Datasets:

tasksource

/

leandojo

like 5

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[ 82, 21 ]

[ 81, 1 ]

[ { "state_after": "no goals", "state_before": "ι : Type ?u.302627\nι' : Type ?u.302630\nα : Type u_1\nβ : Type ?u.302636\nγ : Type ?u.302639\ninst✝¹ : SemilatticeSup α\ninst✝ : Nonempty α\np : α × α → Prop\n⊢ (∀ᶠ (x : α × α) in atTop, p x) ↔ ∃ a, ∀ (k : α), k ≥ a → ∀ (l : α), l ≥ a → p (k, l)", "tactic": "simp only [eventually_atTop_prod_self, ball_cond_comm]" } ]

[ 1485, 57 ]

[ 1483, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type ?u.548949\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommGroup M\ninst✝ : TopologicalAddGroup M\nv w : VectorMeasure α M\ni : Set α\nhi : MeasurableSet i\nh : restrict v i ≤ restrict w i\nj : Set α\nhj₁ : MeasurableSet j\n⊢ ↑(restrict (-w) i) j ≤ ↑(restrict (-v) i) j", "state_before": "α : Type u_1\nβ : Type ?u.548949\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommGroup M\ninst✝ : TopologicalAddGroup M\nv w : VectorMeasure α M\ni : Set α\nhi : MeasurableSet i\nh : restrict v i ≤ restrict w i\n⊢ restrict (-w) i ≤ restrict (-v) i", "tactic": "intro j hj₁" }, { "state_after": "α : Type u_1\nβ : Type ?u.548949\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommGroup M\ninst✝ : TopologicalAddGroup M\nv w : VectorMeasure α M\ni : Set α\nhi : MeasurableSet i\nh : restrict v i ≤ restrict w i\nj : Set α\nhj₁ : MeasurableSet j\n⊢ -↑w (j ∩ i) ≤ -↑v (j ∩ i)", "state_before": "α : Type u_1\nβ : Type ?u.548949\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommGroup M\ninst✝ : TopologicalAddGroup M\nv w : VectorMeasure α M\ni : Set α\nhi : MeasurableSet i\nh : restrict v i ≤ restrict w i\nj : Set α\nhj₁ : MeasurableSet j\n⊢ ↑(restrict (-w) i) j ≤ ↑(restrict (-v) i) j", "tactic": "rw [restrict_apply _ hi hj₁, restrict_apply _ hi hj₁, neg_apply, neg_apply]" }, { "state_after": "α : Type u_1\nβ : Type ?u.548949\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommGroup M\ninst✝ : TopologicalAddGroup M\nv w : VectorMeasure α M\ni : Set α\nhi : MeasurableSet i\nh : restrict v i ≤ restrict w i\nj : Set α\nhj₁ : MeasurableSet j\n⊢ ↑v (j ∩ i) ≤ ↑w (j ∩ i)", "state_before": "α : Type u_1\nβ : Type ?u.548949\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommGroup M\ninst✝ : TopologicalAddGroup M\nv w : VectorMeasure α M\ni : Set α\nhi : MeasurableSet i\nh : restrict v i ≤ restrict w i\nj : Set α\nhj₁ : MeasurableSet j\n⊢ -↑w (j ∩ i) ≤ -↑v (j ∩ i)", "tactic": "refine' neg_le_neg _" }, { "state_after": "α : Type u_1\nβ : Type ?u.548949\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommGroup M\ninst✝ : TopologicalAddGroup M\nv w : VectorMeasure α M\ni : Set α\nhi : MeasurableSet i\nh : restrict v i ≤ restrict w i\nj : Set α\nhj₁ : MeasurableSet j\n⊢ ↑(restrict v i) j ≤ ↑(restrict w i) j", "state_before": "α : Type u_1\nβ : Type ?u.548949\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommGroup M\ninst✝ : TopologicalAddGroup M\nv w : VectorMeasure α M\ni : Set α\nhi : MeasurableSet i\nh : restrict v i ≤ restrict w i\nj : Set α\nhj₁ : MeasurableSet j\n⊢ ↑v (j ∩ i) ≤ ↑w (j ∩ i)", "tactic": "rw [← restrict_apply _ hi hj₁, ← restrict_apply _ hi hj₁]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.548949\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommGroup M\ninst✝ : TopologicalAddGroup M\nv w : VectorMeasure α M\ni : Set α\nhi : MeasurableSet i\nh : restrict v i ≤ restrict w i\nj : Set α\nhj₁ : MeasurableSet j\n⊢ ↑(restrict v i) j ≤ ↑(restrict w i) j", "tactic": "exact h j hj₁" } ]

[ 930, 16 ]

[ 925, 8 ]

[ { "state_after": "no goals", "state_before": "n q : ℕ\n⊢ q = sqrt n ↔ q ^ 2 ≤ n ∧ n < (q + 1) ^ 2", "tactic": "simpa only [pow_two] using eq_sqrt" } ]

[ 127, 37 ]

[ 126, 1 ]

[ { "state_after": "case a.h\nl : Type u_3\nm : Type u_4\nn : Type u_2\no : Type u_5\np : Type ?u.36919\nq : Type ?u.36922\nm' : o → Type ?u.36927\nn' : o → Type ?u.36932\np' : o → Type ?u.36937\nR : Type ?u.36940\nS : Type ?u.36943\nα : Type u_1\nβ : Type ?u.36949\ninst✝ : Add α\nA : Matrix n l α\nB : Matrix n m α\nC : Matrix o l α\nD : Matrix o m α\nA' : Matrix n l α\nB' : Matrix n m α\nC' : Matrix o l α\nD' : Matrix o m α\ni : n ⊕ o\nj : l ⊕ m\n⊢ (fromBlocks A B C D + fromBlocks A' B' C' D') i j = fromBlocks (A + A') (B + B') (C + C') (D + D') i j", "state_before": "l : Type u_3\nm : Type u_4\nn : Type u_2\no : Type u_5\np : Type ?u.36919\nq : Type ?u.36922\nm' : o → Type ?u.36927\nn' : o → Type ?u.36932\np' : o → Type ?u.36937\nR : Type ?u.36940\nS : Type ?u.36943\nα : Type u_1\nβ : Type ?u.36949\ninst✝ : Add α\nA : Matrix n l α\nB : Matrix n m α\nC : Matrix o l α\nD : Matrix o m α\nA' : Matrix n l α\nB' : Matrix n m α\nC' : Matrix o l α\nD' : Matrix o m α\n⊢ fromBlocks A B C D + fromBlocks A' B' C' D' = fromBlocks (A + A') (B + B') (C + C') (D + D')", "tactic": "ext (i j)" }, { "state_after": "no goals", "state_before": "case a.h\nl : Type u_3\nm : Type u_4\nn : Type u_2\no : Type u_5\np : Type ?u.36919\nq : Type ?u.36922\nm' : o → Type ?u.36927\nn' : o → Type ?u.36932\np' : o → Type ?u.36937\nR : Type ?u.36940\nS : Type ?u.36943\nα : Type u_1\nβ : Type ?u.36949\ninst✝ : Add α\nA : Matrix n l α\nB : Matrix n m α\nC : Matrix o l α\nD : Matrix o m α\nA' : Matrix n l α\nB' : Matrix n m α\nC' : Matrix o l α\nD' : Matrix o m α\ni : n ⊕ o\nj : l ⊕ m\n⊢ (fromBlocks A B C D + fromBlocks A' B' C' D') i j = fromBlocks (A + A') (B + B') (C + C') (D + D') i j", "tactic": "rcases i with ⟨⟩ <;> rcases j with ⟨⟩ <;> rfl" } ]

[ 247, 62 ]

[ 243, 1 ]

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[ 400, 60 ]

[ 397, 1 ]

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[ 62, 26 ]

[ 61, 1 ]

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[ 358, 32 ]

[ 357, 1 ]

[]

[ 209, 6 ]

[ 208, 1 ]

[]

[ 1379, 86 ]

[ 1379, 1 ]

[ { "state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v' w' : V\np : Walk G v' w'\nh : u ∈ support (Walk.copy p (_ : v' = v') (_ : w' = w'))\n⊢ u ∈ support p", "state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w v' w' : V\np : Walk G v w\nhv : v = v'\nhw : w = w'\nh : u ∈ support (Walk.copy p hv hw)\n⊢ u ∈ support p", "tactic": "subst_vars" }, { "state_after": "no goals", "state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v' w' : V\np : Walk G v' w'\nh : u ∈ support (Walk.copy p (_ : v' = v') (_ : w' = w'))\n⊢ u ∈ support p", "tactic": "exact h" }, { "state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v' w' : V\np : Walk G v' w'\nh : u ∈ support (Walk.copy p (_ : v' = v') (_ : w' = w'))\n⊢ takeUntil (Walk.copy p (_ : v' = v') (_ : w' = w')) u h =\n Walk.copy (takeUntil p u (_ : u ∈ support p)) (_ : v' = v') (_ : u = u)", "state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w v' w' : V\np : Walk G v w\nhv : v = v'\nhw : w = w'\nh : u ∈ support (Walk.copy p hv hw)\n⊢ takeUntil (Walk.copy p hv hw) u h = Walk.copy (takeUntil p u (_ : u ∈ support p)) hv (_ : u = u)", "tactic": "subst_vars" }, { "state_after": "no goals", "state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v' w' : V\np : Walk G v' w'\nh : u ∈ support (Walk.copy p (_ : v' = v') (_ : w' = w'))\n⊢ takeUntil (Walk.copy p (_ : v' = v') (_ : w' = w')) u h =\n Walk.copy (takeUntil p u (_ : u ∈ support p)) (_ : v' = v') (_ : u = u)", "tactic": "rfl" } ]

[ 1122, 6 ]

[ 1118, 1 ]

[]

[ 823, 61 ]

[ 822, 1 ]

[ { "state_after": "no goals", "state_before": "X : Type u_1\nY : Type ?u.31356\nZ : Type ?u.31359\nα : Type ?u.31362\nι : Type ?u.31365\nπ : ι → Type ?u.31370\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : (i : ι) → TopologicalSpace (π i)\nx y z : X\ns : Set X\nf : X → Y\n⊢ x ⤳ y ∧ y ⤳ x ↔ x ∈ closure {y} ∧ y ∈ closure {x}", "tactic": "simp only [specializes_iff_mem_closure, and_comm]" } ]

[ 303, 96 ]

[ 301, 1 ]

[ { "state_after": "no goals", "state_before": "ι : Type ?u.431878\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nn : ℕ\n⊢ ∏ x in range (n + 1), f x = f n * ∏ x in range n, f x", "tactic": "rw [range_succ, prod_insert not_mem_range_self]" } ]

[ 1213, 50 ]

[ 1211, 1 ]

[]

[ 317, 45 ]

[ 312, 1 ]

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[ 1327, 1 ]

[ { "state_after": "p n : ℕ\nhp : Fact (Nat.Prime p)\nhn : n ≠ 0\n⊢ padicValNat p n < padicValNat p n + 1", "state_before": "p n : ℕ\nhp : Fact (Nat.Prime p)\nhn : n ≠ 0\n⊢ ¬p ^ (padicValNat p n + 1) ∣ n", "tactic": "rw [padicValNat_dvd_iff_le hn, not_le]" }, { "state_after": "no goals", "state_before": "p n : ℕ\nhp : Fact (Nat.Prime p)\nhn : n ≠ 0\n⊢ padicValNat p n < padicValNat p n + 1", "tactic": "exact Nat.lt_succ_self _" } ]

[ 478, 27 ]

[ 475, 1 ]

[]

[ 379, 32 ]

[ 378, 1 ]

[]

[ 546, 71 ]

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[]

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[]

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[ 691, 1 ]

[ { "state_after": "α✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\nx : α\nxs : List (Sigma β)\n⊢ rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x xs) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs", "state_before": "α✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\nx : α\nxs : List (Sigma β)\n⊢ sizeOf (kerase x xs) ≤ sizeOf xs", "tactic": "simp only [SizeOf.sizeOf, _sizeOf_1]" }, { "state_after": "case nil\nα✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\nx : α\n⊢ rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x []) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) []\n\ncase cons\nα✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\nx : α\ny : Sigma β\nys : List (Sigma β)\ntail_ih✝ :\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x ys) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) ys\n⊢ rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x (y :: ys)) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (y :: ys)", "state_before": "α✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\nx : α\nxs : List (Sigma β)\n⊢ rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x xs) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs", "tactic": "induction' xs with y ys" }, { "state_after": "no goals", "state_before": "case nil\nα✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\nx : α\n⊢ rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x []) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) []", "tactic": "simp" }, { "state_after": "no goals", "state_before": "case cons\nα✝ : Type u\nβ✝ : α✝ → Type v\nl l₁ l₂ : List (Sigma β✝)\ninst✝² : DecidableEq α✝\nα : Type u_1\nβ : α → Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : SizeOf (Sigma β)\nx : α\ny : Sigma β\nys : List (Sigma β)\ntail_ih✝ :\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x ys) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) ys\n⊢ rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x (y :: ys)) ≤\n rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (y :: ys)", "tactic": "by_cases x = y.1 <;> simp [*]" } ]

