : \u2115) \u2192 ((m : \u2115) \u2192 m < n \u2192 \u03b3) \u2192 \u03b3 := fun n g => c (range fun k => g \u2191k (_ : \u2191k \u2208 Iio n))\nu : \u2115 \u2192 \u03b3 := fun n => Nat.strongRecOn' n f\nn : \u2115\n\u22a2 u n = c (u '' Iio n)", "state_after": "case h.e'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3\u271d : Type x\n\u03b3 : Type u_1\nP : \u03b3 \u2192 Set \u03b3 \u2192 Prop\nthis : Nonempty \u03b3\nc : Set \u03b3 \u2192 \u03b3\nhc : \u2200 (t : Set \u03b3), Set.Finite t \u2192 P (c t) t\nf : (n : \u2115) \u2192 ((m : \u2115) \u2192 m < n \u2192 \u03b3) \u2192 \u03b3 := fun n g => c (range fun k => g \u2191k (_ : \u2191k \u2208 Iio n))\nu : \u2115 \u2192 \u03b3 := fun n => Nat.strongRecOn' n f\nn : \u2115\n\u22a2 u n = c (range fun x => u \u2191x)"}, {"tactic": "exact Nat.strongRecOn'_beta", "state_before": "case h.e'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03b3\u271d : Type x\n\u03b3 : Type u_1\nP : \u03b3 \u2192 Set \u03b3 \u2192 Prop\nthis : Nonempty \u03b3\nc : Set \u03b3 \u2192 \u03b3\nhc : \u2200 (t : Set \u03b3), Set.Finite t \u2192 P (c t) t\nf : (n : \u2115) \u2192 ((m : \u2115) \u2192 m < n \u2192 \u03b3) \u2192 \u03b3 := fun n g => c (range fun k => g \u2191k (_ : \u2191k \u2208 Iio n))\nu : \u2115 \u2192 \u03b3 := fun n => Nat.strongRecOn' n f\nn : \u2115\n\u22a2 u n = c (range fun x => u \u2191x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/LocallyConstant/Basic.lean", "full_name": "LocallyConstant.coe_inj", "start": [286, 1], "end": [287, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "full_name": "ENNReal.measurable_toNNReal", "start": [1865, 1], "end": [1866, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.filter_zero", "start": [1928, 1], "end": [1929, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.inter_left_comm", "start": [1620, 1], "end": [1621, 55], "traced_tactics": [{"tactic": "simp only [mem_inter, and_left_comm]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.181971\n\u03b3 : Type ?u.181974\ninst\u271d : DecidableEq \u03b1\ns s\u2081\u271d s\u2082\u271d t t\u2081 t\u2082 u v : Finset \u03b1\na b : \u03b1\ns\u2081 s\u2082 s\u2083 : Finset \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 s\u2081 \u2229 (s\u2082 \u2229 s\u2083) \u2194 x\u271d \u2208 s\u2082 \u2229 (s\u2081 \u2229 s\u2083)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Homology/Exact.lean", "full_name": "CategoryTheory.exact_comp_hom_inv_comp_iff", "start": [186, 1], "end": [187, 83], "traced_tactics": [{"tactic": "simpa using exact_comp_inv_hom_comp i h", "state_before": "V : Type u\ninst\u271d\u00b3 : Category V\ninst\u271d\u00b2 : HasImages V\nA B C D : V\nf : A \u27f6 B\ng : B \u27f6 C\nh\u271d : C \u27f6 D\ninst\u271d\u00b9 : HasZeroMorphisms V\ninst\u271d : HasEqualizers V\ni : B \u2245 D\nh : Exact (f \u226b i.hom) (i.inv \u226b g)\n\u22a2 Exact f g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Perm.lean", "full_name": "List.subperm_append_diff_self_of_count_le", "start": [898, 1], "end": [914, 25], "traced_tactics": [{"tactic": "induction' l\u2081 with hd tl IH generalizing l\u2082", "state_before": "\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\n\u22a2 l\u2081 ++ List.diff l\u2082 l\u2081 ~ l\u2082", "state_after": "case nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 [] \u2192 count x [] \u2264 count x l\u2082\n\u22a2 [] ++ List.diff l\u2082 [] ~ l\u2082\n\ncase cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\n\u22a2 hd :: tl ++ List.diff l\u2082 (hd :: tl) ~ l\u2082"}, {"tactic": "simp", "state_before": "case nil\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 [] \u2192 count x [] \u2264 count x l\u2082\n\u22a2 [] ++ List.diff l\u2082 [] ~ l\u2082", "state_after": "no goals"}, {"tactic": "have : hd \u2208 l\u2082 := by\n  rw [\u2190 count_pos]\n  exact lt_of_lt_of_le (count_pos.mpr (mem_cons_self _ _)) (h hd (mem_cons_self _ _))", "state_before": "case cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\n\u22a2 hd :: tl ++ List.diff l\u2082 (hd :: tl) ~ l\u2082", "state_after": "case cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\nthis : hd \u2208 l\u2082\n\u22a2 hd :: tl ++ List.diff l\u2082 (hd :: tl) ~ l\u2082"}, {"tactic": "replace := perm_cons_erase this", "state_before": "case cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\nthis : hd \u2208 l\u2082\n\u22a2 hd :: tl ++ List.diff l\u2082 (hd :: tl) ~ l\u2082", "state_after": "case cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\n\u22a2 hd :: tl ++ List.diff l\u2082 (hd :: tl) ~ l\u2082"}, {"tactic": "refine' Perm.trans _ this.symm", "state_before": "case cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\n\u22a2 hd :: tl ++ List.diff l\u2082 (hd :: tl) ~ l\u2082", "state_after": "case cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\n\u22a2 hd :: tl ++ List.diff l\u2082 (hd :: tl) ~ hd :: List.erase l\u2082 hd"}, {"tactic": "rw [cons_append, diff_cons, perm_cons]", "state_before": "case cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\n\u22a2 hd :: tl ++ List.diff l\u2082 (hd :: tl) ~ hd :: List.erase l\u2082 hd", "state_after": "case cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\n\u22a2 tl ++ List.diff (List.erase l\u2082 hd) tl ~ List.erase l\u2082 hd"}, {"tactic": "refine' IH fun x hx => _", "state_before": "case cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\n\u22a2 tl ++ List.diff (List.erase l\u2082 hd) tl ~ List.erase l\u2082 hd", "state_after": "case cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\nx : \u03b1\nhx : x \u2208 tl\n\u22a2 count x tl \u2264 count x (List.erase l\u2082 hd)"}, {"tactic": "specialize h x (mem_cons_of_mem _ hx)", "state_before": "case cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\nx : \u03b1\nhx : x \u2208 tl\n\u22a2 count x tl \u2264 count x (List.erase l\u2082 hd)", "state_after": "case cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\nx : \u03b1\nhx : x \u2208 tl\nh : count x (hd :: tl) \u2264 count x l\u2082\n\u22a2 count x tl \u2264 count x (List.erase l\u2082 hd)"}, {"tactic": "rw [perm_iff_count.mp this] at h", "state_before": "case cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\nx : \u03b1\nhx : x \u2208 tl\nh : count x (hd :: tl) \u2264 count x l\u2082\n\u22a2 count x tl \u2264 count x (List.erase l\u2082 hd)", "state_after": "case cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\nx : \u03b1\nhx : x \u2208 tl\nh : count x (hd :: tl) \u2264 count x (hd :: List.erase l\u2082 hd)\n\u22a2 count x tl \u2264 count x (List.erase l\u2082 hd)"}, {"tactic": "by_cases hx : x = hd", "state_before": "case cons\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\nx : \u03b1\nhx : x \u2208 tl\nh : count x (hd :: tl) \u2264 count x (hd :: List.erase l\u2082 hd)\n\u22a2 count x tl \u2264 count x (List.erase l\u2082 hd)", "state_after": "case pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\nx : \u03b1\nhx\u271d : x \u2208 tl\nh : count x (hd :: tl) \u2264 count x (hd :: List.erase l\u2082 hd)\nhx : x = hd\n\u22a2 count x tl \u2264 count x (List.erase l\u2082 hd)\n\ncase neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\nx : \u03b1\nhx\u271d : x \u2208 tl\nh : count x (hd :: tl) \u2264 count x (hd :: List.erase l\u2082 hd)\nhx : \u00acx = hd\n\u22a2 count x tl \u2264 count x (List.erase l\u2082 hd)"}, {"tactic": "rw [\u2190 count_pos]", "state_before": "\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\n\u22a2 hd \u2208 l\u2082", "state_after": "\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\n\u22a2 0 < count hd l\u2082"}, {"tactic": "exact lt_of_lt_of_le (count_pos.mpr (mem_cons_self _ _)) (h hd (mem_cons_self _ _))", "state_before": "\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nh : \u2200 (x : \u03b1), x \u2208 hd :: tl \u2192 count x (hd :: tl) \u2264 count x l\u2082\n\u22a2 0 < count hd l\u2082", "state_after": "no goals"}, {"tactic": "subst hd", "state_before": "case pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\nx : \u03b1\nhx\u271d : x \u2208 tl\nh : count x (hd :: tl) \u2264 count x (hd :: List.erase l\u2082 hd)\nhx : x = hd\n\u22a2 count x tl \u2264 count x (List.erase l\u2082 hd)", "state_after": "case pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nx : \u03b1\nhx : x \u2208 tl\nthis : l\u2082 ~ x :: List.erase l\u2082 x\nh : count x (x :: tl) \u2264 count x (x :: List.erase l\u2082 x)\n\u22a2 count x tl \u2264 count x (List.erase l\u2082 x)"}, {"tactic": "simpa [Nat.succ_le_succ_iff] using h", "state_before": "case pos\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nx : \u03b1\nhx : x \u2208 tl\nthis : l\u2082 ~ x :: List.erase l\u2082 x\nh : count x (x :: tl) \u2264 count x (x :: List.erase l\u2082 x)\n\u22a2 count x tl \u2264 count x (List.erase l\u2082 x)", "state_after": "no goals"}, {"tactic": "simpa [hx] using h", "state_before": "case neg\n\u03b1 : Type uu\n\u03b2 : Type vv\nl\u2081\u271d l\u2082\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082\u271d : List \u03b1\nh\u271d : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 count x l\u2081 \u2264 count x l\u2082\u271d\nhd : \u03b1\ntl : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b1}, (\u2200 (x : \u03b1), x \u2208 tl \u2192 count x tl \u2264 count x l\u2082) \u2192 tl ++ List.diff l\u2082 tl ~ l\u2082\nl\u2082 : List \u03b1\nthis : l\u2082 ~ hd :: List.erase l\u2082 hd\nx : \u03b1\nhx\u271d : x \u2208 tl\nh : count x (hd :: tl) \u2264 count x (hd :: List.erase l\u2082 hd)\nhx : \u00acx = hd\n\u22a2 count x tl \u2264 count x (List.erase l\u2082 hd)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/Adjoint.lean", "full_name": "LinearMap.IsSymmetric.isSelfAdjoint", "start": [314, 1], "end": [316, 66], "traced_tactics": [{"tactic": "rwa [\u2190 ContinuousLinearMap.isSelfAdjoint_iff_isSymmetric] at hA", "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.1397379\nG : Type ?u.1397382\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] E\nhA : LinearMap.IsSymmetric \u2191A\n\u22a2 IsSelfAdjoint A", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Regular/Basic.lean", "full_name": "isLeftRegular_of_leftCancelSemigroup", "start": [339, 1], "end": [341, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Category/Basic.lean", "full_name": "CategoryTheory.cancel_mono_id", "start": [300, 1], "end": [302, 7], "traced_tactics": [{"tactic": "convert cancel_mono f", "state_before": "C : Type u\ninst\u271d\u00b9 : Category C\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d : Mono f\ng : X \u27f6 X\n\u22a2 g \u226b f = f \u2194 g = \ud835\udfd9 X", "state_after": "case h.e'_1.h.e'_3.h\nC : Type u\ninst\u271d\u00b9 : Category C\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d : Mono f\ng : X \u27f6 X\n\u22a2 f = \ud835\udfd9 X \u226b f"}, {"tactic": "simp", "state_before": "case h.e'_1.h.e'_3.h\nC : Type u\ninst\u271d\u00b9 : Category C\nX Y Z : C\nf : X \u27f6 Y\ninst\u271d : Mono f\ng : X \u27f6 X\n\u22a2 f = \ud835\udfd9 X \u226b f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Lattice.lean", "full_name": "sup_le", "start": [168, 1], "end": [169, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Parity.lean", "full_name": "Nat.not_even_bit1", "start": [130, 1], "end": [130, 79], "traced_tactics": [{"tactic": "simp [bit1, parity_simps]", "state_before": "m n\u271d n : \u2115\n\u22a2 \u00acEven (bit1 n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.le_add_right", "start": [651, 1], "end": [651, 100], "traced_tactics": [{"tactic": "simpa using add_le_add_left (zero_le t) s", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.41033\n\u03b3 : Type ?u.41036\ns t : Multiset \u03b1\n\u22a2 s \u2264 s + t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Basic.lean", "full_name": "Multiset.toFinset_sum_count_eq", "start": [2136, 1], "end": [2141, 26], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b9 : Type ?u.919655\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\ninst\u271d : DecidableEq \u03b1\ns : Multiset \u03b1\n\u22a2 sum (map (fun x => 1) s) = \u2191card s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicGeometry/Stalks.lean", "full_name": "AlgebraicGeometry.PresheafedSpace.stalkMap_germ", "start": [59, 1], "end": [62, 68], "traced_tactics": [{"tactic": "rw [stalkMap, stalkFunctor_map_germ_assoc, stalkPushforward_germ]", "state_before": "C : Type u\ninst\u271d\u00b9 : Category C\ninst\u271d : HasColimits C\nX Y : PresheafedSpace C\n\u03b1 : X \u27f6 Y\nU : Opens \u2191\u2191Y\nx : { x // x \u2208 (Opens.map \u03b1.base).obj U }\n\u22a2 germ Y.presheaf\n        { val := (CategoryTheory.forget TopCat).map \u03b1.base \u2191x, property := (_ : \u2191x \u2208 (Opens.map \u03b1.base).obj U) } \u226b\n      stalkMap \u03b1 \u2191x =\n    \u03b1.c.app U.op \u226b germ X.presheaf x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Lie/Subalgebra.lean", "full_name": "LieSubalgebra.inf_coe_to_submodule", "start": [554, 1], "end": [556, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/NaturalOps.lean", "full_name": "Ordinal.nadd_le_iff", "start": [221, 1], "end": [223, 22], "traced_tactics": [{"tactic": "rw [nadd_def]", "state_before": "a b c : Ordinal\n\u22a2 b \u266f c \u2264 a \u2194 (\u2200 (b' : Ordinal), b' < b \u2192 b' \u266f c < a) \u2227 \u2200 (c' : Ordinal), c' < c \u2192 b \u266f c' < a", "state_after": "a b c : Ordinal\n\u22a2 max (blsub b fun a' x => a' \u266f c) (blsub c fun b' x => b \u266f b') \u2264 a \u2194\n    (\u2200 (b' : Ordinal), b' < b \u2192 b' \u266f c < a) \u2227 \u2200 (c' : Ordinal), c' < c \u2192 b \u266f c' < a"}, {"tactic": "simp [blsub_le_iff]", "state_before": "a b c : Ordinal\n\u22a2 max (blsub b fun a' x => a' \u266f c) (blsub c fun b' x => b \u266f b') \u2264 a \u2194\n    (\u2200 (b' : Ordinal), b' < b \u2192 b' \u266f c < a) \u2227 \u2200 (c' : Ordinal), c' < c \u2192 b \u266f c' < a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/LinearIndependent.lean", "full_name": "LinearIndependent.eq_of_smul_apply_eq_smul_apply", "start": [592, 1], "end": [604, 18], "traced_tactics": [{"tactic": "let l : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d", "state_before": "\u03b9 : Type u'\n\u03b9' : Type ?u.327136\nR : Type u_2\nK : Type ?u.327142\nM\u271d : Type ?u.327145\nM' : Type ?u.327148\nM'' : Type ?u.327151\nV : Type u\nV' : Type ?u.327156\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_1\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\n\u22a2 i = j", "state_after": "\u03b9 : Type u'\n\u03b9' : Type ?u.327136\nR : Type u_2\nK : Type ?u.327142\nM\u271d : Type ?u.327145\nM' : Type ?u.327148\nM'' : Type ?u.327151\nV : Type u\nV' : Type ?u.327156\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_1\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\n\u22a2 i = j"}, {"tactic": "have h_total : Finsupp.total \u03b9 M R v l = 0 := by\n  simp_rw [LinearMap.map_sub, Finsupp.total_apply]\n  simp [h]", "state_before": "\u03b9 : Type u'\n\u03b9' : Type ?u.327136\nR : Type u_2\nK : Type ?u.327142\nM\u271d : Type ?u.327145\nM' : Type ?u.327148\nM'' : Type ?u.327151\nV : Type u\nV' : Type ?u.327156\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_1\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\n\u22a2 i = j", "state_after": "\u03b9 : Type u'\n\u03b9' : Type ?u.327136\nR : Type u_2\nK : Type ?u.327142\nM\u271d : Type ?u.327145\nM' : Type ?u.327148\nM'' : Type ?u.327151\nV : Type u\nV' : Type ?u.327156\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_1\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\nh_total : \u2191(Finsupp.total \u03b9 M R v) l = 0\n\u22a2 i = j"}, {"tactic": "have h_single_eq : Finsupp.single i c = Finsupp.single j d := by\n  rw [linearIndependent_iff] at li\n  simp [eq_add_of_sub_eq' (li l h_total)]", "state_before": "\u03b9 : Type u'\n\u03b9' : Type ?u.327136\nR : Type u_2\nK : Type ?u.327142\nM\u271d : Type ?u.327145\nM' : Type ?u.327148\nM'' : Type ?u.327151\nV : Type u\nV' : Type ?u.327156\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_1\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\nh_total : \u2191(Finsupp.total \u03b9 M R v) l = 0\n\u22a2 i = j", "state_after": "\u03b9 : Type u'\n\u03b9' : Type ?u.327136\nR : Type u_2\nK : Type ?u.327142\nM\u271d : Type ?u.327145\nM' : Type ?u.327148\nM'' : Type ?u.327151\nV : Type u\nV' : Type ?u.327156\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_1\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\nh_total : \u2191(Finsupp.total \u03b9 M R v) l = 0\nh_single_eq : Finsupp.single i c = Finsupp.single j d\n\u22a2 i = j"}, {"tactic": "rcases (Finsupp.single_eq_single_iff _ _ _ _).mp h_single_eq with (\u27e8H, _\u27e9 | \u27e8hc, _\u27e9)", "state_before": "\u03b9 : Type u'\n\u03b9' : Type ?u.327136\nR : Type u_2\nK : Type ?u.327142\nM\u271d : Type ?u.327145\nM' : Type ?u.327148\nM'' : Type ?u.327151\nV : Type u\nV' : Type ?u.327156\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_1\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\nh_total : \u2191(Finsupp.total \u03b9 M R v) l = 0\nh_single_eq : Finsupp.single i c = Finsupp.single j d\n\u22a2 i = j", "state_after": "case inl.intro\n\u03b9 : Type u'\n\u03b9' : Type ?u.327136\nR : Type u_2\nK : Type ?u.327142\nM\u271d : Type ?u.327145\nM' : Type ?u.327148\nM'' : Type ?u.327151\nV : Type u\nV' : Type ?u.327156\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_1\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\nh_total : \u2191(Finsupp.total \u03b9 M R v) l = 0\nh_single_eq : Finsupp.single i c = Finsupp.single j d\nH : i = j\nright\u271d : c = d\n\u22a2 i = j\n\ncase inr.intro\n\u03b9 : Type u'\n\u03b9' : Type ?u.327136\nR : Type u_2\nK : Type ?u.327142\nM\u271d : Type ?u.327145\nM' : Type ?u.327148\nM'' : Type ?u.327151\nV : Type u\nV' : Type ?u.327156\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_1\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc\u271d : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\nh_total : \u2191(Finsupp.total \u03b9 M R v) l = 0\nh_single_eq : Finsupp.single i c = Finsupp.single j d\nhc : c = 0\nright\u271d : d = 0\n\u22a2 i = j"}, {"tactic": "simp_rw [LinearMap.map_sub, Finsupp.total_apply]", "state_before": "\u03b9 : Type u'\n\u03b9' : Type ?u.327136\nR : Type u_2\nK : Type ?u.327142\nM\u271d : Type ?u.327145\nM' : Type ?u.327148\nM'' : Type ?u.327151\nV : Type u\nV' : Type ?u.327156\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_1\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\n\u22a2 \u2191(Finsupp.total \u03b9 M R v) l = 0", "state_after": "\u03b9 : Type u'\n\u03b9' : Type ?u.327136\nR : Type u_2\nK : Type ?u.327142\nM\u271d : Type ?u.327145\nM' : Type ?u.327148\nM'' : Type ?u.327151\nV : Type u\nV' : Type ?u.327156\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_1\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\n\u22a2 ((Finsupp.sum (Finsupp.single i c) fun i a => a \u2022 v i) - Finsupp.sum (Finsupp.single j d) fun i a => a \u2022 v i) = 0"}, {"tactic": "simp [h]", "state_before": "\u03b9 : Type u'\n\u03b9' : Type ?u.327136\nR : Type u_2\nK : Type ?u.327142\nM\u271d : Type ?u.327145\nM' : Type ?u.327148\nM'' : Type ?u.327151\nV : Type u\nV' : Type ?u.327156\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_1\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\n\u22a2 ((Finsupp.sum (Finsupp.single i c) fun i a => a \u2022 v i) - Finsupp.sum (Finsupp.single j d) fun i a => a \u2022 v i) = 0", "state_after": "no goals"}, {"tactic": "rw [linearIndependent_iff] at li", "state_before": "\u03b9 : Type u'\n\u03b9' : Type ?u.327136\nR : Type u_2\nK : Type ?u.327142\nM\u271d : Type ?u.327145\nM' : Type ?u.327148\nM'' : Type ?u.327151\nV : Type u\nV' : Type ?u.327156\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_1\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\nh_total : \u2191(Finsupp.total \u03b9 M R v) l = 0\n\u22a2 Finsupp.single i c = Finsupp.single j d", "state_after": "\u03b9 : Type u'\n\u03b9' : Type ?u.327136\nR : Type u_2\nK : Type ?u.327142\nM\u271d : Type ?u.327145\nM' : Type ?u.327148\nM'' : Type ?u.327151\nV : Type u\nV' : Type ?u.327156\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_1\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : \u2200 (l : \u03b9 \u2192\u2080 R), \u2191(Finsupp.total \u03b9 M R v) l = 0 \u2192 l = 0\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\nh_total : \u2191(Finsupp.total \u03b9 M R v) l = 0\n\u22a2 Finsupp.single i c = Finsupp.single j d"}, {"tactic": "simp [eq_add_of_sub_eq' (li l h_total)]", "state_before": "\u03b9 : Type u'\n\u03b9' : Type ?u.327136\nR : Type u_2\nK : Type ?u.327142\nM\u271d : Type ?u.327145\nM' : Type ?u.327148\nM'' : Type ?u.327151\nV : Type u\nV' : Type ?u.327156\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_1\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : \u2200 (l : \u03b9 \u2192\u2080 R), \u2191(Finsupp.total \u03b9 M R v) l = 0 \u2192 l = 0\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\nh_total : \u2191(Finsupp.total \u03b9 M R v) l = 0\n\u22a2 Finsupp.single i c = Finsupp.single j d", "state_after": "no goals"}, {"tactic": "exact H", "state_before": "case inl.intro\n\u03b9 : Type u'\n\u03b9' : Type ?u.327136\nR : Type u_2\nK : Type ?u.327142\nM\u271d : Type ?u.327145\nM' : Type ?u.327148\nM'' : Type ?u.327151\nV : Type u\nV' : Type ?u.327156\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_1\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\nh_total : \u2191(Finsupp.total \u03b9 M R v) l = 0\nh_single_eq : Finsupp.single i c = Finsupp.single j d\nH : i = j\nright\u271d : c = d\n\u22a2 i = j", "state_after": "no goals"}, {"tactic": "contradiction", "state_before": "case inr.intro\n\u03b9 : Type u'\n\u03b9' : Type ?u.327136\nR : Type u_2\nK : Type ?u.327142\nM\u271d : Type ?u.327145\nM' : Type ?u.327148\nM'' : Type ?u.327151\nV : Type u\nV' : Type ?u.327156\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_1\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc\u271d : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\nh_total : \u2191(Finsupp.total \u03b9 M R v) l = 0\nh_single_eq : Finsupp.single i c = Finsupp.single j d\nhc : c = 0\nright\u271d : d = 0\n\u22a2 i = j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/Ray.lean", "full_name": "norm_injOn_ray_left", "start": [63, 1], "end": [69, 9], "traced_tactics": [{"tactic": "rintro y hy z hz h", "state_before": "E : Type ?u.17528\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : F\nhx : x \u2260 0\n\u22a2 Set.InjOn Norm.norm {y | SameRay \u211d x y}", "state_after": "E : Type ?u.17528\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y\u271d : F\nhx : x \u2260 0\ny : F\nhy : y \u2208 {y | SameRay \u211d x y}\nz : F\nhz : z \u2208 {y | SameRay \u211d x y}\nh : \u2016y\u2016 = \u2016z\u2016\n\u22a2 y = z"}, {"tactic": "rcases hy.exists_nonneg_left hx with \u27e8r, hr, rfl\u27e9", "state_before": "E : Type ?u.17528\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y\u271d : F\nhx : x \u2260 0\ny : F\nhy : y \u2208 {y | SameRay \u211d x y}\nz : F\nhz : z \u2208 {y | SameRay \u211d x y}\nh : \u2016y\u2016 = \u2016z\u2016\n\u22a2 y = z", "state_after": "case intro.intro\nE : Type ?u.17528\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : F\nhx : x \u2260 0\nz : F\nhz : z \u2208 {y | SameRay \u211d x y}\nr : \u211d\nhr : 0 \u2264 r\nhy : r \u2022 x \u2208 {y | SameRay \u211d x y}\nh : \u2016r \u2022 x\u2016 = \u2016z\u2016\n\u22a2 r \u2022 x = z"}, {"tactic": "rcases hz.exists_nonneg_left hx with \u27e8s, hs, rfl\u27e9", "state_before": "case intro.intro\nE : Type ?u.17528\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : F\nhx : x \u2260 0\nz : F\nhz : z \u2208 {y | SameRay \u211d x y}\nr : \u211d\nhr : 0 \u2264 r\nhy : r \u2022 x \u2208 {y | SameRay \u211d x y}\nh : \u2016r \u2022 x\u2016 = \u2016z\u2016\n\u22a2 r \u2022 x = z", "state_after": "case intro.intro.intro.intro\nE : Type ?u.17528\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : F\nhx : x \u2260 0\nr : \u211d\nhr : 0 \u2264 r\nhy : r \u2022 x \u2208 {y | SameRay \u211d x y}\ns : \u211d\nhs : 0 \u2264 s\nhz : s \u2022 x \u2208 {y | SameRay \u211d x y}\nh : \u2016r \u2022 x\u2016 = \u2016s \u2022 x\u2016\n\u22a2 r \u2022 x = s \u2022 x"}, {"tactic": "rw [norm_smul, norm_smul, mul_left_inj' (norm_ne_zero_iff.2 hx), norm_of_nonneg hr,\n  norm_of_nonneg hs] at h", "state_before": "case intro.intro.intro.intro\nE : Type ?u.17528\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : F\nhx : x \u2260 0\nr : \u211d\nhr : 0 \u2264 r\nhy : r \u2022 x \u2208 {y | SameRay \u211d x y}\ns : \u211d\nhs : 0 \u2264 s\nhz : s \u2022 x \u2208 {y | SameRay \u211d x y}\nh : \u2016r \u2022 x\u2016 = \u2016s \u2022 x\u2016\n\u22a2 r \u2022 x = s \u2022 x", "state_after": "case intro.intro.intro.intro\nE : Type ?u.17528\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : F\nhx : x \u2260 0\nr : \u211d\nhr : 0 \u2264 r\nhy : r \u2022 x \u2208 {y | SameRay \u211d x y}\ns : \u211d\nhs : 0 \u2264 s\nhz : s \u2022 x \u2208 {y | SameRay \u211d x y}\nh : r = s\n\u22a2 r \u2022 x = s \u2022 x"}, {"tactic": "rw [h]", "state_before": "case intro.intro.intro.intro\nE : Type ?u.17528\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : F\nhx : x \u2260 0\nr : \u211d\nhr : 0 \u2264 r\nhy : r \u2022 x \u2208 {y | SameRay \u211d x y}\ns : \u211d\nhs : 0 \u2264 s\nhz : s \u2022 x \u2208 {y | SameRay \u211d x y}\nh : r = s\n\u22a2 r \u2022 x = s \u2022 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Preadditive/Mat.lean", "full_name": "CategoryTheory.Mat_.comp_def", "start": [149, 1], "end": [151, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "closure_closure", "start": [513, 1], "end": [514, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/NAry.lean", "full_name": "Filter.map_map\u2082_antidistrib", "start": [378, 1], "end": [382, 54], "traced_tactics": [{"tactic": "rw [map\u2082_swap m]", "state_before": "\u03b1 : Type u_3\n\u03b1' : Type u_6\n\u03b2 : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_2\n\u03b3' : Type ?u.43638\n\u03b4 : Type u_1\n\u03b4' : Type ?u.43644\n\u03b5 : Type ?u.43647\n\u03b5' : Type ?u.43650\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nn : \u03b3 \u2192 \u03b4\nm' : \u03b2' \u2192 \u03b1' \u2192 \u03b4\nn\u2081 : \u03b2 \u2192 \u03b2'\nn\u2082 : \u03b1 \u2192 \u03b1'\nh_antidistrib : \u2200 (a : \u03b1) (b : \u03b2), n (m a b) = m' (n\u2081 b) (n\u2082 a)\n\u22a2 map n (map\u2082 m f g) = map\u2082 m' (map n\u2081 g) (map n\u2082 f)", "state_after": "\u03b1 : Type u_3\n\u03b1' : Type u_6\n\u03b2 : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_2\n\u03b3' : Type ?u.43638\n\u03b4 : Type u_1\n\u03b4' : Type ?u.43644\n\u03b5 : Type ?u.43647\n\u03b5' : Type ?u.43650\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nn : \u03b3 \u2192 \u03b4\nm' : \u03b2' \u2192 \u03b1' \u2192 \u03b4\nn\u2081 : \u03b2 \u2192 \u03b2'\nn\u2082 : \u03b1 \u2192 \u03b1'\nh_antidistrib : \u2200 (a : \u03b1) (b : \u03b2), n (m a b) = m' (n\u2081 b) (n\u2082 a)\n\u22a2 map n (map\u2082 (fun a b => m b a) g f) = map\u2082 m' (map n\u2081 g) (map n\u2082 f)"}, {"tactic": "exact map_map\u2082_distrib fun _ _ => h_antidistrib _ _", "state_before": "\u03b1 : Type u_3\n\u03b1' : Type u_6\n\u03b2 : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_2\n\u03b3' : Type ?u.43638\n\u03b4 : Type u_1\n\u03b4' : Type ?u.43644\n\u03b5 : Type ?u.43647\n\u03b5' : Type ?u.43650\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nn : \u03b3 \u2192 \u03b4\nm' : \u03b2' \u2192 \u03b1' \u2192 \u03b4\nn\u2081 : \u03b2 \u2192 \u03b2'\nn\u2082 : \u03b1 \u2192 \u03b1'\nh_antidistrib : \u2200 (a : \u03b1) (b : \u03b2), n (m a b) = m' (n\u2081 b) (n\u2082 a)\n\u22a2 map n (map\u2082 (fun a b => m b a) g f) = map\u2082 m' (map n\u2081 g) (map n\u2082 f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.image_inter_subset", "start": [321, 1], "end": [322, 100], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Denumerable.lean", "full_name": "Denumerable.decode_eq_ofNat", "start": [58, 1], "end": [59, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Zip.lean", "full_name": "List.zipWith_append", "start": [498, 1], "end": [507, 20], "traced_tactics": [{"tactic": "induction' l with hd tl hl generalizing l'", "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.233932\n\u03b5 : Type ?u.233935\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u271d : List \u03b1\nl'\u271d : List \u03b2\nn : \u2115\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl la : List \u03b1\nl' lb : List \u03b2\nh : length l = length l'\n\u22a2 zipWith f (l ++ la) (l' ++ lb) = zipWith f l l' ++ zipWith f la lb", "state_after": "case nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.233932\n\u03b5 : Type ?u.233935\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u271d : List \u03b1\nl'\u271d\u00b9 : List \u03b2\nn : \u2115\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl la : List \u03b1\nl'\u271d lb : List \u03b2\nh\u271d : length l = length l'\u271d\nl' : List \u03b2\nh : length [] = length l'\n\u22a2 zipWith f ([] ++ la) (l' ++ lb) = zipWith f [] l' ++ zipWith f la lb\n\ncase cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.233932\n\u03b5 : Type ?u.233935\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u271d : List \u03b1\nl'\u271d\u00b9 : List \u03b2\nn : \u2115\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl la : List \u03b1\nl'\u271d lb : List \u03b2\nh\u271d : length l = length l'\u271d\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), length tl = length l' \u2192 zipWith f (tl ++ la) (l' ++ lb) = zipWith f tl l' ++ zipWith f la lb\nl' : List \u03b2\nh : length (hd :: tl) = length l'\n\u22a2 zipWith f (hd :: tl ++ la) (l' ++ lb) = zipWith f (hd :: tl) l' ++ zipWith f la lb"}, {"tactic": "have : l' = [] := eq_nil_of_length_eq_zero (by simpa using h.symm)", "state_before": "case nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.233932\n\u03b5 : Type ?u.233935\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u271d : List \u03b1\nl'\u271d\u00b9 : List \u03b2\nn : \u2115\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl la : List \u03b1\nl'\u271d lb : List \u03b2\nh\u271d : length l = length l'\u271d\nl' : List \u03b2\nh : length [] = length l'\n\u22a2 zipWith f ([] ++ la) (l' ++ lb) = zipWith f [] l' ++ zipWith f la lb", "state_after": "case nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.233932\n\u03b5 : Type ?u.233935\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u271d : List \u03b1\nl'\u271d\u00b9 : List \u03b2\nn : \u2115\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl la : List \u03b1\nl'\u271d lb : List \u03b2\nh\u271d : length l = length l'\u271d\nl' : List \u03b2\nh : length [] = length l'\nthis : l' = []\n\u22a2 zipWith f ([] ++ la) (l' ++ lb) = zipWith f [] l' ++ zipWith f la lb"}, {"tactic": "simp [this]", "state_before": "case nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.233932\n\u03b5 : Type ?u.233935\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u271d : List \u03b1\nl'\u271d\u00b9 : List \u03b2\nn : \u2115\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl la : List \u03b1\nl'\u271d lb : List \u03b2\nh\u271d : length l = length l'\u271d\nl' : List \u03b2\nh : length [] = length l'\nthis : l' = []\n\u22a2 zipWith f ([] ++ la) (l' ++ lb) = zipWith f [] l' ++ zipWith f la lb", "state_after": "no goals"}, {"tactic": "simpa using h.symm", "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.233932\n\u03b5 : Type ?u.233935\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u271d : List \u03b1\nl'\u271d\u00b9 : List \u03b2\nn : \u2115\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl la : List \u03b1\nl'\u271d lb : List \u03b2\nh\u271d : length l = length l'\u271d\nl' : List \u03b2\nh : length [] = length l'\n\u22a2 length l' = 0", "state_after": "no goals"}, {"tactic": "cases l'", "state_before": "case cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.233932\n\u03b5 : Type ?u.233935\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u271d : List \u03b1\nl'\u271d\u00b9 : List \u03b2\nn : \u2115\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl la : List \u03b1\nl'\u271d lb : List \u03b2\nh\u271d : length l = length l'\u271d\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), length tl = length l' \u2192 zipWith f (tl ++ la) (l' ++ lb) = zipWith f tl l' ++ zipWith f la lb\nl' : List \u03b2\nh : length (hd :: tl) = length l'\n\u22a2 zipWith f (hd :: tl ++ la) (l' ++ lb) = zipWith f (hd :: tl) l' ++ zipWith f la lb", "state_after": "case cons.nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.233932\n\u03b5 : Type ?u.233935\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u271d : List \u03b1\nl'\u271d : List \u03b2\nn : \u2115\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl la : List \u03b1\nl' lb : List \u03b2\nh\u271d : length l = length l'\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), length tl = length l' \u2192 zipWith f (tl ++ la) (l' ++ lb) = zipWith f tl l' ++ zipWith f la lb\nh : length (hd :: tl) = length []\n\u22a2 zipWith f (hd :: tl ++ la) ([] ++ lb) = zipWith f (hd :: tl) [] ++ zipWith f la lb\n\ncase cons.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.233932\n\u03b5 : Type ?u.233935\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u271d : List \u03b1\nl'\u271d : List \u03b2\nn : \u2115\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl la : List \u03b1\nl' lb : List \u03b2\nh\u271d : length l = length l'\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), length tl = length l' \u2192 zipWith f (tl ++ la) (l' ++ lb) = zipWith f tl l' ++ zipWith f la lb\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\nh : length (hd :: tl) = length (head\u271d :: tail\u271d)\n\u22a2 zipWith f (hd :: tl ++ la) (head\u271d :: tail\u271d ++ lb) = zipWith f (hd :: tl) (head\u271d :: tail\u271d) ++ zipWith f la lb"}, {"tactic": "simp at h", "state_before": "case cons.nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.233932\n\u03b5 : Type ?u.233935\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u271d : List \u03b1\nl'\u271d : List \u03b2\nn : \u2115\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl la : List \u03b1\nl' lb : List \u03b2\nh\u271d : length l = length l'\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), length tl = length l' \u2192 zipWith f (tl ++ la) (l' ++ lb) = zipWith f tl l' ++ zipWith f la lb\nh : length (hd :: tl) = length []\n\u22a2 zipWith f (hd :: tl ++ la) ([] ++ lb) = zipWith f (hd :: tl) [] ++ zipWith f la lb", "state_after": "no goals"}, {"tactic": "simp only [add_left_inj, length] at h", "state_before": "case cons.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.233932\n\u03b5 : Type ?u.233935\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u271d : List \u03b1\nl'\u271d : List \u03b2\nn : \u2115\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl la : List \u03b1\nl' lb : List \u03b2\nh\u271d : length l = length l'\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), length tl = length l' \u2192 zipWith f (tl ++ la) (l' ++ lb) = zipWith f tl l' ++ zipWith f la lb\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\nh : length (hd :: tl) = length (head\u271d :: tail\u271d)\n\u22a2 zipWith f (hd :: tl ++ la) (head\u271d :: tail\u271d ++ lb) = zipWith f (hd :: tl) (head\u271d :: tail\u271d) ++ zipWith f la lb", "state_after": "case cons.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.233932\n\u03b5 : Type ?u.233935\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u271d : List \u03b1\nl'\u271d : List \u03b2\nn : \u2115\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl la : List \u03b1\nl' lb : List \u03b2\nh\u271d : length l = length l'\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), length tl = length l' \u2192 zipWith f (tl ++ la) (l' ++ lb) = zipWith f tl l' ++ zipWith f la lb\nhead\u271d : \u03b2\ntail\u271d : List \u03b2\nh : length tl = length tail\u271d\n\u22a2 zipWith f (hd :: tl ++ la) (head\u271d :: tail\u271d ++ lb) = zipWith f (hd :: tl) (head\u271d :: tail\u271d) ++ zipWith f la lb"}, {"tactic": "simp [hl _ h]", "state_before": "case cons.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.233932\n\u03b5 : Type ?u.233935\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u271d : List \u03b1\nl'\u271d : List \u03b2\nn : \u2115\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl la : List \u03b1\nl' lb : List \u03b2\nh\u271d : 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\u2124\nc d : \u2115\nx y : \u2124\n\u22a2 Nonnegg c d x y = Nonnegg d c y x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Module/GradedModule.lean", "full_name": "DirectSum.Gmodule.smulAddMonoidHom_apply_of_of", "start": [100, 1], "end": [103, 26], "traced_tactics": [{"tactic": "simp [smulAddMonoidHom]", "state_before": "\u03b9 : Type u_1\nA : \u03b9 \u2192 Type u_2\nM : \u03b9 \u2192 Type u_3\ninst\u271d\u2075 : AddMonoid \u03b9\ninst\u271d\u2074 : (i : \u03b9) \u2192 AddCommMonoid (A i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : GMonoid A\ninst\u271d : Gmodule A M\ni j : \u03b9\nx : A i\ny : M j\n\u22a2 \u2191(\u2191(smulAddMonoidHom A M) (\u2191(of A i) x)) (\u2191(of M j) y) = \u2191(of M (i + j)) (GSmul.smul x y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Hydra.lean", "full_name": "Relation.cutExpand_le_invImage_lex", "start": [65, 1], "end": [77, 34], "traced_tactics": [{"tactic": "rintro s t \u27e8u, a, hr, he\u27e9", "state_before": "\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsIrrefl \u03b1 r\n\u22a2 CutExpand r \u2264 InvImage (Finsupp.Lex (r\u1d9c \u2293 fun x x_1 => x \u2260 x_1) fun x x_1 => x < x_1) \u2191toFinsupp", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsIrrefl \u03b1 r\ns t u : Multiset \u03b1\na : \u03b1\nhr : \u2200 (a' : \u03b1), a' \u2208 u \u2192 r a' a\nhe : s + {a} = t + u\n\u22a2 InvImage (Finsupp.Lex (r\u1d9c \u2293 fun x x_1 => x \u2260 x_1) fun x x_1 => x < x_1) (\u2191toFinsupp) s t"}, {"tactic": "replace hr := fun a' \u21a6 mt (hr a')", "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsIrrefl \u03b1 r\ns t u : Multiset \u03b1\na : \u03b1\nhr : \u2200 (a' : \u03b1), a' \u2208 u \u2192 r a' a\nhe : s + {a} = t + u\n\u22a2 InvImage (Finsupp.Lex (r\u1d9c \u2293 fun x x_1 => x \u2260 x_1) fun x x_1 => x < x_1) (\u2191toFinsupp) s t", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsIrrefl \u03b1 r\ns t u : Multiset \u03b1\na : \u03b1\nhe : s + {a} = t + u\nhr : \u2200 (a' : \u03b1), \u00acr a' a \u2192 \u00aca' \u2208 u\n\u22a2 InvImage (Finsupp.Lex (r\u1d9c \u2293 fun x x_1 => x \u2260 x_1) fun x x_1 => x < x_1) (\u2191toFinsupp) s t"}, {"tactic": "refine \u27e8a, fun b h \u21a6 ?_, ?_\u27e9 <;> simp_rw [toFinsupp_apply]", "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsIrrefl \u03b1 r\ns t u : Multiset \u03b1\na : \u03b1\nhe : s + {a} = t + u\nhr : \u2200 (a' : \u03b1), \u00acr a' a \u2192 \u00aca' \u2208 u\n\u22a2 InvImage (Finsupp.Lex (r\u1d9c \u2293 fun x x_1 => x \u2260 x_1) fun x x_1 => x < x_1) (\u2191toFinsupp) s t", "state_after": "case intro.intro.intro.refine_1\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsIrrefl \u03b1 r\ns t u : Multiset \u03b1\na : \u03b1\nhe : s + {a} = t + u\nhr : \u2200 (a' : \u03b1), \u00acr a' a \u2192 \u00aca' \u2208 u\nb : \u03b1\nh : (r\u1d9c \u2293 fun x x_1 => x \u2260 x_1) b a\n\u22a2 count b s = count b t\n\ncase intro.intro.intro.refine_2\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsIrrefl \u03b1 r\ns t u : Multiset \u03b1\na : \u03b1\nhe : s + {a} = t + u\nhr : \u2200 (a' : \u03b1), \u00acr a' a \u2192 \u00aca' \u2208 u\n\u22a2 count a s < count a t"}, {"tactic": "apply_fun count b  at he", "state_before": "case intro.intro.intro.refine_1\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsIrrefl \u03b1 r\ns t u : Multiset \u03b1\na : \u03b1\nhe : s + {a} = t + u\nhr : \u2200 (a' : \u03b1), \u00acr a' a \u2192 \u00aca' \u2208 u\nb : \u03b1\nh : (r\u1d9c \u2293 fun x x_1 => x \u2260 x_1) b a\n\u22a2 count b s = count b t", "state_after": "case intro.intro.intro.refine_1\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsIrrefl \u03b1 r\ns t u : Multiset \u03b1\na : \u03b1\nhr : \u2200 (a' : \u03b1), \u00acr a' a \u2192 \u00aca' \u2208 u\nb : \u03b1\nh : (r\u1d9c \u2293 fun x x_1 => x \u2260 x_1) b a\nhe : count b (s + {a}) = count b (t + u)\n\u22a2 count b s = count b t"}, {"tactic": "simpa only [count_add, count_singleton, if_neg h.2, add_zero, count_eq_zero.2 (hr b h.1)]\n  using he", "state_before": "case intro.intro.intro.refine_1\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsIrrefl \u03b1 r\ns t u : Multiset \u03b1\na : \u03b1\nhr : \u2200 (a' : \u03b1), \u00acr a' a \u2192 \u00aca' \u2208 u\nb : \u03b1\nh : (r\u1d9c \u2293 fun x x_1 => x \u2260 x_1) b a\nhe : count b (s + {a}) = count b (t + u)\n\u22a2 count b s = count b t", "state_after": "no goals"}, {"tactic": "apply_fun count a  at he", "state_before": "case intro.intro.intro.refine_2\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsIrrefl \u03b1 r\ns t u : Multiset \u03b1\na : \u03b1\nhe : s + {a} = t + u\nhr : \u2200 (a' : \u03b1), \u00acr a' a \u2192 \u00aca' \u2208 u\n\u22a2 count a s < count a t", "state_after": "case intro.intro.intro.refine_2\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsIrrefl \u03b1 r\ns t u : Multiset \u03b1\na : \u03b1\nhr : \u2200 (a' : \u03b1), \u00acr a' a \u2192 \u00aca' \u2208 u\nhe : count a (s + {a}) = count a (t + u)\n\u22a2 count a s < count a t"}, {"tactic": "simp only [count_add, count_singleton_self, count_eq_zero.2 (hr _ (irrefl_of r a)),\n  add_zero] at he", "state_before": "case intro.intro.intro.refine_2\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsIrrefl \u03b1 r\ns t u : Multiset \u03b1\na : \u03b1\nhr : \u2200 (a' : \u03b1), \u00acr a' a \u2192 \u00aca' \u2208 u\nhe : count a (s + {a}) = count a (t + u)\n\u22a2 count a s < count a t", "state_after": "case intro.intro.intro.refine_2\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsIrrefl \u03b1 r\ns t u : Multiset \u03b1\na : \u03b1\nhr : \u2200 (a' : \u03b1), \u00acr a' a \u2192 \u00aca' \u2208 u\nhe : count a s + 1 = count a t\n\u22a2 count a s < count a t"}, {"tactic": "exact he \u25b8 Nat.lt_succ_self _", "state_before": "case intro.intro.intro.refine_2\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsIrrefl \u03b1 r\ns t u : Multiset \u03b1\na : \u03b1\nhr : \u2200 (a' : \u03b1), \u00acr a' a \u2192 \u00aca' \u2208 u\nhe : count a s + 1 = count a t\n\u22a2 count a s < count a t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Cover.lean", "full_name": "Wcovby.le_of_lt", "start": [430, 1], "end": [431, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/Block.lean", "full_name": "Matrix.det_toBlock", "start": [166, 1], "end": [177, 50], "traced_tactics": [{"tactic": "rw [\u2190 Matrix.det_reindex_self (Equiv.sumCompl p).symm M]", "state_before": "\u03b1 : Type ?u.52561\n\u03b2 : Type ?u.52564\nm : Type u_1\nn : Type ?u.52570\no : Type ?u.52573\nm' : \u03b1 \u2192 Type ?u.52578\nn' : \u03b1 \u2192 Type ?u.52583\nR : Type v\ninst\u271d\u2075 : CommRing R\nM\u271d N : Matrix m m R\nb : m \u2192 \u03b1\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nM : Matrix m m R\np : m \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 det M =\n    det\n      (fromBlocks (toBlock M p p) (toBlock M p fun j => \u00acp j) (toBlock M (fun j => \u00acp j) p)\n        (toBlock M (fun j => \u00acp j) fun j => \u00acp j))", "state_after": "\u03b1 : Type ?u.52561\n\u03b2 : Type ?u.52564\nm : Type u_1\nn : Type ?u.52570\no : Type ?u.52573\nm' : \u03b1 \u2192 Type ?u.52578\nn' : \u03b1 \u2192 Type ?u.52583\nR : Type v\ninst\u271d\u2075 : CommRing R\nM\u271d N : Matrix m m R\nb : m \u2192 \u03b1\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nM : Matrix m m R\np : m \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 det (\u2191(reindex (Equiv.sumCompl p).symm (Equiv.sumCompl p).symm) M) =\n    det\n      (fromBlocks (toBlock M p p) (toBlock M p fun j => \u00acp j) (toBlock M (fun j => \u00acp j) p)\n        (toBlock M (fun j => \u00acp j) fun j => \u00acp j))"}, {"tactic": "rw [det_apply', det_apply']", "state_before": "\u03b1 : Type ?u.52561\n\u03b2 : Type ?u.52564\nm : Type u_1\nn : Type ?u.52570\no : Type ?u.52573\nm' : \u03b1 \u2192 Type ?u.52578\nn' : \u03b1 \u2192 Type ?u.52583\nR : Type v\ninst\u271d\u2075 : CommRing R\nM\u271d N : Matrix m m R\nb : m \u2192 \u03b1\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nM : Matrix m m R\np : m \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 det (\u2191(reindex (Equiv.sumCompl p).symm (Equiv.sumCompl p).symm) M) =\n    det\n      (fromBlocks (toBlock M p p) (toBlock M p fun j => \u00acp j) (toBlock M (fun j => \u00acp j) p)\n        (toBlock M (fun j => \u00acp j) fun j => \u00acp j))", "state_after": "\u03b1 : Type ?u.52561\n\u03b2 : Type ?u.52564\nm : Type u_1\nn : Type ?u.52570\no : Type ?u.52573\nm' : \u03b1 \u2192 Type ?u.52578\nn' : \u03b1 \u2192 Type ?u.52583\nR : Type v\ninst\u271d\u2075 : CommRing R\nM\u271d N : Matrix m m R\nb : m \u2192 \u03b1\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nM : Matrix m m R\np : m \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 \u2211 \u03c3 : Equiv.Perm ({ a // p a } \u2295 { a // \u00acp a }),\n      \u2191\u2191(\u2191Equiv.Perm.sign \u03c3) *\n        \u220f i : { a // p a } \u2295 { a // \u00acp a }, \u2191(reindex (Equiv.sumCompl p).symm (Equiv.sumCompl p).symm) M (\u2191\u03c3 i) i =\n    \u2211 \u03c3 : Equiv.Perm ({ a // p a } \u2295 { a // \u00acp a }),\n      \u2191\u2191(\u2191Equiv.Perm.sign \u03c3) *\n        \u220f i : { a // p a } \u2295 { a // \u00acp a },\n          fromBlocks (toBlock M p p) (toBlock M p fun j => \u00acp j) (toBlock M (fun j => \u00acp j) p)\n            (toBlock M (fun j => \u00acp j) fun j => \u00acp j) (\u2191\u03c3 i) i"}, {"tactic": "congr", "state_before": "\u03b1 : Type ?u.52561\n\u03b2 : Type ?u.52564\nm : Type u_1\nn : Type ?u.52570\no : Type ?u.52573\nm' : \u03b1 \u2192 Type ?u.52578\nn' : \u03b1 \u2192 Type ?u.52583\nR : Type v\ninst\u271d\u2075 : CommRing R\nM\u271d N : Matrix m m R\nb : m \u2192 \u03b1\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nM : Matrix m m R\np : m \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 \u2211 \u03c3 : Equiv.Perm ({ a // p a } \u2295 { a // \u00acp a }),\n      \u2191\u2191(\u2191Equiv.Perm.sign \u03c3) *\n        \u220f i : { a // p a } \u2295 { a // \u00acp a }, \u2191(reindex (Equiv.sumCompl p).symm (Equiv.sumCompl p).symm) M (\u2191\u03c3 i) i =\n    \u2211 \u03c3 : Equiv.Perm ({ a // p a } \u2295 { a // \u00acp a }),\n      \u2191\u2191(\u2191Equiv.Perm.sign \u03c3) *\n        \u220f i : { a // p a } \u2295 { a // \u00acp a },\n          fromBlocks (toBlock M p p) (toBlock M p fun j => \u00acp j) (toBlock M (fun j => \u00acp j) p)\n            (toBlock M (fun j => \u00acp j) fun j => \u00acp j) (\u2191\u03c3 i) i", "state_after": "case e_f\n\u03b1 : Type ?u.52561\n\u03b2 : Type ?u.52564\nm : Type u_1\nn : Type ?u.52570\no : Type ?u.52573\nm' : \u03b1 \u2192 Type ?u.52578\nn' : \u03b1 \u2192 Type ?u.52583\nR : Type v\ninst\u271d\u2075 : CommRing R\nM\u271d N : Matrix m m R\nb : m \u2192 \u03b1\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nM : Matrix m m R\np : m \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 (fun \u03c3 =>\n      \u2191\u2191(\u2191Equiv.Perm.sign \u03c3) *\n        \u220f i : { a // p a } \u2295 { a // \u00acp a }, \u2191(reindex (Equiv.sumCompl p).symm (Equiv.sumCompl p).symm) M (\u2191\u03c3 i) i) =\n    fun \u03c3 =>\n    \u2191\u2191(\u2191Equiv.Perm.sign \u03c3) *\n      \u220f i : { a // p a } \u2295 { a // \u00acp a },\n        fromBlocks (toBlock M p p) (toBlock M p fun j => \u00acp j) (toBlock M (fun j => \u00acp j) p)\n          (toBlock M (fun j => \u00acp j) fun j => \u00acp j) (\u2191\u03c3 i) i"}, {"tactic": "ext \u03c3", "state_before": "case e_f\n\u03b1 : Type ?u.52561\n\u03b2 : Type ?u.52564\nm : Type u_1\nn : Type ?u.52570\no : Type ?u.52573\nm' : \u03b1 \u2192 Type ?u.52578\nn' : \u03b1 \u2192 Type ?u.52583\nR : Type v\ninst\u271d\u2075 : CommRing R\nM\u271d N : Matrix m m R\nb : m \u2192 \u03b1\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nM : Matrix m m R\np : m \u2192 Prop\ninst\u271d : DecidablePred p\n\u22a2 (fun \u03c3 =>\n      \u2191\u2191(\u2191Equiv.Perm.sign \u03c3) *\n        \u220f i : { a // p a } \u2295 { a // \u00acp a }, \u2191(reindex (Equiv.sumCompl p).symm (Equiv.sumCompl p).symm) M (\u2191\u03c3 i) i) =\n    fun \u03c3 =>\n    \u2191\u2191(\u2191Equiv.Perm.sign \u03c3) *\n      \u220f i : { a // p a } \u2295 { a // \u00acp a },\n        fromBlocks (toBlock M p p) (toBlock M p fun j => \u00acp j) (toBlock M (fun j => \u00acp j) p)\n          (toBlock M (fun j => \u00acp j) fun j => \u00acp j) (\u2191\u03c3 i) i", "state_after": "case e_f.h\n\u03b1 : Type ?u.52561\n\u03b2 : Type ?u.52564\nm : Type u_1\nn : Type ?u.52570\no : Type ?u.52573\nm' : \u03b1 \u2192 Type ?u.52578\nn' : \u03b1 \u2192 Type ?u.52583\nR : Type v\ninst\u271d\u2075 : CommRing R\nM\u271d N : Matrix m m R\nb : m \u2192 \u03b1\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nM : Matrix m m R\np : m \u2192 Prop\ninst\u271d : DecidablePred p\n\u03c3 : Equiv.Perm ({ a // p a } \u2295 { a // \u00acp a })\n\u22a2 \u2191\u2191(\u2191Equiv.Perm.sign \u03c3) *\n      \u220f i : { a // p a } \u2295 { a // \u00acp a }, \u2191(reindex (Equiv.sumCompl p).symm (Equiv.sumCompl p).symm) M (\u2191\u03c3 i) i =\n    \u2191\u2191(\u2191Equiv.Perm.sign \u03c3) *\n      \u220f i : { a // p a } \u2295 { a // \u00acp a },\n        fromBlocks (toBlock M p p) (toBlock M p fun j => \u00acp j) (toBlock M (fun j => \u00acp j) p)\n          (toBlock M (fun j => \u00acp j) fun j => \u00acp j) (\u2191\u03c3 i) i"}, {"tactic": "congr", "state_before": "case e_f.h\n\u03b1 : Type ?u.52561\n\u03b2 : Type ?u.52564\nm : Type u_1\nn : Type ?u.52570\no : Type ?u.52573\nm' : \u03b1 \u2192 Type ?u.52578\nn' : \u03b1 \u2192 Type ?u.52583\nR : Type v\ninst\u271d\u2075 : CommRing R\nM\u271d N : Matrix m m R\nb : m \u2192 \u03b1\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nM : Matrix m m R\np : m \u2192 Prop\ninst\u271d : DecidablePred p\n\u03c3 : Equiv.Perm ({ a // p a } \u2295 { a // \u00acp a })\n\u22a2 \u2191\u2191(\u2191Equiv.Perm.sign \u03c3) *\n      \u220f i : { a // p a } \u2295 { a // \u00acp a }, \u2191(reindex (Equiv.sumCompl p).symm (Equiv.sumCompl p).symm) M (\u2191\u03c3 i) i =\n    \u2191\u2191(\u2191Equiv.Perm.sign \u03c3) *\n      \u220f i : { a // p a } \u2295 { a // \u00acp a },\n        fromBlocks (toBlock M p p) (toBlock M p fun j => \u00acp j) (toBlock M (fun j => \u00acp j) p)\n          (toBlock M (fun j => \u00acp j) fun j => \u00acp j) (\u2191\u03c3 i) i", "state_after": "case e_f.h.e_a.e_f\n\u03b1 : Type ?u.52561\n\u03b2 : Type ?u.52564\nm : Type u_1\nn : Type ?u.52570\no : Type ?u.52573\nm' : \u03b1 \u2192 Type ?u.52578\nn' : \u03b1 \u2192 Type ?u.52583\nR : Type v\ninst\u271d\u2075 : CommRing R\nM\u271d N : Matrix m m R\nb : m \u2192 \u03b1\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nM : Matrix m m R\np : m \u2192 Prop\ninst\u271d : DecidablePred p\n\u03c3 : Equiv.Perm ({ a // p a } \u2295 { a // \u00acp a })\n\u22a2 (fun i => \u2191(reindex (Equiv.sumCompl p).symm (Equiv.sumCompl p).symm) M (\u2191\u03c3 i) i) = fun i =>\n    fromBlocks (toBlock M p p) (toBlock M p fun j => \u00acp j) (toBlock M (fun j => \u00acp j) p)\n      (toBlock M (fun j => \u00acp j) fun j => \u00acp j) (\u2191\u03c3 i) i"}, {"tactic": "ext x", "state_before": "case e_f.h.e_a.e_f\n\u03b1 : Type ?u.52561\n\u03b2 : Type ?u.52564\nm : Type u_1\nn : Type ?u.52570\no : Type ?u.52573\nm' : \u03b1 \u2192 Type ?u.52578\nn' : \u03b1 \u2192 Type ?u.52583\nR : Type v\ninst\u271d\u2075 : CommRing R\nM\u271d N : Matrix m m R\nb : m \u2192 \u03b1\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nM : Matrix m m R\np : m \u2192 Prop\ninst\u271d : DecidablePred p\n\u03c3 : Equiv.Perm ({ a // p a } \u2295 { a // \u00acp a })\n\u22a2 (fun i => \u2191(reindex (Equiv.sumCompl p).symm (Equiv.sumCompl p).symm) M (\u2191\u03c3 i) i) = fun i =>\n    fromBlocks (toBlock M p p) (toBlock M p fun j => \u00acp j) (toBlock M (fun j => \u00acp j) p)\n      (toBlock M (fun j => \u00acp j) fun j => \u00acp j) (\u2191\u03c3 i) i", "state_after": "case e_f.h.e_a.e_f.h\n\u03b1 : Type ?u.52561\n\u03b2 : Type ?u.52564\nm : Type u_1\nn : Type ?u.52570\no : Type ?u.52573\nm' : \u03b1 \u2192 Type ?u.52578\nn' : \u03b1 \u2192 Type ?u.52583\nR : Type v\ninst\u271d\u2075 : CommRing R\nM\u271d N : Matrix m m R\nb : m \u2192 \u03b1\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nM : Matrix m m R\np : m \u2192 Prop\ninst\u271d : DecidablePred p\n\u03c3 : Equiv.Perm ({ a // p a } \u2295 { a // \u00acp a })\nx : { a // p a } \u2295 { a // \u00acp a }\n\u22a2 \u2191(reindex (Equiv.sumCompl p).symm (Equiv.sumCompl p).symm) M (\u2191\u03c3 x) x =\n    fromBlocks (toBlock M p p) (toBlock M p fun j => \u00acp j) (toBlock M (fun j => \u00acp j) p)\n      (toBlock M (fun j => \u00acp j) fun j => \u00acp j) (\u2191\u03c3 x) x"}, {"tactic": "generalize hy : \u03c3 x = y", "state_before": "case e_f.h.e_a.e_f.h\n\u03b1 : Type ?u.52561\n\u03b2 : Type ?u.52564\nm : Type u_1\nn : Type ?u.52570\no : Type ?u.52573\nm' : \u03b1 \u2192 Type ?u.52578\nn' : \u03b1 \u2192 Type ?u.52583\nR : Type v\ninst\u271d\u2075 : CommRing R\nM\u271d N : Matrix m m R\nb : m \u2192 \u03b1\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nM : Matrix m m R\np : m \u2192 Prop\ninst\u271d : DecidablePred p\n\u03c3 : Equiv.Perm ({ a // p a } \u2295 { a // \u00acp a })\nx : { a // p a } \u2295 { a // \u00acp a }\n\u22a2 \u2191(reindex (Equiv.sumCompl p).symm (Equiv.sumCompl p).symm) M (\u2191\u03c3 x) x =\n    fromBlocks (toBlock M p p) (toBlock M p fun j => \u00acp j) (toBlock M (fun j => \u00acp j) p)\n      (toBlock M (fun j => \u00acp j) fun j => \u00acp j) (\u2191\u03c3 x) x", "state_after": "case e_f.h.e_a.e_f.h\n\u03b1 : Type ?u.52561\n\u03b2 : Type ?u.52564\nm : Type u_1\nn : Type ?u.52570\no : Type ?u.52573\nm' : \u03b1 \u2192 Type ?u.52578\nn' : \u03b1 \u2192 Type ?u.52583\nR : Type v\ninst\u271d\u2075 : CommRing R\nM\u271d N : Matrix m m R\nb : m \u2192 \u03b1\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nM : Matrix m m R\np : m \u2192 Prop\ninst\u271d : DecidablePred p\n\u03c3 : Equiv.Perm ({ a // p a } \u2295 { a // \u00acp a })\nx y : { a // p a } \u2295 { a // \u00acp a }\nhy : \u2191\u03c3 x = y\n\u22a2 \u2191(reindex (Equiv.sumCompl p).symm (Equiv.sumCompl p).symm) M y x =\n    fromBlocks (toBlock M p p) (toBlock M p fun j => \u00acp j) (toBlock M (fun j => \u00acp j) p)\n      (toBlock M (fun j => \u00acp j) fun j => \u00acp j) y x"}, {"tactic": "cases x <;> cases y <;>\n  simp only [Matrix.reindex_apply, toBlock_apply, Equiv.symm_symm, Equiv.sumCompl_apply_inr,\n    Equiv.sumCompl_apply_inl, fromBlocks_apply\u2081\u2081, fromBlocks_apply\u2081\u2082, fromBlocks_apply\u2082\u2081,\n    fromBlocks_apply\u2082\u2082, Matrix.submatrix_apply]", "state_before": "case e_f.h.e_a.e_f.h\n\u03b1 : Type ?u.52561\n\u03b2 : Type ?u.52564\nm : Type u_1\nn : Type ?u.52570\no : Type ?u.52573\nm' : \u03b1 \u2192 Type ?u.52578\nn' : \u03b1 \u2192 Type ?u.52583\nR : Type v\ninst\u271d\u2075 : CommRing R\nM\u271d N : Matrix m m R\nb : m \u2192 \u03b1\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\nM : Matrix m m R\np : m \u2192 Prop\ninst\u271d : DecidablePred p\n\u03c3 : Equiv.Perm ({ a // p a } \u2295 { a // \u00acp a })\nx y : { a // p a } \u2295 { a // \u00acp a }\nhy : \u2191\u03c3 x = y\n\u22a2 \u2191(reindex (Equiv.sumCompl p).symm (Equiv.sumCompl p).symm) M y x =\n    fromBlocks (toBlock M p p) (toBlock M p fun j => \u00acp j) (toBlock M (fun j => \u00acp j) p)\n      (toBlock M (fun j => \u00acp j) fun j => \u00acp j) y x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Image.lean", "full_name": "Finset.subtype_map_of_mem", "start": [689, 1], "end": [690, 89], "traced_tactics": [{"tactic": "simpa [subtype_map] using h", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.133134\n\u03b3 : Type ?u.133137\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\ns : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 p x\n\u22a2 \u2200 (a : \u03b1), a \u2208 map (Embedding.subtype p) (Finset.subtype p s) \u2194 a \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Digits.lean", "full_name": "Nat.dvd_iff_dvd_ofDigits", "start": [603, 1], "end": [607, 99], "traced_tactics": [{"tactic": "rw [\u2190 Int.coe_nat_dvd]", "state_before": "n\u271d b b' : \u2115\nc : \u2124\nh : \u2191b \u2223 \u2191b' - c\nn : \u2115\n\u22a2 b \u2223 n \u2194 \u2191b \u2223 ofDigits c (digits b' n)", "state_after": "n\u271d b b' : \u2115\nc : \u2124\nh : \u2191b \u2223 \u2191b' - c\nn : \u2115\n\u22a2 \u2191b \u2223 \u2191n \u2194 \u2191b \u2223 ofDigits c (digits b' n)"}, {"tactic": "exact\n  dvd_iff_dvd_of_dvd_sub (zmodeq_ofDigits_digits b b' c (Int.modEq_iff_dvd.2 h).symm _).symm.dvd", "state_before": "n\u271d b b' : \u2115\nc : \u2124\nh : \u2191b \u2223 \u2191b' - c\nn : \u2115\n\u22a2 \u2191b \u2223 \u2191n \u2194 \u2191b \u2223 ofDigits c (digits b' n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.choose_property", "start": [3063, 1], "end": [3064, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Sheaves/Presheaf.lean", "full_name": "TopCat.Presheaf.Pushforward.comp_inv_app", "start": [263, 1], "end": [264, 51], "traced_tactics": [{"tactic": "simp [comp]", "state_before": "C : Type u\ninst\u271d : Category C\nX : TopCat\n\u2131 : Presheaf C X\nY Z : TopCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nU : (Opens \u2191Z)\u1d52\u1d56\n\u22a2 (comp \u2131 f g).inv.app U = \ud835\udfd9 ((g _* (f _* \u2131)).obj U)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/BilinearMap.lean", "full_name": "LinearMap.lcomp\u209b\u2097_apply", "start": [299, 1], "end": [300, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Category/Compactum.lean", "full_name": "Compactum.cl_cl", "start": [224, 9], "end": [281, 46], "traced_tactics": [{"tactic": "rintro _ \u27e8F, hF, rfl\u27e9", "state_before": "X : Compactum\nA : Set X.A\n\u22a2 Compactum.cl (Compactum.cl A) \u2286 Compactum.cl A", "state_after": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\n\u22a2 str X F \u2208 Compactum.cl A"}, {"tactic": "let fsu := Finset (Set (Ultrafilter X))", "state_before": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\n\u22a2 str X F \u2208 Compactum.cl A", "state_after": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\n\u22a2 str X F \u2208 Compactum.cl A"}, {"tactic": "let ssu := Set (Set (Ultrafilter X))", "state_before": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\n\u22a2 str X F \u2208 Compactum.cl A", "state_after": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u22a2 str X F \u2208 Compactum.cl A"}, {"tactic": "let \u03b9 : fsu \u2192 ssu := fun x \u21a6 \u2191x", "state_before": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u22a2 str X F \u2208 Compactum.cl A", "state_after": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\n\u22a2 str X F \u2208 Compactum.cl A"}, {"tactic": "let C0 : ssu := { Z | \u2203 B \u2208 F, X.str \u207b\u00b9' B = Z }", "state_before": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\n\u22a2 str X F \u2208 Compactum.cl A", "state_after": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\n\u22a2 str X F \u2208 Compactum.cl A"}, {"tactic": "let AA := { G : Ultrafilter X | A \u2208 G }", "state_before": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\n\u22a2 str X F \u2208 Compactum.cl A", "state_after": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\n\u22a2 str X F \u2208 Compactum.cl A"}, {"tactic": "let C1 := insert AA C0", "state_before": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\n\u22a2 str X F \u2208 Compactum.cl A", "state_after": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\n\u22a2 str X F \u2208 Compactum.cl A"}, {"tactic": "let C2 := finiteInterClosure C1", "state_before": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\n\u22a2 str X F \u2208 Compactum.cl A", "state_after": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\n\u22a2 str X F \u2208 Compactum.cl A"}, {"tactic": "have claim1 : \u2200 (B) (_ : B \u2208 C0) (C) (_ : C \u2208 C0), B \u2229 C \u2208 C0 := by\n  rintro B \u27e8Q, hQ, rfl\u27e9 C \u27e8R, hR, rfl\u27e9\n  use Q \u2229 R\n  simp only [and_true_iff, eq_self_iff_true, Set.preimage_inter]\n  exact inter_sets _ hQ hR", "state_before": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\n\u22a2 str X F \u2208 Compactum.cl A", "state_after": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\n\u22a2 str X F \u2208 Compactum.cl A"}, {"tactic": "have claim2 : \u2200 B \u2208 C0, Set.Nonempty B := by\n  rintro B \u27e8Q, hQ, rfl\u27e9\n  obtain \u27e8q\u27e9 := Filter.nonempty_of_mem hQ\n  use X.incl q\n  simpa", "state_before": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\n\u22a2 str X F \u2208 Compactum.cl A", "state_after": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\n\u22a2 str X F \u2208 Compactum.cl A"}, {"tactic": "have claim3 : \u2200 B \u2208 C0, (AA \u2229 B).Nonempty := by\n  rintro B \u27e8Q, hQ, rfl\u27e9\n  have : (Q \u2229 cl A).Nonempty := Filter.nonempty_of_mem (inter_mem hQ hF)\n  rcases this with \u27e8q, hq1, P, hq2, hq3\u27e9\n  refine' \u27e8P, hq2, _\u27e9\n  rw [\u2190 hq3] at hq1\n  simpa", "state_before": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\n\u22a2 str X F \u2208 Compactum.cl A", "state_after": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\n\u22a2 str X F \u2208 Compactum.cl A"}, {"tactic": "suffices \u2200 T : fsu, \u03b9 T \u2286 C1 \u2192 (\u22c2\u2080 \u03b9 T).Nonempty by\n  obtain \u27e8G, h1\u27e9 := exists_ultrafilter_of_finite_inter_nonempty _ this\n  use X.join G\n  have : G.map X.str = F := Ultrafilter.coe_le_coe.1 fun S hS => h1 (Or.inr \u27e8S, hS, rfl\u27e9)\n  rw [join_distrib, this]\n  exact \u27e8h1 (Or.inl rfl), rfl\u27e9", "state_before": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\n\u22a2 str X F \u2208 Compactum.cl A", "state_after": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\n\u22a2 \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)"}, {"tactic": "have claim4 := finiteInterClosure_finiteInter C1", "state_before": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\n\u22a2 \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)", "state_after": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\n\u22a2 \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)"}, {"tactic": "have claim5 : FiniteInter C0 := \u27e8\u27e8_, univ_mem, Set.preimage_univ\u27e9, claim1\u27e9", "state_before": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\n\u22a2 \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)", "state_after": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\n\u22a2 \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)"}, {"tactic": "intro T hT", "state_before": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nclaim6 : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 Set.Nonempty P\n\u22a2 \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)", "state_after": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nclaim6 : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 Set.Nonempty P\nT : fsu\nhT : \u03b9 T \u2286 C1\n\u22a2 Set.Nonempty (\u22c2\u2080 \u03b9 T)"}, {"tactic": "suffices \u22c2\u2080 \u03b9 T \u2208 C2 by exact claim6 _ this", "state_before": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nclaim6 : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 Set.Nonempty P\nT : fsu\nhT : \u03b9 T \u2286 C1\n\u22a2 Set.Nonempty (\u22c2\u2080 \u03b9 T)", "state_after": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nclaim6 : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 Set.Nonempty P\nT : fsu\nhT : \u03b9 T \u2286 C1\n\u22a2 \u22c2\u2080 \u03b9 T \u2208 C2"}, {"tactic": "apply claim4.finiteInter_mem T", "state_before": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nclaim6 : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 Set.Nonempty P\nT : fsu\nhT : \u03b9 T \u2286 C1\n\u22a2 \u22c2\u2080 \u03b9 T \u2208 C2", "state_after": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nclaim6 : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 Set.Nonempty P\nT : fsu\nhT : \u03b9 T \u2286 C1\n\u22a2 \u2191T \u2286 finiteInterClosure C1"}, {"tactic": "intro t ht", "state_before": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nclaim6 : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 Set.Nonempty P\nT : fsu\nhT : \u03b9 T \u2286 C1\n\u22a2 \u2191T \u2286 finiteInterClosure C1", "state_after": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nclaim6 : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 Set.Nonempty P\nT : fsu\nhT : \u03b9 T \u2286 C1\nt : Set (Ultrafilter X.A)\nht : t \u2208 \u2191T\n\u22a2 t \u2208 finiteInterClosure C1"}, {"tactic": "refine' finiteInterClosure.basic (@hT t ht)", "state_before": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nclaim6 : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 Set.Nonempty P\nT : fsu\nhT : \u03b9 T \u2286 C1\nt : Set (Ultrafilter X.A)\nht : t \u2208 \u2191T\n\u22a2 t \u2208 finiteInterClosure C1", "state_after": "no goals"}, {"tactic": "rintro B \u27e8Q, hQ, rfl\u27e9 C \u27e8R, hR, rfl\u27e9", "state_before": "X : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\n\u22a2 \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0", "state_after": "case intro.intro.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nQ : Set X.A\nhQ : Q \u2208 F\nR : Set X.A\nhR : R \u2208 F\n\u22a2 str X \u207b\u00b9' Q \u2229 str X \u207b\u00b9' R \u2208 C0"}, {"tactic": "use Q \u2229 R", "state_before": "case intro.intro.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nQ : Set X.A\nhQ : Q \u2208 F\nR : Set X.A\nhR : R \u2208 F\n\u22a2 str X \u207b\u00b9' Q \u2229 str X \u207b\u00b9' R \u2208 C0", "state_after": "case intro.intro.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nQ : Set X.A\nhQ : Q \u2208 F\nR : Set X.A\nhR : R \u2208 F\n\u22a2 Q \u2229 R \u2208 F \u2227 str X \u207b\u00b9' (Q \u2229 R) = str X \u207b\u00b9' Q \u2229 str X \u207b\u00b9' R"}, {"tactic": "simp only [and_true_iff, eq_self_iff_true, Set.preimage_inter]", "state_before": "case intro.intro.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nQ : Set X.A\nhQ : Q \u2208 F\nR : Set X.A\nhR : R \u2208 F\n\u22a2 Q \u2229 R \u2208 F \u2227 str X \u207b\u00b9' (Q \u2229 R) = str X \u207b\u00b9' Q \u2229 str X \u207b\u00b9' R", "state_after": "case intro.intro.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nQ : Set X.A\nhQ : Q \u2208 F\nR : Set X.A\nhR : R \u2208 F\n\u22a2 Q \u2229 R \u2208 F"}, {"tactic": "exact inter_sets _ hQ hR", "state_before": "case intro.intro.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nQ : Set X.A\nhQ : Q \u2208 F\nR : Set X.A\nhR : R \u2208 F\n\u22a2 Q \u2229 R \u2208 F", "state_after": "no goals"}, {"tactic": "rintro B \u27e8Q, hQ, rfl\u27e9", "state_before": "X : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\n\u22a2 \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B", "state_after": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nQ : Set X.A\nhQ : Q \u2208 F\n\u22a2 Set.Nonempty (str X \u207b\u00b9' Q)"}, {"tactic": "obtain \u27e8q\u27e9 := Filter.nonempty_of_mem hQ", "state_before": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nQ : Set X.A\nhQ : Q \u2208 F\n\u22a2 Set.Nonempty (str X \u207b\u00b9' Q)", "state_after": "case intro.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nQ : Set X.A\nhQ : Q \u2208 F\nq : X.A\nh\u271d : q \u2208 Q\n\u22a2 Set.Nonempty (str X \u207b\u00b9' Q)"}, {"tactic": "use X.incl q", "state_before": "case intro.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nQ : Set X.A\nhQ : Q \u2208 F\nq : X.A\nh\u271d : q \u2208 Q\n\u22a2 Set.Nonempty (str X \u207b\u00b9' Q)", "state_after": "case intro.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nQ : Set X.A\nhQ : Q \u2208 F\nq : X.A\nh\u271d : q \u2208 Q\n\u22a2 incl X q \u2208 str X \u207b\u00b9' Q"}, {"tactic": "simpa", "state_before": "case intro.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nQ : Set X.A\nhQ : Q \u2208 F\nq : X.A\nh\u271d : q \u2208 Q\n\u22a2 incl X q \u2208 str X \u207b\u00b9' Q", "state_after": "no goals"}, {"tactic": "rintro B \u27e8Q, hQ, rfl\u27e9", "state_before": "X : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\n\u22a2 \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)", "state_after": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nQ : Set X.A\nhQ : Q \u2208 F\n\u22a2 Set.Nonempty (AA \u2229 str X \u207b\u00b9' Q)"}, {"tactic": "have : (Q \u2229 cl A).Nonempty := Filter.nonempty_of_mem (inter_mem hQ hF)", "state_before": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nQ : Set X.A\nhQ : Q \u2208 F\n\u22a2 Set.Nonempty (AA \u2229 str X \u207b\u00b9' Q)", "state_after": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nQ : Set X.A\nhQ : Q \u2208 F\nthis : Set.Nonempty (Q \u2229 Compactum.cl A)\n\u22a2 Set.Nonempty (AA \u2229 str X \u207b\u00b9' Q)"}, {"tactic": "rcases this with \u27e8q, hq1, P, hq2, hq3\u27e9", "state_before": "case intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nQ : Set X.A\nhQ : Q \u2208 F\nthis : Set.Nonempty (Q \u2229 Compactum.cl A)\n\u22a2 Set.Nonempty (AA \u2229 str X \u207b\u00b9' Q)", "state_after": "case intro.intro.intro.intro.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nQ : Set X.A\nhQ : Q \u2208 F\nq : X.A\nhq1 : q \u2208 Q\nP : Ultrafilter X.A\nhq2 : P \u2208 Compactum.basic A\nhq3 : str X P = q\n\u22a2 Set.Nonempty (AA \u2229 str X \u207b\u00b9' Q)"}, {"tactic": "refine' \u27e8P, hq2, _\u27e9", "state_before": "case intro.intro.intro.intro.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nQ : Set X.A\nhQ : Q \u2208 F\nq : X.A\nhq1 : q \u2208 Q\nP : Ultrafilter X.A\nhq2 : P \u2208 Compactum.basic A\nhq3 : str X P = q\n\u22a2 Set.Nonempty (AA \u2229 str X \u207b\u00b9' Q)", "state_after": "case intro.intro.intro.intro.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nQ : Set X.A\nhQ : Q \u2208 F\nq : X.A\nhq1 : q \u2208 Q\nP : Ultrafilter X.A\nhq2 : P \u2208 Compactum.basic A\nhq3 : str X P = q\n\u22a2 P \u2208 str X \u207b\u00b9' Q"}, {"tactic": "rw [\u2190 hq3] at hq1", "state_before": "case intro.intro.intro.intro.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nQ : Set X.A\nhQ : Q \u2208 F\nq : X.A\nhq1 : q \u2208 Q\nP : Ultrafilter X.A\nhq2 : P \u2208 Compactum.basic A\nhq3 : str X P = q\n\u22a2 P \u2208 str X \u207b\u00b9' Q", "state_after": "case intro.intro.intro.intro.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nQ : Set X.A\nhQ : Q \u2208 F\nq : X.A\nP : Ultrafilter X.A\nhq1 : str X P \u2208 Q\nhq2 : P \u2208 Compactum.basic A\nhq3 : str X P = q\n\u22a2 P \u2208 str X \u207b\u00b9' Q"}, {"tactic": "simpa", "state_before": "case intro.intro.intro.intro.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nQ : Set X.A\nhQ : Q \u2208 F\nq : X.A\nP : Ultrafilter X.A\nhq1 : str X P \u2208 Q\nhq2 : P \u2208 Compactum.basic A\nhq3 : str X P = q\n\u22a2 P \u2208 str X \u207b\u00b9' Q", "state_after": "no goals"}, {"tactic": "obtain \u27e8G, h1\u27e9 := exists_ultrafilter_of_finite_inter_nonempty _ this", "state_before": "X : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nthis : \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)\n\u22a2 str X F \u2208 Compactum.cl A", "state_after": "case intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nthis : \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)\nG : Ultrafilter (Ultrafilter X.A)\nh1 : C1 \u2286 G.sets\n\u22a2 str X F \u2208 Compactum.cl A"}, {"tactic": "use X.join G", "state_before": "case intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nthis : \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)\nG : Ultrafilter (Ultrafilter X.A)\nh1 : C1 \u2286 G.sets\n\u22a2 str X F \u2208 Compactum.cl A", "state_after": "case intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nthis : \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)\nG : Ultrafilter (Ultrafilter X.A)\nh1 : C1 \u2286 G.sets\n\u22a2 join X G \u2208 Compactum.basic A \u2227 str X (join X G) = str X F"}, {"tactic": "have : G.map X.str = F := Ultrafilter.coe_le_coe.1 fun S hS => h1 (Or.inr \u27e8S, hS, rfl\u27e9)", "state_before": "case intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nthis : \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)\nG : Ultrafilter (Ultrafilter X.A)\nh1 : C1 \u2286 G.sets\n\u22a2 join X G \u2208 Compactum.basic A \u2227 str X (join X G) = str X F", "state_after": "case intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nthis\u271d : \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)\nG : Ultrafilter (Ultrafilter X.A)\nh1 : C1 \u2286 G.sets\nthis : Ultrafilter.map (str X) G = F\n\u22a2 join X G \u2208 Compactum.basic A \u2227 str X (join X G) = str X F"}, {"tactic": "rw [join_distrib, this]", "state_before": "case intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nthis\u271d : \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)\nG : Ultrafilter (Ultrafilter X.A)\nh1 : C1 \u2286 G.sets\nthis : Ultrafilter.map (str X) G = F\n\u22a2 join X G \u2208 Compactum.basic A \u2227 str X (join X G) = str X F", "state_after": "case intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nthis\u271d : \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)\nG : Ultrafilter (Ultrafilter X.A)\nh1 : C1 \u2286 G.sets\nthis : Ultrafilter.map (str X) G = F\n\u22a2 join X G \u2208 Compactum.basic A \u2227 str X F = str X F"}, {"tactic": "exact \u27e8h1 (Or.inl rfl), rfl\u27e9", "state_before": "case intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nthis\u271d : \u2200 (T : fsu), \u03b9 T \u2286 C1 \u2192 Set.Nonempty (\u22c2\u2080 \u03b9 T)\nG : Ultrafilter (Ultrafilter X.A)\nh1 : C1 \u2286 G.sets\nthis : Ultrafilter.map (str X) G = F\n\u22a2 join X G \u2208 Compactum.basic A \u2227 str X F = str X F", "state_after": "no goals"}, {"tactic": "intro P hP", "state_before": "X : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\n\u22a2 \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 P \u2208 C0 \u2228 \u2203 Q, Q \u2208 C0 \u2227 P = AA \u2229 Q", "state_after": "X : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nP : Set (Ultrafilter X.A)\nhP : P \u2208 C2\n\u22a2 P \u2208 C0 \u2228 \u2203 Q, Q \u2208 C0 \u2227 P = AA \u2229 Q"}, {"tactic": "exact claim5.finiteInterClosure_insert _ hP", "state_before": "X : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nP : Set (Ultrafilter X.A)\nhP : P \u2208 C2\n\u22a2 P \u2208 C0 \u2228 \u2203 Q, Q \u2208 C0 \u2227 P = AA \u2229 Q", "state_after": "no goals"}, {"tactic": "intro P hP", "state_before": "X : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nthis : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 P \u2208 C0 \u2228 \u2203 Q, Q \u2208 C0 \u2227 P = AA \u2229 Q\n\u22a2 \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 Set.Nonempty P", "state_after": "X : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nthis : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 P \u2208 C0 \u2228 \u2203 Q, Q \u2208 C0 \u2227 P = AA \u2229 Q\nP : Set (Ultrafilter X.A)\nhP : P \u2208 C2\n\u22a2 Set.Nonempty P"}, {"tactic": "cases' this P hP with h h", "state_before": "X : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nthis : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 P \u2208 C0 \u2228 \u2203 Q, Q \u2208 C0 \u2227 P = AA \u2229 Q\nP : Set (Ultrafilter X.A)\nhP : P \u2208 C2\n\u22a2 Set.Nonempty P", "state_after": "case inl\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nthis : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 P \u2208 C0 \u2228 \u2203 Q, Q \u2208 C0 \u2227 P = AA \u2229 Q\nP : Set (Ultrafilter X.A)\nhP : P \u2208 C2\nh : P \u2208 C0\n\u22a2 Set.Nonempty P\n\ncase inr\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nthis : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 P \u2208 C0 \u2228 \u2203 Q, Q \u2208 C0 \u2227 P = AA \u2229 Q\nP : Set (Ultrafilter X.A)\nhP : P \u2208 C2\nh : \u2203 Q, Q \u2208 C0 \u2227 P = AA \u2229 Q\n\u22a2 Set.Nonempty P"}, {"tactic": "exact claim2 _ h", "state_before": "case inl\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nthis : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 P \u2208 C0 \u2228 \u2203 Q, Q \u2208 C0 \u2227 P = AA \u2229 Q\nP : Set (Ultrafilter X.A)\nhP : P \u2208 C2\nh : P \u2208 C0\n\u22a2 Set.Nonempty P", "state_after": "no goals"}, {"tactic": "rcases h with \u27e8Q, hQ, rfl\u27e9", "state_before": "case inr\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nthis : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 P \u2208 C0 \u2228 \u2203 Q, Q \u2208 C0 \u2227 P = AA \u2229 Q\nP : Set (Ultrafilter X.A)\nhP : P \u2208 C2\nh : \u2203 Q, Q \u2208 C0 \u2227 P = AA \u2229 Q\n\u22a2 Set.Nonempty P", "state_after": "case inr.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nthis : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 P \u2208 C0 \u2228 \u2203 Q, Q \u2208 C0 \u2227 P = AA \u2229 Q\nQ : Set (Ultrafilter X.A)\nhQ : Q \u2208 C0\nhP : AA \u2229 Q \u2208 C2\n\u22a2 Set.Nonempty (AA \u2229 Q)"}, {"tactic": "exact claim3 _ hQ", "state_before": "case inr.intro.intro\nX : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nthis : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 P \u2208 C0 \u2228 \u2203 Q, Q \u2208 C0 \u2227 P = AA \u2229 Q\nQ : Set (Ultrafilter X.A)\nhQ : Q \u2208 C0\nhP : AA \u2229 Q \u2208 C2\n\u22a2 Set.Nonempty (AA \u2229 Q)", "state_after": "no goals"}, {"tactic": "exact claim6 _ this", "state_before": "X : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F \u2208 Compactum.basic (Compactum.cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\n\u03b9 : fsu \u2192 ssu := fun x => \u2191x\nC0 : ssu := {Z | \u2203 B, B \u2208 F \u2227 str X \u207b\u00b9' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A \u2208 G}\nC1 : ssu := insert AA C0\nC2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1\nclaim1 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 \u2200 (C : Set (Ultrafilter X.A)), C \u2208 C0 \u2192 B \u2229 C \u2208 C0\nclaim2 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty B\nclaim3 : \u2200 (B : Set (Ultrafilter X.A)), B \u2208 C0 \u2192 Set.Nonempty (AA \u2229 B)\nclaim4 : FiniteInter (finiteInterClosure C1)\nclaim5 : FiniteInter C0\nclaim6 : \u2200 (P : Set (Ultrafilter X.A)), P \u2208 C2 \u2192 Set.Nonempty P\nT : fsu\nhT : \u03b9 T \u2286 C1\nthis : \u22c2\u2080 \u03b9 T \u2208 C2\n\u22a2 Set.Nonempty (\u22c2\u2080 \u03b9 T)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "full_name": "OrthonormalBasis.coe_toBasis_repr", "start": [427, 11], "end": [429, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finsupp/ToDfinsupp.lean", "full_name": "Finsupp.toDfinsupp_sub", "start": [170, 1], "end": [172, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "full_name": "WithTop.coe_bit0", "start": [133, 1], "end": [134, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Prod.lean", "full_name": "Finset.mem_product", "start": [56, 1], "end": [57, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Sym/Basic.lean", "full_name": "Sym.coe_nil", "start": [106, 1], "end": [107, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Units.lean", "full_name": "Units.inv_eq_of_mul_eq_one_left", "start": [347, 11], "end": [350, 47], "traced_tactics": [{"tactic": "rw [one_mul]", "state_before": "\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b c u : \u03b1\u02e3\na : \u03b1\nh : a * \u2191u = 1\n\u22a2 \u2191u\u207b\u00b9 = 1 * \u2191u\u207b\u00b9", "state_after": "no goals"}, {"tactic": "rw [\u2190 h, mul_inv_cancel_right]", "state_before": "\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b c u : \u03b1\u02e3\na : \u03b1\nh : a * \u2191u = 1\n\u22a2 1 * \u2191u\u207b\u00b9 = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/Cotangent.lean", "full_name": "Ideal.cotangentIdeal_square", "start": [129, 1], "end": [135, 41], "traced_tactics": [{"tactic": "rw [eq_bot_iff, pow_two I.cotangentIdeal, \u2190 smul_eq_mul]", "state_before": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\n\u22a2 cotangentIdeal I ^ 2 = \u22a5", "state_after": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\n\u22a2 cotangentIdeal I \u2022 cotangentIdeal I \u2264 \u22a5"}, {"tactic": "intro x hx", "state_before": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\n\u22a2 cotangentIdeal I \u2022 cotangentIdeal I \u2264 \u22a5", "state_after": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx : R \u29f8 I ^ 2\nhx : x \u2208 cotangentIdeal I \u2022 cotangentIdeal I\n\u22a2 x \u2208 \u22a5"}, {"tactic": "refine Submodule.smul_induction_on hx ?_ ?_", "state_before": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx : R \u29f8 I ^ 2\nhx : x \u2208 cotangentIdeal I \u2022 cotangentIdeal I\n\u22a2 x \u2208 \u22a5", "state_after": "case refine_1\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx : R \u29f8 I ^ 2\nhx : x \u2208 cotangentIdeal I \u2022 cotangentIdeal I\n\u22a2 \u2200 (r : R \u29f8 I ^ 2), r \u2208 cotangentIdeal I \u2192 \u2200 (n : R \u29f8 I ^ 2), n \u2208 cotangentIdeal I \u2192 r \u2022 n \u2208 \u22a5\n\ncase refine_2\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx : R \u29f8 I ^ 2\nhx : x \u2208 cotangentIdeal I \u2022 cotangentIdeal I\n\u22a2 \u2200 (x y : R \u29f8 I ^ 2), x \u2208 \u22a5 \u2192 y \u2208 \u22a5 \u2192 x + y \u2208 \u22a5"}, {"tactic": "rintro _ \u27e8x, hx, rfl\u27e9 _ \u27e8y, hy, rfl\u27e9", "state_before": "case refine_1\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx : R \u29f8 I ^ 2\nhx : x \u2208 cotangentIdeal I \u2022 cotangentIdeal I\n\u22a2 \u2200 (r : R \u29f8 I ^ 2), r \u2208 cotangentIdeal I \u2192 \u2200 (n : R \u29f8 I ^ 2), n \u2208 cotangentIdeal I \u2192 r \u2022 n \u2208 \u22a5", "state_after": "case refine_1.intro.intro.intro.intro\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx\u271d : R \u29f8 I ^ 2\nhx\u271d : x\u271d \u2208 cotangentIdeal I \u2022 cotangentIdeal I\nx : R\nhx : x \u2208 \u2191I\ny : R\nhy : y \u2208 \u2191I\n\u22a2 \u2191(RingHom.toSemilinearMap (Quotient.mk (I ^ 2))) x \u2022 \u2191(RingHom.toSemilinearMap (Quotient.mk (I ^ 2))) y \u2208 \u22a5"}, {"tactic": "apply (Submodule.Quotient.eq _).mpr _", "state_before": "case refine_1.intro.intro.intro.intro\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx\u271d : R \u29f8 I ^ 2\nhx\u271d : x\u271d \u2208 cotangentIdeal I \u2022 cotangentIdeal I\nx : R\nhx : x \u2208 \u2191I\ny : R\nhy : y \u2208 \u2191I\n\u22a2 \u2191(RingHom.toSemilinearMap (Quotient.mk (I ^ 2))) x \u2022 \u2191(RingHom.toSemilinearMap (Quotient.mk (I ^ 2))) y \u2208 \u22a5", "state_after": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx\u271d : R \u29f8 I ^ 2\nhx\u271d : x\u271d \u2208 cotangentIdeal I \u2022 cotangentIdeal I\nx : R\nhx : x \u2208 \u2191I\ny : R\nhy : y \u2208 \u2191I\n\u22a2 (fun x x_1 => x * x_1) x y - 0 \u2208 I ^ 2"}, {"tactic": "rw [sub_zero, pow_two]", "state_before": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx\u271d : R \u29f8 I ^ 2\nhx\u271d : x\u271d \u2208 cotangentIdeal I \u2022 cotangentIdeal I\nx : R\nhx : x \u2208 \u2191I\ny : R\nhy : y \u2208 \u2191I\n\u22a2 (fun x x_1 => x * x_1) x y - 0 \u2208 I ^ 2", "state_after": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx\u271d : R \u29f8 I ^ 2\nhx\u271d : x\u271d \u2208 cotangentIdeal I \u2022 cotangentIdeal I\nx : R\nhx : x \u2208 \u2191I\ny : R\nhy : y \u2208 \u2191I\n\u22a2 (fun x x_1 => x * x_1) x y \u2208 I * I"}, {"tactic": "exact Ideal.mul_mem_mul hx hy", "state_before": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx\u271d : R \u29f8 I ^ 2\nhx\u271d : x\u271d \u2208 cotangentIdeal I \u2022 cotangentIdeal I\nx : R\nhx : x \u2208 \u2191I\ny : R\nhy : y \u2208 \u2191I\n\u22a2 (fun x x_1 => x * x_1) x y \u2208 I * I", "state_after": "no goals"}, {"tactic": "intro x y hx hy", "state_before": "case refine_2\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx : R \u29f8 I ^ 2\nhx : x \u2208 cotangentIdeal I \u2022 cotangentIdeal I\n\u22a2 \u2200 (x y : R \u29f8 I ^ 2), x \u2208 \u22a5 \u2192 y \u2208 \u22a5 \u2192 x + y \u2208 \u22a5", "state_after": "case refine_2\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx\u271d : R \u29f8 I ^ 2\nhx\u271d : x\u271d \u2208 cotangentIdeal I \u2022 cotangentIdeal I\nx y : R \u29f8 I ^ 2\nhx : x \u2208 \u22a5\nhy : y \u2208 \u22a5\n\u22a2 x + y \u2208 \u22a5"}, {"tactic": "exact add_mem hx hy", "state_before": "case refine_2\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx\u271d : R \u29f8 I ^ 2\nhx\u271d : x\u271d \u2208 cotangentIdeal I \u2022 cotangentIdeal I\nx y : R \u29f8 I ^ 2\nhx : x \u2208 \u22a5\nhy : y \u2208 \u22a5\n\u22a2 x + y \u2208 \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Subring/Basic.lean", "full_name": "Subring.prod_bot_sup_bot_prod", "start": [1289, 1], "end": [1294, 97], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "full_name": "Polynomial.degree_C_lt_degree_C_mul_X", "start": [1149, 1], "end": [1150, 51], "traced_tactics": [{"tactic": "simpa only [degree_C_mul_X ha] using degree_C_lt", "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.806235\nha : a \u2260 0\n\u22a2 degree (\u2191C b) < degree (\u2191C a * 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_associativity_aux", "start": [455, 1], "end": [469, 98], "traced_tactics": [{"tactic": "slice_rhs 3 5 => rw [\u2190 tensor_comp, \u2190 tensor_comp, hexagon_forward, 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\nW X Y Z : C\n\u22a2 ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Y Z).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ (W \u2297 Y) Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ Z W Y).inv) =\n    (\ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom) \u226b\n      (\u03b1_ (W \u2297 X) Z Y).inv \u226b\n        ((\u03b1_ W X Z).hom \u2297 \ud835\udfd9 Y) \u226b ((\u03b2_ W (X \u2297 Z)).hom \u2297 \ud835\udfd9 Y) \u226b ((\u03b1_ X Z W).hom \u2297 \ud835\udfd9 Y) \u226b (\u03b1_ X (Z \u2297 W) Y).hom", "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\nW X Y Z : C\n\u22a2 ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Y Z).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ (W \u2297 Y) Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ Z W Y).inv) =\n    (\ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom) \u226b\n      (\u03b1_ (W \u2297 X) Z Y).inv \u226b\n        ((((\u03b2_ W X).hom \u2297 \ud835\udfd9 Z) \u2297 \ud835\udfd9 Y) \u226b ((\u03b1_ X W Z).hom \u2297 \ud835\udfd9 Y) \u226b ((\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom) \u2297 \ud835\udfd9 Y)) \u226b (\u03b1_ X (Z \u2297 W) Y).hom"}, {"tactic": "slice_rhs 5 6 => 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\nW X Y Z : C\n\u22a2 ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Y Z).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ (W \u2297 Y) Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ Z W Y).inv) =\n    (\ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom) \u226b\n      (\u03b1_ (W \u2297 X) Z Y).inv \u226b\n        ((((\u03b2_ W X).hom \u2297 \ud835\udfd9 Z) \u2297 \ud835\udfd9 Y) \u226b ((\u03b1_ X W Z).hom \u2297 \ud835\udfd9 Y) \u226b ((\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom) \u2297 \ud835\udfd9 Y)) \u226b (\u03b1_ X (Z \u2297 W) Y).hom", "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\nW X Y Z : C\n\u22a2 ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Y Z).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ (W \u2297 Y) Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ Z W Y).inv) =\n    (\ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom) \u226b\n      (\u03b1_ (W \u2297 X) Z Y).inv \u226b\n        (((\u03b2_ W X).hom \u2297 \ud835\udfd9 Z) \u2297 \ud835\udfd9 Y) \u226b ((\u03b1_ X W Z).hom \u2297 \ud835\udfd9 Y) \u226b (\u03b1_ X (W \u2297 Z) Y).hom \u226b (\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y)"}, {"tactic": "slice_rhs 2 3 => rw [\u2190 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\nW X Y Z : C\n\u22a2 ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Y Z).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ (W \u2297 Y) Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ Z W Y).inv) =\n    (\ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom) \u226b\n      (\u03b1_ (W \u2297 X) Z Y).inv \u226b\n        (((\u03b2_ W X).hom \u2297 \ud835\udfd9 Z) \u2297 \ud835\udfd9 Y) \u226b ((\u03b1_ X W Z).hom \u2297 \ud835\udfd9 Y) \u226b (\u03b1_ X (W \u2297 Z) Y).hom \u226b (\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y)", "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\nW X Y Z : C\n\u22a2 ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Y Z).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ (W \u2297 Y) Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ Z W Y).inv) =\n    (\ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom) \u226b\n      (((((\u03b2_ W X).hom \u2297 \ud835\udfd9 Z \u2297 \ud835\udfd9 Y) \u226b (\u03b1_ (X \u2297 W) Z Y).inv) \u226b ((\u03b1_ X W Z).hom \u2297 \ud835\udfd9 Y)) \u226b (\u03b1_ X (W \u2297 Z) Y).hom) \u226b\n        (\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y)"}, {"tactic": "slice_rhs 3 5 => rw [\u2190 pentagon_hom_inv]", "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\nW X Y Z : C\n\u22a2 ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Y Z).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ (W \u2297 Y) Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ Z W Y).inv) =\n    (\ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom) \u226b\n      (((((\u03b2_ W X).hom \u2297 \ud835\udfd9 Z \u2297 \ud835\udfd9 Y) \u226b (\u03b1_ (X \u2297 W) Z Y).inv) \u226b ((\u03b1_ X W Z).hom \u2297 \ud835\udfd9 Y)) \u226b (\u03b1_ X (W \u2297 Z) Y).hom) \u226b\n        (\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y)", "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\nW X Y Z : C\n\u22a2 ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Y Z).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ (W \u2297 Y) Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ Z W Y).inv) =\n    (\ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom) \u226b\n      ((\u03b2_ W X).hom \u2297 \ud835\udfd9 Z \u2297 \ud835\udfd9 Y) \u226b ((\u03b1_ X W (Z \u2297 Y)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Z Y).inv)) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y)"}, {"tactic": "slice_rhs 1 2 => rw [tensor_id, id_tensor_comp_tensor_id, \u2190 tensor_id_comp_id_tensor]", "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\nW X Y Z : C\n\u22a2 ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Y Z).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ (W \u2297 Y) Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ Z W Y).inv) =\n    (\ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom) \u226b\n      ((\u03b2_ W X).hom \u2297 \ud835\udfd9 Z \u2297 \ud835\udfd9 Y) \u226b ((\u03b1_ X W (Z \u2297 Y)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Z Y).inv)) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y)", "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\nW X Y Z : C\n\u22a2 ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Y Z).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ (W \u2297 Y) Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ Z W Y).inv) =\n    (((((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b (\ud835\udfd9 (X \u2297 W) \u2297 (\u03b2_ Y Z).hom)) \u226b (\u03b1_ X W (Z \u2297 Y)).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Z Y).inv)) \u226b\n      (\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y)"}, {"tactic": "slice_rhs 2 3 => rw [\u2190 tensor_id, 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\nW X Y Z : C\n\u22a2 ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Y Z).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ (W \u2297 Y) Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ Z W Y).inv) =\n    (((((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b (\ud835\udfd9 (X \u2297 W) \u2297 (\u03b2_ Y Z).hom)) \u226b (\u03b1_ X W (Z \u2297 Y)).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Z Y).inv)) \u226b\n      (\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y)", "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\nW X Y Z : C\n\u22a2 ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Y Z).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ (W \u2297 Y) Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ Z W Y).inv) =\n    ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (((\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 \ud835\udfd9 W \u2297 (\u03b2_ Y Z).hom)) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Z Y).inv)) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y)"}, {"tactic": "slice_rhs 3 5 => rw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 hexagon_reverse, 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\nW X Y Z : C\n\u22a2 ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Y Z).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ (W \u2297 Y) Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ Z W Y).inv) =\n    ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (((\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 \ud835\udfd9 W \u2297 (\u03b2_ Y Z).hom)) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Z Y).inv)) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 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C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 (tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (M.mul \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom)) \u226b (\u03c1_ M.X).hom =\n    ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b M.mul", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 (tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (\ud835\udfd9 (M.X \u2297 M.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (M.mul \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b (\u03c1_ M.X).hom =\n    ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul"}, {"tactic": "slice_lhs 3 4 => rw [rightUnitor_naturality]", "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 (tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (\ud835\udfd9 (M.X \u2297 M.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (M.mul \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b (\u03c1_ M.X).hom =\n    ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (\ud835\udfd9 (M.X \u2297 M.