Datasets:

tasksource

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leandojo

like 5

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[ 426, 89 ]

[ 425, 9 ]

[ { "state_after": "no goals", "state_before": "ι : Type ?u.14661329\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nA : Type ?u.14661341\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : CompleteSpace E\ninst✝⁶ : NormedSpace ℝ E\na b : ℝ\nμ : MeasureTheory.Measure ℝ\nf : ℝ → E\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nL : E →L[𝕜] F\nhf : IntervalIntegrable f μ a b\n⊢ (∫ (x : ℝ) in a..b, ↑L (f x) ∂μ) = ↑L (∫ (x : ℝ) in a..b, f x ∂μ)", "tactic": "simp_rw [intervalIntegral, L.integral_comp_comm hf.1, L.integral_comp_comm hf.2, L.map_sub]" } ]

[ 680, 94 ]

[ 678, 1 ]

[ { "state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.2586554\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ ↑re (inner x y) = (‖x‖ * ‖x‖ + 2 * ↑re (inner x y) + ‖y‖ * ‖y‖ - (‖x‖ * ‖x‖ - 2 * ↑re (inner x y) + ‖y‖ * ‖y‖)) / 4", "state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.2586554\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ ↑re (inner x y) = (‖x + y‖ * ‖x + y‖ - ‖x - y‖ * ‖x - y‖) / 4", "tactic": "rw [@norm_add_mul_self 𝕜, @norm_sub_mul_self 𝕜]" }, { "state_after": "no goals", "state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.2586554\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ ↑re (inner x y) = (‖x‖ * ‖x‖ + 2 * ↑re (inner x y) + ‖y‖ * ‖y‖ - (‖x‖ * ‖x‖ - 2 * ↑re (inner x y) + ‖y‖ * ‖y‖)) / 4", "tactic": "ring" } ]

[ 1153, 7 ]

[ 1150, 1 ]

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[ 960, 44 ]

[ 959, 11 ]

[ { "state_after": "no goals", "state_before": "C : Type u₁\ninst✝ : Category C\nZ X Y P : C\nf : Z ⟶ X\ng : Z ⟶ Y\ninl : X ⟶ P\ninr : Y ⟶ P\nc : PushoutCocone f g\nh : IsColimit c\n⊢ PushoutCocone.inl c ≫ (Iso.refl c.pt).hom =\n PushoutCocone.inl\n (PushoutCocone.mk (PushoutCocone.inl c) (PushoutCocone.inr c)\n (_ : f ≫ PushoutCocone.inl c = g ≫ PushoutCocone.inr c))", "tactic": "aesop_cat" }, { "state_after": "no goals", "state_before": "C : Type u₁\ninst✝ : Category C\nZ X Y P : C\nf : Z ⟶ X\ng : Z ⟶ Y\ninl : X ⟶ P\ninr : Y ⟶ P\nc : PushoutCocone f g\nh : IsColimit c\n⊢ PushoutCocone.inr c ≫ (Iso.refl c.pt).hom =\n PushoutCocone.inr\n (PushoutCocone.mk (PushoutCocone.inl c) (PushoutCocone.inr c)\n (_ : f ≫ PushoutCocone.inl c = g ≫ PushoutCocone.inr c))", "tactic": "aesop_cat" } ]

[ 361, 42 ]

[ 357, 1 ]

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[ 1122, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type ?u.49056\ninst✝ : PartialOrder α\na b c : α\ns : Set α\nho : Ioi a ⊆ s\nhc : s ⊆ Ici a\nh : a ∈ s\n⊢ Ioi a ⊆ s ∧ {a} ⊆ s", "state_before": "α : Type u_1\nβ : Type ?u.49056\ninst✝ : PartialOrder α\na b c : α\ns : Set α\nho : Ioi a ⊆ s\nhc : s ⊆ Ici a\nh : a ∈ s\n⊢ Ici a ⊆ s", "tactic": "rw [← Ioi_union_left, union_subset_iff]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.49056\ninst✝ : PartialOrder α\na b c : α\ns : Set α\nho : Ioi a ⊆ s\nhc : s ⊆ Ici a\nh : a ∈ s\n⊢ Ioi a ⊆ s ∧ {a} ⊆ s", "tactic": "simp [*]" } ]

[ 911, 100 ]

[ 905, 1 ]

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[ 263, 1 ]