[ 558, 34 ]

[ 553, 1 ]

[]

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[ 631, 1 ]

[]

[ 96, 6 ]

[ 95, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type u_3\nδ : Type ?u.64152\nε : Type ?u.64155\nζ : Type ?u.64158\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\nι : Type u_1\nκ : Type u_2\nf : ι → β\ng : κ → γ\nhf : DenseRange f\nhg : DenseRange g\n⊢ DenseRange (Prod.map f g)", "tactic": "simpa only [DenseRange, prod_range_range_eq] using hf.prod hg" } ]

[ 801, 64 ]

[ 799, 1 ]

[]

[ 126, 48 ]

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[]

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[]

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[ { "state_after": "𝕜 : Type ?u.510422\nG : Type u_1\nH : Type ?u.510428\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : MeasurableSpace H\ninst✝⁴ : DivisionMonoid G\ninst✝³ : MeasurableMul G\ninst✝² : MeasurableInv G\nμ✝ μ : Measure G\ninst✝¹ : IsInvInvariant μ\ninst✝ : IsMulLeftInvariant μ\ng : G\n⊢ MeasurePreserving fun t => g * t⁻¹", "state_before": "𝕜 : Type ?u.510422\nG : Type u_1\nH : Type ?u.510428\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : MeasurableSpace H\ninst✝⁴ : DivisionMonoid G\ninst✝³ : MeasurableMul G\ninst✝² : MeasurableInv G\nμ✝ μ : Measure G\ninst✝¹ : IsInvInvariant μ\ninst✝ : IsMulLeftInvariant μ\ng : G\n⊢ MeasurePreserving fun t => g / t", "tactic": "simp_rw [div_eq_mul_inv]" }, { "state_after": "no goals", "state_before": "𝕜 : Type ?u.510422\nG : Type u_1\nH : Type ?u.510428\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : MeasurableSpace H\ninst✝⁴ : DivisionMonoid G\ninst✝³ : MeasurableMul G\ninst✝² : MeasurableInv G\nμ✝ μ : Measure G\ninst✝¹ : IsInvInvariant μ\ninst✝ : IsMulLeftInvariant μ\ng : G\n⊢ MeasurePreserving fun t => g * t⁻¹", "tactic": "exact (measurePreserving_mul_left μ g).comp (measurePreserving_inv μ)" } ]

[ 461, 72 ]

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[ { "state_after": "case mp\nι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\n⊢ IsBasis B → ∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us\n\ncase mpr\nι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\n⊢ (∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us) → IsBasis B", "state_before": "ι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\n⊢ IsBasis B ↔ ∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us", "tactic": "constructor" }, { "state_after": "case mp\nι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\nhB : IsBasis B\nU : Opens α\n⊢ ∃ Us, Us ⊆ B ∧ U = sSup Us", "state_before": "case mp\nι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\n⊢ IsBasis B → ∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us", "tactic": "intro hB U" }, { "state_after": "case mp\nι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\nhB : IsBasis B\nU : Opens α\n⊢ ↑U = ↑(sSup {V | V ∈ B ∧ V ≤ U})", "state_before": "case mp\nι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\nhB : IsBasis B\nU : Opens α\n⊢ ∃ Us, Us ⊆ B ∧ U = sSup Us", "tactic": "refine ⟨{ V : Opens α | V ∈ B ∧ V ≤ U }, fun U hU => hU.left, ext ?_⟩" }, { "state_after": "case mp\nι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\nhB : IsBasis B\nU : Opens α\n⊢ ⋃₀ {s | s ∈ SetLike.coe '' B ∧ s ⊆ ↑U} = ⋃ (i : Opens α) (_ : i ∈ {V | V ∈ B ∧ V ≤ U}), ↑i", "state_before": "case mp\nι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\nhB : IsBasis B\nU : Opens α\n⊢ ↑U = ↑(sSup {V | V ∈ B ∧ V ≤ U})", "tactic": "rw [coe_sSup, hB.open_eq_sUnion' U.isOpen]" }, { "state_after": "case mp\nι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\nhB : IsBasis B\nU : Opens α\n⊢ (⨆ (b : Opens α) (_ : b ∈ B) (_ : ↑b ⊆ ↑U), ↑b) = ⨆ (i : Opens α) (_ : i ∈ B) (_ : i ≤ U), ↑i", "state_before": "case mp\nι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\nhB : IsBasis B\nU : Opens α\n⊢ ⋃₀ {s | s ∈ SetLike.coe '' B ∧ s ⊆ ↑U} = ⋃ (i : Opens α) (_ : i ∈ {V | V ∈ B ∧ V ≤ U}), ↑i", "tactic": "simp_rw [sUnion_eq_biUnion, iUnion, mem_setOf_eq, iSup_and, iSup_image]" }, { "state_after": "no goals", "state_before": "case mp\nι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\nhB : IsBasis B\nU : Opens α\n⊢ (⨆ (b : Opens α) (_ : b ∈ B) (_ : ↑b ⊆ ↑U), ↑b) = ⨆ (i : Opens α) (_ : i ∈ B) (_ : i ≤ U), ↑i", "tactic": "rfl" }, { "state_after": "case mpr\nι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\nh : ∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us\n⊢ IsBasis B", "state_before": "case mpr\nι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\n⊢ (∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us) → IsBasis B", "tactic": "intro h" }, { "state_after": "case mpr\nι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\nh : ∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us\n⊢ ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U', U' ∈ B ∧ x ∈ U' ∧ U' ≤ U", "state_before": "case mpr\nι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\nh : ∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us\n⊢ IsBasis B", "tactic": "rw [isBasis_iff_nbhd]" }, { "state_after": "case mpr\nι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\nh : ∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us\nU : Opens α\nx : α\nhx : x ∈ U\n⊢ ∃ U', U' ∈ B ∧ x ∈ U' ∧ U' ≤ U", "state_before": "case mpr\nι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\nh : ∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us\n⊢ ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U', U' ∈ B ∧ x ∈ U' ∧ U' ≤ U", "tactic": "intro U x hx" }, { "state_after": "case mpr.intro.intro\nι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\nh : ∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us\nx : α\nUs : Set (Opens α)\nhUs : Us ⊆ B\nhx : x ∈ sSup Us\n⊢ ∃ U', U' ∈ B ∧ x ∈ U' ∧ U' ≤ sSup Us", "state_before": "case mpr\nι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\nh : ∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us\nU : Opens α\nx : α\nhx : x ∈ U\n⊢ ∃ U', U' ∈ B ∧ x ∈ U' ∧ U' ≤ U", "tactic": "rcases h U with ⟨Us, hUs, rfl⟩" }, { "state_after": "case mpr.intro.intro.intro.intro\nι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\nh : ∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us\nx : α\nUs✝ : Set (Opens α)\nhUs : Us✝ ⊆ B\nhx : x ∈ sSup Us✝\nU : Opens α\nUs : U ∈ Us✝\nxU : x ∈ U\n⊢ ∃ U', U' ∈ B ∧ x ∈ U' ∧ U' ≤ sSup Us✝", "state_before": "case mpr.intro.intro\nι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\nh : ∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us\nx : α\nUs : Set (Opens α)\nhUs : Us ⊆ B\nhx : x ∈ sSup Us\n⊢ ∃ U', U' ∈ B ∧ x ∈ U' ∧ U' ≤ sSup Us", "tactic": "rcases mem_sSup.1 hx with ⟨U, Us, xU⟩" }, { "state_after": "no goals", "state_before": "case mpr.intro.intro.intro.intro\nι : Type ?u.31277\nα : Type u_1\nβ : Type ?u.31283\nγ : Type ?u.31286\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nB : Set (Opens α)\nh : ∀ (U : Opens α), ∃ Us, Us ⊆ B ∧ U = sSup Us\nx : α\nUs✝ : Set (Opens α)\nhUs : Us✝ ⊆ B\nhx : x ∈ sSup Us✝\nU : Opens α\nUs : U ∈ Us✝\nxU : x ∈ U\n⊢ ∃ U', U' ∈ B ∧ x ∈ U' ∧ U' ≤ sSup Us✝", "tactic": "exact ⟨U, hUs Us, xU, le_sSup Us⟩" } ]

[ 325, 38 ]

[ 312, 1 ]

[]

[ 128, 6 ]

[ 127, 1 ]

[]

[ 200, 20 ]

[ 199, 1 ]

[]

[ 442, 12 ]

[ 441, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\nas : List α\na : α\n⊢ concat as a = as ++ a :: nil", "tactic": "induction as <;> simp [concat, *]" } ]

[ 851, 36 ]

[ 850, 1 ]

[ { "state_after": "M : Type u_1\nN : Type ?u.27150\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M\n⊢ infDist x (↑(IsometryEquiv.symm (IsometryEquiv.subLeft x)) ⁻¹' {m | ↑m = ↑x}) = infDist x ↑S", "state_before": "M : Type u_1\nN : Type ?u.27150\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M\n⊢ ‖↑x‖ = infDist x ↑S", "tactic": "rw [norm_eq_infDist, ← infDist_image (IsometryEquiv.subLeft x).isometry,\n IsometryEquiv.subLeft_apply, sub_zero, ← IsometryEquiv.preimage_symm]" }, { "state_after": "case e_s.h\nM : Type u_1\nN : Type ?u.27150\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nx y : M\n⊢ y ∈ ↑(IsometryEquiv.symm (IsometryEquiv.subLeft x)) ⁻¹' {m | ↑m = ↑x} ↔ y ∈ ↑S", "state_before": "M : Type u_1\nN : Type ?u.27150\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M\n⊢ infDist x (↑(IsometryEquiv.symm (IsometryEquiv.subLeft x)) ⁻¹' {m | ↑m = ↑x}) = infDist x ↑S", "tactic": "congr 1 with y" }, { "state_after": "no goals", "state_before": "case e_s.h\nM : Type u_1\nN : Type ?u.27150\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nx y : M\n⊢ y ∈ ↑(IsometryEquiv.symm (IsometryEquiv.subLeft x)) ⁻¹' {m | ↑m = ↑x} ↔ y ∈ ↑S", "tactic": "simp only [mem_preimage, IsometryEquiv.subLeft_symm_apply, mem_setOf_eq, QuotientAddGroup.eq,\n neg_add, neg_neg, neg_add_cancel_right, SetLike.mem_coe]" } ]

[ 127, 61 ]

[ 121, 1 ]

[ { "state_after": "α : Type u_2\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\nβ : Type u_1\nf : α → β\ns : Set β\n⊢ ↑(OuterMeasure.ofFunction m m_empty) (f ⁻¹' s) ≤ m (f ⁻¹' s)", "state_before": "α : Type u_2\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\nβ : Type u_1\nf : α → β\ns : Set β\n⊢ ↑(↑(map f) (OuterMeasure.ofFunction m m_empty)) s ≤ m (f ⁻¹' s)", "tactic": "rw [map_apply]" }, { "state_after": "no goals", "state_before": "α : Type u_2\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\nβ : Type u_1\nf : α → β\ns : Set β\n⊢ ↑(OuterMeasure.ofFunction m m_empty) (f ⁻¹' s) ≤ m (f ⁻¹' s)", "tactic": "apply ofFunction_le" } ]

[ 796, 24 ]

[ 791, 1 ]

[ { "state_after": "R : Type u_1\ninst✝ : Monoid R\na b : R\nh : a * b = 1\n⊢ IsRightRegular 1", "state_before": "R : Type u_1\ninst✝ : Monoid R\na b : R\nh : a * b = 1\n⊢ IsRightRegular (a * ?m.14188 h)", "tactic": "rw [h]" }, { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝ : Monoid R\na b : R\nh : a * b = 1\n⊢ IsRightRegular 1", "tactic": "exact isRegular_one.right" } ]

[ 316, 63 ]

[ 315, 1 ]

[]

[ 48, 10 ]

[ 47, 1 ]

[ { "state_after": "case mk\nL : Language\nL' : Language\nM : Type w\ninst✝ : Structure L M\nϕ : L →ᴸ L'\nL'' : Language\nonFunction✝ : ⦃n : ℕ⦄ → Functions L n → Functions L' n\nonRelation✝ : ⦃n : ℕ⦄ → Relations L n → Relations L' n\n⊢ LHom.id L' ∘' { onFunction := onFunction✝, onRelation := onRelation✝ } =\n { onFunction := onFunction✝, onRelation := onRelation✝ }", "state_before": "L : Language\nL' : Language\nM : Type w\ninst✝ : Structure L M\nϕ : L →ᴸ L'\nL'' : Language\nF : L →ᴸ L'\n⊢ LHom.id L' ∘' F = F", "tactic": "cases F" }, { "state_after": "no goals", "state_before": "case mk\nL : Language\nL' : Language\nM : Type w\ninst✝ : Structure L M\nϕ : L →ᴸ L'\nL'' : Language\nonFunction✝ : ⦃n : ℕ⦄ → Functions L n → Functions L' n\nonRelation✝ : ⦃n : ℕ⦄ → Relations L n → Relations L' n\n⊢ LHom.id L' ∘' { onFunction := onFunction✝, onRelation := onRelation✝ } =\n { onFunction := onFunction✝, onRelation := onRelation✝ }", "tactic": "rfl" } ]