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (\u03c1_ (M.X \u2297 M.X)).hom \u226b M.mul =\n    ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul"}, {"tactic": "slice_lhs 1 3 => rw [\u2190 rightUnitor_monoidal]", "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 tensor_\u03bc C (M.X, \ud835\udfd9_ C) (M.X, \ud835\udfd9_ C) \u226b (\ud835\udfd9 (M.X \u2297 M.X) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (\u03c1_ (M.X \u2297 M.X)).hom \u226b M.mul =\n    ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b M.mul = ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul"}, {"tactic": "simp only [Category.assoc, Category.id_comp]", "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nM : Mon_ C\n\u22a2 ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b M.mul = ((\u03c1_ M.X).hom \u2297 (\u03c1_ M.X).hom) \u226b \ud835\udfd9 (M.X \u2297 M.X) \u226b M.mul", "state_after": "no 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n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\ne : n \u2243 m\nA : Matrix m m R\n\u22a2 \u2211 \u03c3 : Perm n, \u2191\u2191(\u2191sign \u03c3) * \u220f i : n, submatrix A (\u2191e) (\u2191e) (\u2191\u03c3 i) i = \u2211 \u03c3 : Perm m, \u2191\u2191(\u2191sign \u03c3) * \u220f i : m, A (\u2191\u03c3 i) i"}, {"tactic": "apply Fintype.sum_equiv (Equiv.permCongr e)", "state_before": "m : Type u_2\nn : Type u_1\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\ne : n \u2243 m\nA : Matrix m m R\n\u22a2 \u2211 \u03c3 : Perm n, \u2191\u2191(\u2191sign \u03c3) * \u220f i : n, submatrix A (\u2191e) (\u2191e) (\u2191\u03c3 i) i = \u2211 \u03c3 : Perm m, \u2191\u2191(\u2191sign \u03c3) * \u220f i : m, A (\u2191\u03c3 i) i", "state_after": "case h\nm : Type u_2\nn : Type u_1\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\ne : n \u2243 m\nA : Matrix m m R\n\u22a2 \u2200 (x : Perm n),\n    \u2191\u2191(\u2191sign x) * \u220f i : n, submatrix A (\u2191e) (\u2191e) (\u2191x i) i =\n      \u2191\u2191(\u2191sign (\u2191(permCongr e) x)) * \u220f i : m, A (\u2191(\u2191(permCongr e) x) i) i"}, {"tactic": "intro \u03c3", "state_before": "case h\nm : Type u_2\nn : Type u_1\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\ne : n \u2243 m\nA : Matrix m m R\n\u22a2 \u2200 (x : Perm n),\n    \u2191\u2191(\u2191sign x) * \u220f i : n, submatrix A (\u2191e) (\u2191e) (\u2191x i) i =\n      \u2191\u2191(\u2191sign (\u2191(permCongr e) x)) * \u220f i : m, A (\u2191(\u2191(permCongr e) x) i) i", "state_after": "case h\nm : Type u_2\nn : Type u_1\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\ne : n \u2243 m\nA : Matrix m m R\n\u03c3 : Perm n\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f i : n, submatrix A (\u2191e) (\u2191e) (\u2191\u03c3 i) i =\n    \u2191\u2191(\u2191sign (\u2191(permCongr e) \u03c3)) * \u220f i : m, A (\u2191(\u2191(permCongr e) \u03c3) i) i"}, {"tactic": "rw [Equiv.Perm.sign_permCongr e \u03c3]", "state_before": "case h\nm : Type u_2\nn : Type u_1\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\ne : n \u2243 m\nA : Matrix m m R\n\u03c3 : Perm n\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f i : n, submatrix A (\u2191e) (\u2191e) (\u2191\u03c3 i) i =\n    \u2191\u2191(\u2191sign (\u2191(permCongr e) \u03c3)) * \u220f i : m, A (\u2191(\u2191(permCongr e) \u03c3) i) i", "state_after": "case h\nm : Type u_2\nn : Type u_1\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\ne : n \u2243 m\nA : Matrix m m R\n\u03c3 : Perm n\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f i : n, submatrix A (\u2191e) (\u2191e) (\u2191\u03c3 i) i = \u2191\u2191(\u2191sign \u03c3) * \u220f i : m, A (\u2191(\u2191(permCongr e) \u03c3) i) i"}, {"tactic": "congr 1", "state_before": "case h\nm : Type u_2\nn : Type u_1\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\ne : n \u2243 m\nA : Matrix m m R\n\u03c3 : Perm n\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f i : n, submatrix A (\u2191e) (\u2191e) (\u2191\u03c3 i) i = \u2191\u2191(\u2191sign \u03c3) * \u220f i : m, A (\u2191(\u2191(permCongr e) \u03c3) i) i", "state_after": "case h.e_a\nm : Type u_2\nn : Type u_1\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\ne : n \u2243 m\nA : Matrix m m R\n\u03c3 : Perm n\n\u22a2 \u220f i : n, submatrix A (\u2191e) (\u2191e) (\u2191\u03c3 i) i = \u220f i : m, A (\u2191(\u2191(permCongr e) \u03c3) i) i"}, {"tactic": "apply Fintype.prod_equiv e", "state_before": "case h.e_a\nm : Type u_2\nn : Type u_1\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\ne : n \u2243 m\nA : Matrix m m R\n\u03c3 : Perm n\n\u22a2 \u220f i : n, submatrix A (\u2191e) (\u2191e) (\u2191\u03c3 i) i = \u220f i : m, A (\u2191(\u2191(permCongr e) \u03c3) i) i", "state_after": "case h.e_a.h\nm : Type u_2\nn : Type u_1\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\ne : n \u2243 m\nA : Matrix m m R\n\u03c3 : Perm n\n\u22a2 \u2200 (x : n), submatrix A (\u2191e) (\u2191e) (\u2191\u03c3 x) x = A (\u2191(\u2191(permCongr e) \u03c3) (\u2191e x)) (\u2191e x)"}, {"tactic": "intro i", "state_before": "case h.e_a.h\nm : Type u_2\nn : Type u_1\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\ne : n \u2243 m\nA : Matrix m m R\n\u03c3 : Perm n\n\u22a2 \u2200 (x : n), submatrix A (\u2191e) (\u2191e) (\u2191\u03c3 x) x = A (\u2191(\u2191(permCongr e) \u03c3) (\u2191e x)) (\u2191e x)", "state_after": "case h.e_a.h\nm : Type u_2\nn : Type u_1\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\ne : n \u2243 m\nA : Matrix m m R\n\u03c3 : Perm n\ni : n\n\u22a2 submatrix A (\u2191e) (\u2191e) (\u2191\u03c3 i) i = A (\u2191(\u2191(permCongr e) \u03c3) (\u2191e i)) (\u2191e i)"}, {"tactic": "rw [Equiv.permCongr_apply, Equiv.symm_apply_apply, submatrix_apply]", "state_before": "case h.e_a.h\nm : Type u_2\nn : Type u_1\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\ne : n \u2243 m\nA : Matrix m m R\n\u03c3 : Perm n\ni : n\n\u22a2 submatrix A (\u2191e) (\u2191e) (\u2191\u03c3 i) i = A (\u2191(\u2191(permCongr e) \u03c3) (\u2191e i)) (\u2191e i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM2.stmts\u2081_supportsStmt_mono", "start": [2214, 1], "end": [2222, 32], "traced_tactics": [{"tactic": "induction' q\u2082 with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>\n  simp only [stmts\u2081, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]\n    at h hs", "state_before": "K : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh : q\u2081 \u2208 stmts\u2081 q\u2082\nhs : SupportsStmt S q\u2082\n\u22a2 SupportsStmt S q\u2081", "state_after": "case push\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 \u0393 k\u271d\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = push k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n\ncase peek\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = peek k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n\ncase pop\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = pop k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n\ncase load\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n\ncase branch\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 SupportsStmt S a\u271d\u00b9 \u2192 SupportsStmt S q\u2081\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 SupportsStmt S a\u271d \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S a\u271d\u00b9 \u2227 SupportsStmt S a\u271d\nh : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 SupportsStmt S q\u2081\n\ncase goto\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u039b\nhs : \u2200 (v : \u03c3), a\u271d v \u2208 S\nh : q\u2081 = goto a\u271d\n\u22a2 SupportsStmt S q\u2081\n\ncase halt\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081"}, {"tactic": "iterate 4 rcases h with (rfl | h) <;> [exact hs; exact IH h hs]", "state_before": "case push\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 \u0393 k\u271d\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = push k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n\ncase peek\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = peek k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n\ncase pop\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nk\u271d : K\na\u271d : \u03c3 \u2192 Option (\u0393 k\u271d) \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = pop k\u271d a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n\ncase load\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n\ncase branch\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 SupportsStmt S a\u271d\u00b9 \u2192 SupportsStmt S q\u2081\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 SupportsStmt S a\u271d \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S a\u271d\u00b9 \u2227 SupportsStmt S a\u271d\nh : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 SupportsStmt S q\u2081\n\ncase goto\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u039b\nhs : \u2200 (v : \u03c3), a\u271d v \u2208 S\nh : q\u2081 = goto a\u271d\n\u22a2 SupportsStmt S q\u2081\n\ncase halt\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081", "state_after": "case branch\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 SupportsStmt S a\u271d\u00b9 \u2192 SupportsStmt S q\u2081\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 SupportsStmt S a\u271d \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S a\u271d\u00b9 \u2227 SupportsStmt S a\u271d\nh : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 SupportsStmt S q\u2081\n\ncase goto\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u039b\nhs : \u2200 (v : \u03c3), a\u271d v \u2208 S\nh : q\u2081 = goto a\u271d\n\u22a2 SupportsStmt S q\u2081\n\ncase halt\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081"}, {"tactic": "case branch f q\u2081 q\u2082 IH\u2081 IH\u2082 => rcases h with (rfl | h | h); exacts [hs, IH\u2081 h hs.1, IH\u2082 h hs.2]", "state_before": "case branch\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 SupportsStmt S a\u271d\u00b9 \u2192 SupportsStmt S q\u2081\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 SupportsStmt S a\u271d \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S a\u271d\u00b9 \u2227 SupportsStmt S a\u271d\nh : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 SupportsStmt S q\u2081\n\ncase goto\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u039b\nhs : \u2200 (v : \u03c3), a\u271d v \u2208 S\nh : q\u2081 = goto a\u271d\n\u22a2 SupportsStmt S q\u2081\n\ncase halt\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081", "state_after": "case goto\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u039b\nhs : \u2200 (v : \u03c3), a\u271d v \u2208 S\nh : q\u2081 = goto a\u271d\n\u22a2 SupportsStmt S q\u2081\n\ncase halt\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081"}, {"tactic": "case goto l => subst h; exact hs", "state_before": "case goto\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u039b\nhs : \u2200 (v : \u03c3), a\u271d v \u2208 S\nh : q\u2081 = goto a\u271d\n\u22a2 SupportsStmt S q\u2081\n\ncase halt\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081", "state_after": "case halt\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081"}, {"tactic": "case halt => subst h; trivial", "state_before": "case halt\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081", "state_after": "no goals"}, {"tactic": "rcases h with (rfl | h) <;> [exact hs; exact IH h hs]", "state_before": "case load\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u03c3\nq : Stmt\u2082\nIH : q\u2081 \u2208 stmts\u2081 q \u2192 SupportsStmt S q \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S q\nh : q\u2081 = load a\u271d q \u2228 q\u2081 \u2208 stmts\u2081 q\n\u22a2 SupportsStmt S q\u2081\n\ncase branch\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 SupportsStmt S a\u271d\u00b9 \u2192 SupportsStmt S q\u2081\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 SupportsStmt S a\u271d \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S a\u271d\u00b9 \u2227 SupportsStmt S a\u271d\nh : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 SupportsStmt S q\u2081\n\ncase goto\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u039b\nhs : \u2200 (v : \u03c3), a\u271d v \u2208 S\nh : q\u2081 = goto a\u271d\n\u22a2 SupportsStmt S q\u2081\n\ncase halt\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081", "state_after": "case branch\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d\u00b2 : \u03c3 \u2192 Bool\na\u271d\u00b9 a\u271d : Stmt\u2082\na_ih\u271d\u00b9 : q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2192 SupportsStmt S a\u271d\u00b9 \u2192 SupportsStmt S q\u2081\na_ih\u271d : q\u2081 \u2208 stmts\u2081 a\u271d \u2192 SupportsStmt S a\u271d \u2192 SupportsStmt S q\u2081\nhs : SupportsStmt S a\u271d\u00b9 \u2227 SupportsStmt S a\u271d\nh : q\u2081 = branch a\u271d\u00b2 a\u271d\u00b9 a\u271d \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\u00b9 \u2228 q\u2081 \u2208 stmts\u2081 a\u271d\n\u22a2 SupportsStmt S q\u2081\n\ncase goto\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\na\u271d : \u03c3 \u2192 \u039b\nhs : \u2200 (v : \u03c3), a\u271d v \u2208 S\nh : q\u2081 = goto a\u271d\n\u22a2 SupportsStmt S q\u2081\n\ncase halt\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081"}, {"tactic": "rcases h with (rfl | h | h)", "state_before": "K : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nh\u271d : q\u2081\u271d \u2208 stmts\u2081 q\u2082\u271d\nhs\u271d : SupportsStmt S q\u2082\u271d\nf : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 : q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2192 SupportsStmt S q\u2081 \u2192 SupportsStmt S q\u2081\u271d\nIH\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082 \u2192 SupportsStmt S q\u2082 \u2192 SupportsStmt S q\u2081\u271d\nhs : SupportsStmt S q\u2081 \u2227 SupportsStmt S q\u2082\nh : q\u2081\u271d = branch f q\u2081 q\u2082 \u2228 q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2228 q\u2081\u271d \u2208 stmts\u2081 q\u2082\n\u22a2 SupportsStmt S q\u2081\u271d", "state_after": "case inl\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2082\u271d : Stmt\u2082\nhs\u271d : SupportsStmt S q\u2082\u271d\nf : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nhs : SupportsStmt S q\u2081 \u2227 SupportsStmt S q\u2082\nh : branch f q\u2081 q\u2082 \u2208 stmts\u2081 q\u2082\u271d\nIH\u2081 : branch f q\u2081 q\u2082 \u2208 stmts\u2081 q\u2081 \u2192 SupportsStmt S q\u2081 \u2192 SupportsStmt S (branch f q\u2081 q\u2082)\nIH\u2082 : branch f q\u2081 q\u2082 \u2208 stmts\u2081 q\u2082 \u2192 SupportsStmt S q\u2082 \u2192 SupportsStmt S (branch f q\u2081 q\u2082)\n\u22a2 SupportsStmt S (branch f q\u2081 q\u2082)\n\ncase inr.inl\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nh\u271d : q\u2081\u271d \u2208 stmts\u2081 q\u2082\u271d\nhs\u271d : SupportsStmt S q\u2082\u271d\nf : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 : q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2192 SupportsStmt S q\u2081 \u2192 SupportsStmt S q\u2081\u271d\nIH\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082 \u2192 SupportsStmt S q\u2082 \u2192 SupportsStmt S q\u2081\u271d\nhs : SupportsStmt S q\u2081 \u2227 SupportsStmt S q\u2082\nh : q\u2081\u271d \u2208 stmts\u2081 q\u2081\n\u22a2 SupportsStmt S q\u2081\u271d\n\ncase inr.inr\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nh\u271d : q\u2081\u271d \u2208 stmts\u2081 q\u2082\u271d\nhs\u271d : SupportsStmt S q\u2082\u271d\nf : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 : q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2192 SupportsStmt S q\u2081 \u2192 SupportsStmt S q\u2081\u271d\nIH\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082 \u2192 SupportsStmt S q\u2082 \u2192 SupportsStmt S q\u2081\u271d\nhs : SupportsStmt S q\u2081 \u2227 SupportsStmt S q\u2082\nh : q\u2081\u271d \u2208 stmts\u2081 q\u2082\n\u22a2 SupportsStmt S q\u2081\u271d"}, {"tactic": "exacts [hs, IH\u2081 h hs.1, IH\u2082 h hs.2]", "state_before": "case inl\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2082\u271d : Stmt\u2082\nhs\u271d : SupportsStmt S q\u2082\u271d\nf : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nhs : SupportsStmt S q\u2081 \u2227 SupportsStmt S q\u2082\nh : branch f q\u2081 q\u2082 \u2208 stmts\u2081 q\u2082\u271d\nIH\u2081 : branch f q\u2081 q\u2082 \u2208 stmts\u2081 q\u2081 \u2192 SupportsStmt S q\u2081 \u2192 SupportsStmt S (branch f q\u2081 q\u2082)\nIH\u2082 : branch f q\u2081 q\u2082 \u2208 stmts\u2081 q\u2082 \u2192 SupportsStmt S q\u2082 \u2192 SupportsStmt S (branch f q\u2081 q\u2082)\n\u22a2 SupportsStmt S (branch f q\u2081 q\u2082)\n\ncase inr.inl\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nh\u271d : q\u2081\u271d \u2208 stmts\u2081 q\u2082\u271d\nhs\u271d : SupportsStmt S q\u2082\u271d\nf : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 : q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2192 SupportsStmt S q\u2081 \u2192 SupportsStmt S q\u2081\u271d\nIH\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082 \u2192 SupportsStmt S q\u2082 \u2192 SupportsStmt S q\u2081\u271d\nhs : SupportsStmt S q\u2081 \u2227 SupportsStmt S q\u2082\nh : q\u2081\u271d \u2208 stmts\u2081 q\u2081\n\u22a2 SupportsStmt S q\u2081\u271d\n\ncase inr.inr\nK : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081\u271d q\u2082\u271d : Stmt\u2082\nh\u271d : q\u2081\u271d \u2208 stmts\u2081 q\u2082\u271d\nhs\u271d : SupportsStmt S q\u2082\u271d\nf : \u03c3 \u2192 Bool\nq\u2081 q\u2082 : Stmt\u2082\nIH\u2081 : q\u2081\u271d \u2208 stmts\u2081 q\u2081 \u2192 SupportsStmt S q\u2081 \u2192 SupportsStmt S q\u2081\u271d\nIH\u2082 : q\u2081\u271d \u2208 stmts\u2081 q\u2082 \u2192 SupportsStmt S q\u2082 \u2192 SupportsStmt S q\u2081\u271d\nhs : SupportsStmt S q\u2081 \u2227 SupportsStmt S q\u2082\nh : q\u2081\u271d \u2208 stmts\u2081 q\u2082\n\u22a2 SupportsStmt S q\u2081\u271d", "state_after": "no goals"}, {"tactic": "subst h", "state_before": "K : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nl : \u03c3 \u2192 \u039b\nhs : \u2200 (v : \u03c3), l v \u2208 S\nh : q\u2081 = goto l\n\u22a2 SupportsStmt S q\u2081", "state_after": "K : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2082 : Stmt\u2082\nhs\u271d : SupportsStmt S q\u2082\nl : \u03c3 \u2192 \u039b\nhs : \u2200 (v : \u03c3), l v \u2208 S\nh : goto l \u2208 stmts\u2081 q\u2082\n\u22a2 SupportsStmt S (goto l)"}, {"tactic": "exact hs", "state_before": "K : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2082 : Stmt\u2082\nhs\u271d : SupportsStmt S q\u2082\nl : \u03c3 \u2192 \u039b\nhs : \u2200 (v : \u03c3), l v \u2208 S\nh : goto l \u2208 stmts\u2081 q\u2082\n\u22a2 SupportsStmt S (goto l)", "state_after": "no goals"}, {"tactic": "subst h", "state_before": "K : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2081 q\u2082 : Stmt\u2082\nh\u271d : q\u2081 \u2208 stmts\u2081 q\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : q\u2081 = halt\n\u22a2 SupportsStmt S q\u2081", "state_after": "K : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2082 : Stmt\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : halt \u2208 stmts\u2081 q\u2082\n\u22a2 SupportsStmt S halt"}, {"tactic": "trivial", "state_before": "K : Type u_2\ninst\u271d : DecidableEq K\n\u0393 : K \u2192 Type u_3\n\u039b : Type u_1\n\u03c3 : Type u_4\nS : Finset \u039b\nq\u2082 : Stmt\u2082\nhs\u271d : SupportsStmt S q\u2082\nhs : True\nh : halt \u2208 stmts\u2081 q\u2082\n\u22a2 SupportsStmt S halt", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GradedMonoid.lean", "full_name": "List.dProdIndex_nil", "start": [366, 1], "end": [367, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/RootsOfUnity/Basic.lean", "full_name": "IsPrimitiveRoot.neg_one", "start": [657, 1], "end": [661, 30], "traced_tactics": [{"tactic": "convert IsPrimitiveRoot.orderOf (-1 : R)", "state_before": "M : Type ?u.2773733\nN : Type ?u.2773736\nG : Type ?u.2773739\nR : Type u_1\nS : Type ?u.2773745\nF : Type ?u.2773748\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\np : \u2115\ninst\u271d : Nontrivial R\nh : CharP R p\nhp : p \u2260 2\n\u22a2 IsPrimitiveRoot (-1) 2", "state_after": "case h.e'_4\nM : Type ?u.2773733\nN : Type ?u.2773736\nG : Type ?u.2773739\nR : Type u_1\nS : Type ?u.2773745\nF : Type ?u.2773748\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\np : \u2115\ninst\u271d : Nontrivial R\nh : CharP R p\nhp : p \u2260 2\n\u22a2 2 = orderOf (-1)"}, {"tactic": "rw [orderOf_neg_one, if_neg]", "state_before": "case h.e'_4\nM : Type ?u.2773733\nN : Type ?u.2773736\nG : Type ?u.2773739\nR : Type u_1\nS : Type ?u.2773745\nF : Type ?u.2773748\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\np : \u2115\ninst\u271d : Nontrivial R\nh : CharP R p\nhp : p \u2260 2\n\u22a2 2 = orderOf (-1)", "state_after": "case h.e'_4.hnc\nM : Type ?u.2773733\nN : Type ?u.2773736\nG : Type ?u.2773739\nR : Type u_1\nS : Type ?u.2773745\nF : Type ?u.2773748\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\np : \u2115\ninst\u271d : Nontrivial R\nh : CharP R p\nhp : p \u2260 2\n\u22a2 \u00acringChar R = 2"}, {"tactic": "rwa [ringChar.eq_iff.mpr h]", "state_before": "case h.e'_4.hnc\nM : Type ?u.2773733\nN : Type ?u.2773736\nG : Type ?u.2773739\nR : Type u_1\nS : Type ?u.2773745\nF : Type ?u.2773748\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\np : \u2115\ninst\u271d : Nontrivial R\nh : CharP R p\nhp : p \u2260 2\n\u22a2 \u00acringChar R = 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/Padics/PadicNorm.lean", "full_name": "padicNorm.nat_lt_one_iff", "start": [311, 1], "end": [312, 59], "traced_tactics": [{"tactic": "rw [\u2190 Int.coe_nat_dvd, \u2190 int_lt_one_iff, Int.cast_ofNat]", "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2115\n\u22a2 padicNorm p \u2191m < 1 \u2194 p \u2223 m", "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_is_bilimit'", "start": [618, 1], "end": [621, 7], "traced_tactics": [{"tactic": "refine' IsPullback.of_right _ (by simp) (IsPullback.inl_snd' h).flip", "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 IsPullback 0 0 b.inl b.inr", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 IsPullback (0 \u226b 0) 0 0 (b.inr \u226b b.snd)"}, {"tactic": "simp", "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 IsPullback (0 \u226b 0) 0 0 (b.inr \u226b b.snd)", "state_after": "no goals"}, {"tactic": "simp", "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category C\nP X Y Z : C\nfst : P \u27f6 X\nsnd : P \u27f6 Y\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d\u00b9 : HasZeroObject C\ninst\u271d : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n\u22a2 0 \u226b b.inl = 0 \u226b b.inr", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/ENNReal.lean", "full_name": "ENNReal.exists_inv_nat_lt", "start": [1818, 1], "end": [1819, 88], "traced_tactics": [{"tactic": "simp only [ENNReal.inv_lt_inv, ENNReal.exists_nat_gt (inv_ne_top.2 h)]", "state_before": "\u03b1 : Type ?u.339403\n\u03b2 : Type ?u.339406\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\nh : a \u2260 0\n\u22a2 \u2203 n, (\u2191n)\u207b\u00b9 < a\u207b\u00b9\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Rat/Lemmas.lean", "full_name": "Rat.mul_comm", "start": [257, 11], "end": [258, 65], "traced_tactics": [{"tactic": "simp [mul_def, normalize_eq_mkRat, Int.mul_comm, Nat.mul_comm]", "state_before": "a b : Rat\n\u22a2 a * b = b * a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/BilinearForm.lean", "full_name": "BilinForm.comp_id_left", "start": [607, 1], "end": [610, 6], "traced_tactics": [{"tactic": "ext", "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type ?u.507306\nM\u2081 : Type ?u.507309\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.507918\nM\u2082 : Type ?u.507921\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.508108\nM\u2083 : Type ?u.508111\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type ?u.508699\nK : Type ?u.508702\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB : BilinForm R M\nr : M \u2192\u2097[R] M\n\u22a2 comp B LinearMap.id r = compRight B r", "state_after": "case H\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type ?u.507306\nM\u2081 : Type ?u.507309\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.507918\nM\u2082 : Type ?u.507921\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.508108\nM\u2083 : Type ?u.508111\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type ?u.508699\nK : Type ?u.508702\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB : BilinForm R M\nr : M \u2192\u2097[R] M\nx\u271d y\u271d : M\n\u22a2 bilin (comp B LinearMap.id r) x\u271d y\u271d = bilin (compRight B r) x\u271d y\u271d"}, {"tactic": "rfl", "state_before": "case H\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type ?u.507306\nM\u2081 : Type ?u.507309\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.507918\nM\u2082 : Type ?u.507921\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.508108\nM\u2083 : Type ?u.508111\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type ?u.508699\nK : Type ?u.508702\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB : BilinForm R M\nr : M \u2192\u2097[R] M\nx\u271d y\u271d : M\n\u22a2 bilin (comp B LinearMap.id r) x\u271d y\u271d = bilin (compRight B r) x\u271d y\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/EuclideanDomain/Basic.lean", "full_name": "EuclideanDomain.eq_div_of_mul_eq_left", "start": [94, 1], "end": [95, 32], "traced_tactics": [{"tactic": "rw [\u2190 h, mul_div_cancel _ hb]", "state_before": "R : Type u\ninst\u271d : EuclideanDomain R\na b c : R\nhb : b \u2260 0\nh : a * b = c\n\u22a2 a = c / b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Rat/Cast.lean", "full_name": "Rat.cast_coe_nat", "start": [54, 1], "end": [55, 54], "traced_tactics": [{"tactic": "rw [\u2190 Int.cast_ofNat, cast_coe_int, Int.cast_ofNat]", "state_before": "F : Type ?u.1219\n\u03b9 : Type ?u.1222\n\u03b1 : Type u_1\n\u03b2 : Type ?u.1228\ninst\u271d : DivisionRing \u03b1\nn : \u2115\n\u22a2 \u2191\u2191n = \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/RestrictScalars.lean", "full_name": "RestrictScalars.addEquiv_map_smul", "start": [167, 1], "end": [169, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Matrix/Basic.lean", "full_name": "Matrix.conjTranspose_multiset_sum", "start": [2280, 1], "end": [2282, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.disjoint_pure_atBot", "start": [104, 1], "end": [105, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.mul_succ", "start": [721, 1], "end": [722, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Between.lean", "full_name": "wbtw_self_right", "start": [296, 1], "end": [297, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "full_name": "cauchySeq_tendsto_of_complete", "start": [424, 1], "end": [426, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.integral_congr", "start": [881, 1], "end": [885, 72], "traced_tactics": [{"tactic": "cases' le_total a b with hab hab <;>\n  simpa [hab, integral_of_le, integral_of_ge] using\n    set_integral_congr measurableSet_Ioc (h.mono Ioc_subset_Icc_self)", "state_before": "\u03b9 : Type ?u.15077331\n\ud835\udd5c : Type ?u.15077334\nE : Type u_1\nF : Type ?u.15077340\nA : Type ?u.15077343\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\na\u271d b\u271d c d : \u211d\nf g : \u211d \u2192 E\n\u03bc : MeasureTheory.Measure \u211d\na b : \u211d\nh : EqOn f g [[a, b]]\n\u22a2 (\u222b (x : \u211d) in a..b, f x \u2202\u03bc) = \u222b (x : \u211d) in a..b, g x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Analysis/Topology.lean", "full_name": "Ctop.mem_nhds_toTopsp", "start": [101, 1], "end": [105, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Bases.lean", "full_name": "TopologicalSpace.Quotient.secondCountableTopology", "start": [873, 1], "end": [875, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.ppred_to_nat", "start": [846, 1], "end": [851, 8], "traced_tactics": [{"tactic": "rw [ppred, Option.map_some, Nat.ppred_eq_some.2]", "state_before": "\u03b1 : Type ?u.465274\np : PosNum\n\u22a2 castNum <$> ppred (pos p) = Nat.ppred \u2191(pos p)", "state_after": "\u03b1 : Type ?u.465274\np : PosNum\n\u22a2 Nat.succ \u2191(pred' p) = \u2191(pos p)"}, {"tactic": "rw [PosNum.pred'_to_nat, Nat.succ_pred_eq_of_pos (PosNum.to_nat_pos _)]", "state_before": "\u03b1 : Type ?u.465274\np : PosNum\n\u22a2 Nat.succ \u2191(pred' p) = \u2191(pos p)", "state_after": "\u03b1 : Type ?u.465274\np : PosNum\n\u22a2 \u2191p = \u2191(pos p)"}, {"tactic": "rfl", "state_before": "\u03b1 : Type ?u.465274\np : PosNum\n\u22a2 \u2191p = \u2191(pos p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/InfiniteSum/Ring.lean", "full_name": "hasSum_div_const_iff", "start": [100, 1], "end": [101, 73], "traced_tactics": [{"tactic": "simpa only [div_eq_mul_inv] using hasSum_mul_right_iff (inv_ne_zero h)", "state_before": "\u03b9 : Type u_2\n\u03ba : Type ?u.11565\nR : Type ?u.11568\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DivisionSemiring \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nf g : \u03b9 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : a\u2082 \u2260 0\n\u22a2 HasSum (fun i => f i / a\u2082) (a\u2081 / a\u2082) \u2194 HasSum f a\u2081", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Field/Basic.lean", "full_name": "div_le_of_nonneg_of_le_mul", "start": [231, 1], "end": [234, 23], "traced_tactics": [{"tactic": "rcases eq_or_lt_of_le hb with (rfl | hb')", "state_before": "\u03b9 : Type ?u.42038\n\u03b1 : Type u_1\n\u03b2 : Type ?u.42044\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhb : 0 \u2264 b\nhc : 0 \u2264 c\nh : a \u2264 c * b\n\u22a2 a / b \u2264 c", "state_after": "case inl\n\u03b9 : Type ?u.42038\n\u03b1 : Type u_1\n\u03b2 : Type ?u.42044\ninst\u271d : LinearOrderedSemifield \u03b1\na c d e : \u03b1\nm n : \u2124\nhc : 0 \u2264 c\nhb : 0 \u2264 0\nh : a \u2264 c * 0\n\u22a2 a / 0 \u2264 c\n\ncase inr\n\u03b9 : Type ?u.42038\n\u03b1 : Type u_1\n\u03b2 : Type ?u.42044\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhb : 0 \u2264 b\nhc : 0 \u2264 c\nh : a \u2264 c * b\nhb' : 0 < b\n\u22a2 a / b \u2264 c"}, {"tactic": "simp [hc]", "state_before": "case inl\n\u03b9 : Type ?u.42038\n\u03b1 : Type u_1\n\u03b2 : Type ?u.42044\ninst\u271d : LinearOrderedSemifield \u03b1\na c d e : \u03b1\nm n : \u2124\nhc : 0 \u2264 c\nhb : 0 \u2264 0\nh : a \u2264 c * 0\n\u22a2 a / 0 \u2264 c\n\ncase inr\n\u03b9 : Type ?u.42038\n\u03b1 : Type u_1\n\u03b2 : Type ?u.42044\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhb : 0 \u2264 b\nhc : 0 \u2264 c\nh : a \u2264 c * b\nhb' : 0 < b\n\u22a2 a / b \u2264 c", "state_after": "case inr\n\u03b9 : Type ?u.42038\n\u03b1 : Type u_1\n\u03b2 : Type ?u.42044\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhb : 0 \u2264 b\nhc : 0 \u2264 c\nh : a \u2264 c * b\nhb' : 0 < b\n\u22a2 a / b \u2264 c"}, {"tactic": "rwa [div_le_iff hb']", "state_before": "case inr\n\u03b9 : Type ?u.42038\n\u03b1 : Type u_1\n\u03b2 : Type ?u.42044\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nhb : 0 \u2264 b\nhc : 0 \u2264 c\nh : a \u2264 c * b\nhb' : 0 < b\n\u22a2 a / b \u2264 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/FinitePresentation.lean", "full_name": "RingHom.FinitePresentation.of_comp_finiteType", "start": [462, 1], "end": [470, 83], "traced_tactics": [{"tactic": "simp [Algebra.smul_def, mul_assoc]", "state_before": "A : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : CommRing B\ninst\u271d : CommRing C\nf : A \u2192+* B\ng : B \u2192+* C\nhg : FinitePresentation (RingHom.comp g f)\nhf : FiniteType f\nins1 : Algebra A B := toAlgebra f\nins2 : Algebra B C := toAlgebra g\nins3 : Algebra A C := toAlgebra (RingHom.comp g f)\na : A\nb : B\nc : C\n\u22a2 (a \u2022 b) \u2022 c = a \u2022 b \u2022 c", "state_after": "A : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : CommRing B\ninst\u271d : CommRing C\nf : A \u2192+* B\ng : B \u2192+* C\nhg : FinitePresentation (RingHom.comp g f)\nhf : FiniteType f\nins1 : Algebra A B := toAlgebra f\nins2 : Algebra B C := toAlgebra g\nins3 : Algebra A C := toAlgebra (RingHom.comp g f)\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_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : CommRing B\ninst\u271d : CommRing C\nf : A \u2192+* B\ng : B \u2192+* C\nhg : FinitePresentation (RingHom.comp g f)\nhf : FiniteType f\nins1 : Algebra A B := toAlgebra f\nins2 : Algebra B C := toAlgebra g\nins3 : Algebra A C := toAlgebra (RingHom.comp g f)\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/Topology/Algebra/Module/FiniteDimension.lean", "full_name": "LinearMap.isOpenMap_of_finiteDimensional", "start": [334, 1], "end": [343, 81], "traced_tactics": [{"tactic": "rcases f.exists_rightInverse_of_surjective (LinearMap.range_eq_top.2 hf) with \u27e8g, hg\u27e9", "state_before": "\ud835\udd5c : Type u\nhnorm : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2077 : AddCommGroup E\ninst\u271d\u00b9\u2076 : Module \ud835\udd5c E\ninst\u271d\u00b9\u2075 : TopologicalSpace E\ninst\u271d\u00b9\u2074 : TopologicalAddGroup E\ninst\u271d\u00b9\u00b3 : ContinuousSMul \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b2 : AddCommGroup F\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c F\ninst\u271d\u00b9\u2070 : TopologicalSpace F\ninst\u271d\u2079 : TopologicalAddGroup F\ninst\u271d\u2078 : ContinuousSMul \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : T2Space E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : F \u2192\u2097[\ud835\udd5c] E\nhf : Function.Surjective \u2191f\n\u22a2 IsOpenMap \u2191f", "state_after": "case intro\n\ud835\udd5c : Type u\nhnorm : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2077 : AddCommGroup E\ninst\u271d\u00b9\u2076 : Module \ud835\udd5c E\ninst\u271d\u00b9\u2075 : TopologicalSpace E\ninst\u271d\u00b9\u2074 : TopologicalAddGroup E\ninst\u271d\u00b9\u00b3 : ContinuousSMul \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b2 : AddCommGroup F\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c F\ninst\u271d\u00b9\u2070 : TopologicalSpace F\ninst\u271d\u2079 : TopologicalAddGroup F\ninst\u271d\u2078 : ContinuousSMul \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : T2Space E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : F \u2192\u2097[\ud835\udd5c] E\nhf : Function.Surjective \u2191f\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : comp f g = id\n\u22a2 IsOpenMap \u2191f"}, {"tactic": "refine' IsOpenMap.of_sections fun x => \u27e8fun y => g (y - f x) + x, _, _, fun y => _\u27e9", "state_before": "case intro\n\ud835\udd5c : Type u\nhnorm : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2077 : AddCommGroup E\ninst\u271d\u00b9\u2076 : Module \ud835\udd5c E\ninst\u271d\u00b9\u2075 : TopologicalSpace E\ninst\u271d\u00b9\u2074 : TopologicalAddGroup E\ninst\u271d\u00b9\u00b3 : ContinuousSMul \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b2 : AddCommGroup F\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c F\ninst\u271d\u00b9\u2070 : TopologicalSpace F\ninst\u271d\u2079 : TopologicalAddGroup F\ninst\u271d\u2078 : ContinuousSMul \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : T2Space E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : F \u2192\u2097[\ud835\udd5c] E\nhf : Function.Surjective \u2191f\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : comp f g = id\n\u22a2 IsOpenMap \u2191f", "state_after": "case intro.refine'_1\n\ud835\udd5c : Type u\nhnorm : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2077 : AddCommGroup E\ninst\u271d\u00b9\u2076 : Module \ud835\udd5c E\ninst\u271d\u00b9\u2075 : TopologicalSpace E\ninst\u271d\u00b9\u2074 : TopologicalAddGroup E\ninst\u271d\u00b9\u00b3 : ContinuousSMul \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b2 : AddCommGroup F\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c F\ninst\u271d\u00b9\u2070 : TopologicalSpace F\ninst\u271d\u2079 : TopologicalAddGroup F\ninst\u271d\u2078 : ContinuousSMul \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : T2Space E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : F \u2192\u2097[\ud835\udd5c] E\nhf : Function.Surjective \u2191f\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : comp f g = id\nx : F\n\u22a2 ContinuousAt (fun y => \u2191g (y - \u2191f x) + x) (\u2191f x)\n\ncase intro.refine'_2\n\ud835\udd5c : Type u\nhnorm : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2077 : AddCommGroup E\ninst\u271d\u00b9\u2076 : Module \ud835\udd5c E\ninst\u271d\u00b9\u2075 : TopologicalSpace E\ninst\u271d\u00b9\u2074 : TopologicalAddGroup E\ninst\u271d\u00b9\u00b3 : ContinuousSMul \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b2 : AddCommGroup F\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c F\ninst\u271d\u00b9\u2070 : TopologicalSpace F\ninst\u271d\u2079 : TopologicalAddGroup F\ninst\u271d\u2078 : ContinuousSMul \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : T2Space E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : F \u2192\u2097[\ud835\udd5c] E\nhf : Function.Surjective \u2191f\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : comp f g = id\nx : F\n\u22a2 (fun y => \u2191g (y - \u2191f x) + x) (\u2191f x) = x\n\ncase intro.refine'_3\n\ud835\udd5c : Type u\nhnorm : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2077 : AddCommGroup E\ninst\u271d\u00b9\u2076 : Module \ud835\udd5c E\ninst\u271d\u00b9\u2075 : TopologicalSpace E\ninst\u271d\u00b9\u2074 : TopologicalAddGroup E\ninst\u271d\u00b9\u00b3 : ContinuousSMul \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b2 : AddCommGroup F\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c F\ninst\u271d\u00b9\u2070 : TopologicalSpace F\ninst\u271d\u2079 : TopologicalAddGroup F\ninst\u271d\u2078 : ContinuousSMul \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : T2Space E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : F \u2192\u2097[\ud835\udd5c] E\nhf : Function.Surjective \u2191f\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : comp f g = id\nx : F\ny : E\n\u22a2 \u2191f ((fun y => \u2191g (y - \u2191f x) + x) y) = y"}, {"tactic": "exact\n  ((g.continuous_of_finiteDimensional.comp <| continuous_id.sub continuous_const).add\n      continuous_const).continuousAt", "state_before": "case intro.refine'_1\n\ud835\udd5c : Type u\nhnorm : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2077 : AddCommGroup E\ninst\u271d\u00b9\u2076 : Module \ud835\udd5c E\ninst\u271d\u00b9\u2075 : TopologicalSpace E\ninst\u271d\u00b9\u2074 : TopologicalAddGroup E\ninst\u271d\u00b9\u00b3 : ContinuousSMul \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b2 : AddCommGroup F\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c F\ninst\u271d\u00b9\u2070 : TopologicalSpace F\ninst\u271d\u2079 : TopologicalAddGroup F\ninst\u271d\u2078 : ContinuousSMul \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : T2Space E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : F \u2192\u2097[\ud835\udd5c] E\nhf : Function.Surjective \u2191f\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : comp f g = id\nx : F\n\u22a2 ContinuousAt (fun y => \u2191g (y - \u2191f x) + x) (\u2191f x)", "state_after": "no goals"}, {"tactic": "simp only", "state_before": "case intro.refine'_2\n\ud835\udd5c : Type u\nhnorm : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2077 : AddCommGroup E\ninst\u271d\u00b9\u2076 : Module \ud835\udd5c E\ninst\u271d\u00b9\u2075 : TopologicalSpace E\ninst\u271d\u00b9\u2074 : TopologicalAddGroup E\ninst\u271d\u00b9\u00b3 : ContinuousSMul \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b2 : AddCommGroup F\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c F\ninst\u271d\u00b9\u2070 : TopologicalSpace F\ninst\u271d\u2079 : TopologicalAddGroup F\ninst\u271d\u2078 : ContinuousSMul \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : T2Space E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : F \u2192\u2097[\ud835\udd5c] E\nhf : Function.Surjective \u2191f\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : comp f g = id\nx : F\n\u22a2 (fun y => \u2191g (y - \u2191f x) + x) (\u2191f x) = x", "state_after": "case intro.refine'_2\n\ud835\udd5c : Type u\nhnorm : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2077 : AddCommGroup E\ninst\u271d\u00b9\u2076 : Module \ud835\udd5c E\ninst\u271d\u00b9\u2075 : TopologicalSpace E\ninst\u271d\u00b9\u2074 : TopologicalAddGroup E\ninst\u271d\u00b9\u00b3 : ContinuousSMul \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b2 : AddCommGroup F\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c F\ninst\u271d\u00b9\u2070 : TopologicalSpace F\ninst\u271d\u2079 : TopologicalAddGroup F\ninst\u271d\u2078 : ContinuousSMul \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : T2Space E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : F \u2192\u2097[\ud835\udd5c] E\nhf : Function.Surjective \u2191f\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : comp f g = id\nx : F\n\u22a2 \u2191g (\u2191f x - \u2191f x) + x = x"}, {"tactic": "rw [sub_self, map_zero, zero_add]", "state_before": "case intro.refine'_2\n\ud835\udd5c : Type u\nhnorm : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2077 : AddCommGroup E\ninst\u271d\u00b9\u2076 : Module \ud835\udd5c E\ninst\u271d\u00b9\u2075 : TopologicalSpace E\ninst\u271d\u00b9\u2074 : TopologicalAddGroup E\ninst\u271d\u00b9\u00b3 : ContinuousSMul \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b2 : AddCommGroup F\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c F\ninst\u271d\u00b9\u2070 : TopologicalSpace F\ninst\u271d\u2079 : TopologicalAddGroup F\ninst\u271d\u2078 : ContinuousSMul \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : T2Space E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : F \u2192\u2097[\ud835\udd5c] E\nhf : Function.Surjective \u2191f\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : comp f g = id\nx : F\n\u22a2 \u2191g (\u2191f x - \u2191f x) + x = x", "state_after": "no goals"}, {"tactic": "simp only [map_sub, map_add, \u2190 comp_apply f g, hg, id_apply, sub_add_cancel]", "state_before": "case intro.refine'_3\n\ud835\udd5c : Type u\nhnorm : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2077 : AddCommGroup E\ninst\u271d\u00b9\u2076 : Module \ud835\udd5c E\ninst\u271d\u00b9\u2075 : TopologicalSpace E\ninst\u271d\u00b9\u2074 : TopologicalAddGroup E\ninst\u271d\u00b9\u00b3 : ContinuousSMul \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b2 : AddCommGroup F\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c F\ninst\u271d\u00b9\u2070 : TopologicalSpace F\ninst\u271d\u2079 : TopologicalAddGroup F\ninst\u271d\u2078 : ContinuousSMul \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : T2Space E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : F \u2192\u2097[\ud835\udd5c] E\nhf : Function.Surjective \u2191f\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : comp f g = id\nx : F\ny : E\n\u22a2 \u2191f ((fun y => \u2191g (y - \u2191f x) + x) y) = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Bases.lean", "full_name": "Filter.HasBasis.inf_principal_neBot_iff", "start": [641, 1], "end": [643, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.pred_eq_iff_not_succ", "start": [185, 1], "end": [186, 92], "traced_tactics": [{"tactic": "rw [e', pred_succ] at e", "state_before": "\u03b1 : Type ?u.82121\n\u03b2 : Type ?u.82124\n\u03b3 : Type ?u.82127\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal\ne : pred o = o\nx\u271d : \u2203 a, o = succ a\na : Ordinal\ne' : o = succ a\n\u22a2 False", "state_after": "\u03b1 : Type ?u.82121\n\u03b2 : Type ?u.82124\n\u03b3 : Type ?u.82127\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal\nx\u271d : \u2203 a, o = succ a\na : Ordinal\ne : a = succ a\ne' : o = succ a\n\u22a2 False"}, {"tactic": "exact (lt_succ a).ne e", "state_before": "\u03b1 : Type ?u.82121\n\u03b2 : Type ?u.82124\n\u03b3 : Type ?u.82127\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal\nx\u271d : \u2203 a, o = succ a\na : Ordinal\ne : a = succ a\ne' : o = succ a\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "full_name": "SimpleGraph.set_walk_length_succ_eq", "start": [2311, 1], "end": [2323, 10], "traced_tactics": [{"tactic": "ext p", "state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\nn : \u2115\n\u22a2 {p | Walk.length p = Nat.succ n} = \u22c3 (w : V) (h : Adj G u w), Walk.cons h '' {p' | Walk.length p' = n}", "state_after": "case h\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\nn : \u2115\np : Walk G u v\n\u22a2 p \u2208 {p | Walk.length p = Nat.succ n} \u2194 p \u2208 \u22c3 (w : V) (h : Adj G u w), Walk.cons h '' {p' | Walk.length p' = n}"}, {"tactic": "cases' p with _ _ w _ huw pwv", "state_before": "case h\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\nn : \u2115\np : Walk G u v\n\u22a2 p \u2208 {p | Walk.length p = Nat.succ n} \u2194 p \u2208 \u22c3 (w : V) (h : Adj G u w), Walk.cons h '' {p' | Walk.length p' = n}", "state_after": "case h.nil\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu : V\nn : \u2115\n\u22a2 Walk.nil \u2208 {p | Walk.length p = Nat.succ n} \u2194\n    Walk.nil \u2208 \u22c3 (w : V) (h : Adj G u w), Walk.cons h '' {p' | Walk.length p' = n}\n\ncase h.cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\nn : \u2115\nw : V\nhuw : Adj G u w\npwv : Walk G w v\n\u22a2 Walk.cons huw pwv \u2208 {p | Walk.length p = Nat.succ n} \u2194\n    Walk.cons huw pwv \u2208 \u22c3 (w : V) (h : Adj G u w), Walk.cons h '' {p' | Walk.length p' = n}"}, {"tactic": "simp [eq_comm]", "state_before": "case h.nil\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu : V\nn : \u2115\n\u22a2 Walk.nil \u2208 {p | Walk.length p = Nat.succ n} \u2194\n    Walk.nil \u2208 \u22c3 (w : V) (h : Adj G u w), Walk.cons h '' {p' | Walk.length p' = n}", "state_after": "no goals"}, {"tactic": "simp only [Nat.succ_eq_add_one, Set.mem_setOf_eq, Walk.length_cons, add_left_inj,\n  Set.mem_iUnion, Set.mem_image, exists_prop]", "state_before": "case h.cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\nn : \u2115\nw : V\nhuw : Adj G u w\npwv : Walk G w v\n\u22a2 Walk.cons huw pwv \u2208 {p | Walk.length p = Nat.succ n} \u2194\n    Walk.cons huw pwv \u2208 \u22c3 (w : V) (h : Adj G u w), Walk.cons h '' {p' | Walk.length p' = n}", "state_after": "case h.cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\nn : \u2115\nw : V\nhuw : Adj G u w\npwv : Walk G w v\n\u22a2 Walk.length pwv = n \u2194 \u2203 i h x, Walk.length x = n \u2227 Walk.cons (_ : Adj G u i) x = Walk.cons huw pwv"}, {"tactic": "constructor", "state_before": "case h.cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\nn : \u2115\nw : V\nhuw : Adj G u w\npwv : Walk G w v\n\u22a2 Walk.length pwv = n \u2194 \u2203 i h x, Walk.length x = n \u2227 Walk.cons (_ : Adj G u i) x = Walk.cons huw pwv", "state_after": "case h.cons.mp\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\nn : \u2115\nw : V\nhuw : Adj G u w\npwv : Walk G w v\n\u22a2 Walk.length pwv = n \u2192 \u2203 i h x, Walk.length x = n \u2227 Walk.cons (_ : Adj G u i) x = Walk.cons huw pwv\n\ncase h.cons.mpr\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\nn : \u2115\nw : V\nhuw : Adj G u w\npwv : Walk G w v\n\u22a2 (\u2203 i h x, Walk.length x = n \u2227 Walk.cons (_ : Adj G u i) x = Walk.cons huw pwv) \u2192 Walk.length pwv = n"}, {"tactic": "rintro rfl", "state_before": "case h.cons.mp\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\nn : \u2115\nw : V\nhuw : Adj G u w\npwv : Walk G w v\n\u22a2 Walk.length pwv = n \u2192 \u2203 i h x, Walk.length x = n \u2227 Walk.cons (_ : Adj G u i) x = Walk.cons huw pwv", "state_after": "case h.cons.mp\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w : V\nhuw : Adj G u w\npwv : Walk G w v\n\u22a2 \u2203 i h x, Walk.length x = Walk.length pwv \u2227 Walk.cons (_ : Adj G u i) x = Walk.cons huw pwv"}, {"tactic": "exact \u27e8w, huw, pwv, rfl, rfl\u27e9", "state_before": "case h.cons.mp\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w : V\nhuw : Adj G u w\npwv : Walk G w v\n\u22a2 \u2203 i h x, Walk.length x = Walk.length pwv \u2227 Walk.cons (_ : Adj G u i) x = Walk.cons huw pwv", "state_after": "no goals"}, {"tactic": "rintro \u27e8w, huw, pwv, rfl, rfl, rfl\u27e9", "state_before": "case h.cons.mpr\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v : V\nn : \u2115\nw : V\nhuw : Adj G u w\npwv : Walk G w v\n\u22a2 (\u2203 i h x, Walk.length x = n \u2227 Walk.cons (_ : Adj G u i) x = Walk.cons huw pwv) \u2192 Walk.length pwv = n", "state_after": "case h.cons.mpr.intro.intro.intro.intro.refl\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w : V\nhuw\u271d : Adj G u w\npwv : Walk G w v\nhuw : Adj G u w\n\u22a2 Walk.length pwv = Walk.length pwv"}, {"tactic": "rfl", "state_before": "case h.cons.mpr.intro.intro.intro.intro.refl\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w : V\nhuw\u271d : Adj G u w\npwv : Walk G w v\nhuw : Adj G u w\n\u22a2 Walk.length pwv = Walk.length pwv", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.foldr_singleton", "start": [1394, 1], "end": [1395, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/MeasurableSpace.lean", "full_name": "MeasurableSet.pi", "start": [876, 11], "end": [879, 77], "traced_tactics": [{"tactic": "rw [pi_def]", "state_before": "\u03b1 : Type ?u.104676\n\u03b2 : Type ?u.104679\n\u03b3 : Type ?u.104682\n\u03b4 : Type u_1\n\u03b4' : Type ?u.104688\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\nht : \u2200 (i : \u03b4), i \u2208 s \u2192 MeasurableSet (t i)\n\u22a2 MeasurableSet (Set.pi s t)", "state_after": "\u03b1 : Type ?u.104676\n\u03b2 : Type ?u.104679\n\u03b3 : Type ?u.104682\n\u03b4 : Type u_1\n\u03b4' : Type ?u.104688\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\nht : \u2200 (i : \u03b4), i \u2208 s \u2192 MeasurableSet (t i)\n\u22a2 MeasurableSet (\u22c2 (a : \u03b4) (_ : a \u2208 s), eval a \u207b\u00b9' t a)"}, {"tactic": "exact MeasurableSet.biInter hs fun i hi => measurable_pi_apply _ (ht i hi)", "state_before": "\u03b1 : Type ?u.104676\n\u03b2 : Type ?u.104679\n\u03b3 : Type ?u.104682\n\u03b4 : Type u_1\n\u03b4' : Type ?u.104688\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\nht : \u2200 (i : \u03b4), i \u2208 s \u2192 MeasurableSet (t i)\n\u22a2 MeasurableSet (\u22c2 (a : \u03b4) (_ : a \u2208 s), eval a \u207b\u00b9' t a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/Layercake.lean", "full_name": "lintegral_eq_lintegral_meas_lt", "start": [345, 1], "end": [351, 10], "traced_tactics": [{"tactic": "rw [lintegral_eq_lintegral_meas_le \u03bc f_nn f_mble]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc\u271d : MeasureTheory.Measure \u03b1\n\u03b2 : Type ?u.1235354\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\n\u22a2 (\u222b\u207b (\u03c9 : \u03b1), ENNReal.ofReal (f \u03c9) \u2202\u03bc) = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc\u271d : MeasureTheory.Measure \u03b1\n\u03b2 : Type ?u.1235354\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\n\u22a2 (\u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a}) = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}"}, {"tactic": "apply lintegral_congr_ae", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc\u271d : MeasureTheory.Measure \u03b1\n\u03b2 : Type ?u.1235354\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\n\u22a2 (\u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t \u2264 f a}) = \u222b\u207b (t : \u211d) in Ioi 0, \u2191\u2191\u03bc {a | t < f a}", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc\u271d : MeasureTheory.Measure \u03b1\n\u03b2 : Type ?u.1235354\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\n\u22a2 (fun a => \u2191\u2191\u03bc {a_1 | a \u2264 f a_1}) =\u1d50[Measure.restrict volume (Ioi 0)] fun a => \u2191\u2191\u03bc {a_1 | a < f a_1}"}, {"tactic": "filter_upwards [Measure.meas_le_ae_eq_meas_lt \u03bc (volume.restrict (Ioi 0)) f_mble] with t ht", "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc\u271d : MeasureTheory.Measure \u03b1\n\u03b2 : Type ?u.1235354\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\n\u22a2 (fun a => \u2191\u2191\u03bc {a_1 | a \u2264 f a_1}) =\u1d50[Measure.restrict volume (Ioi 0)] fun a => \u2191\u2191\u03bc {a_1 | a < f a_1}", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc\u271d : MeasureTheory.Measure \u03b1\n\u03b2 : Type ?u.1235354\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\nt : \u211d\nht : \u2191\u2191\u03bc {a | t \u2264 f a} = \u2191\u2191\u03bc {a | t < f a}\n\u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} = \u2191\u2191\u03bc {a | t < f a}"}, {"tactic": "rw [ht]", "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc\u271d : MeasureTheory.Measure \u03b1\n\u03b2 : Type ?u.1235354\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSingletonClass \u03b2\nf : \u03b1 \u2192 \u211d\ng : \u211d \u2192 \u211d\ns : Set \u03b1\n\u03bc : MeasureTheory.Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nf_nn : 0 \u2264 f\nf_mble : Measurable f\nt : \u211d\nht : \u2191\u2191\u03bc {a | t \u2264 f a} = \u2191\u2191\u03bc {a | t < f a}\n\u22a2 \u2191\u2191\u03bc {a | t \u2264 f a} = \u2191\u2191\u03bc {a | t < f a}", "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.family_of_elements_compatible", "start": [170, 1], "end": [175, 97], "traced_tactics": [{"tactic": "intro Y\u2081 Y\u2082 Z g\u2081 g\u2082 f\u2081 f\u2082 h\u2081 h\u2082 e", "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\nU : C\u1d52\u1d56\ns : F.obj U\n\u22a2 Presieve.FamilyOfElements.Compatible (familyOfElementsOfSection G s)", "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\nU : C\u1d52\u1d56\ns : F.obj U\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 U.unop\nf\u2082 : Y\u2082 \u27f6 U.unop\nh\u2081 : (sieveOfSection G s).arrows f\u2081\nh\u2082 : (sieveOfSection G s).arrows f\u2082\ne : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\n\u22a2 (toPresheaf G).map g\u2081.op (familyOfElementsOfSection G s f\u2081 h\u2081) =\n    (toPresheaf G).map g\u2082.op (familyOfElementsOfSection G s f\u2082 h\u2082)"}, {"tactic": "refine Subtype.ext ?_", "state_before": "C : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type 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\u2191((toPresheaf G).map g\u2081.op (familyOfElementsOfSection G s f\u2081 h\u2081)) =\n    \u2191((toPresheaf G).map g\u2082.op (familyOfElementsOfSection G s f\u2082 h\u2082))"}, {"tactic": "change F.map g\u2081.op (F.map f\u2081.op s) = F.map g\u2082.op (F.map f\u2082.op s)", "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\nU : C\u1d52\u1d56\ns : F.obj U\nY\u2081 Y\u2082 Z : C\ng\u2081 : Z \u27f6 Y\u2081\ng\u2082 : Z \u27f6 Y\u2082\nf\u2081 : Y\u2081 \u27f6 U.unop\nf\u2082 : Y\u2082 \u27f6 U.unop\nh\u2081 : (sieveOfSection G s).arrows f\u2081\nh\u2082 : (sieveOfSection G s).arrows f\u2082\ne : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082\n\u22a2 \u2191((toPresheaf G).map g\u2081.op (familyOfElementsOfSection G s f\u2081 h\u2081)) =\n    \u2191((toPresheaf G).map g\u2082.op (familyOfElementsOfSection G s f\u2082 h\u2082))", "state_after": "C : Type u\ninst\u271d : Category C\nJ : 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l\u2081\n\u22a2 (Prod.swap (unzip (zip l\u2082 l\u2081))).snd = l\u2082"}, {"tactic": "exact unzip_zip_left h", "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type ?u.86992\n\u03b4 : Type ?u.86995\n\u03b5 : Type ?u.86998\nl\u2081 : List \u03b1\nl\u2082 : List \u03b2\nh : length l\u2082 \u2264 length l\u2081\n\u22a2 (Prod.swap (unzip (zip l\u2082 l\u2081))).snd = l\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "full_name": "mul_neg_iff", "start": [1057, 1], "end": [1058, 72], "traced_tactics": [{"tactic": "rw [\u2190 neg_pos, neg_mul_eq_mul_neg, mul_pos_iff, neg_pos, neg_lt_zero]", "state_before": "\u03b1 : Type u\n\u03b2 : Type ?u.174326\ninst\u271d : LinearOrderedRing \u03b1\na b c : \u03b1\n\u22a2 a * b < 0 \u2194 0 < a \u2227 b < 0 \u2228 a < 0 \u2227 0 < b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "full_name": "MeasureTheory.AEEqFun.coeFn_pair", "start": [286, 1], "end": [288, 17], "traced_tactics": [{"tactic": "rw [pair_eq_mk]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type ?u.393568\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf : \u03b1 \u2192\u2098[\u03bc] \u03b2\ng : \u03b1 \u2192\u2098[\u03bc] \u03b3\n\u22a2 \u2191(pair f g) =\u1d50[\u03bc] fun x => (\u2191f x, \u2191g x)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type ?u.393568\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf : \u03b1 \u2192\u2098[\u03bc] \u03b2\ng : \u03b1 \u2192\u2098[\u03bc] \u03b3\n\u22a2 \u2191(mk (fun x => (\u2191f x, \u2191g x)) (_ : AEStronglyMeasurable (fun x => (\u2191f x, \u2191g x)) \u03bc)) =\u1d50[\u03bc] fun x => (\u2191f x, \u2191g x)"}, {"tactic": "apply coeFn_mk", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type ?u.393568\ninst\u271d\u00b3 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf : \u03b1 \u2192\u2098[\u03bc] \u03b2\ng : \u03b1 \u2192\u2098[\u03bc] \u03b3\n\u22a2 \u2191(mk (fun x => (\u2191f x, \u2191g x)) (_ : AEStronglyMeasurable (fun x => (\u2191f x, \u2191g x)) \u03bc)) =\u1d50[\u03bc] fun x => (\u2191f x, \u2191g x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "full_name": "MeasureTheory.Measure.integrable_measure_prod_mk_left", "start": [159, 1], "end": [168, 58], "traced_tactics": [{"tactic": "refine' \u27e8(measurable_measure_prod_mk_left hs).ennreal_toReal.aemeasurable.aestronglyMeasurable, _\u27e9", "state_before": "\u03b1 : Type u_2\n\u03b1' : Type ?u.2382010\n\u03b2 : Type u_1\n\u03b2' : Type ?u.2382016\n\u03b3 : Type ?u.2382019\nE : Type ?u.2382022\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\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 Integrable fun x => ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s))", "state_after": "\u03b1 : Type u_2\n\u03b1' : Type ?u.2382010\n\u03b2 : Type u_1\n\u03b2' : Type ?u.2382016\n\u03b3 : Type ?u.2382019\nE : Type ?u.2382022\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\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 HasFiniteIntegral fun x => ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s))"}, {"tactic": "simp_rw [HasFiniteIntegral, ennnorm_eq_ofReal toReal_nonneg]", "state_before": "\u03b1 : Type u_2\n\u03b1' : Type ?u.2382010\n\u03b2 : Type u_1\n\u03b2' : Type ?u.2382016\n\u03b3 : Type ?u.2382019\nE : Type ?u.2382022\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\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 HasFiniteIntegral fun x => ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s))", "state_after": "\u03b1 : Type u_2\n\u03b1' : Type ?u.2382010\n\u03b2 : Type u_1\n\u03b2' : Type ?u.2382016\n\u03b3 : Type ?u.2382019\nE : Type ?u.2382022\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\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s))) \u2202\u03bc) < \u22a4"}, {"tactic": "convert h2s.lt_top using 1", "state_before": "\u03b1 : Type u_2\n\u03b1' : Type ?u.2382010\n\u03b2 : Type u_1\n\u03b2' : Type ?u.2382016\n\u03b3 : Type ?u.2382019\nE : Type ?u.2382022\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\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s))) \u2202\u03bc) < \u22a4", "state_after": "case h.e'_3\n\u03b1 : Type u_2\n\u03b1' : Type ?u.2382010\n\u03b2 : Type u_1\n\u03b2' : Type ?u.2382016\n\u03b3 : Type ?u.2382019\nE : Type ?u.2382022\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\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s))) \u2202\u03bc) = \u2191\u2191(Measure.prod \u03bc \u03bd) s"}, {"tactic": "rw [prod_apply hs]", "state_before": "case h.e'_3\n\u03b1 : Type u_2\n\u03b1' : Type ?u.2382010\n\u03b2 : Type u_1\n\u03b2' : Type ?u.2382016\n\u03b3 : Type ?u.2382019\nE : Type ?u.2382022\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\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s))) \u2202\u03bc) = \u2191\u2191(Measure.prod \u03bc \u03bd) s", "state_after": "case h.e'_3\n\u03b1 : Type u_2\n\u03b1' : Type ?u.2382010\n\u03b2 : Type u_1\n\u03b2' : Type ?u.2382016\n\u03b3 : Type ?u.2382019\nE : Type ?u.2382022\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\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s))) \u2202\u03bc) = \u222b\u207b (x : \u03b1), \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) \u2202\u03bc"}, {"tactic": "apply lintegral_congr_ae", "state_before": "case h.e'_3\n\u03b1 : Type u_2\n\u03b1' : Type ?u.2382010\n\u03b2 : Type u_1\n\u03b2' : Type ?u.2382016\n\u03b3 : Type ?u.2382019\nE : Type ?u.2382022\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\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s))) \u2202\u03bc) = \u222b\u207b (x : \u03b1), \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) \u2202\u03bc", "state_after": "case h.e'_3.h\n\u03b1 : Type u_2\n\u03b1' : Type ?u.2382010\n\u03b2 : Type u_1\n\u03b2' : Type ?u.2382016\n\u03b3 : Type ?u.2382019\nE : Type ?u.2382022\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\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 (fun a => ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)))) =\u1da0[ae \u03bc] fun a => \u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)"}, {"tactic": "refine' (ae_measure_lt_top hs h2s).mp _", "state_before": "case h.e'_3.h\n\u03b1 : Type u_2\n\u03b1' : Type ?u.2382010\n\u03b2 : Type u_1\n\u03b2' : Type ?u.2382016\n\u03b3 : Type ?u.2382019\nE : Type ?u.2382022\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\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 (fun a => ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)))) =\u1da0[ae \u03bc] fun a => \u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)", "state_after": "case h.e'_3.h\n\u03b1 : Type u_2\n\u03b1' : Type ?u.2382010\n\u03b2 : Type u_1\n\u03b2' : Type ?u.2382016\n\u03b3 : Type ?u.2382019\nE : Type ?u.2382022\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\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) < \u22a4 \u2192\n      (fun a => ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)))) x = (fun a => \u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)) x"}, {"tactic": "apply eventually_of_forall", "state_before": "case h.e'_3.h\n\u03b1 : Type u_2\n\u03b1' : Type ?u.2382010\n\u03b2 : Type u_1\n\u03b2' : Type ?u.2382016\n\u03b3 : Type ?u.2382019\nE : Type ?u.2382022\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\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) < \u22a4 \u2192\n      (fun a => ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)))) x = (fun a => \u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)) x", "state_after": "case h.e'_3.h.hp\n\u03b1 : Type u_2\n\u03b1' : Type ?u.2382010\n\u03b2 : Type u_1\n\u03b2' : Type ?u.2382016\n\u03b3 : Type ?u.2382019\nE : Type ?u.2382022\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\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 \u2200 (x : \u03b1),\n    \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) < \u22a4 \u2192\n      (fun a => ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)))) x = (fun a => \u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)) x"}, {"tactic": "intro x hx", "state_before": "case h.e'_3.h.hp\n\u03b1 : Type u_2\n\u03b1' : Type ?u.2382010\n\u03b2 : Type u_1\n\u03b2' : Type ?u.2382016\n\u03b3 : Type ?u.2382019\nE : Type ?u.2382022\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\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\n\u22a2 \u2200 (x : \u03b1),\n    \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) < \u22a4 \u2192\n      (fun a => ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)))) x = (fun a => \u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)) x", "state_after": "case h.e'_3.h.hp\n\u03b1 : Type u_2\n\u03b1' : Type ?u.2382010\n\u03b2 : Type u_1\n\u03b2' : Type ?u.2382016\n\u03b3 : Type ?u.2382019\nE : Type ?u.2382022\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\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\nx : \u03b1\nhx : \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) < \u22a4\n\u22a2 (fun a => ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)))) x = (fun a => \u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)) x"}, {"tactic": "rw [lt_top_iff_ne_top] at hx", "state_before": "case h.e'_3.h.hp\n\u03b1 : Type u_2\n\u03b1' : Type ?u.2382010\n\u03b2 : Type u_1\n\u03b2' : Type ?u.2382016\n\u03b3 : Type ?u.2382019\nE : Type ?u.2382022\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\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\nx : \u03b1\nhx : \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) < \u22a4\n\u22a2 (fun a => ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)))) x = (fun a => \u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)) x", "state_after": "case h.e'_3.h.hp\n\u03b1 : Type u_2\n\u03b1' : Type ?u.2382010\n\u03b2 : Type u_1\n\u03b2' : Type ?u.2382016\n\u03b3 : Type ?u.2382019\nE : Type ?u.2382022\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\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\nx : \u03b1\nhx : \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) \u2260 \u22a4\n\u22a2 (fun a => ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)))) x = (fun a => \u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)) x"}, {"tactic": "simp [ofReal_toReal, hx]", "state_before": "case h.e'_3.h.hp\n\u03b1 : Type u_2\n\u03b1' : Type ?u.2382010\n\u03b2 : Type u_1\n\u03b2' : Type ?u.2382016\n\u03b3 : Type ?u.2382019\nE : Type ?u.2382022\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\ns : Set (\u03b1 \u00d7 \u03b2)\nhs : MeasurableSet s\nh2s : \u2191\u2191(Measure.prod \u03bc \u03bd) s \u2260 \u22a4\nx : \u03b1\nhx : \u2191\u2191\u03bd (Prod.mk x \u207b\u00b9' s) \u2260 \u22a4\n\u22a2 (fun a => ENNReal.ofReal (ENNReal.toReal (\u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)))) x = (fun a => \u2191\u2191\u03bd (Prod.mk a \u207b\u00b9' s)) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "full_name": "differentiableOn_univ", "start": [612, 1], "end": [614, 22], "traced_tactics": [{"tactic": "simp only [DifferentiableOn, Differentiable, differentiableWithinAt_univ, mem_univ,\n  forall_true_left]", "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.322615\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type ?u.322710\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\n\u22a2 DifferentiableOn \ud835\udd5c f univ \u2194 Differentiable \ud835\udd5c f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/Circulant.lean", "full_name": "Matrix.Fin.circulant_mul", "start": [137, 1], "end": [140, 34], "traced_tactics": [{"tactic": "simp [Injective]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.20421\nm : Type ?u.20424\nn : Type ?u.20427\nR : Type ?u.20430\ninst\u271d : Semiring \u03b1\n\u22a2 \u2200 (v w : Fin 0 \u2192 \u03b1), circulant v \u2b1d circulant w = circulant (mulVec (circulant v) w)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Lie/Nilpotent.lean", "full_name": "LieIdeal.lcs_zero", "start": [691, 1], "end": [692, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Category/ModuleCat/Colimits.lean", "full_name": "ModuleCat.Colimits.quot_add", "start": [218, 1], "end": [220, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "full_name": "ENNReal.one_rpow", "start": [382, 1], "end": [384, 7], "traced_tactics": [{"tactic": "rw [\u2190 coe_one, coe_rpow_of_ne_zero one_ne_zero]", "state_before": "x : \u211d\n\u22a2 1 ^ x = 1", "state_after": "x : \u211d\n\u22a2 \u2191(1 ^ x) = \u21911"}, {"tactic": "simp", "state_before": "x : \u211d\n\u22a2 \u2191(1 ^ x) = \u21911", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/DFA.lean", "full_name": "DFA.pumping_lemma", "start": [155, 1], "end": [170, 67], "traced_tactics": [{"tactic": "obtain \u27e8_, a, b, c, hx, hlen, hnil, rfl, hb, hc\u27e9 := M.evalFrom_split hlen rfl", "state_before": "\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx : x \u2208 accepts M\nhlen : Fintype.card \u03c3 \u2264 List.length x\n\u22a2 \u2203 a b c, x = a ++ b ++ c \u2227 List.length a + List.length b \u2264 Fintype.card \u03c3 \u2227 b \u2260 [] \u2227 {a} * {b}\u2217 * {c} \u2264 accepts M", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 accepts M\nhlen\u271d : Fintype.card \u03c3 \u2264 List.length x\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : List.length a + List.length b \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : evalFrom M (evalFrom M ?m.8226 a) b = evalFrom M ?m.8226 a\nhc : evalFrom M (evalFrom M ?m.8226 a) c = evalFrom M ?m.8226 x\n\u22a2 \u2203 a b c, x = a ++ b ++ c \u2227 List.length a + List.length b \u2264 Fintype.card \u03c3 \u2227 b \u2260 [] \u2227 {a} * {b}\u2217 * {c} \u2264 accepts M"}, {"tactic": "use a, b, c, hx, hlen, hnil", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 accepts M\nhlen\u271d : Fintype.card \u03c3 \u2264 List.length x\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : List.length a + List.length b \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : evalFrom M (evalFrom M ?m.8226 a) b = evalFrom M ?m.8226 a\nhc : evalFrom M (evalFrom M ?m.8226 a) c = evalFrom M ?m.8226 x\n\u22a2 \u2203 a b c, x = a ++ b ++ c \u2227 List.length a + List.length b \u2264 Fintype.card \u03c3 \u2227 b \u2260 [] \u2227 {a} * {b}\u2217 * {c} \u2264 accepts M", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 accepts M\nhlen\u271d : Fintype.card \u03c3 \u2264 List.length x\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : List.length a + List.length b \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : evalFrom M (evalFrom M ?m.8226 a) b = evalFrom M ?m.8226 a\nhc : evalFrom M (evalFrom M ?m.8226 a) c = evalFrom M ?m.8226 x\n\u22a2 {a} * {b}\u2217 * {c} \u2264 accepts M"}, {"tactic": "intro y hy", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 accepts M\nhlen\u271d : Fintype.card \u03c3 \u2264 List.length x\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : List.length a + List.length b \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : evalFrom M (evalFrom M ?m.8226 a) b = evalFrom M ?m.8226 a\nhc : evalFrom M (evalFrom M ?m.8226 a) c = evalFrom M ?m.8226 x\n\u22a2 {a} * {b}\u2217 * {c} \u2264 accepts M", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 accepts M\nhlen\u271d : Fintype.card \u03c3 \u2264 List.length x\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : List.length a + List.length b \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : evalFrom M (evalFrom M ?m.8226 a) b = evalFrom M ?m.8226 a\nhc : evalFrom M (evalFrom M ?m.8226 a) c = evalFrom M ?m.8226 x\ny : List \u03b1\nhy : y \u2208 {a} * {b}\u2217 * {c}\n\u22a2 y \u2208 accepts M"}, {"tactic": "rw [Language.mem_mul] at hy", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 accepts M\nhlen\u271d : Fintype.card \u03c3 \u2264 List.length x\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : List.length a + List.length b \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : evalFrom M (evalFrom M ?m.8226 a) b = evalFrom M ?m.8226 a\nhc : evalFrom M (evalFrom M ?m.8226 a) c = evalFrom M ?m.8226 x\ny : List \u03b1\nhy : y \u2208 {a} * {b}\u2217 * {c}\n\u22a2 y \u2208 accepts M", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 accepts M\nhlen\u271d : Fintype.card \u03c3 \u2264 List.length x\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : List.length a + List.length b \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : evalFrom M (evalFrom M ?m.8226 a) b = evalFrom M ?m.8226 a\nhc : evalFrom M (evalFrom M ?m.8226 a) c = evalFrom M ?m.8226 x\ny : List \u03b1\nhy : \u2203 a_1 b_1, a_1 \u2208 {a} * {b}\u2217 \u2227 b_1 \u2208 {c} \u2227 a_1 ++ b_1 = y\n\u22a2 y \u2208 accepts M"}, {"tactic": "rcases hy with \u27e8ab, c', hab, hc', rfl\u27e9", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 accepts M\nhlen\u271d : Fintype.card \u03c3 \u2264 List.length x\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : List.length a + List.length b \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : evalFrom M (evalFrom M ?m.8226 a) b = evalFrom M ?m.8226 a\nhc : evalFrom M (evalFrom M ?m.8226 a) c = evalFrom M ?m.8226 x\ny : List \u03b1\nhy : \u2203 a_1 b_1, a_1 \u2208 {a} * {b}\u2217 \u2227 b_1 \u2208 {c} \u2227 a_1 ++ b_1 = y\n\u22a2 y \u2208 accepts M", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 accepts M\nhlen\u271d : Fintype.card \u03c3 \u2264 List.length x\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : List.length a + List.length b \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : evalFrom M (evalFrom M ?m.8226 a) b = evalFrom M ?m.8226 a\nhc : evalFrom M (evalFrom M ?m.8226 a) c = evalFrom M ?m.8226 x\nab c' : List \u03b1\nhab : ab \u2208 {a} * {b}\u2217\nhc' : c' \u2208 {c}\n\u22a2 ab ++ c' \u2208 accepts M"}, {"tactic": "rw [Language.mem_mul] at hab", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 accepts M\nhlen\u271d : Fintype.card \u03c3 \u2264 List.length x\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : List.length a + List.length b \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : evalFrom M (evalFrom M ?m.8226 a) b = evalFrom M ?m.8226 a\nhc : evalFrom M (evalFrom M ?m.8226 a) c = evalFrom M ?m.8226 x\nab c' : List \u03b1\nhab : ab \u2208 {a} * {b}\u2217\nhc' : c' \u2208 {c}\n\u22a2 ab ++ c' \u2208 accepts M", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 accepts M\nhlen\u271d : Fintype.card \u03c3 \u2264 List.length x\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : List.length a + List.length b \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : evalFrom M (evalFrom M ?m.8226 a) b = evalFrom M ?m.8226 a\nhc : evalFrom M (evalFrom M ?m.8226 a) c = evalFrom M ?m.8226 x\nab c' : List \u03b1\nhab : \u2203 a_1 b_1, a_1 \u2208 {a} \u2227 b_1 \u2208 {b}\u2217 \u2227 a_1 ++ b_1 = ab\nhc' : c' \u2208 {c}\n\u22a2 ab ++ c' \u2208 accepts M"}, {"tactic": "rcases hab with \u27e8a', b', ha', hb', rfl\u27e9", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 accepts M\nhlen\u271d : Fintype.card \u03c3 \u2264 List.length x\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : List.length a + List.length b \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : evalFrom M (evalFrom M ?m.8226 a) b = evalFrom M ?m.8226 a\nhc : evalFrom M (evalFrom M ?m.8226 a) c = evalFrom M ?m.8226 x\nab c' : List \u03b1\nhab : \u2203 a_1 b_1, a_1 \u2208 {a} \u2227 b_1 \u2208 {b}\u2217 \u2227 a_1 ++ b_1 = ab\nhc' : c' \u2208 {c}\n\u22a2 ab ++ c' \u2208 accepts M", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 accepts M\nhlen\u271d : Fintype.card \u03c3 \u2264 List.length x\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : List.length a + List.length b \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : evalFrom M (evalFrom M ?m.8226 a) b = evalFrom M ?m.8226 a\nhc : evalFrom M (evalFrom M ?m.8226 a) c = evalFrom M ?m.8226 x\nc' : List \u03b1\nhc' : c' \u2208 {c}\na' b' : List \u03b1\nha' : a' \u2208 {a}\nhb' : b' \u2208 {b}\u2217\n\u22a2 a' ++ b' ++ c' \u2208 accepts M"}, {"tactic": "rw [Set.mem_singleton_iff] at ha' hc'", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 accepts M\nhlen\u271d : Fintype.card \u03c3 \u2264 List.length x\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : List.length a + List.length b \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : evalFrom M (evalFrom M ?m.8226 a) b = evalFrom M ?m.8226 a\nhc : evalFrom M (evalFrom M ?m.8226 a) c = evalFrom M ?m.8226 x\nc' : List \u03b1\nhc' : c' \u2208 {c}\na' b' : List \u03b1\nha' : a' \u2208 {a}\nhb' : b' \u2208 {b}\u2217\n\u22a2 a' ++ b' ++ c' \u2208 accepts M", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 accepts M\nhlen\u271d : Fintype.card \u03c3 \u2264 List.length x\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : List.length a + List.length b \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : evalFrom M (evalFrom M ?m.8226 a) b = evalFrom M ?m.8226 a\nhc : evalFrom M (evalFrom M ?m.8226 a) c = evalFrom M ?m.8226 x\nc' : List \u03b1\nhc' : c' = c\na' b' : List \u03b1\nha' : a' = a\nhb' : b' \u2208 {b}\u2217\n\u22a2 a' ++ b' ++ c' \u2208 accepts M"}, {"tactic": "substs ha' hc'", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 accepts M\nhlen\u271d : Fintype.card \u03c3 \u2264 List.length x\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : List.length a + List.length b \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : evalFrom M (evalFrom M ?m.8226 a) b = evalFrom M ?m.8226 a\nhc : evalFrom M (evalFrom M ?m.8226 a) c = evalFrom M ?m.8226 x\nc' : List \u03b1\nhc' : c' = c\na' b' : List \u03b1\nha' : a' = a\nhb' : b' \u2208 {b}\u2217\n\u22a2 a' ++ b' ++ c' \u2208 accepts M", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 accepts M\nhlen\u271d : Fintype.card \u03c3 \u2264 List.length x\nb : List \u03b1\nhnil : b \u2260 []\nc' a' b' : List \u03b1\nhb' : b' \u2208 {b}\u2217\nhlen : List.length a' + List.length b \u2264 Fintype.card \u03c3\nhb : evalFrom M (evalFrom M ?m.8226 a') b = evalFrom M ?m.8226 a'\nhx : x = a' ++ b ++ c'\nhc : evalFrom M (evalFrom M ?m.8226 a') c' = evalFrom M ?m.8226 x\n\u22a2 a' ++ b' ++ c' \u2208 accepts M"}, {"tactic": "have h := M.evalFrom_of_pow hb hb'", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 accepts M\nhlen\u271d : Fintype.card \u03c3 \u2264 List.length x\nb : List \u03b1\nhnil : b \u2260 []\nc' a' b' : List \u03b1\nhb' : b' \u2208 {b}\u2217\nhlen : List.length a' + List.length b \u2264 Fintype.card \u03c3\nhb : evalFrom M (evalFrom M ?m.8226 a') b = evalFrom M ?m.8226 a'\nhx : x = a' ++ b ++ c'\nhc : evalFrom M (evalFrom M ?m.8226 a') c' = evalFrom M ?m.8226 x\n\u22a2 a' ++ b' ++ c' \u2208 accepts M", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 accepts M\nhlen\u271d : Fintype.card \u03c3 \u2264 List.length x\nb : List \u03b1\nhnil : b \u2260 []\nc' a' b' : List \u03b1\nhb' : b' \u2208 {b}\u2217\nhlen : List.length a' + List.length b \u2264 Fintype.card \u03c3\nhb : evalFrom M (evalFrom M ?m.8226 a') b = evalFrom M ?m.8226 a'\nhx : x = a' ++ b ++ c'\nhc : evalFrom M (evalFrom M ?m.8226 a') c' = evalFrom M ?m.8226 x\nh : evalFrom M (evalFrom M ?m.8226 a') b' = evalFrom M ?m.8226 a'\n\u22a2 a' ++ b' ++ c' \u2208 accepts M"}, {"tactic": "rwa [mem_accepts, evalFrom_of_append, evalFrom_of_append, h, hc]", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 accepts M\nhlen\u271d : Fintype.card \u03c3 \u2264 List.length x\nb : List \u03b1\nhnil : b \u2260 []\nc' a' b' : List \u03b1\nhb' : b' \u2208 {b}\u2217\nhlen : List.length a' + List.length b \u2264 Fintype.card \u03c3\nhb : evalFrom M (evalFrom M ?m.8226 a') b = evalFrom M ?m.8226 a'\nhx : x = a' ++ b ++ c'\nhc : evalFrom M (evalFrom M ?m.8226 a') c' = evalFrom M ?m.8226 x\nh : evalFrom M (evalFrom M ?m.8226 a') b' = evalFrom M ?m.8226 a'\n\u22a2 a' ++ b' ++ c' \u2208 accepts M", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/CompleteBooleanAlgebra.lean", "full_name": "iInf_sup_iInf", "start": [218, 1], "end": [220, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/BoxIntegral/Partition/Split.lean", "full_name": "BoxIntegral.Prepartition.isPartition_splitMany", "start": [264, 1], "end": [266, 96], "traced_tactics": [{"tactic": "simp only [splitMany_empty, isPartitionTop]", "state_before": "\u03b9 : Type u_1\nM : Type ?u.44679\nn : \u2115\nI\u271d J : Box \u03b9\ni : \u03b9\nx : \u211d\nI : Box \u03b9\ns : Finset (\u03b9 \u00d7 \u211d)\n\u22a2 IsPartition (splitMany I \u2205)", "state_after": "no goals"}, {"tactic": "simpa only [splitMany_insert, inf_split] using hs.biUnion fun J _ => isPartitionSplit _ _ _", "state_before": "\u03b9 : Type u_1\nM : Type ?u.44679\nn : \u2115\nI\u271d J : Box \u03b9\ni : \u03b9\nx : \u211d\nI : Box \u03b9\ns\u271d : Finset (\u03b9 \u00d7 \u211d)\na : \u03b9 \u00d7 \u211d\ns : Finset (\u03b9 \u00d7 \u211d)\nx\u271d : \u00aca \u2208 s\nhs : IsPartition (splitMany I s)\n\u22a2 IsPartition (splitMany I (insert a s))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Extr.lean", "full_name": "isMinFilter_dual_iff", "start": [209, 1], "end": [210, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Grade.lean", "full_name": "grade_eq_grade_iff", "start": [189, 1], "end": [190, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Module/LocalizedModule.lean", "full_name": "IsLocalizedModule.mk'_zero", "start": [934, 1], "end": [934, 95], "traced_tactics": [{"tactic": "rw [\u2190 zero_smul R (0 : M), mk'_smul, zero_smul]", "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.820045\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\ns : { x // x \u2208 S }\n\u22a2 mk' f 0 s = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": 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"https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "full_name": "multiplesHom_apply", "start": [869, 1], "end": [870, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Subobject/FactorThru.lean", "full_name": "CategoryTheory.Subobject.factorThru_add", "start": [192, 1], "end": [196, 7], "traced_tactics": [{"tactic": "ext", "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 : Preadditive C\nX Y : C\nP : Subobject Y\nf g : X \u27f6 Y\nw : Factors P (f + g)\nwf : Factors P f\nwg : Factors P g\n\u22a2 factorThru P (f + g) w = factorThru P f wf + factorThru P g wg", "state_after": "case h\nC : 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 : Preadditive C\nX Y : C\nP : Subobject Y\nf g : X \u27f6 Y\nw : Factors P (f + g)\nwf : Factors P f\nwg : Factors P g\n\u22a2 factorThru P (f + g) w \u226b arrow P = (factorThru P f wf + factorThru P g wg) \u226b arrow P"}, {"tactic": "simp", "state_before": "case h\nC : 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 : Preadditive C\nX Y : C\nP : Subobject Y\nf g : X \u27f6 Y\nw : Factors P (f + g)\nwf : Factors P f\nwg : Factors P g\n\u22a2 factorThru P (f + g) w \u226b arrow P = (factorThru P f wf + factorThru P g wg) \u226b arrow P", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Monotone/Monovary.lean", "full_name": "Subsingleton.monovaryOn", "start": [123, 11], "end": [124, 95], "traced_tactics": []}, {"url": 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\u211d\u22650\n\u22a2 toSignedMeasure (r \u2022 \u03bc) = r \u2022 toSignedMeasure \u03bc", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.138627\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nr : \u211d\u22650\ni : Set \u03b1\n\u22a2 MeasurableSet i \u2192 \u2191(toSignedMeasure (r \u2022 \u03bc)) i = \u2191(r \u2022 toSignedMeasure \u03bc) i"}, {"tactic": "intro hi", "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.138627\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nr : \u211d\u22650\ni : Set \u03b1\n\u22a2 MeasurableSet i \u2192 \u2191(toSignedMeasure (r \u2022 \u03bc)) i = \u2191(r \u2022 toSignedMeasure \u03bc) i", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.138627\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nr : \u211d\u22650\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(toSignedMeasure (r \u2022 \u03bc)) i = \u2191(r \u2022 toSignedMeasure \u03bc) i"}, {"tactic": "rw [toSignedMeasure_apply_measurable hi, VectorMeasure.smul_apply,\n  toSignedMeasure_apply_measurable hi, coe_smul, Pi.smul_apply, ENNReal.toReal_smul]", "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.138627\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bc\nr : \u211d\u22650\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(toSignedMeasure (r \u2022 \u03bc)) i = \u2191(r \u2022 toSignedMeasure \u03bc) i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/AdjoinRoot.lean", "full_name": "AdjoinRoot.powerBasisAux'_repr_symm_apply", "start": [508, 1], "end": [510, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": 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< n\n\u22a2 (g \u2218 snoc q y) j = snoc (g \u2218 q) (g y) j\n\ncase neg\nm n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d (last n)\nq\u271d : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (\u2191castSucc i)\ni : Fin n\ny\u271d : \u03b1\u271d (\u2191castSucc i)\nz : \u03b1\u271d (last n)\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b1 \u2192 \u03b2\nq : Fin n \u2192 \u03b1\ny : \u03b1\nj : Fin (n + 1)\nh : \u00ac\u2191j < n\n\u22a2 (g \u2218 snoc q y) j = snoc (g \u2218 q) (g y) j"}, {"tactic": "simp [h, snoc, castSucc_castLT]", "state_before": "case pos\nm n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d (last n)\nq\u271d : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (\u2191castSucc i)\ni : Fin n\ny\u271d : \u03b1\u271d (\u2191castSucc i)\nz : \u03b1\u271d (last n)\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b1 \u2192 \u03b2\nq : Fin n \u2192 \u03b1\ny : \u03b1\nj : Fin (n + 1)\nh : \u2191j < n\n\u22a2 (g \u2218 snoc q y) j = snoc (g \u2218 q) (g y) j", "state_after": "no goals"}, {"tactic": "rw [eq_last_of_not_lt h]", "state_before": "case neg\nm n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d (last n)\nq\u271d : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (\u2191castSucc i)\ni : Fin n\ny\u271d : \u03b1\u271d (\u2191castSucc i)\nz : \u03b1\u271d (last n)\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b1 \u2192 \u03b2\nq : Fin n \u2192 \u03b1\ny : \u03b1\nj : Fin (n + 1)\nh : \u00ac\u2191j < n\n\u22a2 (g \u2218 snoc q y) j = snoc (g \u2218 q) (g y) j", "state_after": "case neg\nm n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d (last n)\nq\u271d : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (\u2191castSucc i)\ni : Fin n\ny\u271d : \u03b1\u271d (\u2191castSucc i)\nz : \u03b1\u271d (last n)\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b1 \u2192 \u03b2\nq : Fin n \u2192 \u03b1\ny : \u03b1\nj : Fin (n + 1)\nh : \u00ac\u2191j < n\n\u22a2 (g \u2218 snoc q y) (last n) = snoc (g \u2218 q) (g y) (last n)"}, {"tactic": "simp", "state_before": "case neg\nm n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d (last n)\nq\u271d : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (\u2191castSucc i)\ni : Fin n\ny\u271d : \u03b1\u271d (\u2191castSucc i)\nz : \u03b1\u271d (last n)\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ng : \u03b1 \u2192 \u03b2\nq : Fin n \u2192 \u03b1\ny : \u03b1\nj : Fin (n + 1)\nh : \u00ac\u2191j < n\n\u22a2 (g \u2218 snoc q y) (last n) = snoc (g \u2218 q) (g y) (last n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Basic.lean", "full_name": "Set.Ico_subset_Ico_iff", "start": [1131, 1], "end": [1135, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "full_name": "mul_right_surjective\u2080", "start": [428, 1], "end": [429, 51], "traced_tactics": [{"tactic": "simp [mul_assoc, inv_mul_cancel h]", "state_before": "\u03b1 : Type ?u.30355\nM\u2080 : Type ?u.30358\nG\u2080 : Type u_1\nM\u2080' : Type ?u.30364\nG\u2080' : Type ?u.30367\nF : Type ?u.30370\nF' : Type ?u.30373\ninst\u271d : GroupWithZero G\u2080\na\u271d b c a : G\u2080\nh : a \u2260 0\ng : G\u2080\n\u22a2 (fun g => g * a) (g * a\u207b\u00b9) = g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Polynomial/Basic.lean", "full_name": "Polynomial.restriction_zero", "start": [322, 1], "end": [323, 58], "traced_tactics": [{"tactic": "simp only [restriction, Finset.sum_empty, support_zero]", "state_before": "R : Type u\nS : Type ?u.76676\ninst\u271d : Ring R\n\u22a2 restriction 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "full_name": "circleIntegral.integral_radius_zero", "start": [362, 1], "end": [363, 31], "traced_tactics": [{"tactic": "simp [circleIntegral, const]", "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\n\u22a2 (\u222e (z : \u2102) in C(c, 0), f z) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "full_name": "PowerSeries.eq_shift_mul_X_add_const", "start": [1646, 1], "end": [1652, 26], "traced_tactics": [{"tactic": "ext (_ | n)", "state_before": "R : Type u_1\ninst\u271d : Semiring R\n\u03c6 : PowerSeries R\n\u22a2 \u03c6 = (mk fun p => \u2191(coeff R (p + 1)) \u03c6) * X + \u2191(C R) (\u2191(constantCoeff R) \u03c6)", "state_after": "case h.zero\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6 : PowerSeries R\n\u22a2 \u2191(coeff R Nat.zero) \u03c6 = \u2191(coeff R Nat.zero) ((mk fun p => \u2191(coeff R (p + 1)) \u03c6) * X + \u2191(C R) (\u2191(constantCoeff R) \u03c6))\n\ncase h.succ\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6 : PowerSeries R\nn : \u2115\n\u22a2 \u2191(coeff R (Nat.succ n)) \u03c6 =\n    \u2191(coeff R (Nat.succ n)) ((mk fun p => \u2191(coeff R (p + 1)) \u03c6) * X + \u2191(C R) (\u2191(constantCoeff R) \u03c6))"}, {"tactic": "simp only [Nat.zero_eq, coeff_zero_eq_constantCoeff, map_add, map_mul, constantCoeff_X,\n  mul_zero, coeff_zero_C, zero_add]", "state_before": "case h.zero\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6 : PowerSeries R\n\u22a2 \u2191(coeff R Nat.zero) \u03c6 = \u2191(coeff R Nat.zero) ((mk fun p => \u2191(coeff R (p + 1)) \u03c6) * X + \u2191(C R) (\u2191(constantCoeff R) \u03c6))", "state_after": "no goals"}, {"tactic": "simp only [coeff_succ_mul_X, coeff_mk, LinearMap.map_add, coeff_C, n.succ_ne_zero, sub_zero,\n  if_false, add_zero]", "state_before": "case h.succ\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6 : PowerSeries R\nn : \u2115\n\u22a2 \u2191(coeff R (Nat.succ n)) \u03c6 =\n    \u2191(coeff R (Nat.succ n)) ((mk fun p => \u2191(coeff R (p + 1)) \u03c6) * X + \u2191(C R) (\u2191(constantCoeff R) \u03c6))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "full_name": "MvPolynomial.coeff_coe", "start": [1075, 1], "end": [1076, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/Operations.lean", "full_name": "RingHom.ker_isPrime", "start": [2063, 1], "end": [2069, 88], "traced_tactics": [{"tactic": "rw [Ne.def, Ideal.eq_top_iff_one]", "state_before": "R : Type u\nS : Type v\nT : Type w\nF : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Ring S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : RingHomClass F R S\nf : F\n\u22a2 ker f \u2260 \u22a4", "state_after": "R : Type u\nS : Type v\nT : Type w\nF : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Ring S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : RingHomClass F R S\nf : F\n\u22a2 \u00ac1 \u2208 ker f"}, {"tactic": "exact not_one_mem_ker f", "state_before": "R : Type u\nS : Type v\nT : Type w\nF : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Ring S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : RingHomClass F R S\nf : F\n\u22a2 \u00ac1 \u2208 ker f", "state_after": "no goals"}, {"tactic": "simpa only [mem_ker, map_mul] using @eq_zero_or_eq_zero_of_mul_eq_zero S _ _ _ _ _", "state_before": "R : Type u\nS : Type v\nT : Type w\nF : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Ring S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : RingHomClass F R S\nf : F\nx y : R\n\u22a2 x * y \u2208 ker f \u2192 x \u2208 ker f \u2228 y \u2208 ker f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean", "full_name": "LinearMap.toMatrix\u2082'_mul", "start": [330, 1], "end": [332, 52], "traced_tactics": [{"tactic": "simp only [B.toMatrix\u2082'_compl\u2082, toMatrix'_toLin']", "state_before": "R : Type u_1\nR\u2081 : Type ?u.1101680\nR\u2082 : Type ?u.1101683\nM\u271d : Type ?u.1101686\nM\u2081 : Type ?u.1101689\nM\u2082 : Type ?u.1101692\nM\u2081' : Type ?u.1101695\nM\u2082' : Type ?u.1101698\nn : Type u_2\nm : Type u_3\nn' : Type ?u.1101707\nm' : Type u_4\n\u03b9 : Type ?u.1101713\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nM : Matrix m m' R\n\u22a2 \u2191toMatrix\u2082' B \u2b1d M = \u2191toMatrix\u2082' (compl\u2082 B (\u2191toLin' M))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Hom/Centroid.lean", "full_name": "CentroidHom.comp_id", "start": [222, 1], "end": [223, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "nhdsWithin_Iio_basis'", "start": [1716, 1], "end": [1718, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Basic.lean", "full_name": "Set.Ioi_inter_Iio", "start": [634, 1], "end": [635, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "round_neg_two_inv", "start": [1458, 1], "end": [1459, 60], "traced_tactics": [{"tactic": "simp only [round_eq, \u2190 one_div, add_left_neg, floor_zero]", "state_before": "F : Type ?u.280809\n\u03b1 : Type u_1\n\u03b2 : Type ?u.280815\ninst\u271d\u00b9 : LinearOrderedField \u03b1\ninst\u271d : FloorRing \u03b1\n\u22a2 round (-2\u207b\u00b9) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Basic.lean", "full_name": "Set.Ioc_union_Ico_eq_Ioo", "start": [1628, 1], "end": [1632, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Prod.lean", "full_name": "SimpleGraph.Preconnected.boxProd", "start": [172, 11], "end": [178, 60], "traced_tactics": [{"tactic": "rintro x y", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.81755\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nhG : Preconnected G\nhH : Preconnected H\n\u22a2 Preconnected (G \u25a1 H)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.81755\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nhG : Preconnected G\nhH : Preconnected H\nx y : \u03b1 \u00d7 \u03b2\n\u22a2 Reachable (G \u25a1 H) x y"}, {"tactic": "obtain \u27e8w\u2081\u27e9 := hG x.1 y.1", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.81755\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nhG : Preconnected G\nhH : Preconnected H\nx y : \u03b1 \u00d7 \u03b2\n\u22a2 Reachable (G \u25a1 H) x y", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.81755\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nhG : Preconnected G\nhH : Preconnected H\nx y : \u03b1 \u00d7 \u03b2\nw\u2081 : Walk G x.fst y.fst\n\u22a2 Reachable (G \u25a1 H) x y"}, {"tactic": "obtain \u27e8w\u2082\u27e9 := hH x.2 y.2", "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.81755\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nhG : Preconnected G\nhH : Preconnected H\nx y : \u03b1 \u00d7 \u03b2\nw\u2081 : Walk G x.fst y.fst\n\u22a2 Reachable (G \u25a1 H) x y", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.81755\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nhG : Preconnected G\nhH : Preconnected H\nx y : \u03b1 \u00d7 \u03b2\nw\u2081 : Walk G x.fst y.fst\nw\u2082 : Walk H x.snd y.snd\n\u22a2 Reachable (G \u25a1 H) x y"}, {"tactic": "rw [\u2190 @Prod.mk.eta _ _ x, \u2190 @Prod.mk.eta _ _ y]", "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.81755\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nhG : Preconnected G\nhH : Preconnected H\nx y : \u03b1 \u00d7 \u03b2\nw\u2081 : Walk G x.fst y.fst\nw\u2082 : Walk H x.snd y.snd\n\u22a2 Reachable (G \u25a1 H) x y", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.81755\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nhG : Preconnected G\nhH : Preconnected H\nx y : \u03b1 \u00d7 \u03b2\nw\u2081 : Walk G x.fst y.fst\nw\u2082 : Walk H x.snd y.snd\n\u22a2 Reachable (G \u25a1 H) (x.fst, x.snd) (y.fst, y.snd)"}, {"tactic": "exact \u27e8(w\u2081.boxProdLeft _ _).append (w\u2082.boxProdRight _ _)\u27e9", "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.81755\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\nhG : Preconnected G\nhH : Preconnected H\nx y : \u03b1 \u00d7 \u03b2\nw\u2081 : Walk G x.fst y.fst\nw\u2082 : Walk H x.snd y.snd\n\u22a2 Reachable (G \u25a1 H) (x.fst, x.snd) (y.fst, y.snd)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/Adjoint.lean", "full_name": "ContinuousLinearMap.adjointAux_norm", "start": [103, 1], "end": [111, 35], "traced_tactics": [{"tactic": "refine' le_antisymm _ _", "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.301007\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\n\u22a2 \u2016\u2191adjointAux A\u2016 = \u2016A\u2016", "state_after": "case refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.301007\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\n\u22a2 \u2016\u2191adjointAux A\u2016 \u2264 \u2016A\u2016\n\ncase refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.301007\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\n\u22a2 \u2016A\u2016 \u2264 \u2016\u2191adjointAux A\u2016"}, {"tactic": "refine' ContinuousLinearMap.op_norm_le_bound _ (norm_nonneg _) fun x => _", "state_before": "case refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.301007\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\n\u22a2 \u2016\u2191adjointAux A\u2016 \u2264 \u2016A\u2016", "state_after": "case refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.301007\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\nx : F\n\u22a2 \u2016\u2191(\u2191adjointAux A) x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016"}, {"tactic": "rw [adjointAux_apply, LinearIsometryEquiv.norm_map]", "state_before": "case refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.301007\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\nx : F\n\u22a2 \u2016\u2191(\u2191adjointAux A) x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016", "state_after": "case refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.301007\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\nx : F\n\u22a2 \u2016\u2191(\u2191toSesqForm A) x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016"}, {"tactic": "exact toSesqForm_apply_norm_le", "state_before": "case refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.301007\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\nx : F\n\u22a2 \u2016\u2191(\u2191toSesqForm A) x\u2016 \u2264 \u2016A\u2016 * \u2016x\u2016", "state_after": "no goals"}, {"tactic": "nth_rw 1 [\u2190 adjointAux_adjointAux A]", "state_before": "case refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.301007\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\n\u22a2 \u2016A\u2016 \u2264 \u2016\u2191adjointAux A\u2016", "state_after": "case refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.301007\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\n\u22a2 \u2016\u2191adjointAux (\u2191adjointAux A)\u2016 \u2264 \u2016\u2191adjointAux A\u2016"}, {"tactic": "refine' ContinuousLinearMap.op_norm_le_bound _ (norm_nonneg _) fun x => _", "state_before": "case refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.301007\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\n\u22a2 \u2016\u2191adjointAux (\u2191adjointAux A)\u2016 \u2264 \u2016\u2191adjointAux A\u2016", "state_after": "case refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.301007\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\nx : E\n\u22a2 \u2016\u2191(\u2191adjointAux (\u2191adjointAux A)) x\u2016 \u2264 \u2016\u2191adjointAux A\u2016 * \u2016x\u2016"}, {"tactic": "rw [adjointAux_apply, LinearIsometryEquiv.norm_map]", "state_before": "case refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.301007\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\nx : E\n\u22a2 \u2016\u2191(\u2191adjointAux (\u2191adjointAux A)) x\u2016 \u2264 \u2016\u2191adjointAux A\u2016 * \u2016x\u2016", "state_after": "case refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.301007\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\nx : E\n\u22a2 \u2016\u2191(\u2191toSesqForm (\u2191adjointAux A)) x\u2016 \u2264 \u2016\u2191adjointAux A\u2016 * \u2016x\u2016"}, {"tactic": "exact toSesqForm_apply_norm_le", "state_before": "case refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.301007\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\nx : E\n\u22a2 \u2016\u2191(\u2191toSesqForm (\u2191adjointAux A)) x\u2016 \u2264 \u2016\u2191adjointAux A\u2016 * \u2016x\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Connected.lean", "full_name": "ConnectedComponents.continuous_coe", "start": [1473, 1], "end": [1474, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/Padics/PadicNumbers.lean", "full_name": "Padic.exi_rat_seq_conv_cauchy", "start": [703, 1], "end": [728, 26], "traced_tactics": [{"tactic": "have h\u03b53 : 0 < \u03b5 / 3 := div_pos h\u03b5 (by norm_num)", "state_before": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 padicNorm p (limSeq f j - limSeq f i) < \u03b5", "state_after": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 padicNorm p (limSeq f j - limSeq f i) < \u03b5"}, {"tactic": "let \u27e8N, hN\u27e9 := exi_rat_seq_conv f h\u03b53", "state_before": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 padicNorm p (limSeq f j - limSeq f i) < \u03b5", "state_after": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 padicNorm p (limSeq f j - limSeq f i) < \u03b5"}, {"tactic": "let \u27e8N2, hN2\u27e9 := f.cauchy\u2082 h\u03b53", "state_before": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 padicNorm p (limSeq f j - limSeq f i) < \u03b5", "state_after": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 padicNorm p (limSeq f j - limSeq f i) < \u03b5"}, {"tactic": "exists max N N2", "state_before": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 padicNorm p (limSeq f j - limSeq f i) < \u03b5", "state_after": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\n\u22a2 \u2200 (j : \u2115), j \u2265 max N N2 \u2192 padicNorm p (limSeq f j - limSeq f (max N N2)) < \u03b5"}, {"tactic": "intro j hj", "state_before": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\n\u22a2 \u2200 (j : \u2115), j \u2265 max N N2 \u2192 padicNorm p (limSeq f j - limSeq f (max N N2)) < \u03b5", "state_after": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 padicNorm p (limSeq f j - limSeq f (max N N2)) < \u03b5"}, {"tactic": "suffices\n  padicNormE (limSeq f j - f (max N N2) + (f (max N N2) - limSeq f (max N N2)) : \u211a_[p]) < \u03b5 by\n  ring_nf at this \u22a2\n  rw [\u2190 padicNormE.eq_padic_norm']\n  exact_mod_cast this", "state_before": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 padicNorm p (limSeq f j - limSeq f (max N N2)) < \u03b5", "state_after": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f (max N N2) + (\u2191f (max N N2) - \u2191(limSeq f (max N N2)))) < \u03b5"}, {"tactic": "norm_num", "state_before": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\n\u22a2 0 < 3", "state_after": "no goals"}, {"tactic": "ring_nf at this \u22a2", "state_before": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\nthis : \u2191padicNormE (\u2191(limSeq f j) - \u2191f (max N N2) + (\u2191f (max N N2) - \u2191(limSeq f (max N N2)))) < \u03b5\n\u22a2 padicNorm p (limSeq f j - limSeq f (max N N2)) < \u03b5", "state_after": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\nthis : \u2191padicNormE (\u2191(limSeq f j) - \u2191(limSeq f (max N N2))) < \u03b5\n\u22a2 padicNorm p (limSeq f j - limSeq f (max N N2)) < \u03b5"}, {"tactic": "rw [\u2190 padicNormE.eq_padic_norm']", "state_before": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\nthis : \u2191padicNormE (\u2191(limSeq f j) - \u2191(limSeq f (max N N2))) < \u03b5\n\u22a2 padicNorm p (limSeq f j - limSeq f (max N N2)) < \u03b5", "state_after": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\nthis : \u2191padicNormE (\u2191(limSeq f j) - \u2191(limSeq f (max N N2))) < \u03b5\n\u22a2 \u2191padicNormE \u2191(limSeq f j - limSeq f (max N N2)) < \u03b5"}, {"tactic": "exact_mod_cast this", "state_before": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\nthis : \u2191padicNormE (\u2191(limSeq f j) - \u2191(limSeq f (max N N2))) < \u03b5\n\u22a2 \u2191padicNormE \u2191(limSeq f j - limSeq f (max N N2)) < \u03b5", "state_after": "no goals"}, {"tactic": "apply lt_of_le_of_lt", "state_before": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f (max N N2) + (\u2191f (max N N2) - \u2191(limSeq f (max N N2)))) < \u03b5", "state_after": "case a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f (max N N2) + (\u2191f (max N N2) - \u2191(limSeq f (max N N2)))) \u2264 ?b\n\ncase a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 ?b < \u03b5\n\ncase b\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 (fun x => \u211a) (\u2191(limSeq f j) - \u2191f (max N N2) + (\u2191f (max N N2) - \u2191(limSeq f (max N N2))))"}, {"tactic": "apply padicNormE.add_le", "state_before": "case a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f (max N N2) + (\u2191f (max N N2) - \u2191(limSeq f (max N N2)))) \u2264 ?b", "state_after": "no goals"}, {"tactic": "rw [\u2190add_thirds \u03b5]", "state_before": "case a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f (max N N2)) + \u2191padicNormE (\u2191f (max N N2) - \u2191(limSeq f (max N N2))) < \u03b5", "state_after": "case a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f (max N N2)) + \u2191padicNormE (\u2191f (max N N2) - \u2191(limSeq f (max N N2))) <\n    \u03b5 / 3 + \u03b5 / 3 + \u03b5 / 3"}, {"tactic": "apply _root_.add_lt_add", "state_before": "case a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f (max N N2)) + \u2191padicNormE (\u2191f (max N N2) - \u2191(limSeq f (max N N2))) <\n    \u03b5 / 3 + \u03b5 / 3 + \u03b5 / 3", "state_after": "case a.h\u2081\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f (max N N2)) < \u03b5 / 3 + \u03b5 / 3\n\ncase a.h\u2082\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191f (max N N2) - \u2191(limSeq f (max N N2))) < \u03b5 / 3"}, {"tactic": "suffices padicNormE (limSeq f j - f j + (f j - f (max N N2)) : \u211a_[p]) < \u03b5 / 3 + \u03b5 / 3 by\n  simpa only [sub_add_sub_cancel]", "state_before": "case a.h\u2081\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f (max N N2)) < \u03b5 / 3 + \u03b5 / 3", "state_after": "case a.h\u2081\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f j + (\u2191f j - \u2191f (max N N2))) < \u03b5 / 3 + \u03b5 / 3"}, {"tactic": "apply lt_of_le_of_lt", "state_before": "case a.h\u2081\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f j + (\u2191f j - \u2191f (max N N2))) < \u03b5 / 3 + \u03b5 / 3", "state_after": "case a.h\u2081.a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f j + (\u2191f j - \u2191f (max N N2))) \u2264 ?a.h\u2081.b\u271d\n\ncase a.h\u2081.a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 ?a.h\u2081.b\u271d < \u03b5 / 3 + \u03b5 / 3\n\ncase a.h\u2081.b\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 (fun x => \u211a) (\u2191(limSeq f j) - \u2191f j + (\u2191f j - \u2191f (max N N2)))"}, {"tactic": "simpa only [sub_add_sub_cancel]", "state_before": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\nthis : \u2191padicNormE (\u2191(limSeq f j) - \u2191f j + (\u2191f j - \u2191f (max N N2))) < \u03b5 / 3 + \u03b5 / 3\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f (max N N2)) < \u03b5 / 3 + \u03b5 / 3", "state_after": "no goals"}, {"tactic": "apply padicNormE.add_le", "state_before": "case a.h\u2081.a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f j + (\u2191f j - \u2191f (max N N2))) \u2264 ?a.h\u2081.b\u271d", "state_after": "no goals"}, {"tactic": "apply _root_.add_lt_add", "state_before": "case a.h\u2081.a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f j) + \u2191padicNormE (\u2191f j - \u2191f (max N N2)) < \u03b5 / 3 + \u03b5 / 3", "state_after": "case a.h\u2081.a.h\u2081\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f j) < \u03b5 / 3\n\ncase a.h\u2081.a.h\u2082\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191f j - \u2191f (max N N2)) < \u03b5 / 3"}, {"tactic": "rw [padicNormE.map_sub]", "state_before": "case a.h\u2081.a.h\u2081\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f j) < \u03b5 / 3", "state_after": "case a.h\u2081.a.h\u2081\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191f j - \u2191(limSeq f j)) < \u03b5 / 3"}, {"tactic": "apply_mod_cast hN j", "state_before": "case a.h\u2081.a.h\u2081\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191f j - \u2191(limSeq f j)) < \u03b5 / 3", "state_after": "case a.h\u2081.a.h\u2081\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 j \u2265 N"}, {"tactic": "exact le_of_max_le_left hj", "state_before": "case a.h\u2081.a.h\u2081\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 j \u2265 N", "state_after": "no goals"}, {"tactic": "exact hN2 _ (le_of_max_le_right hj) _ (le_max_right _ _)", "state_before": "case a.h\u2081.a.h\u2082\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191f j - \u2191f (max N N2)) < \u03b5 / 3", "state_after": "no goals"}, {"tactic": "apply_mod_cast hN (max N N2)", "state_before": "case a.h\u2082\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191f (max N N2) - \u2191(limSeq f (max N N2))) < \u03b5 / 3", "state_after": "case a.h\u2082\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 max N N2 \u2265 N"}, {"tactic": "apply le_max_left", "state_before": "case a.h\u2082\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 max N N2 \u2265 N", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/PartENat.lean", "full_name": "PartENat.add_lt_add_iff_right", "start": [472, 11], "end": [473, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Pairwise.lean", "full_name": "List.pairwise_of_reflexive_of_forall_ne", "start": [307, 1], "end": [309, 81], "traced_tactics": [{"tactic": "classical exact pairwise_of_reflexive_on_dupl_of_forall_ne (fun _ _ => hr _) h", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.44556\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d l : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : Reflexive r\nh : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\n\u22a2 Pairwise r l", "state_after": "no goals"}, {"tactic": "exact pairwise_of_reflexive_on_dupl_of_forall_ne (fun _ _ => hr _) h", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.44556\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d l : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : Reflexive r\nh : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\n\u22a2 Pairwise r l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/Defs.lean", "full_name": "Equiv.congr_fun", "start": [127, 11], "end": [128, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "full_name": "Cardinal.ord_aleph_isLimit", "start": [319, 1], "end": [320, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean", "full_name": "geometric_hahn_banach_point_point", "start": [199, 1], "end": [204, 46], "traced_tactics": [{"tactic": "obtain \u27e8f, s, t, hs, st, ht\u27e9 :=\n  geometric_hahn_banach_compact_closed (convex_singleton x) isCompact_singleton\n    (convex_singleton y) isClosed_singleton (disjoint_singleton.2 hxy)", "state_before": "\ud835\udd5c : Type ?u.125223\nE : Type u_1\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : TopologicalAddGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : ContinuousSMul \u211d E\ns t : Set E\nx y : E\ninst\u271d\u00b9 : LocallyConvexSpace \u211d E\ninst\u271d : T1Space E\nhxy : x \u2260 y\n\u22a2 \u2203 f, \u2191f x < \u2191f y", "state_after": "case intro.intro.intro.intro.intro\n\ud835\udd5c : Type ?u.125223\nE : Type u_1\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : TopologicalAddGroup 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s\nst : s < t\nht : \u2200 (b : E), b \u2208 {y} \u2192 t < \u2191f b\n\u22a2 \u2203 f, \u2191f x < \u2191f y", "state_after": "no goals"}, {"tactic": "linarith [hs x rfl, ht y rfl]", "state_before": "\ud835\udd5c : Type ?u.125223\nE : Type u_1\ninst\u271d\u2076 : TopologicalSpace E\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : TopologicalAddGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : ContinuousSMul \u211d E\ns\u271d t\u271d : Set E\nx y : E\ninst\u271d\u00b9 : LocallyConvexSpace \u211d E\ninst\u271d : T1Space E\nhxy : x \u2260 y\nf : E \u2192L[\u211d] \u211d\ns t : \u211d\nhs : \u2200 (a : E), a \u2208 {x} \u2192 \u2191f a < s\nst : s < t\nht : \u2200 (b : E), b \u2208 {y} \u2192 t < \u2191f b\n\u22a2 \u2191f x < \u2191f y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/BilinearForm.lean", "full_name": "BilinForm.toMatrix_basisFun", "start": [340, 1], "end": [343, 97], "traced_tactics": [{"tactic": "ext B", "state_before": "R : Type ?u.1309864\nM : Type ?u.1309867\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : AddCommMonoid M\ninst\u271d\u00b9\u2075 : Module R M\nR\u2081 : Type ?u.1309903\nM\u2081 : Type ?u.1309906\ninst\u271d\u00b9\u2074 : Ring R\u2081\ninst\u271d\u00b9\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b2 : Module R\u2081 M\u2081\nR\u2082 : Type u_1\nM\u2082 : Type ?u.1310518\ninst\u271d\u00b9\u00b9 : CommSemiring R\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : Module R\u2082 M\u2082\nR\u2083 : Type ?u.1310705\nM\u2083 : Type ?u.1310708\ninst\u271d\u2078 : CommRing R\u2083\ninst\u271d\u2077 : AddCommGroup M\u2083\ninst\u271d\u2076 : Module R\u2083 M\u2083\nV : Type ?u.1311296\nK : Type ?u.1311299\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : 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\no : Type ?u.1312516\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq n\nb : Basis n R\u2082 M\u2082\n\u22a2 toMatrix (Pi.basisFun R\u2082 n) = toMatrix'", "state_after": "case h.a.h\nR : Type ?u.1309864\nM : Type ?u.1309867\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : AddCommMonoid M\ninst\u271d\u00b9\u2075 : Module R M\nR\u2081 : Type ?u.1309903\nM\u2081 : Type ?u.1309906\ninst\u271d\u00b9\u2074 : Ring R\u2081\ninst\u271d\u00b9\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b2 : Module R\u2081 M\u2081\nR\u2082 : Type u_1\nM\u2082 : Type ?u.1310518\ninst\u271d\u00b9\u00b9 : CommSemiring R\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : Module R\u2082 M\u2082\nR\u2083 : Type ?u.1310705\nM\u2083 : Type ?u.1310708\ninst\u271d\u2078 : CommRing R\u2083\ninst\u271d\u2077 : AddCommGroup M\u2083\ninst\u271d\u2076 : Module R\u2083 M\u2083\nV : Type ?u.1311296\nK : Type ?u.1311299\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_2\no : Type ?u.1312516\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq n\nb : Basis n R\u2082 M\u2082\nB : BilinForm R\u2082 (n \u2192 R\u2082)\ni\u271d x\u271d : n\n\u22a2 \u2191(toMatrix (Pi.basisFun R\u2082 n)) B i\u271d x\u271d = \u2191toMatrix' B i\u271d x\u271d"}, {"tactic": "rw [BilinForm.toMatrix_apply, BilinForm.toMatrix'_apply, Pi.basisFun_apply, Pi.basisFun_apply]", "state_before": "case h.a.h\nR : Type ?u.1309864\nM : Type ?u.1309867\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : AddCommMonoid M\ninst\u271d\u00b9\u2075 : Module R M\nR\u2081 : Type ?u.1309903\nM\u2081 : Type ?u.1309906\ninst\u271d\u00b9\u2074 : Ring R\u2081\ninst\u271d\u00b9\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b2 : Module R\u2081 M\u2081\nR\u2082 : Type u_1\nM\u2082 : Type ?u.1310518\ninst\u271d\u00b9\u00b9 : CommSemiring R\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : Module R\u2082 M\u2082\nR\u2083 : Type ?u.1310705\nM\u2083 : Type ?u.1310708\ninst\u271d\u2078 : CommRing R\u2083\ninst\u271d\u2077 : AddCommGroup M\u2083\ninst\u271d\u2076 : Module R\u2083 M\u2083\nV : Type ?u.1311296\nK : Type ?u.1311299\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_2\no : Type ?u.1312516\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq n\nb : Basis n R\u2082 M\u2082\nB : BilinForm R\u2082 (n \u2192 R\u2082)\ni\u271d x\u271d : n\n\u22a2 \u2191(toMatrix (Pi.basisFun R\u2082 n)) B i\u271d x\u271d = \u2191toMatrix' B i\u271d x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Unique.lean", "full_name": "Function.Surjective.subsingleton", "start": [232, 11], "end": [233, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Probability/Kernel/Basic.lean", "full_name": "ProbabilityTheory.kernel.coeFn_add", "start": [86, 1], "end": [87, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Int/Lemmas.lean", "full_name": "Int.lt_irrefl", "start": [621, 11], "end": [626, 42], "traced_tactics": [{"tactic": "rw [hn, Int.add_zero]", "state_before": "a : Int\nH : a < a\nn : Nat\nhn : a + \u2191(succ n) = a\n\u22a2 a + \u2191(succ n) = a + 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/DualQuaternion.lean", "full_name": "Quaternion.snd_imK_dualNumberEquiv_symm", "start": [155, 1], "end": [157, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Preimage.lean", "full_name": "Finset.monotone_preimage", "start": [82, 1], "end": [84, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Module/Injective.lean", "full_name": "Module.Baer.ExtensionOfMaxAdjoin.extensionToFun_wd", "start": [371, 1], "end": [392, 