[ 159, 6 ]

[ 157, 1 ]

[]

[ 949, 32 ]

[ 947, 1 ]

[ { "state_after": "α : Type u_1\ninst✝ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nx : α\ns : Set (Set α)\n⊢ (s ∈ ⨅ (ε : ℝ) (_ : ε ∈ Ioi 0), 𝓟 {a | a ∈ setsAt v x ∧ a ⊆ closedBall x ε}) ↔\n ∃ ε, 0 < ε ∧ ∀ (a : Set α), a ∈ setsAt v x → a ⊆ closedBall x ε → a ∈ s", "state_before": "α : Type u_1\ninst✝ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nx : α\ns : Set (Set α)\n⊢ s ∈ filterAt v x ↔ ∃ ε, ε > 0 ∧ ∀ (a : Set α), a ∈ setsAt v x → a ⊆ closedBall x ε → a ∈ s", "tactic": "simp only [filterAt, exists_prop, gt_iff_lt]" }, { "state_after": "α : Type u_1\ninst✝ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nx : α\ns : Set (Set α)\n⊢ (∃ i, i ∈ Ioi 0 ∧ s ∈ 𝓟 {a | a ∈ setsAt v x ∧ a ⊆ closedBall x i}) ↔\n ∃ ε, 0 < ε ∧ ∀ (a : Set α), a ∈ setsAt v x → a ⊆ closedBall x ε → a ∈ s\n\ncase h\nα : Type u_1\ninst✝ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nx : α\ns : Set (Set α)\n⊢ DirectedOn ((fun ε => 𝓟 {a | a ∈ setsAt v x ∧ a ⊆ closedBall x ε}) ⁻¹'o fun x x_1 => x ≥ x_1) (Ioi 0)\n\ncase ne\nα : Type u_1\ninst✝ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nx : α\ns : Set (Set α)\n⊢ Set.Nonempty (Ioi 0)", "state_before": "α : Type u_1\ninst✝ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nx : α\ns : Set (Set α)\n⊢ (s ∈ ⨅ (ε : ℝ) (_ : ε ∈ Ioi 0), 𝓟 {a | a ∈ setsAt v x ∧ a ⊆ closedBall x ε}) ↔\n ∃ ε, 0 < ε ∧ ∀ (a : Set α), a ∈ setsAt v x → a ⊆ closedBall x ε → a ∈ s", "tactic": "rw [mem_biInf_of_directed]" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nx : α\ns : Set (Set α)\n⊢ (∃ i, i ∈ Ioi 0 ∧ s ∈ 𝓟 {a | a ∈ setsAt v x ∧ a ⊆ closedBall x i}) ↔\n ∃ ε, 0 < ε ∧ ∀ (a : Set α), a ∈ setsAt v x → a ⊆ closedBall x ε → a ∈ s", "tactic": "simp only [subset_def, and_imp, exists_prop, mem_sep_iff, mem_Ioi, mem_principal]" }, { "state_after": "case h\nα : Type u_1\ninst✝ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nx : α\ns : Set (Set α)\n⊢ ∀ (x_1 : ℝ),\n 0 < x_1 →\n ∀ (y : ℝ),\n 0 < y →\n ∃ z,\n 0 < z ∧\n {a | a ∈ setsAt v x ∧ a ⊆ closedBall x z} ⊆ {a | a ∈ setsAt v x ∧ a ⊆ closedBall x x_1} ∧\n {a | a ∈ setsAt v x ∧ a ⊆ closedBall x z} ⊆ {a | a ∈ setsAt v x ∧ a ⊆ closedBall x y}", "state_before": "case h\nα : Type u_1\ninst✝ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nx : α\ns : Set (Set α)\n⊢ DirectedOn ((fun ε => 𝓟 {a | a ∈ setsAt v x ∧ a ⊆ closedBall x ε}) ⁻¹'o fun x x_1 => x ≥ x_1) (Ioi 0)", "tactic": "simp only [DirectedOn, exists_prop, ge_iff_le, le_principal_iff, mem_Ioi, Order.Preimage,\n mem_principal]" }, { "state_after": "case h\nα : Type u_1\ninst✝ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nx✝ : α\ns : Set (Set α)\nx : ℝ\nhx : 0 < x\ny : ℝ\nhy : 0 < y\n⊢ ∃ z,\n 0 < z ∧\n {a | a ∈ setsAt v x✝ ∧ a ⊆ closedBall x✝ z} ⊆ {a | a ∈ setsAt v x✝ ∧ a ⊆ closedBall x✝ x} ∧\n {a | a ∈ setsAt v x✝ ∧ a ⊆ closedBall x✝ z} ⊆ {a | a ∈ setsAt v x✝ ∧ a ⊆ closedBall x✝ y}", "state_before": "case h\nα : Type u_1\ninst✝ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nx : α\ns : Set (Set α)\n⊢ ∀ (x_1 : ℝ),\n 0 < x_1 →\n ∀ (y : ℝ),\n 0 < y →\n ∃ z,\n 0 < z ∧\n {a | a ∈ setsAt v x ∧ a ⊆ closedBall x z} ⊆ {a | a ∈ setsAt v x ∧ a ⊆ closedBall x x_1} ∧\n {a | a ∈ setsAt v x ∧ a ⊆ closedBall x z} ⊆ {a | a ∈ setsAt v x ∧ a ⊆ closedBall x y}", "tactic": "intro x hx y hy" }, { "state_after": "no goals", "state_before": "case h\nα : Type u_1\ninst✝ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nx✝ : α\ns : Set (Set α)\nx : ℝ\nhx : 0 < x\ny : ℝ\nhy : 0 < y\n⊢ ∃ z,\n 0 < z ∧\n {a | a ∈ setsAt v x✝ ∧ a ⊆ closedBall x✝ z} ⊆ {a | a ∈ setsAt v x✝ ∧ a ⊆ closedBall x✝ x} ∧\n {a | a ∈ setsAt v x✝ ∧ a ⊆ closedBall x✝ z} ⊆ {a | a ∈ setsAt v x✝ ∧ a ⊆ closedBall x✝ y}", "tactic": "refine' ⟨min x y, lt_min hx hy,\n fun a ha => ⟨ha.1, ha.2.trans (closedBall_subset_closedBall (min_le_left _ _))⟩,\n fun a ha => ⟨ha.1, ha.2.trans (closedBall_subset_closedBall (min_le_right _ _))⟩⟩" }, { "state_after": "no goals", "state_before": "case ne\nα : Type u_1\ninst✝ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nx : α\ns : Set (Set α)\n⊢ Set.Nonempty (Ioi 0)", "tactic": "exact ⟨(1 : ℝ), mem_Ioi.2 zero_lt_one⟩" } ]

[ 236, 43 ]

[ 225, 1 ]

[ { "state_after": "no goals", "state_before": "n m : ℕ\na : Fin n\n⊢ 0 < succ a", "tactic": "simp [lt_iff_val_lt_val]" } ]

[ 874, 89 ]

[ 874, 1 ]

[]

[ 133, 18 ]

[ 132, 1 ]

[ { "state_after": "case h\nα : Type ?u.241739\nβ : Type ?u.241742\nM✝ : Type ?u.241745\nN : Type ?u.241748\nP : Type ?u.241751\nG : Type ?u.241754\nH : Type ?u.241757\nF : Type ?u.241760\ninst✝⁵ : Group G\ninst✝⁴ : CommGroup H\ninst✝³ : MulOneClass M✝\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝² : MulOneClass M\ninst✝¹ : CommGroup A\ninst✝ : CommGroup B\nφ : A →* B\nψ : M →* A\nx✝ : M\n⊢ ↑(comp φ ψ⁻¹) x✝ = ↑(comp φ ψ)⁻¹ x✝", "state_before": "α : Type ?u.241739\nβ : Type ?u.241742\nM✝ : Type ?u.241745\nN : Type ?u.241748\nP : Type ?u.241751\nG : Type ?u.241754\nH : Type ?u.241757\nF : Type ?u.241760\ninst✝⁵ : Group G\ninst✝⁴ : CommGroup H\ninst✝³ : MulOneClass M✝\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝² : MulOneClass M\ninst✝¹ : CommGroup A\ninst✝ : CommGroup B\nφ : A →* B\nψ : M →* A\n⊢ comp φ ψ⁻¹ = (comp φ ψ)⁻¹", "tactic": "ext" }, { "state_after": "no goals", "state_before": "case h\nα : Type ?u.241739\nβ : Type ?u.241742\nM✝ : Type ?u.241745\nN : Type ?u.241748\nP : Type ?u.241751\nG : Type ?u.241754\nH : Type ?u.241757\nF : Type ?u.241760\ninst✝⁵ : Group G\ninst✝⁴ : CommGroup H\ninst✝³ : MulOneClass M✝\nM : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝² : MulOneClass M\ninst✝¹ : CommGroup A\ninst✝ : CommGroup B\nφ : A →* B\nψ : M →* A\nx✝ : M\n⊢ ↑(comp φ ψ⁻¹) x✝ = ↑(comp φ ψ)⁻¹ x✝", "tactic": "simp only [Function.comp_apply, inv_apply, map_inv, coe_comp]" } ]

[ 1671, 64 ]

[ 1668, 1 ]

[ { "state_after": "case inl\nι : Type ?u.1842380\n𝕜 : Type ?u.1842383\nE : Type u_1\nF : Type ?u.1842389\nA : Type ?u.1842392\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝ : ℝ → E\ng' g φ : ℝ → ℝ\nf f' : ℝ → E\na b : ℝ\nhcont : ContinuousOn f [[a, b]]\nhderiv : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x\nhint : IntervalIntegrable f' volume a b\nhab : a ≤ b\n⊢ (∫ (y : ℝ) in a..b, f' y) = f b - f a\n\ncase inr\nι : Type ?u.1842380\n𝕜 : Type ?u.1842383\nE : Type u_1\nF : Type ?u.1842389\nA : Type ?u.1842392\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝ : ℝ → E\ng' g φ : ℝ → ℝ\nf f' : ℝ → E\na b : ℝ\nhcont : ContinuousOn f [[a, b]]\nhderiv : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x\nhint : IntervalIntegrable f' volume a b\nhab : b ≤ a\n⊢ (∫ (y : ℝ) in a..b, f' y) = f b - f a", "state_before": "ι : Type ?u.1842380\n𝕜 : Type ?u.1842383\nE : Type u_1\nF : Type ?u.1842389\nA : Type ?u.1842392\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝ : ℝ → E\ng' g φ : ℝ → ℝ\nf f' : ℝ → E\na b : ℝ\nhcont : ContinuousOn f [[a, b]]\nhderiv : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x\nhint : IntervalIntegrable f' volume a b\n⊢ (∫ (y : ℝ) in a..b, f' y) = f b - f a", "tactic": "cases' le_total a b with hab hab" }, { "state_after": "case inl\nι : Type ?u.1842380\n𝕜 : Type ?u.1842383\nE : Type u_1\nF : Type ?u.1842389\nA : Type ?u.1842392\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝ : ℝ → E\ng' g φ : ℝ → ℝ\nf f' : ℝ → E\na b : ℝ\nhint : IntervalIntegrable f' volume a b\nhab : a ≤ b\nhcont : ContinuousOn f (Icc a b)\nhderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivWithinAt f (f' x) (Ioi x) x\n⊢ (∫ (y : ℝ) in a..b, f' y) = f b - f a", "state_before": "case inl\nι : Type ?u.1842380\n𝕜 : Type ?u.1842383\nE : Type u_1\nF : Type ?u.1842389\nA : Type ?u.1842392\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝ : ℝ → E\ng' g φ : ℝ → ℝ\nf f' : ℝ → E\na b : ℝ\nhcont : ContinuousOn f [[a, b]]\nhderiv : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x\nhint : IntervalIntegrable f' volume a b\nhab : a ≤ b\n⊢ (∫ (y : ℝ) in a..b, f' y) = f b - f a", "tactic": "simp only [uIcc_of_le, min_eq_left, max_eq_right, hab] at hcont hderiv hint" }, { "state_after": "no goals", "state_before": "case inl\nι : Type ?u.1842380\n𝕜 : Type ?u.1842383\nE : Type u_1\nF : Type ?u.1842389\nA : Type ?u.1842392\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝ : ℝ → E\ng' g φ : ℝ → ℝ\nf f' : ℝ → E\na b : ℝ\nhint : IntervalIntegrable f' volume a b\nhab : a ≤ b\nhcont : ContinuousOn f (Icc a b)\nhderiv : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivWithinAt f (f' x) (Ioi x) x\n⊢ (∫ (y : ℝ) in a..b, f' y) = f b - f a", "tactic": "apply integral_eq_sub_of_hasDeriv_right_of_le hab hcont hderiv hint" }, { "state_after": "case inr\nι : Type ?u.1842380\n𝕜 : Type ?u.1842383\nE : Type u_1\nF : Type ?u.1842389\nA : Type ?u.1842392\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝ : ℝ → E\ng' g φ : ℝ → ℝ\nf f' : ℝ → E\na b : ℝ\nhint : IntervalIntegrable f' volume a b\nhab : b ≤ a\nhcont : ContinuousOn f (Icc b a)\nhderiv : ∀ (x : ℝ), x ∈ Ioo b a → HasDerivWithinAt f (f' x) (Ioi x) x\n⊢ (∫ (y : ℝ) in a..b, f' y) = f b - f a", "state_before": "case inr\nι : Type ?u.1842380\n𝕜 : Type ?u.1842383\nE : Type u_1\nF : Type ?u.1842389\nA : Type ?u.1842392\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝ : ℝ → E\ng' g φ : ℝ → ℝ\nf f' : ℝ → E\na b : ℝ\nhcont : ContinuousOn f [[a, b]]\nhderiv : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x\nhint : IntervalIntegrable f' volume a b\nhab : b ≤ a\n⊢ (∫ (y : ℝ) in a..b, f' y) = f b - f a", "tactic": "simp only [uIcc_of_ge, min_eq_right, max_eq_left, hab] at hcont hderiv" }, { "state_after": "no goals", "state_before": "case inr\nι : Type ?u.1842380\n𝕜 : Type ?u.1842383\nE : Type u_1\nF : Type ?u.1842389\nA : Type ?u.1842392\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝ : ℝ → E\ng' g φ : ℝ → ℝ\nf f' : ℝ → E\na b : ℝ\nhint : IntervalIntegrable f' volume a b\nhab : b ≤ a\nhcont : ContinuousOn f (Icc b a)\nhderiv : ∀ (x : ℝ), x ∈ Ioo b a → HasDerivWithinAt f (f' x) (Ioi x) x\n⊢ (∫ (y : ℝ) in a..b, f' y) = f b - f a", "tactic": "rw [integral_symm, integral_eq_sub_of_hasDeriv_right_of_le hab hcont hderiv hint.symm, neg_sub]" } ]