61], "traced_tactics": [{"tactic": "cases' a with a ha", "state_before": "R : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\na : { x // x \u2208 (extensionOfMax i f).toLinearPMap.domain }\nr : R\neq1 : \u2191x = \u2191a + r \u2022 y\n\u22a2 extensionToFun i f h x = \u2191(extensionOfMax i f).toLinearPMap a + \u2191(extendIdealTo i f h y) r", "state_after": "case mk\nR : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\n\u22a2 extensionToFun i f h x = \u2191(extensionOfMax i f).toLinearPMap { val := a, property := ha } + \u2191(extendIdealTo i f h y) r"}, {"tactic": "have eq2 :\n  (ExtensionOfMaxAdjoin.fst i x - a : N) = (r - ExtensionOfMaxAdjoin.snd i x) \u2022 y := by\n  change x = a + r \u2022 y at eq1\n  rwa [ExtensionOfMaxAdjoin.eqn, \u2190 sub_eq_zero, \u2190 sub_sub_sub_eq, sub_eq_zero, \u2190 sub_smul]\n    at eq1", "state_before": "case mk\nR : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\n\u22a2 extensionToFun i f h x = \u2191(extensionOfMax i f).toLinearPMap { val := a, property := ha } + \u2191(extendIdealTo i f h y) r", "state_after": "case mk\nR : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\neq2 : \u2191(fst i x) - a = (r - snd i x) \u2022 y\n\u22a2 extensionToFun i f h x = \u2191(extensionOfMax i f).toLinearPMap { val := a, property := ha } + \u2191(extendIdealTo i f h y) r"}, {"tactic": "have eq3 :=\n  ExtensionOfMaxAdjoin.extendIdealTo_eq i f h (r - ExtensionOfMaxAdjoin.snd i x)\n    (by rw [\u2190 eq2]; exact Submodule.sub_mem _ (ExtensionOfMaxAdjoin.fst i x).2 ha)", "state_before": "case mk\nR : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\neq2 : \u2191(fst i x) - a = (r - snd i x) \u2022 y\n\u22a2 extensionToFun i f h x = \u2191(extensionOfMax i f).toLinearPMap { val := a, property := ha } + \u2191(extendIdealTo i f h y) r", "state_after": "case mk\nR : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\neq2 : \u2191(fst i x) - a = (r - snd i x) \u2022 y\neq3 :\n  \u2191(extendIdealTo i f h y) (r - snd i x) =\n    \u2191(extensionOfMax i f).toLinearPMap\n      { val := (r - snd i x) \u2022 y, property := (_ : (r - snd i x) \u2022 y \u2208 (extensionOfMax i f).toLinearPMap.domain) }\n\u22a2 extensionToFun i f h x = \u2191(extensionOfMax i f).toLinearPMap { val := a, property := ha } + \u2191(extendIdealTo i f h y) r"}, {"tactic": "simp only [map_sub, sub_smul, sub_eq_iff_eq_add] at eq3", "state_before": "case mk\nR : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\neq2 : \u2191(fst i x) - a = (r - snd i x) \u2022 y\neq3 :\n  \u2191(extendIdealTo i f h y) (r - snd i x) =\n    \u2191(extensionOfMax i f).toLinearPMap\n      { val := (r - snd i x) \u2022 y, property := (_ : (r - snd i x) \u2022 y \u2208 (extensionOfMax i f).toLinearPMap.domain) }\n\u22a2 extensionToFun i f h x = \u2191(extensionOfMax i f).toLinearPMap { val := a, property := ha } + \u2191(extendIdealTo i f h y) r", "state_after": "case mk\nR : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\neq2 : \u2191(fst i x) - a = (r - snd i x) \u2022 y\neq3 :\n  \u2191(extendIdealTo i f h y) r =\n    \u2191(extensionOfMax i f).toLinearPMap\n        { val := r \u2022 y - snd i x \u2022 y,\n          property := (_ : (fun x => x \u2208 (extensionOfMax i f).toLinearPMap.domain) (r \u2022 y - snd i x \u2022 y)) } +\n      \u2191(extendIdealTo i f h y) (snd i x)\n\u22a2 extensionToFun i f h x = \u2191(extensionOfMax i f).toLinearPMap { val := a, property := ha } + \u2191(extendIdealTo i f h y) r"}, {"tactic": "unfold ExtensionOfMaxAdjoin.extensionToFun", "state_before": "case mk\nR : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\neq2 : \u2191(fst i x) - a = (r - snd i x) \u2022 y\neq3 :\n  \u2191(extendIdealTo i f h y) r =\n    \u2191(extensionOfMax i f).toLinearPMap\n        { val := r \u2022 y - snd i x \u2022 y,\n          property := (_ : (fun x => x \u2208 (extensionOfMax i f).toLinearPMap.domain) (r \u2022 y - snd i x \u2022 y)) } +\n      \u2191(extendIdealTo i f h y) (snd i x)\n\u22a2 extensionToFun i f h x = \u2191(extensionOfMax i f).toLinearPMap { val := a, property := ha } + \u2191(extendIdealTo i f h y) r", "state_after": "case mk\nR : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\neq2 : \u2191(fst i x) - a = (r - snd i x) \u2022 y\neq3 :\n  \u2191(extendIdealTo i f h y) r =\n    \u2191(extensionOfMax i f).toLinearPMap\n        { val := r \u2022 y - snd i x \u2022 y,\n          property := (_ : (fun x => x \u2208 (extensionOfMax i f).toLinearPMap.domain) (r \u2022 y - snd i x \u2022 y)) } +\n      \u2191(extendIdealTo i f h y) (snd i x)\n\u22a2 \u2191(extensionOfMax i f).toLinearPMap (fst i x) + \u2191(extendIdealTo i f h y) (snd i x) =\n    \u2191(extensionOfMax i f).toLinearPMap { val := a, property := ha } + \u2191(extendIdealTo i f h y) r"}, {"tactic": "rw [eq3, \u2190 add_assoc, \u2190 (extensionOfMax i f).toLinearPMap.map_add, AddMemClass.mk_add_mk]", "state_before": "case mk\nR : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\neq2 : \u2191(fst i x) - a = (r - snd i x) \u2022 y\neq3 :\n  \u2191(extendIdealTo i f h y) r =\n    \u2191(extensionOfMax i f).toLinearPMap\n        { val := r \u2022 y - snd i x \u2022 y,\n          property := (_ : (fun x => x \u2208 (extensionOfMax i f).toLinearPMap.domain) (r \u2022 y - snd i x \u2022 y)) } +\n      \u2191(extendIdealTo i f h y) (snd i x)\n\u22a2 \u2191(extensionOfMax i f).toLinearPMap (fst i x) + \u2191(extendIdealTo i f h y) (snd i x) =\n    \u2191(extensionOfMax i f).toLinearPMap { val := a, property := ha } + \u2191(extendIdealTo i f h y) r", "state_after": "case mk\nR : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\neq2 : \u2191(fst i x) - a = (r - snd i x) \u2022 y\neq3 :\n  \u2191(extendIdealTo i f h y) r =\n    \u2191(extensionOfMax i f).toLinearPMap\n        { val := r \u2022 y - snd i x \u2022 y,\n          property := (_ : (fun x => x \u2208 (extensionOfMax i f).toLinearPMap.domain) (r \u2022 y - snd i x \u2022 y)) } +\n      \u2191(extendIdealTo i f h y) (snd i x)\n\u22a2 \u2191(extensionOfMax i f).toLinearPMap (fst i x) + \u2191(extendIdealTo i f h y) (snd i x) =\n    \u2191(extensionOfMax i f).toLinearPMap\n        { val := a + (r \u2022 y - snd i x \u2022 y),\n          property := (_ : a + (r \u2022 y - snd i x \u2022 y) \u2208 (extensionOfMax i f).toLinearPMap.domain) } +\n      \u2191(extendIdealTo i f h y) (snd i x)"}, {"tactic": "congr", "state_before": "case mk\nR : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\neq2 : \u2191(fst i x) - a = (r - snd i x) \u2022 y\neq3 :\n  \u2191(extendIdealTo i f h y) r =\n    \u2191(extensionOfMax i f).toLinearPMap\n        { val := r \u2022 y - snd i x \u2022 y,\n          property := (_ : (fun x => x \u2208 (extensionOfMax i f).toLinearPMap.domain) (r \u2022 y - snd i x \u2022 y)) } +\n      \u2191(extendIdealTo i f h y) (snd i x)\n\u22a2 \u2191(extensionOfMax i f).toLinearPMap (fst i x) + \u2191(extendIdealTo i f h y) (snd i x) =\n    \u2191(extensionOfMax i f).toLinearPMap\n        { val := a + (r \u2022 y - snd i x \u2022 y),\n          property := (_ : a + (r \u2022 y - snd i x \u2022 y) \u2208 (extensionOfMax i f).toLinearPMap.domain) } +\n      \u2191(extendIdealTo i f h y) (snd i x)", "state_after": "case mk.e_a.e_a\nR : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\neq2 : \u2191(fst i x) - a = (r - snd i x) \u2022 y\neq3 :\n  \u2191(extendIdealTo i f h y) r =\n    \u2191(extensionOfMax i f).toLinearPMap\n        { val := r \u2022 y - snd i x \u2022 y,\n          property := (_ : (fun x => x \u2208 (extensionOfMax i f).toLinearPMap.domain) (r \u2022 y - snd i x \u2022 y)) } +\n      \u2191(extendIdealTo i f h y) (snd i x)\n\u22a2 fst i x =\n    { val := a + (r \u2022 y - snd i x \u2022 y),\n      property := (_ : a + (r \u2022 y - snd i x \u2022 y) \u2208 (extensionOfMax i f).toLinearPMap.domain) }"}, {"tactic": "ext", "state_before": "case mk.e_a.e_a\nR : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\neq2 : \u2191(fst i x) - a = (r - snd i x) \u2022 y\neq3 :\n  \u2191(extendIdealTo i f h y) r =\n    \u2191(extensionOfMax i f).toLinearPMap\n        { val := r \u2022 y - snd i x \u2022 y,\n          property := (_ : (fun x => x \u2208 (extensionOfMax i f).toLinearPMap.domain) (r \u2022 y - snd i x \u2022 y)) } +\n      \u2191(extendIdealTo i f h y) (snd i x)\n\u22a2 fst i x =\n    { val := a + (r \u2022 y - snd i x \u2022 y),\n      property := (_ : a + (r \u2022 y - snd i x \u2022 y) \u2208 (extensionOfMax i f).toLinearPMap.domain) }", "state_after": "case mk.e_a.e_a.a\nR : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\neq2 : \u2191(fst i x) - a = (r - snd i x) \u2022 y\neq3 :\n  \u2191(extendIdealTo i f h y) r =\n    \u2191(extensionOfMax i f).toLinearPMap\n        { val := r \u2022 y - snd i x \u2022 y,\n          property := (_ : (fun x => x \u2208 (extensionOfMax i f).toLinearPMap.domain) (r \u2022 y - snd i x \u2022 y)) } +\n      \u2191(extendIdealTo i f h y) (snd i x)\n\u22a2 \u2191(fst i x) =\n    \u2191{ val := a + (r \u2022 y - snd i x \u2022 y),\n        property := (_ : a + (r \u2022 y - snd i x \u2022 y) \u2208 (extensionOfMax i f).toLinearPMap.domain) }"}, {"tactic": "dsimp", "state_before": "case mk.e_a.e_a.a\nR : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\neq2 : \u2191(fst i x) - a = (r - snd i x) \u2022 y\neq3 :\n  \u2191(extendIdealTo i f h y) r =\n    \u2191(extensionOfMax i f).toLinearPMap\n        { val := r \u2022 y - snd i x \u2022 y,\n          property := (_ : (fun x => x \u2208 (extensionOfMax i f).toLinearPMap.domain) (r \u2022 y - snd i x \u2022 y)) } +\n      \u2191(extendIdealTo i f h y) (snd i x)\n\u22a2 \u2191(fst i x) =\n    \u2191{ val := a + (r \u2022 y - snd i x \u2022 y),\n        property := (_ : a + (r \u2022 y - snd i x \u2022 y) \u2208 (extensionOfMax i f).toLinearPMap.domain) }", "state_after": "case mk.e_a.e_a.a\nR : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\neq2 : \u2191(fst i x) - a = (r - snd i x) \u2022 y\neq3 :\n  \u2191(extendIdealTo i f h y) r =\n    \u2191(extensionOfMax i f).toLinearPMap\n        { val := r \u2022 y - snd i x \u2022 y,\n          property := (_ : (fun x => x \u2208 (extensionOfMax i f).toLinearPMap.domain) (r \u2022 y - snd i x \u2022 y)) } +\n      \u2191(extendIdealTo i f h y) (snd i x)\n\u22a2 \u2191(fst i x) = a + (r \u2022 y - snd i x \u2022 y)"}, {"tactic": "rw [Subtype.coe_mk, add_sub, \u2190 eq1]", "state_before": "case mk.e_a.e_a.a\nR : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\neq2 : \u2191(fst i x) - a = (r - snd i x) \u2022 y\neq3 :\n  \u2191(extendIdealTo i f h y) r =\n    \u2191(extensionOfMax i f).toLinearPMap\n        { val := r \u2022 y - snd i x \u2022 y,\n          property := (_ : (fun x => x \u2208 (extensionOfMax i f).toLinearPMap.domain) (r \u2022 y - snd i x \u2022 y)) } +\n      \u2191(extendIdealTo i f h y) (snd i x)\n\u22a2 \u2191(fst i x) = a + (r \u2022 y - snd i x \u2022 y)", "state_after": "case mk.e_a.e_a.a\nR : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\neq2 : \u2191(fst i x) - a = (r - snd i x) \u2022 y\neq3 :\n  \u2191(extendIdealTo i f h y) r =\n    \u2191(extensionOfMax i f).toLinearPMap\n        { val := r \u2022 y - snd i x \u2022 y,\n          property := (_ : (fun x => x \u2208 (extensionOfMax i f).toLinearPMap.domain) (r \u2022 y - snd i x \u2022 y)) } +\n      \u2191(extendIdealTo i f h y) (snd i x)\n\u22a2 \u2191(fst i x) = \u2191x - snd i x \u2022 y"}, {"tactic": "exact eq_sub_of_add_eq (ExtensionOfMaxAdjoin.eqn i x).symm", "state_before": "case mk.e_a.e_a.a\nR : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\neq2 : \u2191(fst i x) - a = (r - snd i x) \u2022 y\neq3 :\n  \u2191(extendIdealTo i f h y) r =\n    \u2191(extensionOfMax i f).toLinearPMap\n        { val := r \u2022 y - snd i x \u2022 y,\n          property := (_ : (fun x => x \u2208 (extensionOfMax i f).toLinearPMap.domain) (r \u2022 y - snd i x \u2022 y)) } +\n      \u2191(extendIdealTo i f h y) (snd i x)\n\u22a2 \u2191(fst i x) = \u2191x - snd i x \u2022 y", "state_after": "no goals"}, {"tactic": "change x = a + r \u2022 y at eq1", "state_before": "R : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\n\u22a2 \u2191(fst i x) - a = (r - snd i x) \u2022 y", "state_after": "R : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = a + r \u2022 y\n\u22a2 \u2191(fst i x) - a = (r - snd i x) \u2022 y"}, {"tactic": "rwa [ExtensionOfMaxAdjoin.eqn, \u2190 sub_eq_zero, \u2190 sub_sub_sub_eq, sub_eq_zero, \u2190 sub_smul]\n  at eq1", "state_before": "R : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = a + r \u2022 y\n\u22a2 \u2191(fst i x) - a = (r - snd i x) \u2022 y", "state_after": "no goals"}, {"tactic": "rw [\u2190 eq2]", "state_before": "R : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\neq2 : \u2191(fst i x) - a = (r - snd i x) \u2022 y\n\u22a2 (r - snd i x) \u2022 ?m.219080 \u2208 (extensionOfMax i f).toLinearPMap.domain", "state_after": "R : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\neq2 : \u2191(fst i x) - a = (r - snd i x) \u2022 y\n\u22a2 \u2191(fst i x) - a \u2208 (extensionOfMax i f).toLinearPMap.domain"}, {"tactic": "exact Submodule.sub_mem _ (ExtensionOfMaxAdjoin.fst i x).2 ha", "state_before": "R : Type u\ninst\u271d\u2077 : Ring R\nQ : TypeMax\ninst\u271d\u2076 : AddCommGroup Q\ninst\u271d\u2075 : Module R Q\nM N : Type (max u v)\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R N\ni : M \u2192\u2097[R] N\nf : M \u2192\u2097[R] Q\ninst\u271d : Fact (Function.Injective \u2191i)\nh : Baer R Q\ny : N\nx : { x // x \u2208 supExtensionOfMaxSingleton i f y }\nr : R\na : N\nha : a \u2208 (extensionOfMax i f).toLinearPMap.domain\neq1 : \u2191x = \u2191{ val := a, property := ha } + r \u2022 y\neq2 : \u2191(fst i x) - a = (r - snd i x) \u2022 y\n\u22a2 \u2191(fst i x) - a \u2208 (extensionOfMax i f).toLinearPMap.domain", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/Equicontinuity.lean", "full_name": "Filter.Tendsto.continuousAt_of_equicontinuousAt", "start": [397, 1], "end": [401, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Nat/Gcd.lean", "full_name": "Nat.coprime_iff_gcd_eq_one", "start": [240, 1], "end": [240, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.eval\u2082_zero_apply", "start": [1500, 1], "end": [1502, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Order/LiminfLimsup.lean", "full_name": "Monotone.map_limsup_of_continuousAt", "start": [392, 1], "end": [394, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "full_name": "MeasureTheory.tendsto_Lp_of_tendsto_ae_of_meas", "start": [487, 1], "end": [553, 37], "traced_tactics": [{"tactic": "rw [ENNReal.tendsto_atTop_zero]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 Tendsto (fun n => snorm (f n - g) p \u03bc) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "intro \u03b5 h\u03b5", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 \u2200 (\u03b5 : \u211d\u22650\u221e), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "by_cases \u03b5 < \u221e", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u00ac\u03b5 < \u22a4\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "swap", "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u00ac\u03b5 < \u22a4\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u00ac\u03b5 < \u22a4\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "by_cases h\u03bc : \u03bc = 0", "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u03bc = 0\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "have h\u03b5' : 0 < \u03b5.toReal / 3 :=\n  div_pos (ENNReal.toReal_pos (gt_iff_lt.1 h\u03b5).ne.symm h.ne) (by norm_num)", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "have hdivp : 0 \u2264 1 / p.toReal := by\n  refine' one_div_nonneg.2 _\n  rw [\u2190 ENNReal.zero_toReal, ENNReal.toReal_le_toReal ENNReal.zero_ne_top hp']\n  exact le_trans (zero_le _) hp", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "have hpow : 0 < measureUnivNNReal \u03bc ^ (1 / p.toReal) :=\n  Real.rpow_pos_of_pos (measureUnivNNReal_pos h\u03bc) _", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "obtain \u27e8\u03b4\u2081, h\u03b4\u2081, hsnorm\u2081\u27e9 := hui h\u03b5'", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "obtain \u27e8\u03b4\u2082, h\u03b4\u2082, hsnorm\u2082\u27e9 := hg'.snorm_indicator_le \u03bc hp hp' h\u03b5'", "state_before": "case neg.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "obtain \u27e8t, htm, ht\u2081, ht\u2082\u27e9 := tendstoUniformlyOn_of_ae_tendsto' hf hg hfg (lt_min h\u03b4\u2081 h\u03b4\u2082)", "state_before": "case neg.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nht\u2082 : TendstoUniformlyOn (fun n => f n) g atTop (t\u1d9c)\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "rw [Metric.tendstoUniformlyOn_iff] at ht\u2082", "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nht\u2082 : TendstoUniformlyOn (fun n => f n) g atTop (t\u1d9c)\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nht\u2082 : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < \u03b5\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "specialize ht\u2082 (\u03b5.toReal / (3 * measureUnivNNReal \u03bc ^ (1 / p.toReal)))\n  (div_pos (ENNReal.toReal_pos (gt_iff_lt.1 h\u03b5).ne.symm h.ne) (mul_pos (by norm_num) hpow))", "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nht\u2082 : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < \u03b5\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nht\u2082 :\n  \u2200\u1da0 (n : \u2115) in atTop,\n    \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "obtain \u27e8N, hN\u27e9 := eventually_atTop.1 ht\u2082", "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nht\u2082 :\n  \u2200\u1da0 (n : \u2115) in atTop,\n    \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nht\u2082 :\n  \u2200\u1da0 (n : \u2115) in atTop,\n    \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "clear ht\u2082", "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nht\u2082 :\n  \u2200\u1da0 (n : \u2115) in atTop,\n    \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "refine' \u27e8N, fun n hn => _\u27e9", "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "rw [\u2190 t.indicator_self_add_compl (f n - g)]", "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 snorm (Set.indicator t (f n - g) + Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264 \u03b5"}, {"tactic": "refine' le_trans (snorm_add_le (((hf n).sub hg).indicator htm).aestronglyMeasurable\n  (((hf n).sub hg).indicator htm.compl).aestronglyMeasurable hp) _", "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 snorm (Set.indicator t (f n - g) + Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 snorm (Set.indicator t (f n - g)) p \u03bc + snorm (Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264 \u03b5"}, {"tactic": "rw [sub_eq_add_neg, Set.indicator_add' t, Set.indicator_neg']", "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 snorm (Set.indicator t (f n - g)) p \u03bc + snorm (Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 snorm (Set.indicator t (f n) + -Set.indicator t g) p \u03bc + snorm (Set.indicator (t\u1d9c) (f n + -g)) p \u03bc \u2264 \u03b5"}, {"tactic": "refine' le_trans (add_le_add_right (snorm_add_le ((hf n).indicator htm).aestronglyMeasurable\n  (hg.indicator htm).neg.aestronglyMeasurable hp) _) _", "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 snorm (Set.indicator t (f n) + -Set.indicator t g) p \u03bc + snorm (Set.indicator (t\u1d9c) (f n + -g)) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 snorm (Set.indicator t (f n)) p \u03bc + snorm (-Set.indicator t g) p \u03bc + snorm (Set.indicator (t\u1d9c) (f n + -g)) p \u03bc \u2264 \u03b5"}, {"tactic": "have hnf : snorm (t.indicator (f n)) p \u03bc \u2264 ENNReal.ofReal (\u03b5.toReal / 3) := by\n  refine' hsnorm\u2081 n t htm (le_trans ht\u2081 _)\n  rw [ENNReal.ofReal_le_ofReal_iff h\u03b4\u2081.le]\n  exact min_le_left _ _", "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 snorm (Set.indicator t (f n)) p \u03bc + snorm (-Set.indicator t g) p \u03bc + snorm (Set.indicator (t\u1d9c) (f n + -g)) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 snorm (Set.indicator t (f n)) p \u03bc + snorm (-Set.indicator t g) p \u03bc + snorm (Set.indicator (t\u1d9c) (f n + -g)) p \u03bc \u2264 \u03b5"}, {"tactic": "have hng : snorm (t.indicator g) p \u03bc \u2264 ENNReal.ofReal (\u03b5.toReal / 3) := by\n  refine' hsnorm\u2082 t htm (le_trans ht\u2081 _)\n  rw [ENNReal.ofReal_le_ofReal_iff h\u03b4\u2082.le]\n  exact min_le_right _ _", "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 snorm (Set.indicator t (f n)) p \u03bc + snorm (-Set.indicator t g) p \u03bc + snorm (Set.indicator (t\u1d9c) (f n + -g)) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 snorm (Set.indicator t (f n)) p \u03bc + snorm (-Set.indicator t g) p \u03bc + snorm (Set.indicator (t\u1d9c) (f n + -g)) p \u03bc \u2264 \u03b5"}, {"tactic": "have : ENNReal.ofReal (\u03b5.toReal / 3) = \u03b5 / 3 := by\n  rw [ENNReal.ofReal_div_of_pos (show (0 : \u211d) < 3 by norm_num), ENNReal.ofReal_toReal h.ne]\n  simp", "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhlt : snorm (Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 snorm (Set.indicator t (f n)) p \u03bc + snorm (-Set.indicator t g) p \u03bc + snorm (Set.indicator (t\u1d9c) (f n + -g)) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhlt : snorm (Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nthis : ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) = \u03b5 / 3\n\u22a2 snorm (Set.indicator t (f n)) p \u03bc + snorm (-Set.indicator t g) p \u03bc + snorm (Set.indicator (t\u1d9c) (f n + -g)) p \u03bc \u2264 \u03b5"}, {"tactic": "rw [this] at hnf hng hlt", "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhlt : snorm (Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nthis : ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) = \u03b5 / 3\n\u22a2 snorm (Set.indicator t (f n)) p \u03bc + snorm (-Set.indicator t g) p \u03bc + snorm (Set.indicator (t\u1d9c) (f n + -g)) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 \u03b5 / 3\nhng : snorm (Set.indicator t g) p \u03bc \u2264 \u03b5 / 3\nhlt : snorm (Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264 \u03b5 / 3\nthis : ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) = \u03b5 / 3\n\u22a2 snorm (Set.indicator t (f n)) p \u03bc + snorm (-Set.indicator t g) p \u03bc + snorm (Set.indicator (t\u1d9c) (f n + -g)) p \u03bc \u2264 \u03b5"}, {"tactic": "rw [snorm_neg, \u2190 ENNReal.add_thirds \u03b5, \u2190 sub_eq_add_neg]", "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 \u03b5 / 3\nhng : snorm (Set.indicator t g) p \u03bc \u2264 \u03b5 / 3\nhlt : snorm (Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264 \u03b5 / 3\nthis : ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) = \u03b5 / 3\n\u22a2 snorm (Set.indicator t (f n)) p \u03bc + snorm (-Set.indicator t g) p \u03bc + snorm (Set.indicator (t\u1d9c) (f n + -g)) p \u03bc \u2264 \u03b5", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 \u03b5 / 3\nhng : snorm (Set.indicator t g) p \u03bc \u2264 \u03b5 / 3\nhlt : snorm (Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264 \u03b5 / 3\nthis : ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) = \u03b5 / 3\n\u22a2 snorm (Set.indicator t (f n)) p \u03bc + snorm (Set.indicator t g) p \u03bc + snorm (Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264\n    \u03b5 / 3 + \u03b5 / 3 + \u03b5 / 3"}, {"tactic": "exact add_le_add_three hnf hng hlt", "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 \u03b5 / 3\nhng : snorm (Set.indicator t g) p \u03bc \u2264 \u03b5 / 3\nhlt : snorm (Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264 \u03b5 / 3\nthis : ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) = \u03b5 / 3\n\u22a2 snorm (Set.indicator t (f n)) p \u03bc + snorm (Set.indicator t g) p \u03bc + snorm (Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264\n    \u03b5 / 3 + \u03b5 / 3 + \u03b5 / 3", "state_after": "no goals"}, {"tactic": "rw [not_lt, top_le_iff] at h", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u00ac\u03b5 < \u22a4\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 = \u22a4\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5"}, {"tactic": "exact \u27e80, fun n _ => by simp [h]\u27e9", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 = \u22a4\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "no goals"}, {"tactic": "simp [h]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 = \u22a4\nn : \u2115\nx\u271d : n \u2265 0\n\u22a2 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "no goals"}, {"tactic": "exact \u27e80, fun n _ => by simp [h\u03bc]\u27e9", "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u03bc = 0\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "no goals"}, {"tactic": "simp [h\u03bc]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u03bc = 0\nn : \u2115\nx\u271d : n \u2265 0\n\u22a2 snorm (f n - g) p \u03bc \u2264 \u03b5", "state_after": "no goals"}, {"tactic": "norm_num", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\n\u22a2 0 < 3", "state_after": "no goals"}, {"tactic": "refine' one_div_nonneg.2 _", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\n\u22a2 0 \u2264 1 / ENNReal.toReal p", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\n\u22a2 0 \u2264 ENNReal.toReal p"}, {"tactic": "rw [\u2190 ENNReal.zero_toReal, ENNReal.toReal_le_toReal ENNReal.zero_ne_top hp']", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\n\u22a2 0 \u2264 ENNReal.toReal p", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\n\u22a2 0 \u2264 p"}, {"tactic": "exact le_trans (zero_le _) hp", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\n\u22a2 0 \u2264 p", "state_after": "no goals"}, {"tactic": "norm_num", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nht\u2082 : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < \u03b5\n\u22a2 0 < 3", "state_after": "no goals"}, {"tactic": "refine' hsnorm\u2081 n t htm (le_trans ht\u2081 _)", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) \u2264 ENNReal.ofReal \u03b4\u2081"}, {"tactic": "rw [ENNReal.ofReal_le_ofReal_iff h\u03b4\u2081.le]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) \u2264 ENNReal.ofReal \u03b4\u2081", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 min \u03b4\u2081 \u03b4\u2082 \u2264 \u03b4\u2081"}, {"tactic": "exact min_le_left _ _", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\n\u22a2 min \u03b4\u2081 \u03b4\u2082 \u2264 \u03b4\u2081", "state_after": "no goals"}, {"tactic": "refine' hsnorm\u2082 t htm (le_trans ht\u2081 _)", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) \u2264 ENNReal.ofReal \u03b4\u2082"}, {"tactic": "rw [ENNReal.ofReal_le_ofReal_iff h\u03b4\u2082.le]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082) \u2264 ENNReal.ofReal \u03b4\u2082", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 min \u03b4\u2081 \u03b4\u2082 \u2264 \u03b4\u2082"}, {"tactic": "exact min_le_right _ _", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 min \u03b4\u2081 \u03b4\u2082 \u2264 \u03b4\u2082", "state_after": "no goals"}, {"tactic": "specialize hN n hn", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 snorm (Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 snorm (Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)"}, {"tactic": "have : 0 \u2264 \u03b5.toReal / (3 * measureUnivNNReal \u03bc ^ (1 / p.toReal)) := by\n  rw [div_mul_eq_div_mul_one_div]\n  exact mul_nonneg h\u03b5'.le (one_div_nonneg.2 hpow.le)", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 snorm (Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 snorm (Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)"}, {"tactic": "have := snorm_sub_le_of_dist_bdd \u03bc hp' htm.compl this fun x hx =>\n  (dist_comm (g x) (f n x) \u25b8 (hN x hx).le :\n    dist (f n x) (g x) \u2264 \u03b5.toReal / (3 * measureUnivNNReal \u03bc ^ (1 / p.toReal)))", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 snorm (Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (Set.indicator (t\u1d9c) ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc (t\u1d9c) ^ (1 / ENNReal.toReal p)\n\u22a2 snorm (Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)"}, {"tactic": "refine' le_trans this _", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (Set.indicator (t\u1d9c) ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc (t\u1d9c) ^ (1 / ENNReal.toReal p)\n\u22a2 snorm (Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (Set.indicator (t\u1d9c) ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc (t\u1d9c) ^ (1 / ENNReal.toReal p)\n\u22a2 ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc (t\u1d9c) ^ (1 / ENNReal.toReal p) \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)"}, {"tactic": "rw [div_mul_eq_div_mul_one_div, \u2190 ENNReal.ofReal_toReal (measure_lt_top \u03bc (t\u1d9c)).ne,\n  ENNReal.ofReal_rpow_of_nonneg ENNReal.toReal_nonneg hdivp, \u2190 ENNReal.ofReal_mul, mul_assoc]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (Set.indicator (t\u1d9c) ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc (t\u1d9c) ^ (1 / ENNReal.toReal p)\n\u22a2 ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc (t\u1d9c) ^ (1 / ENNReal.toReal p) \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (Set.indicator (t\u1d9c) ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc (t\u1d9c) ^ (1 / ENNReal.toReal p)\n\u22a2 ENNReal.ofReal\n      (ENNReal.toReal \u03b5 / 3 *\n        (1 / \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) * ENNReal.toReal (\u2191\u2191\u03bc (t\u1d9c)) ^ (1 / ENNReal.toReal p))) \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (Set.indicator (t\u1d9c) ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc (t\u1d9c) ^ (1 / ENNReal.toReal p)\n\u22a2 0 \u2264 ENNReal.toReal \u03b5 / 3 * (1 / \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))"}, {"tactic": "rw [div_mul_eq_div_mul_one_div]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 0 \u2264 ENNReal.toReal \u03b5 / 3 * (1 / \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))"}, {"tactic": "exact mul_nonneg h\u03b5'.le (one_div_nonneg.2 hpow.le)", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\n\u22a2 0 \u2264 ENNReal.toReal \u03b5 / 3 * (1 / \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))", "state_after": "no goals"}, {"tactic": "refine' ENNReal.ofReal_le_ofReal (mul_le_of_le_one_right h\u03b5'.le _)", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (Set.indicator (t\u1d9c) ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc (t\u1d9c) ^ (1 / ENNReal.toReal p)\n\u22a2 ENNReal.ofReal\n      (ENNReal.toReal \u03b5 / 3 *\n        (1 / \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) * ENNReal.toReal (\u2191\u2191\u03bc (t\u1d9c)) ^ (1 / ENNReal.toReal p))) \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (Set.indicator (t\u1d9c) ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc (t\u1d9c) ^ (1 / ENNReal.toReal p)\n\u22a2 1 / \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) * ENNReal.toReal (\u2191\u2191\u03bc (t\u1d9c)) ^ (1 / ENNReal.toReal p) \u2264 1"}, {"tactic": "rw [mul_comm, mul_one_div, div_le_one]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (Set.indicator (t\u1d9c) ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc (t\u1d9c) ^ (1 / ENNReal.toReal p)\n\u22a2 1 / \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p) * ENNReal.toReal (\u2191\u2191\u03bc (t\u1d9c)) ^ (1 / ENNReal.toReal p) \u2264 1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (Set.indicator (t\u1d9c) ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc (t\u1d9c) ^ (1 / ENNReal.toReal p)\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc (t\u1d9c)) ^ (1 / ENNReal.toReal p) \u2264 \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (Set.indicator (t\u1d9c) ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc (t\u1d9c) ^ (1 / ENNReal.toReal p)\n\u22a2 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)"}, {"tactic": "refine' Real.rpow_le_rpow ENNReal.toReal_nonneg\n  (ENNReal.toReal_le_of_le_ofReal (measureUnivNNReal_pos h\u03bc).le _) hdivp", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (Set.indicator (t\u1d9c) ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc (t\u1d9c) ^ (1 / ENNReal.toReal p)\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc (t\u1d9c)) ^ (1 / ENNReal.toReal p) \u2264 \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (Set.indicator (t\u1d9c) ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc (t\u1d9c) ^ (1 / ENNReal.toReal p)\n\u22a2 \u2191\u2191\u03bc (t\u1d9c) \u2264 ENNReal.ofReal \u2191(measureUnivNNReal \u03bc)"}, {"tactic": "rw [ENNReal.ofReal_coe_nnreal, coe_measureUnivNNReal]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (Set.indicator (t\u1d9c) ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc (t\u1d9c) ^ (1 / ENNReal.toReal p)\n\u22a2 \u2191\u2191\u03bc (t\u1d9c) \u2264 ENNReal.ofReal \u2191(measureUnivNNReal \u03bc)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (Set.indicator (t\u1d9c) ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc (t\u1d9c) ^ (1 / ENNReal.toReal p)\n\u22a2 \u2191\u2191\u03bc (t\u1d9c) \u2264 \u2191\u2191\u03bc Set.univ"}, {"tactic": "exact measure_mono (Set.subset_univ _)", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (Set.indicator (t\u1d9c) ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc (t\u1d9c) ^ (1 / ENNReal.toReal p)\n\u22a2 \u2191\u2191\u03bc (t\u1d9c) \u2264 \u2191\u2191\u03bc Set.univ", "state_after": "no goals"}, {"tactic": "exact Real.rpow_pos_of_pos (measureUnivNNReal_pos h\u03bc) _", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (Set.indicator (t\u1d9c) ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc (t\u1d9c) ^ (1 / ENNReal.toReal p)\n\u22a2 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)", "state_after": "no goals"}, {"tactic": "refine' mul_nonneg h\u03b5'.le (one_div_nonneg.2 hpow.le)", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN n : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhN : \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f n x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis\u271d : 0 \u2264 ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nthis :\n  snorm (Set.indicator (t\u1d9c) ((fun x => f n x) - fun x => g x)) p \u03bc \u2264\n    ENNReal.ofReal (ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))) *\n      \u2191\u2191\u03bc (t\u1d9c) ^ (1 / ENNReal.toReal p)\n\u22a2 0 \u2264 ENNReal.toReal \u03b5 / 3 * (1 / \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))", "state_after": "no goals"}, {"tactic": "rw [ENNReal.ofReal_div_of_pos (show (0 : \u211d) < 3 by norm_num), ENNReal.ofReal_toReal h.ne]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhlt : snorm (Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3) = \u03b5 / 3", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhlt : snorm (Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 \u03b5 / ENNReal.ofReal 3 = \u03b5 / 3"}, {"tactic": "simp", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhlt : snorm (Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 \u03b5 / ENNReal.ofReal 3 = \u03b5 / 3", "state_after": "no goals"}, {"tactic": "norm_num", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.432813\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\ninst\u271d : IsFiniteMeasure \u03bc\nhp : 1 \u2264 p\nhp' : p \u2260 \u22a4\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : Mem\u2112p g p\nhui : UnifIntegrable f p \u03bc\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nh : \u03b5 < \u22a4\nh\u03bc : \u00ac\u03bc = 0\nh\u03b5' : 0 < ENNReal.toReal \u03b5 / 3\nhdivp : 0 \u2264 1 / ENNReal.toReal p\nhpow : 0 < \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p)\n\u03b4\u2081 : \u211d\nh\u03b4\u2081 : 0 < \u03b4\u2081\nhsnorm\u2081 :\n  \u2200 (i : \u2115) (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2081 \u2192 snorm (Set.indicator s (f i)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u03b4\u2082 : \u211d\nh\u03b4\u2082 : 0 < \u03b4\u2082\nhsnorm\u2082 :\n  \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\u2082 \u2192 snorm (Set.indicator s g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nt : Set \u03b1\nhtm : MeasurableSet t\nht\u2081 : \u2191\u2191\u03bc t \u2264 ENNReal.ofReal (min \u03b4\u2081 \u03b4\u2082)\nN : \u2115\nhN :\n  \u2200 (b : \u2115),\n    b \u2265 N \u2192\n      \u2200 (x : \u03b1), x \u2208 t\u1d9c \u2192 dist (g x) (f b x) < ENNReal.toReal \u03b5 / (3 * \u2191(measureUnivNNReal \u03bc) ^ (1 / ENNReal.toReal p))\nn : \u2115\nhn : n \u2265 N\nhnf : snorm (Set.indicator t (f n)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhng : snorm (Set.indicator t g) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\nhlt : snorm (Set.indicator (t\u1d9c) (f n - g)) p \u03bc \u2264 ENNReal.ofReal (ENNReal.toReal \u03b5 / 3)\n\u22a2 0 < 3", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Perm/Basic.lean", "full_name": "Equiv.swap_mul_eq_mul_swap", "start": [522, 1], "end": [526, 90], "traced_tactics": [{"tactic": "simp only [Perm.mul_apply, swap_apply_def]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y z : \u03b1\n\u22a2 \u2191(swap x y * f) z = \u2191(f * swap (\u2191f\u207b\u00b9 x) (\u2191f\u207b\u00b9 y)) z", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y z : \u03b1\n\u22a2 (if \u2191f z = x then y else if \u2191f z = y then x else \u2191f z) =\n    \u2191f (if z = \u2191f\u207b\u00b9 x then \u2191f\u207b\u00b9 y else if z = \u2191f\u207b\u00b9 y then \u2191f\u207b\u00b9 x else z)"}, {"tactic": "split_ifs <;>\n  simp_all only [Perm.apply_inv_self, Perm.eq_inv_iff_eq, eq_self_iff_true, not_true]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y z : \u03b1\n\u22a2 (if \u2191f z = x then y else if \u2191f z = y then x else \u2191f z) =\n    \u2191f (if z = \u2191f\u207b\u00b9 x then \u2191f\u207b\u00b9 y else if z = \u2191f\u207b\u00b9 y then \u2191f\u207b\u00b9 x else z)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "full_name": "AEMeasurable.exists_measurable_nonneg", "start": [221, 1], "end": [224, 67], "traced_tactics": [{"tactic": "obtain \u27e8G, hG_meas, hG_mem, hG_ae_eq\u27e9 := hf.exists_ae_eq_range_subset f_nn \u27e80, le_rfl\u27e9", "state_before": "\u03b9 : Type ?u.2730322\n\u03b1 : Type u_2\n\u03b2\u271d : Type ?u.2730328\n\u03b3 : Type ?u.2730331\n\u03b4 : Type ?u.2730334\nR : Type ?u.2730337\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\n\u03bc \u03bd : MeasureTheory.Measure \u03b1\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Zero \u03b2\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\nf_nn : \u2200\u1d50 (t : \u03b1) \u2202\u03bc, 0 \u2264 f t\n\u22a2 \u2203 g, Measurable g \u2227 0 \u2264 g \u2227 f =\u1da0[ae \u03bc] g", "state_after": "case intro.intro.intro\n\u03b9 : Type ?u.2730322\n\u03b1 : Type u_2\n\u03b2\u271d : Type ?u.2730328\n\u03b3 : Type ?u.2730331\n\u03b4 : Type ?u.2730334\nR : Type ?u.2730337\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\n\u03bc \u03bd : MeasureTheory.Measure \u03b1\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Zero \u03b2\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\nf_nn : \u2200\u1d50 (t : \u03b1) \u2202\u03bc, 0 \u2264 f t\nG : \u03b1 \u2192 \u03b2\nhG_meas : Measurable G\nhG_mem : range G \u2286 Preorder.toLE.1 0\nhG_ae_eq : f =\u1da0[ae \u03bc] G\n\u22a2 \u2203 g, Measurable g \u2227 0 \u2264 g \u2227 f =\u1da0[ae \u03bc] g"}, {"tactic": "exact \u27e8G, hG_meas, fun x => hG_mem (mem_range_self x), hG_ae_eq\u27e9", "state_before": "case intro.intro.intro\n\u03b9 : Type ?u.2730322\n\u03b1 : Type u_2\n\u03b2\u271d : Type ?u.2730328\n\u03b3 : Type ?u.2730331\n\u03b4 : Type ?u.2730334\nR : Type ?u.2730337\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : MeasurableSpace \u03b4\nf\u271d g : \u03b1 \u2192 \u03b2\u271d\n\u03bc \u03bd : MeasureTheory.Measure \u03b1\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Zero \u03b2\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f\nf_nn : \u2200\u1d50 (t : \u03b1) \u2202\u03bc, 0 \u2264 f t\nG : \u03b1 \u2192 \u03b2\nhG_meas : Measurable G\nhG_mem : range G \u2286 Preorder.toLE.1 0\nhG_ae_eq : f =\u1da0[ae \u03bc] G\n\u22a2 \u2203 g, Measurable g \u2227 0 \u2264 g \u2227 f =\u1da0[ae \u03bc] g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean", "full_name": "Polynomial.exists_partition_polynomial_aux", "start": [183, 1], "end": [249, 16], "traced_tactics": [{"tactic": "have hb\u03b5 : 0 < cardPowDegree b \u2022 \u03b5 := by\n  rw [Algebra.smul_def, eq_intCast]\n  exact mul_pos (Int.cast_pos.mpr (AbsoluteValue.pos _ hb)) h\u03b5", "state_before": "Fq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA : Fin n \u2192 Fq[X]\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "Fq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA : Fin n \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5"}, {"tactic": "induction' n with n ih", "state_before": "Fq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA : Fin n \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "case zero\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin Nat.zero \u2192 Fq[X]\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : Fin Nat.zero), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\ncase succ\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : Fin (Nat.succ n)), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5"}, {"tactic": "have anti_archim' : \u2200 {i j k} {\u03b5 : \u211d},\n  (cardPowDegree (A i % b - A j % b) : \u211d) < \u03b5 \u2192\n    (cardPowDegree (A j % b - A k % b) : \u211d) < \u03b5 \u2192\n      (cardPowDegree (A i % b - A k % b) : \u211d) < \u03b5 := by\n  intro i j k \u03b5\n  simp_rw [\u2190 Int.lt_ceil]\n  exact cardPowDegree_anti_archimedean", "state_before": "case succ\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : Fin (Nat.succ n)), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "case succ\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : Fin (Nat.succ n)), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5"}, {"tactic": "obtain \u27e8t', ht'\u27e9 := ih (Fin.tail A)", "state_before": "case succ\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : Fin (Nat.succ n)), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "case succ.intro\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : Fin (Nat.succ n)), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5"}, {"tactic": "rsuffices \u27e8j, hj\u27e9 :\n  \u2203 j, \u2200 i, t' i = j \u2194 (cardPowDegree (A 0 % b - A i.succ % b) : \u211d) < cardPowDegree b \u2022 \u03b5", "state_before": "case succ.intro\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : Fin (Nat.succ n)), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "case succ.intro.intro\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : Fin (Nat.succ n)), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 \u2203 j, \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5"}, {"tactic": "by_cases exists_nonempty_j : \u2203 j, (\u2203 i, t' i = j) \u2227\n    \u2200 i, t' i = j \u2192 (cardPowDegree (A 0 % b - A i.succ % b) : \u211d) < cardPowDegree b \u2022 \u03b5", "state_before": "case intro\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 \u2203 j, \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "case pos\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\nexists_nonempty_j :\n  \u2203 j,\n    (\u2203 i, t' i = j) \u2227 \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 \u2203 j, \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\ncase neg\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\nexists_nonempty_j :\n  \u00ac\u2203 j,\n      (\u2203 i, t' i = j) \u2227\n        \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 \u2203 j, \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5"}, {"tactic": "refine' \u27e8j, fun i => \u27e8hj i, fun hi => _\u27e9\u27e9", "state_before": "case neg\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\nexists_nonempty_j :\n  \u00ac\u2203 j,\n      (\u2203 i, t' i = j) \u2227\n        \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 \u2203 j, \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "case neg\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\nexists_nonempty_j :\n  \u00ac\u2203 j,\n      (\u2203 i, t' i = j) \u2227\n        \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni : Fin n\nhi : \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 t' i = j"}, {"tactic": "have := exists_nonempty_j \u27e8t' i, \u27e8i, rfl\u27e9, fun i' hi' => anti_archim' hi ((ht' _ _).mp hi')\u27e9", "state_before": "case neg\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\nexists_nonempty_j :\n  \u00ac\u2203 j,\n      (\u2203 i, t' i = j) \u2227\n        \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni : Fin n\nhi : \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 t' i = j", "state_after": "case neg\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\nexists_nonempty_j :\n  \u00ac\u2203 j,\n      (\u2203 i, t' i = j) \u2227\n        \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni : Fin n\nhi : \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\nthis : False\n\u22a2 t' i = j"}, {"tactic": "contradiction", "state_before": "case neg\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\nexists_nonempty_j :\n  \u00ac\u2203 j,\n      (\u2203 i, t' i = j) \u2227\n        \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni : Fin n\nhi : \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\nthis : False\n\u22a2 t' i = j", "state_after": "no goals"}, {"tactic": "rw [Algebra.smul_def, eq_intCast]", "state_before": "Fq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA : Fin n \u2192 Fq[X]\n\u22a2 0 < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "Fq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA : Fin n \u2192 Fq[X]\n\u22a2 0 < \u2191(\u2191cardPowDegree b) * \u03b5"}, {"tactic": "exact mul_pos (Int.cast_pos.mpr (AbsoluteValue.pos _ hb)) h\u03b5", "state_before": "Fq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA : Fin n \u2192 Fq[X]\n\u22a2 0 < \u2191(\u2191cardPowDegree b) * \u03b5", "state_after": "no goals"}, {"tactic": "refine' \u27e8finZeroElim, finZeroElim\u27e9", "state_before": "case zero\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin Nat.zero \u2192 Fq[X]\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : Fin Nat.zero), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "no goals"}, {"tactic": "intro i j k \u03b5", "state_before": "Fq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\n\u22a2 \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5", "state_after": "Fq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5\u271d : \u211d\nh\u03b5 : 0 < \u03b5\u271d\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\u271d\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\u271d\nA : Fin (Nat.succ n) \u2192 Fq[X]\ni j k : Fin (Nat.succ n)\n\u03b5 : \u211d\n\u22a2 \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n    \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5"}, {"tactic": "simp_rw [\u2190 Int.lt_ceil]", "state_before": "Fq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5\u271d : \u211d\nh\u03b5 : 0 < \u03b5\u271d\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\u271d\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\u271d\nA : Fin (Nat.succ n) \u2192 Fq[X]\ni j k : Fin (Nat.succ n)\n\u03b5 : \u211d\n\u22a2 \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n    \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5", "state_after": "Fq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5\u271d : \u211d\nh\u03b5 : 0 < \u03b5\u271d\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\u271d\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\u271d\nA : Fin (Nat.succ n) \u2192 Fq[X]\ni j k : Fin (Nat.succ n)\n\u03b5 : \u211d\n\u22a2 \u2191cardPowDegree (A i % b - A j % b) < \u2308\u03b5\u2309 \u2192\n    \u2191cardPowDegree (A j % b - A k % b) < \u2308\u03b5\u2309 \u2192 \u2191cardPowDegree (A i % b - A k % b) < \u2308\u03b5\u2309"}, {"tactic": "exact cardPowDegree_anti_archimedean", "state_before": "Fq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5\u271d : \u211d\nh\u03b5 : 0 < \u03b5\u271d\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\u271d\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\u271d\nA : Fin (Nat.succ n) \u2192 Fq[X]\ni j k : Fin (Nat.succ n)\n\u03b5 : \u211d\n\u22a2 \u2191cardPowDegree (A i % b - A j % b) < \u2308\u03b5\u2309 \u2192\n    \u2191cardPowDegree (A j % b - A k % b) < \u2308\u03b5\u2309 \u2192 \u2191cardPowDegree (A i % b - A k % b) < \u2308\u03b5\u2309", "state_after": "no goals"}, {"tactic": "refine' \u27e8Fin.cons j t', fun i\u2080 i\u2081 => _\u27e9", "state_before": "case succ.intro.intro\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : Fin (Nat.succ n)), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "case succ.intro.intro\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni\u2080 i\u2081 : Fin (Nat.succ n)\n\u22a2 Fin.cons j t' i\u2080 = Fin.cons j t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5"}, {"tactic": "refine' Fin.cases _ (fun i\u2080 => _) i\u2080 <;> refine' Fin.cases _ (fun i\u2081 => _) i\u2081", "state_before": "case succ.intro.intro\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni\u2080 i\u2081 : Fin (Nat.succ n)\n\u22a2 Fin.cons j t' i\u2080 = Fin.cons j t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "case succ.intro.intro.refine'_1.refine'_1\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni\u2080 i\u2081 : Fin (Nat.succ n)\n\u22a2 Fin.cons j t' 0 = Fin.cons j t' 0 \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A 0 % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\ncase succ.intro.intro.refine'_1.refine'_2\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni\u2080 i\u2081\u271d : Fin (Nat.succ n)\ni\u2081 : Fin n\n\u22a2 Fin.cons j t' 0 = Fin.cons j t' (Fin.succ i\u2081) \u2194\n    \u2191(\u2191cardPowDegree (A (Fin.succ i\u2081) % b - A 0 % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\ncase succ.intro.intro.refine'_2.refine'_1\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni\u2080\u271d i\u2081 : Fin (Nat.succ n)\ni\u2080 : Fin n\n\u22a2 Fin.cons j t' (Fin.succ i\u2080) = Fin.cons j t' 0 \u2194\n    \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i\u2080) % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\ncase succ.intro.intro.refine'_2.refine'_2\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni\u2080\u271d i\u2081\u271d : Fin (Nat.succ n)\ni\u2080 i\u2081 : Fin n\n\u22a2 Fin.cons j t' (Fin.succ i\u2080) = Fin.cons j t' (Fin.succ i\u2081) \u2194\n    \u2191(\u2191cardPowDegree (A (Fin.succ i\u2081) % b - A (Fin.succ i\u2080) % b)) < \u2191cardPowDegree b \u2022 \u03b5"}, {"tactic": "simpa using hb\u03b5", "state_before": "case succ.intro.intro.refine'_1.refine'_1\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni\u2080 i\u2081 : Fin (Nat.succ n)\n\u22a2 Fin.cons j t' 0 = Fin.cons j t' 0 \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A 0 % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "no goals"}, {"tactic": "rw [Fin.cons_succ, Fin.cons_zero, eq_comm, AbsoluteValue.map_sub]", "state_before": "case succ.intro.intro.refine'_1.refine'_2\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni\u2080 i\u2081\u271d : Fin (Nat.succ n)\ni\u2081 : Fin n\n\u22a2 Fin.cons j t' 0 = Fin.cons j t' (Fin.succ i\u2081) \u2194\n    \u2191(\u2191cardPowDegree (A (Fin.succ i\u2081) % b - A 0 % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "case succ.intro.intro.refine'_1.refine'_2\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni\u2080 i\u2081\u271d : Fin (Nat.succ n)\ni\u2081 : Fin n\n\u22a2 t' i\u2081 = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i\u2081) % b)) < \u2191cardPowDegree b \u2022 \u03b5"}, {"tactic": "exact hj i\u2081", "state_before": "case succ.intro.intro.refine'_1.refine'_2\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni\u2080 i\u2081\u271d : Fin (Nat.succ n)\ni\u2081 : Fin n\n\u22a2 t' i\u2081 = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i\u2081) % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "no goals"}, {"tactic": "rw [Fin.cons_succ, Fin.cons_zero]", "state_before": "case succ.intro.intro.refine'_2.refine'_1\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni\u2080\u271d i\u2081 : Fin (Nat.succ n)\ni\u2080 : Fin n\n\u22a2 Fin.cons j t' (Fin.succ i\u2080) = Fin.cons j t' 0 \u2194\n    \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i\u2080) % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "case succ.intro.intro.refine'_2.refine'_1\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni\u2080\u271d i\u2081 : Fin (Nat.succ n)\ni\u2080 : Fin n\n\u22a2 t' i\u2080 = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i\u2080) % b)) < \u2191cardPowDegree b \u2022 \u03b5"}, {"tactic": "exact hj i\u2080", "state_before": "case succ.intro.intro.refine'_2.refine'_1\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni\u2080\u271d i\u2081 : Fin (Nat.succ n)\ni\u2080 : Fin n\n\u22a2 t' i\u2080 = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i\u2080) % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "no goals"}, {"tactic": "rw [Fin.cons_succ, Fin.cons_succ]", "state_before": "case succ.intro.intro.refine'_2.refine'_2\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni\u2080\u271d i\u2081\u271d : Fin (Nat.succ n)\ni\u2080 i\u2081 : Fin n\n\u22a2 Fin.cons j t' (Fin.succ i\u2080) = Fin.cons j t' (Fin.succ i\u2081) \u2194\n    \u2191(\u2191cardPowDegree (A (Fin.succ i\u2081) % b - A (Fin.succ i\u2080) % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "case succ.intro.intro.refine'_2.refine'_2\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni\u2080\u271d i\u2081\u271d : Fin (Nat.succ n)\ni\u2080 i\u2081 : Fin n\n\u22a2 t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (A (Fin.succ i\u2081) % b - A (Fin.succ i\u2080) % b)) < \u2191cardPowDegree b \u2022 \u03b5"}, {"tactic": "exact ht' i\u2080 i\u2081", "state_before": "case succ.intro.intro.refine'_2.refine'_2\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni\u2080\u271d i\u2081\u271d : Fin (Nat.succ n)\ni\u2080 i\u2081 : Fin n\n\u22a2 t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (A (Fin.succ i\u2081) % b - A (Fin.succ i\u2080) % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "no goals"}, {"tactic": "by_contra hg", "state_before": "Fq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 \u2203 j, \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "Fq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg : \u00ac\u2203 j, \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 False"}, {"tactic": "push_neg at hg", "state_before": "Fq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg : \u00ac\u2203 j, \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 False", "state_after": "Fq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\n\u22a2 False"}, {"tactic": "obtain \u27e8j\u2080, j\u2081, j_ne, approx\u27e9 := exists_approx_polynomial hb h\u03b5\n  (Fin.cons (A 0) fun j => A (Fin.succ (Classical.choose (hg j))))", "state_before": "Fq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\n\u22a2 False", "state_after": "case intro.intro.intro\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080 j\u2081 : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj_ne : j\u2080 \u2260 j\u2081\napprox :\n  \u2191(\u2191cardPowDegree\n        (Fin.cons (A 0)\n              (fun j =>\n                A\n                  (Fin.succ\n                    (Classical.choose\n                      (_ : \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n              j\u2081 %\n            b -\n          Fin.cons (A 0)\n              (fun j =>\n                A\n                  (Fin.succ\n                    (Classical.choose\n                      (_ : \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n              j\u2080 %\n            b)) <\n    \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 False"}, {"tactic": "revert j_ne approx", "state_before": "case intro.intro.intro\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080 j\u2081 : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj_ne : j\u2080 \u2260 j\u2081\napprox :\n  \u2191(\u2191cardPowDegree\n        (Fin.cons (A 0)\n              (fun j =>\n                A\n                  (Fin.succ\n                    (Classical.choose\n                      (_ : \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n              j\u2081 %\n            b -\n          Fin.cons (A 0)\n              (fun j =>\n                A\n                  (Fin.succ\n                    (Classical.choose\n                      (_ : \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n              j\u2080 %\n            b)) <\n    \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 False", "state_after": "case intro.intro.intro\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080 j\u2081 : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\n\u22a2 j\u2080 \u2260 j\u2081 \u2192\n    \u2191(\u2191cardPowDegree\n            (Fin.cons (A 0)\n                  (fun j =>\n                    A\n                      (Fin.succ\n                        (Classical.choose\n                          (_ :\n                            \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n                  j\u2081 %\n                b -\n              Fin.cons (A 0)\n                  (fun j =>\n                    A\n                      (Fin.succ\n                        (Classical.choose\n                          (_ :\n                            \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n                  j\u2080 %\n                b)) <\n        \u2191cardPowDegree b \u2022 \u03b5 \u2192\n      False"}, {"tactic": "refine' Fin.cases _ (fun j\u2080 => _) j\u2080 <;>\n  refine' Fin.cases (fun j_ne approx => _) (fun j\u2081 j_ne approx => _) j\u2081", "state_before": "case intro.intro.intro\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080 j\u2081 : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\n\u22a2 j\u2080 \u2260 j\u2081 \u2192\n    \u2191(\u2191cardPowDegree\n            (Fin.cons (A 0)\n                  (fun j =>\n                    A\n                      (Fin.succ\n                        (Classical.choose\n                          (_ :\n                            \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n                  j\u2081 %\n                b -\n              Fin.cons (A 0)\n                  (fun j =>\n                    A\n                      (Fin.succ\n                        (Classical.choose\n                          (_ :\n                            \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n                  j\u2080 %\n                b)) <\n        \u2191cardPowDegree b \u2022 \u03b5 \u2192\n      False", "state_after": "case intro.intro.intro.refine'_1.refine'_1\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080 j\u2081 : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj_ne : 0 \u2260 0\napprox :\n  \u2191(\u2191cardPowDegree\n        (Fin.cons (A 0)\n              (fun j =>\n                A\n                  (Fin.succ\n                    (Classical.choose\n                      (_ : \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n              0 %\n            b -\n          Fin.cons (A 0)\n              (fun j =>\n                A\n                  (Fin.succ\n                    (Classical.choose\n                      (_ : \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n              0 %\n            b)) <\n    \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 False\n\ncase intro.intro.intro.refine'_1.refine'_2\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080 j\u2081\u271d : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj\u2081 : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nj_ne : 0 \u2260 Fin.succ j\u2081\napprox :\n  \u2191(\u2191cardPowDegree\n        (Fin.cons (A 0)\n              (fun j =>\n                A\n                  (Fin.succ\n                    (Classical.choose\n                      (_ : \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n              (Fin.succ j\u2081) %\n            b -\n          Fin.cons (A 0)\n              (fun j =>\n                A\n                  (Fin.succ\n                    (Classical.choose\n                      (_ : \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n              0 %\n            b)) <\n    \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 False\n\ncase intro.intro.intro.refine'_2.refine'_1\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080\u271d j\u2081 : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj\u2080 : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nj_ne : Fin.succ j\u2080 \u2260 0\napprox :\n  \u2191(\u2191cardPowDegree\n        (Fin.cons (A 0)\n              (fun j =>\n                A\n                  (Fin.succ\n                    (Classical.choose\n                      (_ : \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n              0 %\n            b -\n          Fin.cons (A 0)\n              (fun j =>\n                A\n                  (Fin.succ\n                    (Classical.choose\n                      (_ : \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n              (Fin.succ j\u2080) %\n            b)) <\n    \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 False\n\ncase intro.intro.intro.refine'_2.refine'_2\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080\u271d j\u2081\u271d : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj\u2080 j\u2081 : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nj_ne : Fin.succ j\u2080 \u2260 Fin.succ j\u2081\napprox :\n  \u2191(\u2191cardPowDegree\n        (Fin.cons (A 0)\n              (fun j =>\n                A\n                  (Fin.succ\n                    (Classical.choose\n                      (_ : \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n              (Fin.succ j\u2081) %\n            b -\n          Fin.cons (A 0)\n              (fun j =>\n                A\n                  (Fin.succ\n                    (Classical.choose\n                      (_ : \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n              (Fin.succ j\u2080) %\n            b)) <\n    \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 False"}, {"tactic": "exact absurd rfl j_ne", "state_before": "case intro.intro.intro.refine'_1.refine'_1\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080 j\u2081 : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj_ne : 0 \u2260 0\napprox :\n  \u2191(\u2191cardPowDegree\n        (Fin.cons (A 0)\n              (fun j =>\n                A\n                  (Fin.succ\n                    (Classical.choose\n                      (_ : \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n              0 %\n            b -\n          Fin.cons (A 0)\n              (fun j =>\n                A\n                  (Fin.succ\n                    (Classical.choose\n                      (_ : \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n              0 %\n            b)) <\n    \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rw [Fin.cons_succ, Fin.cons_zero, \u2190 not_le, AbsoluteValue.map_sub] at approx", "state_before": "case intro.intro.intro.refine'_1.refine'_2\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080 j\u2081\u271d : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj\u2081 : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nj_ne : 0 \u2260 Fin.succ j\u2081\napprox :\n  \u2191(\u2191cardPowDegree\n        (Fin.cons (A 0)\n              (fun j =>\n                A\n                  (Fin.succ\n                    (Classical.choose\n                      (_ : \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n              (Fin.succ j\u2081) %\n            b -\n          Fin.cons (A 0)\n              (fun j =>\n                A\n                  (Fin.succ\n                    (Classical.choose\n                      (_ : \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n              0 %\n            b)) <\n    \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 False", "state_after": "case intro.intro.intro.refine'_1.refine'_2\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080 j\u2081\u271d : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj\u2081 : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nj_ne : 0 \u2260 Fin.succ j\u2081\napprox :\n  \u00ac\u2191cardPowDegree b \u2022 \u03b5 \u2264\n      \u2191(\u2191cardPowDegree\n          (A 0 % b -\n            A\n                (Fin.succ\n                  (Classical.choose\n                    (_ : \u2203 i, t' i = j\u2081 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n              b))\n\u22a2 False"}, {"tactic": "have := (Classical.choose_spec (hg j\u2081)).2", "state_before": "case intro.intro.intro.refine'_1.refine'_2\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080 j\u2081\u271d : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj\u2081 : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nj_ne : 0 \u2260 Fin.succ j\u2081\napprox :\n  \u00ac\u2191cardPowDegree b \u2022 \u03b5 \u2264\n      \u2191(\u2191cardPowDegree\n          (A 0 % b -\n            A\n                (Fin.succ\n                  (Classical.choose\n                    (_ : \u2203 i, t' i = j\u2081 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n              b))\n\u22a2 False", "state_after": "case intro.intro.intro.refine'_1.refine'_2\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080 j\u2081\u271d : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj\u2081 : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nj_ne : 0 \u2260 Fin.succ j\u2081\napprox :\n  \u00ac\u2191cardPowDegree b \u2022 \u03b5 \u2264\n      \u2191(\u2191cardPowDegree\n          (A 0 % b -\n            A\n                (Fin.succ\n                  (Classical.choose\n                    (_ : \u2203 i, t' i = j\u2081 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n              b))\nthis :\n  \u2191cardPowDegree b \u2022 \u03b5 \u2264\n    \u2191(\u2191cardPowDegree\n        (A 0 % b -\n          A\n              (Fin.succ\n                (Classical.choose\n                  (_ : \u2203 i, t' i = j\u2081 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n            b))\n\u22a2 False"}, {"tactic": "contradiction", "state_before": "case intro.intro.intro.refine'_1.refine'_2\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080 j\u2081\u271d : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj\u2081 : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nj_ne : 0 \u2260 Fin.succ j\u2081\napprox :\n  \u00ac\u2191cardPowDegree b \u2022 \u03b5 \u2264\n      \u2191(\u2191cardPowDegree\n          (A 0 % b -\n            A\n                (Fin.succ\n                  (Classical.choose\n                    (_ : \u2203 i, t' i = j\u2081 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n              b))\nthis :\n  \u2191cardPowDegree b \u2022 \u03b5 \u2264\n    \u2191(\u2191cardPowDegree\n        (A 0 % b -\n          A\n              (Fin.succ\n                (Classical.choose\n                  (_ : \u2203 i, t' i = j\u2081 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n            b))\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rw [Fin.cons_succ, Fin.cons_zero, \u2190 not_le] at approx", "state_before": "case intro.intro.intro.refine'_2.refine'_1\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080\u271d j\u2081 : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj\u2080 : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nj_ne : Fin.succ j\u2080 \u2260 0\napprox :\n  \u2191(\u2191cardPowDegree\n        (Fin.cons (A 0)\n              (fun j =>\n                A\n                  (Fin.succ\n                    (Classical.choose\n                      (_ : \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n              0 %\n            b -\n          Fin.cons (A 0)\n              (fun j =>\n                A\n                  (Fin.succ\n                    (Classical.choose\n                      (_ : \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n              (Fin.succ j\u2080) %\n            b)) <\n    \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 False", "state_after": "case intro.intro.intro.refine'_2.refine'_1\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080\u271d j\u2081 : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj\u2080 : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nj_ne : Fin.succ j\u2080 \u2260 0\napprox :\n  \u00ac\u2191cardPowDegree b \u2022 \u03b5 \u2264\n      \u2191(\u2191cardPowDegree\n          (A 0 % b -\n            A\n                (Fin.succ\n                  (Classical.choose\n                    (_ : \u2203 i, t' i = j\u2080 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n              b))\n\u22a2 False"}, {"tactic": "have := (Classical.choose_spec (hg j\u2080)).2", "state_before": "case intro.intro.intro.refine'_2.refine'_1\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080\u271d j\u2081 : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj\u2080 : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nj_ne : Fin.succ j\u2080 \u2260 0\napprox :\n  \u00ac\u2191cardPowDegree b \u2022 \u03b5 \u2264\n      \u2191(\u2191cardPowDegree\n          (A 0 % b -\n            A\n                (Fin.succ\n                  (Classical.choose\n                    (_ : \u2203 i, t' i = j\u2080 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n              b))\n\u22a2 False", "state_after": "case intro.intro.intro.refine'_2.refine'_1\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080\u271d j\u2081 : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj\u2080 : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nj_ne : Fin.succ j\u2080 \u2260 0\napprox :\n  \u00ac\u2191cardPowDegree b \u2022 \u03b5 \u2264\n      \u2191(\u2191cardPowDegree\n          (A 0 % b -\n            A\n                (Fin.succ\n                  (Classical.choose\n                    (_ : \u2203 i, t' i = j\u2080 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n              b))\nthis :\n  \u2191cardPowDegree b \u2022 \u03b5 \u2264\n    \u2191(\u2191cardPowDegree\n        (A 0 % b -\n          A\n              (Fin.succ\n                (Classical.choose\n                  (_ : \u2203 i, t' i = j\u2080 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n            b))\n\u22a2 False"}, {"tactic": "contradiction", "state_before": "case intro.intro.intro.refine'_2.refine'_1\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080\u271d j\u2081 : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj\u2080 : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nj_ne : Fin.succ j\u2080 \u2260 0\napprox :\n  \u00ac\u2191cardPowDegree b \u2022 \u03b5 \u2264\n      \u2191(\u2191cardPowDegree\n          (A 0 % b -\n            A\n                (Fin.succ\n                  (Classical.choose\n                    (_ : \u2203 i, t' i = j\u2080 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n              b))\nthis :\n  \u2191cardPowDegree b \u2022 \u03b5 \u2264\n    \u2191(\u2191cardPowDegree\n        (A 0 % b -\n          A\n              (Fin.succ\n                (Classical.choose\n                  (_ : \u2203 i, t' i = j\u2080 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n            b))\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rw [Fin.cons_succ, Fin.cons_succ] at approx", "state_before": "case intro.intro.intro.refine'_2.refine'_2\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080\u271d j\u2081\u271d : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj\u2080 j\u2081 : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nj_ne : Fin.succ j\u2080 \u2260 Fin.succ j\u2081\napprox :\n  \u2191(\u2191cardPowDegree\n        (Fin.cons (A 0)\n              (fun j =>\n                A\n                  (Fin.succ\n                    (Classical.choose\n                      (_ : \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n              (Fin.succ j\u2081) %\n            b -\n          Fin.cons (A 0)\n              (fun j =>\n                A\n                  (Fin.succ\n                    (Classical.choose\n                      (_ : \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))))\n              (Fin.succ j\u2080) %\n            b)) <\n    \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 False", "state_after": "case intro.intro.intro.refine'_2.refine'_2\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080\u271d j\u2081\u271d : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj\u2080 j\u2081 : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nj_ne : Fin.succ j\u2080 \u2260 Fin.succ j\u2081\napprox :\n  \u2191(\u2191cardPowDegree\n        (A\n              (Fin.succ\n                (Classical.choose\n                  (_ : \u2203 i, t' i = j\u2081 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n            b -\n          A\n              (Fin.succ\n                (Classical.choose\n                  (_ : \u2203 i, t' i = j\u2080 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n            b)) <\n    \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 False"}, {"tactic": "rw [Ne.def, Fin.succ_inj] at j_ne", "state_before": "case intro.intro.intro.refine'_2.refine'_2\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080\u271d j\u2081\u271d : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj\u2080 j\u2081 : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nj_ne : Fin.succ j\u2080 \u2260 Fin.succ j\u2081\napprox :\n  \u2191(\u2191cardPowDegree\n        (A\n              (Fin.succ\n                (Classical.choose\n                  (_ : \u2203 i, t' i = j\u2081 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n            b -\n          A\n              (Fin.succ\n                (Classical.choose\n                  (_ : \u2203 i, t' i = j\u2080 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n            b)) <\n    \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 False", "state_after": "case intro.intro.intro.refine'_2.refine'_2\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080\u271d j\u2081\u271d : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj\u2080 j\u2081 : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nj_ne : \u00acj\u2080 = j\u2081\napprox :\n  \u2191(\u2191cardPowDegree\n        (A\n              (Fin.succ\n                (Classical.choose\n                  (_ : \u2203 i, t' i = j\u2081 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n            b -\n          A\n              (Fin.succ\n                (Classical.choose\n                  (_ : \u2203 i, t' i = j\u2080 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n            b)) <\n    \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 False"}, {"tactic": "have : j\u2080 = j\u2081 := (Classical.choose_spec (hg j\u2080)).1.symm.trans\n  (((ht' (Classical.choose (hg j\u2080)) (Classical.choose (hg j\u2081))).mpr approx).trans\n    (Classical.choose_spec (hg j\u2081)).1)", "state_before": "case intro.intro.intro.refine'_2.refine'_2\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080\u271d j\u2081\u271d : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj\u2080 j\u2081 : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nj_ne : \u00acj\u2080 = j\u2081\napprox :\n  \u2191(\u2191cardPowDegree\n        (A\n              (Fin.succ\n                (Classical.choose\n                  (_ : \u2203 i, t' i = j\u2081 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n            b -\n          A\n              (Fin.succ\n                (Classical.choose\n                  (_ : \u2203 i, t' i = j\u2080 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n            b)) <\n    \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 False", "state_after": "case intro.intro.intro.refine'_2.refine'_2\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080\u271d j\u2081\u271d : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj\u2080 j\u2081 : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nj_ne : \u00acj\u2080 = j\u2081\napprox :\n  \u2191(\u2191cardPowDegree\n        (A\n              (Fin.succ\n                (Classical.choose\n                  (_ : \u2203 i, t' i = j\u2081 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n            b -\n          A\n              (Fin.succ\n                (Classical.choose\n                  (_ : \u2203 i, t' i = j\u2080 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n            b)) <\n    \u2191cardPowDegree b \u2022 \u03b5\nthis : j\u2080 = j\u2081\n\u22a2 False"}, {"tactic": "contradiction", "state_before": "case intro.intro.intro.refine'_2.refine'_2\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nhg :\n  \u2200 (j : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)),\n    \u2203 i, t' i = j \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))\nj\u2080\u271d j\u2081\u271d : Fin (Nat.succ (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a))\nj\u2080 j\u2081 : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nj_ne : \u00acj\u2080 = j\u2081\napprox :\n  \u2191(\u2191cardPowDegree\n        (A\n              (Fin.succ\n                (Classical.choose\n                  (_ : \u2203 i, t' i = j\u2081 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n            b -\n          A\n              (Fin.succ\n                (Classical.choose\n                  (_ : \u2203 i, t' i = j\u2080 \u2227 \u2191cardPowDegree b \u2022 \u03b5 \u2264 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b))))) %\n            b)) <\n    \u2191cardPowDegree b \u2022 \u03b5\nthis : j\u2080 = j\u2081\n\u22a2 False", "state_after": "no goals"}, {"tactic": "obtain \u27e8j, \u27e8i, hi\u27e9, hj\u27e9 := exists_nonempty_j", "state_before": "case pos\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\nexists_nonempty_j :\n  \u2203 j,\n    (\u2203 i, t' i = j) \u2227 \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 \u2203 j, \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "case pos.intro.intro.intro\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj\u271d : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj\u271d : \u2200 (i : Fin n), t' i = j\u271d \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni : Fin n\nhi : t' i = j\n\u22a2 \u2203 j, \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5"}, {"tactic": "refine' \u27e8j, fun i' => \u27e8hj i', fun hi' => _root_.trans ((ht' _ _).mpr _) hi\u27e9\u27e9", "state_before": "case pos.intro.intro.intro\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj\u271d : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj\u271d : \u2200 (i : Fin n), t' i = j\u271d \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni : Fin n\nhi : t' i = j\n\u22a2 \u2203 j, \u2200 (i : Fin n), t' i = j \u2194 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "case pos.intro.intro.intro\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj\u271d : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj\u271d : \u2200 (i : Fin n), t' i = j\u271d \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni : Fin n\nhi : t' i = j\ni' : Fin n\nhi' : \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i') % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 \u2191(\u2191cardPowDegree (Fin.tail A i % b - Fin.tail A i' % b)) < \u2191cardPowDegree b \u2022 \u03b5"}, {"tactic": "apply anti_archim' _ hi'", "state_before": "case pos.intro.intro.intro\nFq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj\u271d : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj\u271d : \u2200 (i : Fin n), t' i = j\u271d \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni : Fin n\nhi : t' i = j\ni' : Fin n\nhi' : \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i') % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 \u2191(\u2191cardPowDegree (Fin.tail A i % b - Fin.tail A i' % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "Fq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj\u271d : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj\u271d : \u2200 (i : Fin n), t' i = j\u271d \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni : Fin n\nhi : t' i = j\ni' : Fin n\nhi' : \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i') % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 \u2191(\u2191cardPowDegree (A (Fin.succ i) % b - A 0 % b)) < \u2191cardPowDegree b \u2022 \u03b5"}, {"tactic": "rw [AbsoluteValue.map_sub]", "state_before": "Fq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj\u271d : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj\u271d : \u2200 (i : Fin n), t' i = j\u271d \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni : Fin n\nhi : t' i = j\ni' : Fin n\nhi' : \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i') % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 \u2191(\u2191cardPowDegree (A (Fin.succ i) % b - A 0 % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "Fq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj\u271d : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj\u271d : \u2200 (i : Fin n), t' i = j\u271d \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni : Fin n\nhi : t' i = j\ni' : Fin n\nhi' : \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i') % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5"}, {"tactic": "exact hj _ hi", "state_before": "Fq : Type u_1\ninst\u271d\u00b9 : Fintype Fq\ninst\u271d : Field Fq\nn\u271d : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : Fq[X]\nhb : b \u2260 0\nA\u271d : Fin n\u271d \u2192 Fq[X]\nhb\u03b5 : 0 < \u2191cardPowDegree b \u2022 \u03b5\nn : \u2115\nih :\n  \u2200 (A : Fin n \u2192 Fq[X]),\n    \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2194 \u2191(\u2191cardPowDegree (A i\u2081 % b - A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nA : Fin (Nat.succ n) \u2192 Fq[X]\nanti_archim' :\n  \u2200 {i j k : Fin (Nat.succ n)} {\u03b5 : \u211d},\n    \u2191(\u2191cardPowDegree (A i % b - A j % b)) < \u03b5 \u2192\n      \u2191(\u2191cardPowDegree (A j % b - A k % b)) < \u03b5 \u2192 \u2191(\u2191cardPowDegree (A i % b - A k % b)) < \u03b5\nt' : Fin n \u2192 Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nht' :\n  \u2200 (i\u2080 i\u2081 : Fin n), t' i\u2080 = t' i\u2081 \u2194 \u2191(\u2191cardPowDegree (Fin.tail A i\u2081 % b - Fin.tail A i\u2080 % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj\u271d : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj\u271d : \u2200 (i : Fin n), t' i = j\u271d \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\nj : Fin (Fintype.card Fq ^ \u2308-log \u03b5 / log \u2191(Fintype.card Fq)\u2309\u208a)\nhj : \u2200 (i : Fin n), t' i = j \u2192 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5\ni : Fin n\nhi : t' i = j\ni' : Fin n\nhi' : \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i') % b)) < \u2191cardPowDegree b \u2022 \u03b5\n\u22a2 \u2191(\u2191cardPowDegree (A 0 % b - A (Fin.succ i) % b)) < \u2191cardPowDegree b \u2022 \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/MeasurableSpaceDef.lean", "full_name": "MeasurableSet.iInter", "start": [162, 1], "end": [164, 72], "traced_tactics": [{"tactic": "rw [compl_iInter]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.7084\n\u03b3 : Type ?u.7087\n\u03b4 : Type ?u.7090\n\u03b4' : Type ?u.7093\n\u03b9 : Sort u_1\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u2200 (b : \u03b9), MeasurableSet (f b)\n\u22a2 MeasurableSet ((\u22c2 (b : \u03b9), f b)\u1d9c)", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type ?u.7084\n\u03b3 : Type ?u.7087\n\u03b4 : Type ?u.7090\n\u03b4' : Type ?u.7093\n\u03b9 : Sort u_1\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u2200 (b : \u03b9), MeasurableSet (f b)\n\u22a2 MeasurableSet (\u22c3 (i : \u03b9), f i\u1d9c)"}, {"tactic": "exact .iUnion fun b => (h b).compl", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.7084\n\u03b3 : Type ?u.7087\n\u03b4 : Type ?u.7090\n\u03b4' : Type ?u.7093\n\u03b9 : Sort u_1\ns t u : Set \u03b1\nm : MeasurableSpace \u03b1\ninst\u271d : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u2200 (b : \u03b9), MeasurableSet (f b)\n\u22a2 MeasurableSet (\u22c3 (i : \u03b9), f i\u1d9c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Lie/IdealOperations.lean", "full_name": "LieSubmodule.map_comap_incl", "start": [249, 1], "end": [251, 49], "traced_tactics": [{"tactic": "rw [\u2190 coe_toSubmodule_eq_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\nf : M \u2192\u2097\u2045R,L\u2046 M\u2082\n\u22a2 map (incl N) (comap (incl N) N') = N \u2293 N'", "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\nf : M \u2192\u2097\u2045R,L\u2046 M\u2082\n\u22a2 \u2191(map (incl N) (comap (incl N) N')) = \u2191(N \u2293 N')"}, {"tactic": "exact (N : Submodule R M).map_comap_subtype N'", "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\nf : M \u2192\u2097\u2045R,L\u2046 M\u2082\n\u22a2 \u2191(map (incl N) (comap (incl N) N')) = \u2191(N \u2293 N')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean", "full_name": "Polynomial.hasFDerivWithinAt_aeval", "start": [172, 11], "end": [175, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/Group.lean", "full_name": "Set.pairwise_disjoint_Ioo_zpow", "start": [225, 1], "end": [227, 64], "traced_tactics": [{"tactic": "simpa only [one_mul] using pairwise_disjoint_Ioo_mul_zpow 1 b", "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\n\u22a2 Pairwise (Disjoint on fun n => Ioo (b ^ n) (b ^ (n + 1)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Iso.lean", "full_name": "CategoryTheory.Functor.mapIso_refl", "start": [613, 1], "end": [614, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Quotient.map'_mk", "start": [818, 1], "end": [820, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Cone/Basic.lean", "full_name": "ConvexCone.Blunt.salient", "start": [407, 1], "end": [409, 24], "traced_tactics": [{"tactic": "rw [salient_iff_not_flat, blunt_iff_not_pointed]", "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.79178\nG : Type ?u.79181\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : SMul \ud835\udd5c E\nS\u271d S : ConvexCone \ud835\udd5c E\n\u22a2 Blunt S \u2192 Salient S", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.79178\nG : Type ?u.79181\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : SMul \ud835\udd5c E\nS\u271d S : ConvexCone \ud835\udd5c E\n\u22a2 \u00acPointed S \u2192 \u00acFlat S"}, {"tactic": "exact mt Flat.pointed", "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.79178\nG : Type ?u.79181\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : SMul \ud835\udd5c E\nS\u271d S : ConvexCone \ud835\udd5c E\n\u22a2 \u00acPointed S \u2192 \u00acFlat S", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/Over.lean", "full_name": "Ideal.exists_coeff_mem_comap_sdiff_comap_of_root_mem_sdiff", "start": [206, 1], "end": [225, 34], "traced_tactics": [{"tactic": "obtain \u27e8hrJ, hrI\u27e9 := hr", "state_before": "R : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\nhr : r \u2208 \u2191J \\ \u2191I\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)", "state_after": "case intro\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)"}, {"tactic": "have rbar_ne_zero : Ideal.Quotient.mk I r \u2260 0 := mt (Quotient.mk_eq_zero I).mp hrI", "state_before": "case intro\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)", "state_after": "case intro\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)"}, {"tactic": "have rbar_mem_J : Ideal.Quotient.mk I r \u2208 J.map (Ideal.Quotient.mk I) := mem_map_of_mem _ hrJ", "state_before": "case intro\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)", "state_after": "case intro\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)"}, {"tactic": "have quotient_f : \u2200 x \u2208 I.comap f, (Ideal.Quotient.mk I).comp f x = 0 := by\n  simp [Quotient.eq_zero_iff_mem]", "state_before": "case intro\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)", "state_after": "case intro\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)"}, {"tactic": "have rbar_root :\n  (p.map (Ideal.Quotient.mk (I.comap f))).eval\u2082 (Quotient.lift (I.comap f) _ quotient_f)\n      (Ideal.Quotient.mk I r) =\n    0 := by\n  convert Quotient.eq_zero_iff_mem.mpr hpI\n  exact _root_.trans (eval\u2082_map _ _ _) (hom_eval\u2082 p f (Ideal.Quotient.mk I) r).symm", "state_before": "case intro\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)", "state_after": "case intro\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\nrbar_root :\n  eval\u2082 (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (\u2191(Quotient.mk I) r)\n      (Polynomial.map (Quotient.mk (comap f I)) p) =\n    0\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)"}, {"tactic": "obtain \u27e8i, ne_zero, mem\u27e9 :=\n  exists_coeff_ne_zero_mem_comap_of_root_mem rbar_ne_zero rbar_mem_J p_ne_zero rbar_root", "state_before": "case intro\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\nrbar_root :\n  eval\u2082 (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (\u2191(Quotient.mk I) r)\n      (Polynomial.map (Quotient.mk (comap f I)) p) =\n    0\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)", "state_after": "case intro.intro.intro\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\nrbar_root :\n  eval\u2082 (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (\u2191(Quotient.mk I) r)\n      (Polynomial.map (Quotient.mk (comap f I)) p) =\n    0\ni : \u2115\nne_zero : coeff (Polynomial.map (Quotient.mk (comap f I)) p) i \u2260 0\nmem :\n  coeff (Polynomial.map (Quotient.mk (comap f I)) p) i \u2208\n    comap (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (map (Quotient.mk I) J)\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)"}, {"tactic": "rw [coeff_map] at ne_zero mem", "state_before": "case intro.intro.intro\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\nrbar_root :\n  eval\u2082 (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (\u2191(Quotient.mk I) r)\n      (Polynomial.map (Quotient.mk (comap f I)) p) =\n    0\ni : \u2115\nne_zero : coeff (Polynomial.map (Quotient.mk (comap f I)) p) i \u2260 0\nmem :\n  coeff (Polynomial.map (Quotient.mk (comap f I)) p) i \u2208\n    comap (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (map (Quotient.mk I) J)\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)", "state_after": "case intro.intro.intro\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\nrbar_root :\n  eval\u2082 (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (\u2191(Quotient.mk I) r)\n      (Polynomial.map (Quotient.mk (comap f I)) p) =\n    0\ni : \u2115\nne_zero : \u2191(Quotient.mk (comap f I)) (coeff p i) \u2260 0\nmem :\n  \u2191(Quotient.mk (comap f I)) (coeff p i) \u2208\n    comap (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (map (Quotient.mk I) J)\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)"}, {"tactic": "refine' \u27e8i, (mem_quotient_iff_mem hIJ).mp _, mt _ ne_zero\u27e9", "state_before": "case intro.intro.intro\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\nrbar_root :\n  eval\u2082 (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (\u2191(Quotient.mk I) r)\n      (Polynomial.map (Quotient.mk (comap f I)) p) =\n    0\ni : \u2115\nne_zero : \u2191(Quotient.mk (comap f I)) (coeff p i) \u2260 0\nmem :\n  \u2191(Quotient.mk (comap f I)) (coeff p i) \u2208\n    comap (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (map (Quotient.mk I) J)\n\u22a2 \u2203 i, coeff p i \u2208 \u2191(comap f J) \\ \u2191(comap f I)", "state_after": "case intro.intro.intro.refine'_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\nrbar_root :\n  eval\u2082 (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (\u2191(Quotient.mk I) r)\n      (Polynomial.map (Quotient.mk (comap f I)) p) =\n    0\ni : \u2115\nne_zero : \u2191(Quotient.mk (comap f I)) (coeff p i) \u2260 0\nmem :\n  \u2191(Quotient.mk (comap f I)) (coeff p i) \u2208\n    comap (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (map (Quotient.mk I) J)\n\u22a2 \u2191(Quotient.mk I) (\u2191f (coeff p i)) \u2208 map (Quotient.mk I) J\n\ncase intro.intro.intro.refine'_2\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\nrbar_root :\n  eval\u2082 (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (\u2191(Quotient.mk I) r)\n      (Polynomial.map (Quotient.mk (comap f I)) p) =\n    0\ni : \u2115\nne_zero : \u2191(Quotient.mk (comap f I)) (coeff p i) \u2260 0\nmem :\n  \u2191(Quotient.mk (comap f I)) (coeff p i) \u2208\n    comap (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (map (Quotient.mk I) J)\n\u22a2 coeff p i \u2208 \u2191(comap f I) \u2192 \u2191(Quotient.mk (comap f I)) (coeff p i) = 0"}, {"tactic": "simp [Quotient.eq_zero_iff_mem]", "state_before": "case intro.intro.intro.refine'_2\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\nrbar_root :\n  eval\u2082 (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (\u2191(Quotient.mk I) r)\n      (Polynomial.map (Quotient.mk (comap f I)) p) =\n    0\ni : \u2115\nne_zero : \u2191(Quotient.mk (comap f I)) (coeff p i) \u2260 0\nmem :\n  \u2191(Quotient.mk (comap f I)) (coeff p i) \u2208\n    comap (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (map (Quotient.mk I) J)\n\u22a2 coeff p i \u2208 \u2191(comap f I) \u2192 \u2191(Quotient.mk (comap f I)) (coeff p i) = 0", "state_after": "no goals"}, {"tactic": "simp [Quotient.eq_zero_iff_mem]", "state_before": "R : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\n\u22a2 \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0", "state_after": "no goals"}, {"tactic": "convert Quotient.eq_zero_iff_mem.mpr hpI", "state_before": "R : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\n\u22a2 eval\u2082 (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (\u2191(Quotient.mk I) r)\n      (Polynomial.map (Quotient.mk (comap f I)) p) =\n    0", "state_after": "case h.e'_2\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\n\u22a2 eval\u2082 (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (\u2191(Quotient.mk I) r)\n      (Polynomial.map (Quotient.mk (comap f I)) p) =\n    \u2191(Quotient.mk I) (eval\u2082 f r p)"}, {"tactic": "exact _root_.trans (eval\u2082_map _ _ _) (hom_eval\u2082 p f (Ideal.Quotient.mk I) r).symm", "state_before": "case h.e'_2\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\n\u22a2 eval\u2082 (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (\u2191(Quotient.mk I) r)\n      (Polynomial.map (Quotient.mk (comap f I)) p) =\n    \u2191(Quotient.mk I) (eval\u2082 f r p)", "state_after": "no goals"}, {"tactic": "simpa using mem", "state_before": "case intro.intro.intro.refine'_1\nR : Type u_2\ninst\u271d\u00b2 : CommRing R\nS : Type u_1\ninst\u271d\u00b9 : CommRing S\nf : R \u2192+* S\nI J : Ideal S\ninst\u271d : IsPrime I\nhIJ : I \u2264 J\nr : S\np : R[X]\np_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p \u2260 0\nhpI : eval\u2082 f r p \u2208 I\nhrJ : r \u2208 \u2191J\nhrI : \u00acr \u2208 \u2191I\nrbar_ne_zero : \u2191(Quotient.mk I) r \u2260 0\nrbar_mem_J : \u2191(Quotient.mk I) r \u2208 map (Quotient.mk I) J\nquotient_f : \u2200 (x : R), x \u2208 comap f I \u2192 \u2191(RingHom.comp (Quotient.mk I) f) x = 0\nrbar_root :\n  eval\u2082 (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (\u2191(Quotient.mk I) r)\n      (Polynomial.map (Quotient.mk (comap f I)) p) =\n    0\ni : \u2115\nne_zero : \u2191(Quotient.mk (comap f I)) (coeff p i) \u2260 0\nmem :\n  \u2191(Quotient.mk (comap f I)) (coeff p i) \u2208\n    comap (Quotient.lift (comap f I) (RingHom.comp (Quotient.mk I) f) quotient_f) (map (Quotient.mk I) J)\n\u22a2 \u2191(Quotient.mk I) (\u2191f (coeff p i)) \u2208 map (Quotient.mk I) J", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Module/WeakDual.lean", "full_name": "WeakBilin.tendsto_iff_forall_eval_tendsto", "start": [139, 1], "end": [143, 6], "traced_tactics": [{"tactic": "rw [\u2190 tendsto_pi_nhds, Embedding.tendsto_nhds_iff (embedding hB)]", "state_before": "\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\n\ud835\udd5d : Type ?u.37176\nR : Type ?u.37179\nE : Type u_3\nF : Type u_4\nM : Type ?u.37188\ninst\u271d\u2075 : TopologicalSpace \ud835\udd5c\ninst\u271d\u2074 : CommSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : AddCommMonoid F\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nl : Filter \u03b1\nf : \u03b1 \u2192 WeakBilin B\nx : WeakBilin B\nhB : Function.Injective \u2191B\n\u22a2 Tendsto f l (\ud835\udcdd x) \u2194 \u2200 (y : F), Tendsto (fun i => \u2191(\u2191B (f i)) y) l (\ud835\udcdd (\u2191(\u2191B x) y))", "state_after": "\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\n\ud835\udd5d : Type ?u.37176\nR : Type ?u.37179\nE : Type u_3\nF : Type u_4\nM : Type ?u.37188\ninst\u271d\u2075 : TopologicalSpace \ud835\udd5c\ninst\u271d\u2074 : CommSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : AddCommMonoid F\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nl : Filter \u03b1\nf : \u03b1 \u2192 WeakBilin B\nx : WeakBilin B\nhB : Function.Injective \u2191B\n\u22a2 Tendsto ((fun x y => \u2191(\u2191B x) y) \u2218 f) l (\ud835\udcdd fun y => \u2191(\u2191B x) y) \u2194\n    Tendsto (fun i y => \u2191(\u2191B (f i)) y) l (\ud835\udcdd fun y => \u2191(\u2191B x) y)"}, {"tactic": "rfl", "state_before": "\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\n\ud835\udd5d : Type ?u.37176\nR : Type ?u.37179\nE : Type u_3\nF : Type u_4\nM : Type ?u.37188\ninst\u271d\u2075 : TopologicalSpace \ud835\udd5c\ninst\u271d\u2074 : CommSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : AddCommMonoid F\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nl : Filter \u03b1\nf : \u03b1 \u2192 WeakBilin B\nx : WeakBilin B\nhB : Function.Injective \u2191B\n\u22a2 Tendsto ((fun x y => \u2191(\u2191B x) y) \u2218 f) l (\ud835\udcdd fun y => \u2191(\u2191B x) y) \u2194\n    Tendsto (fun i y => \u2191(\u2191B (f i)) y) l (\ud835\udcdd fun y => \u2191(\u2191B x) y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/Defs.lean", "full_name": "Equiv.forall_congr'", "start": [876, 11], "end": [878, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "full_name": "periodic_circleMap", "start": [96, 1], "end": [97, 44], "traced_tactics": [{"tactic": "simp [circleMap, add_mul, exp_periodic _]", "state_before": "E : Type ?u.2472\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR \u03b8 : \u211d\n\u22a2 circleMap c R (\u03b8 + 2 * \u03c0) = circleMap c R \u03b8", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/Series.lean", "full_name": "differentiable_tsum", "start": [167, 1], "end": [176, 33], "traced_tactics": [{"tactic": "by_cases h : \u2203 x\u2080, Summable fun n => f n x\u2080", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.49306\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : CompleteSpace F\nu : \u03b1 \u2192 \u211d\ninst\u271d : NormedSpace \ud835\udd5c F\nf : \u03b1 \u2192 E \u2192 F\nf' : \u03b1 \u2192 E \u2192 E \u2192L[\ud835\udd5c] F\nv : \u2115 \u2192 \u03b1 \u2192 \u211d\ns : Set E\nx\u2080 x : E\nN : \u2115\u221e\nhu : Summable u\nhf : \u2200 (n : \u03b1) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : \u2200 (n : \u03b1) (x : E), \u2016f' n x\u2016 \u2264 u n\n\u22a2 Differentiable \ud835\udd5c fun y => \u2211' (n : \u03b1), f n y", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.49306\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : CompleteSpace F\nu : \u03b1 \u2192 \u211d\ninst\u271d : NormedSpace \ud835\udd5c F\nf : \u03b1 \u2192 E \u2192 F\nf' : \u03b1 \u2192 E \u2192 E \u2192L[\ud835\udd5c] F\nv : \u2115 \u2192 \u03b1 \u2192 \u211d\ns : Set E\nx\u2080 x : E\nN : \u2115\u221e\nhu : Summable u\nhf : \u2200 (n : \u03b1) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : \u2200 (n : \u03b1) (x : E), \u2016f' n x\u2016 \u2264 u n\nh : \u2203 x\u2080, Summable fun n => f n x\u2080\n\u22a2 Differentiable \ud835\udd5c fun y => \u2211' (n : \u03b1), f n y\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.49306\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : CompleteSpace F\nu : \u03b1 \u2192 \u211d\ninst\u271d : NormedSpace \ud835\udd5c F\nf : \u03b1 \u2192 E \u2192 F\nf' : \u03b1 \u2192 E \u2192 E \u2192L[\ud835\udd5c] F\nv : \u2115 \u2192 \u03b1 \u2192 \u211d\ns : Set E\nx\u2080 x : E\nN : \u2115\u221e\nhu : Summable u\nhf : \u2200 (n : \u03b1) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : \u2200 (n : \u03b1) (x : E), \u2016f' n x\u2016 \u2264 u n\nh : \u00ac\u2203 x\u2080, Summable fun n => f n x\u2080\n\u22a2 Differentiable \ud835\udd5c fun y => \u2211' (n : \u03b1), f n y"}, {"tactic": "rcases h with \u27e8x\u2080, hf0\u27e9", "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.49306\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : CompleteSpace F\nu : \u03b1 \u2192 \u211d\ninst\u271d : NormedSpace \ud835\udd5c F\nf : \u03b1 \u2192 E \u2192 F\nf' : \u03b1 \u2192 E \u2192 E \u2192L[\ud835\udd5c] F\nv : \u2115 \u2192 \u03b1 \u2192 \u211d\ns : Set E\nx\u2080 x : E\nN : \u2115\u221e\nhu : Summable u\nhf : \u2200 (n : \u03b1) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : \u2200 (n : \u03b1) (x : E), \u2016f' n x\u2016 \u2264 u n\nh : \u2203 x\u2080, Summable fun n => f n x\u2080\n\u22a2 Differentiable \ud835\udd5c fun y => \u2211' (n : \u03b1), f n y", "state_after": "case pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.49306\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : CompleteSpace F\nu : \u03b1 \u2192 \u211d\ninst\u271d : NormedSpace \ud835\udd5c F\nf : \u03b1 \u2192 E \u2192 F\nf' : \u03b1 \u2192 E \u2192 E \u2192L[\ud835\udd5c] F\nv : \u2115 \u2192 \u03b1 \u2192 \u211d\ns : Set E\nx\u2080\u271d x : E\nN : \u2115\u221e\nhu : Summable u\nhf : \u2200 (n : \u03b1) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : \u2200 (n : \u03b1) (x : E), \u2016f' n x\u2016 \u2264 u n\nx\u2080 : E\nhf0 : Summable fun n => f n x\u2080\n\u22a2 Differentiable \ud835\udd5c fun y => \u2211' (n : \u03b1), f n y"}, {"tactic": "intro x", "state_before": "case pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.49306\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : CompleteSpace F\nu : \u03b1 \u2192 \u211d\ninst\u271d : NormedSpace \ud835\udd5c F\nf : \u03b1 \u2192 E \u2192 F\nf' : \u03b1 \u2192 E \u2192 E \u2192L[\ud835\udd5c] F\nv : \u2115 \u2192 \u03b1 \u2192 \u211d\ns : Set E\nx\u2080\u271d x : E\nN : \u2115\u221e\nhu : Summable u\nhf : \u2200 (n : \u03b1) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : \u2200 (n : \u03b1) (x : E), \u2016f' n x\u2016 \u2264 u n\nx\u2080 : E\nhf0 : Summable fun n => f n x\u2080\n\u22a2 Differentiable \ud835\udd5c fun y => \u2211' (n : \u03b1), f n y", "state_after": "case pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.49306\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : CompleteSpace F\nu : \u03b1 \u2192 \u211d\ninst\u271d : NormedSpace \ud835\udd5c F\nf : \u03b1 \u2192 E \u2192 F\nf' : \u03b1 \u2192 E \u2192 E \u2192L[\ud835\udd5c] F\nv : \u2115 \u2192 \u03b1 \u2192 \u211d\ns : Set E\nx\u2080\u271d x\u271d : E\nN : \u2115\u221e\nhu : Summable u\nhf : \u2200 (n : \u03b1) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : \u2200 (n : \u03b1) (x : E), \u2016f' n x\u2016 \u2264 u n\nx\u2080 : E\nhf0 : Summable fun n => f n x\u2080\nx : E\n\u22a2 DifferentiableAt \ud835\udd5c (fun y => \u2211' (n : \u03b1), f n y) x"}, {"tactic": "exact (hasFDerivAt_tsum hu hf hf' hf0 x).differentiableAt", "state_before": "case pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.49306\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : CompleteSpace F\nu : \u03b1 \u2192 \u211d\ninst\u271d : NormedSpace \ud835\udd5c F\nf : \u03b1 \u2192 E \u2192 F\nf' : \u03b1 \u2192 E \u2192 E \u2192L[\ud835\udd5c] F\nv : \u2115 \u2192 \u03b1 \u2192 \u211d\ns : Set E\nx\u2080\u271d x\u271d : E\nN : \u2115\u221e\nhu : Summable u\nhf : \u2200 (n : \u03b1) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : \u2200 (n : \u03b1) (x : E), \u2016f' n x\u2016 \u2264 u n\nx\u2080 : E\nhf0 : Summable fun n => f n x\u2080\nx : E\n\u22a2 DifferentiableAt \ud835\udd5c (fun y => \u2211' (n : \u03b1), f n y) x", "state_after": "no goals"}, {"tactic": "push_neg  at h", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.49306\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : CompleteSpace F\nu : \u03b1 \u2192 \u211d\ninst\u271d : NormedSpace \ud835\udd5c F\nf : \u03b1 \u2192 E \u2192 F\nf' : \u03b1 \u2192 E \u2192 E \u2192L[\ud835\udd5c] F\nv : \u2115 \u2192 \u03b1 \u2192 \u211d\ns : Set E\nx\u2080 x : E\nN : \u2115\u221e\nhu : Summable u\nhf : \u2200 (n : \u03b1) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : \u2200 (n : \u03b1) (x : E), \u2016f' n x\u2016 \u2264 u n\nh : \u00ac\u2203 x\u2080, Summable fun n => f n x\u2080\n\u22a2 Differentiable \ud835\udd5c fun y => \u2211' (n : \u03b1), f n y", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.49306\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : CompleteSpace F\nu : \u03b1 \u2192 \u211d\ninst\u271d : NormedSpace \ud835\udd5c F\nf : \u03b1 \u2192 E \u2192 F\nf' : \u03b1 \u2192 E \u2192 E \u2192L[\ud835\udd5c] F\nv : \u2115 \u2192 \u03b1 \u2192 \u211d\ns : Set E\nx\u2080 x : E\nN : \u2115\u221e\nhu : Summable u\nhf : \u2200 (n : \u03b1) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : \u2200 (n : \u03b1) (x : E), \u2016f' n x\u2016 \u2264 u n\nh : \u2200 (x\u2080 : E), \u00acSummable fun n => f n x\u2080\n\u22a2 Differentiable \ud835\udd5c fun y => \u2211' (n : \u03b1), f n y"}, {"tactic": "have : (fun x => \u2211' n, f n x) = 0 := by ext1 x; exact tsum_eq_zero_of_not_summable (h x)", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.49306\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : CompleteSpace F\nu : \u03b1 \u2192 \u211d\ninst\u271d : NormedSpace \ud835\udd5c F\nf : \u03b1 \u2192 E \u2192 F\nf' : \u03b1 \u2192 E \u2192 E \u2192L[\ud835\udd5c] F\nv : \u2115 \u2192 \u03b1 \u2192 \u211d\ns : Set E\nx\u2080 x : E\nN : \u2115\u221e\nhu : Summable u\nhf : \u2200 (n : \u03b1) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : \u2200 (n : \u03b1) (x : E), \u2016f' n x\u2016 \u2264 u n\nh : \u2200 (x\u2080 : E), \u00acSummable fun n => f n x\u2080\n\u22a2 Differentiable \ud835\udd5c fun y => \u2211' (n : \u03b1), f n y", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.49306\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : CompleteSpace F\nu : \u03b1 \u2192 \u211d\ninst\u271d : NormedSpace \ud835\udd5c F\nf : \u03b1 \u2192 E \u2192 F\nf' : \u03b1 \u2192 E \u2192 E \u2192L[\ud835\udd5c] F\nv : \u2115 \u2192 \u03b1 \u2192 \u211d\ns : Set E\nx\u2080 x : E\nN : \u2115\u221e\nhu : Summable u\nhf : \u2200 (n : \u03b1) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : \u2200 (n : \u03b1) (x : E), \u2016f' n x\u2016 \u2264 u n\nh : \u2200 (x\u2080 : E), \u00acSummable fun n => f n x\u2080\nthis : (fun x => \u2211' (n : \u03b1), f n x) = 0\n\u22a2 Differentiable \ud835\udd5c fun y => \u2211' (n : \u03b1), f n y"}, {"tactic": "rw [this]", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.49306\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : CompleteSpace F\nu : \u03b1 \u2192 \u211d\ninst\u271d : NormedSpace \ud835\udd5c F\nf : \u03b1 \u2192 E \u2192 F\nf' : \u03b1 \u2192 E \u2192 E \u2192L[\ud835\udd5c] F\nv : \u2115 \u2192 \u03b1 \u2192 \u211d\ns : Set E\nx\u2080 x : E\nN : \u2115\u221e\nhu : Summable u\nhf : \u2200 (n : \u03b1) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : \u2200 (n : \u03b1) (x : E), \u2016f' n x\u2016 \u2264 u n\nh : \u2200 (x\u2080 : E), \u00acSummable fun n => f n x\u2080\nthis : (fun x => \u2211' (n : \u03b1), f n x) = 0\n\u22a2 Differentiable \ud835\udd5c fun y => \u2211' (n : \u03b1), f n y", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.49306\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : CompleteSpace F\nu : \u03b1 \u2192 \u211d\ninst\u271d : NormedSpace \ud835\udd5c F\nf : \u03b1 \u2192 E \u2192 F\nf' : \u03b1 \u2192 E \u2192 E \u2192L[\ud835\udd5c] F\nv : \u2115 \u2192 \u03b1 \u2192 \u211d\ns : Set E\nx\u2080 x : E\nN : \u2115\u221e\nhu : Summable u\nhf : \u2200 (n : \u03b1) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : \u2200 (n : \u03b1) (x : E), \u2016f' n x\u2016 \u2264 u n\nh : \u2200 (x\u2080 : E), \u00acSummable fun n => f n x\u2080\nthis : (fun x => \u2211' (n : \u03b1), f n x) = 0\n\u22a2 Differentiable \ud835\udd5c 0"}, {"tactic": "exact differentiable_const 0", "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.49306\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : CompleteSpace F\nu : \u03b1 \u2192 \u211d\ninst\u271d : NormedSpace \ud835\udd5c F\nf : \u03b1 \u2192 E \u2192 F\nf' : \u03b1 \u2192 E \u2192 E \u2192L[\ud835\udd5c] F\nv : \u2115 \u2192 \u03b1 \u2192 \u211d\ns : Set E\nx\u2080 x : E\nN : \u2115\u221e\nhu : Summable u\nhf : \u2200 (n : \u03b1) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : \u2200 (n : \u03b1) (x : E), \u2016f' n x\u2016 \u2264 u n\nh : \u2200 (x\u2080 : E), \u00acSummable fun n => f n x\u2080\nthis : (fun x => \u2211' (n : \u03b1), f n x) = 0\n\u22a2 Differentiable \ud835\udd5c 0", "state_after": "no goals"}, {"tactic": "ext1 x", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.49306\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : CompleteSpace F\nu : \u03b1 \u2192 \u211d\ninst\u271d : NormedSpace \ud835\udd5c F\nf : \u03b1 \u2192 E \u2192 F\nf' : \u03b1 \u2192 E \u2192 E \u2192L[\ud835\udd5c] F\nv : \u2115 \u2192 \u03b1 \u2192 \u211d\ns : Set E\nx\u2080 x : E\nN : \u2115\u221e\nhu : Summable u\nhf : \u2200 (n : \u03b1) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : \u2200 (n : \u03b1) (x : E), \u2016f' n x\u2016 \u2264 u n\nh : \u2200 (x\u2080 : E), \u00acSummable fun n => f n x\u2080\n\u22a2 (fun x => \u2211' (n : \u03b1), f n x) = 0", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.49306\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : CompleteSpace F\nu : \u03b1 \u2192 \u211d\ninst\u271d : NormedSpace \ud835\udd5c F\nf : \u03b1 \u2192 E \u2192 F\nf' : \u03b1 \u2192 E \u2192 E \u2192L[\ud835\udd5c] F\nv : \u2115 \u2192 \u03b1 \u2192 \u211d\ns : Set E\nx\u2080 x\u271d : E\nN : \u2115\u221e\nhu : Summable u\nhf : \u2200 (n : \u03b1) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : \u2200 (n : \u03b1) (x : E), \u2016f' n x\u2016 \u2264 u n\nh : \u2200 (x\u2080 : E), \u00acSummable fun n => f n x\u2080\nx : E\n\u22a2 (\u2211' (n : \u03b1), f n x) = OfNat.ofNat 0 x"}, {"tactic": "exact tsum_eq_zero_of_not_summable (h x)", "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.49306\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : CompleteSpace F\nu : \u03b1 \u2192 \u211d\ninst\u271d : NormedSpace \ud835\udd5c F\nf : \u03b1 \u2192 E \u2192 F\nf' : \u03b1 \u2192 E \u2192 E \u2192L[\ud835\udd5c] F\nv : \u2115 \u2192 \u03b1 \u2192 \u211d\ns : Set E\nx\u2080 x\u271d : E\nN : \u2115\u221e\nhu : Summable u\nhf : \u2200 (n : \u03b1) (x : E), HasFDerivAt (f n) (f' n x) x\nhf' : \u2200 (n : \u03b1) (x : E), \u2016f' n x\u2016 \u2264 u n\nh : \u2200 (x\u2080 : E), \u00acSummable fun n => f n x\u2080\nx : E\n\u22a2 (\u2211' (n : \u03b1), f n x) = OfNat.ofNat 0 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "full_name": "Multiset.prod_replicate", "start": [131, 1], "end": [132, 40], "traced_tactics": [{"tactic": "simp [replicate, List.prod_replicate]", "state_before": "\u03b9 : Type ?u.19271\n\u03b1 : Type u_1\n\u03b2 : Type ?u.19277\n\u03b3 : Type ?u.19280\ninst\u271d : CommMonoid \u03b1\ns t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\nn : \u2115\na : \u03b1\n\u22a2 prod (replicate n a) = a ^ n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "full_name": "Asymptotics.IsBigOWith.trans_isLittleO", "start": [500, 1], "end": [505, 76], "traced_tactics": [{"tactic": "simp only [IsLittleO_def] at *", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.88772\nE : Type u_2\nF : Type u_3\nG : Type u_4\nE' : Type ?u.88784\nF' : Type ?u.88787\nG' : Type ?u.88790\nE'' : Type ?u.88793\nF'' : Type ?u.88796\nG'' : Type ?u.88799\nR : Type ?u.88802\nR' : Type ?u.88805\n\ud835\udd5c : Type ?u.88808\n\ud835\udd5c' : Type ?u.88811\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\nhfg : IsBigOWith c l f g\nhgk : g =o[l] k\nhc : 0 < c\n\u22a2 f =o[l] k", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.88772\nE : Type u_2\nF : Type u_3\nG : Type u_4\nE' : Type ?u.88784\nF' : Type ?u.88787\nG' : Type ?u.88790\nE'' : Type ?u.88793\nF'' : Type ?u.88796\nG'' : Type ?u.88799\nR : Type ?u.88802\nR' : Type ?u.88805\n\ud835\udd5c : Type ?u.88808\n\ud835\udd5c' : Type ?u.88811\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\nhfg : IsBigOWith c l f g\nhc : 0 < c\nhgk : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l g k\n\u22a2 \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l f k"}, {"tactic": "intro c' c'pos", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.88772\nE : Type u_2\nF : Type u_3\nG : Type u_4\nE' : Type ?u.88784\nF' : Type ?u.88787\nG' : Type ?u.88790\nE'' : Type ?u.88793\nF'' : Type ?u.88796\nG'' : Type ?u.88799\nR : Type ?u.88802\nR' : Type ?u.88805\n\ud835\udd5c : Type ?u.88808\n\ud835\udd5c' : Type ?u.88811\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\nhfg : IsBigOWith c l f g\nhc : 0 < c\nhgk : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l g k\n\u22a2 \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l f k", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.88772\nE : Type u_2\nF : Type u_3\nG : Type u_4\nE' : Type ?u.88784\nF' : Type ?u.88787\nG' : Type ?u.88790\nE'' : Type ?u.88793\nF'' : Type ?u.88796\nG'' : Type ?u.88799\nR : Type ?u.88802\nR' : Type ?u.88805\n\ud835\udd5c : Type ?u.88808\n\ud835\udd5c' : Type ?u.88811\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'\u271d 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\nhfg : IsBigOWith c l f g\nhc : 0 < c\nhgk : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l g k\nc' : \u211d\nc'pos : 0 < c'\n\u22a2 IsBigOWith c' l f k"}, {"tactic": "have : 0 < c' / c := div_pos c'pos hc", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.88772\nE : Type u_2\nF : Type u_3\nG : Type u_4\nE' : Type ?u.88784\nF' : Type ?u.88787\nG' : Type ?u.88790\nE'' : Type ?u.88793\nF'' : Type ?u.88796\nG'' : Type ?u.88799\nR : Type ?u.88802\nR' : Type ?u.88805\n\ud835\udd5c : Type ?u.88808\n\ud835\udd5c' : Type ?u.88811\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'\u271d 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\nhfg : IsBigOWith c l f g\nhc : 0 < c\nhgk : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l g k\nc' : \u211d\nc'pos : 0 < c'\n\u22a2 IsBigOWith c' l f k", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.88772\nE : Type u_2\nF : Type u_3\nG : Type u_4\nE' : Type ?u.88784\nF' : Type ?u.88787\nG' : Type ?u.88790\nE'' : Type ?u.88793\nF'' : Type ?u.88796\nG'' : Type ?u.88799\nR : Type ?u.88802\nR' : Type ?u.88805\n\ud835\udd5c : Type ?u.88808\n\ud835\udd5c' : Type ?u.88811\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'\u271d 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\nhfg : IsBigOWith c l f g\nhc : 0 < c\nhgk : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l g k\nc' : \u211d\nc'pos : 0 < c'\nthis : 0 < c' / c\n\u22a2 IsBigOWith c' l f k"}, {"tactic": "exact (hfg.trans (hgk this) hc.le).congr_const (mul_div_cancel' _ hc.ne')", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.88772\nE : Type u_2\nF : Type u_3\nG : Type u_4\nE' : Type ?u.88784\nF' : Type ?u.88787\nG' : Type ?u.88790\nE'' : Type ?u.88793\nF'' : Type ?u.88796\nG'' : Type ?u.88799\nR : Type ?u.88802\nR' : Type ?u.88805\n\ud835\udd5c : Type ?u.88808\n\ud835\udd5c' : Type ?u.88811\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'\u271d 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\nhfg : IsBigOWith c l f g\nhc : 0 < c\nhgk : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 IsBigOWith c l g k\nc' : \u211d\nc'pos : 0 < c'\nthis : 0 < c' / c\n\u22a2 IsBigOWith c' l f k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.succAbove_right_inj", "start": [2190, 1], "end": [2191, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/PNat/Defs.lean", "full_name": "PNat.natPred_eq_pred", "start": [66, 1], "end": [67, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/LocallyConvex/WithSeminorms.lean", "full_name": "WithSeminorms.T1_of_separating", "start": [340, 1], "end": [348, 72], "traced_tactics": [{"tactic": "haveI := hp.topologicalAddGroup", "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type ?u.214092\n\ud835\udd5d : Type ?u.214095\n\ud835\udd5d\u2082 : Type ?u.214098\nE : Type u_2\nF : Type ?u.214104\nG : Type ?u.214107\n\u03b9 : Type u_3\n\u03b9' : Type ?u.214113\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : TopologicalSpace E\np : SeminormFamily \ud835\udd5c E \u03b9\nhp : WithSeminorms p\nh : \u2200 (x : E), x \u2260 0 \u2192 \u2203 i, \u2191(p i) x \u2260 0\n\u22a2 T1Space E", "state_after": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type ?u.214092\n\ud835\udd5d : Type ?u.214095\n\ud835\udd5d\u2082 : Type ?u.214098\nE : Type u_2\nF : Type ?u.214104\nG : Type ?u.214107\n\u03b9 : Type u_3\n\u03b9' : Type ?u.214113\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : TopologicalSpace E\np : SeminormFamily \ud835\udd5c E \u03b9\nhp : WithSeminorms p\nh : \u2200 (x : E), x \u2260 0 \u2192 \u2203 i, \u2191(p i) x \u2260 0\nthis : TopologicalAddGroup E\n\u22a2 T1Space E"}, {"tactic": "refine' TopologicalAddGroup.t1Space _ _", "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type ?u.214092\n\ud835\udd5d : Type ?u.214095\n\ud835\udd5d\u2082 : Type ?u.214098\nE : Type u_2\nF : Type ?u.214104\nG : Type ?u.214107\n\u03b9 : Type u_3\n\u03b9' : Type ?u.214113\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : TopologicalSpace E\np : SeminormFamily \ud835\udd5c E \u03b9\nhp : WithSeminorms p\nh : \u2200 (x : E), x \u2260 0 \u2192 \u2203 i, \u2191(p i) x \u2260 0\nthis : TopologicalAddGroup E\n\u22a2 T1Space E", "state_after": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type ?u.214092\n\ud835\udd5d : Type ?u.214095\n\ud835\udd5d\u2082 : Type ?u.214098\nE : Type u_2\nF : Type ?u.214104\nG : Type ?u.214107\n\u03b9 : Type u_3\n\u03b9' : Type ?u.214113\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : TopologicalSpace E\np : SeminormFamily \ud835\udd5c E \u03b9\nhp : WithSeminorms p\nh : \u2200 (x : E), x \u2260 0 \u2192 \u2203 i, \u2191(p i) x \u2260 0\nthis : TopologicalAddGroup E\n\u22a2 IsClosed {0}"}, {"tactic": "rw [\u2190 isOpen_compl_iff, hp.isOpen_iff_mem_balls]", "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type ?u.214092\n\ud835\udd5d : Type ?u.214095\n\ud835\udd5d\u2082 : Type ?u.214098\nE : Type u_2\nF : Type ?u.214104\nG : Type ?u.214107\n\u03b9 : Type u_3\n\u03b9' : Type ?u.214113\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : TopologicalSpace E\np : SeminormFamily \ud835\udd5c E \u03b9\nhp : WithSeminorms p\nh : \u2200 (x : E), x \u2260 0 \u2192 \u2203 i, \u2191(p i) x \u2260 0\nthis : TopologicalAddGroup E\n\u22a2 IsClosed {0}", "state_after": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type ?u.214092\n\ud835\udd5d : Type ?u.214095\n\ud835\udd5d\u2082 : Type ?u.214098\nE : Type u_2\nF : Type ?u.214104\nG : Type ?u.214107\n\u03b9 : Type u_3\n\u03b9' : Type ?u.214113\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : TopologicalSpace E\np : SeminormFamily \ud835\udd5c E \u03b9\nhp : WithSeminorms p\nh : \u2200 (x : E), x \u2260 0 \u2192 \u2203 i, \u2191(p i) x \u2260 0\nthis : TopologicalAddGroup E\n\u22a2 \u2200 (x : E), x \u2208 {0}\u1d9c \u2192 \u2203 s r, r > 0 \u2227 ball (Finset.sup s p) x r \u2286 {0}\u1d9c"}, {"tactic": "rintro x (hx : x \u2260 0)", "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type ?u.214092\n\ud835\udd5d : Type ?u.214095\n\ud835\udd5d\u2082 : Type ?u.214098\nE : Type u_2\nF : Type ?u.214104\nG : Type ?u.214107\n\u03b9 : Type u_3\n\u03b9' : Type ?u.214113\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : TopologicalSpace E\np : SeminormFamily \ud835\udd5c E \u03b9\nhp : WithSeminorms p\nh : \u2200 (x : E), x \u2260 0 \u2192 \u2203 i, \u2191(p i) x \u2260 0\nthis : TopologicalAddGroup E\n\u22a2 \u2200 (x : E), x \u2208 {0}\u1d9c \u2192 \u2203 s r, r > 0 \u2227 ball (Finset.sup s p) x r \u2286 {0}\u1d9c", "state_after": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type ?u.214092\n\ud835\udd5d : Type ?u.214095\n\ud835\udd5d\u2082 : Type ?u.214098\nE : Type u_2\nF : Type ?u.214104\nG : Type ?u.214107\n\u03b9 : Type u_3\n\u03b9' : Type ?u.214113\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : TopologicalSpace E\np : SeminormFamily \ud835\udd5c E \u03b9\nhp : WithSeminorms p\nh : \u2200 (x : E), x \u2260 0 \u2192 \u2203 i, \u2191(p i) x \u2260 0\nthis : TopologicalAddGroup E\nx : E\nhx : x \u2260 0\n\u22a2 \u2203 s r, r > 0 \u2227 ball (Finset.sup s p) x r \u2286 {0}\u1d9c"}, {"tactic": "cases' h x hx with i pi_nonzero", "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type ?u.214092\n\ud835\udd5d : Type ?u.214095\n\ud835\udd5d\u2082 : Type ?u.214098\nE : Type u_2\nF : Type ?u.214104\nG : Type ?u.214107\n\u03b9 : Type u_3\n\u03b9' : Type ?u.214113\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : TopologicalSpace E\np : SeminormFamily \ud835\udd5c E \u03b9\nhp : WithSeminorms p\nh : \u2200 (x : E), x \u2260 0 \u2192 \u2203 i, \u2191(p i) x \u2260 0\nthis : TopologicalAddGroup E\nx : E\nhx : x \u2260 0\n\u22a2 \u2203 s r, r > 0 \u2227 ball (Finset.sup s p) x r \u2286 {0}\u1d9c", "state_after": "case intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type ?u.214092\n\ud835\udd5d : Type ?u.214095\n\ud835\udd5d\u2082 : Type ?u.214098\nE : Type u_2\nF : Type ?u.214104\nG : Type ?u.214107\n\u03b9 : Type u_3\n\u03b9' : Type ?u.214113\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : TopologicalSpace E\np : SeminormFamily \ud835\udd5c E \u03b9\nhp : WithSeminorms p\nh : \u2200 (x : E), x \u2260 0 \u2192 \u2203 i, \u2191(p i) x \u2260 0\nthis : TopologicalAddGroup E\nx : E\nhx : x \u2260 0\ni : \u03b9\npi_nonzero : \u2191(p i) x \u2260 0\n\u22a2 \u2203 s r, r > 0 \u2227 ball (Finset.sup s p) x r \u2286 {0}\u1d9c"}, {"tactic": "refine' \u27e8{i}, p i x, by positivity, subset_compl_singleton_iff.mpr _\u27e9", "state_before": "case intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type ?u.214092\n\ud835\udd5d : Type ?u.214095\n\ud835\udd5d\u2082 : Type ?u.214098\nE : Type u_2\nF : Type ?u.214104\nG : Type ?u.214107\n\u03b9 : Type u_3\n\u03b9' : Type ?u.214113\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : TopologicalSpace E\np : SeminormFamily \ud835\udd5c E \u03b9\nhp : WithSeminorms p\nh : \u2200 (x : E), x \u2260 0 \u2192 \u2203 i, \u2191(p i) x \u2260 0\nthis : TopologicalAddGroup E\nx : E\nhx : x \u2260 0\ni : \u03b9\npi_nonzero : \u2191(p i) x \u2260 0\n\u22a2 \u2203 s r, r > 0 \u2227 ball (Finset.sup s p) x r \u2286 {0}\u1d9c", "state_after": "case intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type ?u.214092\n\ud835\udd5d : Type ?u.214095\n\ud835\udd5d\u2082 : Type ?u.214098\nE : Type u_2\nF : Type ?u.214104\nG : Type ?u.214107\n\u03b9 : Type u_3\n\u03b9' : Type ?u.214113\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : TopologicalSpace E\np : SeminormFamily \ud835\udd5c E \u03b9\nhp : WithSeminorms p\nh : \u2200 (x : E), x \u2260 0 \u2192 \u2203 i, \u2191(p i) x \u2260 0\nthis : TopologicalAddGroup E\nx : E\nhx : x \u2260 0\ni : \u03b9\npi_nonzero : \u2191(p i) x \u2260 0\n\u22a2 \u00ac0 \u2208 ball (Finset.sup {i} p) x (\u2191(p i) x)"}, {"tactic": "rw [Finset.sup_singleton, mem_ball, zero_sub, map_neg_eq_map, not_lt]", "state_before": "case intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type ?u.214092\n\ud835\udd5d : Type ?u.214095\n\ud835\udd5d\u2082 : Type ?u.214098\nE : Type u_2\nF : Type ?u.214104\nG : Type ?u.214107\n\u03b9 : Type u_3\n\u03b9' : Type ?u.214113\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : TopologicalSpace E\np : SeminormFamily \ud835\udd5c E \u03b9\nhp : WithSeminorms p\nh : \u2200 (x : E), x \u2260 0 \u2192 \u2203 i, \u2191(p i) x \u2260 0\nthis : TopologicalAddGroup E\nx : E\nhx : x \u2260 0\ni : \u03b9\npi_nonzero : \u2191(p i) x \u2260 0\n\u22a2 \u00ac0 \u2208 ball (Finset.sup {i} p) x (\u2191(p i) x)", "state_after": "no goals"}, {"tactic": "positivity", "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type ?u.214092\n\ud835\udd5d : Type ?u.214095\n\ud835\udd5d\u2082 : Type ?u.214098\nE : Type u_2\nF : Type ?u.214104\nG : Type ?u.214107\n\u03b9 : Type u_3\n\u03b9' : Type ?u.214113\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : TopologicalSpace E\np : SeminormFamily \ud835\udd5c E \u03b9\nhp : WithSeminorms p\nh : \u2200 (x : E), x \u2260 0 \u2192 \u2203 i, \u2191(p i) x \u2260 0\nthis : TopologicalAddGroup E\nx : E\nhx : x \u2260 0\ni : \u03b9\npi_nonzero : \u2191(p i) x \u2260 0\n\u22a2 \u2191(p i) x > 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.QuasiMeasurePreserving.ae_eq_comp", "start": [2881, 1], "end": [2883, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "full_name": "Polynomial.degree_monomial_le", "start": [302, 1], "end": [304, 38], "traced_tactics": [{"tactic": "rw [h, (monomial n).map_zero]", "state_before": "R : Type u\nS : Type v\na\u271d b c d : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nn : \u2115\na : R\nh : a = 0\n\u22a2 degree (\u2191(monomial n) a) \u2264 \u2191n", "state_after": "R : Type u\nS : Type v\na\u271d b c d : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nn : \u2115\na : R\nh : a = 0\n\u22a2 degree 0 \u2264 \u2191n"}, {"tactic": "exact bot_le", "state_before": "R : Type u\nS : Type v\na\u271d b c d : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nn : \u2115\na : R\nh : a = 0\n\u22a2 degree 0 \u2264 \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean", "full_name": "Finset.card_pow_div_pow_le", "start": [219, 1], "end": [235, 33], "traced_tactics": [{"tactic": "have hA' : A \u2208 A.powerset.erase \u2205 := mem_erase_of_ne_of_mem hA.ne_empty (mem_powerset_self _)", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\n\u22a2 \u2191(card (B ^ m / B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\n\u22a2 \u2191(card (B ^ m / B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A)"}, {"tactic": "obtain \u27e8C, hC, hCA\u27e9 :=\n  exists_min_image (A.powerset.erase \u2205) (fun C \u21a6 (C * B).card / C.card : _ \u2192 \u211a\u22650) \u27e8A, hA'\u27e9", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\n\u22a2 \u2191(card (B ^ m / B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A)", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : C \u2208 erase (powerset A) \u2205\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (B ^ m / B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A)"}, {"tactic": "rw [mem_erase, mem_powerset, \u2190 nonempty_iff_ne_empty] at hC", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : C \u2208 erase (powerset A) \u2205\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (B ^ m / B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A)", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (B ^ m / B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A)"}, {"tactic": "refine' (mul_le_mul_right <| cast_pos.2 hC.1.card_pos).1 _", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (B ^ m / B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A)", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (B ^ m / B ^ n)) * \u2191(card C) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A) * \u2191(card C)"}, {"tactic": "norm_cast", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (B ^ m / B ^ n)) * \u2191(card C) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A) * \u2191(card C)", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (B ^ m / B ^ n) * card C) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A) * \u2191(card C)"}, {"tactic": "refine' (cast_le.2 <| card_div_mul_le_card_mul_mul_card_mul _ _ _).trans _", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (B ^ m / B ^ n) * card C) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A) * \u2191(card C)", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (B ^ m * C) * card (C * B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A) * \u2191(card C)"}, {"tactic": "push_cast", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (B ^ m * C) * card (C * B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A) * \u2191(card C)", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (B ^ m * C)) * \u2191(card (C * B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A) * \u2191(card C)"}, {"tactic": "rw [mul_comm _ C]", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (B ^ m * C)) * \u2191(card (C * B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A) * \u2191(card C)", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (C * B ^ m)) * \u2191(card (C * B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A) * \u2191(card C)"}, {"tactic": "refine' (mul_le_mul (card_mul_pow_le (mul_aux hC.1 hC.2 hCA) _)\n  (card_mul_pow_le (mul_aux hC.1 hC.2 hCA) _) (zero_le _) (zero_le _)).trans _", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (C * B ^ m)) * \u2191(card (C * B ^ n)) \u2264 (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A) * \u2191(card C)", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 (\u2191(card (C * B)) / \u2191(card C)) ^ m * \u2191(card C) * ((\u2191(card (C * B)) / \u2191(card C)) ^ n * \u2191(card C)) \u2264\n    (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A) * \u2191(card C)"}, {"tactic": "rw [mul_mul_mul_comm, \u2190 pow_add, \u2190 mul_assoc]", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 (\u2191(card (C * B)) / \u2191(card C)) ^ m * \u2191(card C) * ((\u2191(card (C * B)) / \u2191(card C)) ^ n * \u2191(card C)) \u2264\n    (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A) * \u2191(card C)", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 (\u2191(card (C * B)) / \u2191(card C)) ^ (m + n) * \u2191(card C) * \u2191(card C) \u2264\n    (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A) * \u2191(card C)"}, {"tactic": "gcongr ((?_ ^ _) * Nat.cast ?_) * _", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 (\u2191(card (C * B)) / \u2191(card C)) ^ (m + n) * \u2191(card C) * \u2191(card C) \u2264\n    (\u2191(card (A * B)) / \u2191(card A)) ^ (m + n) * \u2191(card A) * \u2191(card C)", "state_after": "case intro.intro.bc.h\u2081.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (A * B)) / \u2191(card A)\n\ncase intro.intro.bc.h\u2082.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 card C \u2264 card A"}, {"tactic": "exact hCA _ hA'", "state_before": "case intro.intro.bc.h\u2081.hab\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (A * B)) / \u2191(card A)", "state_after": "no goals"}, {"tactic": "exact card_le_of_subset hC.2", "state_before": "case intro.intro.bc.h\u2082.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : DecidableEq \u03b1\nA B\u271d C\u271d : Finset \u03b1\nhA : Finset.Nonempty A\nB : Finset \u03b1\nm n : \u2115\nhA' : A \u2208 erase (powerset A) \u2205\nC : Finset \u03b1\nhC : Finset.Nonempty C \u2227 C \u2286 A\nhCA : \u2200 (x' : Finset \u03b1), x' \u2208 erase (powerset A) \u2205 \u2192 \u2191(card (C * B)) / \u2191(card C) \u2264 \u2191(card (x' * B)) / \u2191(card x')\n\u22a2 card C \u2264 card A", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/Padics/PadicIntegers.lean", "full_name": "PadicInt.prime_p", "start": [609, 1], "end": [612, 32], "traced_tactics": [{"tactic": "rw [\u2190 Ideal.span_singleton_prime, \u2190 maximalIdeal_eq_span_p]", "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 Prime \u2191p", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 Ideal.IsPrime (maximalIdeal \u2124_[p])\n\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 \u2191p \u2260 0"}, {"tactic": "infer_instance", "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 Ideal.IsPrime (maximalIdeal \u2124_[p])", "state_after": "no goals"}, {"tactic": "exact_mod_cast hp.1.ne_zero", "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 \u2191p \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/ENNReal.lean", "full_name": "ENNReal.coe_le_coe", "start": [330, 20], "end": [330, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/ENNReal.lean", "full_name": "ENNReal.exists_nnreal_pos_mul_lt", "start": [1840, 1], "end": [1843, 34], "traced_tactics": [{"tactic": "rcases exists_nat_pos_inv_mul_lt ha hb with \u27e8n, npos : 0 < n, hn\u27e9", "state_before": "\u03b1 : Type ?u.342393\n\u03b2 : Type ?u.342396\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 \u22a4\nhb : b \u2260 0\n\u22a2 \u2203 n, n > 0 \u2227 \u2191n * a < b", "state_after": "case intro.intro\n\u03b1 : Type ?u.342393\n\u03b2 : Type ?u.342396\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 \u22a4\nhb : b \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : (\u2191n)\u207b\u00b9 * a < b\n\u22a2 \u2203 n, n > 0 \u2227 \u2191n * a < b"}, {"tactic": "use (n : \u211d\u22650)\u207b\u00b9", "state_before": "case intro.intro\n\u03b1 : Type ?u.342393\n\u03b2 : Type ?u.342396\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 \u22a4\nhb : b \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : (\u2191n)\u207b\u00b9 * a < b\n\u22a2 \u2203 n, n > 0 \u2227 \u2191n * a < b", "state_after": "case intro.intro\n\u03b1 : Type ?u.342393\n\u03b2 : Type ?u.342396\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 \u22a4\nhb : b \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : (\u2191n)\u207b\u00b9 * a < b\n\u22a2 (\u2191n)\u207b\u00b9 > 0 \u2227 \u2191(\u2191n)\u207b\u00b9 * a < b"}, {"tactic": "simp [*, npos.ne', zero_lt_one]", "state_before": "case intro.intro\n\u03b1 : Type ?u.342393\n\u03b2 : Type ?u.342396\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nha : a \u2260 \u22a4\nhb : b \u2260 0\nn : \u2115\nnpos : 0 < n\nhn : (\u2191n)\u207b\u00b9 * a < b\n\u22a2 (\u2191n)\u207b\u00b9 > 0 \u2227 \u2191(\u2191n)\u207b\u00b9 * a < b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/InfiniteSum/Module.lean", "full_name": "tsum_smul_const", "start": [32, 1], "end": [33, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Splits.lean", "full_name": "Polynomial.splits_map_iff", "start": [132, 1], "end": [133, 36], "traced_tactics": [{"tactic": "simp [Splits, Polynomial.map_map]", "state_before": "F : Type u\nK : Type v\nL : Type w\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : Field L\ninst\u271d : Field F\ni : K \u2192+* L\nj : L \u2192+* F\nf : K[X]\n\u22a2 Splits j (map i f) \u2194 Splits (RingHom.comp j i) f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Separation.lean", "full_name": "Embedding.t0Space", "start": [329, 11], "end": [331, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Quiver/Path.lean", "full_name": "Quiver.Path.comp_injective_right", "start": [151, 1], "end": [152, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "full_name": "Asymptotics.isBigO_const_mul_right_iff", "start": [1541, 1], "end": [1543, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/RatFunc.lean", "full_name": "RatFunc.num_algebraMap", "start": [1170, 1], "end": [1170, 98], "traced_tactics": [{"tactic": "convert num_div p 1 <;> simp", "state_before": "K : Type u\ninst\u271d : Field K\np : K[X]\n\u22a2 num (\u2191(algebraMap K[X] (RatFunc K)) p) = p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean", "full_name": "Real.arccos_lt_pi_div_two", "start": [437, 1], "end": [437, 84], "traced_tactics": [{"tactic": "simp [arccos]", "state_before": "x : \u211d\n\u22a2 arccos x < \u03c0 / 2 \u2194 0 < x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/BilinearForm.lean", "full_name": "BilinForm.restrictOrthogonalSpanSingletonNondegenerate", "start": [1515, 1], "end": [1524, 77], "traced_tactics": [{"tactic": "refine' fun m hm => Submodule.coe_eq_zero.1 (b\u2081 m.1 fun n => _)", "state_before": "R : Type ?u.1785101\nM : Type ?u.1785104\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type ?u.1785140\nM\u2081 : Type ?u.1785143\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.1785752\nM\u2082 : Type ?u.1785755\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.1785942\nM\u2083 : Type ?u.1785945\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_2\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type ?u.1787750\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nb\u2081 : Nondegenerate B\nb\u2082 : IsRefl B\nx : V\nhx : \u00acIsOrtho B x x\n\u22a2 Nondegenerate (restrict B (orthogonal B (Submodule.span K {x})))", "state_after": "R : Type ?u.1785101\nM : Type ?u.1785104\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type ?u.1785140\nM\u2081 : Type ?u.1785143\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.1785752\nM\u2082 : Type ?u.1785755\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.1785942\nM\u2083 : Type ?u.1785945\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_2\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type ?u.1787750\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nb\u2081 : Nondegenerate B\nb\u2082 : IsRefl B\nx : V\nhx : \u00acIsOrtho B x x\nm : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }\nhm :\n  \u2200 (n : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }),\n    bilin (restrict B (orthogonal B (Submodule.span K {x}))) m n = 0\nn : V\n\u22a2 bilin B (\u2191m) n = 0"}, {"tactic": "have : n \u2208 (K \u2219 x) \u2294 B.orthogonal (K \u2219 x) :=\n  (span_singleton_sup_orthogonal_eq_top hx).symm \u25b8 Submodule.mem_top", "state_before": "R : Type ?u.1785101\nM : Type ?u.1785104\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type ?u.1785140\nM\u2081 : Type ?u.1785143\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.1785752\nM\u2082 : Type ?u.1785755\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.1785942\nM\u2083 : Type ?u.1785945\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_2\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type ?u.1787750\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nb\u2081 : Nondegenerate B\nb\u2082 : IsRefl B\nx : V\nhx : \u00acIsOrtho B x x\nm : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }\nhm :\n  \u2200 (n : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }),\n    bilin (restrict B (orthogonal B (Submodule.span K {x}))) m n = 0\nn : V\n\u22a2 bilin B (\u2191m) n = 0", "state_after": "R : Type ?u.1785101\nM : Type ?u.1785104\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type ?u.1785140\nM\u2081 : Type ?u.1785143\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.1785752\nM\u2082 : Type ?u.1785755\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.1785942\nM\u2083 : Type ?u.1785945\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_2\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type ?u.1787750\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nb\u2081 : Nondegenerate B\nb\u2082 : IsRefl B\nx : V\nhx : \u00acIsOrtho B x x\nm : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }\nhm :\n  \u2200 (n : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }),\n    bilin (restrict B (orthogonal B (Submodule.span K {x}))) m n = 0\nn : V\nthis : n \u2208 Submodule.span K {x} \u2294 orthogonal B (Submodule.span K {x})\n\u22a2 bilin B (\u2191m) n = 0"}, {"tactic": "rcases Submodule.mem_sup.1 this with \u27e8y, hy, z, hz, rfl\u27e9", "state_before": "R : Type ?u.1785101\nM : Type ?u.1785104\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type ?u.1785140\nM\u2081 : Type ?u.1785143\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.1785752\nM\u2082 : Type ?u.1785755\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.1785942\nM\u2083 : Type ?u.1785945\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_2\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type ?u.1787750\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nb\u2081 : Nondegenerate B\nb\u2082 : IsRefl B\nx : V\nhx : \u00acIsOrtho B x x\nm : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }\nhm :\n  \u2200 (n : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }),\n    bilin (restrict B (orthogonal B (Submodule.span K {x}))) m n = 0\nn : V\nthis : n \u2208 Submodule.span K {x} \u2294 orthogonal B (Submodule.span K {x})\n\u22a2 bilin B (\u2191m) n = 0", "state_after": "case intro.intro.intro.intro\nR : Type ?u.1785101\nM : Type ?u.1785104\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type ?u.1785140\nM\u2081 : Type ?u.1785143\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.1785752\nM\u2082 : Type ?u.1785755\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.1785942\nM\u2083 : Type ?u.1785945\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_2\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type ?u.1787750\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nb\u2081 : Nondegenerate B\nb\u2082 : IsRefl B\nx : V\nhx : \u00acIsOrtho B x x\nm : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }\nhm :\n  \u2200 (n : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }),\n    bilin (restrict B (orthogonal B (Submodule.span K {x}))) m n = 0\ny : V\nhy : y \u2208 Submodule.span K {x}\nz : V\nhz : z \u2208 orthogonal B (Submodule.span K {x})\nthis : y + z \u2208 Submodule.span K {x} \u2294 orthogonal B (Submodule.span K {x})\n\u22a2 bilin B (\u2191m) (y + z) = 0"}, {"tactic": "specialize hm \u27e8z, hz\u27e9", "state_before": "case intro.intro.intro.intro\nR : Type ?u.1785101\nM : Type ?u.1785104\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type ?u.1785140\nM\u2081 : Type ?u.1785143\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.1785752\nM\u2082 : Type ?u.1785755\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.1785942\nM\u2083 : Type ?u.1785945\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_2\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type ?u.1787750\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nb\u2081 : Nondegenerate B\nb\u2082 : IsRefl B\nx : V\nhx : \u00acIsOrtho B x x\nm : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }\nhm :\n  \u2200 (n : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }),\n    bilin (restrict B (orthogonal B (Submodule.span K {x}))) m n = 0\ny : V\nhy : y \u2208 Submodule.span K {x}\nz : V\nhz : z \u2208 orthogonal B (Submodule.span K {x})\nthis : y + z \u2208 Submodule.span K {x} \u2294 orthogonal B (Submodule.span K {x})\n\u22a2 bilin B (\u2191m) (y + z) = 0", "state_after": "case intro.intro.intro.intro\nR : Type ?u.1785101\nM : Type ?u.1785104\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type ?u.1785140\nM\u2081 : Type ?u.1785143\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.1785752\nM\u2082 : Type ?u.1785755\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.1785942\nM\u2083 : Type ?u.1785945\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_2\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type ?u.1787750\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nb\u2081 : Nondegenerate B\nb\u2082 : IsRefl B\nx : V\nhx : \u00acIsOrtho B x x\nm : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }\ny : V\nhy : y \u2208 Submodule.span K {x}\nz : V\nhz : z \u2208 orthogonal B (Submodule.span K {x})\nthis : y + z \u2208 Submodule.span K {x} \u2294 orthogonal B (Submodule.span K {x})\nhm : bilin (restrict B (orthogonal B (Submodule.span K {x}))) m { val := z, property := hz } = 0\n\u22a2 bilin B (\u2191m) (y + z) = 0"}, {"tactic": "rw [restrict] at hm", "state_before": "case intro.intro.intro.intro\nR : Type ?u.1785101\nM : Type ?u.1785104\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type ?u.1785140\nM\u2081 : Type ?u.1785143\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.1785752\nM\u2082 : Type ?u.1785755\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.1785942\nM\u2083 : Type ?u.1785945\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_2\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type ?u.1787750\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nb\u2081 : Nondegenerate B\nb\u2082 : IsRefl B\nx : V\nhx : \u00acIsOrtho B x x\nm : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }\ny : V\nhy : y \u2208 Submodule.span K {x}\nz : V\nhz : z \u2208 orthogonal B (Submodule.span K {x})\nthis : y + z \u2208 Submodule.span K {x} \u2294 orthogonal B (Submodule.span K {x})\nhm : bilin (restrict B (orthogonal B (Submodule.span K {x}))) m { val := z, property := hz } = 0\n\u22a2 bilin B (\u2191m) (y + z) = 0", "state_after": "case intro.intro.intro.intro\nR : Type ?u.1785101\nM : Type ?u.1785104\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type ?u.1785140\nM\u2081 : Type ?u.1785143\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.1785752\nM\u2082 : Type ?u.1785755\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.1785942\nM\u2083 : Type ?u.1785945\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_2\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type ?u.1787750\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nb\u2081 : Nondegenerate B\nb\u2082 : IsRefl B\nx : V\nhx : \u00acIsOrtho B x x\nm : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }\ny : V\nhy : y \u2208 Submodule.span K {x}\nz : V\nhz : z \u2208 orthogonal B (Submodule.span K {x})\nthis : y + z \u2208 Submodule.span K {x} \u2294 orthogonal B (Submodule.span K {x})\nhm :\n  bilin\n      { bilin := fun a b => bilin B \u2191a \u2191b,\n        bilin_add_left :=\n          (_ :\n            \u2200 (x_1 x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_1 + \u2191x_2) \u2191x_3 = bilin B \u2191x_1 \u2191x_3 + bilin B \u2191x_2 \u2191x_3),\n        bilin_smul_left :=\n          (_ :\n            \u2200 (x_1 : K) (x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (x_1 \u2022 \u2191x_2) \u2191x_3 = x_1 * bilin B \u2191x_2 \u2191x_3),\n        bilin_add_right :=\n          (_ :\n            \u2200 (x_1 x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_1) (\u2191x_2 + \u2191x_3) = bilin B \u2191x_1 \u2191x_2 + bilin B \u2191x_1 \u2191x_3),\n        bilin_smul_right :=\n          (_ :\n            \u2200 (x_1 : K) (x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_2) (x_1 \u2022 \u2191x_3) = x_1 * bilin B \u2191x_2 \u2191x_3) }\n      m { val := z, property := hz } =\n    0\n\u22a2 bilin B (\u2191m) (y + z) = 0"}, {"tactic": "erw [add_right, show B m.1 y = 0 by rw [b\u2082]; exact m.2 y hy, hm, add_zero]", "state_before": "case intro.intro.intro.intro\nR : Type ?u.1785101\nM : Type ?u.1785104\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type ?u.1785140\nM\u2081 : Type ?u.1785143\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.1785752\nM\u2082 : Type ?u.1785755\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.1785942\nM\u2083 : Type ?u.1785945\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_2\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type ?u.1787750\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nb\u2081 : Nondegenerate B\nb\u2082 : IsRefl B\nx : V\nhx : \u00acIsOrtho B x x\nm : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }\ny : V\nhy : y \u2208 Submodule.span K {x}\nz : V\nhz : z \u2208 orthogonal B (Submodule.span K {x})\nthis : y + z \u2208 Submodule.span K {x} \u2294 orthogonal B (Submodule.span K {x})\nhm :\n  bilin\n      { bilin := fun a b => bilin B \u2191a \u2191b,\n        bilin_add_left :=\n          (_ :\n            \u2200 (x_1 x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_1 + \u2191x_2) \u2191x_3 = bilin B \u2191x_1 \u2191x_3 + bilin B \u2191x_2 \u2191x_3),\n        bilin_smul_left :=\n          (_ :\n            \u2200 (x_1 : K) (x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (x_1 \u2022 \u2191x_2) \u2191x_3 = x_1 * bilin B \u2191x_2 \u2191x_3),\n        bilin_add_right :=\n          (_ :\n            \u2200 (x_1 x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_1) (\u2191x_2 + \u2191x_3) = bilin B \u2191x_1 \u2191x_2 + bilin B \u2191x_1 \u2191x_3),\n        bilin_smul_right :=\n          (_ :\n            \u2200 (x_1 : K) (x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_2) (x_1 \u2022 \u2191x_3) = x_1 * bilin B \u2191x_2 \u2191x_3) }\n      m { val := z, property := hz } =\n    0\n\u22a2 bilin B (\u2191m) (y + z) = 0", "state_after": "no goals"}, {"tactic": "rw [b\u2082]", "state_before": "R : Type ?u.1785101\nM : Type ?u.1785104\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type ?u.1785140\nM\u2081 : Type ?u.1785143\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.1785752\nM\u2082 : Type ?u.1785755\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.1785942\nM\u2083 : Type ?u.1785945\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_2\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type ?u.1787750\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nb\u2081 : Nondegenerate B\nb\u2082 : IsRefl B\nx : V\nhx : \u00acIsOrtho B x x\nm : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }\ny : V\nhy : y \u2208 Submodule.span K {x}\nz : V\nhz : z \u2208 orthogonal B (Submodule.span K {x})\nthis : y + z \u2208 Submodule.span K {x} \u2294 orthogonal B (Submodule.span K {x})\nhm :\n  bilin\n      { bilin := fun a b => bilin B \u2191a \u2191b,\n        bilin_add_left :=\n          (_ :\n            \u2200 (x_1 x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_1 + \u2191x_2) \u2191x_3 = bilin B \u2191x_1 \u2191x_3 + bilin B \u2191x_2 \u2191x_3),\n        bilin_smul_left :=\n          (_ :\n            \u2200 (x_1 : K) (x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (x_1 \u2022 \u2191x_2) \u2191x_3 = x_1 * bilin B \u2191x_2 \u2191x_3),\n        bilin_add_right :=\n          (_ :\n            \u2200 (x_1 x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_1) (\u2191x_2 + \u2191x_3) = bilin B \u2191x_1 \u2191x_2 + bilin B \u2191x_1 \u2191x_3),\n        bilin_smul_right :=\n          (_ :\n            \u2200 (x_1 : K) (x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_2) (x_1 \u2022 \u2191x_3) = x_1 * bilin B \u2191x_2 \u2191x_3) }\n      m { val := z, property := hz } =\n    0\n\u22a2 bilin B (\u2191m) y = 0", "state_after": "case a\nR : Type ?u.1785101\nM : Type ?u.1785104\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type ?u.1785140\nM\u2081 : Type ?u.1785143\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.1785752\nM\u2082 : Type ?u.1785755\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.1785942\nM\u2083 : Type ?u.1785945\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_2\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type ?u.1787750\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nb\u2081 : Nondegenerate B\nb\u2082 : IsRefl B\nx : V\nhx : \u00acIsOrtho B x x\nm : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }\ny : V\nhy : y \u2208 Submodule.span K {x}\nz : V\nhz : z \u2208 orthogonal B (Submodule.span K {x})\nthis : y + z \u2208 Submodule.span K {x} \u2294 orthogonal B (Submodule.span K {x})\nhm :\n  bilin\n      { bilin := fun a b => bilin B \u2191a \u2191b,\n        bilin_add_left :=\n          (_ :\n            \u2200 (x_1 x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_1 + \u2191x_2) \u2191x_3 = bilin B \u2191x_1 \u2191x_3 + bilin B \u2191x_2 \u2191x_3),\n        bilin_smul_left :=\n          (_ :\n            \u2200 (x_1 : K) (x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (x_1 \u2022 \u2191x_2) \u2191x_3 = x_1 * bilin B \u2191x_2 \u2191x_3),\n        bilin_add_right :=\n          (_ :\n            \u2200 (x_1 x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_1) (\u2191x_2 + \u2191x_3) = bilin B \u2191x_1 \u2191x_2 + bilin B \u2191x_1 \u2191x_3),\n        bilin_smul_right :=\n          (_ :\n            \u2200 (x_1 : K) (x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_2) (x_1 \u2022 \u2191x_3) = x_1 * bilin B \u2191x_2 \u2191x_3) }\n      m { val := z, property := hz } =\n    0\n\u22a2 bilin B y \u2191m = 0"}, {"tactic": "exact m.2 y hy", "state_before": "case a\nR : Type ?u.1785101\nM : Type ?u.1785104\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type ?u.1785140\nM\u2081 : Type ?u.1785143\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.1785752\nM\u2082 : Type ?u.1785755\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.1785942\nM\u2083 : Type ?u.1785945\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_2\nK : Type u_1\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM\u2082' : Type ?u.1787750\ninst\u271d\u00b9 : AddCommMonoid M\u2082'\ninst\u271d : Module R\u2082 M\u2082'\nB : BilinForm K V\nb\u2081 : Nondegenerate B\nb\u2082 : IsRefl B\nx : V\nhx : \u00acIsOrtho B x x\nm : { x_1 // x_1 \u2208 orthogonal B (Submodule.span K {x}) }\ny : V\nhy : y \u2208 Submodule.span K {x}\nz : V\nhz : z \u2208 orthogonal B (Submodule.span K {x})\nthis : y + z \u2208 Submodule.span K {x} \u2294 orthogonal B (Submodule.span K {x})\nhm :\n  bilin\n      { bilin := fun a b => bilin B \u2191a \u2191b,\n        bilin_add_left :=\n          (_ :\n            \u2200 (x_1 x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_1 + \u2191x_2) \u2191x_3 = bilin B \u2191x_1 \u2191x_3 + bilin B \u2191x_2 \u2191x_3),\n        bilin_smul_left :=\n          (_ :\n            \u2200 (x_1 : K) (x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (x_1 \u2022 \u2191x_2) \u2191x_3 = x_1 * bilin B \u2191x_2 \u2191x_3),\n        bilin_add_right :=\n          (_ :\n            \u2200 (x_1 x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_1) (\u2191x_2 + \u2191x_3) = bilin B \u2191x_1 \u2191x_2 + bilin B \u2191x_1 \u2191x_3),\n        bilin_smul_right :=\n          (_ :\n            \u2200 (x_1 : K) (x_2 x_3 : { x_3 // x_3 \u2208 orthogonal B (Submodule.span K {x}) }),\n              bilin B (\u2191x_2) (x_1 \u2022 \u2191x_3) = x_1 * bilin B \u2191x_2 \u2191x_3) }\n      m { val := z, property := hz } =\n    0\n\u22a2 bilin B y \u2191m = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subgroup/Finite.lean", "full_name": "Subgroup.card_le_one_iff_eq_bot", "start": [162, 1], "end": [166, 26], "traced_tactics": [{"tactic": "simpa [Subtype.ext_iff] using Fintype.card_le_one_iff.1 h \u27e8x, hx\u27e9 1", "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\nA : Type ?u.15911\ninst\u271d\u00b9 : AddGroup A\nH K : Subgroup G\ninst\u271d : Fintype { x // x \u2208 H }\nh : Fintype.card { x // x \u2208 H } \u2264 1\nx : G\nhx : x \u2208 H\n\u22a2 x = 1", "state_after": "no goals"}, {"tactic": "simp [h]", "state_before": "G : Type 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Type ?u.379456\nR : Type ?u.379459\nS : Type ?u.379462\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type ?u.379469\ninst\u271d\u00b9 : NonUnitalNonAssocRing \u03b1\ninst\u271d : Fintype n\nM : Matrix m n \u03b1\nN : Matrix n o \u03b1\ni\u271d : m\nx\u271d : o\n\u22a2 ((-M) \u2b1d N) i\u271d x\u271d = (-M \u2b1d N) i\u271d x\u271d"}, {"tactic": "apply neg_dotProduct", "state_before": "case a.h\nl : Type ?u.379437\nm : Type u_1\nn : Type u_2\no : Type u_3\nm' : o \u2192 Type ?u.379451\nn' : o \u2192 Type ?u.379456\nR : Type ?u.379459\nS : Type ?u.379462\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type ?u.379469\ninst\u271d\u00b9 : NonUnitalNonAssocRing \u03b1\ninst\u271d : Fintype n\nM : Matrix m n \u03b1\nN : Matrix n o \u03b1\ni\u271d : m\nx\u271d : o\n\u22a2 ((-M) \u2b1d N) i\u271d x\u271d = (-M \u2b1d N) i\u271d x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": 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: Type ?u.376198\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf\u271d g : \u03b1 \u2192 M\na b : \u03b1\ns\u271d t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2 \u2192 M\ns : Finset \u03b1\nt : Finset \u03b2\n\u22a2 (\u220f\u1da0 (i : \u03b1) (_ : i \u2208 \u2191s) (j : \u03b2) (_ : j \u2208 \u2191t), f i j) = \u220f\u1da0 (j : \u03b2) (_ : j \u2208 \u2191t) (i : \u03b1) (_ : i \u2208 \u2191s), f i j"}, {"tactic": "simp only [finprod_mem_coe_finset]", "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type ?u.376189\nG : Type ?u.376192\nM : Type u_3\nN : Type ?u.376198\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf\u271d g : \u03b1 \u2192 M\na b : \u03b1\ns\u271d t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2 \u2192 M\ns : Finset \u03b1\nt : Finset \u03b2\n\u22a2 (\u220f\u1da0 (i : \u03b1) (_ : i \u2208 \u2191s) (j : \u03b2) (_ : j \u2208 \u2191t), f i j) = \u220f\u1da0 (j : \u03b2) (_ : j \u2208 \u2191t) (i : \u03b1) (_ : i 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"full_name": "CompositionAsSet.toComposition_boundaries", "start": [1047, 1], "end": [1059, 81], "traced_tactics": [{"tactic": "ext j", "state_before": "n : \u2115\nc : CompositionAsSet n\n\u22a2 Composition.boundaries (toComposition c) = c.boundaries", "state_after": "case a\nn : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\n\u22a2 j \u2208 Composition.boundaries (toComposition c) \u2194 j \u2208 c.boundaries"}, {"tactic": "simp only [c.mem_boundaries_iff_exists_blocks_sum_take_eq, Composition.boundaries, Finset.mem_map]", "state_before": "case a\nn : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\n\u22a2 j \u2208 Composition.boundaries (toComposition c) \u2194 j \u2208 c.boundaries", "state_after": "case a\nn : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\n\u22a2 (\u2203 a, a \u2208 Finset.univ \u2227 \u2191(Composition.boundary (toComposition c)).toEmbedding a = j) \u2194\n    \u2203 i, i < Finset.card c.boundaries \u2227 sum (take i (blocks c)) = \u2191j"}, {"tactic": "constructor", "state_before": "case a\nn : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\n\u22a2 (\u2203 a, a \u2208 Finset.univ \u2227 \u2191(Composition.boundary (toComposition c)).toEmbedding a = j) \u2194\n    \u2203 i, i < Finset.card c.boundaries \u2227 sum (take i (blocks c)) = \u2191j", "state_after": "case a.mp\nn : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\n\u22a2 (\u2203 a, a \u2208 Finset.univ \u2227 \u2191(Composition.boundary (toComposition c)).toEmbedding a = j) \u2192\n    \u2203 i, i < Finset.card c.boundaries \u2227 sum (take i (blocks c)) = \u2191j\n\ncase a.mpr\nn : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\n\u22a2 (\u2203 i, i < Finset.card c.boundaries \u2227 sum (take i (blocks c)) = \u2191j) \u2192\n    \u2203 a, a \u2208 Finset.univ \u2227 \u2191(Composition.boundary (toComposition c)).toEmbedding a = j"}, {"tactic": "rintro \u27e8i, _, hi\u27e9", "state_before": "case a.mp\nn : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\n\u22a2 (\u2203 a, a \u2208 Finset.univ \u2227 \u2191(Composition.boundary (toComposition c)).toEmbedding a = j) \u2192\n    \u2203 i, i < Finset.card c.boundaries \u2227 sum (take i (blocks c)) = \u2191j", "state_after": "case a.mp.intro.intro\nn : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\ni : Fin (Composition.length (toComposition c) + 1)\nleft\u271d : i \u2208 Finset.univ\nhi : \u2191(Composition.boundary (toComposition c)).toEmbedding i = j\n\u22a2 \u2203 i, i < Finset.card c.boundaries \u2227 sum (take i (blocks c)) = \u2191j"}, {"tactic": "refine' \u27e8i.1, _, _\u27e9", "state_before": "case a.mp.intro.intro\nn : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\ni : Fin (Composition.length (toComposition c) + 1)\nleft\u271d : i \u2208 Finset.univ\nhi : \u2191(Composition.boundary (toComposition c)).toEmbedding i = j\n\u22a2 \u2203 i, i < Finset.card c.boundaries \u2227 sum (take i (blocks c)) = \u2191j", "state_after": "case a.mp.intro.intro.refine'_1\nn : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\ni : Fin (Composition.length (toComposition c) + 1)\nleft\u271d : i \u2208 Finset.univ\nhi : \u2191(Composition.boundary (toComposition c)).toEmbedding i = j\n\u22a2 \u2191i < Finset.card c.boundaries\n\ncase a.mp.intro.intro.refine'_2\nn : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\ni : Fin (Composition.length (toComposition c) + 1)\nleft\u271d : i \u2208 Finset.univ\nhi : \u2191(Composition.boundary (toComposition c)).toEmbedding i = j\n\u22a2 sum (take (\u2191i) (blocks c)) = \u2191j"}, {"tactic": "simpa [c.card_boundaries_eq_succ_length] using i.2", "state_before": "case a.mp.intro.intro.refine'_1\nn : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\ni : Fin (Composition.length (toComposition c) + 1)\nleft\u271d : i \u2208 Finset.univ\nhi : \u2191(Composition.boundary (toComposition c)).toEmbedding i = j\n\u22a2 \u2191i < Finset.card c.boundaries\n\ncase a.mp.intro.intro.refine'_2\nn : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\ni : Fin (Composition.length (toComposition c) + 1)\nleft\u271d : i \u2208 Finset.univ\nhi : \u2191(Composition.boundary (toComposition c)).toEmbedding i = j\n\u22a2 sum (take (\u2191i) (blocks c)) = \u2191j", "state_after": "case a.mp.intro.intro.refine'_2\nn : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\ni : Fin (Composition.length (toComposition c) + 1)\nleft\u271d : i \u2208 Finset.univ\nhi : \u2191(Composition.boundary (toComposition c)).toEmbedding i = j\n\u22a2 sum (take (\u2191i) (blocks c)) = \u2191j"}, {"tactic": "simp [Composition.boundary, Composition.sizeUpTo, \u2190 hi]", "state_before": "case a.mp.intro.intro.refine'_2\nn : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\ni : Fin (Composition.length (toComposition c) + 1)\nleft\u271d : i \u2208 Finset.univ\nhi : \u2191(Composition.boundary (toComposition c)).toEmbedding i = j\n\u22a2 sum (take (\u2191i) (blocks c)) = \u2191j", "state_after": "no goals"}, {"tactic": "rintro \u27e8i, i_lt, hi\u27e9", "state_before": "case a.mpr\nn : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\n\u22a2 (\u2203 i, i < Finset.card c.boundaries \u2227 sum (take i (blocks c)) = \u2191j) \u2192\n    \u2203 a, a \u2208 Finset.univ \u2227 \u2191(Composition.boundary (toComposition c)).toEmbedding a = j", "state_after": "case a.mpr.intro.intro\nn : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\ni : \u2115\ni_lt : i < Finset.card c.boundaries\nhi : sum (take i (blocks c)) = \u2191j\n\u22a2 \u2203 a, a \u2208 Finset.univ \u2227 \u2191(Composition.boundary (toComposition c)).toEmbedding a = j"}, {"tactic": "refine' \u27e8i, by simp, _\u27e9", "state_before": "case a.mpr.intro.intro\nn : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\ni : \u2115\ni_lt : i < Finset.card c.boundaries\nhi : sum (take i (blocks c)) = \u2191j\n\u22a2 \u2203 a, a \u2208 Finset.univ \u2227 \u2191(Composition.boundary (toComposition c)).toEmbedding a = j", "state_after": "case a.mpr.intro.intro\nn : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\ni : \u2115\ni_lt : i < Finset.card c.boundaries\nhi : sum (take i (blocks c)) = \u2191j\n\u22a2 \u2191(Composition.boundary (toComposition c)).toEmbedding \u2191i = j"}, {"tactic": "rw [c.card_boundaries_eq_succ_length] at i_lt", "state_before": "case a.mpr.intro.intro\nn : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\ni : \u2115\ni_lt : i < Finset.card c.boundaries\nhi : sum (take i (blocks c)) = \u2191j\n\u22a2 \u2191(Composition.boundary (toComposition c)).toEmbedding \u2191i = j", "state_after": "case a.mpr.intro.intro\nn : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\ni : \u2115\ni_lt : i < length c + 1\nhi : sum (take i (blocks c)) = \u2191j\n\u22a2 \u2191(Composition.boundary (toComposition c)).toEmbedding \u2191i = j"}, {"tactic": "simp [Composition.boundary, Nat.mod_eq_of_lt i_lt, Composition.sizeUpTo, hi]", "state_before": "case a.mpr.intro.intro\nn : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\ni : \u2115\ni_lt : i < length c + 1\nhi : sum (take i (blocks c)) = \u2191j\n\u22a2 \u2191(Composition.boundary (toComposition c)).toEmbedding \u2191i = j", "state_after": "no goals"}, {"tactic": "simp", "state_before": "n : \u2115\nc : CompositionAsSet n\nj : Fin (n + 1)\ni : \u2115\ni_lt : i < Finset.card c.boundaries\nhi : sum (take i (blocks c)) = \u2191j\n\u22a2 \u2191i \u2208 Finset.univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/Adjoint.lean", "full_name": "IsSelfAdjoint.conj_orthogonalProjection", "start": [325, 1], "end": [331, 28], "traced_tactics": [{"tactic": "rw [\u2190 ContinuousLinearMap.comp_assoc]", "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.1404059\nG : Type ?u.1404062\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace F\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nU : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace { x // x \u2208 U }\n\u22a2 IsSelfAdjoint\n    (comp (Submodule.subtypeL U)\n      (comp (orthogonalProjection U) (comp T (comp (Submodule.subtypeL U) (orthogonalProjection U)))))", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.1404059\nG : Type ?u.1404062\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace F\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nU : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace { x // x \u2208 U }\n\u22a2 IsSelfAdjoint\n    (comp (comp (Submodule.subtypeL U) (orthogonalProjection U))\n      (comp T (comp (Submodule.subtypeL U) (orthogonalProjection U))))"}, {"tactic": "nth_rw 1 [\u2190 (orthogonalProjection_isSelfAdjoint U).adjoint_eq]", "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.1404059\nG : Type ?u.1404062\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace F\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nU : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace { x // x \u2208 U }\n\u22a2 IsSelfAdjoint\n    (comp (comp (Submodule.subtypeL U) (orthogonalProjection U))\n      (comp T (comp (Submodule.subtypeL U) (orthogonalProjection U))))", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.1404059\nG : Type ?u.1404062\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace F\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nU : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace { x // x \u2208 U }\n\u22a2 IsSelfAdjoint\n    (comp (\u2191adjoint (comp (Submodule.subtypeL U) (orthogonalProjection U)))\n      (comp T (comp (Submodule.subtypeL U) (orthogonalProjection U))))"}, {"tactic": "refine' hT.adjoint_conj _", "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.1404059\nG : Type ?u.1404062\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace F\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nU : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace { x // x \u2208 U }\n\u22a2 IsSelfAdjoint\n    (comp (\u2191adjoint (comp (Submodule.subtypeL U) (orthogonalProjection U)))\n      (comp T (comp (Submodule.subtypeL U) (orthogonalProjection U))))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Intervals.lean", "full_name": "Finset.prod_Ico_eq_prod_range", "start": [140, 1], "end": [145, 93], "traced_tactics": [{"tactic": "by_cases h : m \u2264 n", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nm n : \u2115\n\u22a2 \u220f k in Ico m n, f k = \u220f k in range (n - m), f (m + k)", "state_after": "case pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nm n : \u2115\nh : m \u2264 n\n\u22a2 \u220f k in Ico m n, f k = \u220f k in range (n - m), f (m + k)\n\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nm n : \u2115\nh : \u00acm \u2264 n\n\u22a2 \u220f k in Ico m n, f k = \u220f k in range (n - m), f (m + k)"}, {"tactic": "rw [\u2190 Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]", "state_before": "case pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nm n : \u2115\nh : m \u2264 n\n\u22a2 \u220f k in Ico m n, f k = \u220f k in range (n - m), f (m + k)", "state_after": "no goals"}, {"tactic": "replace h : n \u2264 m := le_of_not_ge h", "state_before": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nm n : \u2115\nh : \u00acm \u2264 n\n\u22a2 \u220f k in Ico m n, f k = \u220f k in range (n - m), f (m + k)", "state_after": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nm n : \u2115\nh : n \u2264 m\n\u22a2 \u220f k in Ico m n, f k = \u220f k in range (n - m), f (m + k)"}, {"tactic": "rw [Ico_eq_empty_of_le h, tsub_eq_zero_iff_le.mpr h, range_zero, prod_empty, prod_empty]", "state_before": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nm n : \u2115\nh : n \u2264 m\n\u22a2 \u220f k in Ico m n, f k = \u220f k in range (n - m), f (m + k)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Basic.lean", "full_name": "Polynomial.coeff_neg", "start": [1157, 1], "end": [1160, 62], "traced_tactics": [{"tactic": "rcases p with \u27e8\u27e9", "state_before": "R : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Ring R\np : R[X]\nn : \u2115\n\u22a2 coeff (-p) n = -coeff p n", "state_after": "case ofFinsupp\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Ring R\nn : \u2115\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 coeff (-{ toFinsupp := toFinsupp\u271d }) n = -coeff { toFinsupp := toFinsupp\u271d } n"}, {"tactic": "rw [\u2190 ofFinsupp_neg, coeff, coeff]", "state_before": "case ofFinsupp\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Ring R\nn : \u2115\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 coeff (-{ toFinsupp := toFinsupp\u271d }) n = -coeff { toFinsupp := toFinsupp\u271d } n", "state_after": "case ofFinsupp\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Ring R\nn : \u2115\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 (match (motive := R[X] \u2192 \u2115 \u2192 R) { toFinsupp := -toFinsupp\u271d } with\n      | { toFinsupp := p } => \u2191p)\n      n =\n    -(match (motive := R[X] \u2192 \u2115 \u2192 R) { toFinsupp := toFinsupp\u271d } with\n        | { toFinsupp := p } => \u2191p)\n        n"}, {"tactic": "apply Finsupp.neg_apply", "state_before": "case ofFinsupp\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Ring R\nn : \u2115\ntoFinsupp\u271d : AddMonoidAlgebra R \u2115\n\u22a2 (match (motive := R[X] \u2192 \u2115 \u2192 R) { toFinsupp := -toFinsupp\u271d } with\n      | { toFinsupp := p } => \u2191p)\n      n =\n    -(match (motive := R[X] \u2192 \u2115 \u2192 R) { toFinsupp := toFinsupp\u271d } with\n        | { toFinsupp := p } => \u2191p)\n        n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/EraseLead.lean", "full_name": "Polynomial.eraseLead_ne_zero", "start": [91, 1], "end": [94, 101], "traced_tactics": [{"tactic": "rw [Ne, \u2190 card_support_eq_zero, eraseLead_support]", 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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\nx y z : S\nhx : IsIntegralElem f x\nhy : IsIntegralElem f y\nhz : z \u2208 Subring.closure {x, y}\nthis\u271d : Algebra R S := toAlgebra f\nthis : FG (\u2191Subalgebra.toSubmodule (Algebra.adjoin R {x}) * \u2191Subalgebra.toSubmodule (Algebra.adjoin R {y}))\n\u22a2 IsIntegralElem f z"}, {"tactic": "rw [\u2190 Algebra.adjoin_union_coe_submodule, Set.singleton_union] at this", "state_before": "R : Type u_1\nA : Type ?u.638506\nB : Type ?u.638509\nS : Type u_2\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\nx y z : S\nhx : IsIntegralElem f x\nhy : IsIntegralElem f y\nhz : z \u2208 Subring.closure {x, y}\nthis\u271d : Algebra R S := toAlgebra f\nthis : FG (\u2191Subalgebra.toSubmodule (Algebra.adjoin R {x}) * \u2191Subalgebra.toSubmodule (Algebra.adjoin R {y}))\n\u22a2 IsIntegralElem f z", "state_after": "R : Type u_1\nA : Type ?u.638506\nB : Type ?u.638509\nS : Type u_2\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\nx y z : S\nhx : IsIntegralElem f x\nhy : IsIntegralElem f y\nhz : z \u2208 Subring.closure {x, y}\nthis\u271d : Algebra R S := toAlgebra f\nthis : FG (\u2191Subalgebra.toSubmodule (Algebra.adjoin R {x, y}))\n\u22a2 IsIntegralElem f z"}, {"tactic": "exact\n  isIntegral_of_mem_of_FG (Algebra.adjoin R {x, y}) this z\n    (Algebra.mem_adjoin_iff.2 <| Subring.closure_mono (Set.subset_union_right _ _) hz)", "state_before": "R : Type u_1\nA : Type ?u.638506\nB : Type ?u.638509\nS : Type u_2\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\nx y z : S\nhx : IsIntegralElem f x\nhy : IsIntegralElem f y\nhz : z \u2208 Subring.closure {x, y}\nthis\u271d : Algebra R S := toAlgebra f\nthis : FG (\u2191Subalgebra.toSubmodule (Algebra.adjoin R {x, y}))\n\u22a2 IsIntegralElem f z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subgroup/ZPowers.lean", "full_name": "ofAdd_image_zmultiples_eq_zpowers_ofAdd", "start": [178, 1], "end": [183, 48], "traced_tactics": [{"tactic": "symm", "state_before": "G : Type ?u.36673\ninst\u271d\u00b2 : Group G\nA : Type u_1\ninst\u271d\u00b9 : AddGroup A\nN : Type ?u.36685\ninst\u271d : Group N\nx : A\n\u22a2 \u2191Multiplicative.ofAdd '' \u2191(AddSubgroup.zmultiples x) = \u2191(Subgroup.zpowers (\u2191Multiplicative.ofAdd x))", "state_after": "G : Type ?u.36673\ninst\u271d\u00b2 : Group G\nA : Type u_1\ninst\u271d\u00b9 : AddGroup A\nN : Type ?u.36685\ninst\u271d : Group N\nx : A\n\u22a2 \u2191(Subgroup.zpowers (\u2191Multiplicative.ofAdd x)) = \u2191Multiplicative.ofAdd '' \u2191(AddSubgroup.zmultiples x)"}, {"tactic": "rw [Equiv.eq_image_iff_symm_image_eq]", "state_before": "G : Type ?u.36673\ninst\u271d\u00b2 : Group G\nA : Type u_1\ninst\u271d\u00b9 : AddGroup A\nN : Type ?u.36685\ninst\u271d : Group N\nx : A\n\u22a2 \u2191(Subgroup.zpowers (\u2191Multiplicative.ofAdd x)) = \u2191Multiplicative.ofAdd '' \u2191(AddSubgroup.zmultiples x)", "state_after": "G : Type ?u.36673\ninst\u271d\u00b2 : Group G\nA : Type u_1\ninst\u271d\u00b9 : AddGroup A\nN : Type ?u.36685\ninst\u271d : Group N\nx : A\n\u22a2 \u2191Multiplicative.ofAdd.symm '' \u2191(Subgroup.zpowers (\u2191Multiplicative.ofAdd x)) = \u2191(AddSubgroup.zmultiples x)"}, {"tactic": "exact ofMul_image_zpowers_eq_zmultiples_ofMul", "state_before": "G : Type ?u.36673\ninst\u271d\u00b2 : Group G\nA : Type u_1\ninst\u271d\u00b9 : AddGroup A\nN : Type ?u.36685\ninst\u271d : Group N\nx : A\n\u22a2 \u2191Multiplicative.ofAdd.symm '' \u2191(Subgroup.zpowers (\u2191Multiplicative.ofAdd x)) = \u2191(AddSubgroup.zmultiples x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/AdicCompletion.lean", "full_name": "IsHausdorff.haus", "start": [59, 1], "end": [61, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Zip.lean", "full_name": "List.map_uncurry_zip_eq_zipWith", "start": [447, 1], "end": [454, 16], "traced_tactics": [{"tactic": "rw [zip]", "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.165705\n\u03b5 : Type ?u.165708\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\n\u22a2 map (Function.uncurry f) (zip l l') = zipWith f l l'", "state_after": "\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.165705\n\u03b5 : Type ?u.165708\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\n\u22a2 map (Function.uncurry f) (zipWith Prod.mk l l') = zipWith f l l'"}, {"tactic": "induction' l with hd tl hl generalizing l'", "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.165705\n\u03b5 : Type ?u.165708\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl : List \u03b1\nl' : List \u03b2\n\u22a2 map (Function.uncurry f) (zipWith Prod.mk l l') = zipWith f l l'", "state_after": "case nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.165705\n\u03b5 : Type ?u.165708\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl'\u271d l' : List \u03b2\n\u22a2 map (Function.uncurry f) (zipWith Prod.mk [] l') = zipWith f [] l'\n\ncase cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.165705\n\u03b5 : Type ?u.165708\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl'\u271d : List \u03b2\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), map (Function.uncurry f) (zipWith Prod.mk tl l') = zipWith f tl l'\nl' : List \u03b2\n\u22a2 map (Function.uncurry f) (zipWith Prod.mk (hd :: tl) l') = zipWith f (hd :: tl) l'"}, {"tactic": "simp", "state_before": "case nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.165705\n\u03b5 : Type ?u.165708\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl'\u271d l' : List \u03b2\n\u22a2 map (Function.uncurry f) (zipWith Prod.mk [] l') = zipWith f [] l'", "state_after": "no goals"}, {"tactic": "cases' l' with hd' tl'", "state_before": "case cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.165705\n\u03b5 : Type ?u.165708\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl'\u271d : List \u03b2\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), map (Function.uncurry f) (zipWith Prod.mk tl l') = zipWith f tl l'\nl' : List \u03b2\n\u22a2 map (Function.uncurry f) (zipWith Prod.mk (hd :: tl) l') = zipWith f (hd :: tl) l'", "state_after": "case cons.nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.165705\n\u03b5 : Type ?u.165708\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl' : List \u03b2\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), map (Function.uncurry f) (zipWith Prod.mk tl l') = zipWith f tl l'\n\u22a2 map (Function.uncurry f) (zipWith Prod.mk (hd :: tl) []) = zipWith f (hd :: tl) []\n\ncase cons.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.165705\n\u03b5 : Type ?u.165708\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl' : List \u03b2\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), map (Function.uncurry f) (zipWith Prod.mk tl l') = zipWith f tl l'\nhd' : \u03b2\ntl' : List \u03b2\n\u22a2 map (Function.uncurry f) (zipWith Prod.mk (hd :: tl) (hd' :: tl')) = zipWith f (hd :: tl) (hd' :: tl')"}, {"tactic": "simp", "state_before": "case cons.nil\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.165705\n\u03b5 : Type ?u.165708\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl' : List \u03b2\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), map (Function.uncurry f) (zipWith Prod.mk tl l') = zipWith f tl l'\n\u22a2 map (Function.uncurry f) (zipWith Prod.mk (hd :: tl) []) = zipWith f (hd :: tl) []", "state_after": "no goals"}, {"tactic": "simp [hl]", "state_before": "case cons.cons\n\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type ?u.165705\n\u03b5 : Type ?u.165708\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl' : List \u03b2\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (l' : List \u03b2), map (Function.uncurry f) (zipWith Prod.mk tl l') = zipWith f tl l'\nhd' : \u03b2\ntl' : List \u03b2\n\u22a2 map (Function.uncurry f) (zipWith Prod.mk (hd :: tl) (hd' :: tl')) = zipWith f (hd :: tl) (hd' :: tl')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Data/Nat/Log2.lean", "full_name": "Nat.log2_terminates", "start": [12, 1], "end": [19, 15], "traced_tactics": [{"tactic": "decide", "state_before": "x\u271d : 2 \u2265 2\n\u22a2 2 / 2 < 2", "state_after": "no goals"}, {"tactic": "decide", "state_before": "x\u271d : 3 \u2265 2\n\u22a2 3 / 2 < 3", "state_after": "no goals"}, {"tactic": "rw [div_eq, if_pos]", "state_before": "n : Nat\nx\u271d : n + 4 \u2265 2\n\u22a2 (n + 4) / 2 < n + 4", "state_after": "n : Nat\nx\u271d : n + 4 \u2265 2\n\u22a2 (n + 4 - 2) / 2 + 1 < n + 4\n\ncase hc\nn : Nat\nx\u271d : n + 4 \u2265 2\n\u22a2 0 < 2 \u2227 2 \u2264 n + 4"}, {"tactic": "refine succ_lt_succ (Nat.lt_trans ?_ (lt_succ_self _))", "state_before": "n : Nat\nx\u271d : n + 4 \u2265 2\n\u22a2 (n + 4 - 2) / 2 + 1 < n + 4\n\ncase hc\nn : Nat\nx\u271d : n + 4 \u2265 2\n\u22a2 0 < 2 \u2227 2 \u2264 n + 4", "state_after": "n : Nat\nx\u271d : n + 4 \u2265 2\n\u22a2 (n + 4 - 2) / 2 < n + 2\n\ncase hc\nn : Nat\nx\u271d : n + 4 \u2265 2\n\u22a2 0 < 2 \u2227 2 \u2264 n + 4"}, {"tactic": "exact log2_terminates (n+2) (by simp_arith)", "state_before": "n : Nat\nx\u271d : n + 4 \u2265 2\n\u22a2 (n + 4 - 2) / 2 < n + 2\n\ncase hc\nn : Nat\nx\u271d : n + 4 \u2265 2\n\u22a2 0 < 2 \u2227 2 \u2264 n + 4", "state_after": "case hc\nn : Nat\nx\u271d : n + 4 \u2265 2\n\u22a2 0 < 2 \u2227 2 \u2264 n + 4"}, {"tactic": "simp_arith", "state_before": "case hc\nn : Nat\nx\u271d : n + 4 \u2265 2\n\u22a2 0 < 2 \u2227 2 \u2264 n + 4", "state_after": "no goals"}, {"tactic": "simp_arith", "state_before": "n : Nat\nx\u271d : n + 4 \u2265 2\n\u22a2 n + 2 \u2265 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/MvPolynomial/Equiv.lean", "full_name": "MvPolynomial.iterToSum_C_X", "start": [208, 1], "end": [209, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "HasSubset.Subset.eventuallyLE", "start": [3123, 1], "end": [3124, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/l2Space.lean", "full_name": "IsHilbertSum.hasSum_linearIsometryEquiv_symm", "start": [343, 11], "end": [345, 82], "traced_tactics": [{"tactic": "simp [IsHilbertSum.linearIsometryEquiv, OrthogonalFamily.hasSum_linearIsometry]", "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsHilbertSum \ud835\udd5c G V\nw : { x // x \u2208 lp G 2 }\n\u22a2 HasSum (fun i => \u2191(V i) (\u2191w i)) (\u2191(LinearIsometryEquiv.symm (linearIsometryEquiv hV)) w)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/BoundedOrder.lean", "full_name": "monotone_and", "start": [554, 1], "end": [556, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Max.lean", "full_name": "IsBot.prod_mk", "start": [407, 1], "end": [407, 92], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Bounds/Basic.lean", "full_name": "not_bddBelow_iff'", "start": [128, 1], "end": [129, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/DiffContOnCl.lean", "full_name": "DiffContOnCl.const_add", "start": [101, 1], "end": [102, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/ExtendDeriv.lean", "full_name": "hasDerivAt_of_hasDerivAt_of_ne", "start": [183, 1], "end": [212, 24], "traced_tactics": [{"tactic": "have A : HasDerivWithinAt f (g x) (Ici x) x := by\n  have diff : DifferentiableOn \u211d f (Ioi x) := fun y hy =>\n    (f_diff y (ne_of_gt hy)).differentiableAt.differentiableWithinAt\n  apply\n    has_deriv_at_interval_left_endpoint_of_tendsto_deriv diff hf.continuousWithinAt\n      self_mem_nhdsWithin\n  have : Tendsto g (\ud835\udcdd[>] x) (\ud835\udcdd (g x)) := tendsto_inf_left hg\n  apply this.congr' _\n  apply mem_of_superset self_mem_nhdsWithin fun y hy => _\n  intros y hy\n  exact (f_diff y (ne_of_gt hy)).deriv.symm", "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\n\u22a2 HasDerivAt f (g x) x", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\n\u22a2 HasDerivAt f (g x) x"}, {"tactic": "have B : HasDerivWithinAt f (g x) (Iic x) x := by\n  have diff : DifferentiableOn \u211d f (Iio x) := fun y hy =>\n    (f_diff y (ne_of_lt hy)).differentiableAt.differentiableWithinAt\n  apply\n    has_deriv_at_interval_right_endpoint_of_tendsto_deriv diff hf.continuousWithinAt\n      self_mem_nhdsWithin\n  have : Tendsto g (\ud835\udcdd[<] x) (\ud835\udcdd (g x)) := tendsto_inf_left hg\n  apply this.congr' _\n  apply mem_of_superset self_mem_nhdsWithin fun y hy => _\n  intros y hy\n  exact (f_diff y (ne_of_lt hy)).deriv.symm", "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\n\u22a2 HasDerivAt f (g x) x", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\nB : HasDerivWithinAt f (g x) (Iic x) x\n\u22a2 HasDerivAt f (g x) x"}, {"tactic": "simpa using B.union A", "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\nB : HasDerivWithinAt f (g x) (Iic x) x\n\u22a2 HasDerivAt f (g x) x", "state_after": "no goals"}, {"tactic": "have diff : DifferentiableOn \u211d f (Ioi x) := fun y hy =>\n  (f_diff y (ne_of_gt hy)).differentiableAt.differentiableWithinAt", "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\n\u22a2 HasDerivWithinAt f (g x) (Ici x) x", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\ndiff : DifferentiableOn \u211d f (Ioi x)\n\u22a2 HasDerivWithinAt f (g x) (Ici x) x"}, {"tactic": "apply\n  has_deriv_at_interval_left_endpoint_of_tendsto_deriv diff hf.continuousWithinAt\n    self_mem_nhdsWithin", "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\ndiff : DifferentiableOn \u211d f (Ioi x)\n\u22a2 HasDerivWithinAt f (g x) (Ici x) x", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\ndiff : DifferentiableOn \u211d f (Ioi x)\n\u22a2 Tendsto (fun x => deriv f x) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (g x))"}, {"tactic": "have : Tendsto g (\ud835\udcdd[>] x) (\ud835\udcdd (g x)) := tendsto_inf_left hg", "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\ndiff : DifferentiableOn \u211d f (Ioi x)\n\u22a2 Tendsto (fun x => deriv f x) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (g x))", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\ndiff : DifferentiableOn \u211d f (Ioi x)\nthis : Tendsto g (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (g x))\n\u22a2 Tendsto (fun x => deriv f x) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (g x))"}, {"tactic": "apply this.congr' _", "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\ndiff : DifferentiableOn \u211d f (Ioi x)\nthis : Tendsto g (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (g x))\n\u22a2 Tendsto (fun x => deriv f x) (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (g x))", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\ndiff : DifferentiableOn \u211d f (Ioi x)\nthis : Tendsto g (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (g x))\n\u22a2 g =\u1da0[\ud835\udcdd[Ioi x] x] fun x => deriv f x"}, {"tactic": "apply mem_of_superset self_mem_nhdsWithin fun y hy => _", "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\ndiff : DifferentiableOn \u211d f (Ioi x)\nthis : Tendsto g (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (g x))\n\u22a2 g =\u1da0[\ud835\udcdd[Ioi x] x] fun x => deriv f x", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\ndiff : DifferentiableOn \u211d f (Ioi x)\nthis : Tendsto g (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (g x))\n\u22a2 \u2200 (y : \u211d), y \u2208 Ioi x \u2192 y \u2208 {x | (fun x => g x = (fun x => deriv f x) x) x}"}, {"tactic": "intros y hy", "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\ndiff : DifferentiableOn \u211d f (Ioi x)\nthis : Tendsto g (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (g x))\n\u22a2 \u2200 (y : \u211d), y \u2208 Ioi x \u2192 y \u2208 {x | (fun x => g x = (fun x => deriv f x) x) x}", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\ndiff : DifferentiableOn \u211d f (Ioi x)\nthis : Tendsto g (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (g x))\ny : \u211d\nhy : y \u2208 Ioi x\n\u22a2 y \u2208 {x | (fun x => g x = (fun x => deriv f x) x) x}"}, {"tactic": "exact (f_diff y (ne_of_gt hy)).deriv.symm", "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\ndiff : DifferentiableOn \u211d f (Ioi x)\nthis : Tendsto g (\ud835\udcdd[Ioi x] x) (\ud835\udcdd (g x))\ny : \u211d\nhy : y \u2208 Ioi x\n\u22a2 y \u2208 {x | (fun x => g x = (fun x => deriv f x) x) x}", "state_after": "no goals"}, {"tactic": "have diff : DifferentiableOn \u211d f (Iio x) := fun y hy =>\n  (f_diff y (ne_of_lt hy)).differentiableAt.differentiableWithinAt", "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\n\u22a2 HasDerivWithinAt f (g x) (Iic x) x", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\ndiff : DifferentiableOn \u211d f (Iio x)\n\u22a2 HasDerivWithinAt f (g x) (Iic x) x"}, {"tactic": "apply\n  has_deriv_at_interval_right_endpoint_of_tendsto_deriv diff hf.continuousWithinAt\n    self_mem_nhdsWithin", "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\ndiff : DifferentiableOn \u211d f (Iio x)\n\u22a2 HasDerivWithinAt f (g x) (Iic x) x", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\ndiff : DifferentiableOn \u211d f (Iio x)\n\u22a2 Tendsto (fun x => deriv f x) (\ud835\udcdd[Iio x] x) (\ud835\udcdd (g x))"}, {"tactic": "have : Tendsto g (\ud835\udcdd[<] x) (\ud835\udcdd (g x)) := tendsto_inf_left hg", "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\ndiff : DifferentiableOn \u211d f (Iio x)\n\u22a2 Tendsto (fun x => deriv f x) (\ud835\udcdd[Iio x] x) (\ud835\udcdd (g x))", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\ndiff : DifferentiableOn \u211d f (Iio x)\nthis : Tendsto g (\ud835\udcdd[Iio x] x) (\ud835\udcdd (g x))\n\u22a2 Tendsto (fun x => deriv f x) (\ud835\udcdd[Iio x] x) (\ud835\udcdd (g x))"}, {"tactic": "apply this.congr' _", "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\ndiff : DifferentiableOn \u211d f (Iio x)\nthis : Tendsto g (\ud835\udcdd[Iio x] x) (\ud835\udcdd (g x))\n\u22a2 Tendsto (fun x => deriv f x) (\ud835\udcdd[Iio x] x) (\ud835\udcdd (g x))", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\ndiff : DifferentiableOn \u211d f (Iio x)\nthis : Tendsto g (\ud835\udcdd[Iio x] x) (\ud835\udcdd (g x))\n\u22a2 g =\u1da0[\ud835\udcdd[Iio x] x] fun x => deriv f x"}, {"tactic": "apply mem_of_superset self_mem_nhdsWithin fun y hy => _", "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\ndiff : DifferentiableOn \u211d f (Iio x)\nthis : Tendsto g (\ud835\udcdd[Iio x] x) (\ud835\udcdd (g x))\n\u22a2 g =\u1da0[\ud835\udcdd[Iio x] x] fun x => deriv f x", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\ndiff : DifferentiableOn \u211d f (Iio x)\nthis : Tendsto g (\ud835\udcdd[Iio x] x) (\ud835\udcdd (g x))\n\u22a2 \u2200 (y : \u211d), y \u2208 Iio x \u2192 y \u2208 {x | (fun x => g x = (fun x => deriv f x) x) x}"}, {"tactic": "intros y hy", "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\ndiff : DifferentiableOn \u211d f (Iio x)\nthis : Tendsto g (\ud835\udcdd[Iio x] x) (\ud835\udcdd (g x))\n\u22a2 \u2200 (y : \u211d), y \u2208 Iio x \u2192 y \u2208 {x | (fun x => g x = (fun x => deriv f x) x) x}", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\ndiff : DifferentiableOn \u211d f (Iio x)\nthis : Tendsto g (\ud835\udcdd[Iio x] x) (\ud835\udcdd (g x))\ny : \u211d\nhy : y \u2208 Iio x\n\u22a2 y \u2208 {x | (fun x => g x = (fun x => deriv f x) x) x}"}, {"tactic": "exact (f_diff y (ne_of_lt hy)).deriv.symm", "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type ?u.148628\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \u211d \u2192 E\nx : \u211d\nf_diff : \u2200 (y : \u211d), y \u2260 x \u2192 HasDerivAt f (g y) y\nhf : ContinuousAt f x\nhg : ContinuousAt g x\nA : HasDerivWithinAt f (g x) (Ici x) x\ndiff : DifferentiableOn \u211d f (Iio x)\nthis : Tendsto g (\ud835\udcdd[Iio x] x) (\ud835\udcdd (g x))\ny : \u211d\nhy : y \u2208 Iio x\n\u22a2 y \u2208 {x | (fun x => g x = (fun x => deriv f x) x) x}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/BilinearForm.lean", "full_name": "Matrix.toBilin'_symm", "start": [199, 1], "end": [201, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "Real.abs_sin_eq_sqrt_one_sub_cos_sq", "start": [1305, 1], "end": [1306, 32], "traced_tactics": [{"tactic": "rw [\u2190 sin_sq, sqrt_sq_eq_abs]", "state_before": "x\u271d y x : \u211d\n\u22a2 abs' (sin x) = sqrt (1 - cos x ^ 2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Pairwise.lean", "full_name": "List.pwFilter_nil", "start": [356, 1], "end": [357, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.set_integral_mono_on_ae", "start": [699, 1], "end": [701, 89], "traced_tactics": [{"tactic": "refine' set_integral_mono_ae_restrict hf hg _", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.288198\nE : Type ?u.288201\nF : Type ?u.288204\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns t : Set \u03b1\nhf : IntegrableOn f s\nhg : IntegrableOn g s\nhs : MeasurableSet s\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 g x\n\u22a2 (\u222b (a : \u03b1) in s, f a \u2202\u03bc) \u2264 \u222b (a : \u03b1) in s, g a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.288198\nE : Type ?u.288201\nF : Type ?u.288204\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns t : Set \u03b1\nhf : IntegrableOn f s\nhg : IntegrableOn g s\nhs : MeasurableSet s\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 g x\n\u22a2 (fun a => f a) \u2264\u1d50[Measure.restrict \u03bc s] fun a => g a"}, {"tactic": "rwa [EventuallyLE, ae_restrict_iff' hs]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.288198\nE : Type ?u.288201\nF : Type ?u.288204\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns t : Set \u03b1\nhf : IntegrableOn f s\nhg : IntegrableOn g s\nhs : MeasurableSet s\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x \u2264 g x\n\u22a2 (fun a => f a) \u2264\u1d50[Measure.restrict \u03bc s] fun a => g a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Prod.lean", "full_name": "LinearMap.map_coprod_prod", "start": [468, 1], "end": [475, 45], "traced_tactics": [{"tactic": "refine' le_antisymm _ (sup_le (map_le_iff_le_comap.2 _) (map_le_iff_le_comap.2 _))", "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.203469\nM\u2086 : Type ?u.203472\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 map (coprod f g) (Submodule.prod p q) = map f p \u2294 map g q", "state_after": "case refine'_1\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.203469\nM\u2086 : Type ?u.203472\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 map (coprod f g) (Submodule.prod p q) \u2264 map f p \u2294 map g q\n\ncase refine'_2\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.203469\nM\u2086 : Type ?u.203472\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 p \u2264 comap f (map (coprod f g) (Submodule.prod p q))\n\ncase refine'_3\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.203469\nM\u2086 : Type ?u.203472\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 q \u2264 comap g (map (coprod f g) (Submodule.prod p q))"}, {"tactic": "rw [SetLike.le_def]", "state_before": "case refine'_1\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.203469\nM\u2086 : Type ?u.203472\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 map (coprod f g) (Submodule.prod p q) \u2264 map f p \u2294 map g q", "state_after": "case refine'_1\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.203469\nM\u2086 : Type ?u.203472\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 \u2200 \u2983x : M\u2083\u2984, x \u2208 map (coprod f g) (Submodule.prod p q) \u2192 x \u2208 map f p \u2294 map g q"}, {"tactic": "rintro _ \u27e8x, \u27e8h\u2081, h\u2082\u27e9, rfl\u27e9", "state_before": "case refine'_1\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.203469\nM\u2086 : Type ?u.203472\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 \u2200 \u2983x : M\u2083\u2984, x \u2208 map (coprod f g) (Submodule.prod p q) \u2192 x \u2208 map f p \u2294 map g q", "state_after": "case refine'_1.intro.intro.intro\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.203469\nM\u2086 : Type ?u.203472\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\np : Submodule R M\nq : Submodule R M\u2082\nx : M \u00d7 M\u2082\nh\u2081 : x.fst \u2208 \u2191p\nh\u2082 : x.snd \u2208 \u2191q\n\u22a2 \u2191(coprod f g) x \u2208 map f p \u2294 map g q"}, {"tactic": "exact mem_sup.2 \u27e8_, \u27e8_, h\u2081, rfl\u27e9, _, \u27e8_, h\u2082, rfl\u27e9, rfl\u27e9", "state_before": "case refine'_1.intro.intro.intro\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.203469\nM\u2086 : Type ?u.203472\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\np : Submodule R M\nq : Submodule R M\u2082\nx : M \u00d7 M\u2082\nh\u2081 : x.fst \u2208 \u2191p\nh\u2082 : x.snd \u2208 \u2191q\n\u22a2 \u2191(coprod f g) x \u2208 map f p \u2294 map g q", "state_after": "no goals"}, {"tactic": "exact fun x hx => \u27e8(x, 0), by simp [hx]\u27e9", "state_before": "case refine'_2\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.203469\nM\u2086 : Type ?u.203472\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 p \u2264 comap f (map (coprod f g) (Submodule.prod p q))", "state_after": "no goals"}, {"tactic": "simp [hx]", "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.203469\nM\u2086 : Type ?u.203472\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\np : Submodule R M\nq : Submodule R M\u2082\nx : M\nhx : x \u2208 p\n\u22a2 (x, 0) \u2208 \u2191(Submodule.prod p q) \u2227 \u2191(coprod f g) (x, 0) = \u2191f x", "state_after": "no goals"}, {"tactic": "exact fun x hx => \u27e8(0, x), by simp [hx]\u27e9", "state_before": "case refine'_3\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.203469\nM\u2086 : Type ?u.203472\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 q \u2264 comap g (map (coprod f g) (Submodule.prod p q))", "state_after": "no goals"}, {"tactic": "simp [hx]", "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.203469\nM\u2086 : Type ?u.203472\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\np : Submodule R M\nq : Submodule R M\u2082\nx : M\u2082\nhx : x \u2208 q\n\u22a2 (0, x) \u2208 \u2191(Submodule.prod p q) \u2227 \u2191(coprod f g) (0, x) = \u2191g x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Connected.lean", "full_name": "frontier_eq_empty_iff", "start": [851, 1], "end": [853, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "full_name": "mul_self_lt_mul_self", "start": [540, 1], "end": [541, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "full_name": "tsub_add_eq_add_tsub", "start": [221, 1], "end": [222, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "isClosed_biInter", "start": [226, 1], "end": [228, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/ContDiff.lean", "full_name": "ContDiffAt.hasStrictDerivAt", "start": [1973, 1], "end": [1975, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "full_name": "Asymptotics.isBigOWith_self_const_mul", "start": [1466, 1], "end": [1468, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "aestronglyMeasurable_sum_measure_iff", "start": [1684, 1], "end": [1687, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.nontrivial_iff_exists_lt", "start": [2517, 1], "end": [2519, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Sylow.lean", "full_name": "card_sylow_eq_card_quotient_normalizer", "start": [413, 1], "end": [415, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Heyting/Basic.lean", "full_name": "ofDual_himp", "start": [1141, 1], "end": [1142, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Erased.lean", "full_name": "Erased.nonempty_iff", "start": [94, 1], "end": [95, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Alternating.lean", "full_name": "AlternatingMap.map_eq_zero_of_not_injective", "start": [233, 1], "end": [237, 38], "traced_tactics": [{"tactic": "rw [Function.Injective] at hv", "state_before": "R : Type u_4\ninst\u271d\u00b9\u2070 : Semiring R\nM : Type u_2\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nN : Type u_3\ninst\u271d\u2077 : AddCommMonoid N\ninst\u271d\u2076 : Module R N\nP : Type ?u.72498\ninst\u271d\u2075 : AddCommMonoid P\ninst\u271d\u2074 : Module R P\nM' : Type ?u.72528\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\nN' : Type ?u.72916\ninst\u271d\u00b9 : AddCommGroup N'\ninst\u271d : Module R N'\n\u03b9 : Type u_1\n\u03b9' : Type ?u.73307\n\u03b9'' : Type ?u.73310\nf f' : AlternatingMap R M N \u03b9\ng g\u2082 : AlternatingMap R M N' \u03b9\ng' : AlternatingMap R M' N' \u03b9\nv\u271d : \u03b9 \u2192 M\nv' : \u03b9 \u2192 M'\nv : \u03b9 \u2192 M\nhv : \u00acInjective v\n\u22a2 \u2191f v = 0", "state_after": "R : Type u_4\ninst\u271d\u00b9\u2070 : Semiring R\nM : Type u_2\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nN : Type u_3\ninst\u271d\u2077 : AddCommMonoid N\ninst\u271d\u2076 : Module R N\nP : Type ?u.72498\ninst\u271d\u2075 : AddCommMonoid P\ninst\u271d\u2074 : Module R P\nM' : Type ?u.72528\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\nN' : Type ?u.72916\ninst\u271d\u00b9 : AddCommGroup N'\ninst\u271d : Module R N'\n\u03b9 : Type u_1\n\u03b9' : Type ?u.73307\n\u03b9'' : Type ?u.73310\nf f' : AlternatingMap R M N \u03b9\ng g\u2082 : AlternatingMap R M N' \u03b9\ng' : AlternatingMap R M' N' \u03b9\nv\u271d : \u03b9 \u2192 M\nv' : \u03b9 \u2192 M'\nv : \u03b9 \u2192 M\nhv : \u00ac\u2200 \u2983a\u2081 a\u2082 : \u03b9\u2984, v a\u2081 = v a\u2082 \u2192 a\u2081 = a\u2082\n\u22a2 \u2191f v = 0"}, {"tactic": "push_neg  at hv", "state_before": "R : Type u_4\ninst\u271d\u00b9\u2070 : Semiring R\nM : Type u_2\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nN : Type u_3\ninst\u271d\u2077 : AddCommMonoid N\ninst\u271d\u2076 : Module R N\nP : Type ?u.72498\ninst\u271d\u2075 : AddCommMonoid P\ninst\u271d\u2074 : Module R P\nM' : Type ?u.72528\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\nN' : Type ?u.72916\ninst\u271d\u00b9 : AddCommGroup N'\ninst\u271d : Module R N'\n\u03b9 : Type u_1\n\u03b9' : Type ?u.73307\n\u03b9'' : Type ?u.73310\nf f' : AlternatingMap R M N \u03b9\ng g\u2082 : AlternatingMap R M N' \u03b9\ng' : AlternatingMap R M' N' \u03b9\nv\u271d : \u03b9 \u2192 M\nv' : \u03b9 \u2192 M'\nv : \u03b9 \u2192 M\nhv : \u00ac\u2200 \u2983a\u2081 a\u2082 : \u03b9\u2984, v a\u2081 = v a\u2082 \u2192 a\u2081 = a\u2082\n\u22a2 \u2191f v = 0", "state_after": "R : Type u_4\ninst\u271d\u00b9\u2070 : Semiring R\nM : Type u_2\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nN : Type u_3\ninst\u271d\u2077 : AddCommMonoid N\ninst\u271d\u2076 : Module R N\nP : Type ?u.72498\ninst\u271d\u2075 : AddCommMonoid P\ninst\u271d\u2074 : Module R P\nM' : Type ?u.72528\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\nN' : Type ?u.72916\ninst\u271d\u00b9 : AddCommGroup N'\ninst\u271d : Module R N'\n\u03b9 : Type u_1\n\u03b9' : Type ?u.73307\n\u03b9'' : Type ?u.73310\nf f' : AlternatingMap R M N \u03b9\ng g\u2082 : AlternatingMap R M N' \u03b9\ng' : AlternatingMap R M' N' \u03b9\nv\u271d : \u03b9 \u2192 M\nv' : \u03b9 \u2192 M'\nv : \u03b9 \u2192 M\nhv : Exists fun \u2983a\u2081\u2984 => Exists fun \u2983a\u2082\u2984 => v a\u2081 = v a\u2082 \u2227 a\u2081 \u2260 a\u2082\n\u22a2 \u2191f v = 0"}, {"tactic": "rcases hv with \u27e8i\u2081, i\u2082, heq, hne\u27e9", "state_before": "R : Type u_4\ninst\u271d\u00b9\u2070 : Semiring R\nM : Type u_2\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nN : Type u_3\ninst\u271d\u2077 : AddCommMonoid N\ninst\u271d\u2076 : Module R N\nP : Type ?u.72498\ninst\u271d\u2075 : AddCommMonoid P\ninst\u271d\u2074 : Module R P\nM' : Type ?u.72528\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\nN' : Type ?u.72916\ninst\u271d\u00b9 : AddCommGroup N'\ninst\u271d : Module R N'\n\u03b9 : Type u_1\n\u03b9' : Type ?u.73307\n\u03b9'' : Type ?u.73310\nf f' : AlternatingMap R M N \u03b9\ng g\u2082 : AlternatingMap R M N' \u03b9\ng' : AlternatingMap R M' N' \u03b9\nv\u271d : \u03b9 \u2192 M\nv' : \u03b9 \u2192 M'\nv : \u03b9 \u2192 M\nhv : Exists fun \u2983a\u2081\u2984 => Exists fun \u2983a\u2082\u2984 => v a\u2081 = v a\u2082 \u2227 a\u2081 \u2260 a\u2082\n\u22a2 \u2191f v = 0", "state_after": "case intro.intro.intro\nR : Type u_4\ninst\u271d\u00b9\u2070 : Semiring R\nM : Type u_2\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nN : Type u_3\ninst\u271d\u2077 : AddCommMonoid N\ninst\u271d\u2076 : Module R N\nP : Type ?u.72498\ninst\u271d\u2075 : AddCommMonoid P\ninst\u271d\u2074 : Module R P\nM' : Type ?u.72528\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\nN' : Type ?u.72916\ninst\u271d\u00b9 : AddCommGroup N'\ninst\u271d : Module R N'\n\u03b9 : Type u_1\n\u03b9' : Type ?u.73307\n\u03b9'' : Type ?u.73310\nf f' : AlternatingMap R M N \u03b9\ng g\u2082 : AlternatingMap R M N' \u03b9\ng' : AlternatingMap R M' N' \u03b9\nv\u271d : \u03b9 \u2192 M\nv' : \u03b9 \u2192 M'\nv : \u03b9 \u2192 M\ni\u2081 i\u2082 : \u03b9\nheq : v i\u2081 = v i\u2082\nhne : i\u2081 \u2260 i\u2082\n\u22a2 \u2191f v = 0"}, {"tactic": "exact f.map_eq_zero_of_eq v heq hne", "state_before": "case intro.intro.intro\nR : Type u_4\ninst\u271d\u00b9\u2070 : Semiring R\nM : Type u_2\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nN : Type u_3\ninst\u271d\u2077 : AddCommMonoid N\ninst\u271d\u2076 : Module R N\nP : Type ?u.72498\ninst\u271d\u2075 : AddCommMonoid P\ninst\u271d\u2074 : Module R P\nM' : Type ?u.72528\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\nN' : Type ?u.72916\ninst\u271d\u00b9 : AddCommGroup N'\ninst\u271d : Module R N'\n\u03b9 : Type u_1\n\u03b9' : Type ?u.73307\n\u03b9'' : Type ?u.73310\nf f' : AlternatingMap R M N \u03b9\ng g\u2082 : AlternatingMap R M N' \u03b9\ng' : AlternatingMap R M' N' \u03b9\nv\u271d : \u03b9 \u2192 M\nv' : \u03b9 \u2192 M'\nv : \u03b9 \u2192 M\ni\u2081 i\u2082 : \u03b9\nheq : v i\u2081 = v i\u2082\nhne : i\u2081 \u2260 i\u2082\n\u22a2 \u2191f v = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Function/Basic.lean", "full_name": "Function.update_noteq", "start": [578, 1], "end": [579, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.val_one", "start": [626, 1], "end": [627, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/Fin.lean", "full_name": "finAddFlip_apply_castAdd", "start": [364, 1], "end": [365, 74], "traced_tactics": [{"tactic": "simp [finAddFlip]", "state_before": "m n\u271d : \u2115\nk : Fin m\nn : \u2115\n\u22a2 \u2191finAddFlip (\u2191(Fin.castAdd n) k) = \u2191(Fin.natAdd n) k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/RBMap/WF.lean", "full_name": "Std.RBNode.Balanced.setBlack", "start": [82, 11], "end": [84, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "full_name": "SMulCommClass.of_mul_smul_one", "start": [667, 1], "end": [669, 75], "traced_tactics": [{"tactic": "rw [\u2190 H x z, smul_eq_mul, \u2190 H, smul_eq_mul, mul_assoc]", "state_before": "M\u271d : Type ?u.29809\nN\u271d : Type ?u.29812\nG : Type ?u.29815\nA : Type ?u.29818\nB : Type ?u.29821\n\u03b1 : Type ?u.29824\n\u03b2 : Type ?u.29827\n\u03b3 : Type ?u.29830\n\u03b4 : Type ?u.29833\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : Monoid N\ninst\u271d : SMul M N\nH : \u2200 (x : M) (y : N), y * x \u2022 1 = x \u2022 y\nx : M\ny z : N\n\u22a2 x \u2022 y \u2022 z = y \u2022 x \u2022 z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Nat/Gcd.lean", "full_name": "Nat.coprime_zero_left", "start": [353, 9], "end": [353, 87], "traced_tactics": [{"tactic": "simp [coprime]", "state_before": "n : Nat\n\u22a2 coprime 0 n \u2194 n = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean", "full_name": "Matrix.inv_reindex", "start": [690, 1], "end": [691, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/AlgebraicIndependent.lean", "full_name": "AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_X_none", "start": [450, 1], "end": [454, 22], "traced_tactics": [{"tactic": "rw [AlgebraicIndependent.mvPolynomialOptionEquivPolynomialAdjoin_apply, aeval_X, Option.elim,\n  Polynomial.map_X]", "state_before": "\u03b9 : Type u_1\n\u03b9' : Type ?u.1145486\nR : Type u_2\nK : Type ?u.1145492\nA : Type u_3\nA' : Type ?u.1145498\nA'' : Type ?u.1145501\nV : Type u\nV' : Type ?u.1145506\nx : \u03b9 \u2192 A\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing A'\ninst\u271d\u00b3 : CommRing A''\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Algebra R A'\ninst\u271d : Algebra R A''\na b : R\nhx : AlgebraicIndependent R x\n\u22a2 \u2191(mvPolynomialOptionEquivPolynomialAdjoin hx) (X none) = Polynomial.X", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean", "full_name": "EuclideanGeometry.angle_eq_arcsin_of_angle_eq_pi_div_two", "start": [385, 1], "end": [391, 57], "traced_tactics": [{"tactic": "rw [angle, \u2190 inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, \u2190 neg_eq_zero, \u2190\n  inner_neg_left, neg_vsub_eq_vsub_rev] at h", "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : \u2220 p\u2081 p\u2082 p\u2083 = \u03c0 / 2\nh0 : p\u2081 \u2260 p\u2082 \u2228 p\u2083 \u2260 p\u2082\n\u22a2 \u2220 p\u2082 p\u2083 p\u2081 = Real.arcsin (dist p\u2081 p\u2082 / dist p\u2081 p\u2083)", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : inner (p\u2082 -\u1d65 p\u2083) (p\u2081 -\u1d65 p\u2082) = 0\nh0 : p\u2081 \u2260 p\u2082 \u2228 p\u2083 \u2260 p\u2082\n\u22a2 \u2220 p\u2082 p\u2083 p\u2081 = Real.arcsin (dist p\u2081 p\u2082 / dist p\u2081 p\u2083)"}, {"tactic": "rw [\u2190 @vsub_ne_zero V, @ne_comm _ p\u2083, \u2190 @vsub_ne_zero V _ _ _ p\u2082, or_comm] at h0", "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : inner (p\u2082 -\u1d65 p\u2083) (p\u2081 -\u1d65 p\u2082) = 0\nh0 : p\u2081 \u2260 p\u2082 \u2228 p\u2083 \u2260 p\u2082\n\u22a2 \u2220 p\u2082 p\u2083 p\u2081 = Real.arcsin (dist p\u2081 p\u2082 / dist p\u2081 p\u2083)", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : inner (p\u2082 -\u1d65 p\u2083) (p\u2081 -\u1d65 p\u2082) = 0\nh0 : p\u2082 -\u1d65 p\u2083 \u2260 0 \u2228 p\u2081 -\u1d65 p\u2082 \u2260 0\n\u22a2 \u2220 p\u2082 p\u2083 p\u2081 = Real.arcsin (dist p\u2081 p\u2082 / dist p\u2081 p\u2083)"}, {"tactic": "rw [angle, dist_eq_norm_vsub V p\u2081 p\u2082, dist_eq_norm_vsub V p\u2081 p\u2083, \u2190 vsub_add_vsub_cancel p\u2081 p\u2082 p\u2083,\n  add_comm, angle_add_eq_arcsin_of_inner_eq_zero h h0]", "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : inner (p\u2082 -\u1d65 p\u2083) (p\u2081 -\u1d65 p\u2082) = 0\nh0 : p\u2082 -\u1d65 p\u2083 \u2260 0 \u2228 p\u2081 -\u1d65 p\u2082 \u2260 0\n\u22a2 \u2220 p\u2082 p\u2083 p\u2081 = Real.arcsin (dist p\u2081 p\u2082 / dist p\u2081 p\u2083)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/NAry.lean", "full_name": "Filter.map\u2082_distrib_le_right", "start": [368, 1], "end": [375, 67], "traced_tactics": [{"tactic": "rintro s \u27e8t\u2081, t\u2082, \u27e8u, w\u2081, hu, hw\u2081, ht\u2081\u27e9, \u27e8v, w\u2082, hv, hw\u2082, ht\u2082\u27e9, hs\u27e9", "state_before": "\u03b1 : Type u_4\n\u03b1' : Type u_6\n\u03b2 : Type u_5\n\u03b2' : Type u_7\n\u03b3 : Type u_3\n\u03b3' : Type ?u.42546\n\u03b4 : Type u_2\n\u03b4' : Type ?u.42552\n\u03b5 : Type u_1\n\u03b5' : Type ?u.42558\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nm\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03b1'\nm\u2082 : \u03b2 \u2192 \u03b3 \u2192 \u03b2'\nn' : \u03b1' \u2192 \u03b2' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m (n a b) c = n' (m\u2081 a c) (m\u2082 b c)\n\u22a2 map\u2082 m (map\u2082 n f g) h \u2264 map\u2082 n' (map\u2082 m\u2081 f h) (map\u2082 m\u2082 g h)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_4\n\u03b1' : Type u_6\n\u03b2 : Type u_5\n\u03b2' : Type u_7\n\u03b3 : Type u_3\n\u03b3' : Type ?u.42546\n\u03b4 : Type u_2\n\u03b4' : Type ?u.42552\n\u03b5 : Type u_1\n\u03b5' : Type ?u.42558\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu\u271d : Set \u03b3\nv\u271d : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nm\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03b1'\nm\u2082 : \u03b2 \u2192 \u03b3 \u2192 \u03b2'\nn' : \u03b1' \u2192 \u03b2' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m (n a b) c = n' (m\u2081 a c) (m\u2082 b c)\ns : Set \u03b5\nt\u2081 : Set \u03b1'\nt\u2082 : Set \u03b2'\nu : Set \u03b1\nw\u2081 : Set \u03b3\nhu : u \u2208 f\nhw\u2081 : w\u2081 \u2208 h\nht\u2081 : image2 m\u2081 u w\u2081 \u2286 t\u2081\nhs : image2 n' t\u2081 t\u2082 \u2286 s\nv : Set \u03b2\nw\u2082 : Set \u03b3\nhv : v \u2208 g\nhw\u2082 : w\u2082 \u2208 h\nht\u2082 : image2 m\u2082 v w\u2082 \u2286 t\u2082\n\u22a2 s \u2208 map\u2082 m (map\u2082 n f g) h"}, {"tactic": "refine' \u27e8_, w\u2081 \u2229 w\u2082, image2_mem_map\u2082 hu hv, inter_mem hw\u2081 hw\u2082, _\u27e9", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_4\n\u03b1' : Type u_6\n\u03b2 : Type u_5\n\u03b2' : Type u_7\n\u03b3 : Type u_3\n\u03b3' : Type ?u.42546\n\u03b4 : Type u_2\n\u03b4' : Type ?u.42552\n\u03b5 : Type u_1\n\u03b5' : Type ?u.42558\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu\u271d : Set \u03b3\nv\u271d : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nm\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03b1'\nm\u2082 : \u03b2 \u2192 \u03b3 \u2192 \u03b2'\nn' : \u03b1' \u2192 \u03b2' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m (n a b) c = n' (m\u2081 a c) (m\u2082 b c)\ns : Set \u03b5\nt\u2081 : Set \u03b1'\nt\u2082 : Set \u03b2'\nu : Set \u03b1\nw\u2081 : Set \u03b3\nhu : u \u2208 f\nhw\u2081 : w\u2081 \u2208 h\nht\u2081 : image2 m\u2081 u w\u2081 \u2286 t\u2081\nhs : image2 n' t\u2081 t\u2082 \u2286 s\nv : Set \u03b2\nw\u2082 : Set \u03b3\nhv : v \u2208 g\nhw\u2082 : w\u2082 \u2208 h\nht\u2082 : image2 m\u2082 v w\u2082 \u2286 t\u2082\n\u22a2 s \u2208 map\u2082 m (map\u2082 n f g) h", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_4\n\u03b1' : Type u_6\n\u03b2 : Type u_5\n\u03b2' : Type u_7\n\u03b3 : Type u_3\n\u03b3' : Type ?u.42546\n\u03b4 : Type u_2\n\u03b4' : Type ?u.42552\n\u03b5 : Type u_1\n\u03b5' : Type ?u.42558\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu\u271d : Set \u03b3\nv\u271d : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nm\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03b1'\nm\u2082 : \u03b2 \u2192 \u03b3 \u2192 \u03b2'\nn' : \u03b1' \u2192 \u03b2' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m (n a b) c = n' (m\u2081 a c) (m\u2082 b c)\ns : Set \u03b5\nt\u2081 : Set \u03b1'\nt\u2082 : Set \u03b2'\nu : Set \u03b1\nw\u2081 : Set \u03b3\nhu : u \u2208 f\nhw\u2081 : w\u2081 \u2208 h\nht\u2081 : image2 m\u2081 u w\u2081 \u2286 t\u2081\nhs : image2 n' t\u2081 t\u2082 \u2286 s\nv : Set \u03b2\nw\u2082 : Set \u03b3\nhv : v \u2208 g\nhw\u2082 : w\u2082 \u2208 h\nht\u2082 : image2 m\u2082 v w\u2082 \u2286 t\u2082\n\u22a2 image2 m (image2 n u v) (w\u2081 \u2229 w\u2082) \u2286 s"}, {"tactic": "refine' (image2_distrib_subset_right h_distrib).trans ((image2_subset _ _).trans hs)", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_4\n\u03b1' : Type u_6\n\u03b2 : Type u_5\n\u03b2' : Type u_7\n\u03b3 : Type u_3\n\u03b3' : Type ?u.42546\n\u03b4 : Type u_2\n\u03b4' : Type ?u.42552\n\u03b5 : Type u_1\n\u03b5' : Type ?u.42558\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu\u271d : Set \u03b3\nv\u271d : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nm\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03b1'\nm\u2082 : \u03b2 \u2192 \u03b3 \u2192 \u03b2'\nn' : \u03b1' \u2192 \u03b2' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m (n a b) c = n' (m\u2081 a c) (m\u2082 b c)\ns : Set \u03b5\nt\u2081 : Set \u03b1'\nt\u2082 : Set \u03b2'\nu : Set \u03b1\nw\u2081 : Set \u03b3\nhu : u \u2208 f\nhw\u2081 : w\u2081 \u2208 h\nht\u2081 : image2 m\u2081 u w\u2081 \u2286 t\u2081\nhs : image2 n' t\u2081 t\u2082 \u2286 s\nv : Set \u03b2\nw\u2082 : Set \u03b3\nhv : v \u2208 g\nhw\u2082 : w\u2082 \u2208 h\nht\u2082 : image2 m\u2082 v w\u2082 \u2286 t\u2082\n\u22a2 image2 m (image2 n u v) (w\u2081 \u2229 w\u2082) \u2286 s", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_4\n\u03b1' : Type u_6\n\u03b2 : Type u_5\n\u03b2' : Type u_7\n\u03b3 : Type u_3\n\u03b3' : Type ?u.42546\n\u03b4 : Type u_2\n\u03b4' : Type ?u.42552\n\u03b5 : Type u_1\n\u03b5' : Type ?u.42558\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu\u271d : Set \u03b3\nv\u271d : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nm\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03b1'\nm\u2082 : \u03b2 \u2192 \u03b3 \u2192 \u03b2'\nn' : \u03b1' \u2192 \u03b2' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m (n a b) c = n' (m\u2081 a c) (m\u2082 b c)\ns : Set \u03b5\nt\u2081 : Set \u03b1'\nt\u2082 : Set \u03b2'\nu : Set \u03b1\nw\u2081 : Set \u03b3\nhu : u \u2208 f\nhw\u2081 : w\u2081 \u2208 h\nht\u2081 : image2 m\u2081 u w\u2081 \u2286 t\u2081\nhs : image2 n' t\u2081 t\u2082 \u2286 s\nv : Set \u03b2\nw\u2082 : Set \u03b3\nhv : v \u2208 g\nhw\u2082 : w\u2082 \u2208 h\nht\u2082 : image2 m\u2082 v w\u2082 \u2286 t\u2082\n\u22a2 image2 (fun a c => m\u2081 a c) u (w\u2081 \u2229 w\u2082) \u2286 t\u2081\n\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_4\n\u03b1' : Type u_6\n\u03b2 : Type u_5\n\u03b2' : Type u_7\n\u03b3 : Type u_3\n\u03b3' : Type ?u.42546\n\u03b4 : Type u_2\n\u03b4' : Type ?u.42552\n\u03b5 : Type u_1\n\u03b5' : Type ?u.42558\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu\u271d : Set \u03b3\nv\u271d : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nm\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03b1'\nm\u2082 : \u03b2 \u2192 \u03b3 \u2192 \u03b2'\nn' : \u03b1' \u2192 \u03b2' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m (n a b) c = n' (m\u2081 a c) (m\u2082 b c)\ns : Set \u03b5\nt\u2081 : Set \u03b1'\nt\u2082 : Set \u03b2'\nu : Set \u03b1\nw\u2081 : Set \u03b3\nhu : u \u2208 f\nhw\u2081 : w\u2081 \u2208 h\nht\u2081 : image2 m\u2081 u w\u2081 \u2286 t\u2081\nhs : image2 n' t\u2081 t\u2082 \u2286 s\nv : Set \u03b2\nw\u2082 : Set \u03b3\nhv : v \u2208 g\nhw\u2082 : w\u2082 \u2208 h\nht\u2082 : image2 m\u2082 v w\u2082 \u2286 t\u2082\n\u22a2 image2 (fun b c => m\u2082 b c) v (w\u2081 \u2229 w\u2082) \u2286 t\u2082"}, {"tactic": "exact (image2_subset_left <| inter_subset_left _ _).trans ht\u2081", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_4\n\u03b1' : Type u_6\n\u03b2 : Type u_5\n\u03b2' : Type u_7\n\u03b3 : Type u_3\n\u03b3' : Type ?u.42546\n\u03b4 : Type u_2\n\u03b4' : Type ?u.42552\n\u03b5 : Type u_1\n\u03b5' : Type ?u.42558\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu\u271d : Set \u03b3\nv\u271d : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nm\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03b1'\nm\u2082 : \u03b2 \u2192 \u03b3 \u2192 \u03b2'\nn' : \u03b1' \u2192 \u03b2' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m (n a b) c = n' (m\u2081 a c) (m\u2082 b c)\ns : Set \u03b5\nt\u2081 : Set \u03b1'\nt\u2082 : Set \u03b2'\nu : Set \u03b1\nw\u2081 : Set \u03b3\nhu : u \u2208 f\nhw\u2081 : w\u2081 \u2208 h\nht\u2081 : image2 m\u2081 u w\u2081 \u2286 t\u2081\nhs : image2 n' t\u2081 t\u2082 \u2286 s\nv : Set \u03b2\nw\u2082 : Set \u03b3\nhv : v \u2208 g\nhw\u2082 : w\u2082 \u2208 h\nht\u2082 : image2 m\u2082 v w\u2082 \u2286 t\u2082\n\u22a2 image2 (fun a c => m\u2081 a c) u (w\u2081 \u2229 w\u2082) \u2286 t\u2081", "state_after": "no goals"}, {"tactic": "exact (image2_subset_left <| inter_subset_right _ _).trans ht\u2082", "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_4\n\u03b1' : Type u_6\n\u03b2 : Type u_5\n\u03b2' : Type u_7\n\u03b3 : Type u_3\n\u03b3' : Type ?u.42546\n\u03b4 : Type u_2\n\u03b4' : Type ?u.42552\n\u03b5 : Type u_1\n\u03b5' : Type ?u.42558\nm\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh h\u2081 h\u2082 : Filter \u03b3\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081\u271d t\u2082\u271d : Set \u03b2\nu\u271d : Set \u03b3\nv\u271d : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nm : \u03b4 \u2192 \u03b3 \u2192 \u03b5\nn : \u03b1 \u2192 \u03b2 \u2192 \u03b4\nm\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03b1'\nm\u2082 : \u03b2 \u2192 \u03b3 \u2192 \u03b2'\nn' : \u03b1' \u2192 \u03b2' \u2192 \u03b5\nh_distrib : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m (n a b) c = n' (m\u2081 a c) (m\u2082 b c)\ns : Set \u03b5\nt\u2081 : Set \u03b1'\nt\u2082 : Set \u03b2'\nu : Set \u03b1\nw\u2081 : Set \u03b3\nhu : u \u2208 f\nhw\u2081 : w\u2081 \u2208 h\nht\u2081 : image2 m\u2081 u w\u2081 \u2286 t\u2081\nhs : image2 n' t\u2081 t\u2082 \u2286 s\nv : Set \u03b2\nw\u2082 : Set \u03b3\nhv : v \u2208 g\nhw\u2082 : w\u2082 \u2208 h\nht\u2082 : image2 m\u2082 v w\u2082 \u2286 t\u2082\n\u22a2 image2 (fun b c => m\u2082 b c) v (w\u2081 \u2229 w\u2082) \u2286 t\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Order/IntermediateValue.lean", "full_name": "isPreconnected_uIcc", "start": [424, 1], "end": [425, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/Completion.lean", "full_name": "UniformSpace.Completion.extension_unique", "start": [564, 1], "end": [567, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/ConditionallyCompleteLattice/Finset.lean", "full_name": "Finset.inf'_id_eq_csInf", "start": [111, 1], "end": [113, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean", "full_name": "CategoryTheory.Limits.cokernelIsoOfEq_trans", "start": [853, 1], "end": [856, 50], "traced_tactics": [{"tactic": "cases w\u2081", "state_before": "C : Type u\ninst\u271d\u2075 : Category C\ninst\u271d\u2074 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b3 : HasCokernel f\u271d\nf g h : X \u27f6 Y\ninst\u271d\u00b2 : HasCokernel f\ninst\u271d\u00b9 : HasCokernel g\ninst\u271d : HasCokernel h\nw\u2081 : f = g\nw\u2082 : g = h\n\u22a2 cokernelIsoOfEq w\u2081 \u226a\u226b cokernelIsoOfEq w\u2082 = cokernelIsoOfEq (_ : f = h)", "state_after": "case refl\nC : Type u\ninst\u271d\u2075 : Category C\ninst\u271d\u2074 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b3 : HasCokernel f\u271d\nf h : X \u27f6 Y\ninst\u271d\u00b2 : HasCokernel f\ninst\u271d\u00b9 : HasCokernel h\ninst\u271d : HasCokernel f\nw\u2082 : f = h\n\u22a2 cokernelIsoOfEq (_ : f = f) \u226a\u226b cokernelIsoOfEq w\u2082 = cokernelIsoOfEq (_ : f = h)"}, {"tactic": "cases w\u2082", "state_before": "case refl\nC : Type u\ninst\u271d\u2075 : Category C\ninst\u271d\u2074 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b3 : HasCokernel f\u271d\nf h : X \u27f6 Y\ninst\u271d\u00b2 : HasCokernel f\ninst\u271d\u00b9 : HasCokernel h\ninst\u271d : HasCokernel f\nw\u2082 : f = h\n\u22a2 cokernelIsoOfEq (_ : f = f) \u226a\u226b cokernelIsoOfEq w\u2082 = cokernelIsoOfEq (_ : f = h)", "state_after": "case refl.refl\nC : Type u\ninst\u271d\u2075 : Category C\ninst\u271d\u2074 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b3 : HasCokernel f\u271d\nf : X \u27f6 Y\ninst\u271d\u00b2 inst\u271d\u00b9 inst\u271d : HasCokernel f\n\u22a2 cokernelIsoOfEq (_ : f = f) \u226a\u226b cokernelIsoOfEq (_ : f = f) = cokernelIsoOfEq (_ : f = f)"}, {"tactic": "ext", "state_before": "case refl.refl\nC : Type u\ninst\u271d\u2075 : Category C\ninst\u271d\u2074 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b3 : HasCokernel f\u271d\nf : X \u27f6 Y\ninst\u271d\u00b2 inst\u271d\u00b9 inst\u271d : HasCokernel f\n\u22a2 cokernelIsoOfEq (_ : f = f) \u226a\u226b cokernelIsoOfEq (_ : f = f) = cokernelIsoOfEq (_ : f = f)", "state_after": "case refl.refl.w.h\nC : Type u\ninst\u271d\u2075 : Category C\ninst\u271d\u2074 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b3 : HasCokernel f\u271d\nf : X \u27f6 Y\ninst\u271d\u00b2 inst\u271d\u00b9 inst\u271d : HasCokernel f\n\u22a2 coequalizer.\u03c0 f 0 \u226b (cokernelIsoOfEq (_ : f = f) \u226a\u226b cokernelIsoOfEq (_ : f = f)).hom =\n    coequalizer.\u03c0 f 0 \u226b (cokernelIsoOfEq (_ : f = f)).hom"}, {"tactic": "simp [cokernelIsoOfEq]", "state_before": "case refl.refl.w.h\nC : Type u\ninst\u271d\u2075 : Category C\ninst\u271d\u2074 : HasZeroMorphisms C\nX Y : C\nf\u271d : X \u27f6 Y\ninst\u271d\u00b3 : HasCokernel f\u271d\nf : X \u27f6 Y\ninst\u271d\u00b2 inst\u271d\u00b9 inst\u271d : HasCokernel f\n\u22a2 coequalizer.\u03c0 f 0 \u226b (cokernelIsoOfEq (_ : f = f) \u226a\u226b cokernelIsoOfEq (_ : f = f)).hom =\n    coequalizer.\u03c0 f 0 \u226b (cokernelIsoOfEq (_ : f = f)).hom", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "full_name": "Filter.Tendsto.subseq_mem_entourage", "start": [269, 1], "end": [275, 70], "traced_tactics": [{"tactic": "rcases mem_atTop_sets.1 (hu (ball_mem_nhds a (symm_le_uniformity <| hV 0))) with \u27e8n, hn\u27e9", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nV : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhV : \u2200 (n : \u2115), V n \u2208 \ud835\udce4 \u03b1\nu : \u2115 \u2192 \u03b1\na : \u03b1\nhu : Tendsto u atTop (\ud835\udcdd a)\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 (u (\u03c6 0), a) \u2208 V 0 \u2227 \u2200 (n : \u2115), (u (\u03c6 (n + 1)), u (\u03c6 n)) \u2208 V (n + 1)", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nV : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhV : \u2200 (n : \u2115), V n \u2208 \ud835\udce4 \u03b1\nu : \u2115 \u2192 \u03b1\na : \u03b1\nhu : Tendsto u atTop (\ud835\udcdd a)\nn : \u2115\nhn : \u2200 (b : \u2115), b \u2265 n \u2192 b \u2208 u \u207b\u00b9' ball a (Prod.swap \u207b\u00b9' V 0)\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 (u (\u03c6 0), a) \u2208 V 0 \u2227 \u2200 (n : \u2115), (u (\u03c6 (n + 1)), u (\u03c6 n)) \u2208 V (n + 1)"}, {"tactic": "rcases (hu.comp (tendsto_add_atTop_nat n)).cauchySeq.subseq_mem fun n => hV (n + 1) with\n  \u27e8\u03c6, \u03c6_mono, h\u03c6V\u27e9", "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nV : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhV : \u2200 (n : \u2115), V n \u2208 \ud835\udce4 \u03b1\nu : \u2115 \u2192 \u03b1\na : \u03b1\nhu : Tendsto u atTop (\ud835\udcdd a)\nn : \u2115\nhn : \u2200 (b : \u2115), b \u2265 n \u2192 b \u2208 u \u207b\u00b9' ball a (Prod.swap \u207b\u00b9' V 0)\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 (u (\u03c6 0), a) \u2208 V 0 \u2227 \u2200 (n : \u2115), (u (\u03c6 (n + 1)), u (\u03c6 n)) \u2208 V (n + 1)", "state_after": "case intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nV : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhV : \u2200 (n : \u2115), V n \u2208 \ud835\udce4 \u03b1\nu : \u2115 \u2192 \u03b1\na : \u03b1\nhu : Tendsto u atTop (\ud835\udcdd a)\nn : \u2115\nhn : \u2200 (b : \u2115), b \u2265 n \u2192 b \u2208 u \u207b\u00b9' ball a (Prod.swap \u207b\u00b9' V 0)\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nh\u03c6V : \u2200 (n_1 : \u2115), ((u \u2218 fun a => a + n) (\u03c6 (n_1 + 1)), (u \u2218 fun a => a + n) (\u03c6 n_1)) \u2208 V (n_1 + 1)\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 (u (\u03c6 0), a) \u2208 V 0 \u2227 \u2200 (n : \u2115), (u (\u03c6 (n + 1)), u (\u03c6 n)) \u2208 V (n + 1)"}, {"tactic": "exact \u27e8fun k => \u03c6 k + n, \u03c6_mono.add_const _, hn _ le_add_self, h\u03c6V\u27e9", "state_before": "case intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : UniformSpace \u03b1\nV : \u2115 \u2192 Set (\u03b1 \u00d7 \u03b1)\nhV : \u2200 (n : \u2115), V n \u2208 \ud835\udce4 \u03b1\nu : \u2115 \u2192 \u03b1\na : \u03b1\nhu : Tendsto u atTop (\ud835\udcdd a)\nn : \u2115\nhn : \u2200 (b : \u2115), b \u2265 n \u2192 b \u2208 u \u207b\u00b9' ball a (Prod.swap \u207b\u00b9' V 0)\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_mono : StrictMono \u03c6\nh\u03c6V : \u2200 (n_1 : \u2115), ((u \u2218 fun a => a + n) (\u03c6 (n_1 + 1)), (u \u2218 fun a => a + n) (\u03c6 n_1)) \u2208 V (n_1 + 1)\n\u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 (u (\u03c6 0), a) \u2208 V 0 \u2227 \u2200 (n : \u2115), (u (\u03c6 (n + 1)), u (\u03c6 n)) \u2208 V (n + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Prime.lean", "full_name": "Nat.Prime.even_sub_one", "start": [512, 1], "end": [513, 83], "traced_tactics": [{"tactic": "rw [hn, Nat.add_sub_cancel, two_mul]", "state_before": "p : \u2115\nhp : Prime p\nh2 : p \u2260 2\nn : \u2115\nhn : p = 2 * n + 1\n\u22a2 p - 1 = n + n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.AEStronglyMeasurable.stronglyMeasurable_mk", "start": [1200, 1], "end": [1201, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Ordmap/Ordset.lean", "full_name": "Ordnode.all_node'", "start": [493, 1], "end": [494, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/Basic.lean", "full_name": "Equiv.sumProdDistrib_symm_apply_left", "start": [921, 1], "end": [923, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "full_name": "AddLECancellable.lt_add_of_tsub_lt_right", "start": [318, 11], "end": [323, 17], "traced_tactics": [{"tactic": "rw [lt_iff_le_and_ne, \u2190 tsub_le_iff_right]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.39404\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhc : AddLECancellable c\nh : a - c < b\n\u22a2 a < b + c", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.39404\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhc : AddLECancellable c\nh : a - c < b\n\u22a2 a - c \u2264 b \u2227 a \u2260 b + c"}, {"tactic": "refine' \u27e8h.le, _\u27e9", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.39404\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhc : AddLECancellable c\nh : a - c < b\n\u22a2 a - c \u2264 b \u2227 a \u2260 b + c", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.39404\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhc : AddLECancellable c\nh : a - c < b\n\u22a2 a \u2260 b + c"}, {"tactic": "rintro rfl", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.39404\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhc : AddLECancellable c\nh : a - c < b\n\u22a2 a \u2260 b + c", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.39404\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\nb c d : \u03b1\nhc : AddLECancellable c\nh : b + c - c < b\n\u22a2 False"}, {"tactic": "simp [hc] at h", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.39404\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\nb c d : \u03b1\nhc : AddLECancellable c\nh : b + c - c < b\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Sym/Sym2.lean", "full_name": "Sym2.map_map", "start": [260, 1], "end": [261, 34], "traced_tactics": [{"tactic": "revert x", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_3\n\u03b3 : Type u_2\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\nx : Sym2 \u03b1\n\u22a2 map g (map f x) = map (g \u2218 f) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_3\n\u03b3 : Type u_2\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (x : Sym2 \u03b1), map g (map f x) = map (g \u2218 f) x"}, {"tactic": "apply Sym2.ind", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_3\n\u03b3 : Type u_2\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (x : Sym2 \u03b1), map g (map f x) = map (g \u2218 f) x", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_3\n\u03b3 : Type u_2\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (x y : \u03b1), map g (map f (Quotient.mk (Rel.setoid \u03b1) (x, y))) = map (g \u2218 f) (Quotient.mk (Rel.setoid \u03b1) (x, y))"}, {"tactic": "aesop", "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_3\n\u03b3 : Type u_2\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (x y : \u03b1), map g (map f (Quotient.mk (Rel.setoid \u03b1) (x, y))) = map (g \u2218 f) (Quotient.mk (Rel.setoid \u03b1) (x, y))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "full_name": "MeasureTheory.AECover.mono_ac", "start": [121, 1], "end": [122, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Hall/Finite.lean", "full_name": "HallMarriageTheorem.hall_cond_of_compl", "start": [139, 1], "end": [162, 28], "traced_tactics": [{"tactic": "haveI := Classical.decEq \u03b9", "state_before": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t)", "state_after": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis : DecidableEq \u03b9\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t)"}, {"tactic": "have disj : Disjoint s (s'.image fun z => z.1) := by\n  simp only [disjoint_left, not_exists, mem_image, exists_prop, SetCoe.exists, exists_and_right,\n    exists_eq_right, Subtype.coe_mk]\n  intro x hx hc _\n  exact absurd hx hc", "state_before": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis : DecidableEq \u03b9\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t)", "state_after": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t)"}, {"tactic": "have : s'.card = (s \u222a s'.image fun z => z.1).card - s.card := by\n  simp [disj, card_image_of_injective _ Subtype.coe_injective]", "state_before": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t)", "state_after": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t)"}, {"tactic": "rw [this, hus]", "state_before": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\n\u22a2 card s' \u2264 card (Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t)", "state_after": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\n\u22a2 card (s \u222a image (fun z => \u2191z) s') - card (Finset.biUnion s t) \u2264\n    card (Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t)"}, {"tactic": "refine' (tsub_le_tsub_right (ht _) _).trans _", "state_before": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\n\u22a2 card (s \u222a image (fun z => \u2191z) s') - card (Finset.biUnion s t) \u2264\n    card (Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t)", "state_after": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\n\u22a2 card (Finset.biUnion (s \u222a image (fun z => \u2191z) s') t) - card (Finset.biUnion s t) \u2264\n    card (Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t)"}, {"tactic": "rw [\u2190 card_sdiff]", "state_before": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\n\u22a2 card (Finset.biUnion (s \u222a image (fun z => \u2191z) s') t) - card (Finset.biUnion s t) \u2264\n    card (Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t)", "state_after": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\n\u22a2 card (Finset.biUnion (s \u222a image (fun z => \u2191z) s') t \\ Finset.biUnion s t) \u2264\n    card (Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t)\n\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\n\u22a2 Finset.biUnion s t \u2286 Finset.biUnion (s \u222a image (fun z => \u2191z) s') t"}, {"tactic": "simp only [disjoint_left, not_exists, mem_image, exists_prop, SetCoe.exists, exists_and_right,\n  exists_eq_right, Subtype.coe_mk]", "state_before": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis : DecidableEq \u03b9\n\u22a2 Disjoint s (image (fun z => \u2191z) s')", "state_after": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis : DecidableEq \u03b9\n\u22a2 \u2200 \u2983a : \u03b9\u2984, a \u2208 s \u2192 \u2200 (x : a \u2208 \u2191s\u1d9c), \u00ac{ val := a, property := (_ : a \u2208 \u2191s\u1d9c) } \u2208 s'"}, {"tactic": "intro x hx hc _", "state_before": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis : DecidableEq \u03b9\n\u22a2 \u2200 \u2983a : \u03b9\u2984, a \u2208 s \u2192 \u2200 (x : a \u2208 \u2191s\u1d9c), \u00ac{ val := a, property := (_ : a \u2208 \u2191s\u1d9c) } \u2208 s'", "state_after": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis : DecidableEq \u03b9\nx : \u03b9\nhx : x \u2208 s\nhc : x \u2208 \u2191s\u1d9c\na\u271d : { val := x, property := (_ : x \u2208 \u2191s\u1d9c) } \u2208 s'\n\u22a2 False"}, {"tactic": "exact absurd hx hc", "state_before": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis : DecidableEq \u03b9\nx : \u03b9\nhx : x \u2208 s\nhc : x \u2208 \u2191s\u1d9c\na\u271d : { val := x, property := (_ : x \u2208 \u2191s\u1d9c) } \u2208 s'\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simp [disj, card_image_of_injective _ Subtype.coe_injective]", "state_before": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\n\u22a2 card s' = card (s \u222a image (fun z => \u2191z) s') - card s", "state_after": "no goals"}, {"tactic": "refine' (card_le_of_subset _).trans le_rfl", "state_before": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\n\u22a2 card (Finset.biUnion (s \u222a image (fun z => \u2191z) s') t \\ Finset.biUnion s t) \u2264\n    card (Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t)", "state_after": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\n\u22a2 Finset.biUnion (s \u222a image (fun z => \u2191z) s') t \\ Finset.biUnion s t \u2286\n    Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t"}, {"tactic": "intro t", "state_before": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\n\u22a2 Finset.biUnion (s \u222a image (fun z => \u2191z) s') t \\ Finset.biUnion s t \u2286\n    Finset.biUnion s' fun x' => t \u2191x' \\ Finset.biUnion s t", "state_after": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d\u00b9 : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt\u271d : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t\u271d)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t\u271d)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\nt : \u03b1\n\u22a2 t \u2208 Finset.biUnion (s \u222a image (fun z => \u2191z) s') t\u271d \\ Finset.biUnion s t\u271d \u2192\n    t \u2208 Finset.biUnion s' fun x' => t\u271d \u2191x' \\ Finset.biUnion s t\u271d"}, {"tactic": "simp only [mem_biUnion, mem_sdiff, not_exists, mem_image, and_imp, mem_union, exists_and_right,\n  exists_imp]", "state_before": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d\u00b9 : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt\u271d : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t\u271d)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t\u271d)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\nt : \u03b1\n\u22a2 t \u2208 Finset.biUnion (s \u222a image (fun z => \u2191z) s') t\u271d \\ Finset.biUnion s t\u271d \u2192\n    t \u2208 Finset.biUnion s' fun x' => t\u271d \u2191x' \\ Finset.biUnion s t\u271d", "state_after": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d\u00b9 : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt\u271d : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t\u271d)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t\u271d)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\nt : \u03b1\n\u22a2 \u2200 (x : \u03b9),\n    (x \u2208 s \u2228 \u2203 a, a \u2208 s' \u2227 \u2191a = x) \u2192\n      t \u2208 t\u271d x \u2192 (\u2200 (x : \u03b9), \u00ac(x \u2208 s \u2227 t \u2208 t\u271d x)) \u2192 \u2203 a, a \u2208 s' \u2227 t \u2208 t\u271d \u2191a \u2227 \u2200 (x : \u03b9), \u00ac(x \u2208 s \u2227 t \u2208 t\u271d x)"}, {"tactic": "rintro x (hx | \u27e8x', hx', rfl\u27e9) rat hs", "state_before": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d\u00b9 : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt\u271d : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t\u271d)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t\u271d)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\nt : \u03b1\n\u22a2 \u2200 (x : \u03b9),\n    (x \u2208 s \u2228 \u2203 a, a \u2208 s' \u2227 \u2191a = x) \u2192\n      t \u2208 t\u271d x \u2192 (\u2200 (x : \u03b9), \u00ac(x \u2208 s \u2227 t \u2208 t\u271d x)) \u2192 \u2203 a, a \u2208 s' \u2227 t \u2208 t\u271d \u2191a \u2227 \u2200 (x : \u03b9), \u00ac(x \u2208 s \u2227 t \u2208 t\u271d x)", "state_after": "case inl\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d\u00b9 : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt\u271d : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t\u271d)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t\u271d)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\nt : \u03b1\nx : \u03b9\nhx : x \u2208 s\nrat : t \u2208 t\u271d x\nhs : \u2200 (x : \u03b9), \u00ac(x \u2208 s \u2227 t \u2208 t\u271d x)\n\u22a2 \u2203 a, a \u2208 s' \u2227 t \u2208 t\u271d \u2191a \u2227 \u2200 (x : \u03b9), \u00ac(x \u2208 s \u2227 t \u2208 t\u271d x)\n\ncase inr.intro.intro\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d\u00b9 : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt\u271d : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t\u271d)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t\u271d)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\nt : \u03b1\nx' : \u2191(\u2191s\u1d9c)\nhx' : x' \u2208 s'\nrat : t \u2208 t\u271d \u2191x'\nhs : \u2200 (x : \u03b9), \u00ac(x \u2208 s \u2227 t \u2208 t\u271d x)\n\u22a2 \u2203 a, a \u2208 s' \u2227 t \u2208 t\u271d \u2191a \u2227 \u2200 (x : \u03b9), \u00ac(x \u2208 s \u2227 t \u2208 t\u271d x)"}, {"tactic": "exact False.elim <| (hs x) <| And.intro hx rat", "state_before": "case inl\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d\u00b9 : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt\u271d : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t\u271d)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t\u271d)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\nt : \u03b1\nx : \u03b9\nhx : x \u2208 s\nrat : t \u2208 t\u271d x\nhs : \u2200 (x : \u03b9), \u00ac(x \u2208 s \u2227 t \u2208 t\u271d x)\n\u22a2 \u2203 a, a \u2208 s' \u2227 t \u2208 t\u271d \u2191a \u2227 \u2200 (x : \u03b9), \u00ac(x \u2208 s \u2227 t \u2208 t\u271d x)", "state_after": "no goals"}, {"tactic": "use x'", "state_before": "case inr.intro.intro\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d\u00b9 : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt\u271d : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t\u271d)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t\u271d)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\nt : \u03b1\nx' : \u2191(\u2191s\u1d9c)\nhx' : x' \u2208 s'\nrat : t \u2208 t\u271d \u2191x'\nhs : \u2200 (x : \u03b9), \u00ac(x \u2208 s \u2227 t \u2208 t\u271d x)\n\u22a2 \u2203 a, a \u2208 s' \u2227 t \u2208 t\u271d \u2191a \u2227 \u2200 (x : \u03b9), \u00ac(x \u2208 s \u2227 t \u2208 t\u271d x)", "state_after": "case inr.intro.intro\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d\u00b9 : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt\u271d : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t\u271d)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t\u271d)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\nt : \u03b1\nx' : \u2191(\u2191s\u1d9c)\nhx' : x' \u2208 s'\nrat : t \u2208 t\u271d \u2191x'\nhs : \u2200 (x : \u03b9), \u00ac(x \u2208 s \u2227 t \u2208 t\u271d x)\n\u22a2 x' \u2208 s' \u2227 t \u2208 t\u271d \u2191x' \u2227 \u2200 (x : \u03b9), \u00ac(x \u2208 s \u2227 t \u2208 t\u271d x)"}, {"tactic": "exact And.intro hx' (And.intro rat hs)", "state_before": "case inr.intro.intro\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d\u00b9 : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt\u271d : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t\u271d)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t\u271d)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\nt : \u03b1\nx' : \u2191(\u2191s\u1d9c)\nhx' : x' \u2208 s'\nrat : t \u2208 t\u271d \u2191x'\nhs : \u2200 (x : \u03b9), \u00ac(x \u2208 s \u2227 t \u2208 t\u271d x)\n\u22a2 x' \u2208 s' \u2227 t \u2208 t\u271d \u2191x' \u2227 \u2200 (x : \u03b9), \u00ac(x \u2208 s \u2227 t \u2208 t\u271d x)", "state_after": "no goals"}, {"tactic": "apply biUnion_subset_biUnion_of_subset_left", "state_before": "\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\n\u22a2 Finset.biUnion s t \u2286 Finset.biUnion (s \u222a image (fun z => \u2191z) s') t", "state_after": "case h\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\n\u22a2 s \u2286 s \u222a image (fun z => \u2191z) s'"}, {"tactic": "apply subset_union_left", "state_before": "case h\n\u03b9\u271d : Type u\n\u03b1 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nt\u271d : \u03b9\u271d \u2192 Finset \u03b1\ninst\u271d : Fintype \u03b9\u271d\n\u03b9 : Type u\nt : \u03b9 \u2192 Finset \u03b1\ns : Finset \u03b9\nhus : card s = card (Finset.biUnion s t)\nht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t)\ns' : Finset \u2191(\u2191s\u1d9c)\nthis\u271d : DecidableEq \u03b9\ndisj : Disjoint s (image (fun z => \u2191z) s')\nthis : card s' = card (s \u222a image (fun z => \u2191z) s') - card s\n\u22a2 s \u2286 s \u222a image (fun z => \u2191z) s'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/DualQuaternion.lean", "full_name": "Quaternion.fst_re_dualNumberEquiv_symm", "start": [113, 1], "end": [115, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Torsion.lean", "full_name": "CommGroup.torsion_eq_torsion_submonoid", "start": [325, 1], "end": [326, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Dual.lean", "full_name": "Module.mapEvalEquiv_symm_apply", "start": [626, 1], "end": [628, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "full_name": "Subgroup.normalClosure_eq_self", "start": [2510, 1], "end": [2511, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Antisymmetrization.lean", "full_name": "AntisymmRel.trans", "start": [63, 1], "end": [65, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Dfinsupp/Basic.lean", "full_name": "Dfinsupp.erase_single_ne", "start": [804, 1], "end": [805, 30], "traced_tactics": [{"tactic": "rw [erase_single, if_neg 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 : (i : \u03b9) \u2192 Zero (\u03b2 i)\ns : Finset \u03b9\nx\u271d : (i : \u2191\u2191s) \u2192 \u03b2 \u2191i\ni\u271d i j : \u03b9\nx : \u03b2 i\nh : i \u2260 j\n\u22a2 erase j (single i x) = single i x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Coprime/Basic.lean", "full_name": "IsCoprime.sq_add_sq_ne_zero", "start": [383, 1], "end": [392, 34], "traced_tactics": [{"tactic": "intro h'", "state_before": "R : Type u_1\ninst\u271d : LinearOrderedCommRing R\na b : R\nh : IsCoprime a b\n\u22a2 a ^ 2 + b ^ 2 \u2260 0", "state_after": "R : Type u_1\ninst\u271d : LinearOrderedCommRing R\na b : R\nh : IsCoprime a b\nh' : a ^ 2 + b ^ 2 = 0\n\u22a2 False"}, {"tactic": "obtain \u27e8ha, hb\u27e9 := (add_eq_zero_iff'\n(by rw [pow_two]; exact mul_self_nonneg _)\n  (by rw [pow_two]; exact mul_self_nonneg _)).mp h'", "state_before": "R : Type u_1\ninst\u271d : LinearOrderedCommRing R\na b : R\nh : IsCoprime a b\nh' : a ^ 2 + b ^ 2 = 0\n\u22a2 False", "state_after": "case intro\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na b : R\nh : IsCoprime a b\nh' : a ^ 2 + b ^ 2 = 0\nha : a ^ 2 = 0\nhb : b ^ 2 = 0\n\u22a2 False"}, {"tactic": "obtain rfl := pow_eq_zero ha", "state_before": "case intro\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na b : R\nh : IsCoprime a b\nh' : a ^ 2 + b ^ 2 = 0\nha : a ^ 2 = 0\nhb : b ^ 2 = 0\n\u22a2 False", "state_after": "case intro\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\nb : R\nhb : b ^ 2 = 0\nh : IsCoprime 0 b\nh' : 0 ^ 2 + b ^ 2 = 0\nha : 0 ^ 2 = 0\n\u22a2 False"}, {"tactic": "obtain rfl := pow_eq_zero hb", "state_before": "case intro\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\nb : R\nhb : b ^ 2 = 0\nh : IsCoprime 0 b\nh' : 0 ^ 2 + b ^ 2 = 0\nha : 0 ^ 2 = 0\n\u22a2 False", "state_after": "case intro\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\nha hb : 0 ^ 2 = 0\nh : IsCoprime 0 0\nh' : 0 ^ 2 + 0 ^ 2 = 0\n\u22a2 False"}, {"tactic": "exact not_isCoprime_zero_zero h", "state_before": "case intro\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\nha hb : 0 ^ 2 = 0\nh : IsCoprime 0 0\nh' : 0 ^ 2 + 0 ^ 2 = 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rw [pow_two]", "state_before": "R : Type u_1\ninst\u271d : LinearOrderedCommRing R\na b : R\nh : IsCoprime a b\nh' : a ^ 2 + b ^ 2 = 0\n\u22a2 0 \u2264 a ^ 2", "state_after": "R : Type u_1\ninst\u271d : LinearOrderedCommRing R\na b : R\nh : IsCoprime a b\nh' : a ^ 2 + b ^ 2 = 0\n\u22a2 0 \u2264 a * a"}, {"tactic": "exact mul_self_nonneg _", "state_before": "R : Type u_1\ninst\u271d : LinearOrderedCommRing R\na b : R\nh : IsCoprime a b\nh' : a ^ 2 + b ^ 2 = 0\n\u22a2 0 \u2264 a * a", "state_after": "no goals"}, {"tactic": "rw [pow_two]", "state_before": "R : Type u_1\ninst\u271d : LinearOrderedCommRing R\na b : R\nh : IsCoprime a b\nh' : a ^ 2 + b ^ 2 = 0\n\u22a2 0 \u2264 b ^ 2", "state_after": "R : Type u_1\ninst\u271d : LinearOrderedCommRing R\na b : R\nh : IsCoprime a b\nh' : a ^ 2 + b ^ 2 = 0\n\u22a2 0 \u2264 b * b"}, {"tactic": "exact mul_self_nonneg _", "state_before": "R : Type u_1\ninst\u271d : LinearOrderedCommRing R\na b : R\nh : IsCoprime a b\nh' : a ^ 2 + b ^ 2 = 0\n\u22a2 0 \u2264 b * b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Category/Basic.lean", "full_name": "CategoryTheory.eq_whisker", "start": [191, 1], "end": [191, 86], "traced_tactics": [{"tactic": "rw [w]", "state_before": "C : Type u\ninst\u271d : Category C\nX Y Z : C\nf g : X \u27f6 Y\nw : f = g\nh : Y \u27f6 Z\n\u22a2 f \u226b h = g \u226b h", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/DiffContOnCl.lean", "full_name": "diffContOnCl_univ", "start": [51, 1], "end": [52, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Lex.lean", "full_name": "List.Lex.append_right", "start": [134, 1], "end": [137, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Basic.lean", "full_name": "Finset.prod_ite", "start": [993, 1], "end": [996, 39], "traced_tactics": [{"tactic": "simp [prod_apply_ite _ _ fun x => x]", "state_before": "\u03b9 : Type ?u.398799\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\ns : Finset \u03b1\np : \u03b1 \u2192 Prop\nhp : DecidablePred p\nf g : \u03b1 \u2192 \u03b2\n\u22a2 (\u220f x in s, if p x then f x else g x) = (\u220f x in filter p s, f x) * \u220f x in filter (fun x => \u00acp x) s, g x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/Padics/PadicNumbers.lean", "full_name": "padicNormE.norm_int_lt_one_iff_dvd", "start": [920, 1], "end": [939, 35], "traced_tactics": [{"tactic": "constructor", "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\n\u22a2 \u2016\u2191k\u2016 < 1 \u2194 \u2191p \u2223 k", "state_after": "case mp\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\n\u22a2 \u2016\u2191k\u2016 < 1 \u2192 \u2191p \u2223 k\n\ncase mpr\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\n\u22a2 \u2191p \u2223 k \u2192 \u2016\u2191k\u2016 < 1"}, {"tactic": "intro h", "state_before": "case mp\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\n\u22a2 \u2016\u2191k\u2016 < 1 \u2192 \u2191p \u2223 k", "state_after": "case mp\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nh : \u2016\u2191k\u2016 < 1\n\u22a2 \u2191p \u2223 k"}, {"tactic": "contrapose! h", "state_before": "case mp\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nh : \u2016\u2191k\u2016 < 1\n\u22a2 \u2191p \u2223 k", "state_after": "case mp\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nh : \u00ac\u2191p \u2223 k\n\u22a2 1 \u2264 \u2016\u2191k\u2016"}, {"tactic": "apply le_of_eq", "state_before": "case mp\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nh : \u00ac\u2191p \u2223 k\n\u22a2 1 \u2264 \u2016\u2191k\u2016", "state_after": "case mp.a\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nh : \u00ac\u2191p \u2223 k\n\u22a2 1 = \u2016\u2191k\u2016"}, {"tactic": "rw [eq_comm]", "state_before": "case mp.a\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nh : \u00ac\u2191p \u2223 k\n\u22a2 1 = \u2016\u2191k\u2016", "state_after": "case mp.a\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nh : \u00ac\u2191p \u2223 k\n\u22a2 \u2016\u2191k\u2016 = 1"}, {"tactic": "calc\n  \u2016(k : \u211a_[p])\u2016 = \u2016((k : \u211a) : \u211a_[p])\u2016 := by norm_cast\n  _ = padicNorm p k := (padicNormE.eq_padicNorm _)\n  _ = 1 := by exact_mod_cast (int_eq_one_iff k).mpr h", "state_before": "case mp.a\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nh : \u00ac\u2191p \u2223 k\n\u22a2 \u2016\u2191k\u2016 = 1", "state_after": "no goals"}, {"tactic": "norm_cast", "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nh : \u00ac\u2191p \u2223 k\n\u22a2 \u2016\u2191k\u2016 = \u2016\u2191\u2191k\u2016", "state_after": "no goals"}, {"tactic": "exact_mod_cast (int_eq_one_iff k).mpr h", "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nh : \u00ac\u2191p \u2223 k\n\u22a2 \u2191(padicNorm p \u2191k) = 1", "state_after": "no goals"}, {"tactic": "rintro \u27e8x, rfl\u27e9", "state_before": "case mpr\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\n\u22a2 \u2191p \u2223 k \u2192 \u2016\u2191k\u2016 < 1", "state_after": "case mpr.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124\n\u22a2 \u2016\u2191(\u2191p * x)\u2016 < 1"}, {"tactic": "push_cast", "state_before": "case mpr.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124\n\u22a2 \u2016\u2191(\u2191p * x)\u2016 < 1", "state_after": "case mpr.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124\n\u22a2 \u2016\u2191p * \u2191x\u2016 < 1"}, {"tactic": "rw [padicNormE.mul]", "state_before": "case mpr.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124\n\u22a2 \u2016\u2191p * \u2191x\u2016 < 1", "state_after": "case mpr.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124\n\u22a2 \u2016\u2191p\u2016 * \u2016\u2191x\u2016 < 1"}, {"tactic": "calc\n  _ \u2264 \u2016(p : \u211a_[p])\u2016 * 1 :=\n    mul_le_mul le_rfl (by simpa using norm_int_le_one _) (norm_nonneg _) (norm_nonneg _)\n  _ < 1 := by\n    rw [mul_one, padicNormE.norm_p]\n    apply inv_lt_one\n    exact_mod_cast hp.1.one_lt", "state_before": "case mpr.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124\n\u22a2 \u2016\u2191p\u2016 * \u2016\u2191x\u2016 < 1", "state_after": "no goals"}, {"tactic": "simpa using norm_int_le_one _", "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124\n\u22a2 \u2016\u2191x\u2016 \u2264 1", "state_after": "no goals"}, {"tactic": "rw [mul_one, padicNormE.norm_p]", "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124\n\u22a2 \u2016\u2191p\u2016 * 1 < 1", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124\n\u22a2 (\u2191p)\u207b\u00b9 < 1"}, {"tactic": "apply inv_lt_one", "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124\n\u22a2 (\u2191p)\u207b\u00b9 < 1", "state_after": "case ha\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124\n\u22a2 1 < \u2191p"}, {"tactic": "exact_mod_cast hp.1.one_lt", "state_before": "case ha\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124\n\u22a2 1 < \u2191p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/Reduce.lean", "full_name": "ManyOneEquiv.congr_right", "start": [269, 1], "end": [272, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Adhesive.lean", "full_name": "CategoryTheory.Adhesive.isPullback_of_isPushout_of_mono_right", "start": [244, 1], "end": [246, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.SurjOn.rightInvOn_invFunOn", "start": [1256, 1], "end": [1257, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "full_name": "aemeasurable_of_subsingleton_codomain", "start": [37, 1], "end": [38, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Hom/Monoid.lean", "full_name": "OrderMonoidHom.toOrderHom_eq_coe", "start": [553, 1], "end": [554, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/ContinuousOn.lean", "full_name": "eventually_nhdsWithin_of_forall", "start": [449, 1], "end": [451, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.nthLe_take", "start": [1999, 1], "end": [2001, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.natCast_le", "start": [1350, 1], "end": [1352, 40], "traced_tactics": [{"tactic": "rw [\u2190 lift_mk_fin, \u2190 lift_mk_fin, lift_le, le_def, Function.Embedding.nonempty_iff_card_le,\n  Fintype.card_fin, Fintype.card_fin]", "state_before": "\u03b1 \u03b2 : Type u\nm n : \u2115\n\u22a2 \u2191m \u2264 \u2191n \u2194 m \u2264 n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Option/Basic.lean", "full_name": "Option.map_coe'", "start": [112, 1], "end": [113, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Noetherian.lean", "full_name": "isNoetherian_of_fg_of_noetherian'", "start": [571, 1], "end": [574, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/AlgebraicCard.lean", "full_name": "Algebraic.cardinal_mk_le_mul", "start": [96, 1], "end": [98, 36], "traced_tactics": [{"tactic": "rw [\u2190 lift_id (#_), \u2190 lift_id (#R[X])]", "state_before": "R A : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\n\u22a2 (#{ x // IsAlgebraic R x }) \u2264 (#R[X]) * \u2135\u2080", "state_after": "R A : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\n\u22a2 lift (#{ x // IsAlgebraic R x }) \u2264 lift (#R[X]) * \u2135\u2080"}, {"tactic": "exact cardinal_mk_lift_le_mul R A", "state_before": "R A : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\n\u22a2 lift (#{ x // IsAlgebraic R x }) \u2264 lift (#R[X]) * \u2135\u2080", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Hom/Units.lean", "full_name": "IsUnit.mul_mul_div", "start": [454, 11], "end": [454, 90], "traced_tactics": [{"tactic": "simp [h]", "state_before": "F : Type ?u.61273\nG : Type ?u.61276\n\u03b1 : Type u_1\nM : Type ?u.61282\nN : Type ?u.61285\ninst\u271d : DivisionMonoid \u03b1\na\u271d b c a : \u03b1\nh : IsUnit b\n\u22a2 a * b * (1 / b) = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/Exponential.lean", "full_name": "exp_continuous", "start": [449, 1], "end": [452, 25], "traced_tactics": [{"tactic": "rw [continuous_iff_continuousOn_univ, \u2190 Metric.eball_top_eq_univ (0 : \ud835\udd38), \u2190\n  expSeries_radius_eq_top \ud835\udd42 \ud835\udd38]", "state_before": "\ud835\udd42 : Type u_2\n\ud835\udd38 : Type u_1\n\ud835\udd39 : Type ?u.357251\ninst\u271d\u2075 : IsROrC \ud835\udd42\ninst\u271d\u2074 : NormedRing \ud835\udd38\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d\u00b2 : NormedRing \ud835\udd39\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd39\ninst\u271d : CompleteSpace \ud835\udd38\n\u22a2 Continuous (exp \ud835\udd42)", "state_after": "\ud835\udd42 : Type u_2\n\ud835\udd38 : Type u_1\n\ud835\udd39 : Type ?u.357251\ninst\u271d\u2075 : IsROrC \ud835\udd42\ninst\u271d\u2074 : NormedRing \ud835\udd38\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d\u00b2 : NormedRing \ud835\udd39\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd39\ninst\u271d : CompleteSpace \ud835\udd38\n\u22a2 ContinuousOn (exp \ud835\udd42) (EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries \ud835\udd42 \ud835\udd38)))"}, {"tactic": "exact continuousOn_exp", "state_before": "\ud835\udd42 : Type u_2\n\ud835\udd38 : Type u_1\n\ud835\udd39 : Type ?u.357251\ninst\u271d\u2075 : IsROrC \ud835\udd42\ninst\u271d\u2074 : NormedRing \ud835\udd38\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d\u00b2 : NormedRing \ud835\udd39\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd39\ninst\u271d : CompleteSpace \ud835\udd38\n\u22a2 ContinuousOn (exp \ud835\udd42) (EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries \ud835\udd42 \ud835\udd38)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/Hyperreal.lean", "full_name": "Hyperreal.infiniteNeg_iff_infinite_and_neg", "start": [502, 1], "end": [504, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "PosNum.succ_to_nat", "start": [75, 1], "end": [80, 69], "traced_tactics": [{"tactic": "simp [add_left_comm]", "state_before": "\u03b1 : Type ?u.23313\np : PosNum\n\u22a2 \u2191p + 1 + \u2191p + 1 = \u2191p + \u2191p + 1 + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Int/CharZero.lean", "full_name": "Int.cast_ne_one", "start": [54, 1], "end": [55, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": 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: Type u\n\u03b2 : Type v\nX : Type ?u.213505\n\u03b9 : Type ?u.213508\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\n\u22a2 Tendsto ((fun p => dist p.fst p.snd) \u2218 Prod.map u u) atTop (\ud835\udcdd 0) \u2194\n    Tendsto (fun n => dist (u n.fst) (u n.snd)) atTop (\ud835\udcdd 0)"}, {"tactic": "rfl", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type ?u.213505\n\u03b9 : Type ?u.213508\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b2\ninst\u271d : SemilatticeSup \u03b2\nu : \u03b2 \u2192 \u03b1\n\u22a2 Tendsto ((fun p => dist p.fst p.snd) \u2218 Prod.map u u) atTop (\ud835\udcdd 0) \u2194\n    Tendsto (fun n => dist (u n.fst) (u n.snd)) atTop (\ud835\udcdd 0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Support.lean", "full_name": "HasCompactMulSupport.mono", "start": [196, 1], "end": [198, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Spectral/Hom.lean", "full_name": "SpectralMap.ext", "start": [133, 1], "end": [134, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/ENNReal.lean", "full_name": "ENNReal.bot_eq_zero", "start": [654, 1], "end": [654, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Prod.lean", "full_name": "Finset.diag_union", "start": [392, 1], "end": [394, 48], "traced_tactics": [{"tactic": "ext \u27e8i, j\u27e9", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.123122\n\u03b3 : Type ?u.123125\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nx : \u03b1 \u00d7 \u03b1\n\u22a2 diag (s \u222a t) = diag s \u222a diag t", "state_after": "case a.mk\n\u03b1 : Type u_1\n\u03b2 : Type ?u.123122\n\u03b3 : Type ?u.123125\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nx : \u03b1 \u00d7 \u03b1\ni j : \u03b1\n\u22a2 (i, j) \u2208 diag (s \u222a t) \u2194 (i, j) \u2208 diag s \u222a diag t"}, {"tactic": "simp only [mem_diag, mem_union, or_and_right]", "state_before": "case a.mk\n\u03b1 : Type u_1\n\u03b2 : Type ?u.123122\n\u03b3 : Type ?u.123125\ninst\u271d : DecidableEq \u03b1\ns t : Finset \u03b1\nx : \u03b1 \u00d7 \u03b1\ni j : \u03b1\n\u22a2 (i, j) \u2208 diag (s \u222a t) \u2194 (i, j) \u2208 diag s \u222a diag t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "full_name": "Subgroup.centralizer_top", "start": [2293, 1], "end": [2294, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.erase_le_iff_le_cons", "start": [1098, 1], "end": [1101, 63], "traced_tactics": [{"tactic": "rw [\u2190 cons_erase m] at h", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.101950\n\u03b3 : Type ?u.101953\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Multiset \u03b1\na\u271d b : \u03b1\ns t : Multiset \u03b1\na : \u03b1\nh : s \u2264 a ::\u2098 t\nm : a \u2208 s\n\u22a2 erase s a \u2264 t", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.101950\n\u03b3 : Type ?u.101953\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Multiset \u03b1\na\u271d b : \u03b1\ns t : Multiset \u03b1\na : \u03b1\nh : a ::\u2098 erase s a \u2264 a ::\u2098 t\nm : a \u2208 s\n\u22a2 erase s a \u2264 t"}, {"tactic": "exact (cons_le_cons_iff _).1 h", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.101950\n\u03b3 : Type ?u.101953\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d : Multiset \u03b1\na\u271d b : \u03b1\ns t : Multiset \u03b1\na : \u03b1\nh : a ::\u2098 erase s a \u2264 a ::\u2098 t\nm : a \u2208 s\n\u22a2 erase s a \u2264 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/ModelTheory/ElementaryMaps.lean", "full_name": "FirstOrder.Language.Substructure.realize_boundedFormula_top", "start": [360, 1], "end": [364, 7], "traced_tactics": [{"tactic": "rw [\u2190 Substructure.topEquiv.realize_boundedFormula \u03c6]", "state_before": "L : Language\nM : Type u_4\nN : Type ?u.369446\nP : Type ?u.369449\nQ : Type ?u.369452\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\ninst\u271d : Structure L Q\n\u03b1 : Type u_1\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 { x // x \u2208 \u22a4 }\nxs : Fin n \u2192 { x // x \u2208 \u22a4 }\n\u22a2 BoundedFormula.Realize \u03c6 v xs \u2194 BoundedFormula.Realize \u03c6 (Subtype.val \u2218 v) (Subtype.val \u2218 xs)", "state_after": "L : Language\nM : Type u_4\nN : Type ?u.369446\nP : Type ?u.369449\nQ : Type ?u.369452\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\ninst\u271d : Structure L Q\n\u03b1 : Type u_1\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 { x // x \u2208 \u22a4 }\nxs : Fin n \u2192 { x // x \u2208 \u22a4 }\n\u22a2 BoundedFormula.Realize \u03c6 (\u2191topEquiv \u2218 v) (\u2191topEquiv \u2218 xs) \u2194\n    BoundedFormula.Realize \u03c6 (Subtype.val \u2218 v) (Subtype.val \u2218 xs)"}, {"tactic": "simp", "state_before": "L : Language\nM : Type u_4\nN : Type ?u.369446\nP : Type ?u.369449\nQ : Type ?u.369452\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\ninst\u271d : Structure L Q\n\u03b1 : Type u_1\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 { x // x \u2208 \u22a4 }\nxs : Fin n \u2192 { x // x \u2208 \u22a4 }\n\u22a2 BoundedFormula.Realize \u03c6 (\u2191topEquiv \u2218 v) (\u2191topEquiv \u2218 xs) \u2194\n    BoundedFormula.Realize \u03c6 (Subtype.val \u2218 v) (Subtype.val \u2218 xs)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "iInf_univ", "start": [1411, 1], "end": [1411, 82], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type ?u.113928\n\u03b3 : Type ?u.113931\n\u03b9 : Sort ?u.113934\n\u03b9' : Sort ?u.113937\n\u03ba : \u03b9 \u2192 Sort ?u.113942\n\u03ba' : \u03b9' \u2192 Sort ?u.113947\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b2 \u2192 \u03b1\n\u22a2 (\u2a05 (x : \u03b2) (_ : x \u2208 univ), f x) = \u2a05 (x : \u03b2), f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fin/VecNotation.lean", "full_name": "Matrix.head_neg", "start": [587, 1], "end": [588, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/Average.lean", "full_name": "MeasureTheory.average_add_measure", "start": [130, 1], "end": [137, 37], "traced_tactics": [{"tactic": "simp only [div_eq_inv_mul, mul_smul, measure_smul_average, \u2190 smul_add, \u2190\n  integral_add_measure h\u03bc h\u03bd, \u2190 ENNReal.toReal_add (measure_ne_top \u03bc _) (measure_ne_top \u03bd _)]", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.201513\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\n\u03bc : Measure \u03b1\ns : Set E\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\n\u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nf : \u03b1 \u2192 E\nh\u03bc : Integrable f\nh\u03bd : Integrable f\n\u22a2 (\u2a0d (x : \u03b1), f x \u2202\u03bc + \u03bd) =\n    ((ENNReal.toReal (\u2191\u2191\u03bc univ) / (ENNReal.toReal (\u2191\u2191\u03bc univ) + ENNReal.toReal (\u2191\u2191\u03bd univ))) \u2022 \u2a0d (x : \u03b1), f x \u2202\u03bc) +\n      (ENNReal.toReal (\u2191\u2191\u03bd univ) / (ENNReal.toReal (\u2191\u2191\u03bc univ) + ENNReal.toReal (\u2191\u2191\u03bd univ))) \u2022 \u2a0d (x : \u03b1), f x \u2202\u03bd", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.201513\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\n\u03bc : Measure \u03b1\ns : Set E\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\n\u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nf : \u03b1 \u2192 E\nh\u03bc : Integrable f\nh\u03bd : Integrable f\n\u22a2 (\u2a0d (x : \u03b1), f x \u2202\u03bc + \u03bd) = (ENNReal.toReal (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc + \u03bd"}, {"tactic": "rw [average_eq, Measure.add_apply]", "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type ?u.201513\nm0 : MeasurableSpace \u03b1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\n\u03bc : Measure \u03b1\ns : Set E\ninst\u271d\u00b9 : IsFiniteMeasure \u03bc\n\u03bd : Measure \u03b1\ninst\u271d : IsFiniteMeasure \u03bd\nf : \u03b1 \u2192 E\nh\u03bc : Integrable f\nh\u03bd : Integrable f\n\u22a2 (\u2a0d (x : \u03b1), f x \u2202\u03bc + \u03bd) = (ENNReal.toReal (\u2191\u2191\u03bc univ + \u2191\u2191\u03bd univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc + \u03bd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.zero_smul_subset", "start": [2100, 1], "end": [2100, 99], "traced_tactics": [{"tactic": "simp [subset_iff, mem_smul]", "state_before": "F : Type ?u.911536\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.911545\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMulWithZero \u03b1 \u03b2\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b1\nt\u271d t : Finset \u03b2\n\u22a2 0 \u2022 t \u2286 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/NNReal.lean", "full_name": "NNReal.le_iInf_mul_iInf", "start": [1006, 1], "end": [1008, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Lie/Abelian.lean", "full_name": "LieAlgebra.isLieAbelian_bot", "start": [93, 1], "end": [95, 29], "traced_tactics": [{"tactic": "simp", "state_before": "R : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx\u271d\u00b9 x\u271d : { x // x \u2208 \u2191\u22a5 }\nx : L\nhx : x \u2208 \u2191\u22a5\n\u22a2 \u2045{ val := x, property := hx }, x\u271d\u2046 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "full_name": "ENNReal.lintegral_Lp_add_le_aux", "start": [315, 9], "end": [345, 82], "traced_tactics": [{"tactic": "have hp_not_nonpos : \u00acp \u2264 0 := by simp [hpq.pos]", "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_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "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_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)"}, {"tactic": "have htop_rpow : (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4 := by\n  by_contra h\n  exact h_add_top (@ENNReal.rpow_eq_top_of_nonneg _ (1 / p) (by simp [hpq.nonneg]) h)", "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_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "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_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)"}, {"tactic": "have h0_rpow : (\u222b\u207b a, (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0 := by\n  simp [h_add_zero, h_add_top, hpq.nonneg, hp_not_nonpos, -Pi.add_apply]", "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_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "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_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)"}, {"tactic": "suffices h :\n  1 \u2264\n    (\u222b\u207b a : \u03b1, (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b a : \u03b1, f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b a : \u03b1, g a ^ p \u2202\u03bc) ^ (1 / p))", "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_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "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_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n  1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)\n\ncase h\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\nh_add_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))"}, {"tactic": "have h :\n  (\u222b\u207b a : \u03b1, (f + g) a ^ p \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 + g) a ^ p \u2202\u03bc) ^ (1 / q) :=\n  lintegral_rpow_add_le_add_snorm_mul_lintegral_rpow_add hpq hf hf_top hg hg_top", "state_before": "case h\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\nh_add_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))", "state_after": "case h\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\nh_add_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n  (\u222b\u207b (a : \u03b1), (f + g) a ^ p \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)) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / q)\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))"}, {"tactic": "have h_one_div_q : 1 / q = 1 - 1 / p := by\n  nth_rw 2 [\u2190 hpq.inv_add_inv_conj]\n  ring", "state_before": "case h\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\nh_add_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n  (\u222b\u207b (a : \u03b1), (f + g) a ^ p \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)) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / q)\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))", "state_after": "case h\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\nh_add_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n  (\u222b\u207b (a : \u03b1), (f + g) a ^ p \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)) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / q)\nh_one_div_q : 1 / q = 1 - 1 / p\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))"}, {"tactic": "simp_rw [h_one_div_q, sub_eq_add_neg 1 (1 / p), ENNReal.rpow_add _ _ h_add_zero h_add_top,\n  rpow_one] at h", "state_before": "case h\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\nh_add_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n  (\u222b\u207b (a : \u03b1), (f + g) a ^ p \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)) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / q)\nh_one_div_q : 1 / q = 1 - 1 / p\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))", "state_after": "case h\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\nh_add_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh_one_div_q : 1 / q = 1 - 1 / p\nh :\n  (\u222b\u207b (a : \u03b1), (f + g) a ^ p \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 + g) a ^ p \u2202\u03bc) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)))\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))"}, {"tactic": "conv_rhs at h => enter [2]; rw [mul_comm]", "state_before": "case h\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\nh_add_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh_one_div_q : 1 / q = 1 - 1 / p\nh :\n  (\u222b\u207b (a : \u03b1), (f + g) a ^ p \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 + g) a ^ p \u2202\u03bc) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)))\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))", "state_after": "case h\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\nh_add_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh_one_div_q : 1 / q = 1 - 1 / p\nh :\n  (\u222b\u207b (a : \u03b1), (f + g) a ^ p \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 + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) * \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc)\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))"}, {"tactic": "conv_lhs at h => rw [\u2190 one_mul (\u222b\u207b a : \u03b1, (f + g) a ^ p \u2202\u03bc)]", "state_before": "case h\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\nh_add_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh_one_div_q : 1 / q = 1 - 1 / p\nh :\n  (\u222b\u207b (a : \u03b1), (f + g) a ^ p \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 + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) * \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc)\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))", "state_after": "case h\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\nh_add_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh_one_div_q : 1 / q = 1 - 1 / p\nh :\n  (1 * \u222b\u207b (a : \u03b1), (f + g) a ^ p \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 + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) * \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc)\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))"}, {"tactic": "rwa [\u2190 mul_assoc, ENNReal.mul_le_mul_right h_add_zero h_add_top, mul_comm] at h", "state_before": "case h\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\nh_add_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh_one_div_q : 1 / q = 1 - 1 / p\nh :\n  (1 * \u222b\u207b (a : \u03b1), (f + g) a ^ p \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 + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) * \u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc)\n\u22a2 1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))", "state_after": "no goals"}, {"tactic": "simp [hpq.pos]", "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_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\n\u22a2 \u00acp \u2264 0", "state_after": "no goals"}, {"tactic": "by_contra h", "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_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4", "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_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nh : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) = \u22a4\n\u22a2 False"}, {"tactic": "exact h_add_top (@ENNReal.rpow_eq_top_of_nonneg _ (1 / p) (by simp [hpq.nonneg]) h)", "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_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nh : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) = \u22a4\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simp [hpq.nonneg]", "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_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nh : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) = \u22a4\n\u22a2 0 \u2264 1 / p", "state_after": "no goals"}, {"tactic": "simp [h_add_zero, h_add_top, hpq.nonneg, hp_not_nonpos, -Pi.add_apply]", "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_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0", "state_after": "no goals"}, {"tactic": "rwa [\u2190 mul_le_mul_left h0_rpow htop_rpow, \u2190 mul_assoc, \u2190 rpow_add _ _ h_add_zero h_add_top, \u2190\n  sub_eq_add_neg, _root_.sub_self, rpow_zero, one_mul, mul_one] at h", "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_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n  1 \u2264\n    (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (-(1 / p)) *\n      ((\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p))\n\u22a2 (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) + (\u222b\u207b (a : \u03b1), g a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "no goals"}, {"tactic": "nth_rw 2 [\u2190 hpq.inv_add_inv_conj]", "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_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n  (\u222b\u207b (a : \u03b1), (f + g) a ^ p \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)) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / q)\n\u22a2 1 / q = 1 - 1 / p", "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_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n  (\u222b\u207b (a : \u03b1), (f + g) a ^ p \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)) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / q)\n\u22a2 1 / q = 1 / p + 1 / q - 1 / p"}, {"tactic": "ring", "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_zero : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 0\nh_add_top : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) \u2260 \u22a4\nhp_not_nonpos : \u00acp \u2264 0\nhtop_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 \u22a4\nh0_rpow : (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / p) \u2260 0\nh :\n  (\u222b\u207b (a : \u03b1), (f + g) a ^ p \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)) * (\u222b\u207b (a : \u03b1), (f + g) a ^ p \u2202\u03bc) ^ (1 / q)\n\u22a2 1 / q = 1 / p + 1 / q - 1 / p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/JordanHolder.lean", "full_name": "CompositionSeries.eraseTop_top_le", "start": [405, 1], "end": [406, 84], "traced_tactics": [{"tactic": "simp [eraseTop, top, s.strictMono.le_iff_le, Fin.le_iff_val_le_val, tsub_le_self]", "state_before": "X : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\ns : CompositionSeries X\n\u22a2 top (eraseTop s) \u2264 top s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Game/PGame.lean", "full_name": "PGame.leftMoves_add_cases", "start": [1559, 1], "end": [1565, 15], "traced_tactics": [{"tactic": "rw [\u2190 toLeftMovesAdd.apply_symm_apply k]", "state_before": "x y : PGame\nk : LeftMoves (x + y)\nP : LeftMoves (x + y) \u2192 Prop\nhl : \u2200 (i : LeftMoves x), P (\u2191toLeftMovesAdd (Sum.inl i))\nhr : \u2200 (i : LeftMoves y), P (\u2191toLeftMovesAdd (Sum.inr i))\n\u22a2 P k", "state_after": "x y : PGame\nk : LeftMoves (x + y)\nP : LeftMoves (x + y) \u2192 Prop\nhl : \u2200 (i : LeftMoves x), P (\u2191toLeftMovesAdd (Sum.inl i))\nhr : \u2200 (i : LeftMoves y), P (\u2191toLeftMovesAdd (Sum.inr i))\n\u22a2 P (\u2191toLeftMovesAdd (\u2191toLeftMovesAdd.symm k))"}, {"tactic": "cases' toLeftMovesAdd.symm k with i i", "state_before": "x y : PGame\nk : LeftMoves (x + y)\nP : LeftMoves (x + y) \u2192 Prop\nhl : \u2200 (i : LeftMoves x), P (\u2191toLeftMovesAdd (Sum.inl i))\nhr : \u2200 (i : LeftMoves y), P (\u2191toLeftMovesAdd (Sum.inr i))\n\u22a2 P (\u2191toLeftMovesAdd (\u2191toLeftMovesAdd.symm k))", "state_after": "case inl\nx y : PGame\nk : LeftMoves (x + y)\nP : LeftMoves (x + y) \u2192 Prop\nhl : \u2200 (i : LeftMoves x), P (\u2191toLeftMovesAdd (Sum.inl i))\nhr : \u2200 (i : LeftMoves y), P (\u2191toLeftMovesAdd (Sum.inr i))\ni : LeftMoves x\n\u22a2 P (\u2191toLeftMovesAdd (Sum.inl i))\n\ncase inr\nx y : PGame\nk : LeftMoves (x + y)\nP : LeftMoves (x + y) \u2192 Prop\nhl : \u2200 (i : LeftMoves x), P (\u2191toLeftMovesAdd (Sum.inl i))\nhr : \u2200 (i : LeftMoves y), P 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M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\nc d : R\nx y : M\nb : Basis \u03b9 R M\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : DistribMulAction G R\ninst\u271d\u00b9 : DistribMulAction G M\ninst\u271d : IsScalarTower G R M\nv : \u03b9 \u2192 M\nhv : span R (Set.range v) = \u22a4\nw : \u03b9 \u2192 G\nj : M\nhj : j \u2208 \u22a4\n\u22a2 j \u2208 span R (Set.range (w \u2022 v))", "state_after": "\u03b9 : Type u_4\n\u03b9' : Type ?u.886741\nR : Type u_2\nR\u2082 : Type ?u.886747\nK : Type ?u.886750\nM : Type u_3\nM' : Type ?u.886756\nM'' : Type ?u.886759\nV : Type u\nV' : Type ?u.886764\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : CommRing R\u2082\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : AddCommGroup M''\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\nc d : R\nx y : M\nb : Basis \u03b9 R M\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : DistribMulAction G R\ninst\u271d\u00b9 : DistribMulAction G M\ninst\u271d : IsScalarTower G R M\nv : \u03b9 \u2192 M\nhv : span R (Set.range v) = \u22a4\nw : \u03b9 \u2192 G\nj : M\nhj : j \u2208 span R (Set.range v)\n\u22a2 j \u2208 span R (Set.range (w \u2022 v))"}, {"tactic": "rw [Submodule.mem_span] at hj\u22a2", "state_before": "\u03b9 : Type u_4\n\u03b9' : Type ?u.886741\nR : Type u_2\nR\u2082 : Type ?u.886747\nK : Type ?u.886750\nM : Type u_3\nM' : Type ?u.886756\nM'' : Type ?u.886759\nV : Type u\nV' : Type ?u.886764\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : CommRing R\u2082\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : AddCommGroup M''\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\nc d : R\nx y : M\nb : Basis \u03b9 R M\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : DistribMulAction G R\ninst\u271d\u00b9 : DistribMulAction G M\ninst\u271d : IsScalarTower G R M\nv : \u03b9 \u2192 M\nhv : span R (Set.range v) = \u22a4\nw : \u03b9 \u2192 G\nj : M\nhj : j \u2208 span R (Set.range v)\n\u22a2 j \u2208 span R (Set.range (w \u2022 v))", "state_after": "\u03b9 : Type u_4\n\u03b9' : Type ?u.886741\nR : Type u_2\nR\u2082 : Type ?u.886747\nK : Type ?u.886750\nM : Type u_3\nM' : Type ?u.886756\nM'' : Type ?u.886759\nV : Type u\nV' : Type ?u.886764\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : CommRing R\u2082\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : AddCommGroup M''\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\nc d : R\nx y : M\nb : Basis \u03b9 R M\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : DistribMulAction G R\ninst\u271d\u00b9 : DistribMulAction G M\ninst\u271d : IsScalarTower G R M\nv : \u03b9 \u2192 M\nhv : span R (Set.range v) = \u22a4\nw : \u03b9 \u2192 G\nj : M\nhj : \u2200 (p : Submodule R M), Set.range v \u2286 \u2191p \u2192 j \u2208 p\n\u22a2 \u2200 (p : Submodule R M), Set.range (w \u2022 v) \u2286 \u2191p \u2192 j \u2208 p"}, {"tactic": "refine' fun p hp => hj p fun u hu => _", "state_before": "\u03b9 : Type u_4\n\u03b9' : Type ?u.886741\nR : Type u_2\nR\u2082 : Type ?u.886747\nK : Type ?u.886750\nM : Type u_3\nM' : Type ?u.886756\nM'' : Type ?u.886759\nV : Type u\nV' : Type ?u.886764\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : CommRing R\u2082\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : AddCommGroup M''\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\nc d : R\nx y : M\nb : Basis \u03b9 R M\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : DistribMulAction G R\ninst\u271d\u00b9 : DistribMulAction G M\ninst\u271d : IsScalarTower G R M\nv : \u03b9 \u2192 M\nhv : span R (Set.range v) = \u22a4\nw : \u03b9 \u2192 G\nj : M\nhj : \u2200 (p : Submodule R M), Set.range v \u2286 \u2191p \u2192 j \u2208 p\n\u22a2 \u2200 (p : Submodule R M), Set.range (w \u2022 v) \u2286 \u2191p \u2192 j \u2208 p", "state_after": "\u03b9 : Type u_4\n\u03b9' : Type ?u.886741\nR : Type u_2\nR\u2082 : Type ?u.886747\nK : Type ?u.886750\nM : Type u_3\nM' : Type ?u.886756\nM'' : Type ?u.886759\nV : Type u\nV' : Type ?u.886764\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : CommRing R\u2082\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : AddCommGroup M''\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\nc d : R\nx y : M\nb : Basis \u03b9 R M\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : DistribMulAction G R\ninst\u271d\u00b9 : DistribMulAction G M\ninst\u271d : IsScalarTower G R M\nv : \u03b9 \u2192 M\nhv : span R (Set.range v) = \u22a4\nw : \u03b9 \u2192 G\nj : M\nhj : \u2200 (p : Submodule R M), Set.range v \u2286 \u2191p \u2192 j \u2208 p\np : Submodule R M\nhp : Set.range (w \u2022 v) \u2286 \u2191p\nu : M\nhu : u \u2208 Set.range v\n\u22a2 u \u2208 \u2191p"}, {"tactic": "obtain \u27e8i, rfl\u27e9 := hu", "state_before": "\u03b9 : Type u_4\n\u03b9' : Type ?u.886741\nR : Type u_2\nR\u2082 : Type ?u.886747\nK : Type ?u.886750\nM : Type u_3\nM' : Type ?u.886756\nM'' : Type ?u.886759\nV : Type u\nV' : Type ?u.886764\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : CommRing R\u2082\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : AddCommGroup M''\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\nc d : R\nx y : M\nb : Basis \u03b9 R M\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : DistribMulAction G R\ninst\u271d\u00b9 : DistribMulAction G M\ninst\u271d : IsScalarTower G R M\nv : \u03b9 \u2192 M\nhv : span R (Set.range v) = \u22a4\nw : \u03b9 \u2192 G\nj : M\nhj : \u2200 (p : Submodule R M), Set.range v \u2286 \u2191p \u2192 j \u2208 p\np : Submodule R M\nhp : Set.range (w \u2022 v) \u2286 \u2191p\nu : M\nhu : u \u2208 Set.range v\n\u22a2 u \u2208 \u2191p", "state_after": "case intro\n\u03b9 : Type u_4\n\u03b9' : Type ?u.886741\nR : Type u_2\nR\u2082 : Type ?u.886747\nK : Type ?u.886750\nM : Type u_3\nM' : Type ?u.886756\nM'' : Type ?u.886759\nV : Type u\nV' : Type ?u.886764\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : CommRing R\u2082\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : AddCommGroup M''\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\nc d : R\nx y : M\nb : Basis \u03b9 R M\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : DistribMulAction G R\ninst\u271d\u00b9 : DistribMulAction G M\ninst\u271d : IsScalarTower G R M\nv : \u03b9 \u2192 M\nhv : span R (Set.range v) = \u22a4\nw : \u03b9 \u2192 G\nj : M\nhj : \u2200 (p : Submodule R M), Set.range v \u2286 \u2191p \u2192 j \u2208 p\np : Submodule R M\nhp : Set.range (w \u2022 v) \u2286 \u2191p\ni : \u03b9\n\u22a2 v i \u2208 \u2191p"}, {"tactic": "have : ((w i)\u207b\u00b9 \u2022 (1 : R)) \u2022 w i \u2022 v i \u2208 p := p.smul_mem ((w i)\u207b\u00b9 \u2022 (1 : R)) (hp \u27e8i, rfl\u27e9)", "state_before": "case intro\n\u03b9 : Type u_4\n\u03b9' : Type ?u.886741\nR : Type u_2\nR\u2082 : Type ?u.886747\nK : Type ?u.886750\nM : Type u_3\nM' : Type ?u.886756\nM'' : Type ?u.886759\nV : Type u\nV' : Type ?u.886764\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : CommRing R\u2082\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : AddCommGroup M''\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\nc d : R\nx y : M\nb : Basis \u03b9 R M\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : DistribMulAction G R\ninst\u271d\u00b9 : DistribMulAction G M\ninst\u271d : IsScalarTower G R M\nv : \u03b9 \u2192 M\nhv : span R (Set.range v) = \u22a4\nw : \u03b9 \u2192 G\nj : M\nhj : \u2200 (p : Submodule R M), Set.range v \u2286 \u2191p \u2192 j \u2208 p\np : Submodule R M\nhp : Set.range (w \u2022 v) \u2286 \u2191p\ni : \u03b9\n\u22a2 v i \u2208 \u2191p", "state_after": "case intro\n\u03b9 : Type u_4\n\u03b9' : Type ?u.886741\nR : Type u_2\nR\u2082 : Type ?u.886747\nK : Type ?u.886750\nM : Type u_3\nM' : Type ?u.886756\nM'' : Type ?u.886759\nV : Type u\nV' : Type ?u.886764\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : CommRing R\u2082\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : AddCommGroup M''\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\nc d : R\nx y : M\nb : Basis \u03b9 R M\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : DistribMulAction G R\ninst\u271d\u00b9 : DistribMulAction G M\ninst\u271d : IsScalarTower G R M\nv : \u03b9 \u2192 M\nhv : span R (Set.range v) = \u22a4\nw : \u03b9 \u2192 G\nj : M\nhj : \u2200 (p : Submodule R M), Set.range v \u2286 \u2191p \u2192 j \u2208 p\np : Submodule R M\nhp : Set.range (w \u2022 v) \u2286 \u2191p\ni : \u03b9\nthis : ((w i)\u207b\u00b9 \u2022 1) \u2022 w i \u2022 v i \u2208 p\n\u22a2 v i \u2208 \u2191p"}, {"tactic": "rwa [smul_one_smul, inv_smul_smul] at this", "state_before": "case intro\n\u03b9 : Type u_4\n\u03b9' : Type ?u.886741\nR : Type u_2\nR\u2082 : Type ?u.886747\nK : Type ?u.886750\nM : Type u_3\nM' : Type ?u.886756\nM'' : Type ?u.886759\nV : Type u\nV' : Type ?u.886764\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u00b9\u00b2 : Ring R\ninst\u271d\u00b9\u00b9 : CommRing R\u2082\ninst\u271d\u00b9\u2070 : AddCommGroup M\ninst\u271d\u2079 : AddCommGroup M'\ninst\u271d\u2078 : AddCommGroup M''\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\nc d : R\nx y : M\nb : Basis \u03b9 R M\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : DistribMulAction G R\ninst\u271d\u00b9 : DistribMulAction G M\ninst\u271d : IsScalarTower G R M\nv : \u03b9 \u2192 M\nhv : span R (Set.range v) = \u22a4\nw : \u03b9 \u2192 G\nj : M\nhj : \u2200 (p : Submodule R M), Set.range v \u2286 \u2191p \u2192 j \u2208 p\np : Submodule R M\nhp : Set.range (w \u2022 v) \u2286 \u2191p\ni : \u03b9\nthis : ((w i)\u207b\u00b9 \u2022 1) \u2022 w i \u2022 v i \u2208 p\n\u22a2 v i \u2208 \u2191p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.trStmts\u2081_trans", "start": [1734, 1], "end": [1752, 52], "traced_tactics": [{"tactic": "induction' q with _ _ _ q q_ih _ _ q q_ih q q_ih _ _ q q_ih q q_ih q q_ih q\u2081 q\u2082 q\u2081_ih q\u2082_ih _ <;>\n  simp (config := { contextual := true }) only [trStmts\u2081, Finset.mem_insert, Finset.mem_union,\n    or_imp, Finset.mem_singleton, Finset.Subset.refl, imp_true_iff, true_and_iff]", "state_before": "q q' : \u039b'\n\u22a2 q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q", "state_after": "case move\nq' : \u039b'\np\u271d : \u0393' \u2192 Bool\nk\u2081\u271d k\u2082\u271d : K'\nq : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\n\u22a2 q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q) (trStmts\u2081 q)\n\ncase clear\nq' : \u039b'\np\u271d : \u0393' \u2192 Bool\nk\u271d : K'\nq : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\n\u22a2 q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.clear p\u271d k\u271d q) (trStmts\u2081 q)\n\ncase copy\nq' q : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\n\u22a2 q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.copy q) (trStmts\u2081 q)\n\ncase push\nq' : \u039b'\nk\u271d : K'\ns\u271d : Option \u0393' \u2192 Option \u0393'\nq : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\n\u22a2 q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.push k\u271d s\u271d q) (trStmts\u2081 q)\n\ncase read\nq' : \u039b'\nq : Option \u0393' \u2192 \u039b'\nq_ih : \u2200 (a : Option \u0393'), q' \u2208 trStmts\u2081 (q a) \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 (q a)\n\u22a2 (q' \u2208 Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s)) \u2192\n    trStmts\u2081 q' \u2286 insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s))\n\ncase succ\nq' q : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\n\u22a2 (q' = unrev q \u2192\n      insert (\u039b'.move (fun x => false) rev main q) (trStmts\u2081 q) \u2286 insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q))) \u2227\n    (q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q)))\n\ncase pred\nq' q\u2081 q\u2082 : \u039b'\nq\u2081_ih : q' \u2208 trStmts\u2081 q\u2081 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2081\nq\u2082_ih : q' \u2208 trStmts\u2081 q\u2082 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2082\n\u22a2 (q' \u2208 trStmts\u2081 q\u2081 \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.pred q\u2081 q\u2082) (trStmts\u2081 q\u2081 \u222a insert (unrev q\u2082) (trStmts\u2081 q\u2082))) \u2227\n    (q' = unrev q\u2082 \u2192\n        insert (\u039b'.move (fun x => false) rev main q\u2082) (trStmts\u2081 q\u2082) \u2286\n          insert (\u039b'.pred q\u2081 q\u2082) (trStmts\u2081 q\u2081 \u222a insert (unrev q\u2082) (trStmts\u2081 q\u2082))) \u2227\n      (q' \u2208 trStmts\u2081 q\u2082 \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.pred q\u2081 q\u2082) (trStmts\u2081 q\u2081 \u222a insert (unrev q\u2082) (trStmts\u2081 q\u2082)))"}, {"tactic": "iterate 4 exact fun h => Finset.Subset.trans (q_ih h) (Finset.subset_insert _ _)", "state_before": "case move\nq' : \u039b'\np\u271d : \u0393' \u2192 Bool\nk\u2081\u271d k\u2082\u271d : K'\nq : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\n\u22a2 q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.move p\u271d k\u2081\u271d k\u2082\u271d q) (trStmts\u2081 q)\n\ncase clear\nq' : \u039b'\np\u271d : \u0393' \u2192 Bool\nk\u271d : K'\nq : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\n\u22a2 q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.clear p\u271d k\u271d q) (trStmts\u2081 q)\n\ncase copy\nq' q : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\n\u22a2 q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.copy q) (trStmts\u2081 q)\n\ncase push\nq' : \u039b'\nk\u271d : K'\ns\u271d : Option \u0393' \u2192 Option \u0393'\nq : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\n\u22a2 q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.push k\u271d s\u271d q) (trStmts\u2081 q)\n\ncase read\nq' : \u039b'\nq : Option \u0393' \u2192 \u039b'\nq_ih : \u2200 (a : Option \u0393'), q' \u2208 trStmts\u2081 (q a) \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 (q a)\n\u22a2 (q' \u2208 Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s)) \u2192\n    trStmts\u2081 q' \u2286 insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s))\n\ncase succ\nq' q : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\n\u22a2 (q' = unrev q \u2192\n      insert (\u039b'.move (fun x => false) rev main q) (trStmts\u2081 q) \u2286 insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q))) \u2227\n    (q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q)))\n\ncase pred\nq' q\u2081 q\u2082 : \u039b'\nq\u2081_ih : q' \u2208 trStmts\u2081 q\u2081 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2081\nq\u2082_ih : q' \u2208 trStmts\u2081 q\u2082 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2082\n\u22a2 (q' \u2208 trStmts\u2081 q\u2081 \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.pred q\u2081 q\u2082) (trStmts\u2081 q\u2081 \u222a insert (unrev q\u2082) (trStmts\u2081 q\u2082))) \u2227\n    (q' = unrev q\u2082 \u2192\n        insert (\u039b'.move (fun x => false) rev main q\u2082) (trStmts\u2081 q\u2082) \u2286\n          insert (\u039b'.pred q\u2081 q\u2082) (trStmts\u2081 q\u2081 \u222a insert (unrev q\u2082) (trStmts\u2081 q\u2082))) \u2227\n      (q' \u2208 trStmts\u2081 q\u2082 \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.pred q\u2081 q\u2082) (trStmts\u2081 q\u2081 \u222a insert (unrev q\u2082) (trStmts\u2081 q\u2082)))", "state_after": "case read\nq' : \u039b'\nq : Option \u0393' \u2192 \u039b'\nq_ih : \u2200 (a : Option \u0393'), q' \u2208 trStmts\u2081 (q a) \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 (q a)\n\u22a2 (q' \u2208 Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s)) \u2192\n    trStmts\u2081 q' \u2286 insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s))\n\ncase succ\nq' q : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\n\u22a2 (q' = unrev q \u2192\n      insert (\u039b'.move (fun x => false) rev main q) (trStmts\u2081 q) \u2286 insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q))) \u2227\n    (q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q)))\n\ncase pred\nq' q\u2081 q\u2082 : \u039b'\nq\u2081_ih : q' \u2208 trStmts\u2081 q\u2081 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2081\nq\u2082_ih : q' \u2208 trStmts\u2081 q\u2082 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2082\n\u22a2 (q' \u2208 trStmts\u2081 q\u2081 \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.pred q\u2081 q\u2082) (trStmts\u2081 q\u2081 \u222a insert (unrev q\u2082) (trStmts\u2081 q\u2082))) \u2227\n    (q' = unrev q\u2082 \u2192\n        insert (\u039b'.move (fun x => false) rev main q\u2082) (trStmts\u2081 q\u2082) \u2286\n          insert (\u039b'.pred q\u2081 q\u2082) (trStmts\u2081 q\u2081 \u222a insert (unrev q\u2082) (trStmts\u2081 q\u2082))) \u2227\n      (q' \u2208 trStmts\u2081 q\u2082 \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.pred q\u2081 q\u2082) (trStmts\u2081 q\u2081 \u222a insert (unrev q\u2082) (trStmts\u2081 q\u2082)))"}, {"tactic": "exact fun h => Finset.Subset.trans (q_ih h) (Finset.subset_insert _ _)", "state_before": "case push\nq' : \u039b'\nk\u271d : K'\ns\u271d : Option \u0393' \u2192 Option \u0393'\nq : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\n\u22a2 q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.push k\u271d s\u271d q) (trStmts\u2081 q)\n\ncase read\nq' : \u039b'\nq : Option \u0393' \u2192 \u039b'\nq_ih : \u2200 (a : Option \u0393'), q' \u2208 trStmts\u2081 (q a) \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 (q a)\n\u22a2 (q' \u2208 Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s)) \u2192\n    trStmts\u2081 q' \u2286 insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s))\n\ncase succ\nq' q : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\n\u22a2 (q' = unrev q \u2192\n      insert (\u039b'.move (fun x => false) rev main q) (trStmts\u2081 q) \u2286 insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q))) \u2227\n    (q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q)))\n\ncase pred\nq' q\u2081 q\u2082 : \u039b'\nq\u2081_ih : q' \u2208 trStmts\u2081 q\u2081 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2081\nq\u2082_ih : q' \u2208 trStmts\u2081 q\u2082 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2082\n\u22a2 (q' \u2208 trStmts\u2081 q\u2081 \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.pred q\u2081 q\u2082) (trStmts\u2081 q\u2081 \u222a insert (unrev q\u2082) (trStmts\u2081 q\u2082))) \u2227\n    (q' = unrev q\u2082 \u2192\n        insert (\u039b'.move (fun x => false) rev main q\u2082) (trStmts\u2081 q\u2082) \u2286\n          insert (\u039b'.pred q\u2081 q\u2082) (trStmts\u2081 q\u2081 \u222a insert (unrev q\u2082) (trStmts\u2081 q\u2082))) \u2227\n      (q' \u2208 trStmts\u2081 q\u2082 \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.pred q\u2081 q\u2082) (trStmts\u2081 q\u2081 \u222a insert (unrev q\u2082) (trStmts\u2081 q\u2082)))", "state_after": "case read\nq' : \u039b'\nq : Option \u0393' \u2192 \u039b'\nq_ih : \u2200 (a : Option \u0393'), q' \u2208 trStmts\u2081 (q a) \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 (q a)\n\u22a2 (q' \u2208 Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s)) \u2192\n    trStmts\u2081 q' \u2286 insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s))\n\ncase succ\nq' q : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\n\u22a2 (q' = unrev q \u2192\n      insert (\u039b'.move (fun x => false) rev main q) (trStmts\u2081 q) \u2286 insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q))) \u2227\n    (q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q)))\n\ncase pred\nq' q\u2081 q\u2082 : \u039b'\nq\u2081_ih : q' \u2208 trStmts\u2081 q\u2081 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2081\nq\u2082_ih : q' \u2208 trStmts\u2081 q\u2082 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2082\n\u22a2 (q' \u2208 trStmts\u2081 q\u2081 \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.pred q\u2081 q\u2082) (trStmts\u2081 q\u2081 \u222a insert (unrev q\u2082) (trStmts\u2081 q\u2082))) \u2227\n    (q' = unrev q\u2082 \u2192\n        insert (\u039b'.move (fun x => false) rev main q\u2082) (trStmts\u2081 q\u2082) \u2286\n          insert (\u039b'.pred q\u2081 q\u2082) (trStmts\u2081 q\u2081 \u222a insert (unrev q\u2082) (trStmts\u2081 q\u2082))) \u2227\n      (q' \u2208 trStmts\u2081 q\u2082 \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.pred q\u2081 q\u2082) (trStmts\u2081 q\u2081 \u222a insert (unrev q\u2082) (trStmts\u2081 q\u2082)))"}, {"tactic": "simp", "state_before": "case read\nq' : \u039b'\nq : Option \u0393' \u2192 \u039b'\nq_ih : \u2200 (a : Option \u0393'), q' \u2208 trStmts\u2081 (q a) \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 (q a)\n\u22a2 (q' \u2208 Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s)) \u2192\n    trStmts\u2081 q' \u2286 insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s))", "state_after": "case read\nq' : \u039b'\nq : Option \u0393' \u2192 \u039b'\nq_ih : \u2200 (a : Option \u0393'), q' \u2208 trStmts\u2081 (q a) \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 (q a)\n\u22a2 \u2200 (x : Option \u0393'),\n    q' \u2208 trStmts\u2081 (q x) \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s))"}, {"tactic": "intro s h x h'", "state_before": "case read\nq' : \u039b'\nq : Option \u0393' \u2192 \u039b'\nq_ih : \u2200 (a : Option \u0393'), q' \u2208 trStmts\u2081 (q a) \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 (q a)\n\u22a2 \u2200 (x : Option \u0393'),\n    q' \u2208 trStmts\u2081 (q x) \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s))", "state_after": "case read\nq' : \u039b'\nq : Option \u0393' \u2192 \u039b'\nq_ih : \u2200 (a : Option \u0393'), q' \u2208 trStmts\u2081 (q a) \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 (q a)\ns : Option \u0393'\nh : q' \u2208 trStmts\u2081 (q s)\nx : \u039b'\nh' : x \u2208 trStmts\u2081 q'\n\u22a2 x \u2208 insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s))"}, {"tactic": "simp", "state_before": "case read\nq' : \u039b'\nq : Option \u0393' \u2192 \u039b'\nq_ih : \u2200 (a : Option \u0393'), q' \u2208 trStmts\u2081 (q a) \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 (q a)\ns : Option \u0393'\nh : q' \u2208 trStmts\u2081 (q s)\nx : \u039b'\nh' : x \u2208 trStmts\u2081 q'\n\u22a2 x \u2208 insert (\u039b'.read q) (Finset.biUnion Finset.univ fun s => trStmts\u2081 (q s))", "state_after": "case read\nq' : \u039b'\nq : Option \u0393' \u2192 \u039b'\nq_ih : \u2200 (a : Option \u0393'), q' \u2208 trStmts\u2081 (q a) \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 (q a)\ns : Option \u0393'\nh : q' \u2208 trStmts\u2081 (q s)\nx : \u039b'\nh' : x \u2208 trStmts\u2081 q'\n\u22a2 x = \u039b'.read q \u2228 \u2203 a, x \u2208 trStmts\u2081 (q a)"}, {"tactic": "exact Or.inr \u27e8_, q_ih s h h'\u27e9", "state_before": "case read\nq' : \u039b'\nq : Option \u0393' \u2192 \u039b'\nq_ih : \u2200 (a : Option \u0393'), q' \u2208 trStmts\u2081 (q a) \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 (q a)\ns : Option \u0393'\nh : q' \u2208 trStmts\u2081 (q s)\nx : \u039b'\nh' : x \u2208 trStmts\u2081 q'\n\u22a2 x = \u039b'.read q \u2228 \u2203 a, x \u2208 trStmts\u2081 (q a)", "state_after": "no goals"}, {"tactic": "constructor", "state_before": "case succ\nq' q : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\n\u22a2 (q' = unrev q \u2192\n      insert (\u039b'.move (fun x => false) rev main q) (trStmts\u2081 q) \u2286 insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q))) \u2227\n    (q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q)))", "state_after": "case succ.left\nq' q : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\n\u22a2 q' = unrev q \u2192\n    insert (\u039b'.move (fun x => false) rev main q) (trStmts\u2081 q) \u2286 insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q))\n\ncase succ.right\nq' q : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\n\u22a2 q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q))"}, {"tactic": "rintro rfl", "state_before": "case succ.left\nq' q : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\n\u22a2 q' = unrev q \u2192\n    insert (\u039b'.move (fun x => false) rev main q) (trStmts\u2081 q) \u2286 insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q))", "state_after": "case succ.left\nq : \u039b'\nq_ih : unrev q \u2208 trStmts\u2081 q \u2192 trStmts\u2081 (unrev q) \u2286 trStmts\u2081 q\n\u22a2 insert (\u039b'.move (fun x => false) rev main q) (trStmts\u2081 q) \u2286 insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q))"}, {"tactic": "apply Finset.subset_insert", "state_before": "case succ.left\nq : \u039b'\nq_ih : unrev q \u2208 trStmts\u2081 q \u2192 trStmts\u2081 (unrev q) \u2286 trStmts\u2081 q\n\u22a2 insert (\u039b'.move (fun x => false) rev main q) (trStmts\u2081 q) \u2286 insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q))", "state_after": "no goals"}, {"tactic": "intro h x h'", "state_before": "case succ.right\nq' q : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\n\u22a2 q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q))", "state_after": "case succ.right\nq' q : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\nh : q' \u2208 trStmts\u2081 q\nx : \u039b'\nh' : x \u2208 trStmts\u2081 q'\n\u22a2 x \u2208 insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q))"}, {"tactic": "simp", "state_before": "case succ.right\nq' q : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\nh : q' \u2208 trStmts\u2081 q\nx : \u039b'\nh' : x \u2208 trStmts\u2081 q'\n\u22a2 x \u2208 insert (\u039b'.succ q) (insert (unrev q) (trStmts\u2081 q))", "state_after": "case succ.right\nq' q : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\nh : q' \u2208 trStmts\u2081 q\nx : \u039b'\nh' : x \u2208 trStmts\u2081 q'\n\u22a2 x = \u039b'.succ q \u2228 x = unrev q \u2228 x \u2208 trStmts\u2081 q"}, {"tactic": "exact Or.inr (Or.inr <| q_ih h h')", "state_before": "case succ.right\nq' q : \u039b'\nq_ih : q' \u2208 trStmts\u2081 q \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\nh : q' \u2208 trStmts\u2081 q\nx : \u039b'\nh' : x \u2208 trStmts\u2081 q'\n\u22a2 x = \u039b'.succ q \u2228 x = unrev q \u2228 x \u2208 trStmts\u2081 q", "state_after": "no goals"}, {"tactic": "refine' \u27e8fun h x h' => _, fun _ x h' => _, fun h x h' => _\u27e9 <;> simp", "state_before": "case pred\nq' q\u2081 q\u2082 : \u039b'\nq\u2081_ih : q' \u2208 trStmts\u2081 q\u2081 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2081\nq\u2082_ih : q' \u2208 trStmts\u2081 q\u2082 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2082\n\u22a2 (q' \u2208 trStmts\u2081 q\u2081 \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.pred q\u2081 q\u2082) (trStmts\u2081 q\u2081 \u222a insert (unrev q\u2082) (trStmts\u2081 q\u2082))) \u2227\n    (q' = unrev q\u2082 \u2192\n        insert (\u039b'.move (fun x => false) rev main q\u2082) (trStmts\u2081 q\u2082) \u2286\n          insert (\u039b'.pred q\u2081 q\u2082) (trStmts\u2081 q\u2081 \u222a insert (unrev q\u2082) (trStmts\u2081 q\u2082))) \u2227\n      (q' \u2208 trStmts\u2081 q\u2082 \u2192 trStmts\u2081 q' \u2286 insert (\u039b'.pred q\u2081 q\u2082) (trStmts\u2081 q\u2081 \u222a insert (unrev q\u2082) (trStmts\u2081 q\u2082)))", "state_after": "case pred.refine'_1\nq' q\u2081 q\u2082 : \u039b'\nq\u2081_ih : q' \u2208 trStmts\u2081 q\u2081 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2081\nq\u2082_ih : q' \u2208 trStmts\u2081 q\u2082 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2082\nh : q' \u2208 trStmts\u2081 q\u2081\nx : \u039b'\nh' : x \u2208 trStmts\u2081 q'\n\u22a2 x = \u039b'.pred q\u2081 q\u2082 \u2228 x = unrev q\u2082 \u2228 x \u2208 trStmts\u2081 q\u2081 \u2228 x \u2208 trStmts\u2081 q\u2082\n\ncase pred.refine'_2\nq' q\u2081 q\u2082 : \u039b'\nq\u2081_ih : q' \u2208 trStmts\u2081 q\u2081 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2081\nq\u2082_ih : q' \u2208 trStmts\u2081 q\u2082 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2082\nx\u271d : q' = unrev q\u2082\nx : \u039b'\nh' : x \u2208 insert (\u039b'.move (fun x => false) rev main q\u2082) (trStmts\u2081 q\u2082)\n\u22a2 x = \u039b'.pred q\u2081 q\u2082 \u2228 x = unrev q\u2082 \u2228 x \u2208 trStmts\u2081 q\u2081 \u2228 x \u2208 trStmts\u2081 q\u2082\n\ncase pred.refine'_3\nq' q\u2081 q\u2082 : \u039b'\nq\u2081_ih : q' \u2208 trStmts\u2081 q\u2081 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2081\nq\u2082_ih : q' \u2208 trStmts\u2081 q\u2082 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2082\nh : q' \u2208 trStmts\u2081 q\u2082\nx : \u039b'\nh' : x \u2208 trStmts\u2081 q'\n\u22a2 x = \u039b'.pred q\u2081 q\u2082 \u2228 x = unrev q\u2082 \u2228 x \u2208 trStmts\u2081 q\u2081 \u2228 x \u2208 trStmts\u2081 q\u2082"}, {"tactic": "exact Or.inr (Or.inr <| Or.inl <| q\u2081_ih h h')", "state_before": "case pred.refine'_1\nq' q\u2081 q\u2082 : \u039b'\nq\u2081_ih : q' \u2208 trStmts\u2081 q\u2081 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2081\nq\u2082_ih : q' \u2208 trStmts\u2081 q\u2082 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2082\nh : q' \u2208 trStmts\u2081 q\u2081\nx : \u039b'\nh' : x \u2208 trStmts\u2081 q'\n\u22a2 x = \u039b'.pred q\u2081 q\u2082 \u2228 x = unrev q\u2082 \u2228 x \u2208 trStmts\u2081 q\u2081 \u2228 x \u2208 trStmts\u2081 q\u2082", "state_after": "no goals"}, {"tactic": "cases' Finset.mem_insert.1 h' with h' h' <;> simp [h', unrev]", "state_before": "case pred.refine'_2\nq' q\u2081 q\u2082 : \u039b'\nq\u2081_ih : q' \u2208 trStmts\u2081 q\u2081 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2081\nq\u2082_ih : q' \u2208 trStmts\u2081 q\u2082 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2082\nx\u271d : q' = unrev q\u2082\nx : \u039b'\nh' : x \u2208 insert (\u039b'.move (fun x => false) rev main q\u2082) (trStmts\u2081 q\u2082)\n\u22a2 x = \u039b'.pred q\u2081 q\u2082 \u2228 x = unrev q\u2082 \u2228 x \u2208 trStmts\u2081 q\u2081 \u2228 x \u2208 trStmts\u2081 q\u2082", "state_after": "no goals"}, {"tactic": "exact Or.inr (Or.inr <| Or.inr <| q\u2082_ih h h')", "state_before": "case pred.refine'_3\nq' q\u2081 q\u2082 : \u039b'\nq\u2081_ih : q' \u2208 trStmts\u2081 q\u2081 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2081\nq\u2082_ih : q' \u2208 trStmts\u2081 q\u2082 \u2192 trStmts\u2081 q' \u2286 trStmts\u2081 q\u2082\nh : q' \u2208 trStmts\u2081 q\u2082\nx : \u039b'\nh' : x \u2208 trStmts\u2081 q'\n\u22a2 x = \u039b'.pred q\u2081 q\u2082 \u2228 x = unrev q\u2082 \u2228 x \u2208 trStmts\u2081 q\u2081 \u2228 x \u2208 trStmts\u2081 q\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/PiTensorProduct.lean", "full_name": "PiTensorProduct.liftAux_tprod", "start": [371, 1], "end": [376, 16], "traced_tactics": [{"tactic": "simp only [liftAux, liftAddHom, tprod_eq_tprodCoeff_one, tprodCoeff, AddCon.coe_mk']", "state_before": "\u03b9 : Type u_4\n\u03b9\u2082 : Type ?u.250369\n\u03b9\u2083 : Type ?u.250372\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type ?u.250381\nR\u2082 : Type ?u.250384\ns : \u03b9 \u2192 Type u_2\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type ?u.250574\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_3\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type ?u.250842\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nf : (i : \u03b9) \u2192 s i\n\u22a2 \u2191(liftAux \u03c6) (\u2191(tprod R) f) = \u2191\u03c6 f", "state_after": "\u03b9 : Type u_4\n\u03b9\u2082 : Type ?u.250369\n\u03b9\u2083 : Type ?u.250372\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type ?u.250381\nR\u2082 : Type ?u.250384\ns : \u03b9 \u2192 Type u_2\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type ?u.250574\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_3\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type ?u.250842\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nf : (i : \u03b9) \u2192 s i\n\u22a2 \u2191(AddCon.lift (addConGen (Eqv R fun i => s i)) (\u2191FreeAddMonoid.lift fun p => p.fst \u2022 \u2191\u03c6 p.snd)\n          (_ : addConGen (Eqv R fun i => s i) \u2264 AddCon.ker (\u2191FreeAddMonoid.lift fun p => p.fst \u2022 \u2191\u03c6 p.snd)))\n      \u2191(FreeAddMonoid.of (1, f)) =\n    \u2191\u03c6 f"}, {"tactic": "rw [FreeAddMonoid.of, FreeAddMonoid.ofList, Equiv.refl_apply, AddCon.lift_coe]", "state_before": "\u03b9 : Type u_4\n\u03b9\u2082 : Type ?u.250369\n\u03b9\u2083 : Type ?u.250372\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type ?u.250381\nR\u2082 : Type ?u.250384\ns : \u03b9 \u2192 Type u_2\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type ?u.250574\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_3\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type ?u.250842\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nf : (i : \u03b9) \u2192 s i\n\u22a2 \u2191(AddCon.lift (addConGen (Eqv R fun i => s i)) (\u2191FreeAddMonoid.lift fun p => p.fst \u2022 \u2191\u03c6 p.snd)\n          (_ : addConGen (Eqv R fun i => s i) \u2264 AddCon.ker (\u2191FreeAddMonoid.lift fun p => p.fst \u2022 \u2191\u03c6 p.snd)))\n      \u2191(FreeAddMonoid.of (1, f)) =\n    \u2191\u03c6 f", "state_after": "\u03b9 : Type u_4\n\u03b9\u2082 : Type ?u.250369\n\u03b9\u2083 : Type ?u.250372\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type ?u.250381\nR\u2082 : Type ?u.250384\ns : \u03b9 \u2192 Type u_2\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type ?u.250574\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_3\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type ?u.250842\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nf : (i : \u03b9) \u2192 s i\n\u22a2 \u2191(\u2191FreeAddMonoid.lift fun p => p.fst \u2022 \u2191\u03c6 p.snd) [(1, f)] = \u2191\u03c6 f"}, {"tactic": "dsimp [FreeAddMonoid.lift, FreeAddMonoid.sumAux]", "state_before": "\u03b9 : Type u_4\n\u03b9\u2082 : Type ?u.250369\n\u03b9\u2083 : Type ?u.250372\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type ?u.250381\nR\u2082 : Type ?u.250384\ns : \u03b9 \u2192 Type u_2\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type ?u.250574\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_3\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type ?u.250842\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nf : (i : \u03b9) \u2192 s i\n\u22a2 \u2191(\u2191FreeAddMonoid.lift fun p => p.fst \u2022 \u2191\u03c6 p.snd) [(1, f)] = \u2191\u03c6 f", "state_after": "\u03b9 : Type u_4\n\u03b9\u2082 : Type ?u.250369\n\u03b9\u2083 : Type ?u.250372\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type ?u.250381\nR\u2082 : Type ?u.250384\ns : \u03b9 \u2192 Type u_2\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type ?u.250574\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_3\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type ?u.250842\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nf : (i : \u03b9) \u2192 s i\n\u22a2 (FreeAddMonoid.sumAux.match_1 (fun x => E) [1 \u2022 \u2191\u03c6 f] (fun _ => 0) fun x xs =>\n      List.foldl (fun x x_1 => x + x_1) x xs) =\n    \u2191\u03c6 f"}, {"tactic": "show _ \u2022 _ = _", "state_before": "\u03b9 : Type u_4\n\u03b9\u2082 : Type ?u.250369\n\u03b9\u2083 : Type ?u.250372\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type ?u.250381\nR\u2082 : Type ?u.250384\ns : \u03b9 \u2192 Type u_2\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type ?u.250574\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_3\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type ?u.250842\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nf : (i : \u03b9) \u2192 s i\n\u22a2 (FreeAddMonoid.sumAux.match_1 (fun x => E) [1 \u2022 \u2191\u03c6 f] (fun _ => 0) fun x xs =>\n      List.foldl (fun x x_1 => x + x_1) x xs) =\n    \u2191\u03c6 f", "state_after": "\u03b9 : Type u_4\n\u03b9\u2082 : Type ?u.250369\n\u03b9\u2083 : Type ?u.250372\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type ?u.250381\nR\u2082 : Type ?u.250384\ns : \u03b9 \u2192 Type u_2\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type ?u.250574\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_3\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type ?u.250842\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nf : (i : \u03b9) \u2192 s i\n\u22a2 1 \u2022 \u2191\u03c6 f = \u2191\u03c6 f"}, {"tactic": "rw [one_smul]", "state_before": "\u03b9 : Type u_4\n\u03b9\u2082 : Type ?u.250369\n\u03b9\u2083 : Type ?u.250372\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type ?u.250381\nR\u2082 : Type ?u.250384\ns : \u03b9 \u2192 Type u_2\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type ?u.250574\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_3\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type ?u.250842\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nf : (i : \u03b9) \u2192 s i\n\u22a2 1 \u2022 \u2191\u03c6 f = \u2191\u03c6 f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.EventuallyLE.refl", "start": [1652, 1], "end": [1653, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.singleton_pi'", "start": [736, 1], "end": [737, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Composition.lean", "full_name": "Composition.sizeUpTo_le", "start": [232, 1], "end": [234, 29], "traced_tactics": [{"tactic": "conv_rhs => rw [\u2190 c.blocks_sum, \u2190 sum_take_add_sum_drop _ i]", "state_before": "n : \u2115\nc : Composition n\ni : \u2115\n\u22a2 sizeUpTo c i \u2264 n", "state_after": "n : \u2115\nc : Composition n\ni : \u2115\n\u22a2 sizeUpTo c i \u2264 sum (take i c.blocks) + sum (drop i c.blocks)"}, {"tactic": "exact Nat.le_add_right _ _", "state_before": "n : \u2115\nc : Composition n\ni : \u2115\n\u22a2 sizeUpTo c i \u2264 sum (take i c.blocks) + sum (drop i c.blocks)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Prelude.lean", "full_name": "ULift.down_up", "start": [794, 1], "end": [794, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/Multilinear.lean", "full_name": "ContinuousMultilinearMap.continuous_eval", "start": [636, 1], "end": [663, 11], "traced_tactics": [{"tactic": "apply continuous_iff_continuousAt.2 fun p => ?