[ 1192, 100 ]

[ 1185, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type ?u.37158\nβ : Type ?u.37161\nn d✝ k N : ℕ\nx : Fin n → ℕ\nd : ℕ\na : Fin 0 → ℕ\n⊢ ↑(map d) a = 0", "tactic": "simp [map]" } ]

[ 127, 72 ]

[ 127, 1 ]

[]

[ 2207, 12 ]

[ 2206, 1 ]

[]

[ 288, 93 ]

[ 287, 1 ]

[ { "state_after": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\n⊢ Submodule.comap (↑f) (Submodule.span R {m | ∃ x n, ⁅↑x, ↑n⁆ = m}) =\n Submodule.comap (↑f) (Submodule.span R (↑↑f '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}))", "state_before": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\n⊢ comap f ⁅LieHom.idealRange f ⊓ J₁, LieHom.idealRange f ⊓ J₂⁆ = ⁅comap f J₁, comap f J₂⁆ ⊔ LieHom.ker f", "tactic": "rw [← LieSubmodule.coe_toSubmodule_eq_iff, comap_coeSubmodule,\n LieSubmodule.sup_coe_toSubmodule, f.ker_coeSubmodule, ← Submodule.comap_map_eq,\n LieSubmodule.lieIdeal_oper_eq_linear_span, LieSubmodule.lieIdeal_oper_eq_linear_span,\n LinearMap.map_span]" }, { "state_after": "case e_p.e_s\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\n⊢ {m | ∃ x n, ⁅↑x, ↑n⁆ = m} = ↑↑f '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "state_before": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\n⊢ Submodule.comap (↑f) (Submodule.span R {m | ∃ x n, ⁅↑x, ↑n⁆ = m}) =\n Submodule.comap (↑f) (Submodule.span R (↑↑f '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}))", "tactic": "congr" }, { "state_after": "case e_p.e_s\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\n⊢ {m | ∃ x n, ⁅↑x, ↑n⁆ = m} = (fun a => ↑f a) '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "state_before": "case e_p.e_s\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\n⊢ {m | ∃ x n, ⁅↑x, ↑n⁆ = m} = ↑↑f '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "tactic": "simp only [LieHom.coe_toLinearMap, Set.mem_setOf_eq]" }, { "state_after": "case e_p.e_s.h\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny : L'\n⊢ y ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m} ↔ y ∈ (fun a => ↑f a) '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "state_before": "case e_p.e_s\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\n⊢ {m | ∃ x n, ⁅↑x, ↑n⁆ = m} = (fun a => ↑f a) '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "tactic": "ext y" }, { "state_after": "case e_p.e_s.h.mp\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny : L'\n⊢ y ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m} → y ∈ (fun a => ↑f a) '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}\n\ncase e_p.e_s.h.mpr\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny : L'\n⊢ y ∈ (fun a => ↑f a) '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m} → y ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "state_before": "case e_p.e_s.h\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny : L'\n⊢ y ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m} ↔ y ∈ (fun a => ↑f a) '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "tactic": "constructor" }, { "state_after": "case e_p.e_s.h.mp.intro.mk.intro.mk\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny x₁ : L'\nhx₁ : x₁ ∈ LieHom.idealRange f ⊓ J₁\nx₂ : L'\nhx₂ : x₂ ∈ LieHom.idealRange f ⊓ J₂\nhy : ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂ }⁆ = y\n⊢ y ∈ (fun a => ↑f a) '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "state_before": "case e_p.e_s.h.mp\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny : L'\n⊢ y ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m} → y ∈ (fun a => ↑f a) '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "tactic": "rintro ⟨⟨x₁, hx₁⟩, ⟨x₂, hx₂⟩, hy⟩" }, { "state_after": "case e_p.e_s.h.mp.intro.mk.intro.mk\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny x₁ : L'\nhx₁ : x₁ ∈ LieHom.idealRange f ⊓ J₁\nx₂ : L'\nhx₂ : x₂ ∈ LieHom.idealRange f ⊓ J₂\nhy : ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂ }⁆ = y\n⊢ ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂ }⁆ ∈ (fun a => ↑f a) '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "state_before": "case e_p.e_s.h.mp.intro.mk.intro.mk\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny x₁ : L'\nhx₁ : x₁ ∈ LieHom.idealRange f ⊓ J₁\nx₂ : L'\nhx₂ : x₂ ∈ LieHom.idealRange f ⊓ J₂\nhy : ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂ }⁆ = y\n⊢ y ∈ (fun a => ↑f a) '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "tactic": "rw [← hy]" }, { "state_after": "case e_p.e_s.h.mp.intro.mk.intro.mk\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny x₁ : L'\nhx₁✝ : x₁ ∈ LieHom.idealRange f ⊓ J₁\nhx₁ : (∃ x, ↑f x = x₁) ∧ x₁ ∈ J₁\nx₂ : L'\nhx₂✝ : x₂ ∈ LieHom.idealRange f ⊓ J₂\nhx₂ : (∃ x, ↑f x = x₂) ∧ x₂ ∈ J₂\nhy : ⁅↑{ val := x₁, property := hx₁✝ }, ↑{ val := x₂, property := hx₂✝ }⁆ = y\n⊢ ⁅↑{ val := x₁, property := hx₁✝ }, ↑{ val := x₂, property := hx₂✝ }⁆ ∈ (fun a => ↑f a) '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "state_before": "case e_p.e_s.h.mp.intro.mk.intro.mk\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny x₁ : L'\nhx₁ : x₁ ∈ LieHom.idealRange f ⊓ J₁\nx₂ : L'\nhx₂ : x₂ ∈ LieHom.idealRange f ⊓ J₂\nhy : ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂ }⁆ = y\n⊢ ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂ }⁆ ∈ (fun a => ↑f a) '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "tactic": "erw [LieSubmodule.mem_inf, f.mem_idealRange_iff h] at hx₁ hx₂" }, { "state_after": "case e_p.e_s.h.mp.intro.mk.intro.mk.intro.intro\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny x₁ : L'\nhx₁ : x₁ ∈ LieHom.idealRange f ⊓ J₁\nx₂ : L'\nhx₂✝ : x₂ ∈ LieHom.idealRange f ⊓ J₂\nhx₂ : (∃ x, ↑f x = x₂) ∧ x₂ ∈ J₂\nhy : ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂✝ }⁆ = y\nhz₁' : x₁ ∈ J₁\nz₁ : L\nhz₁ : ↑f z₁ = x₁\n⊢ ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂✝ }⁆ ∈ (fun a => ↑f a) '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "state_before": "case e_p.e_s.h.mp.intro.mk.intro.mk\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny x₁ : L'\nhx₁✝ : x₁ ∈ LieHom.idealRange f ⊓ J₁\nhx₁ : (∃ x, ↑f x = x₁) ∧ x₁ ∈ J₁\nx₂ : L'\nhx₂✝ : x₂ ∈ LieHom.idealRange f ⊓ J₂\nhx₂ : (∃ x, ↑f x = x₂) ∧ x₂ ∈ J₂\nhy : ⁅↑{ val := x₁, property := hx₁✝ }, ↑{ val := x₂, property := hx₂✝ }⁆ = y\n⊢ ⁅↑{ val := x₁, property := hx₁✝ }, ↑{ val := x₂, property := hx₂✝ }⁆ ∈ (fun a => ↑f a) '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "tactic": "obtain ⟨⟨z₁, hz₁⟩, hz₁'⟩ := hx₁" }, { "state_after": "case e_p.e_s.h.mp.intro.mk.intro.mk.intro.intro\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny x₁ : L'\nhx₁ : x₁ ∈ LieHom.idealRange f ⊓ J₁\nx₂ : L'\nhx₂✝ : x₂ ∈ LieHom.idealRange f ⊓ J₂\nhx₂ : (∃ x, ↑f x = x₂) ∧ x₂ ∈ J₂\nhy : ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂✝ }⁆ = y\nz₁ : L\nhz₁' : ↑f z₁ ∈ J₁\nhz₁ : ↑f z₁ = x₁\n⊢ ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂✝ }⁆ ∈ (fun a => ↑f a) '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "state_before": "case e_p.e_s.h.mp.intro.mk.intro.mk.intro.intro\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny x₁ : L'\nhx₁ : x₁ ∈ LieHom.idealRange f ⊓ J₁\nx₂ : L'\nhx₂✝ : x₂ ∈ LieHom.idealRange f ⊓ J₂\nhx₂ : (∃ x, ↑f x = x₂) ∧ x₂ ∈ J₂\nhy : ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂✝ }⁆ = y\nhz₁' : x₁ ∈ J₁\nz₁ : L\nhz₁ : ↑f z₁ = x₁\n⊢ ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂✝ }⁆ ∈ (fun a => ↑f a) '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "tactic": "rw [← hz₁] at hz₁'" }, { "state_after": "case e_p.e_s.h.mp.intro.mk.intro.mk.intro.intro.intro.intro\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny x₁ : L'\nhx₁ : x₁ ∈ LieHom.idealRange f ⊓ J₁\nx₂ : L'\nhx₂ : x₂ ∈ LieHom.idealRange f ⊓ J₂\nhy : ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂ }⁆ = y\nz₁ : L\nhz₁' : ↑f z₁ ∈ J₁\nhz₁ : ↑f z₁ = x₁\nhz₂' : x₂ ∈ J₂\nz₂ : L\nhz₂ : ↑f z₂ = x₂\n⊢ ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂ }⁆ ∈ (fun a => ↑f a) '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "state_before": "case e_p.e_s.h.mp.intro.mk.intro.mk.intro.intro\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny x₁ : L'\nhx₁ : x₁ ∈ LieHom.idealRange f ⊓ J₁\nx₂ : L'\nhx₂✝ : x₂ ∈ LieHom.idealRange f ⊓ J₂\nhx₂ : (∃ x, ↑f x = x₂) ∧ x₂ ∈ J₂\nhy : ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂✝ }⁆ = y\nz₁ : L\nhz₁' : ↑f z₁ ∈ J₁\nhz₁ : ↑f z₁ = x₁\n⊢ ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂✝ }⁆ ∈ (fun a => ↑f a) '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "tactic": "obtain ⟨⟨z₂, hz₂⟩, hz₂'⟩ := hx₂" }, { "state_after": "case e_p.e_s.h.mp.intro.mk.intro.mk.intro.intro.intro.intro\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny x₁ : L'\nhx₁ : x₁ ∈ LieHom.idealRange f ⊓ J₁\nx₂ : L'\nhx₂ : x₂ ∈ LieHom.idealRange f ⊓ J₂\nhy : ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂ }⁆ = y\nz₁ : L\nhz₁' : ↑f z₁ ∈ J₁\nhz₁ : ↑f z₁ = x₁\nz₂ : L\nhz₂' : ↑f z₂ ∈ J₂\nhz₂ : ↑f z₂ = x₂\n⊢ ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂ }⁆ ∈ (fun a => ↑f a) '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "state_before": "case e_p.e_s.h.mp.intro.mk.intro.mk.intro.intro.intro.intro\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny x₁ : L'\nhx₁ : x₁ ∈ LieHom.idealRange f ⊓ J₁\nx₂ : L'\nhx₂ : x₂ ∈ LieHom.idealRange f ⊓ J₂\nhy : ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂ }⁆ = y\nz₁ : L\nhz₁' : ↑f z₁ ∈ J₁\nhz₁ : ↑f z₁ = x₁\nhz₂' : x₂ ∈ J₂\nz₂ : L\nhz₂ : ↑f z₂ = x₂\n⊢ ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂ }⁆ ∈ (fun a => ↑f a) '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "tactic": "rw [← hz₂] at hz₂'" }, { "state_after": "case e_p.e_s.h.mp.intro.mk.intro.mk.intro.intro.intro.intro\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny x₁ : L'\nhx₁ : x₁ ∈ LieHom.idealRange f ⊓ J₁\nx₂ : L'\nhx₂ : x₂ ∈ LieHom.idealRange f ⊓ J₂\nhy : ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂ }⁆ = y\nz₁ : L\nhz₁' : ↑f z₁ ∈ J₁\nhz₁ : ↑f z₁ = x₁\nz₂ : L\nhz₂' : ↑f z₂ ∈ J₂\nhz₂ : ↑f z₂ = x₂\n⊢ (fun a => ↑f a) ⁅z₁, z₂⁆ = ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂ }⁆", "state_before": "case e_p.e_s.h.mp.intro.mk.intro.mk.intro.intro.intro.intro\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny x₁ : L'\nhx₁ : x₁ ∈ LieHom.idealRange f ⊓ J₁\nx₂ : L'\nhx₂ : x₂ ∈ LieHom.idealRange f ⊓ J₂\nhy : ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂ }⁆ = y\nz₁ : L\nhz₁' : ↑f z₁ ∈ J₁\nhz₁ : ↑f z₁ = x₁\nz₂ : L\nhz₂' : ↑f z₂ ∈ J₂\nhz₂ : ↑f z₂ = x₂\n⊢ ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂ }⁆ ∈ (fun a => ↑f a) '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "tactic": "refine ⟨⁅z₁, z₂⁆, ⟨⟨z₁, hz₁'⟩, ⟨z₂, hz₂'⟩, rfl⟩, ?_⟩" }, { "state_after": "no goals", "state_before": "case e_p.e_s.h.mp.intro.mk.intro.mk.intro.intro.intro.intro\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny x₁ : L'\nhx₁ : x₁ ∈ LieHom.idealRange f ⊓ J₁\nx₂ : L'\nhx₂ : x₂ ∈ LieHom.idealRange f ⊓ J₂\nhy : ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂ }⁆ = y\nz₁ : L\nhz₁' : ↑f z₁ ∈ J₁\nhz₁ : ↑f z₁ = x₁\nz₂ : L\nhz₂' : ↑f z₂ ∈ J₂\nhz₂ : ↑f z₂ = x₂\n⊢ (fun a => ↑f a) ⁅z₁, z₂⁆ = ⁅↑{ val := x₁, property := hx₁ }, ↑{ val := x₂, property := hx₂ }⁆", "tactic": "simp only [hz₁, hz₂, Submodule.coe_mk, LieHom.map_lie]" }, { "state_after": "case e_p.e_s.h.mpr.intro.intro.intro.mk.intro.mk\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny : L'\nx : L\nhy : (fun a => ↑f a) x = y\nz₁ : L\nhz₁ : z₁ ∈ comap f J₁\nz₂ : L\nhz₂ : z₂ ∈ comap f J₂\nhx : ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆ = x\n⊢ y ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "state_before": "case e_p.e_s.h.mpr\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny : L'\n⊢ y ∈ (fun a => ↑f a) '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m} → y ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "tactic": "rintro ⟨x, ⟨⟨z₁, hz₁⟩, ⟨z₂, hz₂⟩, hx⟩, hy⟩" }, { "state_after": "case e_p.e_s.h.mpr.intro.intro.intro.mk.intro.mk\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny : L'\nx : L\nhy : (fun a => ↑f a) x = y\nz₁ : L\nhz₁ : z₁ ∈ comap f J₁\nz₂ : L\nhz₂ : z₂ ∈ comap f J₂\nhx : ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆ = x\n⊢ (fun a => ↑f a) ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆ ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "state_before": "case e_p.e_s.h.mpr.intro.intro.intro.mk.intro.mk\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny : L'\nx : L\nhy : (fun a => ↑f a) x = y\nz₁ : L\nhz₁ : z₁ ∈ comap f J₁\nz₂ : L\nhz₂ : z₂ ∈ comap f J₂\nhx : ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆ = x\n⊢ y ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "tactic": "rw [← hy, ← hx]" }, { "state_after": "case e_p.e_s.h.mpr.intro.intro.intro.mk.intro.mk\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny : L'\nx : L\nhy : (fun a => ↑f a) x = y\nz₁ : L\nhz₁ : z₁ ∈ comap f J₁\nz₂ : L\nhz₂ : z₂ ∈ comap f J₂\nhx : ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆ = x\nhz₁' : ↑f z₁ ∈ LieHom.idealRange f ⊓ J₁\n⊢ (fun a => ↑f a) ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆ ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "state_before": "case e_p.e_s.h.mpr.intro.intro.intro.mk.intro.mk\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny : L'\nx : L\nhy : (fun a => ↑f a) x = y\nz₁ : L\nhz₁ : z₁ ∈ comap f J₁\nz₂ : L\nhz₂ : z₂ ∈ comap f J₂\nhx : ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆ = x\n⊢ (fun a => ↑f a) ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆ ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "tactic": "have hz₁' : f z₁ ∈ f.idealRange ⊓ J₁ := by\n rw [LieSubmodule.mem_inf]; exact ⟨f.mem_idealRange, hz₁⟩" }, { "state_after": "case e_p.e_s.h.mpr.intro.intro.intro.mk.intro.mk\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny : L'\nx : L\nhy : (fun a => ↑f a) x = y\nz₁ : L\nhz₁ : z₁ ∈ comap f J₁\nz₂ : L\nhz₂ : z₂ ∈ comap f J₂\nhx : ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆ = x\nhz₁' : ↑f z₁ ∈ LieHom.idealRange f ⊓ J₁\nhz₂' : ↑f z₂ ∈ LieHom.idealRange f ⊓ J₂\n⊢ (fun a => ↑f a) ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆ ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "state_before": "case e_p.e_s.h.mpr.intro.intro.intro.mk.intro.mk\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny : L'\nx : L\nhy : (fun a => ↑f a) x = y\nz₁ : L\nhz₁ : z₁ ∈ comap f J₁\nz₂ : L\nhz₂ : z₂ ∈ comap f J₂\nhx : ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆ = x\nhz₁' : ↑f z₁ ∈ LieHom.idealRange f ⊓ J₁\n⊢ (fun a => ↑f a) ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆ ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "tactic": "have hz₂' : f z₂ ∈ f.idealRange ⊓ J₂ := by\n rw [LieSubmodule.mem_inf]; exact ⟨f.mem_idealRange, hz₂⟩" }, { "state_after": "case e_p.e_s.h.mpr.intro.intro.intro.mk.intro.mk\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny : L'\nx : L\nhy : (fun a => ↑f a) x = y\nz₁ : L\nhz₁ : z₁ ∈ comap f J₁\nz₂ : L\nhz₂ : z₂ ∈ comap f J₂\nhx : ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆ = x\nhz₁' : ↑f z₁ ∈ LieHom.idealRange f ⊓ J₁\nhz₂' : ↑f z₂ ∈ LieHom.idealRange f ⊓ J₂\n⊢ ⁅↑{ val := ↑f z₁, property := hz₁' }, ↑{ val := ↑f z₂, property := hz₂' }⁆ =\n (fun a => ↑f a) ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆", "state_before": "case e_p.e_s.h.mpr.intro.intro.intro.mk.intro.mk\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny : L'\nx : L\nhy : (fun a => ↑f a) x = y\nz₁ : L\nhz₁ : z₁ ∈ comap f J₁\nz₂ : L\nhz₂ : z₂ ∈ comap f J₂\nhx : ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆ = x\nhz₁' : ↑f z₁ ∈ LieHom.idealRange f ⊓ J₁\nhz₂' : ↑f z₂ ∈ LieHom.idealRange f ⊓ J₂\n⊢ (fun a => ↑f a) ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆ ∈ {m | ∃ x n, ⁅↑x, ↑n⁆ = m}", "tactic": "use ⟨f z₁, hz₁'⟩, ⟨f z₂, hz₂'⟩" }, { "state_after": "no goals", "state_before": "case e_p.e_s.h.mpr.intro.intro.intro.mk.intro.mk\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny : L'\nx : L\nhy : (fun a => ↑f a) x = y\nz₁ : L\nhz₁ : z₁ ∈ comap f J₁\nz₂ : L\nhz₂ : z₂ ∈ comap f J₂\nhx : ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆ = x\nhz₁' : ↑f z₁ ∈ LieHom.idealRange f ⊓ J₁\nhz₂' : ↑f z₂ ∈ LieHom.idealRange f ⊓ J₂\n⊢ ⁅↑{ val := ↑f z₁, property := hz₁' }, ↑{ val := ↑f z₂, property := hz₂' }⁆ =\n (fun a => ↑f a) ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆", "tactic": "simp only [Submodule.coe_mk, LieHom.map_lie]" }, { "state_after": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny : L'\nx : L\nhy : (fun a => ↑f a) x = y\nz₁ : L\nhz₁ : z₁ ∈ comap f J₁\nz₂ : L\nhz₂ : z₂ ∈ comap f J₂\nhx : ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆ = x\n⊢ ↑f z₁ ∈ LieHom.idealRange f ∧ ↑f z₁ ∈ J₁", "state_before": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny : L'\nx : L\nhy : (fun a => ↑f a) x = y\nz₁ : L\nhz₁ : z₁ ∈ comap f J₁\nz₂ : L\nhz₂ : z₂ ∈ comap f J₂\nhx : ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆ = x\n⊢ ↑f z₁ ∈ LieHom.idealRange f ⊓ J₁", "tactic": "rw [LieSubmodule.mem_inf]" }, { "state_after": "no goals", "state_before": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny : L'\nx : L\nhy : (fun a => ↑f a) x = y\nz₁ : L\nhz₁ : z₁ ∈ comap f J₁\nz₂ : L\nhz₂ : z₂ ∈ comap f J₂\nhx : ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆ = x\n⊢ ↑f z₁ ∈ LieHom.idealRange f ∧ ↑f z₁ ∈ J₁", "tactic": "exact ⟨f.mem_idealRange, hz₁⟩" }, { "state_after": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny : L'\nx : L\nhy : (fun a => ↑f a) x = y\nz₁ : L\nhz₁ : z₁ ∈ comap f J₁\nz₂ : L\nhz₂ : z₂ ∈ comap f J₂\nhx : ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆ = x\nhz₁' : ↑f z₁ ∈ LieHom.idealRange f ⊓ J₁\n⊢ ↑f z₂ ∈ LieHom.idealRange f ∧ ↑f z₂ ∈ J₂", "state_before": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny : L'\nx : L\nhy : (fun a => ↑f a) x = y\nz₁ : L\nhz₁ : z₁ ∈ comap f J₁\nz₂ : L\nhz₂ : z₂ ∈ comap f J₂\nhx : ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆ = x\nhz₁' : ↑f z₁ ∈ LieHom.idealRange f ⊓ J₁\n⊢ ↑f z₂ ∈ LieHom.idealRange f ⊓ J₂", "tactic": "rw [LieSubmodule.mem_inf]" }, { "state_after": "no goals", "state_before": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ J₁ J₂ : LieIdeal R L'\nh : LieHom.IsIdealMorphism f\ny : L'\nx : L\nhy : (fun a => ↑f a) x = y\nz₁ : L\nhz₁ : z₁ ∈ comap f J₁\nz₂ : L\nhz₂ : z₂ ∈ comap f J₂\nhx : ⁅↑{ val := z₁, property := hz₁ }, ↑{ val := z₂, property := hz₂ }⁆ = x\nhz₁' : ↑f z₁ ∈ LieHom.idealRange f ⊓ J₁\n⊢ ↑f z₂ ∈ LieHom.idealRange f ∧ ↑f z₂ ∈ J₂", "tactic": "exact ⟨f.mem_idealRange, hz₂⟩" } ]

[ 325, 81 ]

[ 306, 1 ]