_", "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\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\n\u22a2 Continuous fun p => \u2191p.fst p.snd", "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\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\n\u22a2 ContinuousAt (fun p => \u2191p.fst p.snd) p"}, {"tactic": "apply\n  continuousAt_of_locally_lipschitz zero_lt_one\n    ((\u2016p\u2016 + 1) * Fintype.card \u03b9 * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i, \u2016p.2 i\u2016) fun q hq => ?_", "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\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\n\u22a2 ContinuousAt (fun p => \u2191p.fst p.snd) p", "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\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\n\u22a2 dist (\u2191q.fst q.snd) (\u2191p.fst p.snd) \u2264\n    ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * dist q p"}, {"tactic": "have : 0 \u2264 max \u2016q.2\u2016 \u2016p.2\u2016 := by simp", "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\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\n\u22a2 dist (\u2191q.fst q.snd) (\u2191p.fst p.snd) \u2264\n    ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * dist q p", "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\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\n\u22a2 dist (\u2191q.fst q.snd) (\u2191p.fst p.snd) \u2264\n    ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * dist q p"}, {"tactic": "have : 0 \u2264 \u2016p\u2016 + 1 := zero_le_one.trans ((le_add_iff_nonneg_left 1).2 <| norm_nonneg p)", "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\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\n\u22a2 dist (\u2191q.fst q.snd) (\u2191p.fst p.snd) \u2264\n    ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * dist q p", "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\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis : 0 \u2264 \u2016p\u2016 + 1\n\u22a2 dist (\u2191q.fst q.snd) (\u2191p.fst p.snd) \u2264\n    ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * dist q p"}, {"tactic": "have A : \u2016q\u2016 \u2264 \u2016p\u2016 + 1 := norm_le_of_mem_closedBall hq.le", "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\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis : 0 \u2264 \u2016p\u2016 + 1\n\u22a2 dist (\u2191q.fst q.snd) (\u2191p.fst p.snd) \u2264\n    ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * dist q p", "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\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis : 0 \u2264 \u2016p\u2016 + 1\nA : \u2016q\u2016 \u2264 \u2016p\u2016 + 1\n\u22a2 dist (\u2191q.fst q.snd) (\u2191p.fst p.snd) \u2264\n    ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * dist q p"}, {"tactic": "have : max \u2016q.2\u2016 \u2016p.2\u2016 \u2264 \u2016p\u2016 + 1 :=\n  (max_le_max (norm_snd_le q) (norm_snd_le p)).trans (by simp [A, zero_le_one])", "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\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis : 0 \u2264 \u2016p\u2016 + 1\nA : \u2016q\u2016 \u2264 \u2016p\u2016 + 1\n\u22a2 dist (\u2191q.fst q.snd) (\u2191p.fst p.snd) \u2264\n    ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * dist q p", "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\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d\u00b9 : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis\u271d : 0 \u2264 \u2016p\u2016 + 1\nA : \u2016q\u2016 \u2264 \u2016p\u2016 + 1\nthis : max \u2016q.snd\u2016 \u2016p.snd\u2016 \u2264 \u2016p\u2016 + 1\n\u22a2 dist (\u2191q.fst q.snd) (\u2191p.fst p.snd) \u2264\n    ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * dist q p"}, {"tactic": "have : \u2200 i : \u03b9, i \u2208 univ \u2192 0 \u2264 \u2016p.2 i\u2016 := fun i _ => norm_nonneg _", "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\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d\u00b9 : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis\u271d : 0 \u2264 \u2016p\u2016 + 1\nA : \u2016q\u2016 \u2264 \u2016p\u2016 + 1\nthis : max \u2016q.snd\u2016 \u2016p.snd\u2016 \u2264 \u2016p\u2016 + 1\n\u22a2 dist (\u2191q.fst q.snd) (\u2191p.fst p.snd) \u2264\n    ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * dist q p", "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\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d\u00b2 : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis\u271d\u00b9 : 0 \u2264 \u2016p\u2016 + 1\nA : \u2016q\u2016 \u2264 \u2016p\u2016 + 1\nthis\u271d : max \u2016q.snd\u2016 \u2016p.snd\u2016 \u2264 \u2016p\u2016 + 1\nthis : \u2200 (i : \u03b9), i \u2208 univ \u2192 0 \u2264 \u2016Prod.snd p i\u2016\n\u22a2 dist (\u2191q.fst q.snd) (\u2191p.fst p.snd) \u2264\n    ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * dist q p"}, {"tactic": "calc\n  dist (q.1 q.2) (p.1 p.2) \u2264 dist (q.1 q.2) (q.1 p.2) + dist (q.1 p.2) (p.1 p.2) :=\n    dist_triangle _ _ _\n  _ = \u2016q.1 q.2 - q.1 p.2\u2016 + \u2016q.1 p.2 - p.1 p.2\u2016 := by rw [dist_eq_norm, dist_eq_norm]\n  _ \u2264 \u2016q.1\u2016 * Fintype.card \u03b9 * max \u2016q.2\u2016 \u2016p.2\u2016 ^ (Fintype.card \u03b9 - 1) * \u2016q.2 - p.2\u2016 +\n      \u2016q.1 - p.1\u2016 * \u220f i, \u2016p.2 i\u2016 :=\n    (add_le_add (norm_image_sub_le _ _ _) ((q.1 - p.1).le_op_norm p.2))\n  _ \u2264 (\u2016p\u2016 + 1) * Fintype.card \u03b9 * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) * \u2016q - p\u2016 +\n      \u2016q - p\u2016 * \u220f i, \u2016p.2 i\u2016 := by\n    apply_rules [add_le_add, mul_le_mul, le_refl, le_trans (norm_fst_le q) A, Nat.cast_nonneg,\n      mul_nonneg, pow_le_pow_of_le_left, pow_nonneg, norm_snd_le (q - p), norm_nonneg,\n      norm_fst_le (q - p), prod_nonneg]\n  _ = ((\u2016p\u2016 + 1) * Fintype.card \u03b9 * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i, \u2016p.2 i\u2016)\n        * dist q p := by\n    rw [dist_eq_norm]\n    ring", "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\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d\u00b2 : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis\u271d\u00b9 : 0 \u2264 \u2016p\u2016 + 1\nA : \u2016q\u2016 \u2264 \u2016p\u2016 + 1\nthis\u271d : max \u2016q.snd\u2016 \u2016p.snd\u2016 \u2264 \u2016p\u2016 + 1\nthis : \u2200 (i : \u03b9), i \u2208 univ \u2192 0 \u2264 \u2016Prod.snd p i\u2016\n\u22a2 dist (\u2191q.fst q.snd) (\u2191p.fst p.snd) \u2264\n    ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * dist q p", "state_after": "no goals"}, {"tactic": "simp", "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\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\n\u22a2 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016", "state_after": "no goals"}, {"tactic": "simp [A, zero_le_one]", "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\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis : 0 \u2264 \u2016p\u2016 + 1\nA : \u2016q\u2016 \u2264 \u2016p\u2016 + 1\n\u22a2 max \u2016q\u2016 \u2016p\u2016 \u2264 \u2016p\u2016 + 1", "state_after": "no goals"}, {"tactic": "rw [dist_eq_norm, 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\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d\u00b2 : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis\u271d\u00b9 : 0 \u2264 \u2016p\u2016 + 1\nA : \u2016q\u2016 \u2264 \u2016p\u2016 + 1\nthis\u271d : max \u2016q.snd\u2016 \u2016p.snd\u2016 \u2264 \u2016p\u2016 + 1\nthis : \u2200 (i : \u03b9), i \u2208 univ \u2192 0 \u2264 \u2016Prod.snd p i\u2016\n\u22a2 dist (\u2191q.fst q.snd) (\u2191q.fst p.snd) + dist (\u2191q.fst p.snd) (\u2191p.fst p.snd) =\n    \u2016\u2191q.fst q.snd - \u2191q.fst p.snd\u2016 + \u2016\u2191q.fst p.snd - \u2191p.fst p.snd\u2016", "state_after": "no goals"}, {"tactic": "apply_rules [add_le_add, mul_le_mul, le_refl, le_trans (norm_fst_le q) A, Nat.cast_nonneg,\n  mul_nonneg, pow_le_pow_of_le_left, pow_nonneg, norm_snd_le (q - p), norm_nonneg,\n  norm_fst_le (q - p), prod_nonneg]", "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\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d\u00b2 : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis\u271d\u00b9 : 0 \u2264 \u2016p\u2016 + 1\nA : \u2016q\u2016 \u2264 \u2016p\u2016 + 1\nthis\u271d : max \u2016q.snd\u2016 \u2016p.snd\u2016 \u2264 \u2016p\u2016 + 1\nthis : \u2200 (i : \u03b9), i \u2208 univ \u2192 0 \u2264 \u2016Prod.snd p i\u2016\n\u22a2 \u2016q.fst\u2016 * \u2191(Fintype.card \u03b9) * max \u2016q.snd\u2016 \u2016p.snd\u2016 ^ (Fintype.card \u03b9 - 1) * \u2016q.snd - p.snd\u2016 +\n      \u2016q.fst - p.fst\u2016 * \u220f i : \u03b9, \u2016Prod.snd p i\u2016 \u2264\n    (\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) * \u2016q - p\u2016 + \u2016q - p\u2016 * \u220f i : \u03b9, \u2016Prod.snd p i\u2016", "state_after": "no goals"}, {"tactic": "rw [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\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d\u00b2 : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis\u271d\u00b9 : 0 \u2264 \u2016p\u2016 + 1\nA : \u2016q\u2016 \u2264 \u2016p\u2016 + 1\nthis\u271d : max \u2016q.snd\u2016 \u2016p.snd\u2016 \u2264 \u2016p\u2016 + 1\nthis : \u2200 (i : \u03b9), i \u2208 univ \u2192 0 \u2264 \u2016Prod.snd p i\u2016\n\u22a2 (\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) * \u2016q - p\u2016 + \u2016q - p\u2016 * \u220f i : \u03b9, \u2016Prod.snd p i\u2016 =\n    ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * dist q p", "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\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d\u00b2 : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis\u271d\u00b9 : 0 \u2264 \u2016p\u2016 + 1\nA : \u2016q\u2016 \u2264 \u2016p\u2016 + 1\nthis\u271d : max \u2016q.snd\u2016 \u2016p.snd\u2016 \u2264 \u2016p\u2016 + 1\nthis : \u2200 (i : \u03b9), i \u2208 univ \u2192 0 \u2264 \u2016Prod.snd p i\u2016\n\u22a2 (\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) * \u2016q - p\u2016 + \u2016q - p\u2016 * \u220f i : \u03b9, \u2016Prod.snd p i\u2016 =\n    ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * \u2016q - p\u2016"}, {"tactic": "ring", "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\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : Fintype \u03b9'\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nc : \ud835\udd5c\nf g : ContinuousMultilinearMap \ud835\udd5c E G\nm : (i : \u03b9) \u2192 E i\np q : ContinuousMultilinearMap \ud835\udd5c E G \u00d7 ((i : \u03b9) \u2192 E i)\nhq : dist q p < 1\nthis\u271d\u00b2 : 0 \u2264 max \u2016q.snd\u2016 \u2016p.snd\u2016\nthis\u271d\u00b9 : 0 \u2264 \u2016p\u2016 + 1\nA : \u2016q\u2016 \u2264 \u2016p\u2016 + 1\nthis\u271d : max \u2016q.snd\u2016 \u2016p.snd\u2016 \u2264 \u2016p\u2016 + 1\nthis : \u2200 (i : \u03b9), i \u2208 univ \u2192 0 \u2264 \u2016Prod.snd p i\u2016\n\u22a2 (\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) * \u2016q - p\u2016 + \u2016q - p\u2016 * \u220f i : \u03b9, \u2016Prod.snd p i\u2016 =\n    ((\u2016p\u2016 + 1) * \u2191(Fintype.card \u03b9) * (\u2016p\u2016 + 1) ^ (Fintype.card \u03b9 - 1) + \u220f i : \u03b9, \u2016Prod.snd p i\u2016) * \u2016q - p\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/DirectSum/Basic.lean", "full_name": "DirectSum.of_injective", "start": [160, 1], "end": [161, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Hom/Ring.lean", "full_name": "NonUnitalRingHom.coe_addMonoidHom_injective", "start": [193, 1], "end": [194, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.insert_inter_of_not_mem", "start": [2014, 1], "end": [2015, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Star/Unitary.lean", "full_name": "unitary.coe_inv", "start": [169, 1], "end": [170, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/List.lean", "full_name": "Denumerable.raise'_chain", "start": [374, 1], "end": [377, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/LocallyConvex/WithSeminorms.lean", "full_name": "SeminormFamily.basisSets_singleton_mem", "start": [91, 1], "end": [92, 67], "traced_tactics": [{"tactic": "rw [Finset.sup_singleton]", "state_before": "\ud835\udd5c : Type u_2\n\ud835\udd5c\u2082 : Type ?u.15394\n\ud835\udd5d : Type ?u.15397\n\ud835\udd5d\u2082 : Type ?u.15400\nE : Type u_1\nF : Type ?u.15406\nG : Type ?u.15409\n\u03b9 : Type u_3\n\u03b9' : Type ?u.15415\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\np : SeminormFamily \ud835\udd5c E \u03b9\ni : \u03b9\nr : \u211d\nhr : 0 < r\n\u22a2 ball (p i) 0 r = ball (Finset.sup {i} p) 0 r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Prod.lean", "full_name": "DifferentiableAt.fst", "start": [181, 11], "end": [183, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/PFunctor/Univariate/M.lean", "full_name": "PFunctor.M.mk_dest", "start": [341, 1], "end": [364, 8], "traced_tactics": [{"tactic": "apply ext'", "state_before": "F : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\n\u22a2 M.mk (dest x) = x", "state_after": "case H\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\n\u22a2 \u2200 (i : \u2115), MIntl.approx (M.mk (dest x)) i = MIntl.approx x i"}, {"tactic": "intro n", "state_before": "case H\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\n\u22a2 \u2200 (i : \u2115), MIntl.approx (M.mk (dest x)) i = MIntl.approx x i", "state_after": "case H\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\nn : \u2115\n\u22a2 MIntl.approx (M.mk (dest x)) n = MIntl.approx x n"}, {"tactic": "dsimp only [M.mk]", "state_before": "case H\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\nn : \u2115\n\u22a2 MIntl.approx (M.mk (dest x)) n = MIntl.approx x n", "state_after": "case H\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\nn : \u2115\n\u22a2 Approx.sMk (dest x) n = MIntl.approx x n"}, {"tactic": "induction' n with n", "state_before": "case H\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\nn : \u2115\n\u22a2 Approx.sMk (dest x) n = MIntl.approx x n", "state_after": "case H.zero\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\n\u22a2 Approx.sMk (dest x) zero = MIntl.approx x zero\n\ncase H.succ\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\nn : \u2115\nn_ih\u271d : Approx.sMk (dest x) n = MIntl.approx x n\n\u22a2 Approx.sMk (dest x) (succ n) = MIntl.approx x (succ n)"}, {"tactic": "dsimp only [Approx.sMk, dest, head]", "state_before": "case H.succ\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\nn : \u2115\nn_ih\u271d : Approx.sMk (dest x) n = MIntl.approx x n\n\u22a2 Approx.sMk (dest x) (succ n) = MIntl.approx x (succ n)", "state_after": "case H.succ\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\nn : \u2115\nn_ih\u271d : Approx.sMk (dest x) n = MIntl.approx x n\n\u22a2 (CofixA.intro (head' (MIntl.approx x 1)) fun i => MIntl.approx (children x i) n) = MIntl.approx x (succ n)"}, {"tactic": "cases' h : x.approx (succ n) with _ hd ch", "state_before": "case H.succ\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\nn : \u2115\nn_ih\u271d : Approx.sMk (dest x) n = MIntl.approx x n\n\u22a2 (CofixA.intro (head' (MIntl.approx x 1)) fun i => MIntl.approx (children x i) n) = MIntl.approx x (succ n)", "state_after": "case H.succ.intro\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\nn : \u2115\nn_ih\u271d : Approx.sMk (dest x) n = MIntl.approx x n\nhd : F.A\nch : B F hd \u2192 CofixA F n\nh : MIntl.approx x (succ n) = CofixA.intro hd ch\n\u22a2 (CofixA.intro (head' (MIntl.approx x 1)) fun i => MIntl.approx (children x i) n) = CofixA.intro hd ch"}, {"tactic": "have h' : hd = head' (x.approx 1) := by\n  rw [\u2190 head_succ' n, h, head']\n  apply x.consistent", "state_before": "case H.succ.intro\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\nn : \u2115\nn_ih\u271d : Approx.sMk (dest x) n = MIntl.approx x n\nhd : F.A\nch : B F hd \u2192 CofixA F n\nh : MIntl.approx x (succ n) = CofixA.intro hd ch\n\u22a2 (CofixA.intro (head' (MIntl.approx x 1)) fun i => MIntl.approx (children x i) n) = CofixA.intro hd ch", "state_after": "case H.succ.intro\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\nn : \u2115\nn_ih\u271d : Approx.sMk (dest x) n = MIntl.approx x n\nhd : F.A\nch : B F hd \u2192 CofixA F n\nh : MIntl.approx x (succ n) = CofixA.intro hd ch\nh' : hd = head' (MIntl.approx x 1)\n\u22a2 (CofixA.intro (head' (MIntl.approx x 1)) fun i => MIntl.approx (children x i) n) = CofixA.intro hd ch"}, {"tactic": "revert ch", "state_before": "case H.succ.intro\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\nn : \u2115\nn_ih\u271d : Approx.sMk (dest x) n = MIntl.approx x n\nhd : F.A\nch : B F hd \u2192 CofixA F n\nh : MIntl.approx x (succ n) = CofixA.intro hd ch\nh' : hd = head' (MIntl.approx x 1)\n\u22a2 (CofixA.intro (head' (MIntl.approx x 1)) fun i => MIntl.approx (children x i) n) = CofixA.intro hd ch", "state_after": "case H.succ.intro\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\nn : \u2115\nn_ih\u271d : Approx.sMk (dest x) n = MIntl.approx x n\nhd : F.A\nh' : hd = head' (MIntl.approx x 1)\n\u22a2 \u2200 (ch : B F hd \u2192 CofixA F n),\n    MIntl.approx x (succ n) = CofixA.intro hd ch \u2192\n      (CofixA.intro (head' (MIntl.approx x 1)) fun i => MIntl.approx (children x i) n) = CofixA.intro hd ch"}, {"tactic": "rw [h']", "state_before": "case H.succ.intro\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\nn : \u2115\nn_ih\u271d : Approx.sMk (dest x) n = MIntl.approx x n\nhd : F.A\nh' : hd = head' (MIntl.approx x 1)\n\u22a2 \u2200 (ch : B F hd \u2192 CofixA F n),\n    MIntl.approx x (succ n) = CofixA.intro hd ch \u2192\n      (CofixA.intro (head' (MIntl.approx x 1)) fun i => MIntl.approx (children x i) n) = CofixA.intro hd ch", "state_after": "case H.succ.intro\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\nn : \u2115\nn_ih\u271d : Approx.sMk (dest x) n = MIntl.approx x n\nhd : F.A\nh' : hd = head' (MIntl.approx x 1)\n\u22a2 \u2200 (ch : B F (head' (MIntl.approx x 1)) \u2192 CofixA F n),\n    MIntl.approx x (succ n) = CofixA.intro (head' (MIntl.approx x 1)) ch \u2192\n      (CofixA.intro (head' (MIntl.approx x 1)) fun i => MIntl.approx (children x i) n) =\n        CofixA.intro (head' (MIntl.approx x 1)) ch"}, {"tactic": "intros ch h", "state_before": "case H.succ.intro\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\nn : \u2115\nn_ih\u271d : Approx.sMk (dest x) n = MIntl.approx x n\nhd : F.A\nh' : hd = head' (MIntl.approx x 1)\n\u22a2 \u2200 (ch : B F (head' (MIntl.approx x 1)) \u2192 CofixA F n),\n    MIntl.approx x (succ n) = CofixA.intro (head' (MIntl.approx x 1)) ch \u2192\n      (CofixA.intro (head' (MIntl.approx x 1)) fun i => MIntl.approx (children x i) n) =\n        CofixA.intro (head' (MIntl.approx x 1)) ch", "state_after": "case H.succ.intro\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\nn : \u2115\nn_ih\u271d : Approx.sMk (dest x) n = MIntl.approx x n\nhd : F.A\nh' : hd = head' (MIntl.approx x 1)\nch : B F (head' (MIntl.approx x 1)) \u2192 CofixA F n\nh : MIntl.approx x (succ n) = CofixA.intro (head' (MIntl.approx x 1)) ch\n\u22a2 (CofixA.intro (head' (MIntl.approx x 1)) fun i => MIntl.approx (children x i) n) =\n    CofixA.intro (head' (MIntl.approx x 1)) ch"}, {"tactic": "congr", "state_before": "case H.succ.intro\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\nn : \u2115\nn_ih\u271d : Approx.sMk (dest x) n = MIntl.approx x n\nhd : F.A\nh' : hd = head' (MIntl.approx x 1)\nch : B F (head' (MIntl.approx x 1)) \u2192 CofixA F n\nh : MIntl.approx x (succ n) = CofixA.intro (head' (MIntl.approx x 1)) ch\n\u22a2 (CofixA.intro (head' (MIntl.approx x 1)) fun i => MIntl.approx (children x i) n) =\n    CofixA.intro (head' (MIntl.approx x 1)) ch", "state_after": "case H.succ.intro.e_a\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\nn : \u2115\nn_ih\u271d : Approx.sMk (dest x) n = MIntl.approx x n\nhd : F.A\nh' : hd = head' (MIntl.approx x 1)\nch : B F (head' (MIntl.approx x 1)) \u2192 CofixA F n\nh : MIntl.approx x (succ n) = CofixA.intro (head' (MIntl.approx x 1)) ch\n\u22a2 (fun i => MIntl.approx (children x i) n) = ch"}, {"tactic": "apply @Subsingleton.elim _ CofixA.instSubsingleton", "state_before": "case H.zero\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\n\u22a2 Approx.sMk (dest x) zero = MIntl.approx x zero", "state_after": "no goals"}, {"tactic": "rw [\u2190 head_succ' n, h, head']", "state_before": "F : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\nn : \u2115\nn_ih\u271d : Approx.sMk (dest x) n = MIntl.approx x n\nhd : F.A\nch : B F hd \u2192 CofixA F n\nh : MIntl.approx x (succ n) = CofixA.intro hd ch\n\u22a2 hd = head' (MIntl.approx x 1)", "state_after": "case Hconsistent\nF : PFunctor\nX : Type ?u.14923\nf : X \u2192 Obj F X\nx : M F\nn : \u2115\nn_ih\u271d : Approx.sMk (dest x) n = MIntl.approx x n\nhd : F.A\nch : B F hd \u2192 CofixA F n\nh : MIntl.approx x 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(l\u2081 i) * l\u2082 i", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.2093366\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\n\u03b9 : Type u_3\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl\u2081 l\u2082 : \u03b9 \u2192 \ud835\udd5c\ns : Finset \u03b9\n\u22a2 \u2211 x in s, \u2191(starRingEnd \ud835\udd5c) (l\u2081 x) * inner (v x) (\u2211 i in s, l\u2082 i \u2022 v i) = \u2211 x in s, \u2191(starRingEnd \ud835\udd5c) (l\u2081 x) * l\u2082 x"}, {"tactic": "refine' Finset.sum_congr rfl fun i hi => _", "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.2093366\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\n\u03b9 : Type u_3\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl\u2081 l\u2082 : \u03b9 \u2192 \ud835\udd5c\ns : Finset \u03b9\n\u22a2 \u2211 x in s, \u2191(starRingEnd \ud835\udd5c) (l\u2081 x) * inner (v x) (\u2211 i in s, l\u2082 i \u2022 v i) = \u2211 x in s, \u2191(starRingEnd \ud835\udd5c) (l\u2081 x) * l\u2082 x", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.2093366\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\n\u03b9 : Type u_3\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl\u2081 l\u2082 : \u03b9 \u2192 \ud835\udd5c\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2191(starRingEnd \ud835\udd5c) (l\u2081 i) * inner (v i) (\u2211 i in s, l\u2082 i \u2022 v i) = \u2191(starRingEnd \ud835\udd5c) (l\u2081 i) * l\u2082 i"}, {"tactic": "rw [hv.inner_right_sum l\u2082 hi]", "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type ?u.2093366\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\n\u03b9 : Type u_3\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl\u2081 l\u2082 : \u03b9 \u2192 \ud835\udd5c\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2191(starRingEnd \ud835\udd5c) (l\u2081 i) * inner (v i) (\u2211 i in s, l\u2082 i \u2022 v i) = \u2191(starRingEnd \ud835\udd5c) (l\u2081 i) * l\u2082 i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Function/Basic.lean", "full_name": "Function.Surjective.hasRightInverse", "start": [502, 1], "end": [503, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/Subfield.lean", "full_name": "Subfield.closure_sUnion", "start": [783, 1], "end": [784, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean", "full_name": "gramSchmidtOrthonormalBasis_inv_blockTriangular", "start": [399, 1], "end": [401, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/UnitTrinomial.lean", "full_name": "Polynomial.trinomial_leadingCoeff", "start": [98, 1], "end": [100, 86], "traced_tactics": [{"tactic": "rw [leadingCoeff, trinomial_natDegree hkm hmn hw, trinomial_leading_coeff' hkm hmn]", "state_before": "R : Type u_1\ninst\u271d : Semiring R\nk m n : \u2115\nu v w : R\nhkm : k < m\nhmn : m < n\nhw : w \u2260 0\n\u22a2 leadingCoeff (trinomial k m n u v w) = w", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Logic.lean", "full_name": "Decidable.and_or_imp", "start": [615, 1], "end": [617, 61], "traced_tactics": [{"tactic": "simp only [ha, true_and, true_imp_iff]", "state_before": "a b c : Prop\ninst\u271d : Decidable a\nha : a\n\u22a2 a \u2227 b \u2228 (a \u2192 c) \u2194 a \u2192 b \u2228 c", "state_after": "no goals"}, {"tactic": "simp only [ha, false_or, false_and, false_imp_iff]", "state_before": "a b c : Prop\ninst\u271d : Decidable a\nha : \u00aca\n\u22a2 a \u2227 b \u2228 (a \u2192 c) \u2194 a \u2192 b \u2228 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/LazyList/Basic.lean", "full_name": "Thunk.ext", "start": [33, 1], "end": [37, 26], "traced_tactics": [{"tactic": "have \u27e8_\u27e9 := a", "state_before": "\u03b1 : Type u\na b : Thunk \u03b1\neq : Thunk.get a = Thunk.get b\n\u22a2 a = b", "state_after": "\u03b1 : Type u\na b : Thunk \u03b1\nfn\u271d : Unit \u2192 \u03b1\neq : Thunk.get { fn := fn\u271d } = Thunk.get b\n\u22a2 { fn := fn\u271d } = b"}, {"tactic": "have \u27e8_\u27e9 := b", "state_before": "\u03b1 : Type u\na b : Thunk \u03b1\nfn\u271d : Unit \u2192 \u03b1\neq : Thunk.get { fn := fn\u271d } = Thunk.get b\n\u22a2 { fn := fn\u271d } = b", "state_after": "\u03b1 : Type u\na b : Thunk \u03b1\nfn\u271d\u00b9 fn\u271d : Unit \u2192 \u03b1\neq : Thunk.get { fn := fn\u271d\u00b9 } = Thunk.get { fn := fn\u271d }\n\u22a2 { fn := fn\u271d\u00b9 } = { fn := fn\u271d }"}, {"tactic": "congr", "state_before": "\u03b1 : Type u\na b : Thunk \u03b1\nfn\u271d\u00b9 fn\u271d : Unit \u2192 \u03b1\neq : Thunk.get { fn := fn\u271d\u00b9 } = Thunk.get { fn := fn\u271d }\n\u22a2 { fn := fn\u271d\u00b9 } = { fn := fn\u271d }", "state_after": "case e_fn\n\u03b1 : Type u\na b : Thunk \u03b1\nfn\u271d\u00b9 fn\u271d : Unit \u2192 \u03b1\neq : Thunk.get { fn := fn\u271d\u00b9 } = Thunk.get { fn := fn\u271d }\n\u22a2 fn\u271d\u00b9 = fn\u271d"}, {"tactic": "exact funext fun _ \u21a6 eq", "state_before": "case e_fn\n\u03b1 : Type u\na b : Thunk \u03b1\nfn\u271d\u00b9 fn\u271d : Unit \u2192 \u03b1\neq : Thunk.get { fn := fn\u271d\u00b9 } = Thunk.get { fn := fn\u271d }\n\u22a2 fn\u271d\u00b9 = fn\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/ModelTheory/Substructures.lean", "full_name": "FirstOrder.Language.Substructure.closure_empty", "start": [385, 1], "end": [386, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/SuccPred/Basic.lean", "full_name": "Order.le_of_lt_succ", "start": [228, 1], "end": [229, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup.lean", "full_name": "Matrix.SpecialLinearGroup.coe_GLPos_coe_GL_coe_matrix", "start": [276, 1], "end": [278, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finsupp/Basic.lean", "full_name": "Finsupp.mapDomain_support_of_injOn", "start": [561, 1], "end": [572, 26], "traced_tactics": [{"tactic": "intro x hx", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.199345\n\u03b9 : Type ?u.199348\nM : Type u_3\nM' : Type ?u.199354\nN : Type ?u.199357\nP : Type ?u.199360\nG : Type ?u.199363\nH : Type ?u.199366\nR : Type ?u.199369\nS : Type ?u.199372\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\n\u22a2 image f s.support \u2286 (mapDomain f s).support", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.199345\n\u03b9 : Type ?u.199348\nM : Type u_3\nM' : Type ?u.199354\nN : Type ?u.199357\nP : Type ?u.199360\nG : Type ?u.199363\nH : Type ?u.199366\nR : Type ?u.199369\nS : Type ?u.199372\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nx : \u03b2\nhx : x \u2208 image f s.support\n\u22a2 x \u2208 (mapDomain f s).support"}, {"tactic": "simp only [mem_image, exists_prop, mem_support_iff, Ne.def] at hx", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.199345\n\u03b9 : Type ?u.199348\nM : Type u_3\nM' : Type ?u.199354\nN : Type ?u.199357\nP : Type ?u.199360\nG : Type ?u.199363\nH : Type ?u.199366\nR : Type ?u.199369\nS : Type ?u.199372\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nx : \u03b2\nhx : x \u2208 image f s.support\n\u22a2 x \u2208 (mapDomain f s).support", "state_after": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.199345\n\u03b9 : Type ?u.199348\nM : Type u_3\nM' : Type ?u.199354\nN : Type ?u.199357\nP : Type ?u.199360\nG : Type ?u.199363\nH : Type ?u.199366\nR : Type ?u.199369\nS : Type ?u.199372\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nx : \u03b2\nhx : \u2203 a, \u00ac\u2191s a = 0 \u2227 f a = x\n\u22a2 x \u2208 (mapDomain f s).support"}, {"tactic": "rcases hx with \u27e8hx_w, hx_h_left, rfl\u27e9", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.199345\n\u03b9 : Type ?u.199348\nM : Type u_3\nM' : Type ?u.199354\nN : Type ?u.199357\nP : Type ?u.199360\nG : Type ?u.199363\nH : Type ?u.199366\nR : Type ?u.199369\nS : Type ?u.199372\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nx : \u03b2\nhx : \u2203 a, \u00ac\u2191s a = 0 \u2227 f a = x\n\u22a2 x \u2208 (mapDomain f s).support", "state_after": "case intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.199345\n\u03b9 : Type ?u.199348\nM : Type u_3\nM' : Type ?u.199354\nN : Type ?u.199357\nP : Type ?u.199360\nG : Type ?u.199363\nH : Type ?u.199366\nR : Type ?u.199369\nS : Type ?u.199372\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nhx_w : \u03b1\nhx_h_left : \u00ac\u2191s hx_w = 0\n\u22a2 f hx_w \u2208 (mapDomain f s).support"}, {"tactic": "simp only [mem_support_iff, Ne.def]", "state_before": "case intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.199345\n\u03b9 : Type ?u.199348\nM : Type u_3\nM' : Type ?u.199354\nN : Type ?u.199357\nP : Type ?u.199360\nG : Type ?u.199363\nH : Type ?u.199366\nR : Type ?u.199369\nS : Type ?u.199372\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nhx_w : \u03b1\nhx_h_left : \u00ac\u2191s hx_w = 0\n\u22a2 f hx_w \u2208 (mapDomain f s).support", "state_after": "case intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.199345\n\u03b9 : Type ?u.199348\nM : Type u_3\nM' : Type ?u.199354\nN : Type ?u.199357\nP : Type ?u.199360\nG : Type ?u.199363\nH : Type ?u.199366\nR : Type ?u.199369\nS : Type ?u.199372\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nhx_w : \u03b1\nhx_h_left : \u00ac\u2191s hx_w = 0\n\u22a2 \u00ac\u2191(mapDomain f s) (f hx_w) = 0"}, {"tactic": "rw [mapDomain_apply' (\u2191s.support : Set _) _ _ hf]", "state_before": "case intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.199345\n\u03b9 : Type ?u.199348\nM : Type u_3\nM' : Type ?u.199354\nN : Type ?u.199357\nP : Type ?u.199360\nG : Type ?u.199363\nH : Type ?u.199366\nR : Type ?u.199369\nS : Type ?u.199372\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nhx_w : \u03b1\nhx_h_left : \u00ac\u2191s hx_w = 0\n\u22a2 \u00ac\u2191(mapDomain f s) (f hx_w) = 0", "state_after": "case intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.199345\n\u03b9 : Type ?u.199348\nM : Type u_3\nM' : Type ?u.199354\nN : Type ?u.199357\nP : Type ?u.199360\nG : Type ?u.199363\nH : Type ?u.199366\nR : Type ?u.199369\nS : Type ?u.199372\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nhx_w : \u03b1\nhx_h_left : \u00ac\u2191s hx_w = 0\n\u22a2 \u00ac\u2191s hx_w = 0\n\ncase intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.199345\n\u03b9 : Type ?u.199348\nM : Type u_3\nM' : Type ?u.199354\nN : Type ?u.199357\nP : Type ?u.199360\nG : Type ?u.199363\nH : Type ?u.199366\nR : Type ?u.199369\nS : Type ?u.199372\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nhx_w : \u03b1\nhx_h_left : \u00ac\u2191s hx_w = 0\n\u22a2 hx_w \u2208 \u2191s.support\n\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.199345\n\u03b9 : Type ?u.199348\nM : Type u_3\nM' : Type ?u.199354\nN : Type ?u.199357\nP : Type ?u.199360\nG : Type ?u.199363\nH : Type ?u.199366\nR : Type ?u.199369\nS : Type ?u.199372\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nhx_w : \u03b1\nhx_h_left : \u00ac\u2191s hx_w = 0\n\u22a2 \u2191s.support \u2286 \u2191s.support"}, {"tactic": "exact hx_h_left", "state_before": "case intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.199345\n\u03b9 : Type ?u.199348\nM : Type u_3\nM' : Type ?u.199354\nN : Type ?u.199357\nP : Type ?u.199360\nG : Type ?u.199363\nH : Type ?u.199366\nR : Type ?u.199369\nS : Type ?u.199372\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nhx_w : \u03b1\nhx_h_left : \u00ac\u2191s hx_w = 0\n\u22a2 \u00ac\u2191s hx_w = 0", "state_after": "no goals"}, {"tactic": "simp only [mem_coe, mem_support_iff, Ne.def]", "state_before": "case intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.199345\n\u03b9 : Type ?u.199348\nM : Type u_3\nM' : Type ?u.199354\nN : Type ?u.199357\nP : Type ?u.199360\nG : Type ?u.199363\nH : Type ?u.199366\nR : Type ?u.199369\nS : Type ?u.199372\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nhx_w : \u03b1\nhx_h_left : \u00ac\u2191s hx_w = 0\n\u22a2 hx_w \u2208 \u2191s.support", "state_after": "case intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.199345\n\u03b9 : Type ?u.199348\nM : Type u_3\nM' : Type ?u.199354\nN : Type ?u.199357\nP : Type ?u.199360\nG : Type ?u.199363\nH : Type ?u.199366\nR : Type ?u.199369\nS : Type ?u.199372\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nhx_w : \u03b1\nhx_h_left : \u00ac\u2191s hx_w = 0\n\u22a2 \u00ac\u2191s hx_w = 0"}, {"tactic": "exact hx_h_left", "state_before": "case intro.intro\n\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.199345\n\u03b9 : Type ?u.199348\nM : Type u_3\nM' : Type ?u.199354\nN : Type ?u.199357\nP : Type ?u.199360\nG : Type ?u.199363\nH : Type ?u.199366\nR : Type ?u.199369\nS : Type ?u.199372\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nhx_w : \u03b1\nhx_h_left : \u00ac\u2191s hx_w = 0\n\u22a2 \u00ac\u2191s hx_w = 0", "state_after": "no goals"}, {"tactic": "exact Subset.refl _", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.199345\n\u03b9 : Type ?u.199348\nM : Type u_3\nM' : Type ?u.199354\nN : Type ?u.199357\nP : Type ?u.199360\nG : Type ?u.199363\nH : Type ?u.199366\nR : Type ?u.199369\nS : Type ?u.199372\ninst\u271d\u00b9 : AddCommMonoid M\nv v\u2081 v\u2082 : \u03b1 \u2192\u2080 M\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : \u03b1 \u2192\u2080 M\nhf : Set.InjOn f \u2191s.support\nhx_w : \u03b1\nhx_h_left : \u00ac\u2191s hx_w = 0\n\u22a2 \u2191s.support \u2286 \u2191s.support", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "full_name": "normalize_eq_normalize_iff", "start": [187, 1], "end": [189, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Finsupp.lean", "full_name": "Finsupp.mapRange.linearMap_apply", "start": [823, 1], "end": [824, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Pigeonhole.lean", "full_name": "Finset.exists_lt_card_fiber_of_nsmul_lt_card_of_maps_to", "start": [233, 1], "end": [236, 61], "traced_tactics": [{"tactic": "simp_rw [cast_card] at ht\u22a2", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nM : Type w\ninst\u271d\u00b9 : DecidableEq \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nw : \u03b1 \u2192 M\nb : M\nn : \u2115\ninst\u271d : LinearOrderedCommSemiring M\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nht : card t \u2022 b < \u2191(card s)\n\u22a2 \u2203 y, y \u2208 t \u2227 b < \u2191(card (filter (fun x => f x = y) s))", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nM : Type w\ninst\u271d\u00b9 : DecidableEq \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nw : \u03b1 \u2192 M\nb : M\nn : \u2115\ninst\u271d : LinearOrderedCommSemiring M\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nht : card t \u2022 b < \u2211 a in s, 1\n\u22a2 \u2203 y, y \u2208 t \u2227 b < \u2211 a in filter (fun x => f x = y) s, 1"}, {"tactic": "exact exists_lt_sum_fiber_of_maps_to_of_nsmul_lt_sum hf ht", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nM : Type w\ninst\u271d\u00b9 : DecidableEq \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nw : \u03b1 \u2192 M\nb : M\nn : \u2115\ninst\u271d : LinearOrderedCommSemiring M\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nht : card t \u2022 b < \u2211 a in s, 1\n\u22a2 \u2203 y, y \u2208 t \u2227 b < \u2211 a in filter (fun x => f x = y) s, 1", "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_sub_two_pi", "start": [330, 1], "end": [331, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Fintype/Basic.lean", "full_name": "Function.Embedding.invOfMemRange_surjective", "start": [549, 1], "end": [550, 41], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type u_1\n\u03b3 : Type ?u.73841\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u21aa \u03b2\nb : \u2191(Set.range \u2191f)\na : \u03b1\n\u22a2 invOfMemRange f { val := \u2191f a, property := (_ : \u2191f a \u2208 Set.range \u2191f) } = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Skeletal.lean", "full_name": "CategoryTheory.ThinSkeleton.map_comp_eq", "start": [305, 1], "end": [307, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finite/Card.lean", "full_name": "Finite.card_pos", "start": [65, 1], "end": [66, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "full_name": "inv_mem_iff", "start": [131, 1], "end": [133, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/Partrec.lean", "full_name": "Primrec.to_comp", "start": [255, 1], "end": [258, 43], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03c3\nhf : Primrec f\nn : \u2115\n\u22a2 (do\n      let n \u2190 (\u2191fun n => encode (Option.map f (decode n))) n\n      \u2191(Nat.ppred n)) =\n    Part.bind \u2191(decode n) fun a => map encode (\u2191f a)", "state_after": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03c3\nhf : Primrec f\nn : \u2115\n\u22a2 \u2191(Nat.ppred (encode (Option.map f (decode n)))) = Part.bind \u2191(decode n) fun a => Part.some (encode (f a))"}, {"tactic": "cases decode (\u03b1 := \u03b1) n <;> simp", "state_before": "\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03c3\nhf : Primrec f\nn : \u2115\n\u22a2 \u2191(Nat.ppred (encode (Option.map f (decode n)))) = Part.bind \u2191(decode n) fun a => Part.some (encode (f a))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean", "full_name": "Path.Homotopy.trans_assoc_reparam", "start": [231, 1], "end": [271, 9], "traced_tactics": [{"tactic": "continuity", "state_before": "X : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\n\u22a2 Continuous fun t => { val := transAssocReparamAux t, property := (_ : transAssocReparamAux t \u2208 I) }", "state_after": "no goals"}, {"tactic": "ext x", "state_before": "X : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\n\u22a2 Path.trans (Path.trans p q) r =\n    Path.reparam (Path.trans p (Path.trans q r))\n      (fun t => { val := transAssocReparamAux t, property := (_ : transAssocReparamAux t \u2208 I) })\n      (_ : Continuous fun x => { val := transAssocReparamAux x, property := (_ : transAssocReparamAux x \u2208 I) })\n      (_ : (fun t => { val := transAssocReparamAux t, property := (_ : transAssocReparamAux t \u2208 I) }) 0 = 0)\n      (_ : (fun t => { val := transAssocReparamAux t, property := (_ : transAssocReparamAux t \u2208 I) }) 1 = 1)", "state_after": "case a.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\n\u22a2 \u2191(Path.trans (Path.trans p q) r) x =\n    \u2191(Path.reparam (Path.trans p (Path.trans q r))\n          (fun t => { val := transAssocReparamAux t, property := (_ : transAssocReparamAux t \u2208 I) })\n          (_ : Continuous fun x => { val := transAssocReparamAux x, property := (_ : transAssocReparamAux x \u2208 I) })\n          (_ : (fun t => { val := transAssocReparamAux t, property := (_ : transAssocReparamAux t \u2208 I) }) 0 = 0)\n          (_ : (fun t => { val := transAssocReparamAux t, property := (_ : transAssocReparamAux t \u2208 I) }) 1 = 1))\n      x"}, {"tactic": "simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left\u2080, not_le,\n  Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,\n  Path.coe_reparam]", "state_before": "case a.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\n\u22a2 \u2191(Path.trans (Path.trans p q) r) x =\n    \u2191(Path.reparam (Path.trans p (Path.trans q r))\n          (fun t => { val := transAssocReparamAux t, property := (_ : transAssocReparamAux t \u2208 I) })\n          (_ : Continuous fun x => { val := transAssocReparamAux x, property := (_ : transAssocReparamAux x \u2208 I) })\n          (_ : (fun t => { val := transAssocReparamAux t, property := (_ : transAssocReparamAux t \u2208 I) }) 0 = 0)\n          (_ : (fun t => { val := transAssocReparamAux t, property := (_ : transAssocReparamAux t \u2208 I) }) 1 = 1))\n      x", "state_after": "case a.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\n\u22a2 (if h : \u2191x \u2264 1 / 2 then\n      if h_1 : 2 * \u2191x \u2264 1 / 2 then\n        \u2191p { val := 2 * (2 * \u2191x), property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } \u2208 I) }\n      else\n        \u2191q { val := 2 * (2 * \u2191x) - 1, property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } - 1 \u2208 I) }\n    else \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) }) =\n    if h : (if \u2191x \u2264 1 / 4 then 2 * \u2191x else if \u2191x \u2264 1 / 2 then \u2191x + 1 / 4 else 1 / 2 * (\u2191x + 1)) \u2264 1 / 2 then\n      \u2191p\n        { val := if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x) else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1)),\n          property :=\n            (_ :\n              (fun x => x \u2208 I)\n                (if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x)\n                else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1)))) }\n    else\n      if h_1 :\n          (if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x) else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1))) - 1 \u2264\n            1 / 2 then\n        \u2191q\n          {\n            val :=\n              2 *\n                ((if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x)\n                  else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1))) -\n                  1),\n            property :=\n              (_ :\n                2 *\n                    \u2191{\n                        val :=\n                          (if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x)\n                            else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1))) -\n                            1,\n                        property :=\n                          (_ :\n                            (fun x => x \u2208 I)\n                              ((if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x)\n                                else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1))) -\n                                1)) } \u2208\n                  I) }\n      else\n        \u2191r\n          {\n            val :=\n              2 *\n                  ((if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x)\n                    else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1))) -\n                    1) -\n                1,\n            property :=\n              (_ :\n                2 *\n                      \u2191{\n                          val :=\n                            (if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x)\n                              else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1))) -\n                              1,\n                          property :=\n                            (_ :\n                              (fun x => x \u2208 I)\n                                ((if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x)\n                                  else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1))) -\n                                  1)) } -\n                    1 \u2208\n                  I) }"}, {"tactic": "split_ifs with h\u2081 h\u2082 h\u2083 h\u2084 h\u2085", "state_before": "case a.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\n\u22a2 (if h : \u2191x \u2264 1 / 2 then\n      if h_1 : 2 * \u2191x \u2264 1 / 2 then\n        \u2191p { val := 2 * (2 * \u2191x), property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } \u2208 I) }\n      else\n        \u2191q { val := 2 * (2 * \u2191x) - 1, property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } - 1 \u2208 I) }\n    else \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) }) =\n    if h : (if \u2191x \u2264 1 / 4 then 2 * \u2191x else if \u2191x \u2264 1 / 2 then \u2191x + 1 / 4 else 1 / 2 * (\u2191x + 1)) \u2264 1 / 2 then\n      \u2191p\n        { val := if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x) else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1)),\n          property :=\n            (_ :\n              (fun x => x \u2208 I)\n                (if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x)\n                else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1)))) }\n    else\n      if h_1 :\n          (if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x) else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1))) - 1 \u2264\n            1 / 2 then\n        \u2191q\n          {\n            val :=\n              2 *\n                ((if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x)\n                  else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1))) -\n                  1),\n            property :=\n              (_ :\n                2 *\n                    \u2191{\n                        val :=\n                          (if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x)\n                            else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1))) -\n                            1,\n                        property :=\n                          (_ :\n                            (fun x => x \u2208 I)\n                              ((if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x)\n                                else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1))) -\n                                1)) } \u2208\n                  I) }\n      else\n        \u2191r\n          {\n            val :=\n              2 *\n                  ((if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x)\n                    else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1))) -\n                    1) -\n                1,\n            property :=\n              (_ :\n                2 *\n                      \u2191{\n                          val :=\n                            (if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x)\n                              else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1))) -\n                              1,\n                          property :=\n                            (_ :\n                              (fun x => x \u2208 I)\n                                ((if \u2191x \u2264 1 / 4 then 2 * (2 * \u2191x)\n                                  else if \u2191x \u2264 1 / 2 then 2 * (\u2191x + 1 / 4) else 2 * (1 / 2 * (\u2191x + 1))) -\n                                  1)) } -\n                    1 \u2208\n                  I) }", "state_after": "case a.h.inl.inl.inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : 2 * \u2191x \u2264 1 / 2\nh\u2083 : \u2191x \u2264 1 / 4\n\u22a2 \u2191p { val := 2 * (2 * \u2191x), property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } \u2208 I) } =\n    \u2191p { val := 2 * (2 * \u2191x), property := (_ : (fun x => x \u2208 I) (2 * (2 * \u2191x))) }\n\ncase a.h.inl.inl.inr.inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : 2 * \u2191x \u2264 1 / 2\nh\u2083 : \u00ac\u2191x \u2264 1 / 4\nh\u2084 : \u2191x + 1 / 4 \u2264 1 / 2\n\u22a2 \u2191p { val := 2 * (2 * \u2191x), property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } \u2208 I) } =\n    \u2191p { val := 2 * (\u2191x + 1 / 4), property := (_ : (fun x => x \u2208 I) (2 * (\u2191x + 1 / 4))) }\n\ncase a.h.inl.inl.inr.inr.inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : 2 * \u2191x \u2264 1 / 2\nh\u2083 : \u00ac\u2191x \u2264 1 / 4\nh\u2084 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u2085 : 2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 \u2191p { val := 2 * (2 * \u2191x), property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } \u2208 I) } =\n    \u2191q { val := 2 * (2 * (\u2191x + 1 / 4) - 1), property := (_ : (fun x => x \u2208 I) (2 * (2 * (\u2191x + 1 / 4) - 1))) }\n\ncase a.h.inl.inl.inr.inr.inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : 2 * \u2191x \u2264 1 / 2\nh\u2083 : \u00ac\u2191x \u2264 1 / 4\nh\u2084 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u2085 : \u00ac2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 \u2191p { val := 2 * (2 * \u2191x), property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } \u2208 I) } =\n    \u2191r { val := 2 * (2 * (\u2191x + 1 / 4) - 1) - 1, property := (_ : (fun x => x \u2208 I) (2 * (2 * (\u2191x + 1 / 4) - 1) - 1)) }\n\ncase a.h.inl.inr.inl.inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u2191x \u2264 1 / 4\nh\u271d : 2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 \u2191q { val := 2 * (2 * \u2191x) - 1, property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } - 1 \u2208 I) } =\n    \u2191q { val := 2 * (2 * (2 * \u2191x) - 1), property := (_ : (fun x => x \u2208 I) (2 * (2 * (2 * \u2191x) - 1))) }\n\ncase a.h.inl.inr.inl.inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u2191x \u2264 1 / 4\nh\u271d : \u00ac2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 \u2191q { val := 2 * (2 * \u2191x) - 1, property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } - 1 \u2208 I) } =\n    \u2191r { val := 2 * (2 * (2 * \u2191x) - 1) - 1, property := (_ : (fun x => x \u2208 I) (2 * (2 * (2 * \u2191x) - 1) - 1)) }\n\ncase a.h.inl.inr.inr.inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u00ac\u2191x \u2264 1 / 4\nh\u271d : \u2191x + 1 / 4 \u2264 1 / 2\n\u22a2 \u2191q { val := 2 * (2 * \u2191x) - 1, property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } - 1 \u2208 I) } =\n    \u2191p { val := 2 * (\u2191x + 1 / 4), property := (_ : (fun x => x \u2208 I) (2 * (\u2191x + 1 / 4))) }\n\ncase a.h.inl.inr.inr.inr.inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u271d : 2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 \u2191q { val := 2 * (2 * \u2191x) - 1, property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } - 1 \u2208 I) } =\n    \u2191q { val := 2 * (2 * (\u2191x + 1 / 4) - 1), property := (_ : (fun x => x \u2208 I) (2 * (2 * (\u2191x + 1 / 4) - 1))) }\n\ncase a.h.inl.inr.inr.inr.inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u271d : \u00ac2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 \u2191q { val := 2 * (2 * \u2191x) - 1, property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } - 1 \u2208 I) } =\n    \u2191r { val := 2 * (2 * (\u2191x + 1 / 4) - 1) - 1, property := (_ : (fun x => x \u2208 I) (2 * (2 * (\u2191x + 1 / 4) - 1) - 1)) }\n\ncase a.h.inr.inl.inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u2191x \u2264 1 / 4\nh\u271d : 2 * \u2191x \u2264 1 / 2\n\u22a2 \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) } =\n    \u2191p { val := 2 * (2 * \u2191x), property := (_ : (fun x => x \u2208 I) (2 * (2 * \u2191x))) }\n\ncase a.h.inr.inl.inr.inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d : 2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) } =\n    \u2191q { val := 2 * (2 * (2 * \u2191x) - 1), property := (_ : (fun x => x \u2208 I) (2 * (2 * (2 * \u2191x) - 1))) }\n\ncase a.h.inr.inl.inr.inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d : \u00ac2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) } =\n    \u2191r { val := 2 * (2 * (2 * \u2191x) - 1) - 1, property := (_ : (fun x => x \u2208 I) (2 * (2 * (2 * \u2191x) - 1) - 1)) }\n\ncase a.h.inr.inr.inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u00ac\u2191x \u2264 1 / 4\nh\u271d : 1 / 2 * (\u2191x + 1) \u2264 1 / 2\n\u22a2 \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) } =\n    \u2191p { val := 2 * (1 / 2 * (\u2191x + 1)), property := (_ : (fun x => x \u2208 I) (2 * (1 / 2 * (\u2191x + 1)))) }\n\ncase a.h.inr.inr.inr.inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac1 / 2 * (\u2191x + 1) \u2264 1 / 2\nh\u271d : 2 * (1 / 2 * (\u2191x + 1)) - 1 \u2264 1 / 2\n\u22a2 \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) } =\n    \u2191q\n      { val := 2 * (2 * (1 / 2 * (\u2191x + 1)) - 1), property := (_ : (fun x => x \u2208 I) (2 * (2 * (1 / 2 * (\u2191x + 1)) - 1))) }\n\ncase a.h.inr.inr.inr.inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac1 / 2 * (\u2191x + 1) \u2264 1 / 2\nh\u271d : \u00ac2 * (1 / 2 * (\u2191x + 1)) - 1 \u2264 1 / 2\n\u22a2 \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) } =\n    \u2191r\n      { val := 2 * (2 * (1 / 2 * (\u2191x + 1)) - 1) - 1,\n        property := (_ : (fun x => x \u2208 I) (2 * (2 * (1 / 2 * (\u2191x + 1)) - 1) - 1)) }"}, {"tactic": "rfl", "state_before": "case a.h.inl.inl.inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : 2 * \u2191x \u2264 1 / 2\nh\u2083 : \u2191x \u2264 1 / 4\n\u22a2 \u2191p { val := 2 * (2 * \u2191x), property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } \u2208 I) } =\n    \u2191p { val := 2 * (2 * \u2191x), property := (_ : (fun x => x \u2208 I) (2 * (2 * \u2191x))) }", "state_after": "no goals"}, {"tactic": "exfalso", "state_before": "case a.h.inl.inl.inr.inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : 2 * \u2191x \u2264 1 / 2\nh\u2083 : \u00ac\u2191x \u2264 1 / 4\nh\u2084 : \u2191x + 1 / 4 \u2264 1 / 2\n\u22a2 \u2191p { val := 2 * (2 * \u2191x), property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } \u2208 I) } =\n    \u2191p { val := 2 * (\u2191x + 1 / 4), property := (_ : (fun x => x \u2208 I) (2 * (\u2191x + 1 / 4))) }", "state_after": "case a.h.inl.inl.inr.inl.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : 2 * \u2191x \u2264 1 / 2\nh\u2083 : \u00ac\u2191x \u2264 1 / 4\nh\u2084 : \u2191x + 1 / 4 \u2264 1 / 2\n\u22a2 False"}, {"tactic": "linarith", "state_before": "case a.h.inl.inl.inr.inl.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : 2 * \u2191x \u2264 1 / 2\nh\u2083 : \u00ac\u2191x \u2264 1 / 4\nh\u2084 : \u2191x + 1 / 4 \u2264 1 / 2\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exfalso", "state_before": "case a.h.inl.inl.inr.inr.inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : 2 * \u2191x \u2264 1 / 2\nh\u2083 : \u00ac\u2191x \u2264 1 / 4\nh\u2084 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u2085 : 2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 \u2191p { val := 2 * (2 * \u2191x), property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } \u2208 I) } =\n    \u2191q { val := 2 * (2 * (\u2191x + 1 / 4) - 1), property := (_ : (fun x => x \u2208 I) (2 * (2 * (\u2191x + 1 / 4) - 1))) }", "state_after": "case a.h.inl.inl.inr.inr.inl.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : 2 * \u2191x \u2264 1 / 2\nh\u2083 : \u00ac\u2191x \u2264 1 / 4\nh\u2084 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u2085 : 2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 False"}, {"tactic": "linarith", "state_before": "case a.h.inl.inl.inr.inr.inl.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : 2 * \u2191x \u2264 1 / 2\nh\u2083 : \u00ac\u2191x \u2264 1 / 4\nh\u2084 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u2085 : 2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exfalso", "state_before": "case a.h.inl.inl.inr.inr.inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : 2 * \u2191x \u2264 1 / 2\nh\u2083 : \u00ac\u2191x \u2264 1 / 4\nh\u2084 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u2085 : \u00ac2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 \u2191p { val := 2 * (2 * \u2191x), property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } \u2208 I) } =\n    \u2191r { val := 2 * (2 * (\u2191x + 1 / 4) - 1) - 1, property := (_ : (fun x => x \u2208 I) (2 * (2 * (\u2191x + 1 / 4) - 1) - 1)) }", "state_after": "case a.h.inl.inl.inr.inr.inr.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : 2 * \u2191x \u2264 1 / 2\nh\u2083 : \u00ac\u2191x \u2264 1 / 4\nh\u2084 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u2085 : \u00ac2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 False"}, {"tactic": "linarith", "state_before": "case a.h.inl.inl.inr.inr.inr.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : 2 * \u2191x \u2264 1 / 2\nh\u2083 : \u00ac\u2191x \u2264 1 / 4\nh\u2084 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u2085 : \u00ac2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exfalso", "state_before": "case a.h.inl.inr.inl.inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u2191x \u2264 1 / 4\nh\u271d : 2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 \u2191q { val := 2 * (2 * \u2191x) - 1, property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } - 1 \u2208 I) } =\n    \u2191q { val := 2 * (2 * (2 * \u2191x) - 1), property := (_ : (fun x => x \u2208 I) (2 * (2 * (2 * \u2191x) - 1))) }", "state_after": "case a.h.inl.inr.inl.inl.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u2191x \u2264 1 / 4\nh\u271d : 2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 False"}, {"tactic": "linarith", "state_before": "case a.h.inl.inr.inl.inl.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u2191x \u2264 1 / 4\nh\u271d : 2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exfalso", "state_before": "case a.h.inl.inr.inl.inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u2191x \u2264 1 / 4\nh\u271d : \u00ac2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 \u2191q { val := 2 * (2 * \u2191x) - 1, property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } - 1 \u2208 I) } =\n    \u2191r { val := 2 * (2 * (2 * \u2191x) - 1) - 1, property := (_ : (fun x => x \u2208 I) (2 * (2 * (2 * \u2191x) - 1) - 1)) }", "state_after": "case a.h.inl.inr.inl.inr.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u2191x \u2264 1 / 4\nh\u271d : \u00ac2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 False"}, {"tactic": "linarith", "state_before": "case a.h.inl.inr.inl.inr.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u2191x \u2264 1 / 4\nh\u271d : \u00ac2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exfalso", "state_before": "case a.h.inl.inr.inr.inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u00ac\u2191x \u2264 1 / 4\nh\u271d : \u2191x + 1 / 4 \u2264 1 / 2\n\u22a2 \u2191q { val := 2 * (2 * \u2191x) - 1, property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } - 1 \u2208 I) } =\n    \u2191p { val := 2 * (\u2191x + 1 / 4), property := (_ : (fun x => x \u2208 I) (2 * (\u2191x + 1 / 4))) }", "state_after": "case a.h.inl.inr.inr.inl.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u00ac\u2191x \u2264 1 / 4\nh\u271d : \u2191x + 1 / 4 \u2264 1 / 2\n\u22a2 False"}, {"tactic": "linarith", "state_before": "case a.h.inl.inr.inr.inl.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u00ac\u2191x \u2264 1 / 4\nh\u271d : \u2191x + 1 / 4 \u2264 1 / 2\n\u22a2 False", "state_after": "no goals"}, {"tactic": "have h : 2 * (2 * (x : \u211d)) - 1 = 2 * (2 * (\u2191x + 1 / 4) - 1) := by linarith", "state_before": "case a.h.inl.inr.inr.inr.inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u271d : 2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 \u2191q { val := 2 * (2 * \u2191x) - 1, property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } - 1 \u2208 I) } =\n    \u2191q { val := 2 * (2 * (\u2191x + 1 / 4) - 1), property := (_ : (fun x => x \u2208 I) (2 * (2 * (\u2191x + 1 / 4) - 1))) }", "state_after": "case a.h.inl.inr.inr.inr.inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u271d : 2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\nh : 2 * (2 * \u2191x) - 1 = 2 * (2 * (\u2191x + 1 / 4) - 1)\n\u22a2 \u2191q { val := 2 * (2 * \u2191x) - 1, property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } - 1 \u2208 I) } =\n    \u2191q { val := 2 * (2 * (\u2191x + 1 / 4) - 1), property := (_ : (fun x => x \u2208 I) (2 * (2 * (\u2191x + 1 / 4) - 1))) }"}, {"tactic": "simp [h\u2082, h\u2081, h, dif_neg (show \u00acFalse from id), dif_pos True.intro, if_false, if_true]", "state_before": "case a.h.inl.inr.inr.inr.inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u271d : 2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\nh : 2 * (2 * \u2191x) - 1 = 2 * (2 * (\u2191x + 1 / 4) - 1)\n\u22a2 \u2191q { val := 2 * (2 * \u2191x) - 1, property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } - 1 \u2208 I) } =\n    \u2191q { val := 2 * (2 * (\u2191x + 1 / 4) - 1), property := (_ : (fun x => x \u2208 I) (2 * (2 * (\u2191x + 1 / 4) - 1))) }", "state_after": "no goals"}, {"tactic": "linarith", "state_before": "X : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u271d : 2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 2 * (2 * \u2191x) - 1 = 2 * (2 * (\u2191x + 1 / 4) - 1)", "state_after": "no goals"}, {"tactic": "exfalso", "state_before": "case a.h.inl.inr.inr.inr.inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u271d : \u00ac2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 \u2191q { val := 2 * (2 * \u2191x) - 1, property := (_ : 2 * \u2191{ val := 2 * \u2191x, property := (_ : 2 * \u2191x \u2208 I) } - 1 \u2208 I) } =\n    \u2191r { val := 2 * (2 * (\u2191x + 1 / 4) - 1) - 1, property := (_ : (fun x => x \u2208 I) (2 * (2 * (\u2191x + 1 / 4) - 1) - 1)) }", "state_after": "case a.h.inl.inr.inr.inr.inr.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u271d : \u00ac2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 False"}, {"tactic": "linarith", "state_before": "case a.h.inl.inr.inr.inr.inr.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u2191x \u2264 1 / 2\nh\u2082 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac\u2191x + 1 / 4 \u2264 1 / 2\nh\u271d : \u00ac2 * (\u2191x + 1 / 4) - 1 \u2264 1 / 2\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exfalso", "state_before": "case a.h.inr.inl.inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u2191x \u2264 1 / 4\nh\u271d : 2 * \u2191x \u2264 1 / 2\n\u22a2 \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) } =\n    \u2191p { val := 2 * (2 * \u2191x), property := (_ : (fun x => x \u2208 I) (2 * (2 * \u2191x))) }", "state_after": "case a.h.inr.inl.inl.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u2191x \u2264 1 / 4\nh\u271d : 2 * \u2191x \u2264 1 / 2\n\u22a2 False"}, {"tactic": "linarith", "state_before": "case a.h.inr.inl.inl.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u2191x \u2264 1 / 4\nh\u271d : 2 * \u2191x \u2264 1 / 2\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exfalso", "state_before": "case a.h.inr.inl.inr.inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d : 2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) } =\n    \u2191q { val := 2 * (2 * (2 * \u2191x) - 1), property := (_ : (fun x => x \u2208 I) (2 * (2 * (2 * \u2191x) - 1))) }", "state_after": "case a.h.inr.inl.inr.inl.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d : 2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 False"}, {"tactic": "linarith", "state_before": "case a.h.inr.inl.inr.inl.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d : 2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exfalso", "state_before": "case a.h.inr.inl.inr.inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d : \u00ac2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) } =\n    \u2191r { val := 2 * (2 * (2 * \u2191x) - 1) - 1, property := (_ : (fun x => x \u2208 I) (2 * (2 * (2 * \u2191x) - 1) - 1)) }", "state_after": "case a.h.inr.inl.inr.inr.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d : \u00ac2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 False"}, {"tactic": "linarith", "state_before": "case a.h.inr.inl.inr.inr.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac2 * \u2191x \u2264 1 / 2\nh\u271d : \u00ac2 * (2 * \u2191x) - 1 \u2264 1 / 2\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exfalso", "state_before": "case a.h.inr.inr.inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u00ac\u2191x \u2264 1 / 4\nh\u271d : 1 / 2 * (\u2191x + 1) \u2264 1 / 2\n\u22a2 \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) } =\n    \u2191p { val := 2 * (1 / 2 * (\u2191x + 1)), property := (_ : (fun x => x \u2208 I) (2 * (1 / 2 * (\u2191x + 1)))) }", "state_after": "case a.h.inr.inr.inl.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u00ac\u2191x \u2264 1 / 4\nh\u271d : 1 / 2 * (\u2191x + 1) \u2264 1 / 2\n\u22a2 False"}, {"tactic": "linarith", "state_before": "case a.h.inr.inr.inl.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b9 : \u00ac\u2191x \u2264 1 / 4\nh\u271d : 1 / 2 * (\u2191x + 1) \u2264 1 / 2\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exfalso", "state_before": "case a.h.inr.inr.inr.inl\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac1 / 2 * (\u2191x + 1) \u2264 1 / 2\nh\u271d : 2 * (1 / 2 * (\u2191x + 1)) - 1 \u2264 1 / 2\n\u22a2 \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) } =\n    \u2191q\n      { val := 2 * (2 * (1 / 2 * (\u2191x + 1)) - 1), property := (_ : (fun x => x \u2208 I) (2 * (2 * (1 / 2 * (\u2191x + 1)) - 1))) }", "state_after": "case a.h.inr.inr.inr.inl.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac1 / 2 * (\u2191x + 1) \u2264 1 / 2\nh\u271d : 2 * (1 / 2 * (\u2191x + 1)) - 1 \u2264 1 / 2\n\u22a2 False"}, {"tactic": "linarith", "state_before": "case a.h.inr.inr.inr.inl.h\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac1 / 2 * (\u2191x + 1) \u2264 1 / 2\nh\u271d : 2 * (1 / 2 * (\u2191x + 1)) - 1 \u2264 1 / 2\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simp only [h\u2081, if_false, dif_neg (show \u00acFalse from id)]", "state_before": "case a.h.inr.inr.inr.inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac1 / 2 * (\u2191x + 1) \u2264 1 / 2\nh\u271d : \u00ac2 * (1 / 2 * (\u2191x + 1)) - 1 \u2264 1 / 2\n\u22a2 \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) } =\n    \u2191r\n      { val := 2 * (2 * (1 / 2 * (\u2191x + 1)) - 1) - 1,\n        property := (_ : (fun x => x \u2208 I) (2 * (2 * (1 / 2 * (\u2191x + 1)) - 1) - 1)) }", "state_after": "case a.h.inr.inr.inr.inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac1 / 2 * (\u2191x + 1) \u2264 1 / 2\nh\u271d : \u00ac2 * (1 / 2 * (\u2191x + 1)) - 1 \u2264 1 / 2\n\u22a2 \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) } =\n    \u2191r\n      { val := 2 * (2 * (1 / 2 * (\u2191x + 1)) - 1) - 1,\n        property := (_ : (fun x => x \u2208 I) (2 * (2 * (1 / 2 * (\u2191x + 1)) - 1) - 1)) }"}, {"tactic": "congr", "state_before": "case a.h.inr.inr.inr.inr\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac1 / 2 * (\u2191x + 1) \u2264 1 / 2\nh\u271d : \u00ac2 * (1 / 2 * (\u2191x + 1)) - 1 \u2264 1 / 2\n\u22a2 \u2191r { val := 2 * \u2191x - 1, property := (_ : 2 * \u2191x - 1 \u2208 I) } =\n    \u2191r\n      { val := 2 * (2 * (1 / 2 * (\u2191x + 1)) - 1) - 1,\n        property := (_ : (fun x => x \u2208 I) (2 * (2 * (1 / 2 * (\u2191x + 1)) - 1) - 1)) }", "state_after": "case a.h.inr.inr.inr.inr.h.e_6.h.e_val.e_a.e_a\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac1 / 2 * (\u2191x + 1) \u2264 1 / 2\nh\u271d : \u00ac2 * (1 / 2 * (\u2191x + 1)) - 1 \u2264 1 / 2\n\u22a2 \u2191x = 2 * (1 / 2 * (\u2191x + 1)) - 1"}, {"tactic": "ring", "state_before": "case a.h.inr.inr.inr.inr.h.e_6.h.e_val.e_a.e_a\nX : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080\u271d x\u2081\u271d x\u2080 x\u2081 x\u2082 x\u2083 : X\np : Path x\u2080 x\u2081\nq : Path x\u2081 x\u2082\nr : Path x\u2082 x\u2083\nx : \u2191I\nh\u2081 : \u00ac\u2191x \u2264 1 / 2\nh\u271d\u00b2 : \u00ac\u2191x \u2264 1 / 4\nh\u271d\u00b9 : \u00ac1 / 2 * (\u2191x + 1) \u2264 1 / 2\nh\u271d : \u00ac2 * (1 / 2 * (\u2191x + 1)) - 1 \u2264 1 / 2\n\u22a2 \u2191x = 2 * (1 / 2 * (\u2191x + 1)) - 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Topology.lean", "full_name": "Set.Finite.isCompact_convexHull", "start": [288, 1], "end": [293, 35], "traced_tactics": 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u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs : Set.Finite s\n\u22a2 IsCompact (\u2191(Finset.sum Finset.univ fun x => LinearMap.smulRight (LinearMap.proj x) \u2191x) '' stdSimplex \u211d \u2191s)", "state_after": "\u03b9 : Type ?u.158241\n\ud835\udd5c : Type ?u.158244\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs : Set.Finite s\n\u22a2 Continuous \u2191(Finset.sum Finset.univ fun x => LinearMap.smulRight (LinearMap.proj x) \u2191x)"}, {"tactic": "haveI := hs.fintype", "state_before": "\u03b9 : Type ?u.158241\n\ud835\udd5c : Type ?u.158244\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs : Set.Finite s\n\u22a2 Continuous \u2191(Finset.sum Finset.univ fun x => LinearMap.smulRight (LinearMap.proj x) \u2191x)", "state_after": "\u03b9 : Type ?u.158241\n\ud835\udd5c : Type ?u.158244\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs : Set.Finite s\nthis : Fintype \u2191s\n\u22a2 Continuous \u2191(Finset.sum Finset.univ fun x => LinearMap.smulRight (LinearMap.proj x) \u2191x)"}, {"tactic": "apply LinearMap.continuous_on_pi", "state_before": "\u03b9 : Type ?u.158241\n\ud835\udd5c : Type ?u.158244\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : TopologicalAddGroup E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nhs : Set.Finite s\nthis : Fintype \u2191s\n\u22a2 Continuous \u2191(Finset.sum Finset.univ fun x => LinearMap.smulRight (LinearMap.proj x) \u2191x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Polynomial/Splits.lean", "full_name": "Polynomial.roots_map", "start": [330, 1], "end": [333, 24], "traced_tactics": [{"tactic": "convert(natDegree_eq_card_roots hf).symm", "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]\nhf : Splits (RingHom.id K) f\n\u22a2 \u2191Multiset.card (roots f) = natDegree f", "state_after": "case h.e'_2.h.e'_6.h.e'_4\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]\nhf : Splits (RingHom.id K) f\n\u22a2 f = map (RingHom.id K) f"}, {"tactic": "rw [map_id]", "state_before": "case h.e'_2.h.e'_6.h.e'_4\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]\nhf : Splits (RingHom.id K) f\n\u22a2 f = map (RingHom.id K) f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean", "full_name": "CategoryTheory.Limits.parallelFamily_obj_one", "start": [146, 1], "end": [147, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "full_name": "circleIntegral.integral_sub", "start": [385, 1], "end": [388, 84], "traced_tactics": [{"tactic": "simp only [circleIntegral, smul_sub, intervalIntegral.integral_sub hf.out hg.out]", "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf g : \u2102 \u2192 E\nc : \u2102\nR : \u211d\nhf : CircleIntegrable f c R\nhg : CircleIntegrable g c R\n\u22a2 (\u222e (z : \u2102) in C(c, R), f z - g z) = (\u222e (z : \u2102) in C(c, R), f z) - \u222e (z : \u2102) in C(c, R), g z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "full_name": "CategoryTheory.Limits.limit.pre_\u03c0", "start": [416, 1], "end": [418, 6], "traced_tactics": [{"tactic": "erw [IsLimit.fac]", "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\ninst\u271d\u00b9 : HasLimit F\nE : K \u2964 J\ninst\u271d : HasLimit (E \u22d9 F)\nk : K\n\u22a2 pre F E \u226b \u03c0 (E \u22d9 F) k = \u03c0 F (E.obj k)", "state_after": "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\ninst\u271d\u00b9 : HasLimit F\nE : K \u2964 J\ninst\u271d : HasLimit (E \u22d9 F)\nk : K\n\u22a2 (Cone.whisker E (cone F)).\u03c0.app k = \u03c0 F (E.obj k)"}, {"tactic": "rfl", "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\ninst\u271d\u00b9 : HasLimit F\nE : K \u2964 J\ninst\u271d : HasLimit (E \u22d9 F)\nk : K\n\u22a2 (Cone.whisker E (cone F)).\u03c0.app k = \u03c0 F (E.obj k)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "full_name": "MeasureTheory.subset_toMeasurable", "start": [599, 1], "end": [601, 99], "traced_tactics": [{"tactic": "rw [toMeasurable_def]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.141314\n\u03b3 : Type ?u.141317\n\u03b4 : Type ?u.141320\n\u03b9 : Type ?u.141323\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\n\u22a2 s \u2286 toMeasurable \u03bc s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.141314\n\u03b3 : Type ?u.141317\n\u03b4 : Type ?u.141320\n\u03b9 : Type ?u.141323\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\n\u22a2 s \u2286\n    if h : \u2203 t x, MeasurableSet t \u2227 t =\u1d50[\u03bc] s then Exists.choose h\n    else\n      if h' : \u2203 t x, MeasurableSet t \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u2191\u2191\u03bc (t \u2229 u) = \u2191\u2191\u03bc (s \u2229 u) then Exists.choose h'\n      else Exists.choose (_ : \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191\u2191\u03bc t = \u2191\u2191\u03bc s)"}, {"tactic": "split_ifs with hs h's", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.141314\n\u03b3 : Type ?u.141317\n\u03b4 : Type ?u.141320\n\u03b9 : Type ?u.141323\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\n\u22a2 s \u2286\n    if h : \u2203 t x, MeasurableSet t \u2227 t =\u1d50[\u03bc] s then Exists.choose h\n    else\n      if h' : \u2203 t x, MeasurableSet t \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u2191\u2191\u03bc (t \u2229 u) = \u2191\u2191\u03bc (s \u2229 u) then Exists.choose h'\n      else Exists.choose (_ : \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191\u2191\u03bc t = \u2191\u2191\u03bc s)", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type ?u.141314\n\u03b3 : Type ?u.141317\n\u03b4 : Type ?u.141320\n\u03b9 : Type ?u.141323\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : \u2203 t x, MeasurableSet t \u2227 t =\u1d50[\u03bc] s\n\u22a2 s \u2286 Exists.choose hs\n\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type ?u.141314\n\u03b3 : Type ?u.141317\n\u03b4 : Type ?u.141320\n\u03b9 : Type ?u.141323\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : \u00ac\u2203 t x, MeasurableSet t \u2227 t =\u1d50[\u03bc] s\nh's : \u2203 t x, MeasurableSet t \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u2191\u2191\u03bc (t \u2229 u) = \u2191\u2191\u03bc (s \u2229 u)\n\u22a2 s \u2286 Exists.choose h's\n\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type ?u.141314\n\u03b3 : Type ?u.141317\n\u03b4 : Type ?u.141320\n\u03b9 : Type ?u.141323\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : \u00ac\u2203 t x, MeasurableSet t \u2227 t =\u1d50[\u03bc] s\nh's : \u00ac\u2203 t x, MeasurableSet t \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u2191\u2191\u03bc (t \u2229 u) = \u2191\u2191\u03bc (s \u2229 u)\n\u22a2 s \u2286 Exists.choose (_ : \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191\u2191\u03bc t = \u2191\u2191\u03bc s)"}, {"tactic": "exacts [hs.choose_spec.fst, h's.choose_spec.fst, (exists_measurable_superset \u03bc s).choose_spec.1]", "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type ?u.141314\n\u03b3 : Type ?u.141317\n\u03b4 : Type ?u.141320\n\u03b9 : Type ?u.141323\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : \u2203 t x, MeasurableSet t \u2227 t =\u1d50[\u03bc] s\n\u22a2 s \u2286 Exists.choose hs\n\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type ?u.141314\n\u03b3 : Type ?u.141317\n\u03b4 : Type ?u.141320\n\u03b9 : Type ?u.141323\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : \u00ac\u2203 t x, MeasurableSet t \u2227 t =\u1d50[\u03bc] s\nh's : \u2203 t x, MeasurableSet t \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u2191\u2191\u03bc (t \u2229 u) = \u2191\u2191\u03bc (s \u2229 u)\n\u22a2 s \u2286 Exists.choose h's\n\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type ?u.141314\n\u03b3 : Type ?u.141317\n\u03b4 : Type ?u.141320\n\u03b9 : Type ?u.141323\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\n\u03bc : Measure \u03b1\ns : Set \u03b1\nhs : \u00ac\u2203 t x, MeasurableSet t \u2227 t =\u1d50[\u03bc] s\nh's : \u00ac\u2203 t x, MeasurableSet t \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u2191\u2191\u03bc (t \u2229 u) = \u2191\u2191\u03bc (s \u2229 u)\n\u22a2 s \u2286 Exists.choose (_ : \u2203 t, s \u2286 t \u2227 MeasurableSet t \u2227 \u2191\u2191\u03bc t = \u2191\u2191\u03bc s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/MonoidAlgebra/Grading.lean", "full_name": "AddMonoidAlgebra.single_mem_gradeBy", "start": [88, 1], "end": [91, 66], "traced_tactics": [{"tactic": "intro x hx", "state_before": "M : Type u_2\n\u03b9 : Type u_3\nR\u271d : Type ?u.12927\ninst\u271d\u00b2 : DecidableEq M\ninst\u271d\u00b9 : CommSemiring R\u271d\nR : Type u_1\ninst\u271d : CommSemiring R\nf : M \u2192 \u03b9\nm : M\nr : R\n\u22a2 Finsupp.single m r 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u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 F\nf' : E \u2192L[\u211d] F\nx : E\nL : Filter E\n\u22a2 Tendsto (fun x' => \u2016x' - x\u2016\u207b\u00b9 * \u2016f x' - f x - \u2191f' (x' - x)\u2016) L (\ud835\udcdd 0) \u2194\n    Tendsto (fun x' => \u2016x' - x\u2016\u207b\u00b9 \u2022 (f x' - f x - \u2191f' (x' - x))) L (\ud835\udcdd 0)", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 F\nf' : E \u2192L[\u211d] F\nx : E\nL : Filter E\n\u22a2 Tendsto (fun x' => \u2016x' - x\u2016\u207b\u00b9 \u2022 (f x' - f x - \u2191f' (x' - x))) L (\ud835\udcdd 0) \u2194\n    Tendsto (fun x' => \u2016x' - x\u2016\u207b\u00b9 * \u2016f x' - f x - \u2191f' (x' - x)\u2016) L (\ud835\udcdd 0)"}, {"tactic": "rw [tendsto_iff_norm_tendsto_zero]", "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 F\nf' : E \u2192L[\u211d] F\nx : E\nL : Filter E\n\u22a2 Tendsto (fun x' => \u2016x' - x\u2016\u207b\u00b9 \u2022 (f x' - f x - \u2191f' (x' - x))) L (\ud835\udcdd 0) \u2194\n    Tendsto (fun x' => \u2016x' - x\u2016\u207b\u00b9 * \u2016f x' - f x - \u2191f' (x' - x)\u2016) L (\ud835\udcdd 0)", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 F\nf' : E \u2192L[\u211d] F\nx : E\nL : Filter E\n\u22a2 Tendsto (fun e => \u2016\u2016e - x\u2016\u207b\u00b9 \u2022 (f e - f x - \u2191f' (e - x)) - 0\u2016) L (\ud835\udcdd 0) \u2194\n    Tendsto (fun x' => \u2016x' - x\u2016\u207b\u00b9 * \u2016f x' - f x - \u2191f' (x' - x)\u2016) L (\ud835\udcdd 0)"}, {"tactic": "refine' tendsto_congr fun x' => _", "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 F\nf' : E \u2192L[\u211d] F\nx : E\nL : Filter E\n\u22a2 Tendsto (fun e => \u2016\u2016e - x\u2016\u207b\u00b9 \u2022 (f e - f x - \u2191f' (e - x)) - 0\u2016) L (\ud835\udcdd 0) \u2194\n    Tendsto (fun x' => \u2016x' - x\u2016\u207b\u00b9 * \u2016f x' - f x - \u2191f' (x' - x)\u2016) L (\ud835\udcdd 0)", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 F\nf' : E \u2192L[\u211d] F\nx : E\nL : Filter E\nx' : E\n\u22a2 \u2016\u2016x' - x\u2016\u207b\u00b9 \u2022 (f x' - f x - \u2191f' (x' - x)) - 0\u2016 = \u2016x' - x\u2016\u207b\u00b9 * \u2016f x' - f x - \u2191f' (x' - 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1], "end": [47, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "full_name": "Metric.continuous_infNndist_pt", "start": [678, 1], "end": [679, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.one_le_iff_ne_zero", "start": [1402, 1], "end": [1403, 39], "traced_tactics": [{"tactic": "rw [one_le_iff_pos, pos_iff_ne_zero]", "state_before": "\u03b1 \u03b2 : Type u\nc : Cardinal\n\u22a2 1 \u2264 c \u2194 c \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Game/PGame.lean", "full_name": "PGame.add_lf_add_of_le_of_lf", "start": [1767, 1], "end": [1768, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "full_name": "PiNat.inter_cylinder_longestPrefix_nonempty", "start": [545, 1], "end": [557, 10], "traced_tactics": [{"tactic": "by_cases hx : x \u2208 s", "state_before": "E : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\n\u22a2 Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))", "state_after": "case pos\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : x \u2208 s\n\u22a2 Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))\n\ncase neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\n\u22a2 Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))"}, {"tactic": "have A := exists_disjoint_cylinder hs hx", "state_before": "case neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\n\u22a2 Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))", "state_after": "case neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\n\u22a2 Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))"}, {"tactic": "have B : longestPrefix x s < shortestPrefixDiff x s :=\n  Nat.pred_lt (shortestPrefixDiff_pos hs hne hx).ne'", "state_before": "case neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\n\u22a2 Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))", "state_after": "case neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\nB : longestPrefix x s < shortestPrefixDiff x s\n\u22a2 Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))"}, {"tactic": "rw [longestPrefix, shortestPrefixDiff, dif_pos A] at B \u22a2", "state_before": "case neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\nB : longestPrefix x s < shortestPrefixDiff x s\n\u22a2 Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))", "state_after": "case neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\nB : Nat.find A - 1 < Nat.find A\n\u22a2 Set.Nonempty (s \u2229 cylinder x (Nat.find A - 1))"}, {"tactic": "obtain \u27e8y, ys, hy\u27e9 : \u2203 y : \u2200 n : \u2115, E n, y \u2208 s \u2227 x \u2208 cylinder y (Nat.find A - 1) := by\n  simpa only [not_disjoint_iff, mem_cylinder_comm] using Nat.find_min A B", "state_before": "case neg\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\nB : Nat.find A - 1 < Nat.find A\n\u22a2 Set.Nonempty (s \u2229 cylinder x (Nat.find A - 1))", "state_after": "case neg.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\nB : Nat.find A - 1 < Nat.find A\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : x \u2208 cylinder y (Nat.find A - 1)\n\u22a2 Set.Nonempty (s \u2229 cylinder x (Nat.find A - 1))"}, {"tactic": "refine' \u27e8y, ys, _\u27e9", "state_before": "case neg.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\nB : Nat.find A - 1 < Nat.find A\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : x \u2208 cylinder y (Nat.find A - 1)\n\u22a2 Set.Nonempty (s \u2229 cylinder x (Nat.find A - 1))", "state_after": "case neg.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\nB : Nat.find A - 1 < Nat.find A\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : x \u2208 cylinder y (Nat.find A - 1)\n\u22a2 y \u2208 cylinder x (Nat.find A - 1)"}, {"tactic": "rw [mem_cylinder_iff_eq] at hy \u22a2", "state_before": "case neg.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\nB : Nat.find A - 1 < Nat.find A\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : x \u2208 cylinder y (Nat.find A - 1)\n\u22a2 y \u2208 cylinder x (Nat.find A - 1)", "state_after": "case neg.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\nB : Nat.find A - 1 < Nat.find A\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : cylinder x (Nat.find A - 1) = cylinder y (Nat.find A - 1)\n\u22a2 cylinder y (Nat.find A - 1) = cylinder x (Nat.find A - 1)"}, {"tactic": "rw [hy]", "state_before": "case neg.intro.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\nB : Nat.find A - 1 < Nat.find A\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : cylinder x (Nat.find A - 1) = cylinder y (Nat.find A - 1)\n\u22a2 cylinder y (Nat.find A - 1) = cylinder x (Nat.find A - 1)", "state_after": "no goals"}, {"tactic": "exact \u27e8x, hx, self_mem_cylinder _ _\u27e9", "state_before": "case pos\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : x \u2208 s\n\u22a2 Set.Nonempty (s \u2229 cylinder x (longestPrefix x s))", "state_after": "no goals"}, {"tactic": "simpa only [not_disjoint_iff, mem_cylinder_comm] using Nat.find_min A B", "state_before": "E : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : Set.Nonempty s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nA : \u2203 n, Disjoint s (cylinder x n)\nB : Nat.find A - 1 < Nat.find A\n\u22a2 \u2203 y, y \u2208 s \u2227 x \u2208 cylinder y (Nat.find A - 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.erase_eq_of_not_mem", "start": [1897, 1], "end": [1898, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Sign.lean", "full_name": "SignType.pos_eq_one", "start": [66, 1], "end": [67, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/LocalEquiv.lean", "full_name": "LocalEquiv.prod_symm", "start": [962, 1], "end": [964, 33], "traced_tactics": [{"tactic": "ext x <;> simp [prod_coe_symm]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ne\u271d : LocalEquiv \u03b1 \u03b2\ne'\u271d : LocalEquiv \u03b2 \u03b3\ne : LocalEquiv \u03b1 \u03b2\ne' : LocalEquiv \u03b3 \u03b4\n\u22a2 LocalEquiv.symm (prod e e') = prod (LocalEquiv.symm e) (LocalEquiv.symm e')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/ENNReal.lean", "full_name": "ENNReal.toNNReal_nat", "start": [699, 1], "end": [700, 49], "traced_tactics": [{"tactic": "rw [\u2190 ENNReal.coe_nat n, ENNReal.toNNReal_coe]", "state_before": "\u03b1 : Type ?u.103128\n\u03b2 : Type ?u.103131\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nn : \u2115\n\u22a2 ENNReal.toNNReal \u2191n = \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "full_name": "PowerSeries.zero_inv", "start": [2140, 1], "end": [2141, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "full_name": "Finsupp.sum_sub", "start": [366, 1], "end": [368, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Real/ConjugateExponents.lean", "full_name": "Real.IsConjugateExponent.inv_add_inv_conj_nnreal", "start": [109, 1], "end": [111, 86], "traced_tactics": [{"tactic": "rw [\u2190 Real.toNNReal_one, \u2190 Real.toNNReal_div' h.nonneg, \u2190 Real.toNNReal_div' h.symm.nonneg,\n    \u2190 Real.toNNReal_add h.one_div_nonneg h.symm.one_div_nonneg, h.inv_add_inv_conj]", "state_before": "p q : \u211d\nh : IsConjugateExponent p q\n\u22a2 1 / toNNReal p + 1 / toNNReal q = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Hom/CompleteLattice.lean", "full_name": "CompleteLatticeHom.toFun_eq_coe_aux", "start": [693, 1], "end": [694, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "TopologicalGroup.of_nhds_one", "start": [937, 1], "end": [947, 32], "traced_tactics": [{"tactic": "refine' TopologicalGroup.of_nhds_one' hmul hinv hleft fun x\u2080 => _", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nG\u271d : Type w\nH : Type x\ninst\u271d\u2075 : TopologicalSpace G\u271d\ninst\u271d\u2074 : Group G\u271d\ninst\u271d\u00b3 : TopologicalGroup G\u271d\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf : \u03b1 \u2192 G\u271d\ns : Set \u03b1\nx : \u03b1\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalSpace G\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhinv : Tendsto (fun x => x\u207b\u00b9) (\ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : G), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhconj : \u2200 (x\u2080 : G), Tendsto (fun x => x\u2080 * x * x\u2080\u207b\u00b9) (\ud835\udcdd 1) (\ud835\udcdd 1)\n\u22a2 TopologicalGroup G", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nG\u271d : Type w\nH : Type x\ninst\u271d\u2075 : TopologicalSpace G\u271d\ninst\u271d\u2074 : Group G\u271d\ninst\u271d\u00b3 : TopologicalGroup G\u271d\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf : \u03b1 \u2192 G\u271d\ns : Set \u03b1\nx : \u03b1\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalSpace G\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhinv : Tendsto (fun x => x\u207b\u00b9) (\ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : G), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhconj : \u2200 (x\u2080 : G), Tendsto (fun x => x\u2080 * x * x\u2080\u207b\u00b9) (\ud835\udcdd 1) (\ud835\udcdd 1)\nx\u2080 : G\n\u22a2 \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1)"}, {"tactic": "replace hconj : \u2200 x\u2080 : G, map (fun x => x\u2080 * x * x\u2080\u207b\u00b9) (\ud835\udcdd 1) = \ud835\udcdd 1", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nG\u271d : Type w\nH : Type x\ninst\u271d\u2075 : TopologicalSpace G\u271d\ninst\u271d\u2074 : Group G\u271d\ninst\u271d\u00b3 : TopologicalGroup G\u271d\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf : \u03b1 \u2192 G\u271d\ns : Set \u03b1\nx : \u03b1\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalSpace G\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhinv : Tendsto (fun x => x\u207b\u00b9) (\ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : G), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhconj : \u2200 (x\u2080 : G), Tendsto (fun x => x\u2080 * x * x\u2080\u207b\u00b9) (\ud835\udcdd 1) (\ud835\udcdd 1)\nx\u2080 : G\n\u22a2 \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1)", "state_after": "case hconj\n\u03b1 : Type u\n\u03b2 : Type v\nG\u271d : Type w\nH : Type x\ninst\u271d\u2075 : TopologicalSpace G\u271d\ninst\u271d\u2074 : Group G\u271d\ninst\u271d\u00b3 : TopologicalGroup G\u271d\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf : \u03b1 \u2192 G\u271d\ns : Set \u03b1\nx : \u03b1\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalSpace G\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhinv : Tendsto (fun x => x\u207b\u00b9) (\ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : G), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhconj : \u2200 (x\u2080 : G), Tendsto (fun x => x\u2080 * x * x\u2080\u207b\u00b9) (\ud835\udcdd 1) (\ud835\udcdd 1)\nx\u2080 : G\n\u22a2 \u2200 (x\u2080 : G), map (fun x => x\u2080 * x * x\u2080\u207b\u00b9) (\ud835\udcdd 1) = \ud835\udcdd 1\n\n\u03b1 : Type u\n\u03b2 : Type v\nG\u271d : Type w\nH : Type x\ninst\u271d\u2075 : TopologicalSpace G\u271d\ninst\u271d\u2074 : Group G\u271d\ninst\u271d\u00b3 : TopologicalGroup G\u271d\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf : \u03b1 \u2192 G\u271d\ns : Set \u03b1\nx : \u03b1\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalSpace G\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhinv : Tendsto (fun x => x\u207b\u00b9) (\ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : G), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nx\u2080 : G\nhconj : \u2200 (x\u2080 : G), map (fun x => x\u2080 * x * x\u2080\u207b\u00b9) (\ud835\udcdd 1) = \ud835\udcdd 1\n\u22a2 \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1)"}, {"tactic": "rw [\u2190 hconj x\u2080]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nG\u271d : Type w\nH : Type x\ninst\u271d\u2075 : TopologicalSpace G\u271d\ninst\u271d\u2074 : Group G\u271d\ninst\u271d\u00b3 : TopologicalGroup G\u271d\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf : \u03b1 \u2192 G\u271d\ns : Set \u03b1\nx : \u03b1\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalSpace G\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhinv : Tendsto (fun x => x\u207b\u00b9) (\ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : G), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nx\u2080 : G\nhconj : \u2200 (x\u2080 : G), map (fun x => x\u2080 * x * x\u2080\u207b\u00b9) (\ud835\udcdd 1) = \ud835\udcdd 1\n\u22a2 \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nG\u271d : Type w\nH : Type x\ninst\u271d\u2075 : TopologicalSpace G\u271d\ninst\u271d\u2074 : Group G\u271d\ninst\u271d\u00b3 : TopologicalGroup G\u271d\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf : \u03b1 \u2192 G\u271d\ns : Set \u03b1\nx : \u03b1\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalSpace G\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhinv : Tendsto (fun x => x\u207b\u00b9) (\ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : G), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nx\u2080 : G\nhconj : \u2200 (x\u2080 : G), map (fun x => x\u2080 * x * x\u2080\u207b\u00b9) (\ud835\udcdd 1) = \ud835\udcdd 1\n\u22a2 \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (map (fun x => x\u2080 * x * x\u2080\u207b\u00b9) (\ud835\udcdd 1))"}, {"tactic": "exact fun x\u2080 =>\n  map_eq_of_inverse (fun x => x\u2080\u207b\u00b9 * x * x\u2080\u207b\u00b9\u207b\u00b9) (by ext; simp [mul_assoc]) (hconj _) (hconj _)", "state_before": "case hconj\n\u03b1 : Type u\n\u03b2 : Type v\nG\u271d : Type w\nH : Type x\ninst\u271d\u2075 : TopologicalSpace G\u271d\ninst\u271d\u2074 : Group G\u271d\ninst\u271d\u00b3 : TopologicalGroup G\u271d\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf : \u03b1 \u2192 G\u271d\ns : Set \u03b1\nx : \u03b1\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalSpace G\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhinv : Tendsto (fun x => x\u207b\u00b9) (\ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : G), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhconj : \u2200 (x\u2080 : G), Tendsto (fun x => x\u2080 * x * x\u2080\u207b\u00b9) (\ud835\udcdd 1) (\ud835\udcdd 1)\nx\u2080 : G\n\u22a2 \u2200 (x\u2080 : G), map (fun x => x\u2080 * x * x\u2080\u207b\u00b9) (\ud835\udcdd 1) = \ud835\udcdd 1", "state_after": "no goals"}, {"tactic": "ext", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nG\u271d : Type w\nH : Type x\ninst\u271d\u2075 : TopologicalSpace G\u271d\ninst\u271d\u2074 : Group G\u271d\ninst\u271d\u00b3 : TopologicalGroup G\u271d\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf : \u03b1 \u2192 G\u271d\ns : Set \u03b1\nx : \u03b1\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalSpace G\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhinv : Tendsto (fun x => x\u207b\u00b9) (\ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : G), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhconj : \u2200 (x\u2080 : G), Tendsto (fun x => x\u2080 * x * x\u2080\u207b\u00b9) (\ud835\udcdd 1) (\ud835\udcdd 1)\nx\u2080\u271d x\u2080 : G\n\u22a2 ((fun x => x\u2080 * x * x\u2080\u207b\u00b9) \u2218 fun x => x\u2080\u207b\u00b9 * x * x\u2080\u207b\u00b9\u207b\u00b9) = id", "state_after": "case h\n\u03b1 : Type u\n\u03b2 : Type v\nG\u271d : Type w\nH : Type x\ninst\u271d\u2075 : TopologicalSpace G\u271d\ninst\u271d\u2074 : Group G\u271d\ninst\u271d\u00b3 : TopologicalGroup G\u271d\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf : \u03b1 \u2192 G\u271d\ns : Set \u03b1\nx : \u03b1\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalSpace G\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhinv : Tendsto (fun x => x\u207b\u00b9) (\ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : G), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhconj : \u2200 (x\u2080 : G), Tendsto (fun x => x\u2080 * x * x\u2080\u207b\u00b9) (\ud835\udcdd 1) (\ud835\udcdd 1)\nx\u2080\u271d x\u2080 x\u271d : G\n\u22a2 ((fun x => x\u2080 * x * x\u2080\u207b\u00b9) \u2218 fun x => x\u2080\u207b\u00b9 * x * x\u2080\u207b\u00b9\u207b\u00b9) x\u271d = id x\u271d"}, {"tactic": "simp [mul_assoc]", "state_before": "case h\n\u03b1 : Type u\n\u03b2 : Type v\nG\u271d : Type w\nH : Type x\ninst\u271d\u2075 : TopologicalSpace G\u271d\ninst\u271d\u2074 : Group G\u271d\ninst\u271d\u00b3 : TopologicalGroup G\u271d\ninst\u271d\u00b2 : TopologicalSpace \u03b1\nf : \u03b1 \u2192 G\u271d\ns : Set \u03b1\nx : \u03b1\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : TopologicalSpace G\nhmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nhinv : Tendsto (fun x => x\u207b\u00b9) (\ud835\udcdd 1) (\ud835\udcdd 1)\nhleft : \u2200 (x\u2080 : G), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1)\nhconj : \u2200 (x\u2080 : G), Tendsto (fun x => x\u2080 * x * x\u2080\u207b\u00b9) (\ud835\udcdd 1) (\ud835\udcdd 1)\nx\u2080\u271d x\u2080 x\u271d : G\n\u22a2 ((fun x => x\u2080 * x * x\u2080\u207b\u00b9) \u2218 fun x => x\u2080\u207b\u00b9 * x * x\u2080\u207b\u00b9\u207b\u00b9) x\u271d = id x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean", "full_name": "Ideal.IsHomogeneous.mul", "start": [455, 1], "end": [460, 88], "traced_tactics": [{"tactic": "rw [Ideal.IsHomogeneous.iff_exists] at HI HJ\u22a2", "state_before": "\u03b9 : Type u_2\n\u03c3 : Type u_3\nR : Type ?u.185047\nA : Type u_1\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI\u271d I J : Ideal A\nHI : IsHomogeneous \ud835\udc9c I\nHJ : IsHomogeneous \ud835\udc9c J\n\u22a2 IsHomogeneous \ud835\udc9c (I * J)", "state_after": "\u03b9 : Type u_2\n\u03c3 : Type u_3\nR : Type ?u.185047\nA : Type u_1\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI\u271d I J : Ideal A\nHI : \u2203 S, I = span (Subtype.val '' S)\nHJ : \u2203 S, J = span (Subtype.val '' S)\n\u22a2 \u2203 S, I * J = span (Subtype.val '' S)"}, {"tactic": "obtain \u27e8\u27e8s\u2081, rfl\u27e9, \u27e8s\u2082, rfl\u27e9\u27e9 := HI, HJ", "state_before": "\u03b9 : Type u_2\n\u03c3 : Type u_3\nR : Type ?u.185047\nA : Type u_1\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI\u271d I J : Ideal A\nHI : \u2203 S, I = span (Subtype.val '' S)\nHJ : \u2203 S, J = span (Subtype.val '' S)\n\u22a2 \u2203 S, I * J = span (Subtype.val '' S)", "state_after": "case intro.intro\n\u03b9 : Type u_2\n\u03c3 : Type u_3\nR : Type ?u.185047\nA : Type u_1\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ns\u2081 s\u2082 : Set { x // x \u2208 homogeneousSubmonoid \ud835\udc9c }\n\u22a2 \u2203 S, span (Subtype.val '' s\u2081) * span (Subtype.val '' s\u2082) = span (Subtype.val '' S)"}, {"tactic": "rw [Ideal.span_mul_span']", "state_before": "case intro.intro\n\u03b9 : Type u_2\n\u03c3 : Type u_3\nR : Type ?u.185047\nA : Type u_1\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ns\u2081 s\u2082 : Set { x // x \u2208 homogeneousSubmonoid \ud835\udc9c }\n\u22a2 \u2203 S, span (Subtype.val '' s\u2081) * span (Subtype.val '' s\u2082) = span (Subtype.val '' S)", "state_after": "case intro.intro\n\u03b9 : Type u_2\n\u03c3 : Type u_3\nR : Type ?u.185047\nA : Type u_1\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ns\u2081 s\u2082 : Set { x // x \u2208 homogeneousSubmonoid \ud835\udc9c }\n\u22a2 \u2203 S, span (Subtype.val '' s\u2081 * Subtype.val '' s\u2082) = span (Subtype.val '' S)"}, {"tactic": "exact \u27e8s\u2081 * s\u2082, congr_arg _ <| (Set.image_mul (homogeneousSubmonoid \ud835\udc9c).subtype).symm\u27e9", "state_before": "case intro.intro\n\u03b9 : Type u_2\n\u03c3 : Type u_3\nR : Type ?u.185047\nA : Type u_1\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : DecidableEq \u03b9\ninst\u271d\u00b3 : AddMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\ns\u2081 s\u2082 : Set { x // x \u2208 homogeneousSubmonoid \ud835\udc9c }\n\u22a2 \u2203 S, span (Subtype.val '' s\u2081 * Subtype.val '' s\u2082) = span (Subtype.val '' S)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Order/MonotoneContinuity.lean", "full_name": "StrictMonoOn.continuousWithinAt_right_of_exists_between", "start": [46, 1], "end": [58, 57], "traced_tactics": [{"tactic": "have ha : a \u2208 Ici a := left_mem_Ici", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[Ici a] a\nhfs : \u2200 (b : \u03b2), b > f a \u2192 \u2203 c, c \u2208 s \u2227 f c \u2208 Ioc (f a) b\n\u22a2 ContinuousWithinAt f (Ici a) a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[Ici a] a\nhfs : \u2200 (b : \u03b2), b > f a \u2192 \u2203 c, c \u2208 s \u2227 f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\n\u22a2 ContinuousWithinAt f (Ici a) a"}, {"tactic": "have has : a \u2208 s := mem_of_mem_nhdsWithin ha hs", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[Ici a] a\nhfs : \u2200 (b : \u03b2), b > f a \u2192 \u2203 c, c \u2208 s \u2227 f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\n\u22a2 ContinuousWithinAt f (Ici a) a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[Ici a] a\nhfs : \u2200 (b : \u03b2), b > f a \u2192 \u2203 c, c \u2208 s \u2227 f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\n\u22a2 ContinuousWithinAt f (Ici a) a"}, {"tactic": "refine' tendsto_order.2 \u27e8fun b hb => _, fun b hb => _\u27e9", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[Ici a] a\nhfs : \u2200 (b : \u03b2), b > f a \u2192 \u2203 c, c \u2208 s \u2227 f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\n\u22a2 ContinuousWithinAt f (Ici a) a", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[Ici a] a\nhfs : \u2200 (b : \u03b2), b > f a \u2192 \u2203 c, c \u2208 s \u2227 f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b < f a\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[Ici a] a, b < f b_1\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[Ici a] a\nhfs : \u2200 (b : \u03b2), b > f a \u2192 \u2203 c, c \u2208 s \u2227 f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[Ici a] a, f b_1 < b"}, {"tactic": "filter_upwards [hs, @self_mem_nhdsWithin _ _ a (Ici a)] with _ hxs hxa using hb.trans_le\n  ((h_mono.le_iff_le has hxs).2 hxa)", "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[Ici a] a\nhfs : \u2200 (b : \u03b2), b > f a \u2192 \u2203 c, c \u2208 s \u2227 f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b < f a\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[Ici a] a, b < f b_1", "state_after": "no goals"}, {"tactic": "rcases hfs b hb with \u27e8c, hcs, hac, hcb\u27e9", "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[Ici a] a\nhfs : \u2200 (b : \u03b2), b > f a \u2192 \u2203 c, c \u2208 s \u2227 f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[Ici a] a, f b_1 < b", "state_after": "case refine'_2.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[Ici a] a\nhfs : \u2200 (b : \u03b2), b > f a \u2192 \u2203 c, c \u2208 s \u2227 f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : f a < f c\nhcb : f c \u2264 b\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[Ici a] a, f b_1 < b"}, {"tactic": "rw [h_mono.lt_iff_lt has hcs] at hac", "state_before": "case refine'_2.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[Ici a] a\nhfs : \u2200 (b : \u03b2), b > f a \u2192 \u2203 c, c \u2208 s \u2227 f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : f a < f c\nhcb : f c \u2264 b\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[Ici a] a, f b_1 < b", "state_after": "case refine'_2.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[Ici a] a\nhfs : \u2200 (b : \u03b2), b > f a \u2192 \u2203 c, c \u2208 s \u2227 f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : a < c\nhcb : f c \u2264 b\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[Ici a] a, f b_1 < b"}, {"tactic": "filter_upwards [hs, Ico_mem_nhdsWithin_Ici (left_mem_Ico.2 hac)]", "state_before": "case refine'_2.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[Ici a] a\nhfs : \u2200 (b : \u03b2), b > f a \u2192 \u2203 c, c \u2208 s \u2227 f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : a < c\nhcb : f c \u2264 b\n\u22a2 \u2200\u1da0 (b_1 : \u03b1) in \ud835\udcdd[Ici a] a, f b_1 < b", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[Ici a] a\nhfs : \u2200 (b : \u03b2), b > f a \u2192 \u2203 c, c \u2208 s \u2227 f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : a < c\nhcb : f c \u2264 b\n\u22a2 \u2200 (a_1 : \u03b1), a_1 \u2208 s \u2192 a_1 \u2208 Ico a c \u2192 f a_1 < b"}, {"tactic": "rintro x hx \u27e8_, hxc\u27e9", "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[Ici a] a\nhfs : \u2200 (b : \u03b2), b > f a \u2192 \u2203 c, c \u2208 s \u2227 f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : a < c\nhcb : f c \u2264 b\n\u22a2 \u2200 (a_1 : \u03b1), a_1 \u2208 s \u2192 a_1 \u2208 Ico a c \u2192 f a_1 < b", "state_after": "case h.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[Ici a] a\nhfs : \u2200 (b : \u03b2), b > f a \u2192 \u2203 c, c \u2208 s \u2227 f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : a < c\nhcb : f c \u2264 b\nx : \u03b1\nhx : x \u2208 s\nleft\u271d : a \u2264 x\nhxc : x < c\n\u22a2 f x < b"}, {"tactic": "exact ((h_mono.lt_iff_lt hx hcs).2 hxc).trans_le hcb", "state_before": "case h.