[ { "state_after": "V : Type u\ninst✝³ : Category V\ninst✝² : HasImages V\nA B C D : V\nf : A ⟶ B\ng : B ⟶ C\nh✝ : C ⟶ D\ninst✝¹ : HasZeroMorphisms V\ninst✝ : HasEqualizers V\nh : Exact 0 g\nthis : Epi (imageToKernel 0 g (_ : 0 ≫ g = 0))\n⊢ Subobject.arrow (kernelSubobject g) = 0", "state_before": "V : Type u\ninst✝³ : Category V\ninst✝² : HasImages V\nA B C D : V\nf : A ⟶ B\ng : B ⟶ C\nh✝ : C ⟶ D\ninst✝¹ : HasZeroMorphisms V\ninst✝ : HasEqualizers V\nh : Exact 0 g\n⊢ Subobject.arrow (kernelSubobject g) = 0", "tactic": "haveI := h.epi" }, { "state_after": "V : Type u\ninst✝³ : Category V\ninst✝² : HasImages V\nA B C D : V\nf : A ⟶ B\ng : B ⟶ C\nh✝ : C ⟶ D\ninst✝¹ : HasZeroMorphisms V\ninst✝ : HasEqualizers V\nh : Exact 0 g\nthis : Epi (imageToKernel 0 g (_ : 0 ≫ g = 0))\n⊢ factorThruImageSubobject 0 ≫ imageToKernel 0 g (_ : 0 ≫ g = 0) ≫ Subobject.arrow (kernelSubobject g) =\n factorThruImageSubobject 0 ≫ imageToKernel 0 g (_ : 0 ≫ g = 0) ≫ 0", "state_before": "V : Type u\ninst✝³ : Category V\ninst✝² : HasImages V\nA B C D : V\nf : A ⟶ B\ng : B ⟶ C\nh✝ : C ⟶ D\ninst✝¹ : HasZeroMorphisms V\ninst✝ : HasEqualizers V\nh : Exact 0 g\nthis : Epi (imageToKernel 0 g (_ : 0 ≫ g = 0))\n⊢ Subobject.arrow (kernelSubobject g) = 0", "tactic": "rw [← cancel_epi (imageToKernel (0 : A ⟶ B) g h.w), ←\n cancel_epi (factorThruImageSubobject (0 : A ⟶ B))]" }, { "state_after": "no goals", "state_before": "V : Type u\ninst✝³ : Category V\ninst✝² : HasImages V\nA B C D : V\nf : A ⟶ B\ng : B ⟶ C\nh✝ : C ⟶ D\ninst✝¹ : HasZeroMorphisms V\ninst✝ : HasEqualizers V\nh : Exact 0 g\nthis : Epi (imageToKernel 0 g (_ : 0 ≫ g = 0))\n⊢ factorThruImageSubobject 0 ≫ imageToKernel 0 g (_ : 0 ≫ g = 0) ≫ Subobject.arrow (kernelSubobject g) =\n factorThruImageSubobject 0 ≫ imageToKernel 0 g (_ : 0 ≫ g = 0) ≫ 0", "tactic": "simp" } ]

[ 255, 7 ]

[ 250, 1 ]

[]

[ 598, 6 ]

[ 597, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_3\nι : Type ?u.32313\nγ : Type ?u.32316\nA : Type ?u.32319\nB : Type ?u.32322\nC : Type ?u.32325\ninst✝⁵ : AddCommMonoid A\ninst✝⁴ : AddCommMonoid B\ninst✝³ : AddCommMonoid C\nt : ι → A → C\nh0✝ : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf✝ : α → ι →₀ A\ni : ι\ng✝ : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type ?u.35455\nM : Type u_2\nM' : Type u_1\nN : Type u_4\nP : Type ?u.35467\nG : Type ?u.35470\nH✝ : Type ?u.35473\nR : Type ?u.35476\nS : Type ?u.35479\ninst✝² : Zero M\ninst✝¹ : Zero M'\ninst✝ : CommMonoid N\nf : M → M'\nhf : f 0 = 0\ng : α →₀ M\nh : α → M' → N\nh0 : ∀ (a : α), h a 0 = 1\nx✝¹ : α\nx✝ : x✝¹ ∈ g.support\nH : ¬x✝¹ ∈ (mapRange f hf g).support\n⊢ h x✝¹ (↑(mapRange f hf g) x✝¹) = 1", "tactic": "rw [not_mem_support_iff.1 H, h0]" } ]

[ 89, 87 ]

[ 87, 1 ]

[]

[ 1153, 6 ]

[ 1151, 1 ]

[]

[ 1246, 24 ]

[ 1245, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\nA : Type w\nB : Type ?u.321762\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Semiring A\ninst✝⁵ : Algebra R A\ninst✝⁴ : Semiring B\ninst✝³ : Algebra R B\nα : Type u_1\ninst✝² : Monoid α\ninst✝¹ : MulDistribMulAction α A\ninst✝ : SMulCommClass α R A\na : α\nr : R\n⊢ a • ↑(algebraMap R A) r = ↑(algebraMap R A) r", "tactic": "rw [algebraMap_eq_smul_one, smul_comm a r (1 : A), smul_one]" } ]

[ 406, 63 ]

[ 404, 1 ]

[]

[ 240, 6 ]

[ 239, 1 ]

[]

[ 148, 10 ]

[ 147, 1 ]

[]

[ 233, 14 ]

[ 232, 1 ]

[ { "state_after": "l : Type ?u.21103\nm : Type u_1\nn : Type u_2\nR : Type ?u.21112\nα : Type u_3\ninst✝³ : DecidableEq l\ninst✝² : DecidableEq m\ninst✝¹ : DecidableEq n\ninst✝ : Semiring α\ni : m\nj : n\nc : α\ni' : m\nj' : n\nh : ¬(i = i' ∧ j = j')\n⊢ i = i' → j = j' → c = 0", "state_before": "l : Type ?u.21103\nm : Type u_1\nn : Type u_2\nR : Type ?u.21112\nα : Type u_3\ninst✝³ : DecidableEq l\ninst✝² : DecidableEq m\ninst✝¹ : DecidableEq n\ninst✝ : Semiring α\ni : m\nj : n\nc : α\ni' : m\nj' : n\nh : ¬(i = i' ∧ j = j')\n⊢ stdBasisMatrix i j c i' j' = 0", "tactic": "simp only [stdBasisMatrix, and_imp, ite_eq_right_iff]" }, { "state_after": "no goals", "state_before": "l : Type ?u.21103\nm : Type u_1\nn : Type u_2\nR : Type ?u.21112\nα : Type u_3\ninst✝³ : DecidableEq l\ninst✝² : DecidableEq m\ninst✝¹ : DecidableEq n\ninst✝ : Semiring α\ni : m\nj : n\nc : α\ni' : m\nj' : n\nh : ¬(i = i' ∧ j = j')\n⊢ i = i' → j = j' → c = 0", "tactic": "tauto" } ]

[ 132, 8 ]

[ 130, 1 ]

[]

[ 799, 11 ]

[ 798, 1 ]

[]

[ 600, 10 ]

[ 599, 1 ]

[]

[ 52, 97 ]

[ 52, 9 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nw x y✝ z✝ : α\nhwx : w ≠ x\nhyz✝ : y✝ ≠ z✝\ny z : α\nhyz : y ≠ z\nhwz : w ≠ z\n⊢ swap w y * swap x z * swap w x * (swap w y * swap x z)⁻¹ = swap y z", "tactic": "rw [mul_inv_rev, swap_inv, swap_inv, mul_assoc (swap w y), mul_assoc (swap w y), ←\n mul_assoc _ (swap x z), swap_mul_swap_mul_swap hwx hwz, ← mul_assoc,\n swap_mul_swap_mul_swap hwz.symm hyz.symm]" }, { "state_after": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nw x y z : α\nhwx : w ≠ x\nhyz : y ≠ z\nh : ∀ {y z : α}, y ≠ z → w ≠ z → swap w y * swap x z * swap w x * (swap w y * swap x z)⁻¹ = swap y z\nhwz : w = z\n⊢ z ≠ y", "state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nw x y z : α\nhwx : w ≠ x\nhyz : y ≠ z\nh : ∀ {y z : α}, y ≠ z → w ≠ z → swap w y * swap x z * swap w x * (swap w y * swap x z)⁻¹ = swap y z\nhwz : w = z\n⊢ w ≠ y", "tactic": "rw [hwz]" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nw x y z : α\nhwx : w ≠ x\nhyz : y ≠ z\nh : ∀ {y z : α}, y ≠ z → w ≠ z → swap w y * swap x z * swap w x * (swap w y * swap x z)⁻¹ = swap y z\nhwz : w = z\n⊢ z ≠ y", "tactic": "exact hyz.symm" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nw x y z : α\nhwx : w ≠ x\nhyz : y ≠ z\nh : ∀ {y z : α}, y ≠ z → w ≠ z → swap w y * swap x z * swap w x * (swap w y * swap x z)⁻¹ = swap y z\nhwz : w = z\nhwy : w ≠ y\n⊢ swap w z * swap x y * swap w x * (swap w z * swap x y)⁻¹ = swap y z", "tactic": "rw [swap_comm y z, h hyz.symm hwy]" } ]

[ 300, 43 ]

[ 288, 1 ]

[]

[ 56, 22 ]

[ 55, 1 ]

[]

[ 163, 33 ]

[ 162, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\np : α → Prop\ninst✝ : DecidablePred p\nx : α\n⊢ ↑(countp p) (of x) = if p x = (true = true) then 1 else 0", "tactic": "simp [countp, List.countp, List.countp.go]" } ]

[ 36, 45 ]

[ 35, 1 ]

[]

[ 956, 11 ]

[ 955, 1 ]

[ { "state_after": "case intro.intro\nα✝ : Type ?u.168313\nβ : Type ?u.168316\nγ : Type ?u.168319\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\ninst✝ : IsSymm α r\nh : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → ∃ y, P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α\nhf : ∀ (n : ℕ), P (f n)\nhf' : ∀ (m n : ℕ), m < n → r (f m) (f n)\n⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m ≠ n → r (f m) (f n)", "state_before": "α✝ : Type ?u.168313\nβ : Type ?u.168316\nγ : Type ?u.168319\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\ninst✝ : IsSymm α r\nh : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → ∃ y, P y ∧ ∀ (x : α), x ∈ s → r x y\n⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m ≠ n → r (f m) (f n)", "tactic": "rcases exists_seq_of_forall_finset_exists P r h with ⟨f, hf, hf'⟩" }, { "state_after": "case intro.intro\nα✝ : Type ?u.168313\nβ : Type ?u.168316\nγ : Type ?u.168319\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\ninst✝ : IsSymm α r\nh : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → ∃ y, P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α\nhf : ∀ (n : ℕ), P (f n)\nhf' : ∀ (m n : ℕ), m < n → r (f m) (f n)\nm n : ℕ\nhmn : m ≠ n\n⊢ r (f m) (f n)", "state_before": "case intro.intro\nα✝ : Type ?u.168313\nβ : Type ?u.168316\nγ : Type ?u.168319\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\ninst✝ : IsSymm α r\nh : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → ∃ y, P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α\nhf : ∀ (n : ℕ), P (f n)\nhf' : ∀ (m n : ℕ), m < n → r (f m) (f n)\n⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m ≠ n → r (f m) (f n)", "tactic": "refine' ⟨f, hf, fun m n hmn => _⟩" }, { "state_after": "case intro.intro.inl\nα✝ : Type ?u.168313\nβ : Type ?u.168316\nγ : Type ?u.168319\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\ninst✝ : IsSymm α r\nh✝ : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → ∃ y, P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α\nhf : ∀ (n : ℕ), P (f n)\nhf' : ∀ (m n : ℕ), m < n → r (f m) (f n)\nm n : ℕ\nhmn : m ≠ n\nh : m < n\n⊢ r (f m) (f n)\n\ncase intro.intro.inr.inl\nα✝ : Type ?u.168313\nβ : Type ?u.168316\nγ : Type ?u.168319\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\ninst✝ : IsSymm α r\nh : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → ∃ y, P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α\nhf : ∀ (n : ℕ), P (f n)\nhf' : ∀ (m n : ℕ), m < n → r (f m) (f n)\nm : ℕ\nhmn : m ≠ m\n⊢ r (f m) (f m)\n\ncase intro.intro.inr.inr\nα✝ : Type ?u.168313\nβ : Type ?u.168316\nγ : Type ?u.168319\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\ninst✝ : IsSymm α r\nh✝ : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → ∃ y, P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α\nhf : ∀ (n : ℕ), P (f n)\nhf' : ∀ (m n : ℕ), m < n → r (f m) (f n)\nm n : ℕ\nhmn : m ≠ n\nh : n < m\n⊢ r (f m) (f n)", "state_before": "case intro.intro\nα✝ : Type ?u.168313\nβ : Type ?u.168316\nγ : Type ?u.168319\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\ninst✝ : IsSymm α r\nh : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → ∃ y, P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α\nhf : ∀ (n : ℕ), P (f n)\nhf' : ∀ (m n : ℕ), m < n → r (f m) (f n)\nm n : ℕ\nhmn : m ≠ n\n⊢ r (f m) (f n)", "tactic": "rcases lt_trichotomy m n with (h | rfl | h)" }, { "state_after": "no goals", "state_before": "case intro.intro.inl\nα✝ : Type ?u.168313\nβ : Type ?u.168316\nγ : Type ?u.168319\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\ninst✝ : IsSymm α r\nh✝ : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → ∃ y, P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α\nhf : ∀ (n : ℕ), P (f n)\nhf' : ∀ (m n : ℕ), m < n → r (f m) (f n)\nm n : ℕ\nhmn : m ≠ n\nh : m < n\n⊢ r (f m) (f n)", "tactic": "exact hf' m n h" }, { "state_after": "no goals", "state_before": "case intro.intro.inr.inl\nα✝ : Type ?u.168313\nβ : Type ?u.168316\nγ : Type ?u.168319\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\ninst✝ : IsSymm α r\nh : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → ∃ y, P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α\nhf : ∀ (n : ℕ), P (f n)\nhf' : ∀ (m n : ℕ), m < n → r (f m) (f n)\nm : ℕ\nhmn : m ≠ m\n⊢ r (f m) (f m)", "tactic": "exact (hmn rfl).elim" }, { "state_after": "case intro.intro.inr.inr.a\nα✝ : Type ?u.168313\nβ : Type ?u.168316\nγ : Type ?u.168319\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\ninst✝ : IsSymm α r\nh✝ : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → ∃ y, P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α\nhf : ∀ (n : ℕ), P (f n)\nhf' : ∀ (m n : ℕ), m < n → r (f m) (f n)\nm n : ℕ\nhmn : m ≠ n\nh : n < m\n⊢ r (f n) (f m)", "state_before": "case intro.intro.inr.inr\nα✝ : Type ?u.168313\nβ : Type ?u.168316\nγ : Type ?u.168319\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\ninst✝ : IsSymm α r\nh✝ : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → ∃ y, P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α\nhf : ∀ (n : ℕ), P (f n)\nhf' : ∀ (m n : ℕ), m < n → r (f m) (f n)\nm n : ℕ\nhmn : m ≠ n\nh : n < m\n⊢ r (f m) (f n)", "tactic": "apply symm" }, { "state_after": "no goals", "state_before": "case intro.intro.inr.inr.a\nα✝ : Type ?u.168313\nβ : Type ?u.168316\nγ : Type ?u.168319\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\ninst✝ : IsSymm α r\nh✝ : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → ∃ y, P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α\nhf : ∀ (n : ℕ), P (f n)\nhf' : ∀ (m n : ℕ), m < n → r (f m) (f n)\nm n : ℕ\nhmn : m ≠ n\nh : n < m\n⊢ r (f n) (f m)", "tactic": "exact hf' n m h" } ]

[ 1261, 20 ]

[ 1252, 1 ]

[ { "state_after": "case h\nG : Type u_1\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : TopologicalGroup G\ninst✝ : CompactSpace G\n⊢ Continuous fun p => p.fst / p.snd", "state_before": "G : Type u_1\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : TopologicalGroup G\ninst✝ : CompactSpace G\n⊢ UniformContinuous fun p => p.fst / p.snd", "tactic": "apply CompactSpace.uniformContinuous_of_continuous" }, { "state_after": "no goals", "state_before": "case h\nG : Type u_1\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : TopologicalGroup G\ninst✝ : CompactSpace G\n⊢ Continuous fun p => p.fst / p.snd", "tactic": "exact continuous_div'" } ]

[ 589, 27 ]

[ 586, 1 ]

[]

[ 774, 82 ]

[ 773, 1 ]

[]

[ 699, 40 ]

[ 695, 1 ]

[]

[ 186, 66 ]

[ 177, 1 ]

[ { "state_after": "case refine'_1\nι : Type u_1\nα : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ns✝ t u v : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Pairwise (AEDisjoint μ on s)\ni : ι\n⊢ ↑↑μ ((fun i => toMeasurable μ (s i ∩ ⋃ (j : ι) (_ : j ∈ {i}ᶜ), s j)) i) = 0\n\ncase refine'_2\nι : Type u_1\nα : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ns✝ t u v : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Pairwise (AEDisjoint μ on s)\n⊢ Pairwise (Disjoint on fun i => s i \\ (fun i => toMeasurable μ (s i ∩ ⋃ (j : ι) (_ : j ∈ {i}ᶜ), s j)) i)", "state_before": "ι : Type u_1\nα : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ns✝ t u v : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Pairwise (AEDisjoint μ on s)\n⊢ ∃ t, (∀ (i : ι), MeasurableSet (t i)) ∧ (∀ (i : ι), ↑↑μ (t i) = 0) ∧ Pairwise (Disjoint on fun i => s i \\ t i)", "tactic": "refine' ⟨fun i => toMeasurable μ (s i ∩ ⋃ j ∈ ({i}ᶜ : Set ι), s j), fun i =>\n measurableSet_toMeasurable _ _, fun i => _, _⟩" }, { "state_after": "case refine'_1\nι : Type u_1\nα : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ns✝ t u v : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Pairwise (AEDisjoint μ on s)\ni : ι\n⊢ ↑↑μ (⋃ (i_1 : ι) (_ : i_1 ∈ {i}ᶜ), s i ∩ s i_1) = 0", "state_before": "case refine'_1\nι : Type u_1\nα : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ns✝ t u v : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Pairwise (AEDisjoint μ on s)\ni : ι\n⊢ ↑↑μ ((fun i => toMeasurable μ (s i ∩ ⋃ (j : ι) (_ : j ∈ {i}ᶜ), s j)) i) = 0", "tactic": "simp only [measure_toMeasurable, inter_iUnion]" }, { "state_after": "no goals", "state_before": "case refine'_1\nι : Type u_1\nα : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ns✝ t u v : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Pairwise (AEDisjoint μ on s)\ni : ι\n⊢ ↑↑μ (⋃ (i_1 : ι) (_ : i_1 ∈ {i}ᶜ), s i ∩ s i_1) = 0", "tactic": "exact (measure_biUnion_null_iff <| to_countable _).2 fun j hj => hd (Ne.symm hj)" }, { "state_after": "case refine'_2\nι : Type u_1\nα : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ns✝ t u v : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Pairwise (AEDisjoint μ on s)\n⊢ ∀ ⦃i j : ι⦄,\n i ≠ j →\n ∀ ⦃a : α⦄,\n a ∈ s i →\n ¬a ∈ toMeasurable μ (s i ∩ ⋃ (j : ι) (_ : j ∈ {i}ᶜ), s j) →\n a ∈ s j → a ∈ toMeasurable μ (s j ∩ ⋃ (j_1 : ι) (_ : j_1 ∈ {j}ᶜ), s j_1)", "state_before": "case refine'_2\nι : Type u_1\nα : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ns✝ t u v : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Pairwise (AEDisjoint μ on s)\n⊢ Pairwise (Disjoint on fun i => s i \\ (fun i => toMeasurable μ (s i ∩ ⋃ (j : ι) (_ : j ∈ {i}ᶜ), s j)) i)", "tactic": "simp only [Pairwise, disjoint_left, onFun, mem_diff, not_and, and_imp, Classical.not_not]" }, { "state_after": "case refine'_2\nι : Type u_1\nα : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ns✝ t u v : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Pairwise (AEDisjoint μ on s)\ni j : ι\nhne : i ≠ j\nx : α\nhi : x ∈ s i\nhU : ¬x ∈ toMeasurable μ (s i ∩ ⋃ (j : ι) (_ : j ∈ {i}ᶜ), s j)\nhj : x ∈ s j\n⊢ x ∈ toMeasurable μ (s j ∩ ⋃ (j_1 : ι) (_ : j_1 ∈ {j}ᶜ), s j_1)", "state_before": "case refine'_2\nι : Type u_1\nα : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ns✝ t u v : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Pairwise (AEDisjoint μ on s)\n⊢ ∀ ⦃i j : ι⦄,\n i ≠ j →\n ∀ ⦃a : α⦄,\n a ∈ s i →\n ¬a ∈ toMeasurable μ (s i ∩ ⋃ (j : ι) (_ : j ∈ {i}ᶜ), s j) →\n a ∈ s j → a ∈ toMeasurable μ (s j ∩ ⋃ (j_1 : ι) (_ : j_1 ∈ {j}ᶜ), s j_1)", "tactic": "intro i j hne x hi hU hj" }, { "state_after": "case refine'_2\nι : Type u_1\nα : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ns✝ t u v : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Pairwise (AEDisjoint μ on s)\ni j : ι\nhne : i ≠ j\nx : α\nhi : x ∈ s i\nhj : x ∈ s j\nhU : ¬x ∈ s i ∩ ⋃ (j : ι) (_ : j ∈ {i}ᶜ), s j\n⊢ x ∈ toMeasurable μ (s j ∩ ⋃ (j_1 : ι) (_ : j_1 ∈ {j}ᶜ), s j_1)", "state_before": "case refine'_2\nι : Type u_1\nα : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ns✝ t u v : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Pairwise (AEDisjoint μ on s)\ni j : ι\nhne : i ≠ j\nx : α\nhi : x ∈ s i\nhU : ¬x ∈ toMeasurable μ (s i ∩ ⋃ (j : ι) (_ : j ∈ {i}ᶜ), s j)\nhj : x ∈ s j\n⊢ x ∈ toMeasurable μ (s j ∩ ⋃ (j_1 : ι) (_ : j_1 ∈ {j}ᶜ), s j_1)", "tactic": "replace hU : x ∉ s i ∩ iUnion λ j => iUnion λ _ => s j := λ h => hU (subset_toMeasurable _ _ h)" }, { "state_after": "case refine'_2\nι : Type u_1\nα : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ns✝ t u v : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Pairwise (AEDisjoint μ on s)\ni j : ι\nhne : i ≠ j\nx : α\nhi : x ∈ s i\nhj : x ∈ s j\nhU : x ∈ s i → ∀ (x_1 : ι), x_1 ∈ {i}ᶜ → ¬x ∈ s x_1\n⊢ x ∈ toMeasurable μ (s j ∩ ⋃ (j_1 : ι) (_ : j_1 ∈ {j}ᶜ), s j_1)", "state_before": "case refine'_2\nι : Type u_1\nα : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ns✝ t u v : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Pairwise (AEDisjoint μ on s)\ni j : ι\nhne : i ≠ j\nx : α\nhi : x ∈ s i\nhj : x ∈ s j\nhU : ¬x ∈ s i ∩ ⋃ (j : ι) (_ : j ∈ {i}ᶜ), s j\n⊢ x ∈ toMeasurable μ (s j ∩ ⋃ (j_1 : ι) (_ : j_1 ∈ {j}ᶜ), s j_1)", "tactic": "simp only [mem_inter_iff, mem_iUnion, not_and, not_exists] at hU" }, { "state_after": "no goals", "state_before": "case refine'_2\nι : Type u_1\nα : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ns✝ t u v : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Pairwise (AEDisjoint μ on s)\ni j : ι\nhne : i ≠ j\nx : α\nhi : x ∈ s i\nhj : x ∈ s j\nhU : x ∈ s i → ∀ (x_1 : ι), x_1 ∈ {i}ᶜ → ¬x ∈ s x_1\n⊢ x ∈ toMeasurable μ (s j ∩ ⋃ (j_1 : ι) (_ : j_1 ∈ {j}ᶜ), s j_1)", "tactic": "exact (hU hi j hne.symm hj).elim" } ]

[ 48, 37 ]

[ 37, 1 ]

[]

[ 81, 10 ]

[ 80, 1 ]

[]

[ 447, 37 ]

[ 444, 1 ]

[]

[ 1433, 97 ]

[ 1430, 1 ]

[ { "state_after": "no goals", "state_before": "θ : Angle\n⊢ 2 • θ = ↑π ↔ θ = ↑(π / 2) ∨ θ = ↑(-π / 2)", "tactic": "rw [two_zsmul, ← two_nsmul, two_nsmul_eq_pi_iff]" } ]

[ 240, 51 ]

[ 239, 1 ]

[]

[ 397, 22 ]

[ 395, 1 ]

[]

[ 1480, 36 ]

[ 1479, 1 ]

[]

[ 289, 59 ]

[ 288, 1 ]

[]

[ 2302, 60 ]

[ 2300, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.97707\nγ : Type ?u.97710\ninst✝ : DecidableEq α\ns✝ t✝ : Multiset α\na✝ b a : α\ns t : Multiset α\nh : a ∈ t\n⊢ erase (s + t) a = s + erase t a", "tactic": "rw [add_comm, erase_add_left_pos s h, add_comm]" } ]

[ 1056, 90 ]

[ 1055, 1 ]

[ { "state_after": "no goals", "state_before": "a : ℕ\na1 : 1 < a\n⊢ xn a1 1 = a", "tactic": "simp" } ]

[ 152, 40 ]

[ 152, 1 ]

[]

[ 540, 6 ]

[ 538, 1 ]

[]

[ 991, 85 ]

[ 989, 1 ]

[]

[ 138, 10 ]

[ 137, 1 ]

[ { "state_after": "no goals", "state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nG : Cᵒᵖ\n⊢ IsCodetector G.unop ↔ IsDetector G", "tactic": "rw [IsDetector, IsCodetector, ← isCodetecting_unop_iff, Set.singleton_unop]" } ]

[ 431, 78 ]

[ 430, 1 ]