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : LinearOrder \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : OrderTopology \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : OrderTopology \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nh_mono : StrictMonoOn f s\nhs : s \u2208 \ud835\udcdd[Ici a] a\nhfs : \u2200 (b : \u03b2), b > f a \u2192 \u2203 c, c \u2208 s \u2227 f c \u2208 Ioc (f a) b\nha : a \u2208 Ici a\nhas : a \u2208 s\nb : \u03b2\nhb : b > f a\nc : \u03b1\nhcs : c \u2208 s\nhac : a < c\nhcb : f c \u2264 b\nx : \u03b1\nhx : x \u2208 s\nleft\u271d : a \u2264 x\nhxc : x < c\n\u22a2 f x < b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/MvPolynomial/Rename.lean", "full_name": "MvPolynomial.coeff_rename_mapDomain", "start": [296, 1], "end": [305, 45], "traced_tactics": [{"tactic": "apply \u03c6.induction_on' (P := fun \u03c8 => coeff (Finsupp.mapDomain f d) ((rename f) \u03c8) = coeff d \u03c8)", "state_before": "\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03b1 : Type ?u.911003\nR : Type u_3\nS : Type ?u.911009\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : \u03c3 \u2192 \u03c4\nhf : Injective f\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n\u22a2 coeff (Finsupp.mapDomain f d) (\u2191(rename f) \u03c6) = coeff d \u03c6", "state_after": "case h1\n\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03b1 : Type ?u.911003\nR : Type u_3\nS : Type ?u.911009\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : \u03c3 \u2192 \u03c4\nhf : Injective f\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n\u22a2 \u2200 (u : \u03c3 \u2192\u2080 \u2115) (a : R), coeff (Finsupp.mapDomain f d) (\u2191(rename f) (\u2191(monomial u) a)) = coeff d (\u2191(monomial u) a)\n\ncase h2\n\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03b1 : Type ?u.911003\nR : Type u_3\nS : Type ?u.911009\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : \u03c3 \u2192 \u03c4\nhf : Injective f\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n\u22a2 \u2200 (p q : MvPolynomial \u03c3 R),\n    coeff (Finsupp.mapDomain f d) (\u2191(rename f) p) = coeff d p \u2192\n      coeff (Finsupp.mapDomain f d) (\u2191(rename f) q) = coeff d q \u2192\n        coeff (Finsupp.mapDomain f d) (\u2191(rename f) (p + q)) = coeff d (p + q)"}, {"tactic": "intro u r", "state_before": "case h1\n\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03b1 : Type ?u.911003\nR : Type u_3\nS : Type ?u.911009\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : \u03c3 \u2192 \u03c4\nhf : Injective f\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n\u22a2 \u2200 (u : \u03c3 \u2192\u2080 \u2115) (a : R), coeff (Finsupp.mapDomain f d) (\u2191(rename f) (\u2191(monomial u) a)) = coeff d (\u2191(monomial u) a)", "state_after": "case h1\n\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03b1 : Type ?u.911003\nR : Type u_3\nS : Type ?u.911009\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : \u03c3 \u2192 \u03c4\nhf : Injective f\n\u03c6 : MvPolynomial \u03c3 R\nd u : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 coeff (Finsupp.mapDomain f d) (\u2191(rename f) (\u2191(monomial u) r)) = coeff d (\u2191(monomial u) r)"}, {"tactic": "rw [rename_monomial, coeff_monomial, coeff_monomial]", "state_before": "case h1\n\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03b1 : Type ?u.911003\nR : Type u_3\nS : Type ?u.911009\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : \u03c3 \u2192 \u03c4\nhf : Injective f\n\u03c6 : MvPolynomial \u03c3 R\nd u : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 coeff (Finsupp.mapDomain f d) (\u2191(rename f) (\u2191(monomial u) r)) = coeff d (\u2191(monomial u) r)", "state_after": "case h1\n\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03b1 : Type ?u.911003\nR : Type u_3\nS : Type ?u.911009\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : \u03c3 \u2192 \u03c4\nhf : Injective f\n\u03c6 : MvPolynomial \u03c3 R\nd u : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 (if Finsupp.mapDomain f u = Finsupp.mapDomain f d then r else 0) = if u = d then r else 0"}, {"tactic": "simp only [(Finsupp.mapDomain_injective hf).eq_iff]", "state_before": "case h1\n\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03b1 : Type ?u.911003\nR : Type u_3\nS : Type ?u.911009\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : \u03c3 \u2192 \u03c4\nhf : Injective f\n\u03c6 : MvPolynomial \u03c3 R\nd u : \u03c3 \u2192\u2080 \u2115\nr : R\n\u22a2 (if Finsupp.mapDomain f u = Finsupp.mapDomain f d then r else 0) = if u = d then r else 0", "state_after": "no goals"}, {"tactic": "intros", "state_before": "case h2\n\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03b1 : Type ?u.911003\nR : Type u_3\nS : Type ?u.911009\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : \u03c3 \u2192 \u03c4\nhf : Injective f\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\n\u22a2 \u2200 (p q : MvPolynomial \u03c3 R),\n    coeff (Finsupp.mapDomain f d) (\u2191(rename f) p) = coeff d p \u2192\n      coeff (Finsupp.mapDomain f d) (\u2191(rename f) q) = coeff d q \u2192\n        coeff (Finsupp.mapDomain f d) (\u2191(rename f) (p + q)) = coeff d (p + q)", "state_after": "case h2\n\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03b1 : Type ?u.911003\nR : Type u_3\nS : Type ?u.911009\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : \u03c3 \u2192 \u03c4\nhf : Injective f\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\np\u271d q\u271d : MvPolynomial \u03c3 R\na\u271d\u00b9 : coeff (Finsupp.mapDomain f d) (\u2191(rename f) p\u271d) = coeff d p\u271d\na\u271d : coeff (Finsupp.mapDomain f d) (\u2191(rename f) q\u271d) = coeff d q\u271d\n\u22a2 coeff (Finsupp.mapDomain f d) (\u2191(rename f) (p\u271d + q\u271d)) = coeff d (p\u271d + q\u271d)"}, {"tactic": "simp only [*, AlgHom.map_add, coeff_add]", "state_before": "case h2\n\u03c3 : Type u_1\n\u03c4 : Type u_2\n\u03b1 : Type ?u.911003\nR : Type u_3\nS : Type ?u.911009\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : \u03c3 \u2192 \u03c4\nhf : Injective f\n\u03c6 : MvPolynomial \u03c3 R\nd : \u03c3 \u2192\u2080 \u2115\np\u271d q\u271d : MvPolynomial \u03c3 R\na\u271d\u00b9 : coeff (Finsupp.mapDomain f d) (\u2191(rename f) p\u271d) = coeff d p\u271d\na\u271d : coeff (Finsupp.mapDomain f d) (\u2191(rename f) q\u271d) = coeff d q\u271d\n\u22a2 coeff (Finsupp.mapDomain f d) (\u2191(rename f) (p\u271d + q\u271d)) = coeff d (p\u271d + q\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Equitable.lean", "full_name": "Finset.equitableOn_iff", "start": [127, 1], "end": [130, 72], "traced_tactics": [{"tactic": "simp_rw [equitableOn_iff_le_le_add_one, Nat.le_and_le_add_one_iff]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.11967\ns : Finset \u03b1\nf : \u03b1 \u2192 \u2115\na : \u03b1\n\u22a2 EquitableOn (\u2191s) f \u2194 \u2200 (a : \u03b1), a \u2208 s \u2192 f a = (\u2211 i in s, f i) / card s \u2228 f a = (\u2211 i in s, f i) / card s + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "indicator_ae_eq_restrict", "start": [4663, 1], "end": [4664, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "full_name": "Equiv.tsum_eq", "start": [568, 1], "end": [569, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Pi.lean", "full_name": "Filter.tendsto_eval_pi", "start": [48, 1], "end": [49, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean", "full_name": "Asymptotics.superpolynomialDecay_iff_isBigO", "start": [335, 1], "end": [351, 8], "traced_tactics": [{"tactic": "refine' (superpolynomialDecay_iff_zpow_tendsto_zero f hk).trans _", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\n\u22a2 SuperpolynomialDecay l k f \u2194 \u2200 (z : \u2124), f =O[l] fun a => k a ^ z", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\n\u22a2 (\u2200 (z : \u2124), Tendsto (fun a => k a ^ z * f a) l (\ud835\udcdd 0)) \u2194 \u2200 (z : \u2124), f =O[l] fun a => k a ^ z"}, {"tactic": "have hk0 : \u2200\u1da0 x in l, k x \u2260 0 := hk.eventually_ne_atTop 0", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\n\u22a2 (\u2200 (z : \u2124), Tendsto (fun a => k a ^ z * f a) l (\ud835\udcdd 0)) \u2194 \u2200 (z : \u2124), f =O[l] fun a => k a ^ z", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\n\u22a2 (\u2200 (z : \u2124), Tendsto (fun a => k a ^ z * f a) l (\ud835\udcdd 0)) \u2194 \u2200 (z : \u2124), f =O[l] fun a => k a ^ z"}, {"tactic": "refine' \u27e8fun h z => _, fun h z => _\u27e9", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\n\u22a2 (\u2200 (z : \u2124), Tendsto (fun a => k a ^ z * f a) l (\ud835\udcdd 0)) \u2194 \u2200 (z : \u2124), f =O[l] fun a => k a ^ z", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\nh : \u2200 (z : \u2124), Tendsto (fun a => k a ^ z * f a) l (\ud835\udcdd 0)\nz : \u2124\n\u22a2 f =O[l] fun a => k a ^ z\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\nh : \u2200 (z : \u2124), f =O[l] fun a => k a ^ z\nz : \u2124\n\u22a2 Tendsto (fun a => k a ^ z * f a) l (\ud835\udcdd 0)"}, {"tactic": "refine' isBigO_of_div_tendsto_nhds (hk0.mono fun x hx hxz => absurd (zpow_eq_zero hxz) hx) 0 _", "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\nh : \u2200 (z : \u2124), Tendsto (fun a => k a ^ z * f a) l (\ud835\udcdd 0)\nz : \u2124\n\u22a2 f =O[l] fun a => k a ^ z", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\nh : \u2200 (z : \u2124), Tendsto (fun a => k a ^ z * f a) l (\ud835\udcdd 0)\nz : \u2124\n\u22a2 Tendsto (f / fun a => k a ^ z) l (\ud835\udcdd 0)"}, {"tactic": "have : (fun a : \u03b1 => k a ^ z)\u207b\u00b9 = fun a : \u03b1 => k a ^ (-z) := funext fun x => by simp", "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\nh : \u2200 (z : \u2124), Tendsto (fun a => k a ^ z * f a) l (\ud835\udcdd 0)\nz : \u2124\n\u22a2 Tendsto (f / fun a => k a ^ z) l (\ud835\udcdd 0)", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\nh : \u2200 (z : \u2124), Tendsto (fun a => k a ^ z * f a) l (\ud835\udcdd 0)\nz : \u2124\nthis : (fun a => k a ^ z)\u207b\u00b9 = fun a => k a ^ (-z)\n\u22a2 Tendsto (f / fun a => k a ^ z) l (\ud835\udcdd 0)"}, {"tactic": "rw [div_eq_mul_inv, mul_comm f, this]", "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\nh : \u2200 (z : \u2124), Tendsto (fun a => k a ^ z * f a) l (\ud835\udcdd 0)\nz : \u2124\nthis : (fun a => k a ^ z)\u207b\u00b9 = fun a => k a ^ (-z)\n\u22a2 Tendsto (f / fun a => k a ^ z) l (\ud835\udcdd 0)", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\nh : \u2200 (z : \u2124), Tendsto (fun a => k a ^ z * f a) l (\ud835\udcdd 0)\nz : \u2124\nthis : (fun a => k a ^ z)\u207b\u00b9 = fun a => k a ^ (-z)\n\u22a2 Tendsto ((fun a => k a ^ (-z)) * f) l (\ud835\udcdd 0)"}, {"tactic": "exact h (-z)", "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\nh : \u2200 (z : \u2124), Tendsto (fun a => k a ^ z * f a) l (\ud835\udcdd 0)\nz : \u2124\nthis : (fun a => k a ^ z)\u207b\u00b9 = fun a => k a ^ (-z)\n\u22a2 Tendsto ((fun a => k a ^ (-z)) * f) l (\ud835\udcdd 0)", "state_after": "no goals"}, {"tactic": "simp", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\nh : \u2200 (z : \u2124), Tendsto (fun a => k a ^ z * f a) l (\ud835\udcdd 0)\nz : \u2124\nx : \u03b1\n\u22a2 (fun a => k a ^ z)\u207b\u00b9 x = k x ^ (-z)", "state_after": "no goals"}, {"tactic": "suffices : (fun a : \u03b1 => k a ^ z * f a) =O[l] fun a : \u03b1 => (k a)\u207b\u00b9", "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\nh : \u2200 (z : \u2124), f =O[l] fun a => k a ^ z\nz : \u2124\n\u22a2 Tendsto (fun a => k a ^ z * f a) l (\ud835\udcdd 0)", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\nh : \u2200 (z : \u2124), f =O[l] fun a => k a ^ z\nz : \u2124\nthis : (fun a => k a ^ z * f a) =O[l] fun a => (k a)\u207b\u00b9\n\u22a2 Tendsto (fun a => k a ^ z * f a) l (\ud835\udcdd 0)\n\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\nh : \u2200 (z : \u2124), f =O[l] fun a => k a ^ z\nz : \u2124\n\u22a2 (fun a => k a ^ z * f a) =O[l] fun a => (k a)\u207b\u00b9"}, {"tactic": "exact IsBigO.trans_tendsto this hk.inv_tendsto_atTop", "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\nh : \u2200 (z : \u2124), f =O[l] fun a => k a ^ z\nz : \u2124\nthis : (fun a => k a ^ z * f a) =O[l] fun a => (k a)\u207b\u00b9\n\u22a2 Tendsto (fun a => k a ^ z * f a) l (\ud835\udcdd 0)\n\ncase this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\nh : \u2200 (z : \u2124), f =O[l] fun a => k a ^ z\nz : \u2124\n\u22a2 (fun a => k a ^ z * f a) =O[l] fun a => (k a)\u207b\u00b9", "state_after": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\nh : \u2200 (z : \u2124), f =O[l] fun a => k a ^ z\nz : \u2124\n\u22a2 (fun a => k a ^ z * f a) =O[l] fun a => (k a)\u207b\u00b9"}, {"tactic": "refine'\n  ((isBigO_refl (fun a => k a ^ z) l).mul (h (-(z + 1)))).trans\n    (IsBigO.of_bound 1 <| hk0.mono fun a ha0 => _)", "state_before": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\nh : \u2200 (z : \u2124), f =O[l] fun a => k a ^ z\nz : \u2124\n\u22a2 (fun a => k a ^ z * f a) =O[l] fun a => (k a)\u207b\u00b9", "state_after": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\nh : \u2200 (z : \u2124), f =O[l] fun a => k a ^ z\nz : \u2124\na : \u03b1\nha0 : k a \u2260 0\n\u22a2 \u2016k a ^ z * k a ^ (-(z + 1))\u2016 \u2264 1 * \u2016(k a)\u207b\u00b9\u2016"}, {"tactic": "simp only [one_mul, neg_add z 1, zpow_add\u2080 ha0, \u2190 mul_assoc, zpow_neg,\n  mul_inv_cancel (zpow_ne_zero z ha0), zpow_one]", "state_before": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\nh : \u2200 (z : \u2124), f =O[l] fun a => k a ^ z\nz : \u2124\na : \u03b1\nha0 : k a \u2260 0\n\u22a2 \u2016k a ^ z * k a ^ (-(z + 1))\u2016 \u2264 1 * \u2016(k a)\u207b\u00b9\u2016", "state_after": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\nh : \u2200 (z : \u2124), f =O[l] fun a => k a ^ z\nz : \u2124\na : \u03b1\nha0 : k a \u2260 0\n\u22a2 \u2016(k a)\u207b\u00b9\u2016 \u2264 \u2016(k a)\u207b\u00b9\u2016"}, {"tactic": "rfl", "state_before": "case this\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : Filter \u03b1\nk f g g' : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : NormedLinearOrderedField \u03b2\ninst\u271d : OrderTopology \u03b2\nhk : Tendsto k l atTop\nhk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0\nh : \u2200 (z : \u2124), f =O[l] fun a => k a ^ z\nz : \u2124\na : \u03b1\nha0 : k a \u2260 0\n\u22a2 \u2016(k a)\u207b\u00b9\u2016 \u2264 \u2016(k a)\u207b\u00b9\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/ModelTheory/Substructures.lean", "full_name": "FirstOrder.Language.Hom.range_id", "start": [847, 1], "end": [848, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Iterate.lean", "full_name": "Function.Commute.iterate_pos_eq_iff_map_eq", "start": [219, 1], "end": [222, 48], "traced_tactics": [{"tactic": "simp only [le_antisymm_iff, h.iterate_pos_le_iff_map_le hf hg hn,\n  h.symm.iterate_pos_le_iff_map_le' hg hf hn]", "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf g : \u03b1 \u2192 \u03b1\nh : Commute f g\nhf : Monotone f\nhg : StrictMono g\nx : \u03b1\nn : \u2115\nhn : 0 < n\n\u22a2 (f^[n]) x = (g^[n]) x \u2194 f x = g x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/LiminfLimsup.lean", "full_name": "Filter.bliminf_or_le_inf_aux_left", "start": [939, 1], "end": [940, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Dual.lean", "full_name": "Subspace.dualPairing_nondegenerate", "start": [1393, 1], "end": [1401, 47], "traced_tactics": [{"tactic": "constructor", "state_before": "K : Type u\ninst\u271d\u2074 : Field K\nV\u2081 : Type v'\nV\u2082 : Type v''\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nW : Subspace K V\u2081\n\u22a2 LinearMap.Nondegenerate (Submodule.dualPairing W)", "state_after": "case left\nK : Type u\ninst\u271d\u2074 : Field K\nV\u2081 : Type v'\nV\u2082 : Type v''\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nW : Subspace K V\u2081\n\u22a2 LinearMap.SeparatingLeft (Submodule.dualPairing W)\n\ncase right\nK : Type u\ninst\u271d\u2074 : Field K\nV\u2081 : Type v'\nV\u2082 : Type v''\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nW : Subspace K V\u2081\n\u22a2 LinearMap.SeparatingRight (Submodule.dualPairing W)"}, {"tactic": "rw [LinearMap.separatingLeft_iff_ker_eq_bot, dualPairing_eq]", "state_before": "case left\nK : Type u\ninst\u271d\u2074 : Field K\nV\u2081 : Type v'\nV\u2082 : Type v''\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nW : Subspace K V\u2081\n\u22a2 LinearMap.SeparatingLeft (Submodule.dualPairing W)", "state_after": "case left\nK : Type u\ninst\u271d\u2074 : Field K\nV\u2081 : Type v'\nV\u2082 : Type v''\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nW : Subspace K V\u2081\n\u22a2 LinearMap.ker \u2191(quotAnnihilatorEquiv W) = \u22a5"}, {"tactic": "apply LinearEquiv.ker", "state_before": "case left\nK : Type u\ninst\u271d\u2074 : Field K\nV\u2081 : Type v'\nV\u2082 : Type v''\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nW : Subspace K V\u2081\n\u22a2 LinearMap.ker \u2191(quotAnnihilatorEquiv W) = \u22a5", "state_after": "no goals"}, {"tactic": "intro x h", "state_before": "case right\nK : Type u\ninst\u271d\u2074 : Field K\nV\u2081 : Type v'\nV\u2082 : Type v''\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nW : Subspace K V\u2081\n\u22a2 LinearMap.SeparatingRight (Submodule.dualPairing W)", "state_after": "case right\nK : Type u\ninst\u271d\u2074 : Field K\nV\u2081 : Type v'\nV\u2082 : Type v''\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nW : Subspace K V\u2081\nx : { x // x \u2208 W }\nh : \u2200 (x_1 : Dual K V\u2081 \u29f8 dualAnnihilator W), \u2191(\u2191(Submodule.dualPairing W) x_1) x = 0\n\u22a2 x = 0"}, {"tactic": "rw [\u2190 forall_dual_apply_eq_zero_iff K x]", "state_before": "case right\nK : Type u\ninst\u271d\u2074 : Field K\nV\u2081 : Type v'\nV\u2082 : Type v''\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nW : Subspace K V\u2081\nx : { x // x \u2208 W }\nh : \u2200 (x_1 : Dual K V\u2081 \u29f8 dualAnnihilator W), \u2191(\u2191(Submodule.dualPairing W) x_1) x = 0\n\u22a2 x = 0", "state_after": "case right\nK : Type u\ninst\u271d\u2074 : Field K\nV\u2081 : Type v'\nV\u2082 : Type v''\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nW : Subspace K V\u2081\nx : { x // x \u2208 W }\nh : \u2200 (x_1 : Dual K V\u2081 \u29f8 dualAnnihilator W), \u2191(\u2191(Submodule.dualPairing W) x_1) x = 0\n\u22a2 \u2200 (\u03c6 : Dual K { x // x \u2208 W }), \u2191\u03c6 x = 0"}, {"tactic": "intro \u03c6", "state_before": "case right\nK : Type u\ninst\u271d\u2074 : Field K\nV\u2081 : Type v'\nV\u2082 : Type v''\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nW : Subspace K V\u2081\nx : { x // x \u2208 W }\nh : \u2200 (x_1 : Dual K V\u2081 \u29f8 dualAnnihilator W), \u2191(\u2191(Submodule.dualPairing W) x_1) x = 0\n\u22a2 \u2200 (\u03c6 : Dual K { x // x \u2208 W }), \u2191\u03c6 x = 0", "state_after": "case right\nK : Type u\ninst\u271d\u2074 : Field K\nV\u2081 : Type v'\nV\u2082 : Type v''\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nW : Subspace K V\u2081\nx : { x // x \u2208 W }\nh : \u2200 (x_1 : Dual K V\u2081 \u29f8 dualAnnihilator W), \u2191(\u2191(Submodule.dualPairing W) x_1) x = 0\n\u03c6 : Dual K { x // x \u2208 W }\n\u22a2 \u2191\u03c6 x = 0"}, {"tactic": "simpa only [Submodule.dualPairing_apply, dualLift_of_subtype] using\n  h (Submodule.Quotient.mk (W.dualLift \u03c6))", "state_before": "case right\nK : Type u\ninst\u271d\u2074 : Field K\nV\u2081 : Type v'\nV\u2082 : Type v''\ninst\u271d\u00b3 : AddCommGroup V\u2081\ninst\u271d\u00b2 : Module K V\u2081\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nW : Subspace K V\u2081\nx : { x // x \u2208 W }\nh : \u2200 (x_1 : Dual K V\u2081 \u29f8 dualAnnihilator W), \u2191(\u2191(Submodule.dualPairing W) x_1) x = 0\n\u03c6 : Dual K { x // x \u2208 W }\n\u22a2 \u2191\u03c6 x = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Log.lean", "full_name": "Nat.log_div_mul_self", "start": [220, 1], "end": [226, 59], "traced_tactics": [{"tactic": "cases' le_or_lt b 1 with hb hb", "state_before": "b n : \u2115\n\u22a2 log b (n / b * b) = log b n", "state_after": "case inl\nb n : \u2115\nhb : b \u2264 1\n\u22a2 log b (n / b * b) = log b n\n\ncase inr\nb n : \u2115\nhb : 1 < b\n\u22a2 log b (n / b * b) = log b n"}, {"tactic": "cases' lt_or_le n b with h h", "state_before": "case inr\nb n : \u2115\nhb : 1 < b\n\u22a2 log b (n / b * b) = log b n", "state_after": "case inr.inl\nb n : \u2115\nhb : 1 < b\nh : n < b\n\u22a2 log b (n / b * b) = log b n\n\ncase inr.inr\nb n : \u2115\nhb : 1 < b\nh : b \u2264 n\n\u22a2 log b (n / b * b) = log b n"}, {"tactic": "rw [log_mul_base hb (Nat.div_pos h (zero_le_one.trans_lt hb)).ne', log_div_base,\n  tsub_add_cancel_of_le (succ_le_iff.2 <| log_pos hb h)]", "state_before": "case inr.inr\nb n : \u2115\nhb : 1 < b\nh : b \u2264 n\n\u22a2 log b (n / b * b) = log b n", "state_after": "no goals"}, {"tactic": "rw [log_of_left_le_one hb, log_of_left_le_one hb]", "state_before": "case inl\nb n : \u2115\nhb : b \u2264 1\n\u22a2 log b (n / b * b) = log b n", "state_after": "no goals"}, {"tactic": "rw [div_eq_of_lt h, zero_mul, log_zero_right, log_of_lt h]", "state_before": "case inr.inl\nb n : \u2115\nhb : 1 < b\nh : n < b\n\u22a2 log b (n / b * b) = log b n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Group/Measure.lean", "full_name": "MeasureTheory.forall_measure_preimage_mul_iff", "start": [181, 1], "end": [188, 37], "traced_tactics": [{"tactic": "exact \u27e8fun h => \u27e8h\u27e9, fun h => h.1\u27e9", "state_before": "\ud835\udd5c : Type ?u.128647\nG : Type u_1\nH : Type ?u.128653\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSpace H\ninst\u271d\u00b9 : Mul G\n\u03bc\u271d : Measure G\ninst\u271d : MeasurableMul G\n\u03bc : Measure G\n\u22a2 (\u2200 (g : G), Measure.map (fun x => g * x) \u03bc = \u03bc) \u2194 IsMulLeftInvariant \u03bc", "state_after": "no goals"}, {"tactic": "simp_rw [Measure.ext_iff]", "state_before": "\ud835\udd5c : Type ?u.128647\nG : Type u_1\nH : Type ?u.128653\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSpace H\ninst\u271d\u00b9 : Mul G\n\u03bc\u271d : Measure G\ninst\u271d : MeasurableMul G\n\u03bc : Measure G\n\u22a2 (\u2200 (g : G) (A : Set G), MeasurableSet A \u2192 \u2191\u2191\u03bc ((fun h => g * h) \u207b\u00b9' A) = \u2191\u2191\u03bc A) \u2194\n    \u2200 (g : G), Measure.map (fun x => g * x) \u03bc = \u03bc", "state_after": "\ud835\udd5c : Type ?u.128647\nG : Type u_1\nH : Type ?u.128653\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSpace H\ninst\u271d\u00b9 : Mul G\n\u03bc\u271d : Measure G\ninst\u271d : MeasurableMul G\n\u03bc : Measure G\n\u22a2 (\u2200 (g : G) (A : Set G), MeasurableSet A \u2192 \u2191\u2191\u03bc ((fun h => g * h) \u207b\u00b9' A) = \u2191\u2191\u03bc A) \u2194\n    \u2200 (g : G) (s : Set G), MeasurableSet s \u2192 \u2191\u2191(Measure.map (fun x => g * x) \u03bc) s = \u2191\u2191\u03bc s"}, {"tactic": "refine' forall_congr' fun g => forall_congr' fun A => forall_congr' fun hA => _", "state_before": "\ud835\udd5c : Type ?u.128647\nG : Type u_1\nH : Type ?u.128653\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSpace H\ninst\u271d\u00b9 : Mul G\n\u03bc\u271d : Measure G\ninst\u271d : MeasurableMul G\n\u03bc : Measure G\n\u22a2 (\u2200 (g : G) (A : Set G), MeasurableSet A \u2192 \u2191\u2191\u03bc ((fun h => g * h) \u207b\u00b9' A) = \u2191\u2191\u03bc A) \u2194\n    \u2200 (g : G) (s : Set G), MeasurableSet s \u2192 \u2191\u2191(Measure.map (fun x => g * x) \u03bc) s = \u2191\u2191\u03bc s", "state_after": "\ud835\udd5c : Type ?u.128647\nG : Type u_1\nH : Type ?u.128653\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSpace H\ninst\u271d\u00b9 : Mul G\n\u03bc\u271d : Measure G\ninst\u271d : MeasurableMul G\n\u03bc : Measure G\ng : G\nA : Set G\nhA : MeasurableSet A\n\u22a2 \u2191\u2191\u03bc ((fun h => g * h) \u207b\u00b9' A) = \u2191\u2191\u03bc A \u2194 \u2191\u2191(Measure.map (fun x => g * x) \u03bc) A = \u2191\u2191\u03bc A"}, {"tactic": "rw [map_apply (measurable_const_mul g) hA]", "state_before": "\ud835\udd5c : Type ?u.128647\nG : Type u_1\nH : Type ?u.128653\ninst\u271d\u00b3 : MeasurableSpace G\ninst\u271d\u00b2 : MeasurableSpace H\ninst\u271d\u00b9 : Mul G\n\u03bc\u271d : Measure G\ninst\u271d : MeasurableMul G\n\u03bc : Measure G\ng : G\nA : Set G\nhA : MeasurableSet A\n\u22a2 \u2191\u2191\u03bc ((fun h => g * h) \u207b\u00b9' A) = \u2191\u2191\u03bc A \u2194 \u2191\u2191(Measure.map (fun x => g * x) \u03bc) A = \u2191\u2191\u03bc A", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "full_name": "sup_mul", "start": [81, 1], "end": [82, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Hull.lean", "full_name": "Convex.convex_remove_iff_not_mem_convexHull_remove", "start": [152, 1], "end": [166, 21], "traced_tactics": [{"tactic": "constructor", "state_before": "\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.27823\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\n\u22a2 Convex \ud835\udd5c (s \\ {x}) \u2194 \u00acx \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x})", "state_after": "case mp\n\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.27823\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\n\u22a2 Convex \ud835\udd5c (s \\ {x}) \u2192 \u00acx \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x})\n\ncase mpr\n\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.27823\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\n\u22a2 \u00acx \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x}) \u2192 Convex \ud835\udd5c (s \\ {x})"}, {"tactic": "rintro hx", "state_before": "case mpr\n\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.27823\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\n\u22a2 \u00acx \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x}) \u2192 Convex \ud835\udd5c (s \\ {x})", "state_after": "case mpr\n\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.27823\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nhx : \u00acx \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x})\n\u22a2 Convex \ud835\udd5c (s \\ {x})"}, {"tactic": "suffices h : s \\ {x} = convexHull \ud835\udd5c (s \\ {x})", "state_before": "case mpr\n\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.27823\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nhx : \u00acx \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x})\n\u22a2 Convex \ud835\udd5c (s \\ {x})", "state_after": "case mpr\n\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.27823\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nhx : \u00acx \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x})\nh : s \\ {x} = \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x})\n\u22a2 Convex \ud835\udd5c (s \\ {x})\n\ncase h\n\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.27823\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nhx : \u00acx \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x})\n\u22a2 s \\ {x} = \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x})"}, {"tactic": "exact\n  Subset.antisymm (subset_convexHull \ud835\udd5c _) fun y hy =>\n    \u27e8convexHull_min (diff_subset _ _) hs hy, by\n      rintro (rfl : y = x)\n      exact hx hy\u27e9", "state_before": "case h\n\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.27823\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nhx : \u00acx \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x})\n\u22a2 s \\ {x} = \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x})", "state_after": "no goals"}, {"tactic": "rintro hsx hx", "state_before": "case mp\n\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.27823\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\n\u22a2 Convex \ud835\udd5c (s \\ {x}) \u2192 \u00acx \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x})", "state_after": "case mp\n\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.27823\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nhsx : Convex \ud835\udd5c (s \\ {x})\nhx : x \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x})\n\u22a2 False"}, {"tactic": "rw [hsx.convexHull_eq] at hx", "state_before": "case mp\n\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.27823\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nhsx : Convex \ud835\udd5c (s \\ {x})\nhx : x \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x})\n\u22a2 False", "state_after": "case mp\n\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.27823\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nhsx : Convex \ud835\udd5c (s \\ {x})\nhx : x \u2208 s \\ {x}\n\u22a2 False"}, {"tactic": "exact hx.2 (mem_singleton _)", "state_before": "case mp\n\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.27823\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nhsx : Convex \ud835\udd5c (s \\ {x})\nhx : x \u2208 s \\ {x}\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rw [h]", "state_before": "case mpr\n\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.27823\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nhx : \u00acx \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x})\nh : s \\ {x} = \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x})\n\u22a2 Convex \ud835\udd5c (s \\ {x})", "state_after": "case mpr\n\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.27823\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nhx : \u00acx \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x})\nh : s \\ {x} = \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x})\n\u22a2 Convex \ud835\udd5c (\u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x}))"}, {"tactic": "exact convex_convexHull \ud835\udd5c _", "state_before": "case mpr\n\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.27823\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nhx : \u00acx \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x})\nh : s \\ {x} = \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x})\n\u22a2 Convex \ud835\udd5c (\u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x}))", "state_after": "no goals"}, {"tactic": "rintro (rfl : y = x)", "state_before": "\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.27823\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx\u271d y\u271d : E\ns : Set E\nhs : Convex \ud835\udd5c s\nx : E\nhx : \u00acx \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x})\ny : E\nhy : y \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {x})\n\u22a2 \u00acy \u2208 {x}", "state_after": "\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.27823\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx y\u271d : E\ns : Set E\nhs : Convex \ud835\udd5c s\ny : E\nhx : \u00acy \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {y})\nhy : y \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {y})\n\u22a2 False"}, {"tactic": "exact hx hy", "state_before": "\ud835\udd5c : Type u_2\nE : Type u_1\nF : Type ?u.27823\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns\u271d t : Set E\nx y\u271d : E\ns : Set E\nhs : Convex \ud835\udd5c s\ny : E\nhx : \u00acy \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {y})\nhy : y \u2208 \u2191(convexHull \ud835\udd5c).toOrderHom (s \\ {y})\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/LocalRing.lean", "full_name": "LocalRing.isUnit_one_sub_self_of_mem_nonunits", "start": [180, 1], "end": [181, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Bounded.lean", "full_name": "Set.unbounded_lt_of_forall_exists_le", "start": [50, 1], "end": [53, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Localization/Construction.lean", "full_name": "CategoryTheory.Localization.Construction.NatTransExtension.app_eq", "start": [284, 1], "end": [286, 6], "traced_tactics": [{"tactic": "simp only [app, eqToHom_refl, comp_id, id_comp]", "state_before": "C : Type u_3\ninst\u271d\u00b9 : Category C\nW : MorphismProperty C\nD : Type u_2\ninst\u271d : Category D\nG : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\nF\u2081 F\u2082 : MorphismProperty.Localization W \u2964 D\n\u03c4 : MorphismProperty.Q W \u22d9 F\u2081 \u27f6 MorphismProperty.Q W \u22d9 F\u2082\nX : C\n\u22a2 app \u03c4 ((MorphismProperty.Q W).obj X) = \u03c4.app X", "state_after": "C : Type u_3\ninst\u271d\u00b9 : Category C\nW : MorphismProperty C\nD : Type u_2\ninst\u271d : Category D\nG : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\nF\u2081 F\u2082 : MorphismProperty.Localization W \u2964 D\n\u03c4 : MorphismProperty.Q W \u22d9 F\u2081 \u27f6 MorphismProperty.Q W \u22d9 F\u2082\nX : C\n\u22a2 \u03c4.app (Equiv.invFun (objEquiv W) ((MorphismProperty.Q W).obj X)) = \u03c4.app X"}, {"tactic": "rfl", "state_before": "C : Type u_3\ninst\u271d\u00b9 : Category C\nW : MorphismProperty C\nD : Type u_2\ninst\u271d : Category D\nG : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\nF\u2081 F\u2082 : MorphismProperty.Localization W \u2964 D\n\u03c4 : MorphismProperty.Q W \u22d9 F\u2081 \u27f6 MorphismProperty.Q W \u22d9 F\u2082\nX : C\n\u22a2 \u03c4.app (Equiv.invFun (objEquiv W) ((MorphismProperty.Q W).obj X)) = \u03c4.app X", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Data/Nat/Basic.lean", "full_name": "Nat.le_of_mul_le_mul_left", "start": [459, 11], "end": [462, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/ContinuousFunction/Compact.lean", "full_name": "ContinuousMap.norm_lt_iff_of_nonempty", "start": [242, 1], "end": [243, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/SimpleGraph/Prod.lean", "full_name": "SimpleGraph.boxProd_adj_right", "start": [67, 1], "end": [68, 77], "traced_tactics": [{"tactic": "simp only [boxProd_adj, SimpleGraph.irrefl, false_and, and_true, false_or]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.9451\nG : SimpleGraph \u03b1\nH : SimpleGraph \u03b2\na : \u03b1\nb\u2081 b\u2082 : \u03b2\n\u22a2 Adj (G \u25a1 H) (a, b\u2081) (a, b\u2082) \u2194 Adj H b\u2081 b\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Submonoid/Basic.lean", "full_name": "MonoidHom.coe_ofClosureMEqTopLeft", "start": [675, 1], "end": [677, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Field/Power.lean", "full_name": "Even.zpow_nonneg", "start": [181, 11], "end": [182, 52], "traced_tactics": [{"tactic": "obtain \u27e8k, rfl\u27e9 := hn", "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u271d b c d : \u03b1\nn : \u2124\nhn : Even n\na : \u03b1\n\u22a2 0 \u2264 a ^ n", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u271d b c d a : \u03b1\nk : \u2124\n\u22a2 0 \u2264 a ^ (k + k)"}, {"tactic": "exact zpow_bit0_nonneg _ _", "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na\u271d b c d a : \u03b1\nk : \u2124\n\u22a2 0 \u2264 a ^ (k + k)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subgroup/Finite.lean", "full_name": "Subgroup.pi_le_iff", "start": [233, 1], "end": [239, 97], "traced_tactics": [{"tactic": "constructor", "state_before": "G : Type ?u.39010\ninst\u271d\u2074 : Group G\nA : Type ?u.39016\ninst\u271d\u00b3 : AddGroup A\n\u03b7 : Type u_1\nf : \u03b7 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d\u00b9 : DecidableEq \u03b7\ninst\u271d : Finite \u03b7\nH : (i : \u03b7) \u2192 Subgroup (f i)\nJ : Subgroup ((i : \u03b7) \u2192 f i)\n\u22a2 pi univ H \u2264 J \u2194 \u2200 (i : \u03b7), map (MonoidHom.single f i) (H i) \u2264 J", "state_after": "case mp\nG : Type ?u.39010\ninst\u271d\u2074 : Group G\nA : Type ?u.39016\ninst\u271d\u00b3 : AddGroup A\n\u03b7 : Type u_1\nf : \u03b7 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d\u00b9 : DecidableEq \u03b7\ninst\u271d : Finite \u03b7\nH : (i : \u03b7) \u2192 Subgroup (f i)\nJ : Subgroup ((i : \u03b7) \u2192 f i)\n\u22a2 pi univ H \u2264 J \u2192 \u2200 (i : \u03b7), map (MonoidHom.single f i) (H i) \u2264 J\n\ncase mpr\nG : Type ?u.39010\ninst\u271d\u2074 : Group G\nA : Type ?u.39016\ninst\u271d\u00b3 : AddGroup A\n\u03b7 : Type u_1\nf : \u03b7 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d\u00b9 : DecidableEq \u03b7\ninst\u271d : Finite \u03b7\nH : (i : \u03b7) \u2192 Subgroup (f i)\nJ : Subgroup ((i : \u03b7) \u2192 f i)\n\u22a2 (\u2200 (i : \u03b7), map (MonoidHom.single f i) (H i) \u2264 J) \u2192 pi univ H \u2264 J"}, {"tactic": "rintro h i _ \u27e8x, hx, rfl\u27e9", "state_before": "case mp\nG : Type ?u.39010\ninst\u271d\u2074 : Group G\nA : Type ?u.39016\ninst\u271d\u00b3 : AddGroup A\n\u03b7 : Type u_1\nf : \u03b7 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d\u00b9 : DecidableEq \u03b7\ninst\u271d : Finite \u03b7\nH : (i : \u03b7) \u2192 Subgroup (f i)\nJ : Subgroup ((i : \u03b7) \u2192 f i)\n\u22a2 pi univ H \u2264 J \u2192 \u2200 (i : \u03b7), map (MonoidHom.single f i) (H i) \u2264 J", "state_after": "case mp.intro.intro\nG : Type ?u.39010\ninst\u271d\u2074 : Group G\nA : Type ?u.39016\ninst\u271d\u00b3 : AddGroup A\n\u03b7 : Type u_1\nf : \u03b7 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d\u00b9 : DecidableEq \u03b7\ninst\u271d : Finite \u03b7\nH : (i : \u03b7) \u2192 Subgroup (f i)\nJ : Subgroup ((i : \u03b7) \u2192 f i)\nh : pi univ H \u2264 J\ni : \u03b7\nx : f i\nhx : x \u2208 \u2191(H i)\n\u22a2 \u2191(MonoidHom.single f i) x \u2208 J"}, {"tactic": "apply h", "state_before": "case mp.intro.intro\nG : Type ?u.39010\ninst\u271d\u2074 : Group G\nA : Type ?u.39016\ninst\u271d\u00b3 : AddGroup A\n\u03b7 : Type u_1\nf : \u03b7 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d\u00b9 : DecidableEq \u03b7\ninst\u271d : Finite \u03b7\nH : (i : \u03b7) \u2192 Subgroup (f i)\nJ : Subgroup ((i : \u03b7) \u2192 f i)\nh : pi univ H \u2264 J\ni : \u03b7\nx : f i\nhx : x \u2208 \u2191(H i)\n\u22a2 \u2191(MonoidHom.single f i) x \u2208 J", "state_after": "case mp.intro.intro.a\nG : Type ?u.39010\ninst\u271d\u2074 : Group G\nA : Type ?u.39016\ninst\u271d\u00b3 : AddGroup A\n\u03b7 : Type u_1\nf : \u03b7 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d\u00b9 : DecidableEq \u03b7\ninst\u271d : Finite \u03b7\nH : (i : \u03b7) \u2192 Subgroup (f i)\nJ : Subgroup ((i : \u03b7) \u2192 f i)\nh : pi univ H \u2264 J\ni : \u03b7\nx : f i\nhx : x \u2208 \u2191(H i)\n\u22a2 \u2191(MonoidHom.single f i) x \u2208 pi univ H"}, {"tactic": "simpa using hx", "state_before": "case mp.intro.intro.a\nG : Type ?u.39010\ninst\u271d\u2074 : Group G\nA : Type ?u.39016\ninst\u271d\u00b3 : AddGroup A\n\u03b7 : Type u_1\nf : \u03b7 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d\u00b9 : DecidableEq \u03b7\ninst\u271d : Finite \u03b7\nH : (i : \u03b7) \u2192 Subgroup (f i)\nJ : Subgroup ((i : \u03b7) \u2192 f i)\nh : pi univ H \u2264 J\ni : \u03b7\nx : f i\nhx : x \u2208 \u2191(H i)\n\u22a2 \u2191(MonoidHom.single f i) x \u2208 pi univ H", "state_after": "no goals"}, {"tactic": "exact fun h x hx => pi_mem_of_mulSingle_mem x fun i => h i (mem_map_of_mem _ (hx i trivial))", "state_before": "case mpr\nG : Type ?u.39010\ninst\u271d\u2074 : Group G\nA : Type ?u.39016\ninst\u271d\u00b3 : AddGroup A\n\u03b7 : Type u_1\nf : \u03b7 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b7) \u2192 Group (f i)\ninst\u271d\u00b9 : DecidableEq \u03b7\ninst\u271d : Finite \u03b7\nH : (i : \u03b7) \u2192 Subgroup (f i)\nJ : Subgroup ((i : \u03b7) \u2192 f i)\n\u22a2 (\u2200 (i : \u03b7), map (MonoidHom.single f i) (H i) \u2264 J) \u2192 pi univ H \u2264 J", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Mul.lean", "full_name": "fderiv_mul_const", "start": [458, 1], "end": [460, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Group/MinMax.lean", "full_name": "max_div_div_left'", "start": [73, 1], "end": [74, 61], "traced_tactics": [{"tactic": "simp only [div_eq_mul_inv, max_mul_mul_left, max_inv_inv']", "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedCommGroup \u03b1\na\u271d b\u271d c\u271d a b c : \u03b1\n\u22a2 max (a / b) (a / c) = a / min b c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.sInter_union", "start": [1134, 1], "end": [1135, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "Real.map_linearMap_volume_pi_eq_smul_volume_pi", "start": [449, 1], "end": [459, 18], "traced_tactics": [{"tactic": "classical\n  let M := LinearMap.toMatrix' f\n  have A : LinearMap.det f = det M := by simp only [LinearMap.det_toMatrix']\n  have B : f = toLin' M := by simp only [toLin'_toMatrix']\n  rw [A, B]\n  apply map_matrix_volume_pi_eq_smul_volume_pi\n  rwa [A] at hf", "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\n\u22a2 Measure.map (\u2191f) volume = ofReal (abs (\u2191LinearMap.det f)\u207b\u00b9) \u2022 volume", "state_after": "no goals"}, {"tactic": "let M := LinearMap.toMatrix' f", "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\n\u22a2 Measure.map (\u2191f) volume = ofReal (abs (\u2191LinearMap.det f)\u207b\u00b9) \u2022 volume", "state_after": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\n\u22a2 Measure.map (\u2191f) volume = ofReal (abs (\u2191LinearMap.det f)\u207b\u00b9) \u2022 volume"}, {"tactic": "have A : LinearMap.det f = det M := by simp only [LinearMap.det_toMatrix']", "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\n\u22a2 Measure.map (\u2191f) volume = ofReal (abs (\u2191LinearMap.det f)\u207b\u00b9) \u2022 volume", "state_after": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\nA : \u2191LinearMap.det f = det M\n\u22a2 Measure.map (\u2191f) volume = ofReal (abs (\u2191LinearMap.det f)\u207b\u00b9) \u2022 volume"}, {"tactic": "have B : f = toLin' M := by simp only [toLin'_toMatrix']", "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\nA : \u2191LinearMap.det f = det M\n\u22a2 Measure.map (\u2191f) volume = ofReal (abs (\u2191LinearMap.det f)\u207b\u00b9) \u2022 volume", "state_after": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\nA : \u2191LinearMap.det f = det M\nB : f = \u2191toLin' M\n\u22a2 Measure.map (\u2191f) volume = ofReal (abs (\u2191LinearMap.det f)\u207b\u00b9) \u2022 volume"}, {"tactic": "rw [A, B]", "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\nA : \u2191LinearMap.det f = det M\nB : f = \u2191toLin' M\n\u22a2 Measure.map (\u2191f) volume = ofReal (abs (\u2191LinearMap.det f)\u207b\u00b9) \u2022 volume", "state_after": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\nA : \u2191LinearMap.det f = det M\nB : f = \u2191toLin' M\n\u22a2 Measure.map (\u2191(\u2191toLin' M)) volume = ofReal (abs (det M)\u207b\u00b9) \u2022 volume"}, {"tactic": "apply map_matrix_volume_pi_eq_smul_volume_pi", "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\nA : \u2191LinearMap.det f = det M\nB : f = \u2191toLin' M\n\u22a2 Measure.map (\u2191(\u2191toLin' M)) volume = ofReal (abs (det M)\u207b\u00b9) \u2022 volume", "state_after": "case hM\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\nA : \u2191LinearMap.det f = det M\nB : f = \u2191toLin' M\n\u22a2 det M \u2260 0"}, {"tactic": "rwa [A] at hf", "state_before": "case hM\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\nA : \u2191LinearMap.det f = det M\nB : f = \u2191toLin' M\n\u22a2 det M \u2260 0", "state_after": "no goals"}, {"tactic": "simp only [LinearMap.det_toMatrix']", "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\n\u22a2 \u2191LinearMap.det f = det M", "state_after": "no goals"}, {"tactic": "simp only [toLin'_toMatrix']", "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\nA : \u2191LinearMap.det f = det M\n\u22a2 f = \u2191toLin' M", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "full_name": "smul_div'", "start": [1062, 1], "end": [1063, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Nontrivial.lean", "full_name": "Subsingleton.le", "start": [204, 11], "end": [205, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Basic.lean", "full_name": "Nat.mul_ne_mul_left", "start": [345, 1], "end": [346, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Localization/Basic.lean", "full_name": "IsLocalization.mk'_eq_of_eq'", "start": [380, 1], "end": [382, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Nat/Order/Basic.lean", "full_name": "Nat.findGreatest_mono_left", "start": [656, 1], "end": [664, 64], "traced_tactics": [{"tactic": "intro n", "state_before": "m n k l : \u2115\nP Q : \u2115 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred P\ninst\u271d : DecidablePred Q\nhPQ : P \u2264 Q\n\u22a2 Nat.findGreatest P \u2264 Nat.findGreatest Q", "state_after": "m n\u271d k l : \u2115\nP Q : \u2115 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred P\ninst\u271d : DecidablePred Q\nhPQ : P \u2264 Q\nn : \u2115\n\u22a2 Nat.findGreatest P n \u2264 Nat.findGreatest Q n"}, {"tactic": "induction' n with n hn", "state_before": "m n\u271d k l : \u2115\nP Q : \u2115 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred P\ninst\u271d : DecidablePred Q\nhPQ : P \u2264 Q\nn : \u2115\n\u22a2 Nat.findGreatest P n \u2264 Nat.findGreatest Q n", "state_after": "case zero\nm n k l : \u2115\nP Q : \u2115 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred P\ninst\u271d : DecidablePred Q\nhPQ : P \u2264 Q\n\u22a2 Nat.findGreatest P zero \u2264 Nat.findGreatest Q zero\n\ncase succ\nm n\u271d k l : \u2115\nP Q : \u2115 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred P\ninst\u271d : DecidablePred Q\nhPQ : P \u2264 Q\nn : \u2115\nhn : Nat.findGreatest P n \u2264 Nat.findGreatest Q n\n\u22a2 Nat.findGreatest P (succ n) \u2264 Nat.findGreatest Q (succ n)"}, {"tactic": "by_cases h : P (n + 1)", "state_before": "case succ\nm n\u271d k l : \u2115\nP Q : \u2115 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred P\ninst\u271d : DecidablePred Q\nhPQ : P \u2264 Q\nn : \u2115\nhn : Nat.findGreatest P n \u2264 Nat.findGreatest Q n\n\u22a2 Nat.findGreatest P (succ n) \u2264 Nat.findGreatest Q (succ n)", "state_after": "case pos\nm n\u271d k l : \u2115\nP Q : \u2115 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred P\ninst\u271d : DecidablePred Q\nhPQ : P \u2264 Q\nn : \u2115\nhn : Nat.findGreatest P n \u2264 Nat.findGreatest Q n\nh : P (n + 1)\n\u22a2 Nat.findGreatest P (succ n) \u2264 Nat.findGreatest Q (succ n)\n\ncase neg\nm n\u271d k l : \u2115\nP Q : \u2115 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred P\ninst\u271d : DecidablePred Q\nhPQ : P \u2264 Q\nn : \u2115\nhn : Nat.findGreatest P n \u2264 Nat.findGreatest Q n\nh : \u00acP (n + 1)\n\u22a2 Nat.findGreatest P (succ n) \u2264 Nat.findGreatest Q (succ n)"}, {"tactic": "rfl", "state_before": "case zero\nm n k l : \u2115\nP Q : \u2115 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred P\ninst\u271d : DecidablePred Q\nhPQ : P \u2264 Q\n\u22a2 Nat.findGreatest P zero \u2264 Nat.findGreatest Q zero", "state_after": "no goals"}, {"tactic": "rw [findGreatest_eq h, findGreatest_eq (hPQ _ h)]", "state_before": "case pos\nm n\u271d k l : \u2115\nP Q : \u2115 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred P\ninst\u271d : DecidablePred Q\nhPQ : P \u2264 Q\nn : \u2115\nhn : Nat.findGreatest P n \u2264 Nat.findGreatest Q n\nh : P (n + 1)\n\u22a2 Nat.findGreatest P (succ n) \u2264 Nat.findGreatest Q (succ n)", "state_after": "no goals"}, {"tactic": "rw [findGreatest_of_not h]", "state_before": "case neg\nm n\u271d k l : \u2115\nP Q : \u2115 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred P\ninst\u271d : DecidablePred Q\nhPQ : P \u2264 Q\nn : \u2115\nhn : Nat.findGreatest P n \u2264 Nat.findGreatest Q n\nh : \u00acP (n + 1)\n\u22a2 Nat.findGreatest P (succ n) \u2264 Nat.findGreatest Q (succ n)", "state_after": "case neg\nm n\u271d k l : \u2115\nP Q : \u2115 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred P\ninst\u271d : DecidablePred Q\nhPQ : P \u2264 Q\nn : \u2115\nhn : Nat.findGreatest P n \u2264 Nat.findGreatest Q n\nh : \u00acP (n + 1)\n\u22a2 Nat.findGreatest P n \u2264 Nat.findGreatest Q (succ n)"}, {"tactic": "exact hn.trans (Nat.findGreatest_mono_right _ <| le_succ _)", "state_before": "case neg\nm n\u271d k l : \u2115\nP Q : \u2115 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred P\ninst\u271d : DecidablePred Q\nhPQ : P \u2264 Q\nn : \u2115\nhn : Nat.findGreatest P n \u2264 Nat.findGreatest Q n\nh : \u00acP (n + 1)\n\u22a2 Nat.findGreatest P n \u2264 Nat.findGreatest Q (succ n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/FractionalIdeal.lean", "full_name": "FractionalIdeal.mul_induction_on", "start": [598, 11], "end": [602, 44], "traced_tactics": [{"tactic": "simp only [mul_def] at hr", "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\nC : P \u2192 Prop\nr : P\nhr : r \u2208 I * J\nhm : \u2200 (i : P), i \u2208 I \u2192 \u2200 (j : P), j \u2208 J \u2192 C (i * j)\nha : \u2200 (x y : P), C x \u2192 C y \u2192 C (x + y)\n\u22a2 C r", "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\nC : P \u2192 Prop\nr : P\nhm : \u2200 (i : P), i \u2208 I \u2192 \u2200 (j : P), j \u2208 J \u2192 C (i * j)\nha : \u2200 (x y : P), C x \u2192 C y \u2192 C (x + y)\nhr : r \u2208 { val := \u2191I * \u2191J, property := (_ : IsFractional S (\u2191I * \u2191J)) }\n\u22a2 C r"}, {"tactic": "exact Submodule.mul_induction_on hr hm ha", "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\nC : P \u2192 Prop\nr : P\nhm : \u2200 (i : P), i \u2208 I \u2192 \u2200 (j : P), j \u2208 J \u2192 C (i * j)\nha : \u2200 (x y : P), C x \u2192 C y \u2192 C (x + y)\nhr : r \u2208 { val := \u2191I * \u2191J, property := (_ : IsFractional S (\u2191I * \u2191J)) }\n\u22a2 C r", "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_le", "start": [130, 1], "end": [131, 62], "traced_tactics": [{"tactic": "simpa using degree_list_prod_le", "state_before": "R : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf : \u03b9 \u2192 R[X]\nt : Multiset R[X]\n\u22a2 \u2200 (a : List R[X]),\n    degree (prod (Quotient.mk (List.isSetoid R[X]) a)) \u2264\n      Multiset.sum (Multiset.map degree (Quotient.mk (List.isSetoid R[X]) a))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Constructions/Polish.lean", "full_name": "MeasurableSet.isClopenable", "start": [221, 1], "end": [227, 49], "traced_tactics": [{"tactic": "revert s", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type ?u.62603\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 IsClopenable s", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type ?u.62603\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\n\u22a2 \u2200 {s : Set \u03b1}, MeasurableSet s \u2192 IsClopenable s"}, {"tactic": "apply MeasurableSet.induction_on_open", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type ?u.62603\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\n\u22a2 \u2200 {s : Set \u03b1}, MeasurableSet s \u2192 IsClopenable s", "state_after": "case h_open\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type ?u.62603\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\n\u22a2 \u2200 (U : Set \u03b1), IsOpen U \u2192 IsClopenable U\n\ncase h_compl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type ?u.62603\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\n\u22a2 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 IsClopenable t \u2192 IsClopenable (t\u1d9c)\n\ncase h_union\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type ?u.62603\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u03b1),\n    Pairwise (Disjoint on f) \u2192\n      (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 (\u2200 (i : \u2115), IsClopenable (f i)) \u2192 IsClopenable (\u22c3 (i : \u2115), f i)"}, {"tactic": "exact fun u hu => hu.isClopenable", "state_before": "case h_open\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type ?u.62603\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\n\u22a2 \u2200 (U : Set \u03b1), IsOpen U \u2192 IsClopenable U", "state_after": "no goals"}, {"tactic": "exact fun u _ h'u => h'u.compl", "state_before": "case h_compl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type ?u.62603\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\n\u22a2 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 IsClopenable t \u2192 IsClopenable (t\u1d9c)", "state_after": "no goals"}, {"tactic": "exact fun f _ _ hf => IsClopenable.iUnion hf", "state_before": "case h_union\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type ?u.62603\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u03b1),\n    Pairwise (Disjoint on f) \u2192\n      (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 (\u2200 (i : \u2115), IsClopenable (f i)) \u2192 IsClopenable (\u22c3 (i : \u2115), f i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "full_name": "mul_lt_mul_left'", "start": [123, 1], "end": [125, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Polynomial/Chebyshev.lean", "full_name": "Polynomial.Chebyshev.T_two", "start": [88, 1], "end": [88, 87], "traced_tactics": [{"tactic": "simp only [T, sub_left_inj, sq, mul_assoc]", "state_before": "R : Type u_1\nS : Type ?u.3233\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 T R 2 = 2 * X ^ 2 - 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean", "full_name": "Real.arccos_zero", "start": [383, 1], "end": [383, 59], "traced_tactics": [{"tactic": "simp [arccos]", "state_before": "\u22a2 arccos 0 = \u03c0 / 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Fintype.lean", "full_name": "Multiset.map_toEnumFinset_fst", "start": [221, 1], "end": [224, 78], "traced_tactics": [{"tactic": "ext x", "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nm\u271d m : Multiset \u03b1\n\u22a2 map Prod.fst (toEnumFinset m).val = m", "state_after": "case a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nm\u271d m : Multiset \u03b1\nx : \u03b1\n\u22a2 count x (map Prod.fst (toEnumFinset m).val) = count x m"}, {"tactic": "simp only [Multiset.count_map, \u2190 Finset.filter_val, Multiset.toEnumFinset_filter_eq,\n  Finset.map_val, Finset.range_val, Multiset.card_map, Multiset.card_range]", "state_before": "case a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nm\u271d m : Multiset \u03b1\nx : \u03b1\n\u22a2 count x (map Prod.fst (toEnumFinset m).val) = count x m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "Complex.exp_conj", "start": [564, 1], "end": [571, 57], "traced_tactics": [{"tactic": "dsimp [exp]", "state_before": "x y : \u2102\n\u22a2 exp (\u2191(starRingEnd \u2102) x) = \u2191(starRingEnd \u2102) (exp x)", "state_after": "x y : \u2102\n\u22a2 CauSeq.lim (exp' (\u2191(starRingEnd \u2102) x)) = \u2191(starRingEnd \u2102) (CauSeq.lim (exp' x))"}, {"tactic": "rw [\u2190 lim_conj]", "state_before": "x y : \u2102\n\u22a2 CauSeq.lim (exp' (\u2191(starRingEnd \u2102) x)) = \u2191(starRingEnd \u2102) (CauSeq.lim (exp' x))", "state_after": "x y : \u2102\n\u22a2 CauSeq.lim (exp' (\u2191(starRingEnd \u2102) x)) = CauSeq.lim (cauSeqConj (exp' x))"}, {"tactic": "refine' congr_arg CauSeq.lim (CauSeq.ext fun _ => _)", "state_before": "x y : \u2102\n\u22a2 CauSeq.lim (exp' (\u2191(starRingEnd \u2102) x)) = CauSeq.lim (cauSeqConj (exp' x))", "state_after": "x y : \u2102\nx\u271d : \u2115\n\u22a2 \u2191(exp' (\u2191(starRingEnd \u2102) x)) x\u271d = \u2191(cauSeqConj (exp' x)) x\u271d"}, {"tactic": "dsimp [exp', Function.comp, isCauSeq_conj, cauSeqConj]", "state_before": "x y : \u2102\nx\u271d : \u2115\n\u22a2 \u2191(exp' (\u2191(starRingEnd \u2102) x)) x\u271d = \u2191(cauSeqConj (exp' x)) x\u271d", "state_after": "x y : \u2102\nx\u271d : \u2115\n\u22a2 \u2211 m in range x\u271d, \u2191(starRingEnd \u2102) x ^ m / \u2191(Nat.factorial m) =\n    \u2191(starRingEnd \u2102) (\u2211 m in range x\u271d, x ^ m / \u2191(Nat.factorial m))"}, {"tactic": "rw [(starRingEnd _).map_sum]", "state_before": "x y : \u2102\nx\u271d : \u2115\n\u22a2 \u2211 m in range x\u271d, \u2191(starRingEnd \u2102) x ^ m / \u2191(Nat.factorial m) =\n    \u2191(starRingEnd \u2102) (\u2211 m in range x\u271d, x ^ m / \u2191(Nat.factorial m))", "state_after": "x y : \u2102\nx\u271d : \u2115\n\u22a2 \u2211 m in range x\u271d, \u2191(starRingEnd \u2102) x ^ m / \u2191(Nat.factorial m) =\n    \u2211 x_1 in range x\u271d, \u2191(starRingEnd \u2102) (x ^ x_1 / \u2191(Nat.factorial x_1))"}, {"tactic": "refine' sum_congr rfl fun n _ => _", "state_before": "x y : \u2102\nx\u271d : \u2115\n\u22a2 \u2211 m in range x\u271d, \u2191(starRingEnd \u2102) x ^ m / \u2191(Nat.factorial m) =\n    \u2211 x_1 in range x\u271d, \u2191(starRingEnd \u2102) (x ^ x_1 / \u2191(Nat.factorial x_1))", "state_after": "x y : \u2102\nx\u271d\u00b9 n : \u2115\nx\u271d : n \u2208 range x\u271d\u00b9\n\u22a2 \u2191(starRingEnd \u2102) x ^ n / \u2191(Nat.factorial n) = \u2191(starRingEnd \u2102) (x ^ n / \u2191(Nat.factorial n))"}, {"tactic": "rw [map_div\u2080, map_pow, \u2190 ofReal_nat_cast, conj_ofReal]", "state_before": "x y : \u2102\nx\u271d\u00b9 n : \u2115\nx\u271d : n \u2208 range x\u271d\u00b9\n\u22a2 \u2191(starRingEnd \u2102) x ^ n / \u2191(Nat.factorial n) = \u2191(starRingEnd \u2102) (x ^ n / \u2191(Nat.factorial n))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.inclusion_self", "start": [2771, 1], "end": [2773, 6], "traced_tactics": [{"tactic": "cases x", "state_before": "\u03b1 : Type u_1\ns t u : Set \u03b1\nx : \u2191s\n\u22a2 inclusion (_ : s \u2286 s) x = x", "state_after": "case mk\n\u03b1 : Type u_1\ns t u : Set \u03b1\nval\u271d : \u03b1\nproperty\u271d : val\u271d \u2208 s\n\u22a2 inclusion (_ : s \u2286 s) { val := val\u271d, property := property\u271d } = { val := val\u271d, property := property\u271d }"}, {"tactic": "rfl", "state_before": "case mk\n\u03b1 : Type u_1\ns t u : Set \u03b1\nval\u271d : \u03b1\nproperty\u271d : val\u271d \u2208 s\n\u22a2 inclusion (_ : s \u2286 s) { val := val\u271d, property := property\u271d } = { val := val\u271d, property := property\u271d }", "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": [356, 11], "end": [360, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Sober.lean", "full_name": "IsGenericPoint.disjoint_iff", "start": [95, 1], "end": [96, 83], "traced_tactics": [{"tactic": "rw [h.mem_open_set_iff hU, \u2190 not_disjoint_iff_nonempty_inter, Classical.not_not]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.2254\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nx y : \u03b1\nS U Z : Set \u03b1\nh : IsGenericPoint x S\nhU : IsOpen U\n\u22a2 Disjoint S U \u2194 \u00acx \u2208 U", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/Nat/Lemmas.lean", "full_name": "Nat.le_of_sub_eq_zero", "start": [148, 11], "end": [152, 78], "traced_tactics": [{"tactic": "rw [Nat.sub_zero] at H", "state_before": "n : Nat\nH : n - 0 = 0\n\u22a2 n \u2264 0", "state_after": "n : Nat\nH : n = 0\n\u22a2 n \u2264 0"}, {"tactic": "simp [H]", "state_before": "n : Nat\nH : n = 0\n\u22a2 n \u2264 0", "state_after": "no goals"}, {"tactic": "simp [Nat.add_sub_add_right] at H", "state_before": "n m : Nat\nH : n + 1 - (m + 1) = 0\n\u22a2 n - m = 0", "state_after": "n m : Nat\nH : n - m = 0\n\u22a2 n - m = 0"}, {"tactic": "exact H", "state_before": "n m : Nat\nH : n - m = 0\n\u22a2 n - m = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/TrivSqZeroExt.lean", "full_name": "TrivSqZeroExt.eq_smul_exp_of_invertible", "start": [162, 1], "end": [165, 65], "traced_tactics": [{"tactic": "rw [\u2190 inr_smul, exp_inr, smul_add, \u2190 inl_one, \u2190 inl_smul, \u2190 inr_smul, smul_eq_mul, mul_one,\n  smul_smul, mul_invOf_self, one_smul, inl_fst_add_inr_snd_eq]", "state_before": "\ud835\udd5c : Type u_3\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedCommRing R\ninst\u271d\u00b9\u2074 : AddCommGroup M\ninst\u271d\u00b9\u00b3 : NormedAlgebra \ud835\udd5c R\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module R\u1d50\u1d52\u1d56 M\ninst\u271d\u00b9\u2070 : IsCentralScalar R M\ninst\u271d\u2079 : Module \ud835\udd5c M\ninst\u271d\u2078 : IsScalarTower \ud835\udd5c R M\ninst\u271d\u2077 : TopologicalSpace M\ninst\u271d\u2076 : TopologicalRing R\ninst\u271d\u2075 : TopologicalAddGroup M\ninst\u271d\u2074 : ContinuousSMul R M\ninst\u271d\u00b3 : CompleteSpace R\ninst\u271d\u00b2 : T2Space R\ninst\u271d\u00b9 : T2Space M\nx : tsze R M\ninst\u271d : Invertible (fst x)\n\u22a2 x = fst x \u2022 exp \ud835\udd5c (\u215f(fst x) \u2022 inr (snd x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Bounds/Basic.lean", "full_name": "upperBounds_Iic", "start": [526, 1], "end": [527, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Polynomial/Basic.lean", "full_name": "MvPolynomial.noZeroDivisors_of_finite", "start": [1105, 1], "end": [1109, 97], "traced_tactics": [{"tactic": "cases nonempty_fintype \u03c3", "state_before": "R\u271d : Type u\nS : Type ?u.794295\n\u03c3\u271d : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\u271d\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R\u271d M\nR : Type u\n\u03c3 : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Finite \u03c3\ninst\u271d : NoZeroDivisors R\n\u22a2 NoZeroDivisors (MvPolynomial \u03c3 R)", "state_after": "case intro\nR\u271d : Type u\nS : Type ?u.794295\n\u03c3\u271d : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\u271d\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R\u271d M\nR : Type u\n\u03c3 : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Finite \u03c3\ninst\u271d : NoZeroDivisors R\nval\u271d : Fintype \u03c3\n\u22a2 NoZeroDivisors (MvPolynomial \u03c3 R)"}, {"tactic": "haveI := noZeroDivisors_fin R (Fintype.card \u03c3)", "state_before": "case intro\nR\u271d : Type u\nS : Type ?u.794295\n\u03c3\u271d : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\u271d\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R\u271d M\nR : Type u\n\u03c3 : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Finite \u03c3\ninst\u271d : NoZeroDivisors R\nval\u271d : Fintype \u03c3\n\u22a2 NoZeroDivisors (MvPolynomial \u03c3 R)", "state_after": "case intro\nR\u271d : Type u\nS : Type ?u.794295\n\u03c3\u271d : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\u271d\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R\u271d M\nR : Type u\n\u03c3 : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Finite \u03c3\ninst\u271d : NoZeroDivisors R\nval\u271d : Fintype \u03c3\nthis : NoZeroDivisors (MvPolynomial (Fin (Fintype.card \u03c3)) R)\n\u22a2 NoZeroDivisors (MvPolynomial \u03c3 R)"}, {"tactic": "exact (renameEquiv R (Fintype.equivFin \u03c3)).injective.noZeroDivisors _ (map_zero _) (map_mul _)", "state_before": "case intro\nR\u271d : Type u\nS : Type ?u.794295\n\u03c3\u271d : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\u271d\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R\u271d M\nR : Type u\n\u03c3 : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Finite \u03c3\ninst\u271d : NoZeroDivisors R\nval\u271d : Fintype \u03c3\nthis : NoZeroDivisors (MvPolynomial (Fin (Fintype.card \u03c3)) R)\n\u22a2 NoZeroDivisors (MvPolynomial \u03c3 R)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/Divisors.lean", "full_name": "Nat.properDivisors_one", "start": [183, 1], "end": [183, 100], "traced_tactics": [{"tactic": "rw [properDivisors, Ico_self, filter_empty]", "state_before": "n : \u2115\n\u22a2 properDivisors 1 = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.union_subset_union_right", "start": [1388, 1], "end": [1389, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Ring/BooleanRing.lean", "full_name": "toBoolAlg_inj", "start": [167, 1], "end": [168, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Game/PGame.lean", "full_name": "PGame.lf_def", "start": [632, 1], "end": [639, 33], "traced_tactics": [{"tactic": "rw [lf_iff_exists_le]", "state_before": "x y : PGame\n\u22a2 x \u29cf y \u2194\n    (\u2203 i,\n        (\u2200 (i' : LeftMoves x), moveLeft x i' \u29cf moveLeft y i) \u2227\n          \u2200 (j : RightMoves (moveLeft y i)), x \u29cf moveRight (moveLeft y i) j) \u2228\n      \u2203 j,\n        (\u2200 (i : LeftMoves (moveRight x j)), moveLeft (moveRight x j) i \u29cf y) \u2227\n          \u2200 (j' : RightMoves y), moveRight x j \u29cf moveRight y j'", "state_after": "x y : PGame\n\u22a2 ((\u2203 i, x \u2264 moveLeft y i) \u2228 \u2203 j, moveRight x j \u2264 y) \u2194\n    (\u2203 i,\n        (\u2200 (i' : LeftMoves x), moveLeft x i' \u29cf moveLeft y i) \u2227\n          \u2200 (j : RightMoves (moveLeft y i)), x \u29cf moveRight (moveLeft y i) j) \u2228\n      \u2203 j,\n        (\u2200 (i : LeftMoves (moveRight x j)), moveLeft (moveRight x j) i \u29cf y) \u2227\n          \u2200 (j' : RightMoves y), moveRight x j \u29cf moveRight y j'"}, {"tactic": "conv =>\n  lhs\n  simp only [le_iff_forall_lf]", "state_before": "x y : PGame\n\u22a2 ((\u2203 i, x \u2264 moveLeft y i) \u2228 \u2203 j, moveRight x j \u2264 y) \u2194\n    (\u2203 i,\n        (\u2200 (i' : LeftMoves x), moveLeft x i' \u29cf moveLeft y i) \u2227\n          \u2200 (j : RightMoves (moveLeft y i)), x \u29cf moveRight (moveLeft y i) j) \u2228\n      \u2203 j,\n        (\u2200 (i : LeftMoves (moveRight x j)), moveLeft (moveRight x j) i \u29cf y) \u2227\n          \u2200 (j' : RightMoves y), moveRight x j \u29cf moveRight y j'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Computability/Partrec.lean", "full_name": "Partrec.const'", "start": [452, 1], "end": [454, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.sdiff_sdiff_left'", "start": [2290, 1], "end": [2291, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/IsROrC/Basic.lean", "full_name": "IsROrC.ofReal_natCast", "start": [631, 1], "end": [632, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Multiset/Lattice.lean", "full_name": "Multiset.inf_union", "start": [171, 1], "end": [172, 58], "traced_tactics": [{"tactic": "rw [\u2190 inf_dedup, dedup_ext.2, inf_dedup, inf_add]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : Multiset \u03b1\n\u22a2 inf (s\u2081 \u222a s\u2082) = inf s\u2081 \u2293 inf s\u2082", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : Multiset \u03b1\n\u22a2 \u2200 (a : \u03b1), a \u2208 s\u2081 \u222a s\u2082 \u2194 a \u2208 s\u2081 + s\u2082"}, {"tactic": "simp", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : Multiset \u03b1\n\u22a2 \u2200 (a : \u03b1), a \u2208 s\u2081 \u222a s\u2082 \u2194 a \u2208 s\u2081 + s\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Group/Basic.lean", "full_name": "mul_div_right_comm", "start": [555, 1], "end": [555, 62], "traced_tactics": [{"tactic": "simp", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.38312\nG : Type ?u.38315\ninst\u271d : DivisionCommMonoid \u03b1\na b c d : \u03b1\n\u22a2 a * b / c = a / c * b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Bases.lean", "full_name": "Filter.HasAntitoneBasis.mem_iff", "start": [1050, 11], "end": [1052, 70], "traced_tactics": [{"tactic": "simp only [exists_prop, true_and]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.95817\n\u03b3 : Type ?u.95820\n\u03b9 : Type u_1\n\u03b9' : Sort ?u.95826\ninst\u271d : Preorder \u03b9\nl : Filter \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : HasAntitoneBasis l s\nt : Set \u03b1\n\u22a2 (\u2203 i, True \u2227 s i \u2286 t) \u2194 \u2203 i, s i \u2286 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Order/Group/Defs.lean", "full_name": "div_lt_div_iff_right", "start": [878, 1], "end": [879, 59], "traced_tactics": [{"tactic": "simpa only [div_eq_mul_inv] using mul_lt_mul_iff_right _", "state_before": "\u03b1 : Type u\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c\u271d d c : \u03b1\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/Data/Matrix/Basic.lean", "full_name": "RingEquiv.mapMatrix_refl", "start": [1565, 1], "end": [1566, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/MeanInequalities.lean", "full_name": "NNReal.Lp_add_le_tsum", "start": [495, 1], "end": [517, 34], "traced_tactics": [{"tactic": "have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp", "state_before": "\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\n\u22a2 (Summable fun i => (f i + g i) ^ p) \u2227\n    (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)", "state_after": "\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\n\u22a2 (Summable fun i => (f i + g i) ^ p) \u2227\n    (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)"}, {"tactic": "have H\u2081 : \u2200 s : Finset \u03b9,\n    (\u2211 i in s, (f i + g i) ^ p) \u2264\n      ((\u2211' i, f i ^ p) ^ (1 / p) + (\u2211' i, g i ^ p) ^ (1 / p)) ^ p := by\n  intro s\n  rw [\u2190 NNReal.rpow_one_div_le_iff pos]\n  refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>\n      rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>\n    refine' sum_le_tsum _ (fun _ _ => zero_le _) _\n  exacts [hf, hg]", "state_before": "\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\n\u22a2 (Summable fun i => (f i + g i) ^ p) \u2227\n    (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)", "state_after": "\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\n\u22a2 (Summable fun i => (f i + g i) ^ p) \u2227\n    (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)"}, {"tactic": "have bdd : BddAbove (Set.range fun s => \u2211 i in s, (f i + g i) ^ p) := by\n  refine' \u27e8((\u2211' i, f i ^ p) ^ (1 / p) + (\u2211' i, g i ^ p) ^ (1 / p)) ^ p, _\u27e9\n  rintro a \u27e8s, rfl\u27e9\n  exact H\u2081 s", "state_before": "\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\n\u22a2 (Summable fun i => (f i + g i) ^ p) \u2227\n    (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)", "state_after": "\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\nbdd : BddAbove (Set.range fun s => \u2211 i in s, (f i + g i) ^ p)\n\u22a2 (Summable fun i => (f i + g i) ^ p) \u2227\n    (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)"}, {"tactic": "have H\u2082 : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable", "state_before": "\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\nbdd : BddAbove (Set.range fun s => \u2211 i in s, (f i + g i) ^ p)\n\u22a2 (Summable fun i => (f i + g i) ^ p) \u2227\n    (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)", "state_after": "\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\nbdd : BddAbove (Set.range fun s => \u2211 i in s, (f i + g i) ^ p)\nH\u2082 : Summable fun i => (f i + g i) ^ p\n\u22a2 (Summable fun i => (f i + g i) ^ p) \u2227\n    (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)"}, {"tactic": "refine' \u27e8H\u2082, _\u27e9", "state_before": "\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\nbdd : BddAbove (Set.range fun s => \u2211 i in s, (f i + g i) ^ p)\nH\u2082 : Summable fun i => (f i + g i) ^ p\n\u22a2 (Summable fun i => (f i + g i) ^ p) \u2227\n    (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)", "state_after": "\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\nbdd : BddAbove (Set.range fun s => \u2211 i in s, (f i + g i) ^ p)\nH\u2082 : Summable fun i => (f i + g i) ^ p\n\u22a2 (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)"}, {"tactic": "rw [NNReal.rpow_one_div_le_iff pos]", "state_before": "\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\nbdd : BddAbove (Set.range fun s => \u2211 i in s, (f i + g i) ^ p)\nH\u2082 : Summable fun i => (f i + g i) ^ p\n\u22a2 (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)", "state_after": "\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\nbdd : BddAbove (Set.range fun s => \u2211 i in s, (f i + g i) ^ p)\nH\u2082 : Summable fun i => (f i + g i) ^ p\n\u22a2 (\u2211' (i : \u03b9), (f i + g i) ^ p) \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p"}, {"tactic": "refine' tsum_le_of_sum_le H\u2082 H\u2081", "state_before": "\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\nbdd : BddAbove (Set.range fun s => \u2211 i in s, (f i + g i) ^ p)\nH\u2082 : Summable fun i => (f i + g i) ^ p\n\u22a2 (\u2211' (i : \u03b9), (f i + g i) ^ p) \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p", "state_after": "no goals"}, {"tactic": "intro s", "state_before": "\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\n\u22a2 \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p", "state_after": "\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\ns : Finset \u03b9\n\u22a2 \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p"}, {"tactic": "rw [\u2190 NNReal.rpow_one_div_le_iff pos]", "state_before": "\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\ns : Finset \u03b9\n\u22a2 \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p", "state_after": "\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\ns : Finset \u03b9\n\u22a2 (\u2211 i in s, (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)"}, {"tactic": "refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;>\n    rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>\n  refine' sum_le_tsum _ (fun _ _ => zero_le _) _", "state_before": "\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\ns : Finset \u03b9\n\u22a2 (\u2211 i in s, (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)", "state_after": "case refine'_1\n\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\ns : Finset \u03b9\n\u22a2 Summable fun i => f i ^ p\n\ncase refine'_2\n\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\ns : Finset \u03b9\n\u22a2 Summable fun i => g i ^ p"}, {"tactic": "exacts [hf, hg]", "state_before": "case refine'_1\n\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\ns : Finset \u03b9\n\u22a2 Summable fun i => f i ^ p\n\ncase refine'_2\n\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\ns : Finset \u03b9\n\u22a2 Summable fun i => g i ^ p", "state_after": "no goals"}, {"tactic": "refine' \u27e8((\u2211' i, f i ^ p) ^ (1 / p) + (\u2211' i, g i ^ p) ^ (1 / p)) ^ p, _\u27e9", "state_before": "\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\n\u22a2 BddAbove (Set.range fun s => \u2211 i in s, (f i + g i) ^ p)", "state_after": "\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\n\u22a2 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p \u2208\n    upperBounds (Set.range fun s => \u2211 i in s, (f i + g i) ^ p)"}, {"tactic": "rintro a \u27e8s, rfl\u27e9", "state_before": "\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\n\u22a2 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p \u2208\n    upperBounds (Set.range fun s => \u2211 i in s, (f i + g i) ^ p)", "state_after": "case intro\n\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\ns : Finset \u03b9\n\u22a2 (fun s => \u2211 i in s, (f i + g i) ^ p) s \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p"}, {"tactic": "exact H\u2081 s", "state_before": "case intro\n\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\ns : Finset \u03b9\n\u22a2 (fun s => \u2211 i in s, (f i + g i) ^ p) s \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean", "full_name": "HasDerivAtFilter.const_smul", "start": [130, 8], "end": [132, 49], "traced_tactics": [{"tactic": "simpa using (hf.const_smul c).hasDerivAtFilter", "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : 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_1\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F\ninst\u271d : ContinuousConstSMul R F\nc : R\nhf : HasDerivAtFilter f f' x L\n\u22a2 HasDerivAtFilter (fun y => c \u2022 f y) (c \u2022 f') x L", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.to_nat_to_int", "start": [481, 1], "end": [482, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Mul.lean", "full_name": "fderiv_mul_const'", "start": [453, 1], "end": [455, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/BigOperators/Basic.lean", "full_name": "Multiset.disjoint_sum_left", "start": [2067, 1], "end": [2071, 33], "traced_tactics": [{"tactic": "rw [quot_mk_to_coe, Multiset.coe_sum]", "state_before": "\u03b9 : Type ?u.895595\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nf g : \u03b1 \u2192 \u03b2\na : Multiset \u03b1\ni : Multiset (Multiset \u03b1)\nl : List (Multiset \u03b1)\n\u22a2 Disjoint (sum (Quotient.mk (List.isSetoid (Multiset \u03b1)) l)) a \u2194\n    \u2200 (b : Multiset \u03b1), b \u2208 Quotient.mk (List.isSetoid (Multiset \u03b1)) l \u2192 Disjoint b a", "state_after": "\u03b9 : Type ?u.895595\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nf g : \u03b1 \u2192 \u03b2\na : Multiset \u03b1\ni : Multiset (Multiset \u03b1)\nl : List (Multiset \u03b1)\n\u22a2 Disjoint (List.sum l) a \u2194 \u2200 (b : Multiset \u03b1), b \u2208 \u2191l \u2192 Disjoint b a"}, {"tactic": "exact disjoint_list_sum_left", "state_before": "\u03b9 : Type ?u.895595\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nf g : \u03b1 \u2192 \u03b2\na : Multiset \u03b1\ni : Multiset (Multiset \u03b1)\nl : List (Multiset \u03b1)\n\u22a2 Disjoint (List.sum l) a \u2194 \u2200 (b : Multiset \u03b1), b \u2208 \u2191l \u2192 Disjoint b a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "e68aa8f5fe47aad78987df45f99094afbcb5e936", "file_path": "Std/Data/List/Lemmas.lean", "full_name": "List.get_singleton", "start": [584, 9], "end": [585, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Algebra/Monoid.lean", "full_name": "Filter.Tendsto.mul", "start": [123, 1], "end": [125, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/ProjectiveSpace/Independence.lean", "full_name": "Projectivization.dependent_pair_iff_eq", "start": [111, 1], "end": [116, 31], "traced_tactics": [{"tactic": "rw [dependent_iff_not_independent, independent_iff, linearIndependent_fin2,\n  Function.comp_apply, Matrix.cons_val_one, Matrix.head_cons, Ne.def]", "state_before": "\u03b9 : Type ?u.38028\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nu v : \u2119 K V\n\u22a2 Dependent ![u, v] \u2194 u = v", "state_after": "\u03b9 : Type ?u.38028\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nu v : \u2119 K V\n\u22a2 \u00ac(\u00acProjectivization.rep v = 0 \u2227 \u2200 (a : K), a \u2022 Projectivization.rep v \u2260 (Projectivization.rep \u2218 ![u, v]) 0) \u2194 u = v"}, {"tactic": "simp only [Matrix.cons_val_zero, not_and, not_forall, Classical.not_not, Function.comp_apply,\n  \u2190 mk_eq_mk_iff' K _ _ (rep_nonzero u) (rep_nonzero v), mk_rep, imp_iff_right_iff]", "state_before": "\u03b9 : Type ?u.38028\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nu v : \u2119 K V\n\u22a2 \u00ac(\u00acProjectivization.rep v = 0 \u2227 \u2200 (a : K), a \u2022 Projectivization.rep v \u2260 (Projectivization.rep \u2218 ![u, v]) 0) \u2194 u = v", "state_after": "\u03b9 : Type ?u.38028\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nu v : \u2119 K V\n\u22a2 \u00acProjectivization.rep v = 0 \u2228 u = v"}, {"tactic": "exact Or.inl (rep_nonzero v)", "state_before": "\u03b9 : Type ?u.38028\nK : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nu v : \u2119 K V\n\u22a2 \u00acProjectivization.rep v = 0 \u2228 u = v", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Young/YoungDiagram.lean", "full_name": "YoungDiagram.length_rowLens", "start": [425, 1], "end": [426, 58], "traced_tactics": [{"tactic": "simp only [rowLens, List.length_map, List.length_range]", "state_before": "\u03bc : YoungDiagram\n\u22a2 List.length (rowLens \u03bc) = colLen \u03bc 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm.lean", "full_name": "ContinuousLinearMap.sSup_closed_unit_ball_eq_nnnorm", "start": [565, 1], "end": [575, 48], "traced_tactics": [{"tactic": "have hbdd : \u2200 y \u2208 (fun x => \u2016f x\u2016\u208a) '' closedBall 0 1, y \u2264 \u2016f\u2016\u208a := by\n  rintro - \u27e8x, hx, rfl\u27e9\n  exact f.unit_le_op_norm x (mem_closedBall_zero_iff.1 hx)", "state_before": "\ud835\udd5c\u271d : Type ?u.825981\n\ud835\udd5c\u2082\u271d : Type ?u.825984\n\ud835\udd5c\u2083 : Type ?u.825987\nE\u271d : Type ?u.825990\nE\u2097 : Type ?u.825993\nF\u271d : Type ?u.825996\nF\u2097 : Type ?u.825999\nG : Type ?u.826002\nG\u2097 : Type ?u.826005\n\ud835\udcd5 : Type ?u.826008\ninst\u271d\u00b2\u00b3 : SeminormedAddCommGroup E\u271d\ninst\u271d\u00b2\u00b2 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b2\u00b9 : SeminormedAddCommGroup F\u271d\ninst\u271d\u00b2\u2070 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2079 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\u271d\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\u271d\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u271d E\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u271d E\u2097\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082\u271d F\u271d\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c\u271d F\u2097\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2079 : NormedSpace \ud835\udd5c\u271d G\u2097\n\u03c3\u2081\u2082\u271d : \ud835\udd5c\u271d \u2192+* \ud835\udd5c\u2082\u271d\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082\u271d \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c\u271d \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2078 : RingHomCompTriple \u03c3\u2081\u2082\u271d \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2077 : RingHomIsometric \u03c3\u2081\u2082\u271d\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SeminormedAddCommGroup F\ninst\u271d\u2074 : DenselyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\nf : E \u2192SL[\u03c3\u2081\u2082] F\n\u22a2 sSup ((fun x => \u2016\u2191f x\u2016\u208a) '' closedBall 0 1) = \u2016f\u2016\u208a", "state_after": "\ud835\udd5c\u271d : Type ?u.825981\n\ud835\udd5c\u2082\u271d : Type ?u.825984\n\ud835\udd5c\u2083 : Type ?u.825987\nE\u271d : Type ?u.825990\nE\u2097 : Type ?u.825993\nF\u271d : Type ?u.825996\nF\u2097 : Type ?u.825999\nG : Type ?u.826002\nG\u2097 : Type ?u.826005\n\ud835\udcd5 : Type ?u.826008\ninst\u271d\u00b2\u00b3 : SeminormedAddCommGroup E\u271d\ninst\u271d\u00b2\u00b2 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b2\u00b9 : SeminormedAddCommGroup F\u271d\ninst\u271d\u00b2\u2070 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2079 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\u271d\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\u271d\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u271d E\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u271d E\u2097\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082\u271d F\u271d\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c\u271d F\u2097\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2079 : NormedSpace \ud835\udd5c\u271d G\u2097\n\u03c3\u2081\u2082\u271d : \ud835\udd5c\u271d \u2192+* \ud835\udd5c\u2082\u271d\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082\u271d \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c\u271d \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2078 : RingHomCompTriple \u03c3\u2081\u2082\u271d \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2077 : RingHomIsometric \u03c3\u2081\u2082\u271d\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SeminormedAddCommGroup F\ninst\u271d\u2074 : DenselyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\nf : E \u2192SL[\u03c3\u2081\u2082] F\nhbdd : \u2200 (y : \u211d\u22650), y \u2208 (fun x => \u2016\u2191f x\u2016\u208a) '' closedBall 0 1 \u2192 y \u2264 \u2016f\u2016\u208a\n\u22a2 sSup ((fun x => \u2016\u2191f x\u2016\u208a) '' closedBall 0 1) = \u2016f\u2016\u208a"}, {"tactic": "refine' le_antisymm (csSup_le ((nonempty_closedBall.mpr zero_le_one).image _) hbdd) _", "state_before": "\ud835\udd5c\u271d : Type ?u.825981\n\ud835\udd5c\u2082\u271d : Type ?u.825984\n\ud835\udd5c\u2083 : Type ?u.825987\nE\u271d : Type ?u.825990\nE\u2097 : Type ?u.825993\nF\u271d : Type ?u.825996\nF\u2097 : Type ?u.825999\nG : Type ?u.826002\nG\u2097 : Type ?u.826005\n\ud835\udcd5 : Type ?u.826008\ninst\u271d\u00b2\u00b3 : SeminormedAddCommGroup E\u271d\ninst\u271d\u00b2\u00b2 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b2\u00b9 : SeminormedAddCommGroup F\u271d\ninst\u271d\u00b2\u2070 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2079 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\u271d\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\u271d\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u271d E\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u271d E\u2097\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082\u271d F\u271d\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c\u271d F\u2097\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2079 : NormedSpace \ud835\udd5c\u271d G\u2097\n\u03c3\u2081\u2082\u271d : \ud835\udd5c\u271d \u2192+* \ud835\udd5c\u2082\u271d\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082\u271d \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c\u271d \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2078 : RingHomCompTriple \u03c3\u2081\u2082\u271d \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2077 : RingHomIsometric \u03c3\u2081\u2082\u271d\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SeminormedAddCommGroup F\ninst\u271d\u2074 : DenselyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\nf : E \u2192SL[\u03c3\u2081\u2082] F\nhbdd : \u2200 (y : \u211d\u22650), y \u2208 (fun x => \u2016\u2191f x\u2016\u208a) '' closedBall 0 1 \u2192 y \u2264 \u2016f\u2016\u208a\n\u22a2 sSup ((fun x => \u2016\u2191f x\u2016\u208a) '' closedBall 0 1) = \u2016f\u2016\u208a", "state_after": "\ud835\udd5c\u271d : Type ?u.825981\n\ud835\udd5c\u2082\u271d : Type ?u.825984\n\ud835\udd5c\u2083 : Type ?u.825987\nE\u271d : Type ?u.825990\nE\u2097 : Type ?u.825993\nF\u271d : Type ?u.825996\nF\u2097 : Type ?u.825999\nG : Type ?u.826002\nG\u2097 : Type ?u.826005\n\ud835\udcd5 : Type ?u.826008\ninst\u271d\u00b2\u00b3 : SeminormedAddCommGroup E\u271d\ninst\u271d\u00b2\u00b2 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b2\u00b9 : SeminormedAddCommGroup F\u271d\ninst\u271d\u00b2\u2070 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2079 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\u271d\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\u271d\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u271d E\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u271d E\u2097\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082\u271d F\u271d\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c\u271d F\u2097\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2079 : NormedSpace \ud835\udd5c\u271d G\u2097\n\u03c3\u2081\u2082\u271d : \ud835\udd5c\u271d \u2192+* \ud835\udd5c\u2082\u271d\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082\u271d \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c\u271d \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2078 : RingHomCompTriple \u03c3\u2081\u2082\u271d \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2077 : RingHomIsometric \u03c3\u2081\u2082\u271d\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SeminormedAddCommGroup F\ninst\u271d\u2074 : DenselyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\nf : E \u2192SL[\u03c3\u2081\u2082] F\nhbdd : \u2200 (y : \u211d\u22650), y \u2208 (fun x => \u2016\u2191f x\u2016\u208a) '' closedBall 0 1 \u2192 y \u2264 \u2016f\u2016\u208a\n\u22a2 \u2016f\u2016\u208a \u2264 sSup ((fun x => \u2016\u2191f x\u2016\u208a) '' closedBall 0 1)"}, {"tactic": "rw [\u2190 sSup_unit_ball_eq_nnnorm]", "state_before": "\ud835\udd5c\u271d : Type ?u.825981\n\ud835\udd5c\u2082\u271d : Type ?u.825984\n\ud835\udd5c\u2083 : Type ?u.825987\nE\u271d : Type ?u.825990\nE\u2097 : Type ?u.825993\nF\u271d : Type ?u.825996\nF\u2097 : Type ?u.825999\nG : Type ?u.826002\nG\u2097 : Type ?u.826005\n\ud835\udcd5 : Type ?u.826008\ninst\u271d\u00b2\u00b3 : SeminormedAddCommGroup E\u271d\ninst\u271d\u00b2\u00b2 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b2\u00b9 : SeminormedAddCommGroup F\u271d\ninst\u271d\u00b2\u2070 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2079 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\u271d\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\u271d\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u271d E\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u271d E\u2097\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082\u271d F\u271d\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c\u271d F\u2097\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2079 : NormedSpace \ud835\udd5c\u271d G\u2097\n\u03c3\u2081\u2082\u271d : \ud835\udd5c\u271d \u2192+* \ud835\udd5c\u2082\u271d\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082\u271d \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c\u271d \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2078 : RingHomCompTriple \u03c3\u2081\u2082\u271d \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2077 : RingHomIsometric \u03c3\u2081\u2082\u271d\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SeminormedAddCommGroup F\ninst\u271d\u2074 : DenselyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\nf : E \u2192SL[\u03c3\u2081\u2082] F\nhbdd : \u2200 (y : \u211d\u22650), y \u2208 (fun x => \u2016\u2191f x\u2016\u208a) '' closedBall 0 1 \u2192 y \u2264 \u2016f\u2016\u208a\n\u22a2 \u2016f\u2016\u208a \u2264 sSup ((fun x => \u2016\u2191f x\u2016\u208a) '' closedBall 0 1)", "state_after": "\ud835\udd5c\u271d : Type ?u.825981\n\ud835\udd5c\u2082\u271d : Type ?u.825984\n\ud835\udd5c\u2083 : Type ?u.825987\nE\u271d : Type ?u.825990\nE\u2097 : Type ?u.825993\nF\u271d : Type ?u.825996\nF\u2097 : Type ?u.825999\nG : Type ?u.826002\nG\u2097 : Type ?u.826005\n\ud835\udcd5 : Type ?u.826008\ninst\u271d\u00b2\u00b3 : SeminormedAddCommGroup E\u271d\ninst\u271d\u00b2\u00b2 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b2\u00b9 : SeminormedAddCommGroup F\u271d\ninst\u271d\u00b2\u2070 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2079 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\u271d\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\u271d\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u271d E\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u271d E\u2097\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082\u271d F\u271d\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c\u271d F\u2097\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2079 : NormedSpace \ud835\udd5c\u271d G\u2097\n\u03c3\u2081\u2082\u271d : \ud835\udd5c\u271d \u2192+* \ud835\udd5c\u2082\u271d\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082\u271d \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c\u271d \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2078 : RingHomCompTriple \u03c3\u2081\u2082\u271d \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2077 : RingHomIsometric \u03c3\u2081\u2082\u271d\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SeminormedAddCommGroup F\ninst\u271d\u2074 : DenselyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\nf : E \u2192SL[\u03c3\u2081\u2082] F\nhbdd : \u2200 (y : \u211d\u22650), y \u2208 (fun x => \u2016\u2191f x\u2016\u208a) '' closedBall 0 1 \u2192 y \u2264 \u2016f\u2016\u208a\n\u22a2 sSup ((fun x => \u2016\u2191f x\u2016\u208a) '' ball 0 1) \u2264 sSup ((fun x => \u2016\u2191f x\u2016\u208a) '' closedBall 0 1)"}, {"tactic": "exact csSup_le_csSup \u27e8\u2016f\u2016\u208a, hbdd\u27e9 ((nonempty_ball.2 zero_lt_one).image _)\n  (Set.image_subset _ ball_subset_closedBall)", "state_before": "\ud835\udd5c\u271d : Type ?u.825981\n\ud835\udd5c\u2082\u271d : Type ?u.825984\n\ud835\udd5c\u2083 : Type ?u.825987\nE\u271d : Type ?u.825990\nE\u2097 : Type ?u.825993\nF\u271d : Type ?u.825996\nF\u2097 : Type ?u.825999\nG : Type ?u.826002\nG\u2097 : Type ?u.826005\n\ud835\udcd5 : Type ?u.826008\ninst\u271d\u00b2\u00b3 : SeminormedAddCommGroup E\u271d\ninst\u271d\u00b2\u00b2 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b2\u00b9 : SeminormedAddCommGroup F\u271d\ninst\u271d\u00b2\u2070 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2079 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\u271d\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\u271d\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u271d E\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u271d E\u2097\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082\u271d F\u271d\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c\u271d F\u2097\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2079 : NormedSpace \ud835\udd5c\u271d G\u2097\n\u03c3\u2081\u2082\u271d : \ud835\udd5c\u271d \u2192+* \ud835\udd5c\u2082\u271d\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082\u271d \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c\u271d \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2078 : RingHomCompTriple \u03c3\u2081\u2082\u271d \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2077 : RingHomIsometric \u03c3\u2081\u2082\u271d\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SeminormedAddCommGroup F\ninst\u271d\u2074 : DenselyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\nf : E \u2192SL[\u03c3\u2081\u2082] F\nhbdd : \u2200 (y : \u211d\u22650), y \u2208 (fun x => \u2016\u2191f x\u2016\u208a) '' closedBall 0 1 \u2192 y \u2264 \u2016f\u2016\u208a\n\u22a2 sSup ((fun x => \u2016\u2191f x\u2016\u208a) '' ball 0 1) \u2264 sSup ((fun x => \u2016\u2191f x\u2016\u208a) '' closedBall 0 1)", "state_after": "no goals"}, {"tactic": "rintro - \u27e8x, hx, rfl\u27e9", "state_before": "\ud835\udd5c\u271d : Type ?u.825981\n\ud835\udd5c\u2082\u271d : Type ?u.825984\n\ud835\udd5c\u2083 : Type ?u.825987\nE\u271d : Type ?u.825990\nE\u2097 : Type ?u.825993\nF\u271d : Type ?u.825996\nF\u2097 : Type ?u.825999\nG : Type ?u.826002\nG\u2097 : Type ?u.826005\n\ud835\udcd5 : Type ?u.826008\ninst\u271d\u00b2\u00b3 : SeminormedAddCommGroup E\u271d\ninst\u271d\u00b2\u00b2 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b2\u00b9 : SeminormedAddCommGroup F\u271d\ninst\u271d\u00b2\u2070 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2079 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\u271d\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\u271d\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u271d E\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u271d E\u2097\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082\u271d F\u271d\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c\u271d F\u2097\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2079 : NormedSpace \ud835\udd5c\u271d G\u2097\n\u03c3\u2081\u2082\u271d : \ud835\udd5c\u271d \u2192+* \ud835\udd5c\u2082\u271d\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082\u271d \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c\u271d \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2078 : RingHomCompTriple \u03c3\u2081\u2082\u271d \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2077 : RingHomIsometric \u03c3\u2081\u2082\u271d\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SeminormedAddCommGroup F\ninst\u271d\u2074 : DenselyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\nf : E \u2192SL[\u03c3\u2081\u2082] F\n\u22a2 \u2200 (y : \u211d\u22650), y \u2208 (fun x => \u2016\u2191f x\u2016\u208a) '' closedBall 0 1 \u2192 y \u2264 \u2016f\u2016\u208a", "state_after": "case intro.intro\n\ud835\udd5c\u271d : Type ?u.825981\n\ud835\udd5c\u2082\u271d : Type ?u.825984\n\ud835\udd5c\u2083 : Type ?u.825987\nE\u271d : Type ?u.825990\nE\u2097 : Type ?u.825993\nF\u271d : Type ?u.825996\nF\u2097 : Type ?u.825999\nG : Type ?u.826002\nG\u2097 : Type ?u.826005\n\ud835\udcd5 : Type ?u.826008\ninst\u271d\u00b2\u00b3 : SeminormedAddCommGroup E\u271d\ninst\u271d\u00b2\u00b2 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b2\u00b9 : SeminormedAddCommGroup F\u271d\ninst\u271d\u00b2\u2070 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2079 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\u271d\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\u271d\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u271d E\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u271d E\u2097\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082\u271d F\u271d\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c\u271d F\u2097\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2079 : NormedSpace \ud835\udd5c\u271d G\u2097\n\u03c3\u2081\u2082\u271d : \ud835\udd5c\u271d \u2192+* \ud835\udd5c\u2082\u271d\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082\u271d \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c\u271d \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2078 : RingHomCompTriple \u03c3\u2081\u2082\u271d \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2077 : RingHomIsometric \u03c3\u2081\u2082\u271d\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SeminormedAddCommGroup F\ninst\u271d\u2074 : DenselyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\nf : E \u2192SL[\u03c3\u2081\u2082] F\nx : E\nhx : x \u2208 closedBall 0 1\n\u22a2 (fun x => \u2016\u2191f x\u2016\u208a) x \u2264 \u2016f\u2016\u208a"}, {"tactic": "exact f.unit_le_op_norm x (mem_closedBall_zero_iff.1 hx)", "state_before": "case intro.intro\n\ud835\udd5c\u271d : Type ?u.825981\n\ud835\udd5c\u2082\u271d : Type ?u.825984\n\ud835\udd5c\u2083 : Type ?u.825987\nE\u271d : Type ?u.825990\nE\u2097 : Type ?u.825993\nF\u271d : Type ?u.825996\nF\u2097 : Type ?u.825999\nG : Type ?u.826002\nG\u2097 : Type ?u.826005\n\ud835\udcd5 : Type ?u.826008\ninst\u271d\u00b2\u00b3 : SeminormedAddCommGroup E\u271d\ninst\u271d\u00b2\u00b2 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b2\u00b9 : SeminormedAddCommGroup F\u271d\ninst\u271d\u00b2\u2070 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u2079 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u2078 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u2077 : NontriviallyNormedField \ud835\udd5c\u271d\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\u2082\u271d\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c\u271d E\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c\u271d E\u2097\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2082\u271d F\u271d\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c\u271d F\u2097\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u2079 : NormedSpace \ud835\udd5c\u271d G\u2097\n\u03c3\u2081\u2082\u271d : \ud835\udd5c\u271d \u2192+* \ud835\udd5c\u2082\u271d\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082\u271d \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c\u271d \u2192+* \ud835\udd5c\u2083\ninst\u271d\u2078 : RingHomCompTriple \u03c3\u2081\u2082\u271d \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2077 : RingHomIsometric \u03c3\u2081\u2082\u271d\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : SeminormedAddCommGroup F\ninst\u271d\u2074 : DenselyNormedField \ud835\udd5c\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\nf : E \u2192SL[\u03c3\u2081\u2082] F\nx : E\nhx : x \u2208 closedBall 0 1\n\u22a2 (fun x => \u2016\u2191f x\u2016\u208a) x \u2264 \u2016f\u2016\u208a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/FreeGroup.lean", "full_name": "FreeGroup.reduce.Step.eq", "start": [1211, 1], "end": [1213, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "ContinuousLinearMap.integrable_comp", "start": [1445, 1], "end": [1449, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Hom.lean", "full_name": "AlgHom.map_finsupp_prod", "start": [471, 11], "end": [473, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/Prod.lean", "full_name": "Filter.prod_comm'", "start": [250, 1], "end": [251, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "isClosed_iff_nhds", "start": [1371, 1], "end": [1372, 74], "traced_tactics": [{"tactic": "simp_rw [isClosed_iff_clusterPt, ClusterPt, inf_principal_neBot_iff]", "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 s \u2194 \u2200 (x : \u03b1), (\u2200 (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 Set.Nonempty (U \u2229 s)) \u2192 x \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "full_name": "Subgroup.ker_subtype", "start": [2849, 1], "end": [2850, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Sites/Subsheaf.lean", "full_name": "CategoryTheory.GrothendieckTopology.Subpresheaf.to_sheafifyLift", "start": [303, 1], "end": [311, 83], "traced_tactics": [{"tactic": "ext (U s)", "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\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\n\u22a2 homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G f h = f", "state_after": "case w.h.h\nC : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU : C\u1d52\u1d56\ns : (toPresheaf G).obj U\n\u22a2 (homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G f h).app U s = f.app U s"}, {"tactic": "apply (h _ ((Subpresheaf.homOfLe (G.le_sheafify J)).app U s).prop).isSeparatedFor.ext", "state_before": "case w.h.h\nC : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU : C\u1d52\u1d56\ns : (toPresheaf G).obj U\n\u22a2 (homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G f h).app U s = f.app U s", "state_after": "case w.h.h\nC : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU : C\u1d52\u1d56\ns : (toPresheaf G).obj U\n\u22a2 \u2200 \u2983Y : C\u2984 \u2983f_1 : Y \u27f6 U.unop\u2984,\n    (sieveOfSection G \u2191((homOfLe (_ : G \u2264 sheafify J G)).app U s)).arrows f_1 \u2192\n      F'.map f_1.op ((homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G f h).app U s) = F'.map f_1.op (f.app U s)"}, {"tactic": "intro V i hi", "state_before": "case w.h.h\nC : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU : C\u1d52\u1d56\ns : (toPresheaf G).obj U\n\u22a2 \u2200 \u2983Y : C\u2984 \u2983f_1 : Y \u27f6 U.unop\u2984,\n    (sieveOfSection G \u2191((homOfLe (_ : G \u2264 sheafify J G)).app U s)).arrows f_1 \u2192\n      F'.map f_1.op ((homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G f h).app U s) = F'.map f_1.op (f.app U s)", "state_after": "case w.h.h\nC : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU : C\u1d52\u1d56\ns : (toPresheaf G).obj U\nV : C\ni : V \u27f6 U.unop\nhi : (sieveOfSection G \u2191((homOfLe (_ : G \u2264 sheafify J G)).app U s)).arrows i\n\u22a2 F'.map i.op ((homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G f h).app U s) = F'.map i.op (f.app U s)"}, {"tactic": "have := elementwise_of% f.naturality", "state_before": "case w.h.h\nC : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU : C\u1d52\u1d56\ns : (toPresheaf G).obj U\nV : C\ni : V \u27f6 U.unop\nhi : (sieveOfSection G \u2191((homOfLe (_ : G \u2264 sheafify J G)).app U s)).arrows i\n\u22a2 F'.map i.op ((homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G f h).app U s) = F'.map i.op (f.app U s)", "state_after": "case w.h.h\nC : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU : C\u1d52\u1d56\ns : (toPresheaf G).obj U\nV : C\ni : V \u27f6 U.unop\nhi : (sieveOfSection G \u2191((homOfLe (_ : G \u2264 sheafify J G)).app U s)).arrows i\nthis :\n  \u2200 \u2983X Y : C\u1d52\u1d56\u2984 (f_1 : X \u27f6 Y) (x : (toPresheaf G).obj X), f.app Y ((toPresheaf G).map f_1 x) = F'.map f_1 (f.app X x)\n\u22a2 F'.map i.op ((homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G f h).app U s) = F'.map i.op (f.app U s)"}, {"tactic": "exact (Presieve.IsSheafFor.valid_glue (h _ ((homOfLe (_ : G \u2264 sheafify J G)).app U s).2)\n  ((G.family_of_elements_compatible _).compPresheafMap _) _ hi).trans (this _ _)", "state_before": "case w.h.h\nC : Type u\ninst\u271d : Category C\nJ : GrothendieckTopology C\nF F' F'' : C\u1d52\u1d56 \u2964 Type w\nG G' : Subpresheaf F\nf : toPresheaf G \u27f6 F'\nh : Presieve.IsSheaf J F'\nU : C\u1d52\u1d56\ns : (toPresheaf G).obj U\nV : C\ni : V \u27f6 U.unop\nhi : (sieveOfSection G \u2191((homOfLe (_ : G \u2264 sheafify J G)).app U s)).arrows i\nthis :\n  \u2200 \u2983X Y : C\u1d52\u1d56\u2984 (f_1 : X \u27f6 Y) (x : (toPresheaf G).obj X), f.app Y ((toPresheaf G).map f_1 x) = F'.map f_1 (f.app X x)\n\u22a2 F'.map i.op ((homOfLe (_ : G \u2264 sheafify J G) \u226b sheafifyLift G f h).app U s) = F'.map i.op (f.app U s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/SuccPred/Basic.lean", "full_name": "WithBot.pred_coe_of_ne_bot", "start": [1281, 1], "end": [1282, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Hom/Open.lean", "full_name": "ContinuousOpenMap.coe_toContinuousMap", "start": [82, 1], "end": [82, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/LocalEquiv.lean", "full_name": "LocalEquiv.IsImage.symm_apply_mem_iff", "start": [366, 1], "end": [367, 66], "traced_tactics": [{"tactic": "rw [e.left_inv hx, h hx]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.26025\n\u03b4 : Type ?u.26028\ne : LocalEquiv \u03b1 \u03b2\ne' : LocalEquiv \u03b2 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nx\u271d : \u03b1\ny : \u03b2\nh : IsImage e s t\nx : \u03b1\nhx : x \u2208 e.source\n\u22a2 \u2191(LocalEquiv.symm e) (\u2191e x) \u2208 s \u2194 \u2191e x \u2208 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.max_of_nonempty", "start": [1185, 1], "end": [1187, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Basic.lean", "full_name": "convex_uIcc", "start": [320, 1], "end": [321, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "full_name": "MeasureTheory.ae_le_set_inter", "start": [459, 1], "end": [461, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/FunctorCategory.lean", "full_name": "CategoryTheory.Limits.colimit.\u03b9_desc_app", "start": [46, 1], "end": [48, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/UrysohnsLemma.lean", "full_name": "exists_continuous_zero_one_of_closed", "start": [308, 1], "end": [313, 19], "traced_tactics": [{"tactic": "set c : Urysohns.CU X := \u27e8s, t\u1d9c, hs, ht.isOpen_compl, disjoint_left.1 hd\u27e9", "state_before": "X : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\ns t : Set X\nhs : IsClosed s\nht : IsClosed t\nhd : Disjoint s t\n\u22a2 \u2203 f, EqOn (\u2191f) 0 s \u2227 EqOn (\u2191f) 1 t \u2227 \u2200 (x : X), \u2191f x \u2208 Icc 0 1", "state_after": "X : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\ns t : Set X\nhs : IsClosed s\nht : IsClosed t\nhd : Disjoint s t\nc : Urysohns.CU X :=\n  { C := s, U := t\u1d9c, closed_C := hs, open_U := (_ : IsOpen (t\u1d9c)), subset := (_ : \u2200 \u2983a : X\u2984, a \u2208 s \u2192 \u00aca \u2208 t) }\n\u22a2 \u2203 f, EqOn (\u2191f) 0 s \u2227 EqOn (\u2191f) 1 t \u2227 \u2200 (x : X), \u2191f x \u2208 Icc 0 1"}, {"tactic": "exact \u27e8\u27e8c.lim, c.continuous_lim\u27e9, c.lim_of_mem_C, fun x hx => c.lim_of_nmem_U _ fun h => h hx,\n  c.lim_mem_Icc\u27e9", "state_before": "X : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : NormalSpace X\ns t : Set X\nhs : IsClosed s\nht : IsClosed t\nhd : Disjoint s t\nc : Urysohns.CU X :=\n  { C := s, U := t\u1d9c, closed_C := hs, open_U := (_ : IsOpen (t\u1d9c)), subset := (_ : \u2200 \u2983a : X\u2984, a \u2208 s \u2192 \u00aca \u2208 t) }\n\u22a2 \u2203 f, EqOn (\u2191f) 0 s \u2227 EqOn (\u2191f) 1 t \u2227 \u2200 (x : X), \u2191f x \u2208 Icc 0 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.mk_union_le", "start": [2097, 1], "end": [2098, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/QuotientOperations.lean", "full_name": "RingHom.kerLift_mk", "start": [34, 1], "end": [35, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Opposites.lean", "full_name": "MulOpposite.unop_neg", "start": [279, 1], "end": [280, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Int/Order/Basic.lean", "full_name": "Int.le_ediv_of_mul_le", "start": [437, 11], "end": [439, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.vsub_singleton", "start": [649, 1], "end": [650, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.Mem\u2112p.integrable", "start": [794, 1], "end": [796, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/SchurZassenhaus.lean", "full_name": "Subgroup.SchurZassenhausInduction.step5", "start": [245, 9], "end": [247, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Matrix/ZPow.lean", "full_name": "Matrix.zpow_neg_coe_nat", "start": [117, 1], "end": [120, 37], "traced_tactics": [{"tactic": "cases n", "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 : \u2115\n\u22a2 A ^ (-\u2191n) = (A ^ n)\u207b\u00b9", "state_after": "case zero\nn' : Type u_1\ninst\u271d\u00b2 : DecidableEq n'\ninst\u271d\u00b9 : Fintype n'\nR : Type u_2\ninst\u271d : CommRing R\nA : M\n\u22a2 A ^ (-\u2191Nat.zero) = (A ^ Nat.zero)\u207b\u00b9\n\ncase succ\nn' : Type u_1\ninst\u271d\u00b2 : DecidableEq n'\ninst\u271d\u00b9 : Fintype n'\nR : Type u_2\ninst\u271d : CommRing R\nA : M\nn\u271d : \u2115\n\u22a2 A ^ (-\u2191(Nat.succ n\u271d)) = (A ^ Nat.succ n\u271d)\u207b\u00b9"}, {"tactic": "simp", "state_before": "case zero\nn' : Type u_1\ninst\u271d\u00b2 : DecidableEq n'\ninst\u271d\u00b9 : Fintype n'\nR : Type u_2\ninst\u271d : CommRing R\nA : M\n\u22a2 A ^ (-\u2191Nat.zero) = (A ^ Nat.zero)\u207b\u00b9", "state_after": "no goals"}, {"tactic": "exact DivInvMonoid.zpow_neg' _ _", "state_before": "case succ\nn' : Type u_1\ninst\u271d\u00b2 : DecidableEq n'\ninst\u271d\u00b9 : Fintype n'\nR : Type u_2\ninst\u271d : CommRing R\nA : M\nn\u271d : \u2115\n\u22a2 A ^ (-\u2191(Nat.succ n\u271d)) = (A ^ Nat.succ n\u271d)\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Lie/Submodule.lean", "full_name": "LieSubmodule.mem_lieSpan", "start": [611, 1], "end": [612, 76], "traced_tactics": [{"tactic": "change x \u2208 (lieSpan R L s : Set M) \u2194 _", "state_before": "R : 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\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\ns : Set M\nx : M\n\u22a2 x \u2208 lieSpan R L s \u2194 \u2200 (N : LieSubmodule R L M), s \u2286 \u2191N \u2192 x \u2208 N", "state_after": "R : 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\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\ns : Set M\nx : M\n\u22a2 x \u2208 \u2191(lieSpan R L s) \u2194 \u2200 (N : LieSubmodule R L M), s \u2286 \u2191N \u2192 x \u2208 N"}, {"tactic": "erw [sInf_coe]", "state_before": "R : 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\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\ns : Set M\nx : M\n\u22a2 x \u2208 \u2191(lieSpan R L s) \u2194 \u2200 (N : LieSubmodule R L M), s \u2286 \u2191N \u2192 x \u2208 N", "state_after": "R : 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\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\ns : Set M\nx : M\n\u22a2 (x \u2208 \u22c2 (s_1 : LieSubmodule R L M) (_ : s_1 \u2208 {N | s \u2286 \u2191N}), \u2191s_1) \u2194 \u2200 (N : LieSubmodule R L M), s \u2286 \u2191N \u2192 x \u2208 N"}, {"tactic": "exact mem_iInter\u2082", "state_before": "R : 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\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\ns : Set M\nx : M\n\u22a2 (x \u2208 \u22c2 (s_1 : LieSubmodule R L M) (_ : s_1 \u2208 {N | s \u2286 \u2191N}), \u2191s_1) \u2194 \u2200 (N : LieSubmodule R L M), s \u2286 \u2191N \u2192 x \u2208 N", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/NumberTheory/PellMatiyasevic.lean", "full_name": "Pell.matiyasevic", "start": [841, 1], "end": [926, 85], "traced_tactics": [{"tactic": "rw [\u2190 hx, \u2190 hy]", "state_before": "a k x y : \u2115\nx\u271d : \u2203 a1, xn a1 k = x \u2227 yn a1 k = y\na1 : 1 < a\nhx : xn a1 k = x\nhy : yn a1 k = y\n\u22a2 1 < a \u2227\n    k \u2264 y \u2227\n      (x = 1 \u2227 y = 0 \u2228\n        \u2203 u v s t b,\n          x * x - (a * a - 1) * y * y = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227 b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y])", "state_after": "a k x y : \u2115\nx\u271d : \u2203 a1, xn a1 k = x \u2227 yn a1 k = y\na1 : 1 < a\nhx : xn a1 k = x\nhy : yn a1 k = y\n\u22a2 1 < a \u2227\n    k \u2264 yn a1 k \u2227\n      (xn a1 k = 1 \u2227 yn a1 k = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 k * xn a1 k - (a * a - 1) * yn a1 k * yn a1 k = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 k] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 k * yn a1 k \u2223 v \u2227 s \u2261 xn a1 k [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 k])"}, {"tactic": "refine' \u27e8a1,\n    (Nat.eq_zero_or_pos k).elim (fun k0 => by rw [k0]; exact \u27e8le_rfl, Or.inl \u27e8rfl, rfl\u27e9\u27e9)\n      fun kpos => _\u27e9", "state_before": "a k x y : \u2115\nx\u271d : \u2203 a1, xn a1 k = x \u2227 yn a1 k = y\na1 : 1 < a\nhx : xn a1 k = x\nhy : yn a1 k = y\n\u22a2 1 < a \u2227\n    k \u2264 yn a1 k \u2227\n      (xn a1 k = 1 \u2227 yn a1 k = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 k * xn a1 k - (a * a - 1) * yn a1 k * yn a1 k = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 k] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 k * yn a1 k \u2223 v \u2227 s \u2261 xn a1 k [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 k])", "state_after": "a k x y : \u2115\nx\u271d : \u2203 a1, xn a1 k = x \u2227 yn a1 k = y\na1 : 1 < a\nhx : xn a1 k = x\nhy : yn a1 k = y\nkpos : k > 0\n\u22a2 k \u2264 yn a1 k \u2227\n    (xn a1 k = 1 \u2227 yn a1 k = 0 \u2228\n      \u2203 u v s t b,\n        xn a1 k * xn a1 k - (a * a - 1) * yn a1 k * yn a1 k = 1 \u2227\n          u * u - (a * a - 1) * v * v = 1 \u2227\n            s * s - (b * b - 1) * t * t = 1 \u2227\n              1 < b \u2227\n                b \u2261 1 [MOD 4 * yn a1 k] \u2227\n                  b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 k * yn a1 k \u2223 v \u2227 s \u2261 xn a1 k [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 k])"}, {"tactic": "exact\n  let x := xn a1 k\n  let y := yn a1 k\n  let m := 2 * (k * y)\n  let u := xn a1 m\n  let v := yn a1 m\n  have ky : k \u2264 y := yn_ge_n a1 k\n  have yv : y * y \u2223 v := (ysq_dvd_yy a1 k).trans <| (y_dvd_iff _ _ _).2 <| dvd_mul_left _ _\n  have uco : Nat.coprime u (4 * y) :=\n    have : 2 \u2223 v :=\n      modEq_zero_iff_dvd.1 <| (yn_modEq_two _ _).trans (dvd_mul_right _ _).modEq_zero_nat\n    have : Nat.coprime u 2 := (xy_coprime a1 m).coprime_dvd_right this\n    (this.mul_right this).mul_right <|\n      (xy_coprime _ _).coprime_dvd_right (dvd_of_mul_left_dvd yv)\n  let \u27e8b, ba, bm1\u27e9 := chineseRemainder uco a 1\n  have m1 : 1 < m :=\n    have : 0 < k * y := mul_pos kpos (strictMono_y a1 kpos)\n    Nat.mul_le_mul_left 2 this\n  have vp : 0 < v := strictMono_y a1 (lt_trans zero_lt_one m1)\n  have b1 : 1 < b :=\n    have : xn a1 1 < u := strictMono_x a1 m1\n    have : a < u := by simp at this; exact this\n    lt_of_lt_of_le a1 <| by\n      delta ModEq at ba; rw [Nat.mod_eq_of_lt this] at ba; rw [\u2190 ba]\n      apply Nat.mod_le\n  let s := xn b1 k\n  let t := yn b1 k\n  have sx : s \u2261 x [MOD u] := (xy_modEq_of_modEq b1 a1 ba k).left\n  have tk : t \u2261 k [MOD 4 * y] :=\n    have : 4 * y \u2223 b - 1 :=\n      Int.coe_nat_dvd.1 <| by rw [Int.ofNat_sub (le_of_lt b1)]; exact bm1.symm.dvd\n    (yn_modEq_a_sub_one _ _).of_dvd this\n  \u27e8ky,\n    Or.inr\n      \u27e8u, v, s, t, b, pell_eq _ _, pell_eq _ _, pell_eq _ _, b1, bm1, ba, vp, yv, sx, tk\u27e9\u27e9", "state_before": "a k x y : \u2115\nx\u271d : \u2203 a1, xn a1 k = x \u2227 yn a1 k = y\na1 : 1 < a\nhx : xn a1 k = x\nhy : yn a1 k = y\nkpos : k > 0\n\u22a2 k \u2264 yn a1 k \u2227\n    (xn a1 k = 1 \u2227 yn a1 k = 0 \u2228\n      \u2203 u v s t b,\n        xn a1 k * xn a1 k - (a * a - 1) * yn a1 k * yn a1 k = 1 \u2227\n          u * u - (a * a - 1) * v * v = 1 \u2227\n            s * s - (b * b - 1) * t * t = 1 \u2227\n              1 < b \u2227\n                b \u2261 1 [MOD 4 * yn a1 k] \u2227\n                  b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 k * yn a1 k \u2223 v \u2227 s \u2261 xn a1 k [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 k])", "state_after": "no goals"}, {"tactic": "rw [k0]", "state_before": "a k x y : \u2115\nx\u271d : \u2203 a1, xn a1 k = x \u2227 yn a1 k = y\na1 : 1 < a\nhx : xn a1 k = x\nhy : yn a1 k = y\nk0 : k = 0\n\u22a2 k \u2264 yn a1 k \u2227\n    (xn a1 k = 1 \u2227 yn a1 k = 0 \u2228\n      \u2203 u v s t b,\n        xn a1 k * xn a1 k - (a * a - 1) * yn a1 k * yn a1 k = 1 \u2227\n          u * u - (a * a - 1) * v * v = 1 \u2227\n            s * s - (b * b - 1) * t * t = 1 \u2227\n              1 < b \u2227\n                b \u2261 1 [MOD 4 * yn a1 k] \u2227\n                  b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 k * yn a1 k \u2223 v \u2227 s \u2261 xn a1 k [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 k])", "state_after": "a k x y : \u2115\nx\u271d : \u2203 a1, xn a1 k = x \u2227 yn a1 k = y\na1 : 1 < a\nhx : xn a1 k = x\nhy : yn a1 k = y\nk0 : k = 0\n\u22a2 0 \u2264 yn a1 0 \u2227\n    (xn a1 0 = 1 \u2227 yn a1 0 = 0 \u2228\n      \u2203 u v s t b,\n        xn a1 0 * xn a1 0 - (a * a - 1) * yn a1 0 * yn a1 0 = 1 \u2227\n          u * u - (a * a - 1) * v * v = 1 \u2227\n            s * s - (b * b - 1) * t * t = 1 \u2227\n              1 < b \u2227\n                b \u2261 1 [MOD 4 * yn a1 0] \u2227\n                  b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 0 * yn a1 0 \u2223 v \u2227 s \u2261 xn a1 0 [MOD u] \u2227 t \u2261 0 [MOD 4 * yn a1 0])"}, {"tactic": "exact \u27e8le_rfl, Or.inl \u27e8rfl, rfl\u27e9\u27e9", "state_before": "a k x y : \u2115\nx\u271d : \u2203 a1, xn a1 k = x \u2227 yn a1 k = y\na1 : 1 < a\nhx : xn a1 k = x\nhy : yn a1 k = y\nk0 : k = 0\n\u22a2 0 \u2264 yn a1 0 \u2227\n    (xn a1 0 = 1 \u2227 yn a1 0 = 0 \u2228\n      \u2203 u v s t b,\n        xn a1 0 * xn a1 0 - (a * a - 1) * yn a1 0 * yn a1 0 = 1 \u2227\n          u * u - (a * a - 1) * v * v = 1 \u2227\n            s * s - (b * b - 1) * t * t = 1 \u2227\n              1 < b \u2227\n                b \u2261 1 [MOD 4 * yn a1 0] \u2227\n                  b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 0 * yn a1 0 \u2223 v \u2227 s \u2261 xn a1 0 [MOD u] \u2227 t \u2261 0 [MOD 4 * yn a1 0])", "state_after": "no goals"}, {"tactic": "simp at this", "state_before": "a k x\u271d\u00b9 y\u271d : \u2115\nx\u271d : \u2203 a1, xn a1 k = x\u271d\u00b9 \u2227 yn a1 k = y\u271d\na1 : 1 < a\nhx : xn a1 k = x\u271d\u00b9\nhy : yn a1 k = y\u271d\nkpos : k > 0\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nm : \u2115 := 2 * (k * y)\nu : \u2115 := xn a1 m\nv : \u2115 := yn a1 m\nky : k \u2264 y\nyv : y * y \u2223 v\nuco : coprime u (4 * y)\nb : \u2115\nba : b \u2261 a [MOD u]\nbm1 : b \u2261 1 [MOD 4 * y]\nm1 : 1 < m\nvp : 0 < v\nthis : xn a1 1 < u\n\u22a2 a < u", "state_after": "a k x\u271d\u00b9 y\u271d : \u2115\nx\u271d : \u2203 a1, xn a1 k = x\u271d\u00b9 \u2227 yn a1 k = y\u271d\na1 : 1 < a\nhx : xn a1 k = x\u271d\u00b9\nhy : yn a1 k = y\u271d\nkpos : k > 0\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nm : \u2115 := 2 * (k * y)\nu : \u2115 := xn a1 m\nv : \u2115 := yn a1 m\nky : k \u2264 y\nyv : y * y \u2223 v\nuco : coprime u (4 * y)\nb : \u2115\nba : b \u2261 a [MOD u]\nbm1 : b \u2261 1 [MOD 4 * y]\nm1 : 1 < m\nvp : 0 < v\nthis : a < xn a1 (2 * (k * yn a1 k))\n\u22a2 a < u"}, {"tactic": "exact this", "state_before": "a k x\u271d\u00b9 y\u271d : \u2115\nx\u271d : \u2203 a1, xn a1 k = x\u271d\u00b9 \u2227 yn a1 k = y\u271d\na1 : 1 < a\nhx : xn a1 k = x\u271d\u00b9\nhy : yn a1 k = y\u271d\nkpos : k > 0\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nm : \u2115 := 2 * (k * y)\nu : \u2115 := xn a1 m\nv : \u2115 := yn a1 m\nky : k \u2264 y\nyv : y * y \u2223 v\nuco : coprime u (4 * y)\nb : \u2115\nba : b \u2261 a [MOD u]\nbm1 : b \u2261 1 [MOD 4 * y]\nm1 : 1 < m\nvp : 0 < v\nthis : a < xn a1 (2 * (k * yn a1 k))\n\u22a2 a < u", "state_after": "no goals"}, {"tactic": "delta ModEq at ba", "state_before": "a k x\u271d\u00b9 y\u271d : \u2115\nx\u271d : \u2203 a1, xn a1 k = x\u271d\u00b9 \u2227 yn a1 k = y\u271d\na1 : 1 < a\nhx : xn a1 k = x\u271d\u00b9\nhy : yn a1 k = y\u271d\nkpos : k > 0\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nm : \u2115 := 2 * (k * y)\nu : \u2115 := xn a1 m\nv : \u2115 := yn a1 m\nky : k \u2264 y\nyv : y * y \u2223 v\nuco : coprime u (4 * y)\nb : \u2115\nba : b \u2261 a [MOD u]\nbm1 : b \u2261 1 [MOD 4 * y]\nm1 : 1 < m\nvp : 0 < v\nthis\u271d : xn a1 1 < u\nthis : a < u\n\u22a2 a \u2264 b", "state_after": "a k x\u271d\u00b9 y\u271d : \u2115\nx\u271d : \u2203 a1, xn a1 k = x\u271d\u00b9 \u2227 yn a1 k = y\u271d\na1 : 1 < a\nhx : xn a1 k = x\u271d\u00b9\nhy : yn a1 k = y\u271d\nkpos : k > 0\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nm : \u2115 := 2 * (k * y)\nu : \u2115 := xn a1 m\nv : \u2115 := yn a1 m\nky : k \u2264 y\nyv : y * y \u2223 v\nuco : coprime u (4 * y)\nb : \u2115\nba : b % u = a % u\nbm1 : b \u2261 1 [MOD 4 * y]\nm1 : 1 < m\nvp : 0 < v\nthis\u271d : xn a1 1 < u\nthis : a < u\n\u22a2 a \u2264 b"}, {"tactic": "rw [Nat.mod_eq_of_lt this] at ba", "state_before": "a k x\u271d\u00b9 y\u271d : \u2115\nx\u271d : \u2203 a1, xn a1 k = x\u271d\u00b9 \u2227 yn a1 k = y\u271d\na1 : 1 < a\nhx : xn a1 k = x\u271d\u00b9\nhy : yn a1 k = y\u271d\nkpos : k > 0\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nm : \u2115 := 2 * (k * y)\nu : \u2115 := xn a1 m\nv : \u2115 := yn a1 m\nky : k \u2264 y\nyv : y * y \u2223 v\nuco : coprime u (4 * y)\nb : \u2115\nba : b % u = a % u\nbm1 : b \u2261 1 [MOD 4 * y]\nm1 : 1 < m\nvp : 0 < v\nthis\u271d : xn a1 1 < u\nthis : a < u\n\u22a2 a \u2264 b", "state_after": "a k x\u271d\u00b9 y\u271d : \u2115\nx\u271d : \u2203 a1, xn a1 k = x\u271d\u00b9 \u2227 yn a1 k = y\u271d\na1 : 1 < a\nhx : xn a1 k = x\u271d\u00b9\nhy : yn a1 k = y\u271d\nkpos : k > 0\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nm : \u2115 := 2 * (k * y)\nu : \u2115 := xn a1 m\nv : \u2115 := yn a1 m\nky : k \u2264 y\nyv : y * y \u2223 v\nuco : coprime u (4 * y)\nb : \u2115\nba : b % u = a\nbm1 : b \u2261 1 [MOD 4 * y]\nm1 : 1 < m\nvp : 0 < v\nthis\u271d : xn a1 1 < u\nthis : a < u\n\u22a2 a \u2264 b"}, {"tactic": "rw [\u2190 ba]", "state_before": "a k x\u271d\u00b9 y\u271d : \u2115\nx\u271d : \u2203 a1, xn a1 k = x\u271d\u00b9 \u2227 yn a1 k = y\u271d\na1 : 1 < a\nhx : xn a1 k = x\u271d\u00b9\nhy : yn a1 k = y\u271d\nkpos : k > 0\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nm : \u2115 := 2 * (k * y)\nu : \u2115 := xn a1 m\nv : \u2115 := yn a1 m\nky : k \u2264 y\nyv : y * y \u2223 v\nuco : coprime u (4 * y)\nb : \u2115\nba : b % u = a\nbm1 : b \u2261 1 [MOD 4 * y]\nm1 : 1 < m\nvp : 0 < v\nthis\u271d : xn a1 1 < u\nthis : a < u\n\u22a2 a \u2264 b", "state_after": "a k x\u271d\u00b9 y\u271d : \u2115\nx\u271d : \u2203 a1, xn a1 k = x\u271d\u00b9 \u2227 yn a1 k = y\u271d\na1 : 1 < a\nhx : xn a1 k = x\u271d\u00b9\nhy : yn a1 k = y\u271d\nkpos : k > 0\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nm : \u2115 := 2 * (k * y)\nu : \u2115 := xn a1 m\nv : \u2115 := yn a1 m\nky : k \u2264 y\nyv : y * y \u2223 v\nuco : coprime u (4 * y)\nb : \u2115\nba : b % u = a\nbm1 : b \u2261 1 [MOD 4 * y]\nm1 : 1 < m\nvp : 0 < v\nthis\u271d : xn a1 1 < u\nthis : a < u\n\u22a2 b % u \u2264 b"}, {"tactic": "apply Nat.mod_le", "state_before": "a k x\u271d\u00b9 y\u271d : \u2115\nx\u271d : \u2203 a1, xn a1 k = x\u271d\u00b9 \u2227 yn a1 k = y\u271d\na1 : 1 < a\nhx : xn a1 k = x\u271d\u00b9\nhy : yn a1 k = y\u271d\nkpos : k > 0\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nm : \u2115 := 2 * (k * y)\nu : \u2115 := xn a1 m\nv : \u2115 := yn a1 m\nky : k \u2264 y\nyv : y * y \u2223 v\nuco : coprime u (4 * y)\nb : \u2115\nba : b % u = a\nbm1 : b \u2261 1 [MOD 4 * y]\nm1 : 1 < m\nvp : 0 < v\nthis\u271d : xn a1 1 < u\nthis : a < u\n\u22a2 b % u \u2264 b", "state_after": "no goals"}, {"tactic": "rw [Int.ofNat_sub (le_of_lt b1)]", "state_before": "a k x\u271d\u00b9 y\u271d : \u2115\nx\u271d : \u2203 a1, xn a1 k = x\u271d\u00b9 \u2227 yn a1 k = y\u271d\na1 : 1 < a\nhx : xn a1 k = x\u271d\u00b9\nhy : yn a1 k = y\u271d\nkpos : k > 0\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nm : \u2115 := 2 * (k * y)\nu : \u2115 := xn a1 m\nv : \u2115 := yn a1 m\nky : k \u2264 y\nyv : y * y \u2223 v\nuco : coprime u (4 * y)\nb : \u2115\nba : b \u2261 a [MOD u]\nbm1 : b \u2261 1 [MOD 4 * y]\nm1 : 1 < m\nvp : 0 < v\nb1 : 1 < b\ns : \u2115 := xn b1 k\nt : \u2115 := yn b1 k\nsx : s \u2261 x [MOD u]\n\u22a2 \u2191(4 * y) \u2223 \u2191(b - 1)", "state_after": "a k x\u271d\u00b9 y\u271d : \u2115\nx\u271d : \u2203 a1, xn a1 k = x\u271d\u00b9 \u2227 yn a1 k = y\u271d\na1 : 1 < a\nhx : xn a1 k = x\u271d\u00b9\nhy : yn a1 k = y\u271d\nkpos : k > 0\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nm : \u2115 := 2 * (k * y)\nu : \u2115 := xn a1 m\nv : \u2115 := yn a1 m\nky : k \u2264 y\nyv : y * y \u2223 v\nuco : coprime u (4 * y)\nb : \u2115\nba : b \u2261 a [MOD u]\nbm1 : b \u2261 1 [MOD 4 * y]\nm1 : 1 < m\nvp : 0 < v\nb1 : 1 < b\ns : \u2115 := xn b1 k\nt : \u2115 := yn b1 k\nsx : s \u2261 x [MOD u]\n\u22a2 \u2191(4 * y) \u2223 \u2191b - \u21911"}, {"tactic": "exact bm1.symm.dvd", "state_before": "a k x\u271d\u00b9 y\u271d : \u2115\nx\u271d : \u2203 a1, xn a1 k = x\u271d\u00b9 \u2227 yn a1 k = y\u271d\na1 : 1 < a\nhx : xn a1 k = x\u271d\u00b9\nhy : yn a1 k = y\u271d\nkpos : k > 0\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nm : \u2115 := 2 * (k * y)\nu : \u2115 := xn a1 m\nv : \u2115 := yn a1 m\nky : k \u2264 y\nyv : y * y \u2223 v\nuco : coprime u (4 * y)\nb : \u2115\nba : b \u2261 a [MOD u]\nbm1 : b \u2261 1 [MOD 4 * y]\nm1 : 1 < m\nvp : 0 < v\nb1 : 1 < b\ns : \u2115 := xn b1 k\nt : \u2115 := yn b1 k\nsx : s \u2261 x [MOD u]\n\u22a2 \u2191(4 * y) \u2223 \u2191b - \u21911", "state_after": "no goals"}, {"tactic": "rw [y0] at ky", "state_before": "a k x y : \u2115\nx\u271d :\n  1 < a \u2227\n    k \u2264 y \u2227\n      (x = 1 \u2227 y = 0 \u2228\n        \u2203 u v s t b,\n          x * x - (a * a - 1) * y * y = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227 b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y])\na1 : 1 < a\nky : k \u2264 y\no :\n  x = 1 \u2227 y = 0 \u2228\n    \u2203 u v s t b,\n      x * x - (a * a - 1) * y * y = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227 b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\nx1 : x = 1\ny0 : y = 0\n\u22a2 xn a1 k = x \u2227 yn a1 k = y", "state_after": "a k x y : \u2115\nx\u271d :\n  1 < a \u2227\n    k \u2264 y \u2227\n      (x = 1 \u2227 y = 0 \u2228\n        \u2203 u v s t b,\n          x * x - (a * a - 1) * y * y = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227 b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y])\na1 : 1 < a\nky : k \u2264 0\no :\n  x = 1 \u2227 y = 0 \u2228\n    \u2203 u v s t b,\n      x * x - (a * a - 1) * y * y = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227 b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\nx1 : x = 1\ny0 : y = 0\n\u22a2 xn a1 k = x \u2227 yn a1 k = y"}, {"tactic": "rw [Nat.eq_zero_of_le_zero ky, x1, y0]", "state_before": "a k x y : \u2115\nx\u271d :\n  1 < a \u2227\n    k \u2264 y \u2227\n      (x = 1 \u2227 y = 0 \u2228\n        \u2203 u v s t b,\n          x * x - (a * a - 1) * y * y = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227 b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y])\na1 : 1 < a\nky : k \u2264 0\no :\n  x = 1 \u2227 y = 0 \u2228\n    \u2203 u v s t b,\n      x * x - (a * a - 1) * y * y = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227 b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\nx1 : x = 1\ny0 : y = 0\n\u22a2 xn a1 k = x \u2227 yn a1 k = y", "state_after": "a k x y : \u2115\nx\u271d :\n  1 < a \u2227\n    k \u2264 y \u2227\n      (x = 1 \u2227 y = 0 \u2228\n        \u2203 u v s t b,\n          x * x - (a * a - 1) * y * y = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227 b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y])\na1 : 1 < a\nky : k \u2264 0\no :\n  x = 1 \u2227 y = 0 \u2228\n    \u2203 u v s t b,\n      x * x - (a * a - 1) * y * y = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227 b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\nx1 : x = 1\ny0 : y = 0\n\u22a2 xn a1 0 = 1 \u2227 yn a1 0 = 0"}, {"tactic": "exact \u27e8rfl, rfl\u27e9", "state_before": "a k x y : \u2115\nx\u271d :\n  1 < a \u2227\n    k \u2264 y \u2227\n      (x = 1 \u2227 y = 0 \u2228\n        \u2203 u v s t b,\n          x * x - (a * a - 1) * y * y = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227 b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y])\na1 : 1 < a\nky : k \u2264 0\no :\n  x = 1 \u2227 y = 0 \u2228\n    \u2203 u v s t b,\n      x * x - (a * a - 1) * y * y = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227 b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\nx1 : x = 1\ny0 : y = 0\n\u22a2 xn a1 0 = 1 \u2227 yn a1 0 = 0", "state_after": "no goals"}, {"tactic": "simp [i0] at ky", "state_before": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\nrem : b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n    \u2203 u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227\n              b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\ni0 : i = 0\n\u22a2 xn a1 k = xn a1 i \u2227 yn a1 k = yn a1 i", "state_after": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\nrem : b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n    \u2203 u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227\n              b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\ni0 : i = 0\nky : k = 0\n\u22a2 xn a1 k = xn a1 i \u2227 yn a1 k = yn a1 i"}, {"tactic": "rw [i0, ky]", "state_before": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\nrem : b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n    \u2203 u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227\n              b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\ni0 : i = 0\nky : k = 0\n\u22a2 xn a1 k = xn a1 i \u2227 yn a1 k = yn a1 i", "state_after": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\nrem : b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n    \u2203 u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227\n              b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\ni0 : i = 0\nky : k = 0\n\u22a2 xn a1 0 = xn a1 0 \u2227 yn a1 0 = yn a1 0"}, {"tactic": "exact \u27e8rfl, rfl\u27e9", "state_before": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\nrem : b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n    \u2203 u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227\n              b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\ni0 : i = 0\nky : k = 0\n\u22a2 xn a1 0 = xn a1 0 \u2227 yn a1 0 = yn a1 0", "state_after": "no goals"}, {"tactic": "suffices i = k by rw [this]; exact \u27e8rfl, rfl\u27e9", "state_before": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\nrem : b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n    \u2203 u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227\n              b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\nipos : i > 0\n\u22a2 xn a1 k = xn a1 i \u2227 yn a1 k = yn a1 i", "state_after": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\nrem : b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n    \u2203 u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227\n              b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\nipos : i > 0\n\u22a2 i = k"}, {"tactic": "clear o rem xy uv st", "state_before": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\nrem : b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n    \u2203 u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227\n              b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\nipos : i > 0\n\u22a2 i = k", "state_after": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\n\u22a2 i = k"}, {"tactic": "have iln : i \u2264 n :=\n  le_of_not_gt fun hin =>\n    not_lt_of_ge (Nat.le_of_dvd vp (dvd_of_mul_left_dvd yv)) (strictMono_y a1 hin)", "state_before": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\n\u22a2 i = k", "state_after": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\n\u22a2 i = k"}, {"tactic": "have yd : 4 * yn a1 i \u2223 4 * n := mul_dvd_mul_left _ <| dvd_of_ysq_dvd a1 yv", "state_before": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\n\u22a2 i = k", "state_after": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\n\u22a2 i = k"}, {"tactic": "have jk : j \u2261 k [MOD 4 * yn a1 i] :=\n  have : 4 * yn a1 i \u2223 b - 1 :=\n    Int.coe_nat_dvd.1 <| by rw [Int.ofNat_sub (le_of_lt b1)]; exact bm1.symm.dvd\n  ((yn_modEq_a_sub_one b1 _).of_dvd this).symm.trans tk", "state_before": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\n\u22a2 i = k", "state_after": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\njk : j \u2261 k [MOD 4 * yn a1 i]\n\u22a2 i = k"}, {"tactic": "have ki : k + i < 4 * yn a1 i :=\n  lt_of_le_of_lt (_root_.add_le_add ky (yn_ge_n a1 i)) <| by\n    rw [\u2190 two_mul]\n    exact Nat.mul_lt_mul_of_pos_right (by decide) (strictMono_y a1 ipos)", "state_before": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\njk : j \u2261 k [MOD 4 * yn a1 i]\n\u22a2 i = k", "state_after": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\njk : j \u2261 k [MOD 4 * yn a1 i]\nki : k + i < 4 * yn a1 i\n\u22a2 i = k"}, {"tactic": "have ji : j \u2261 i [MOD 4 * n] :=\n  have : xn a1 j \u2261 xn a1 i [MOD xn a1 n] :=\n    (xy_modEq_of_modEq b1 a1 ba j).left.symm.trans sx\n  (modEq_of_xn_modEq a1 ipos iln this).resolve_right\n    fun ji : j + i \u2261 0 [MOD 4 * n] =>\n    not_le_of_gt ki <|\n      Nat.le_of_dvd (lt_of_lt_of_le ipos <| Nat.le_add_left _ _) <|\n        modEq_zero_iff_dvd.1 <| (jk.symm.add_right i).trans <| ji.of_dvd yd", "state_before": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\njk : j \u2261 k [MOD 4 * yn a1 i]\nki : k + i < 4 * yn a1 i\n\u22a2 i = k", "state_after": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\njk : j \u2261 k [MOD 4 * yn a1 i]\nki : k + i < 4 * yn a1 i\nji : j \u2261 i [MOD 4 * n]\n\u22a2 i = k"}, {"tactic": "have : i % (4 * yn a1 i) = k % (4 * yn a1 i) := (ji.of_dvd yd).symm.trans jk", "state_before": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\njk : j \u2261 k [MOD 4 * yn a1 i]\nki : k + i < 4 * yn a1 i\nji : j \u2261 i [MOD 4 * n]\n\u22a2 i = k", "state_after": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\njk : j \u2261 k [MOD 4 * yn a1 i]\nki : k + i < 4 * yn a1 i\nji : j \u2261 i [MOD 4 * n]\nthis : i % (4 * yn a1 i) = k % (4 * yn a1 i)\n\u22a2 i = k"}, {"tactic": "rwa [Nat.mod_eq_of_lt (lt_of_le_of_lt (Nat.le_add_left _ _) ki),\n  Nat.mod_eq_of_lt (lt_of_le_of_lt (Nat.le_add_right _ _) ki)] at this", "state_before": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\njk : j \u2261 k [MOD 4 * yn a1 i]\nki : k + i < 4 * yn a1 i\nji : j \u2261 i [MOD 4 * n]\nthis : i % (4 * yn a1 i) = k % (4 * yn a1 i)\n\u22a2 i = k", "state_after": "no goals"}, {"tactic": "rw [this]", "state_before": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\nrem : b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n    \u2203 u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227\n              b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\nipos : i > 0\nthis : i = k\n\u22a2 xn a1 k = xn a1 i \u2227 yn a1 k = yn a1 i", "state_after": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\nrem : b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n    \u2203 u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227\n              b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\nipos : i > 0\nthis : i = k\n\u22a2 xn a1 k = xn a1 k \u2227 yn a1 k = yn a1 k"}, {"tactic": "exact \u27e8rfl, rfl\u27e9", "state_before": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\nrem : b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n    \u2203 u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227\n              b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\nipos : i > 0\nthis : i = k\n\u22a2 xn a1 k = xn a1 k \u2227 yn a1 k = yn a1 k", "state_after": "no goals"}, {"tactic": "rw [Int.ofNat_sub (le_of_lt b1)]", "state_before": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\n\u22a2 \u2191(4 * yn a1 i) \u2223 \u2191(b - 1)", "state_after": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\n\u22a2 \u2191(4 * yn a1 i) \u2223 \u2191b - \u21911"}, {"tactic": "exact bm1.symm.dvd", "state_before": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\n\u22a2 \u2191(4 * yn a1 i) \u2223 \u2191b - \u21911", "state_after": "no goals"}, {"tactic": "rw [\u2190 two_mul]", "state_before": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\njk : j \u2261 k [MOD 4 * yn a1 i]\n\u22a2 yn a1 i + yn a1 i < 4 * yn a1 i", "state_after": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\njk : j \u2261 k [MOD 4 * yn a1 i]\n\u22a2 2 * yn a1 i < 4 * yn a1 i"}, {"tactic": "exact Nat.mul_lt_mul_of_pos_right (by decide) (strictMono_y a1 ipos)", "state_before": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\njk : j \u2261 k [MOD 4 * yn a1 i]\n\u22a2 2 * yn a1 i < 4 * yn a1 i", "state_after": "no goals"}, {"tactic": "decide", "state_before": "a k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\njk : j \u2261 k [MOD 4 * yn a1 i]\n\u22a2 2 < 4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/ModelTheory/Satisfiability.lean", "full_name": "FirstOrder.Language.Theory.IsMaximal.mem_iff_models", "start": [410, 1], "end": [411, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/LiminfLimsup.lean", "full_name": "Filter.bliminf_or_le_inf_aux_right", "start": [943, 1], "end": [944, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Basic.lean", "full_name": "and_or_imp", "start": [180, 9], "end": [180, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Hom/CompleteLattice.lean", "full_name": "sSupHom.comp_assoc", "start": [328, 1], "end": [330, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Sym/Sym2.lean", "full_name": "Sym2.filter_image_quotient_mk''_isDiag", "start": [732, 1], "end": [743, 39], "traced_tactics": [{"tactic": "ext z", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.115916\n\u03b3 : Type ?u.115919\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\n\u22a2 filter IsDiag (image Quotient.mk'' (s \u00d7\u02e2 s)) = image Quotient.mk'' (Finset.diag s)", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type ?u.115916\n\u03b3 : Type ?u.115919\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nz : Sym2 \u03b1\n\u22a2 z \u2208 filter IsDiag (image Quotient.mk'' (s \u00d7\u02e2 s)) \u2194 z \u2208 image Quotient.mk'' (Finset.diag s)"}, {"tactic": "induction' z using Sym2.inductionOn", "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type ?u.115916\n\u03b3 : Type ?u.115919\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nz : Sym2 \u03b1\n\u22a2 z \u2208 filter IsDiag (image Quotient.mk'' (s \u00d7\u02e2 s)) \u2194 z \u2208 image Quotient.mk'' (Finset.diag s)", "state_after": "case a.hf\n\u03b1 : Type u_1\n\u03b2 : Type ?u.115916\n\u03b3 : Type ?u.115919\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d : \u03b1\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d) \u2208 filter IsDiag (image Quotient.mk'' (s \u00d7\u02e2 s)) \u2194\n    Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d) \u2208 image Quotient.mk'' (Finset.diag s)"}, {"tactic": "simp only [mem_image, mem_diag, exists_prop, mem_filter, Prod.exists, mem_product]", "state_before": "case a.hf\n\u03b1 : Type u_1\n\u03b2 : Type ?u.115916\n\u03b3 : Type ?u.115919\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d : \u03b1\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d) \u2208 filter IsDiag (image Quotient.mk'' (s \u00d7\u02e2 s)) \u2194\n    Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d) \u2208 image Quotient.mk'' (Finset.diag s)", "state_after": "case a.hf\n\u03b1 : Type u_1\n\u03b2 : Type ?u.115916\n\u03b3 : Type ?u.115919\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d : \u03b1\n\u22a2 (\u2203 a b, (a \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2227\n      IsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2194\n    \u2203 a b, (a \u2208 s \u2227 a = b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)"}, {"tactic": "constructor", "state_before": "case a.hf\n\u03b1 : Type u_1\n\u03b2 : Type ?u.115916\n\u03b3 : Type ?u.115919\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d : \u03b1\n\u22a2 (\u2203 a b, (a \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2227\n      IsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2194\n    \u2203 a b, (a \u2208 s \u2227 a = b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)", "state_after": "case a.hf.mp\n\u03b1 : Type u_1\n\u03b2 : Type ?u.115916\n\u03b3 : Type ?u.115919\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d : \u03b1\n\u22a2 (\u2203 a b, (a \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2227\n      IsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2192\n    \u2203 a b, (a \u2208 s \u2227 a = b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\n\ncase a.hf.mpr\n\u03b1 : Type u_1\n\u03b2 : Type ?u.115916\n\u03b3 : Type ?u.115919\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d : \u03b1\n\u22a2 (\u2203 a b, (a \u2208 s \u2227 a = b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2192\n    (\u2203 a b, (a \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2227\n      IsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d))"}, {"tactic": "rintro \u27e8\u27e8a, b, \u27e8ha, hb\u27e9, (h : Quotient.mk _ _ = _)\u27e9, hab\u27e9", "state_before": "case a.hf.mp\n\u03b1 : Type u_1\n\u03b2 : Type ?u.115916\n\u03b3 : Type ?u.115919\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d : \u03b1\n\u22a2 (\u2203 a b, (a \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2227\n      IsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2192\n    \u2203 a b, (a \u2208 s \u2227 a = b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)", "state_after": "case a.hf.mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.115916\n\u03b3 : Type ?u.115919\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d : \u03b1\nhab : IsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d))\na b : \u03b1\nh : Quotient.mk (Rel.setoid \u03b1) (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\nha : a \u2208 s\nhb : b \u2208 s\n\u22a2 \u2203 a b, (a \u2208 s \u2227 a = b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)"}, {"tactic": "rw [\u2190 h, Sym2.mk''_isDiag_iff] at hab", "state_before": "case a.hf.mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.115916\n\u03b3 : Type ?u.115919\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d : \u03b1\nhab : IsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d))\na b : \u03b1\nh : Quotient.mk (Rel.setoid \u03b1) (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\nha : a \u2208 s\nhb : b \u2208 s\n\u22a2 \u2203 a b, (a \u2208 s \u2227 a = b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)", "state_after": "case a.hf.mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.115916\n\u03b3 : Type ?u.115919\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d a b : \u03b1\nhab : a = b\nh : Quotient.mk (Rel.setoid \u03b1) (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\nha : a \u2208 s\nhb : b \u2208 s\n\u22a2 \u2203 a b, (a \u2208 s \u2227 a = b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)"}, {"tactic": "exact \u27e8a, b, \u27e8ha, hab\u27e9, h\u27e9", "state_before": "case a.hf.mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.115916\n\u03b3 : Type ?u.115919\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d a b : \u03b1\nhab : a = b\nh : Quotient.mk (Rel.setoid \u03b1) (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\nha : a \u2208 s\nhb : b \u2208 s\n\u22a2 \u2203 a b, (a \u2208 s \u2227 a = b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)", "state_after": "no goals"}, {"tactic": "rintro \u27e8a, b, \u27e8ha, rfl\u27e9, h\u27e9", "state_before": "case a.hf.mpr\n\u03b1 : Type u_1\n\u03b2 : Type ?u.115916\n\u03b3 : Type ?u.115919\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d : \u03b1\n\u22a2 (\u2203 a b, (a \u2208 s \u2227 a = b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2192\n    (\u2203 a b, (a \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2227\n      IsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d))", "state_after": "case a.hf.mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.115916\n\u03b3 : Type ?u.115919\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d a : \u03b1\nha : a \u2208 s\nh : Quotient.mk'' (a, a) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\n\u22a2 (\u2203 a b, (a \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2227\n    IsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d))"}, {"tactic": "rw [\u2190 h]", "state_before": "case a.hf.mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.115916\n\u03b3 : Type ?u.115919\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d a : \u03b1\nha : a \u2208 s\nh : Quotient.mk'' (a, a) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\n\u22a2 (\u2203 a b, (a \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2227\n    IsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d))", "state_after": "case a.hf.mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.115916\n\u03b3 : Type ?u.115919\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d a : \u03b1\nha : a \u2208 s\nh : Quotient.mk'' (a, a) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\n\u22a2 (\u2203 a_1 b, (a_1 \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a_1, b) = Quotient.mk'' (a, a)) \u2227 IsDiag (Quotient.mk'' (a, a))"}, {"tactic": "exact \u27e8\u27e8a, a, \u27e8ha, ha\u27e9, rfl\u27e9, rfl\u27e9", "state_before": "case a.hf.mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.115916\n\u03b3 : Type ?u.115919\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nx\u271d y\u271d a : \u03b1\nha : a \u2208 s\nh : Quotient.mk'' (a, a) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)\n\u22a2 (\u2203 a_1 b, (a_1 \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a_1, b) = Quotient.mk'' (a, a)) \u2227 IsDiag (Quotient.mk'' (a, a))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Quaternion.lean", "full_name": "QuaternionAlgebra.algebraMap_eq", "start": [494, 1], "end": [495, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Polynomial/Vieta.lean", "full_name": "Polynomial.coeff_eq_esymm_roots_of_card", "start": [142, 1], "end": [148, 66], "traced_tactics": [{"tactic": "conv_lhs => rw [\u2190 C_leadingCoeff_mul_prod_multiset_X_sub_C hroots]", "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : R[X]\nhroots : \u2191card (roots p) = natDegree p\nk : \u2115\nh : k \u2264 natDegree p\n\u22a2 coeff p k = leadingCoeff p * (-1) ^ (natDegree p - k) * esymm (roots p) (natDegree p - k)", "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : R[X]\nhroots : \u2191card (roots p) = natDegree p\nk : \u2115\nh : k \u2264 natDegree p\n\u22a2 coeff (\u2191C (leadingCoeff p) * prod (map (fun a => X - \u2191C a) (roots p))) k =\n    leadingCoeff p * (-1) ^ (natDegree p - k) * esymm (roots p) (natDegree p - k)"}, {"tactic": "rw [coeff_C_mul, mul_assoc]", "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : R[X]\nhroots : \u2191card (roots p) = natDegree p\nk : \u2115\nh : k \u2264 natDegree p\n\u22a2 coeff (\u2191C (leadingCoeff p) * prod (map (fun a => X - \u2191C a) (roots p))) k =\n    leadingCoeff p * (-1) ^ (natDegree p - k) * esymm (roots p) (natDegree p - k)", "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : R[X]\nhroots : \u2191card (roots p) = natDegree p\nk : \u2115\nh : k \u2264 natDegree p\n\u22a2 leadingCoeff p * coeff (prod (map (fun a => X - \u2191C a) (roots p))) k =\n    leadingCoeff p * ((-1) ^ (natDegree p - k) * esymm (roots p) (natDegree p - k))"}, {"tactic": "congr", "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : R[X]\nhroots : \u2191card (roots p) = natDegree p\nk : \u2115\nh : k \u2264 natDegree p\n\u22a2 leadingCoeff p * coeff (prod (map (fun a => X - \u2191C a) (roots p))) k =\n    leadingCoeff p * ((-1) ^ (natDegree p - k) * esymm (roots p) (natDegree p - k))", "state_after": "case e_a\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : R[X]\nhroots : \u2191card (roots p) = natDegree p\nk : \u2115\nh : k \u2264 natDegree p\n\u22a2 coeff (prod (map (fun a => X - \u2191C a) (roots p))) k = (-1) ^ (natDegree p - k) * esymm (roots p) (natDegree p - k)"}, {"tactic": "have : k \u2264 card (roots p) := by rw [hroots]; exact h", "state_before": "case e_a\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : R[X]\nhroots : \u2191card (roots p) = natDegree p\nk : \u2115\nh : k \u2264 natDegree p\n\u22a2 coeff (prod (map (fun a => X - \u2191C a) (roots p))) k = (-1) ^ (natDegree p - k) * esymm (roots p) (natDegree p - k)", "state_after": "case e_a\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : R[X]\nhroots : \u2191card (roots p) = natDegree p\nk : \u2115\nh : k \u2264 natDegree p\nthis : k \u2264 \u2191card (roots p)\n\u22a2 coeff (prod (map (fun a => X - \u2191C a) (roots p))) k = (-1) ^ (natDegree p - k) * esymm (roots p) (natDegree p - k)"}, {"tactic": "convert p.roots.prod_X_sub_C_coeff this using 3 <;> rw [hroots]", "state_before": "case e_a\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : R[X]\nhroots : \u2191card (roots p) = natDegree p\nk : \u2115\nh : k \u2264 natDegree p\nthis : k \u2264 \u2191card (roots p)\n\u22a2 coeff (prod (map (fun a => X - \u2191C a) (roots p))) k = (-1) ^ (natDegree p - k) * esymm (roots p) (natDegree p - k)", "state_after": "no goals"}, {"tactic": "rw [hroots]", "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : R[X]\nhroots : \u2191card (roots p) = natDegree p\nk : \u2115\nh : k \u2264 natDegree p\n\u22a2 k \u2264 \u2191card (roots p)", "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : R[X]\nhroots : \u2191card (roots p) = natDegree p\nk : \u2115\nh : k \u2264 natDegree p\n\u22a2 k \u2264 natDegree p"}, {"tactic": "exact h", "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np : R[X]\nhroots : \u2191card (roots p) = natDegree p\nk : \u2115\nh : k \u2264 natDegree p\n\u22a2 k \u2264 natDegree p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Dynamics/PeriodicPts.lean", "full_name": "Function.iterate_add_minimalPeriod_eq", "start": [290, 1], "end": [293, 39], "traced_tactics": [{"tactic": "rw [iterate_add_apply]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.19831\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\n\u22a2 (f^[n + minimalPeriod f x]) x = (f^[n]) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type ?u.19831\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\n\u22a2 (f^[n]) ((f^[minimalPeriod f x]) x) = (f^[n]) x"}, {"tactic": "congr", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.19831\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\n\u22a2 (f^[n]) ((f^[minimalPeriod f x]) x) = (f^[n]) x", "state_after": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type ?u.19831\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\n\u22a2 (f^[minimalPeriod f x]) x = x"}, {"tactic": "exact isPeriodicPt_minimalPeriod f x", "state_before": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type ?u.19831\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\n\u22a2 (f^[minimalPeriod f x]) x = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "full_name": "CategoryTheory.Limits.cospanExt_app_right", "start": [405, 1], "end": [406, 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).app WalkingCospan.right = iY", "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.cos_add_pi_div_two", "start": [464, 1], "end": [466, 34], "traced_tactics": [{"tactic": "induction \u03b8 using Real.Angle.induction_on", "state_before": "\u03b8 : Angle\n\u22a2 cos (\u03b8 + \u2191(\u03c0 / 2)) = -sin \u03b8", "state_after": "case h\nx\u271d : \u211d\n\u22a2 cos (\u2191x\u271d + \u2191(\u03c0 / 2)) = -sin \u2191x\u271d"}, {"tactic": "exact Real.cos_add_pi_div_two _", "state_before": "case h\nx\u271d : \u211d\n\u22a2 cos (\u2191x\u271d + \u2191(\u03c0 / 2)) = -sin \u2191x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.lift_succ", "start": [80, 1], "end": [82, 6], "traced_tactics": [{"tactic": "rw [\u2190 add_one_eq_succ, lift_add, lift_one]", "state_before": "\u03b1 : Type ?u.839\n\u03b2 : Type ?u.842\n\u03b3 : Type ?u.845\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na : Ordinal\n\u22a2 lift (succ a) = succ (lift a)", "state_after": "\u03b1 : Type ?u.839\n\u03b2 : Type ?u.842\n\u03b3 : Type ?u.845\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na : Ordinal\n\u22a2 lift a + 1 = succ (lift a)"}, {"tactic": "rfl", "state_before": "\u03b1 : Type ?u.839\n\u03b2 : Type ?u.842\n\u03b3 : Type ?u.845\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na : Ordinal\n\u22a2 lift a + 1 = succ (lift a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "full_name": "CategoryTheory.Limits.hasPushout_of_left_iso", "start": [1812, 1], "end": [1813, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "full_name": "PowerSeries.constantCoeff_inv", "start": [2131, 1], "end": [2132, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/TensorProduct.lean", "full_name": "LinearMap.baseChange_eq_ltensor", "start": [270, 1], "end": [271, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/ODE/PicardLindelof.lean", "full_name": "PicardLindelof.FunSpace.intervalIntegrable_vComp", "start": [261, 1], "end": [262, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Add.lean", "full_name": "DifferentiableAt.neg", "start": [428, 1], "end": [429, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/FieldTheory/Finite/Basic.lean", "full_name": "Nat.sq_add_sq_modEq", "start": [345, 1], "end": [347, 70], "traced_tactics": [{"tactic": "simpa only [\u2190 Int.coe_nat_modEq_iff] using Nat.sq_add_sq_zmodEq p x", "state_before": "K : Type ?u.856220\nR : Type ?u.856223\np : \u2115\ninst\u271d : Fact (Prime p)\nx : \u2115\n\u22a2 \u2203 a b, a \u2264 p / 2 \u2227 b \u2264 p / 2 \u2227 a ^ 2 + b ^ 2 \u2261 x [MOD p]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Option/Basic.lean", "full_name": "Option.bind_pmap", "start": [206, 1], "end": [208, 74], "traced_tactics": [{"tactic": "cases x <;> simp only [pmap, bind_eq_bind, none_bind, some_bind, pbind]", "state_before": "\u03b1\u271d 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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\nhf : DifferentiableOn \ud835\udd5c f s\nh : ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderivWithin \ud835\udd5c f s y) s\n\u22a2 ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s", "state_after": "\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\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhf : DifferentiableOn \ud835\udd5c f s\nh : ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderivWithin \ud835\udd5c f s y) s\nx : E\nhx : x \u2208 s\n\u22a2 ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x"}, {"tactic": "rw [contDiffWithinAt_succ_iff_hasFDerivWithinAt, insert_eq_of_mem hx]", "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\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhf : DifferentiableOn \ud835\udd5c f s\nh : ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderivWithin \ud835\udd5c f s y) s\nx : E\nhx : x \u2208 s\n\u22a2 ContDiffWithinAt \ud835\udd5c (\u2191(n + 1)) f s x", "state_after": "\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\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhf : DifferentiableOn \ud835\udd5c f s\nh : ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderivWithin \ud835\udd5c f s y) s\nx : E\nhx : x \u2208 s\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x"}, {"tactic": "exact\n  \u27e8s, self_mem_nhdsWithin, fderivWithin \ud835\udd5c f s, fun y hy => (hf y hy).hasFDerivWithinAt, h x hx\u27e9", "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\u271d x\u2080 : E\nc : F\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nhf : DifferentiableOn \ud835\udd5c f s\nh : ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderivWithin \ud835\udd5c f s y) s\nx : E\nhx : x \u2208 s\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 \u2203 f', (\u2200 (x : E), x \u2208 u \u2192 HasFDerivWithinAt f (f' x) u x) \u2227 ContDiffWithinAt \ud835\udd5c (\u2191n) f' u x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/BilinearForm.lean", "full_name": "BilinForm.neg_left", "start": [125, 1], "end": [126, 54], "traced_tactics": [{"tactic": "rw [\u2190 @neg_one_smul R\u2081 _ _, smul_left, neg_one_mul]", "state_before": "R : Type ?u.48014\nM : Type ?u.48017\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : Module R M\nR\u2081 : Type u_1\nM\u2081 : Type u_2\ninst\u271d\u00b9\u00b9 : Ring R\u2081\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2081\ninst\u271d\u2079 : Module R\u2081 M\u2081\nR\u2082 : Type ?u.48665\nM\u2082 : Type ?u.48668\ninst\u271d\u2078 : CommSemiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\nR\u2083 : Type ?u.48855\nM\u2083 : Type ?u.48858\ninst\u271d\u2075 : CommRing R\u2083\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R\u2083 M\u2083\nV : Type ?u.49446\nK : Type ?u.49449\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nx y : M\u2081\n\u22a2 bilin B\u2081 (-x) y = -bilin B\u2081 x y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Hom/Basic.lean", "full_name": "OrderHom.dual_comp", "start": [582, 1], "end": [584, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/WittVector/Teichmuller.lean", "full_name": "WittVector.teichmuller_coeff_zero", "start": [114, 1], "end": [115, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/MeasurableSpaceDef.lean", "full_name": "MeasurableSpace.measurableSet_generateFrom", "start": [374, 1], "end": [376, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/Basic.lean", "full_name": "Equiv.swap_apply_ne_self_iff", "start": [1645, 1], "end": [1655, 47], "traced_tactics": [{"tactic": "by_cases hab : a = b", "state_before": "\u03b1 : Sort u_1\ninst\u271d : DecidableEq \u03b1\na b x : \u03b1\n\u22a2 \u2191(swap a b) x \u2260 x \u2194 a \u2260 b \u2227 (x = a \u2228 x = b)", "state_after": "case pos\n\u03b1 : Sort u_1\ninst\u271d : DecidableEq \u03b1\na b x : \u03b1\nhab : a = b\n\u22a2 \u2191(swap a b) x \u2260 x \u2194 a \u2260 b \u2227 (x = a \u2228 x = b)\n\ncase neg\n\u03b1 : Sort u_1\ninst\u271d : DecidableEq \u03b1\na b x : \u03b1\nhab : \u00aca = b\n\u22a2 \u2191(swap a b) x \u2260 x \u2194 a \u2260 b \u2227 (x = a \u2228 x = b)"}, {"tactic": "by_cases hax : x = a", "state_before": "case neg\n\u03b1 : Sort u_1\ninst\u271d : DecidableEq \u03b1\na b x : \u03b1\nhab : \u00aca = b\n\u22a2 \u2191(swap a b) x \u2260 x \u2194 a \u2260 b \u2227 (x = a \u2228 x = b)", "state_after": "case pos\n\u03b1 : Sort u_1\ninst\u271d : DecidableEq \u03b1\na b x : \u03b1\nhab : \u00aca = b\nhax : x = a\n\u22a2 \u2191(swap a b) x \u2260 x \u2194 a \u2260 b \u2227 (x = a \u2228 x = b)\n\ncase neg\n\u03b1 : Sort u_1\ninst\u271d : DecidableEq \u03b1\na b x : \u03b1\nhab : \u00aca = b\nhax : \u00acx = a\n\u22a2 \u2191(swap a b) x \u2260 x \u2194 a \u2260 b \u2227 (x = a \u2228 x = b)"}, {"tactic": "by_cases hbx : x = b", "state_before": "case neg\n\u03b1 : Sort u_1\ninst\u271d : DecidableEq \u03b1\na b x : \u03b1\nhab : \u00aca = b\nhax : \u00acx = a\n\u22a2 \u2191(swap a b) x \u2260 x \u2194 a \u2260 b \u2227 (x = a \u2228 x = b)", "state_after": "case pos\n\u03b1 : Sort u_1\ninst\u271d : DecidableEq \u03b1\na b x : \u03b1\nhab : \u00aca = b\nhax : \u00acx = a\nhbx : x = b\n\u22a2 \u2191(swap a b) x \u2260 x \u2194 a \u2260 b \u2227 (x = a \u2228 x = b)\n\ncase neg\n\u03b1 : Sort u_1\ninst\u271d : DecidableEq \u03b1\na b x : \u03b1\nhab : \u00aca = b\nhax : \u00acx = a\nhbx : \u00acx = b\n\u22a2 \u2191(swap a b) x \u2260 x \u2194 a \u2260 b \u2227 (x = a \u2228 x = b)"}, {"tactic": "simp [hab, hax, hbx, swap_apply_of_ne_of_ne]", "state_before": "case neg\n\u03b1 : Sort u_1\ninst\u271d : DecidableEq \u03b1\na b x : \u03b1\nhab : \u00aca = b\nhax : \u00acx = a\nhbx : \u00acx = b\n\u22a2 \u2191(swap a b) x \u2260 x \u2194 a \u2260 b \u2227 (x = a \u2228 x = b)", "state_after": "no goals"}, {"tactic": "simp [hab]", "state_before": "case pos\n\u03b1 : Sort u_1\ninst\u271d : DecidableEq \u03b1\na b x : \u03b1\nhab : a = b\n\u22a2 \u2191(swap a b) x \u2260 x \u2194 a \u2260 b \u2227 (x = a \u2228 x = b)", "state_after": "no goals"}, {"tactic": "simp [hax, eq_comm]", "state_before": "case pos\n\u03b1 : Sort u_1\ninst\u271d : DecidableEq \u03b1\na b x : \u03b1\nhab : \u00aca = b\nhax : x = a\n\u22a2 \u2191(swap a b) x \u2260 x \u2194 a \u2260 b \u2227 (x = a \u2228 x = b)", "state_after": "no goals"}, {"tactic": "simp [hbx]", "state_before": "case pos\n\u03b1 : Sort u_1\ninst\u271d : DecidableEq \u03b1\na b x : \u03b1\nhab : \u00aca = b\nhax : \u00acx = a\nhbx : x = b\n\u22a2 \u2191(swap a b) x \u2260 x \u2194 a \u2260 b \u2227 (x = a \u2228 x = b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/OrderOfElement.lean", "full_name": "IsOfFinOrder.of_mem_zpowers", "start": [598, 1], "end": [601, 15], "traced_tactics": [{"tactic": "obtain \u27e8k, rfl\u27e9 := Subgroup.mem_zpowers_iff.mp h'", "state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn m : \u2115\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ni : \u2124\nh : IsOfFinOrder x\nh' : y \u2208 Subgroup.zpowers x\n\u22a2 IsOfFinOrder y", "state_after": "case intro\nG : Type u\nA : Type v\nx : G\na b : A\nn m : \u2115\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ni : \u2124\nh : IsOfFinOrder x\nk : \u2124\nh' : x ^ k \u2208 Subgroup.zpowers x\n\u22a2 IsOfFinOrder (x ^ k)"}, {"tactic": "exact h.zpow", "state_before": "case intro\nG : Type u\nA : Type v\nx : G\na b : A\nn m : \u2115\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ni : \u2124\nh : IsOfFinOrder x\nk : \u2124\nh' : x ^ k \u2208 Subgroup.zpowers x\n\u22a2 IsOfFinOrder (x ^ k)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Coset.lean", "full_name": "orbit_subgroup_one_eq_self", "start": [247, 1], "end": [248, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Instances/ENNReal.lean", "full_name": "ENNReal.tsum_prod", "start": [814, 11], "end": [815, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Combinatorics/Quiver/Path.lean", "full_name": "Quiver.Path.length_nil", "start": [76, 1], "end": [77, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.prod_eq_biUnion_left", "start": [1931, 1], "end": [1932, 44], "traced_tactics": [{"tactic": "rw [iUnion_image_left, image2_mk_eq_prod]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.257641\n\u03b9 : Sort ?u.257644\n\u03b9' : Sort ?u.257647\n\u03b9\u2082 : Sort ?u.257650\n\u03ba : \u03b9 \u2192 Sort ?u.257655\n\u03ba\u2081 : \u03b9 \u2192 Sort ?u.257660\n\u03ba\u2082 : \u03b9 \u2192 Sort ?u.257665\n\u03ba' : \u03b9' \u2192 Sort ?u.257670\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 s \u00d7\u02e2 t = \u22c3 (a : \u03b1) (_ : a \u2208 s), (fun b => (a, b)) '' t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Basic.lean", "full_name": "Submodule.ker_ofLe", "start": [1646, 1], "end": [1647, 42], "traced_tactics": [{"tactic": "rw [ofLe, ker_codRestrict, ker_subtype]", "state_before": "R : Type u_1\nR\u2081 : Type ?u.1544682\nR\u2082 : Type ?u.1544685\nR\u2083 : Type ?u.1544688\nR\u2084 : Type ?u.1544691\nS : Type ?u.1544694\nK : Type ?u.1544697\nK\u2082 : Type ?u.1544700\nM : Type u_2\nM' : Type ?u.1544706\nM\u2081 : Type ?u.1544709\nM\u2082 : Type ?u.1544712\nM\u2083 : Type ?u.1544715\nM\u2084 : Type ?u.1544718\nN : Type ?u.1544721\nN\u2082 : Type ?u.1544724\n\u03b9 : Type ?u.1544727\nV : Type ?u.1544730\nV\u2082 : Type ?u.1544733\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'\u271d : Submodule R M\nq : Submodule R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\nF : Type ?u.1544907\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\np p' : Submodule R M\nh : p \u2264 p'\n\u22a2 ker (ofLe h) = \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Add.lean", "full_name": "DifferentiableWithinAt.sub_const", "start": [547, 1], "end": [549, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.measure_iUnion_eq_iSup", "start": [451, 1], "end": [483, 46], "traced_tactics": [{"tactic": "cases nonempty_encodable \u03b9", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.46518\n\u03b3 : Type ?u.46521\n\u03b4 : Type ?u.46524\n\u03b9 : Type u_1\nR : Type ?u.46530\nR' : Type ?u.46533\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) s\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), s i) = \u2a06 (i : \u03b9), \u2191\u2191\u03bc (s i)", "state_after": "case intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.46518\n\u03b3 : Type ?u.46521\n\u03b4 : Type ?u.46524\n\u03b9 : Type u_1\nR : Type ?u.46530\nR' : Type ?u.46533\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) s\nval\u271d : Encodable \u03b9\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), s i) = \u2a06 (i : \u03b9), \u2191\u2191\u03bc (s i)"}, {"tactic": "generalize ht : Function.extend Encodable.encode s \u22a5 = t", "state_before": "case intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.46518\n\u03b3 : Type ?u.46521\n\u03b4 : Type ?u.46524\n\u03b9 : Type u_1\nR : Type ?u.46530\nR' : Type ?u.46533\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) s\nval\u271d : Encodable \u03b9\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), s i) = \u2a06 (i : \u03b9), \u2191\u2191\u03bc (s i)", "state_after": "case intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.46518\n\u03b3 : Type ?u.46521\n\u03b4 : Type ?u.46524\n\u03b9 : Type u_1\nR : Type ?u.46530\nR' : Type ?u.46533\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) s\nval\u271d : Encodable \u03b9\nt : \u2115 \u2192 Set \u03b1\nht : Function.extend Encodable.encode s \u22a5 = t\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), s i) = \u2a06 (i : \u03b9), \u2191\u2191\u03bc (s i)"}, {"tactic": "clear! \u03b9", "state_before": "case intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.46518\n\u03b3 : Type ?u.46521\n\u03b4 : Type ?u.46524\n\u03b9 : Type u_1\nR : Type ?u.46530\nR' : Type ?u.46533\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nval\u271d : Encodable \u03b9\nt : \u2115 \u2192 Set \u03b1\nht : Function.extend Encodable.encode s \u22a5 = t\nhd : Directed (fun x x_1 => x \u2286 x_1) t\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), t n) = \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)", "state_after": "case intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.46518\n\u03b3 : Type ?u.46521\n\u03b4 : Type ?u.46524\nR : Type ?u.46530\nR' : Type ?u.46533\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), t n) = \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)"}, {"tactic": "refine' le_antisymm _ (iSup_le fun i => measure_mono <| subset_iUnion _ _)", "state_before": "case intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.46518\n\u03b3 : Type ?u.46521\n\u03b4 : Type ?u.46524\nR : Type ?u.46530\nR' : Type ?u.46533\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), t n) = \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)", "state_after": "case intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.46518\n\u03b3 : Type ?u.46521\n\u03b4 : Type ?u.46524\nR : Type ?u.46530\nR' : Type ?u.46533\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), t n) \u2264 \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)"}, {"tactic": "set T : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)", "state_before": "case intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.46518\n\u03b3 : Type ?u.46521\n\u03b4 : Type ?u.46524\nR : Type ?u.46530\nR' : Type ?u.46533\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), t n) \u2264 \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)", "state_after": "case intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.46518\n\u03b3 : Type ?u.46521\n\u03b4 : Type ?u.46524\nR : Type ?u.46530\nR' : Type ?u.46533\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), t n) \u2264 \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)"}, {"tactic": "set Td : \u2115 \u2192 Set \u03b1 := disjointed T", "state_before": "case intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.46518\n\u03b3 : Type ?u.46521\n\u03b4 : Type ?u.46524\nR : Type ?u.46530\nR' : Type ?u.46533\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), t n) \u2264 \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)", "state_after": "case intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.46518\n\u03b3 : Type ?u.46521\n\u03b4 : Type ?u.46524\nR : Type ?u.46530\nR' : Type ?u.46533\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\nTd : \u2115 \u2192 Set \u03b1 := disjointed T\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), t n) \u2264 \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)"}, {"tactic": "have hm : \u2200 n, MeasurableSet (Td n) :=\n  MeasurableSet.disjointed fun n => measurableSet_toMeasurable _ _", "state_before": "case intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.46518\n\u03b3 : Type ?u.46521\n\u03b4 : Type ?u.46524\nR : Type ?u.46530\nR' : Type ?u.46533\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\nTd : \u2115 \u2192 Set \u03b1 := disjointed T\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), t n) \u2264 \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)", "state_after": "case intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.46518\n\u03b3 : Type ?u.46521\n\u03b4 : Type ?u.46524\nR : Type ?u.46530\nR' : Type ?u.46533\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\nTd : \u2115 \u2192 Set \u03b1 := disjointed T\nhm : \u2200 (n : \u2115), MeasurableSet (Td n)\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), t n) \u2264 \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)"}, {"tactic": "calc\n  \u03bc (\u22c3 n, t n) \u2264 \u03bc (\u22c3 n, T n) := measure_mono (iUnion_mono fun i => subset_toMeasurable _ _)\n  _ = \u03bc (\u22c3 n, Td n) := by rw [iUnion_disjointed]\n  _ \u2264 \u2211' n, \u03bc (Td n) := (measure_iUnion_le _)\n  _ = \u2a06 I : Finset \u2115, \u2211 n in I, \u03bc (Td n) := ENNReal.tsum_eq_iSup_sum\n  _ \u2264 \u2a06 n, \u03bc (t n) := iSup_le fun I => by\n    rcases hd.finset_le I with \u27e8N, hN\u27e9\n    calc\n      (\u2211 n in I, \u03bc (Td n)) = \u03bc (\u22c3 n \u2208 I, Td n) :=\n        (measure_biUnion_finset ((disjoint_disjointed T).set_pairwise I) fun n _ => hm n).symm\n      _ \u2264 \u03bc (\u22c3 n \u2208 I, T n) := (measure_mono (iUnion\u2082_mono fun n _hn => disjointed_subset _ _))\n      _ = \u03bc (\u22c3 n \u2208 I, t n) := (measure_biUnion_toMeasurable I.countable_toSet _)\n      _ \u2264 \u03bc (t N) := (measure_mono (iUnion\u2082_subset hN))\n      _ \u2264 \u2a06 n, \u03bc (t n) := le_iSup (\u03bc \u2218 t) N", "state_before": "case intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.46518\n\u03b3 : Type ?u.46521\n\u03b4 : Type ?u.46524\nR : Type ?u.46530\nR' : Type ?u.46533\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\nTd : \u2115 \u2192 Set \u03b1 := disjointed T\nhm : \u2200 (n : \u2115), MeasurableSet (Td n)\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), t n) \u2264 \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)", "state_after": "no goals"}, {"tactic": "exact this.trans (iSup_extend_bot Encodable.encode_injective _)", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.46518\n\u03b3 : Type ?u.46521\n\u03b4 : Type ?u.46524\n\u03b9 : Type u_1\nR : Type ?u.46530\nR' : Type ?u.46533\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nval\u271d : Encodable \u03b9\nt : \u2115 \u2192 Set \u03b1\nht : Function.extend Encodable.encode s \u22a5 = t\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nthis : \u2191\u2191\u03bc (\u2a06 (i : \u03b9), s i) = \u2a06 (n : \u2115), Function.extend Encodable.encode (fun x => \u2191\u2191\u03bc (s x)) (fun x => 0) n\n\u22a2 \u2191\u2191\u03bc (\u22c3 (i : \u03b9), s i) = \u2a06 (i : \u03b9), \u2191\u2191\u03bc (s i)", "state_after": "no goals"}, {"tactic": "rw [iUnion_disjointed]", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.46518\n\u03b3 : Type ?u.46521\n\u03b4 : Type ?u.46524\nR : Type ?u.46530\nR' : Type ?u.46533\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\nTd : \u2115 \u2192 Set \u03b1 := disjointed T\nhm : \u2200 (n : \u2115), MeasurableSet (Td n)\n\u22a2 \u2191\u2191\u03bc (\u22c3 (n : \u2115), T n) = \u2191\u2191\u03bc (\u22c3 (n : \u2115), Td n)", "state_after": "no goals"}, {"tactic": "rcases hd.finset_le I with \u27e8N, hN\u27e9", "state_before": "\u03b1 : Type u_2\n\u03b2 : Type ?u.46518\n\u03b3 : Type ?u.46521\n\u03b4 : Type ?u.46524\nR : Type ?u.46530\nR' : Type ?u.46533\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\nTd : \u2115 \u2192 Set \u03b1 := disjointed T\nhm : \u2200 (n : \u2115), MeasurableSet (Td n)\nI : Finset \u2115\n\u22a2 \u2211 n in I, \u2191\u2191\u03bc (Td n) \u2264 \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)", "state_after": "case intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.46518\n\u03b3 : Type ?u.46521\n\u03b4 : Type ?u.46524\nR : Type ?u.46530\nR' : Type ?u.46533\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\nTd : \u2115 \u2192 Set \u03b1 := disjointed T\nhm : \u2200 (n : \u2115), MeasurableSet (Td n)\nI : Finset \u2115\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2208 I \u2192 t i \u2286 t N\n\u22a2 \u2211 n in I, \u2191\u2191\u03bc (Td n) \u2264 \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)"}, {"tactic": "calc\n  (\u2211 n in I, \u03bc (Td n)) = \u03bc (\u22c3 n \u2208 I, Td n) :=\n    (measure_biUnion_finset ((disjoint_disjointed T).set_pairwise I) fun n _ => hm n).symm\n  _ \u2264 \u03bc (\u22c3 n \u2208 I, T n) := (measure_mono (iUnion\u2082_mono fun n _hn => disjointed_subset _ _))\n  _ = \u03bc (\u22c3 n \u2208 I, t n) := (measure_biUnion_toMeasurable I.countable_toSet _)\n  _ \u2264 \u03bc (t N) := (measure_mono (iUnion\u2082_subset hN))\n  _ \u2264 \u2a06 n, \u03bc (t n) := le_iSup (\u03bc \u2218 t) N", "state_before": "case intro\n\u03b1 : Type u_2\n\u03b2 : Type ?u.46518\n\u03b3 : Type ?u.46521\n\u03b4 : Type ?u.46524\nR : Type ?u.46530\nR' : Type ?u.46533\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nt : \u2115 \u2192 Set \u03b1\nhd : Directed (fun x x_1 => x \u2286 x_1) t\nT : \u2115 \u2192 Set \u03b1 := fun n => toMeasurable \u03bc (t n)\nTd : \u2115 \u2192 Set \u03b1 := disjointed T\nhm : \u2200 (n : \u2115), MeasurableSet (Td n)\nI : Finset \u2115\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2208 I \u2192 t i \u2286 t N\n\u22a2 \u2211 n in I, \u2191\u2191\u03bc (Td n) \u2264 \u2a06 (n : \u2115), \u2191\u2191\u03bc (t n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/IsROrC/Basic.lean", "full_name": "IsROrC.re_le_norm", "start": [732, 1], "end": [733, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "full_name": "Asymptotics.div_isBoundedUnder_of_isBigO", "start": [1987, 1], "end": [1992, 57], "traced_tactics": [{"tactic": "obtain \u27e8c, h\u2080, hc\u27e9 := h.exists_nonneg", "state_before": "\u03b1\u271d : Type ?u.635937\n\u03b2 : Type ?u.635940\nE : Type ?u.635943\nF : Type ?u.635946\nG : Type ?u.635949\nE' : Type ?u.635952\nF' : Type ?u.635955\nG' : Type ?u.635958\nE'' : Type ?u.635961\nF'' : Type ?u.635964\nG'' : Type ?u.635967\nR : Type ?u.635970\nR' : Type ?u.635973\n\ud835\udd5c : Type u_2\n\ud835\udd5c' : Type ?u.635979\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\u271d : \u03b1\u271d \u2192 E\ng\u271d : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl\u271d l' : Filter \u03b1\u271d\n\u03b1 : Type u_1\nl : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nh : f =O[l] g\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2264 x_1) l fun x => \u2016f x / g x\u2016", "state_after": "case intro.intro\n\u03b1\u271d : Type ?u.635937\n\u03b2 : Type ?u.635940\nE : Type ?u.635943\nF : Type ?u.635946\nG : Type ?u.635949\nE' : Type ?u.635952\nF' : Type ?u.635955\nG' : Type ?u.635958\nE'' : Type ?u.635961\nF'' : Type ?u.635964\nG'' : Type ?u.635967\nR : Type ?u.635970\nR' : Type ?u.635973\n\ud835\udd5c : Type u_2\n\ud835\udd5c' : Type ?u.635979\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\u271d : \u03b1\u271d \u2192 E\ng\u271d : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl\u271d l' : Filter \u03b1\u271d\n\u03b1 : Type u_1\nl : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nh : f =O[l] g\nc : \u211d\nh\u2080 : 0 \u2264 c\nhc : IsBigOWith c l f g\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2264 x_1) l fun x => \u2016f x / g x\u2016"}, {"tactic": "refine' \u27e8c, eventually_map.2 (hc.bound.mono fun x hx => _)\u27e9", "state_before": "case intro.intro\n\u03b1\u271d : Type ?u.635937\n\u03b2 : Type ?u.635940\nE : Type ?u.635943\nF : Type ?u.635946\nG : Type ?u.635949\nE' : Type ?u.635952\nF' : Type ?u.635955\nG' : Type ?u.635958\nE'' : Type ?u.635961\nF'' : Type ?u.635964\nG'' : Type ?u.635967\nR : Type ?u.635970\nR' : Type ?u.635973\n\ud835\udd5c : Type u_2\n\ud835\udd5c' : Type ?u.635979\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\u271d : \u03b1\u271d \u2192 E\ng\u271d : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl\u271d l' : Filter \u03b1\u271d\n\u03b1 : Type u_1\nl : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nh : f =O[l] g\nc : \u211d\nh\u2080 : 0 \u2264 c\nhc : IsBigOWith c l f g\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2264 x_1) l fun x => \u2016f x / g x\u2016", "state_after": "case intro.intro\n\u03b1\u271d : Type ?u.635937\n\u03b2 : Type ?u.635940\nE : Type ?u.635943\nF : Type ?u.635946\nG : Type ?u.635949\nE' : Type ?u.635952\nF' : Type ?u.635955\nG' : Type ?u.635958\nE'' : Type ?u.635961\nF'' : Type ?u.635964\nG'' : Type ?u.635967\nR : Type ?u.635970\nR' : Type ?u.635973\n\ud835\udd5c : Type u_2\n\ud835\udd5c' : Type ?u.635979\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\u271d : \u03b1\u271d \u2192 E\ng\u271d : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl\u271d l' : Filter \u03b1\u271d\n\u03b1 : Type u_1\nl : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nh : f =O[l] g\nc : \u211d\nh\u2080 : 0 \u2264 c\nhc : IsBigOWith c l f g\nx : \u03b1\nhx : \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 (fun x x_1 => x \u2264 x_1) \u2016f x / g x\u2016 c"}, {"tactic": "rw [norm_div]", "state_before": "case intro.intro\n\u03b1\u271d : Type ?u.635937\n\u03b2 : Type ?u.635940\nE : Type ?u.635943\nF : Type ?u.635946\nG : Type ?u.635949\nE' : Type ?u.635952\nF' : Type ?u.635955\nG' : Type ?u.635958\nE'' : Type ?u.635961\nF'' : Type ?u.635964\nG'' : Type ?u.635967\nR : Type ?u.635970\nR' : Type ?u.635973\n\ud835\udd5c : Type u_2\n\ud835\udd5c' : Type ?u.635979\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\u271d : \u03b1\u271d \u2192 E\ng\u271d : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl\u271d l' : Filter \u03b1\u271d\n\u03b1 : Type u_1\nl : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nh : f =O[l] g\nc : \u211d\nh\u2080 : 0 \u2264 c\nhc : IsBigOWith c l f g\nx : \u03b1\nhx : \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 (fun x x_1 => x \u2264 x_1) \u2016f x / g x\u2016 c", "state_after": "case intro.intro\n\u03b1\u271d : Type ?u.635937\n\u03b2 : Type ?u.635940\nE : Type ?u.635943\nF : Type ?u.635946\nG : Type ?u.635949\nE' : Type ?u.635952\nF' : Type ?u.635955\nG' : Type ?u.635958\nE'' : Type ?u.635961\nF'' : Type ?u.635964\nG'' : Type ?u.635967\nR : Type ?u.635970\nR' : Type ?u.635973\n\ud835\udd5c : Type u_2\n\ud835\udd5c' : Type ?u.635979\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\u271d : \u03b1\u271d \u2192 E\ng\u271d : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl\u271d l' : Filter \u03b1\u271d\n\u03b1 : Type u_1\nl : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nh : f =O[l] g\nc : \u211d\nh\u2080 : 0 \u2264 c\nhc : IsBigOWith c l f g\nx : \u03b1\nhx : \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 (fun x x_1 => x \u2264 x_1) (\u2016f x\u2016 / \u2016g x\u2016) c"}, {"tactic": "exact div_le_of_nonneg_of_le_mul (norm_nonneg _) h\u2080 hx", "state_before": "case intro.intro\n\u03b1\u271d : Type ?u.635937\n\u03b2 : Type ?u.635940\nE : Type ?u.635943\nF : Type ?u.635946\nG : Type ?u.635949\nE' : Type ?u.635952\nF' : Type ?u.635955\nG' : Type ?u.635958\nE'' : Type ?u.635961\nF'' : Type ?u.635964\nG'' : Type ?u.635967\nR : Type ?u.635970\nR' : Type ?u.635973\n\ud835\udd5c : Type u_2\n\ud835\udd5c' : Type ?u.635979\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\u271d : \u03b1\u271d \u2192 E\ng\u271d : \u03b1\u271d \u2192 F\nk : \u03b1\u271d \u2192 G\nf' : \u03b1\u271d \u2192 E'\ng' : \u03b1\u271d \u2192 F'\nk' : \u03b1\u271d \u2192 G'\nf'' : \u03b1\u271d \u2192 E''\ng'' : \u03b1\u271d \u2192 F''\nk'' : \u03b1\u271d \u2192 G''\nl\u271d l' : Filter \u03b1\u271d\n\u03b1 : Type u_1\nl : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nh : f =O[l] g\nc : \u211d\nh\u2080 : 0 \u2264 c\nhc : IsBigOWith c l f g\nx : \u03b1\nhx : \u2016f x\u2016 \u2264 c * \u2016g x\u2016\n\u22a2 (fun x x_1 => x \u2264 x_1) (\u2016f x\u2016 / \u2016g x\u2016) c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Equiv.lean", "full_name": "AlgEquiv.map_add", "start": [207, 11], "end": [208, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Logic/Equiv/Defs.lean", "full_name": "Equiv.psigmaCongrRight_symm", "start": [715, 1], "end": [716, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Localization/Predicate.lean", "full_name": "CategoryTheory.Localization.whiskeringLeftFunctor'_obj", "start": [268, 1], "end": [269, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/SemidirectProduct.lean", "full_name": "SemidirectProduct.inv_left", "start": [97, 1], "end": [97, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.Eventually.exists_forall_of_atBot", "start": [318, 1], "end": [320, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/UpperLower/Basic.lean", "full_name": "isLowerSet_iff_Iio_subset", "start": [379, 1], "end": [380, 68], "traced_tactics": [{"tactic": "simp [isLowerSet_iff_forall_lt, subset_def, @forall_swap (_ \u2208 s)]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.34448\n\u03b3 : Type ?u.34451\n\u03b9 : Sort ?u.34454\n\u03ba : \u03b9 \u2192 Sort ?u.34459\ninst\u271d : PartialOrder \u03b1\ns : Set \u03b1\n\u22a2 IsLowerSet s \u2194 \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2192 Iio a \u2286 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Deprecated/Group.lean", "full_name": "IsMulHom.inv", "start": [91, 1], "end": [92, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "continuousWithinAt_Ioc_iff_Iic", "start": [609, 1], "end": [611, 69], "traced_tactics": [{"tactic": "simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Iic h]", "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderClosedTopology \u03b1\na\u271d b\u271d : \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b2\na b : \u03b1\nf : \u03b1 \u2192 \u03b2\nh : a < b\n\u22a2 ContinuousWithinAt f (Ioc a b) b \u2194 ContinuousWithinAt f (Iic b) b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Type.lean", "full_name": "Equiv.Perm.exists_fixed_point_of_prime'", "start": [345, 1], "end": [353, 33], "traced_tactics": [{"tactic": "classical\n  have h : \u2200 b : \u03b1, b \u2208 \u03c3.support\u1d9c \u2194 \u03c3 b = b := fun b => by\n    rw [Finset.mem_compl, mem_support, Classical.not_not]\n  obtain \u27e8b, hb1, hb2\u27e9 := Finset.exists_ne_of_one_lt_card (hp.out.one_lt.trans_le\n    (Nat.le_of_dvd (Finset.card_pos.mpr \u27e8a, (h a).mpr ha\u27e9) (Nat.modEq_zero_iff_dvd.mp\n      ((card_compl_support_modEq h\u03c3).trans (Nat.modEq_zero_iff_dvd.mpr h\u03b1))))) a\n  exact \u27e8b, (h b).mp hb1, hb2\u27e9", "state_before": "\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\nh\u03b1 : p \u2223 Fintype.card \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\na : \u03b1\nha : \u2191\u03c3 a = a\n\u22a2 \u2203 b, \u2191\u03c3 b = b \u2227 b \u2260 a", "state_after": "no goals"}, {"tactic": "have h : \u2200 b : \u03b1, b \u2208 \u03c3.support\u1d9c \u2194 \u03c3 b = b := fun b => by\n  rw [Finset.mem_compl, mem_support, Classical.not_not]", "state_before": "\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\nh\u03b1 : p \u2223 Fintype.card \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\na : \u03b1\nha : \u2191\u03c3 a = a\n\u22a2 \u2203 b, \u2191\u03c3 b = b \u2227 b \u2260 a", "state_after": "\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\nh\u03b1 : p \u2223 Fintype.card \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\na : \u03b1\nha : \u2191\u03c3 a = a\nh : \u2200 (b : \u03b1), b \u2208 support \u03c3\u1d9c \u2194 \u2191\u03c3 b = b\n\u22a2 \u2203 b, \u2191\u03c3 b = b \u2227 b \u2260 a"}, {"tactic": "obtain \u27e8b, hb1, hb2\u27e9 := Finset.exists_ne_of_one_lt_card (hp.out.one_lt.trans_le\n  (Nat.le_of_dvd (Finset.card_pos.mpr \u27e8a, (h a).mpr ha\u27e9) (Nat.modEq_zero_iff_dvd.mp\n    ((card_compl_support_modEq h\u03c3).trans (Nat.modEq_zero_iff_dvd.mpr h\u03b1))))) a", "state_before": "\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\nh\u03b1 : p \u2223 Fintype.card \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\na : \u03b1\nha : \u2191\u03c3 a = a\nh : \u2200 (b : \u03b1), b \u2208 support \u03c3\u1d9c \u2194 \u2191\u03c3 b = b\n\u22a2 \u2203 b, \u2191\u03c3 b = b \u2227 b \u2260 a", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\nh\u03b1 : p \u2223 Fintype.card \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\na : \u03b1\nha : \u2191\u03c3 a = a\nh : \u2200 (b : \u03b1), b \u2208 support \u03c3\u1d9c \u2194 \u2191\u03c3 b = b\nb : \u03b1\nhb1 : b \u2208 support \u03c3\u1d9c\nhb2 : b \u2260 a\n\u22a2 \u2203 b, \u2191\u03c3 b = b \u2227 b \u2260 a"}, {"tactic": "exact \u27e8b, (h b).mp hb1, hb2\u27e9", "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\nh\u03b1 : p \u2223 Fintype.card \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\na : \u03b1\nha : \u2191\u03c3 a = a\nh : \u2200 (b : \u03b1), b \u2208 support \u03c3\u1d9c \u2194 \u2191\u03c3 b = b\nb : \u03b1\nhb1 : b \u2208 support \u03c3\u1d9c\nhb2 : b \u2260 a\n\u22a2 \u2203 b, \u2191\u03c3 b = b \u2227 b \u2260 a", "state_after": "no goals"}, {"tactic": "rw [Finset.mem_compl, mem_support, Classical.not_not]", "state_before": "\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\np n : \u2115\nhp : Fact (Nat.Prime p)\nh\u03b1 : p \u2223 Fintype.card \u03b1\n\u03c3 : Perm \u03b1\nh\u03c3 : \u03c3 ^ p ^ n = 1\na : \u03b1\nha : \u2191\u03c3 a = a\nb : \u03b1\n\u22a2 b \u2208 support \u03c3\u1d9c \u2194 \u2191\u03c3 b = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/GroupTheory/Commutator.lean", "full_name": "commutatorElement_one_left", "start": [52, 1], "end": [53, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Monoidal/End.lean", "full_name": "CategoryTheory.\u03bc_naturality\u2082", "start": [145, 1], "end": [150, 19], "traced_tactics": [{"tactic": "have := congr_app (F.toLaxMonoidalFunctor.\u03bc_natural f g) X", "state_before": "C : Type u\ninst\u271d\u00b2 : Category C\nM : Type u_2\ninst\u271d\u00b9 : Category M\ninst\u271d : MonoidalCategory M\nF : MonoidalFunctor M (C \u2964 C)\nm n m' n' : M\nf : m \u27f6 m'\ng : n \u27f6 n'\nX : C\n\u22a2 (F.map g).app ((F.obj m).obj X) \u226b\n      (F.obj n').map ((F.map f).app X) \u226b (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor m' n').app X =\n    (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor m n).app X \u226b (F.map (f \u2297 g)).app X", "state_after": "C : Type u\ninst\u271d\u00b2 : Category C\nM : Type u_2\ninst\u271d\u00b9 : Category M\ninst\u271d : MonoidalCategory M\nF : MonoidalFunctor M (C \u2964 C)\nm n m' n' : M\nf : m \u27f6 m'\ng : n \u27f6 n'\nX : C\nthis :\n  ((F.map f \u2297 F.map g) \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor m' n').app X =\n    (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor m n \u226b F.map (f \u2297 g)).app X\n\u22a2 (F.map g).app ((F.obj m).obj X) \u226b\n      (F.obj n').map ((F.map f).app X) \u226b (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor m' n').app X =\n    (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor m n).app X \u226b (F.map (f \u2297 g)).app X"}, {"tactic": "dsimp [endofunctorMonoidalCategory] at this", "state_before": "C : Type u\ninst\u271d\u00b2 : Category C\nM : Type u_2\ninst\u271d\u00b9 : Category M\ninst\u271d : MonoidalCategory M\nF : MonoidalFunctor M (C \u2964 C)\nm n m' n' : M\nf : m \u27f6 m'\ng : n \u27f6 n'\nX : C\nthis :\n  ((F.map f \u2297 F.map g) \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor m' n').app X =\n    (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor m n \u226b F.map (f \u2297 g)).app X\n\u22a2 (F.map g).app ((F.obj m).obj X) \u226b\n      (F.obj n').map ((F.map f).app X) \u226b (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor m' n').app X =\n    (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor m n).app X \u226b (F.map (f \u2297 g)).app X", "state_after": "C : Type u\ninst\u271d\u00b2 : Category C\nM : Type u_2\ninst\u271d\u00b9 : Category M\ninst\u271d : MonoidalCategory M\nF : MonoidalFunctor M (C \u2964 C)\nm n m' n' : M\nf : m \u27f6 m'\ng : n \u27f6 n'\nX : C\nthis :\n  ((F.map g).app ((F.obj m).obj X) \u226b (F.obj n').map ((F.map f).app X)) \u226b\n      (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor m' n').app X =\n    (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor m n).app X \u226b (F.map (f \u2297 g)).app X\n\u22a2 (F.map g).app ((F.obj m).obj X) \u226b\n      (F.obj n').map ((F.map f).app X) \u226b (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor m' n').app X =\n    (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor m n).app X \u226b (F.map (f \u2297 g)).app X"}, {"tactic": "simpa using this", "state_before": "C : Type u\ninst\u271d\u00b2 : Category C\nM : Type u_2\ninst\u271d\u00b9 : Category M\ninst\u271d : MonoidalCategory M\nF : MonoidalFunctor M (C \u2964 C)\nm n m' n' : M\nf : m \u27f6 m'\ng : n \u27f6 n'\nX : C\nthis :\n  ((F.map g).app ((F.obj m).obj X) \u226b (F.obj n').map ((F.map f).app X)) \u226b\n      (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor m' n').app X =\n    (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor m n).app X \u226b (F.map (f \u2297 g)).app X\n\u22a2 (F.map g).app ((F.obj m).obj X) \u226b\n      (F.obj n').map ((F.map f).app X) \u226b (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor m' n').app X =\n    (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor m n).app X \u226b (F.map (f \u2297 g)).app X", "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.mapPair_left", "start": [164, 1], "end": [165, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Core.lean", "full_name": "Subtype.existsOfSubtype", "start": [955, 1], "end": [956, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/Convex/Side.lean", "full_name": "AffineSubspace.not_wSameSide_bot", "start": [225, 1], "end": [226, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "full_name": "MeasureTheory.IntegrableOn.congr_fun", "start": [156, 1], "end": [158, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/LinearAlgebra/Span.lean", "full_name": "Submodule.span_span_of_tower", "start": [519, 1], "end": [521, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.le_mul_right", "start": [751, 1], "end": [753, 17], "traced_tactics": [{"tactic": "convert mul_le_mul_right' (one_le_iff_pos.2 hb) a", "state_before": "\u03b1 : Type ?u.194451\n\u03b2 : Type ?u.194454\n\u03b3 : Type ?u.194457\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal\nhb : 0 < b\n\u22a2 a \u2264 b * a", "state_after": "case h.e'_3\n\u03b1 : Type ?u.194451\n\u03b2 : Type ?u.194454\n\u03b3 : Type ?u.194457\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal\nhb : 0 < b\n\u22a2 a = 1 * a"}, {"tactic": "rw [one_mul a]", "state_before": "case h.e'_3\n\u03b1 : Type ?u.194451\n\u03b2 : Type ?u.194454\n\u03b3 : Type ?u.194457\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b : Ordinal\nhb : 0 < b\n\u22a2 a = 1 * a", "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_Ico", "start": [91, 1], "end": [95, 98], "traced_tactics": [{"tactic": "simpa only [Equiv.neg_apply, mem_Ico, neg_zsmul, \u2190 sub_eq_add_neg, le_sub_iff_add_le, zero_add,\n  add_comm c, sub_lt_iff_lt_add', add_assoc] using existsUnique_zsmul_near_of_pos' ha (b - c)", "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.neg \u2124) x \u2022 a \u2208 Ico c (c + a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "full_name": "Nat.cast_pow", "start": [534, 1], "end": [537, 62], "traced_tactics": [{"tactic": "induction' m with m ih", "state_before": "\u03b1 : Type ?u.266183\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 : Semiring R\nn m : \u2115\n\u22a2 \u2191(n ^ m) = \u2191n ^ m", "state_after": "case zero\n\u03b1 : Type ?u.266183\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 : Semiring R\nn : \u2115\n\u22a2 \u2191(n ^ zero) = \u2191n ^ zero\n\ncase succ\n\u03b1 : Type ?u.266183\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 : Semiring R\nn m : \u2115\nih : \u2191(n ^ m) = \u2191n ^ m\n\u22a2 \u2191(n ^ succ m) = \u2191n ^ succ m"}, {"tactic": "simp", "state_before": "case zero\n\u03b1 : Type ?u.266183\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 : Semiring R\nn : \u2115\n\u22a2 \u2191(n ^ zero) = \u2191n ^ zero", "state_after": "no goals"}, {"tactic": "rw [_root_.pow_succ', _root_.pow_succ', Nat.cast_mul, ih]", "state_before": "case succ\n\u03b1 : Type ?u.266183\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 : Semiring R\nn m : \u2115\nih : \u2191(n ^ m) = \u2191n ^ m\n\u22a2 \u2191(n ^ succ m) = \u2191n ^ succ m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.prod_le_prod", "start": [1057, 1], "end": [1059, 88], "traced_tactics": [{"tactic": "have := H i", "state_before": "\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal\nH : \u2200 (i : \u03b9), f i \u2264 g i\ni : \u03b9\n\u22a2 Nonempty (Quotient.out (f i) \u21aa Quotient.out (g i))", "state_after": "\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal\nH : \u2200 (i : \u03b9), f i \u2264 g i\ni : \u03b9\nthis : f i \u2264 g i\n\u22a2 Nonempty (Quotient.out (f i) \u21aa Quotient.out (g i))"}, {"tactic": "rwa [\u2190 mk_out (f i), \u2190 mk_out (g i)] at this", "state_before": "\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf g : \u03b9 \u2192 Cardinal\nH : \u2200 (i : \u03b9), f i \u2264 g i\ni : \u03b9\nthis : f i \u2264 g i\n\u22a2 Nonempty (Quotient.out (f i) \u21aa Quotient.out (g i))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Chain.lean", "full_name": "IsChain.exists3", "start": [131, 1], "end": [136, 63], "traced_tactics": [{"tactic": "rcases directedOn_iff_directed.mpr (IsChain.directed hchain) a mem1 b mem2 with \u27e8z, mem4, H1, H2\u27e9", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.5666\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc\u271d c\u2081 c\u2082 c\u2083 s t : Set \u03b1\na\u271d b\u271d x y : \u03b1\ninst\u271d\u00b9 : IsRefl \u03b1 r\nhchain : IsChain r s\ninst\u271d : IsTrans \u03b1 r\na b c : \u03b1\nmem1 : a \u2208 s\nmem2 : b \u2208 s\nmem3 : c \u2208 s\n\u22a2 \u2203 z x, r a z \u2227 r b z \u2227 r c z", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.5666\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc\u271d c\u2081 c\u2082 c\u2083 s t : Set \u03b1\na\u271d b\u271d x y : \u03b1\ninst\u271d\u00b9 : IsRefl \u03b1 r\nhchain : IsChain r s\ninst\u271d : IsTrans \u03b1 r\na b c : \u03b1\nmem1 : a \u2208 s\nmem2 : b \u2208 s\nmem3 : c \u2208 s\nz : \u03b1\nmem4 : z \u2208 s\nH1 : r a z\nH2 : r b z\n\u22a2 \u2203 z x, r a z \u2227 r b z \u2227 r c z"}, {"tactic": "rcases directedOn_iff_directed.mpr (IsChain.directed hchain) z mem4 c mem3 with\n  \u27e8z', mem5, H3, H4\u27e9", "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.5666\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc\u271d c\u2081 c\u2082 c\u2083 s t : Set \u03b1\na\u271d b\u271d x y : \u03b1\ninst\u271d\u00b9 : IsRefl \u03b1 r\nhchain : IsChain r s\ninst\u271d : IsTrans \u03b1 r\na b c : \u03b1\nmem1 : a \u2208 s\nmem2 : b \u2208 s\nmem3 : c \u2208 s\nz : \u03b1\nmem4 : z \u2208 s\nH1 : r a z\nH2 : r b z\n\u22a2 \u2203 z x, r a z \u2227 r b z \u2227 r c z", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.5666\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc\u271d c\u2081 c\u2082 c\u2083 s t : Set \u03b1\na\u271d b\u271d x y : \u03b1\ninst\u271d\u00b9 : IsRefl \u03b1 r\nhchain : IsChain r s\ninst\u271d : IsTrans \u03b1 r\na b c : \u03b1\nmem1 : a \u2208 s\nmem2 : b \u2208 s\nmem3 : c \u2208 s\nz : \u03b1\nmem4 : z \u2208 s\nH1 : r a z\nH2 : r b z\nz' : \u03b1\nmem5 : z' \u2208 s\nH3 : r z z'\nH4 : r c z'\n\u22a2 \u2203 z x, r a z \u2227 r b z \u2227 r c z"}, {"tactic": "exact \u27e8z', mem5, _root_.trans H1 H3, _root_.trans H2 H3, H4\u27e9", "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type ?u.5666\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nc\u271d c\u2081 c\u2082 c\u2083 s t : Set \u03b1\na\u271d b\u271d x y : \u03b1\ninst\u271d\u00b9 : IsRefl \u03b1 r\nhchain : IsChain r s\ninst\u271d : IsTrans \u03b1 r\na b c : \u03b1\nmem1 : a \u2208 s\nmem2 : b \u2208 s\nmem3 : c \u2208 s\nz : \u03b1\nmem4 : z \u2208 s\nH1 : r a z\nH2 : r b z\nz' : \u03b1\nmem5 : z' \u2208 s\nH3 : r z z'\nH4 : r c z'\n\u22a2 \u2203 z x, r a z \u2227 r b z \u2227 r c z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", "full_name": "Set.ordConnectedComponent_eq_empty", "start": [65, 1], "end": [66, 67], "traced_tactics": [{"tactic": "rw [\u2190 not_nonempty_iff_eq_empty, nonempty_ordConnectedComponent]", "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\ns t : Set \u03b1\nx y z : \u03b1\n\u22a2 ordConnectedComponent s x = \u2205 \u2194 \u00acx \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Antichain.lean", "full_name": "IsAntichain.image_iso_iff", "start": [173, 1], "end": [175, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/PNat/Xgcd.lean", "full_name": "PNat.gcd_a_eq", "start": [526, 1], "end": [527, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean", "full_name": "Complex.mul_cpow_ofReal_nonneg", "start": [167, 1], "end": [178, 78], "traced_tactics": [{"tactic": "rcases eq_or_ne r 0 with (rfl | hr)", "state_before": "a b : \u211d\nha : 0 \u2264 a\nhb : 0 \u2264 b\nr : \u2102\n\u22a2 (\u2191a * \u2191b) ^ r = \u2191a ^ r * \u2191b ^ r", "state_after": "case inl\na b : \u211d\nha : 0 \u2264 a\nhb : 0 \u2264 b\n\u22a2 (\u2191a * \u2191b) ^ 0 = \u2191a ^ 0 * \u2191b ^ 0\n\ncase inr\na b : \u211d\nha : 0 \u2264 a\nhb : 0 \u2264 b\nr : \u2102\nhr : r \u2260 0\n\u22a2 (\u2191a * \u2191b) ^ r = \u2191a ^ r * \u2191b ^ r"}, {"tactic": "rcases eq_or_lt_of_le ha with (rfl | ha')", "state_before": "case inr\na b : \u211d\nha : 0 \u2264 a\nhb : 0 \u2264 b\nr : \u2102\nhr : r \u2260 0\n\u22a2 (\u2191a * \u2191b) ^ r = \u2191a ^ r * \u2191b ^ r", "state_after": "case inr.inl\nb : \u211d\nhb : 0 \u2264 b\nr : \u2102\nhr : r \u2260 0\nha : 0 \u2264 0\n\u22a2 (\u21910 * \u2191b) ^ r = \u21910 ^ r * \u2191b ^ r\n\ncase inr.inr\na b : \u211d\nha : 0 \u2264 a\nhb : 0 \u2264 b\nr : \u2102\nhr : r \u2260 0\nha' : 0 < a\n\u22a2 (\u2191a * \u2191b) ^ r = \u2191a ^ r * \u2191b ^ r"}, {"tactic": "rcases eq_or_lt_of_le hb with (rfl | hb')", "state_before": "case inr.inr\na b : \u211d\nha : 0 \u2264 a\nhb : 0 \u2264 b\nr : \u2102\nhr : r \u2260 0\nha' : 0 < a\n\u22a2 (\u2191a * \u2191b) ^ r = \u2191a ^ r * \u2191b ^ r", "state_after": "case inr.inr.inl\na : \u211d\nha : 0 \u2264 a\nr : \u2102\nhr : r \u2260 0\nha' : 0 < a\nhb : 0 \u2264 0\n\u22a2 (\u2191a * \u21910) ^ r = \u2191a ^ r * \u21910 ^ r\n\ncase inr.inr.inr\na b : \u211d\nha : 0 \u2264 a\nhb : 0 \u2264 b\nr : \u2102\nhr : r \u2260 0\nha' : 0 < a\nhb' : 0 < b\n\u22a2 (\u2191a * \u2191b) ^ r = \u2191a ^ r * \u2191b ^ r"}, {"tactic": "have ha'' : (a : \u2102) \u2260 0 := ofReal_ne_zero.mpr ha'.ne'", "state_before": "case inr.inr.inr\na b : \u211d\nha : 0 \u2264 a\nhb : 0 \u2264 b\nr : \u2102\nhr : r \u2260 0\nha' : 0 < a\nhb' : 0 < b\n\u22a2 (\u2191a * \u2191b) ^ r = \u2191a ^ r * \u2191b ^ r", "state_after": "case inr.inr.inr\na b : \u211d\nha : 0 \u2264 a\nhb : 0 \u2264 b\nr : \u2102\nhr : r \u2260 0\nha' : 0 < a\nhb' : 0 < b\nha'' : \u2191a \u2260 0\n\u22a2 (\u2191a * \u2191b) ^ r = \u2191a ^ r * \u2191b ^ r"}, {"tactic": "have hb'' : (b : \u2102) \u2260 0 := ofReal_ne_zero.mpr hb'.ne'", "state_before": "case inr.inr.inr\na b : \u211d\nha : 0 \u2264 a\nhb : 0 \u2264 b\nr : \u2102\nhr : r \u2260 0\nha' : 0 < a\nhb' : 0 < b\nha'' : \u2191a \u2260 0\n\u22a2 (\u2191a * \u2191b) ^ r = \u2191a ^ r * \u2191b ^ r", "state_after": "case inr.inr.inr\na b : \u211d\nha : 0 \u2264 a\nhb : 0 \u2264 b\nr : \u2102\nhr : r \u2260 0\nha' : 0 < a\nhb' : 0 < b\nha'' : \u2191a \u2260 0\nhb'' : \u2191b \u2260 0\n\u22a2 (\u2191a * \u2191b) ^ r = \u2191a ^ r * \u2191b ^ r"}, {"tactic": "rw [cpow_def_of_ne_zero (mul_ne_zero ha'' hb''), log_ofReal_mul ha' hb'', ofReal_log ha,\n  add_mul, exp_add, \u2190 cpow_def_of_ne_zero ha'', \u2190 cpow_def_of_ne_zero hb'']", "state_before": "case inr.inr.inr\na b : \u211d\nha : 0 \u2264 a\nhb : 0 \u2264 b\nr : \u2102\nhr : r \u2260 0\nha' : 0 < a\nhb' : 0 < b\nha'' : \u2191a \u2260 0\nhb'' : \u2191b \u2260 0\n\u22a2 (\u2191a * \u2191b) ^ r = \u2191a ^ r * \u2191b ^ r", "state_after": "no goals"}, {"tactic": "simp only [cpow_zero, mul_one]", "state_before": "case inl\na b : \u211d\nha : 0 \u2264 a\nhb : 0 \u2264 b\n\u22a2 (\u2191a * \u2191b) ^ 0 = \u2191a ^ 0 * \u2191b ^ 0", "state_after": "no goals"}, {"tactic": "rw [ofReal_zero, MulZeroClass.zero_mul, zero_cpow hr, MulZeroClass.zero_mul]", "state_before": "case inr.inl\nb : \u211d\nhb : 0 \u2264 b\nr : \u2102\nhr : r \u2260 0\nha : 0 \u2264 0\n\u22a2 (\u21910 * \u2191b) ^ r = \u21910 ^ r * \u2191b ^ r", "state_after": "no goals"}, {"tactic": "rw [ofReal_zero, MulZeroClass.mul_zero, zero_cpow hr, MulZeroClass.mul_zero]", "state_before": "case inr.inr.inl\na : \u211d\nha : 0 \u2264 a\nr : \u2102\nhr : r \u2260 0\nha' : 0 < a\nhb : 0 \u2264 0\n\u22a2 (\u2191a * \u21910) ^ r = \u2191a ^ r * \u21910 ^ r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/CategoryTheory/Limits/Constructions/Over/Products.lean", "full_name": "CategoryTheory.Over.ConstructProducts.over_product_of_widePullback", "start": [144, 1], "end": [146, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Module/Zlattice.lean", "full_name": "Zspan.coe_floor_self", "start": [189, 1], "end": [190, 97], "traced_tactics": [{"tactic": "rw [repr_floor_apply, Basis.singleton_repr, Basis.singleton_repr]", "state_before": "E : Type ?u.329491\n\u03b9 : Type u_2\nK : Type u_1\ninst\u271d\u2075 : NormedLinearOrderedField K\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace K E\nb : Basis \u03b9 K E\ninst\u271d\u00b2 : FloorRing K\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d 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"5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Bounded.lean", "full_name": "Set.unbounded_inter_not", "start": [311, 1], "end": [313, 51], "traced_tactics": [{"tactic": "simp_rw [\u2190 not_bounded_iff, bounded_inter_not H]", "state_before": "\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\nH : \u2200 (a b : \u03b1), \u2203 m, \u2200 (c : \u03b1), r c a \u2228 r c b \u2192 r c m\na : \u03b1\n\u22a2 Unbounded r (s \u2229 {b | \u00acr b a}) \u2194 Unbounded r s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Operations.lean", "full_name": "Submodule.mul_comm", "start": [599, 11], "end": [601, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.prod_image_image_eq", "start": [277, 1], "end": [280, 88], "traced_tactics": [{"tactic": "simp [-exists_and_right, exists_and_right.symm, and_left_comm, and_assoc, and_comm]", "state_before": "\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_1\n\u03b4 : Type u_2\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nm\u2081 : \u03b1 \u2192 \u03b3\nm\u2082 : \u03b2 \u2192 \u03b4\n\u22a2 \u2200 (x : \u03b3 \u00d7 \u03b4), x \u2208 (m\u2081 '' s) \u00d7\u02e2 (m\u2082 '' t) \u2194 x \u2208 (fun p => (m\u2081 p.fst, m\u2082 p.snd)) '' s \u00d7\u02e2 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/lean4", "commit": "d5348dfac847a56a4595fb6230fd0708dcb4e7e9", "file_path": "src/lean/Init/Data/Nat/Div.lean", "full_name": "Nat.zero_mod", "start": [147, 9], "end": [152, 14], "traced_tactics": [{"tactic": "rw [mod_eq]", "state_before": "b : Nat\n\u22a2 0 % b = 0", "state_after": "b : Nat\n\u22a2 ite (0 < b \u2227 b \u2264 0) ((0 - b) % b) 0 = 0"}, {"tactic": "have : \u00ac (0 < b \u2227 b = 0) := by\n  intro \u27e8h\u2081, h\u2082\u27e9\n  simp_all", "state_before": "b : Nat\n\u22a2 ite (0 < b \u2227 b \u2264 0) ((0 - b) % b) 0 = 0", "state_after": "b : Nat\nthis : \u00ac(0 < b \u2227 b = 0)\n\u22a2 ite (0 < b \u2227 b \u2264 0) ((0 - b) % b) 0 = 0"}, {"tactic": "simp [this]", "state_before": "b : Nat\nthis : \u00ac(0 < b \u2227 b = 0)\n\u22a2 ite (0 < b \u2227 b \u2264 0) ((0 - b) % b) 0 = 0", "state_after": "no goals"}, {"tactic": "intro \u27e8h\u2081, h\u2082\u27e9", "state_before": "b : Nat\n\u22a2 \u00ac(0 < b \u2227 b = 0)", "state_after": "b : Nat\nh\u2081 : 0 < b\nh\u2082 : b = 0\n\u22a2 False"}, {"tactic": "simp_all", "state_before": "b : Nat\nh\u2081 : 0 < b\nh\u2082 : b = 0\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasurableEmbedding.ae_map_iff", "start": [4209, 1], "end": [4210, 54], "traced_tactics": [{"tactic": "simp only [ae_iff, hf.map_apply, preimage_setOf_eq]", "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type ?u.1169876\n\u03b4 : Type ?u.1169879\n\u03b9 : Type ?u.1169882\nR : Type ?u.1169885\nR' : Type ?u.1169888\nm0 : MeasurableSpace \u03b1\nm1 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\np : \u03b2 \u2192 Prop\n\u03bc : MeasureTheory.Measure \u03b1\n\u22a2 (\u2200\u1d50 (x : \u03b2) \u2202Measure.map f \u03bc, p x) \u2194 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p (f x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Order/Bounded.lean", "full_name": "Set.bounded_gt_Ioc", "start": [249, 1], "end": [250, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/Algebra/Tower.lean", "full_name": "IsScalarTower.toAlgHom_apply", "start": [125, 1], "end": [125, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.smul_univ\u2080", "start": [1062, 1], "end": [1064, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/IsROrC/Basic.lean", "full_name": "IsROrC.norm_im_le_norm", "start": [728, 1], "end": [729, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Sign.lean", "full_name": "Right.sign_neg", "start": [448, 1], "end": [455, 9], "traced_tactics": [{"tactic": "simp_rw [sign_apply, Right.neg_pos_iff, Right.neg_neg_iff]", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1\na : \u03b1\n\u22a2 \u2191sign (-a) = -\u2191sign a", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1\na : \u03b1\n\u22a2 (if a < 0 then 1 else if 0 < a then -1 else 0) = -if 0 < a then 1 else if a < 0 then -1 else 0"}, {"tactic": "split_ifs with h h'", "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1\na : \u03b1\n\u22a2 (if a < 0 then 1 else if 0 < a then -1 else 0) = -if 0 < a then 1 else if a < 0 then -1 else 0", "state_after": "case inl.inl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1\na : \u03b1\nh : a < 0\nh' : 0 < a\n\u22a2 False\n\ncase inl.inr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1\na : \u03b1\nh : a < 0\nh' : \u00ac0 < a\n\u22a2 1 = - -1\n\ncase inr.inl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1\na : \u03b1\nh : \u00aca < 0\nh\u271d : 0 < a\n\u22a2 -1 = -1\n\ncase inr.inr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1\na : \u03b1\nh : \u00aca < 0\nh\u271d : \u00ac0 < a\n\u22a2 0 = -0"}, {"tactic": "exact False.elim (lt_asymm h h')", "state_before": "case inl.inl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1\na : \u03b1\nh : a < 0\nh' : 0 < a\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simp", "state_before": "case inl.inr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1\na : \u03b1\nh : a < 0\nh' : \u00ac0 < a\n\u22a2 1 = - -1", "state_after": "no goals"}, {"tactic": "simp", "state_before": "case inr.inl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1\na : \u03b1\nh : \u00aca < 0\nh\u271d : 0 < a\n\u22a2 -1 = -1", "state_after": "no goals"}, {"tactic": "simp", "state_before": "case inr.inr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1\na : \u03b1\nh : \u00aca < 0\nh\u271d : \u00ac0 < a\n\u22a2 0 = -0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Algebra/IndicatorFunction.lean", "full_name": "Set.mulIndicator_compl_mul_self", "start": [424, 1], "end": [426, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/Data/Complex/Basic.lean", "full_name": "Complex.sub_im", "start": [704, 1], "end": [705, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "5a919533f110b7d76410134a237ee374f24eaaad", "file_path": "Mathlib/RingTheory/Ideal/Over.lean", "full_name": "Ideal.injective_quotient_le_comap_map", "start": [80, 1], "end": [92, 97], "traced_tactics": [{"tactic": "refine' quotientMap_injective' (le_of_eq _)", "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type ?u.9808\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\n\u22a2 Function.Injective\n    \u2191(quotientMap (map (mapRingHom (Quotient.mk (comap C P))) P) (mapRingHom (Quotient.mk (comap C P)))\n        (_ : P \u2264 comap (mapRingHom (Quotient.mk (comap C P))) (map (mapRingHom (Quotient.mk (comap C P))) P)))", "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type ?u.9808\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\n\u22a2 comap (mapRingHom (Quotient.mk (comap C P))) (map (mapRingHom (Quotient.mk (comap C P))) P) = P"}, {"tactic": "rw [comap_map_of_surjective (mapRingHom (Ideal.Quotient.mk (P.comap (C : R \u2192+* R[X]))))\n    (map_surjective (Ideal.Quotient.mk (P.comap (C : R \u2192+* R[X]))) Ideal.Quotient.mk_surjective)]", "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type ?u.9808\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\n\u22a2 comap (mapRingHom (Quotient.mk (comap C P))) (map (mapRingHom (Quotient.mk (comap C P))) P) = P", "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type ?u.9808\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\n\u22a2 P \u2294 comap (mapRingHom (Quotient.mk (comap C P))) \u22a5 = P"}, {"tactic": "refine' le_antisymm (sup_le le_rfl _) (le_sup_of_le_left le_rfl)", "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type ?u.9808\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\n\u22a2 P \u2294 comap (mapRingHom (Quotient.mk (comap C P))) \u22a5 = P", "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type ?u.9808\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\n\u22a2 comap (mapRingHom (Quotient.mk (comap C P))) \u22a5 \u2264 P"}, {"tactic": "refine' fun p hp =>\n  polynomial_mem_ideal_of_coeff_mem_ideal P p fun n => Ideal.Quotient.eq_zero_iff_mem.mp _", "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type ?u.9808\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\n\u22a2 comap (mapRingHom (Quotient.mk (comap C P))) \u22a5 \u2264 P", "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type ?u.9808\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\np : R[X]\nhp : p \u2208 comap (mapRingHom (Quotient.mk (comap C P))) \u22a5\nn : \u2115\n\u22a2 \u2191(Quotient.mk (comap C P)) (coeff p n) = 0"}, {"tactic": "simpa only [coeff_map, coe_mapRingHom] using ext_iff.mp (Ideal.mem_bot.mp (mem_comap.mp hp)) n", "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type ?u.9808\ninst\u271d : CommRing S\nf : R \u2192+* S\nI J : Ideal S\nP : Ideal R[X]\np : R[X]\nhp : p \u2208 comap (mapRingHom (Quotient.mk (comap C P))) \u22a5\nn : \u2115\n\u22a2 \u2191(Quotient.mk (comap C P)) (coeff p n) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": 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u_1\nl\u271d : List \u03b1\np q : \u03b1 \u2192 Bool\nx : \u03b1\nl : List \u03b1\nih : countp p l = length (filter p l)\nh : p x = true\n\u22a2 countp p (x :: l) = length (filter p (x :: l))\n\ncase neg\n\u03b1 : Type u_1\nl\u271d : List \u03b1\np q : \u03b1 \u2192 Bool\nx : \u03b1\nl : List \u03b1\nih : countp p l = length (filter p l)\nh : \u00acp x = true\n\u22a2 countp p (x :: l) = length (filter p (x :: l))"}, {"tactic": "rfl", "state_before": "case nil\n\u03b1 : Type u_1\nl : List \u03b1\np q : \u03b1 \u2192 Bool\n\u22a2 countp p [] = length (filter p [])", "state_after": "no goals"}, {"tactic": "rw [countp_cons_of_pos p l h, ih, filter_cons_of_pos l h, length]", "state_before": "case pos\n\u03b1 : Type u_1\nl\u271d : List \u03b1\np q : \u03b1 \u2192 Bool\nx : \u03b1\nl : List \u03b1\nih : countp p l = length (filter p l)\nh : p x = true\n\u22a2 countp p (x :: l) = length (filter p (x :: l))", "state_after": "no goals"}, {"tactic": "rw [countp_cons_of_neg p l h, ih, 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