[ { "state_after": "case nil\nι : Type ?u.8615\nα : Type ?u.8618\nM : Type ?u.8621\nN : Type ?u.8624\nP : Type ?u.8627\nM₀ : Type ?u.8630\nG : Type ?u.8633\nR : Type ?u.8636\nL : List ℕ\nh✝ : ∀ (i : ℕ), i ∈ L → 1 ≤ i\nh : ∀ (i : ℕ), i ∈ [] → 1 ≤ i\n⊢ length [] ≤ sum []\n\ncase cons\nι : Type ?u.8615\nα : Type ?u.8618\nM : Type ?u.8621\nN : Type ?u.8624\nP : Type ?u.8627\nM₀ : Type ?u.8630\nG : Type ?u.8633\nR : Type ?u.8636\nL✝ : List ℕ\nh✝ : ∀ (i : ℕ), i ∈ L✝ → 1 ≤ i\nj : ℕ\nL : List ℕ\nIH : (∀ (i : ℕ), i ∈ L → 1 ≤ i) → length L ≤ sum L\nh : ∀ (i : ℕ), i ∈ j :: L → 1 ≤ i\n⊢ length (j :: L) ≤ sum (j :: L)", "state_before": "ι : Type ?u.8615\nα : Type ?u.8618\nM : Type ?u.8621\nN : Type ?u.8624\nP : Type ?u.8627\nM₀ : Type ?u.8630\nG : Type ?u.8633\nR : Type ?u.8636\nL : List ℕ\nh : ∀ (i : ℕ), i ∈ L → 1 ≤ i\n⊢ length L ≤ sum L", "tactic": "induction' L with j L IH h" }, { "state_after": "case cons\nι : Type ?u.8615\nα : Type ?u.8618\nM : Type ?u.8621\nN : Type ?u.8624\nP : Type ?u.8627\nM₀ : Type ?u.8630\nG : Type ?u.8633\nR : Type ?u.8636\nL✝ : List ℕ\nh✝ : ∀ (i : ℕ), i ∈ L✝ → 1 ≤ i\nj : ℕ\nL : List ℕ\nIH : (∀ (i : ℕ), i ∈ L → 1 ≤ i) → length L ≤ sum L\nh : ∀ (i : ℕ), i ∈ j :: L → 1 ≤ i\n⊢ 1 + length L ≤ j + sum L", "state_before": "case cons\nι : Type ?u.8615\nα : Type ?u.8618\nM : Type ?u.8621\nN : Type ?u.8624\nP : Type ?u.8627\nM₀ : Type ?u.8630\nG : Type ?u.8633\nR : Type ?u.8636\nL✝ : List ℕ\nh✝ : ∀ (i : ℕ), i ∈ L✝ → 1 ≤ i\nj : ℕ\nL : List ℕ\nIH : (∀ (i : ℕ), i ∈ L → 1 ≤ i) → length L ≤ sum L\nh : ∀ (i : ℕ), i ∈ j :: L → 1 ≤ i\n⊢ length (j :: L) ≤ sum (j :: L)", "tactic": "rw [sum_cons, length, add_comm]" }, { "state_after": "no goals", "state_before": "case cons\nι : Type ?u.8615\nα : Type ?u.8618\nM : Type ?u.8621\nN : Type ?u.8624\nP : Type ?u.8627\nM₀ : Type ?u.8630\nG : Type ?u.8633\nR : Type ?u.8636\nL✝ : List ℕ\nh✝ : ∀ (i : ℕ), i ∈ L✝ → 1 ≤ i\nj : ℕ\nL : List ℕ\nIH : (∀ (i : ℕ), i ∈ L → 1 ≤ i) → length L ≤ sum L\nh : ∀ (i : ℕ), i ∈ j :: L → 1 ≤ i\n⊢ 1 + length L ≤ j + sum L", "tactic": "exact add_le_add (h _ (mem_cons_self _ _)) (IH fun i hi => h i (mem_cons.2 (Or.inr hi)))" }, { "state_after": "no goals", "state_before": "case nil\nι : Type ?u.8615\nα : Type ?u.8618\nM : Type ?u.8621\nN : Type ?u.8624\nP : Type ?u.8627\nM₀ : Type ?u.8630\nG : Type ?u.8633\nR : Type ?u.8636\nL : List ℕ\nh✝ : ∀ (i : ℕ), i ∈ L → 1 ≤ i\nh : ∀ (i : ℕ), i ∈ [] → 1 ≤ i\n⊢ length [] ≤ sum []", "tactic": "simp" } ]

[ 81, 91 ]

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[ { "state_after": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI J : FractionalIdeal S P\n⊢ ↑{ val := ↑I * ↑J, property := (_ : IsFractional S (↑I * ↑J)) } = ↑I * ↑J", "state_before": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI J : FractionalIdeal S P\n⊢ ↑(I * J) = ↑I * ↑J", "tactic": "simp only [mul_def]" }, { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI J : FractionalIdeal S P\n⊢ ↑{ val := ↑I * ↑J, property := (_ : IsFractional S (↑I * ↑J)) } = ↑I * ↑J", "tactic": "rfl" } ]

[ 557, 6 ]

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[ { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\nf g : α → ℝ\ns : Set α\nhf : Measurable f\n⊢ MeasurableSet {p | p.snd = f p.fst}", "tactic": "simpa using measurableSet_region_between_cc hf hf MeasurableSet.univ" } ]

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[ { "state_after": "case h\nα : Type u_1\nβ : Type ?u.317389\nγ : Type ?u.317392\nδ : Type ?u.317395\nι : Type ?u.317398\nR : Type ?u.317401\nR' : Type ?u.317404\nm0✝ : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nm0 : MeasurableSpace α\nm : Set (Measure α)\nhm : Set.Nonempty m\nht : MeasurableSet t\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑(restrict (sInf m) t) s = ↑↑(sInf ((fun μ => restrict μ t) '' m)) s", "state_before": "α : Type u_1\nβ : Type ?u.317389\nγ : Type ?u.317392\nδ : Type ?u.317395\nι : Type ?u.317398\nR : Type ?u.317401\nR' : Type ?u.317404\nm0✝ : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nm0 : MeasurableSpace α\nm : Set (Measure α)\nhm : Set.Nonempty m\nht : MeasurableSet t\n⊢ restrict (sInf m) t = sInf ((fun μ => restrict μ t) '' m)", "tactic": "ext1 s hs" }, { "state_after": "no goals", "state_before": "case h\nα : Type u_1\nβ : Type ?u.317389\nγ : Type ?u.317392\nδ : Type ?u.317395\nι : Type ?u.317398\nR : Type ?u.317401\nR' : Type ?u.317404\nm0✝ : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\nm0 : MeasurableSpace α\nm : Set (Measure α)\nhm : Set.Nonempty m\nht : MeasurableSet t\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑(restrict (sInf m) t) s = ↑↑(sInf ((fun μ => restrict μ t) '' m)) s", "tactic": "simp_rw [sInf_apply hs, restrict_apply hs, sInf_apply (MeasurableSet.inter hs ht),\n Set.image_image, restrict_toOuterMeasure_eq_toOuterMeasure_restrict ht, ←\n Set.image_image _ toOuterMeasure, ← OuterMeasure.restrict_sInf_eq_sInf_restrict _ (hm.image _),\n OuterMeasure.restrict_apply]" } ]

[ 1877, 33 ]

[ 1870, 1 ]

[]

[ 541, 12 ]

[ 539, 8 ]

[]

[ 352, 24 ]

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[ { "state_after": "case mp\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\n⊢ LocallySmall C → LocallySmall D\n\ncase mpr\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\n⊢ LocallySmall D → LocallySmall C", "state_before": "C✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\n⊢ LocallySmall C ↔ LocallySmall D", "tactic": "fconstructor" }, { "state_after": "case mp.mk\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\nL : ∀ (X Y : C), Small (X ⟶ Y)\n⊢ LocallySmall D", "state_before": "case mp\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\n⊢ LocallySmall C → LocallySmall D", "tactic": "rintro ⟨L⟩" }, { "state_after": "case mp.mk.hom_small\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\nL : ∀ (X Y : C), Small (X ⟶ Y)\n⊢ autoParam (∀ (X Y : D), Small (X ⟶ Y)) _auto✝", "state_before": "case mp.mk\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\nL : ∀ (X Y : C), Small (X ⟶ Y)\n⊢ LocallySmall D", "tactic": "fconstructor" }, { "state_after": "case mp.mk.hom_small\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\nL : ∀ (X Y : C), Small (X ⟶ Y)\nX Y : D\n⊢ Small (X ⟶ Y)", "state_before": "case mp.mk.hom_small\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\nL : ∀ (X Y : C), Small (X ⟶ Y)\n⊢ autoParam (∀ (X Y : D), Small (X ⟶ Y)) _auto✝", "tactic": "intro X Y" }, { "state_after": "case mp.mk.hom_small\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\nX Y : D\nL : Small (e.inverse.obj X ⟶ e.inverse.obj Y)\n⊢ Small (X ⟶ Y)", "state_before": "case mp.mk.hom_small\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\nL : ∀ (X Y : C), Small (X ⟶ Y)\nX Y : D\n⊢ Small (X ⟶ Y)", "tactic": "specialize L (e.inverse.obj X) (e.inverse.obj Y)" }, { "state_after": "case mp.mk.hom_small\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\nX Y : D\nL : Small (e.inverse.obj X ⟶ e.inverse.obj Y)\n⊢ (X ⟶ Y) ≃ (e.inverse.obj X ⟶ e.inverse.obj Y)", "state_before": "case mp.mk.hom_small\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\nX Y : D\nL : Small (e.inverse.obj X ⟶ e.inverse.obj Y)\n⊢ Small (X ⟶ Y)", "tactic": "refine' (small_congr _).mpr L" }, { "state_after": "no goals", "state_before": "case mp.mk.hom_small\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\nX Y : D\nL : Small (e.inverse.obj X ⟶ e.inverse.obj Y)\n⊢ (X ⟶ Y) ≃ (e.inverse.obj X ⟶ e.inverse.obj Y)", "tactic": "exact equivOfFullyFaithful e.inverse" }, { "state_after": "case mpr.mk\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\nL : ∀ (X Y : D), Small (X ⟶ Y)\n⊢ LocallySmall C", "state_before": "case mpr\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\n⊢ LocallySmall D → LocallySmall C", "tactic": "rintro ⟨L⟩" }, { "state_after": "case mpr.mk.hom_small\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\nL : ∀ (X Y : D), Small (X ⟶ Y)\n⊢ autoParam (∀ (X Y : C), Small (X ⟶ Y)) _auto✝", "state_before": "case mpr.mk\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\nL : ∀ (X Y : D), Small (X ⟶ Y)\n⊢ LocallySmall C", "tactic": "fconstructor" }, { "state_after": "case mpr.mk.hom_small\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\nL : ∀ (X Y : D), Small (X ⟶ Y)\nX Y : C\n⊢ Small (X ⟶ Y)", "state_before": "case mpr.mk.hom_small\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\nL : ∀ (X Y : D), Small (X ⟶ Y)\n⊢ autoParam (∀ (X Y : C), Small (X ⟶ Y)) _auto✝", "tactic": "intro X Y" }, { "state_after": "case mpr.mk.hom_small\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\nX Y : C\nL : Small (e.functor.obj X ⟶ e.functor.obj Y)\n⊢ Small (X ⟶ Y)", "state_before": "case mpr.mk.hom_small\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\nL : ∀ (X Y : D), Small (X ⟶ Y)\nX Y : C\n⊢ Small (X ⟶ Y)", "tactic": "specialize L (e.functor.obj X) (e.functor.obj Y)" }, { "state_after": "case mpr.mk.hom_small\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\nX Y : C\nL : Small (e.functor.obj X ⟶ e.functor.obj Y)\n⊢ (X ⟶ Y) ≃ (e.functor.obj X ⟶ e.functor.obj Y)", "state_before": "case mpr.mk.hom_small\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\nX Y : C\nL : Small (e.functor.obj X ⟶ e.functor.obj Y)\n⊢ Small (X ⟶ Y)", "tactic": "refine' (small_congr _).mpr L" }, { "state_after": "no goals", "state_before": "case mpr.mk.hom_small\nC✝ : Type u\ninst✝² : Category C✝\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\ne : C ≌ D\nX Y : C\nL : Small (e.functor.obj X ⟶ e.functor.obj Y)\n⊢ (X ⟶ Y) ≃ (e.functor.obj X ⟶ e.functor.obj Y)", "tactic": "exact equivOfFullyFaithful e.functor" } ]

[ 116, 41 ]

[ 102, 1 ]

[ { "state_after": "α : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\n⊢ tail (corec (fun s => head s) (fun s => tail (tail s)) s) =\n corec (fun s => head s) (fun s => tail (tail s)) (tail (tail s))", "state_before": "α : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\n⊢ tail (even s) = even (tail (tail s))", "tactic": "unfold even" }, { "state_after": "α : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\n⊢ tail (head s :: corec (fun s => head s) (fun s => tail (tail s)) (tail (tail s))) =\n corec (fun s => head s) (fun s => tail (tail s)) (tail (tail s))", "state_before": "α : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\n⊢ tail (corec (fun s => head s) (fun s => tail (tail s)) s) =\n corec (fun s => head s) (fun s => tail (tail s)) (tail (tail s))", "tactic": "rw [corec_eq]" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\n⊢ tail (head s :: corec (fun s => head s) (fun s => tail (tail s)) (tail (tail s))) =\n corec (fun s => head s) (fun s => tail (tail s)) (tail (tail s))", "tactic": "rfl" } ]

[ 472, 6 ]

[ 469, 1 ]

[]

[ 508, 57 ]

[ 507, 1 ]

[]

[ 73, 47 ]

[ 72, 1 ]

[]

[ 370, 6 ]

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