[ { "state_after": "case hz\n\n⊢ card (treesOfNumNodesEq 0) = catalan 0\n\ncase hi\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\n⊢ card (treesOfNumNodesEq (Nat.succ n)) = catalan (Nat.succ n)", "state_before": "n : ℕ\n⊢ card (treesOfNumNodesEq n) = catalan n", "tactic": "induction' n using Nat.case_strong_induction_on with n ih" }, { "state_after": "case hi\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\n⊢ ∑ u in Nat.antidiagonal n, card (pairwiseNode (treesOfNumNodesEq u.fst) (treesOfNumNodesEq u.snd)) =\n ∑ ij in Nat.antidiagonal n, catalan ij.fst * catalan ij.snd\n\ncase hi\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\n⊢ ∀ (x : ℕ × ℕ),\n x ∈ Nat.antidiagonal n →\n ∀ (y : ℕ × ℕ),\n y ∈ Nat.antidiagonal n →\n x ≠ y →\n Disjoint (pairwiseNode (treesOfNumNodesEq x.fst) (treesOfNumNodesEq x.snd))\n (pairwiseNode (treesOfNumNodesEq y.fst) (treesOfNumNodesEq y.snd))", "state_before": "case hi\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\n⊢ card (treesOfNumNodesEq (Nat.succ n)) = catalan (Nat.succ n)", "tactic": "rw [treesOfNumNodesEq_succ, card_biUnion, catalan_succ']" }, { "state_after": "no goals", "state_before": "case hz\n\n⊢ card (treesOfNumNodesEq 0) = catalan 0", "tactic": "simp" }, { "state_after": "case hi\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\n⊢ ∀ (x : ℕ × ℕ),\n x ∈ Nat.antidiagonal n →\n card (pairwiseNode (treesOfNumNodesEq x.fst) (treesOfNumNodesEq x.snd)) = catalan x.fst * catalan x.snd", "state_before": "case hi\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\n⊢ ∑ u in Nat.antidiagonal n, card (pairwiseNode (treesOfNumNodesEq u.fst) (treesOfNumNodesEq u.snd)) =\n ∑ ij in Nat.antidiagonal n, catalan ij.fst * catalan ij.snd", "tactic": "apply sum_congr rfl" }, { "state_after": "case hi.mk\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\nH : (i, j) ∈ Nat.antidiagonal n\n⊢ card (pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd)) =\n catalan (i, j).fst * catalan (i, j).snd", "state_before": "case hi\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\n⊢ ∀ (x : ℕ × ℕ),\n x ∈ Nat.antidiagonal n →\n card (pairwiseNode (treesOfNumNodesEq x.fst) (treesOfNumNodesEq x.snd)) = catalan x.fst * catalan x.snd", "tactic": "rintro ⟨i, j⟩ H" }, { "state_after": "no goals", "state_before": "case hi.mk\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\nH : (i, j) ∈ Nat.antidiagonal n\n⊢ card (pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd)) =\n catalan (i, j).fst * catalan (i, j).snd", "tactic": "rw [card_map, card_product, ih _ (fst_le H), ih _ (snd_le H)]" }, { "state_after": "case hi\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\n⊢ ∀ (x : ℕ × ℕ),\n x ∈ Nat.antidiagonal n →\n ∀ (y : ℕ × ℕ),\n y ∈ Nat.antidiagonal n →\n x ≠ y →\n ∀ ⦃a : Tree Unit⦄,\n a ∈ pairwiseNode (treesOfNumNodesEq x.fst) (treesOfNumNodesEq x.snd) →\n ¬a ∈ pairwiseNode (treesOfNumNodesEq y.fst) (treesOfNumNodesEq y.snd)", "state_before": "case hi\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\n⊢ ∀ (x : ℕ × ℕ),\n x ∈ Nat.antidiagonal n →\n ∀ (y : ℕ × ℕ),\n y ∈ Nat.antidiagonal n →\n x ≠ y →\n Disjoint (pairwiseNode (treesOfNumNodesEq x.fst) (treesOfNumNodesEq x.snd))\n (pairwiseNode (treesOfNumNodesEq y.fst) (treesOfNumNodesEq y.snd))", "tactic": "simp_rw [disjoint_left]" }, { "state_after": "case hi.mk.mk\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\na✝¹ : (i, j) ∈ Nat.antidiagonal n\ni' j' : ℕ\na✝ : (i', j') ∈ Nat.antidiagonal n\n⊢ (i, j) ≠ (i', j') →\n ∀ ⦃a : Tree Unit⦄,\n a ∈ pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd) →\n ¬a ∈ pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)", "state_before": "case hi\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\n⊢ ∀ (x : ℕ × ℕ),\n x ∈ Nat.antidiagonal n →\n ∀ (y : ℕ × ℕ),\n y ∈ Nat.antidiagonal n →\n x ≠ y →\n ∀ ⦃a : Tree Unit⦄,\n a ∈ pairwiseNode (treesOfNumNodesEq x.fst) (treesOfNumNodesEq x.snd) →\n ¬a ∈ pairwiseNode (treesOfNumNodesEq y.fst) (treesOfNumNodesEq y.snd)", "tactic": "rintro ⟨i, j⟩ _ ⟨i', j'⟩ _" }, { "state_after": "case hi.mk.mk\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\na✝¹ : (i, j) ∈ Nat.antidiagonal n\ni' j' : ℕ\na✝ : (i', j') ∈ Nat.antidiagonal n\nh : (i, j) ≠ (i', j')\na : Tree Unit\n⊢ a ∈ pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd) →\n ¬a ∈ pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)", "state_before": "case hi.mk.mk\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\na✝¹ : (i, j) ∈ Nat.antidiagonal n\ni' j' : ℕ\na✝ : (i', j') ∈ Nat.antidiagonal n\n⊢ (i, j) ≠ (i', j') →\n ∀ ⦃a : Tree Unit⦄,\n a ∈ pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd) →\n ¬a ∈ pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)", "tactic": "intros h a" }, { "state_after": "case hi.mk.mk.nil\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\na✝¹ : (i, j) ∈ Nat.antidiagonal n\ni' j' : ℕ\na✝ : (i', j') ∈ Nat.antidiagonal n\nh : (i, j) ≠ (i', j')\n⊢ nil ∈ pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd) →\n ¬nil ∈ pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)\n\ncase hi.mk.mk.node\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\na✝¹ : (i, j) ∈ Nat.antidiagonal n\ni' j' : ℕ\na✝ : (i', j') ∈ Nat.antidiagonal n\nh : (i, j) ≠ (i', j')\na : Unit\nl r : Tree Unit\n⊢ node a l r ∈ pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd) →\n ¬node a l r ∈ pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)", "state_before": "case hi.mk.mk\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\na✝¹ : (i, j) ∈ Nat.antidiagonal n\ni' j' : ℕ\na✝ : (i', j') ∈ Nat.antidiagonal n\nh : (i, j) ≠ (i', j')\na : Tree Unit\n⊢ a ∈ pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd) →\n ¬a ∈ pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)", "tactic": "cases' a with a l r" }, { "state_after": "case hi.mk.mk.nil\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\na✝¹ : (i, j) ∈ Nat.antidiagonal n\ni' j' : ℕ\na✝ : (i', j') ∈ Nat.antidiagonal n\nh✝ : (i, j) ≠ (i', j')\nh : nil ∈ pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd)\n⊢ ¬nil ∈ pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)", "state_before": "case hi.mk.mk.nil\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\na✝¹ : (i, j) ∈ Nat.antidiagonal n\ni' j' : ℕ\na✝ : (i', j') ∈ Nat.antidiagonal n\nh : (i, j) ≠ (i', j')\n⊢ nil ∈ pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd) →\n ¬nil ∈ pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)", "tactic": "intro h" }, { "state_after": "no goals", "state_before": "case hi.mk.mk.nil\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\na✝¹ : (i, j) ∈ Nat.antidiagonal n\ni' j' : ℕ\na✝ : (i', j') ∈ Nat.antidiagonal n\nh✝ : (i, j) ≠ (i', j')\nh : nil ∈ pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd)\n⊢ ¬nil ∈ pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)", "tactic": "simp at h" }, { "state_after": "case hi.mk.mk.node\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\na✝¹ : (i, j) ∈ Nat.antidiagonal n\ni' j' : ℕ\na✝ : (i', j') ∈ Nat.antidiagonal n\nh : (i, j) ≠ (i', j')\na : Unit\nl r : Tree Unit\nh1 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd)\nh2 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)\n⊢ False", "state_before": "case hi.mk.mk.node\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\na✝¹ : (i, j) ∈ Nat.antidiagonal n\ni' j' : ℕ\na✝ : (i', j') ∈ Nat.antidiagonal n\nh : (i, j) ≠ (i', j')\na : Unit\nl r : Tree Unit\n⊢ node a l r ∈ pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd) →\n ¬node a l r ∈ pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)", "tactic": "intro h1 h2" }, { "state_after": "case hi.mk.mk.node\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\na✝¹ : (i, j) ∈ Nat.antidiagonal n\ni' j' : ℕ\na✝ : (i', j') ∈ Nat.antidiagonal n\nh : (i, j) ≠ (i', j')\na : Unit\nl r : Tree Unit\nh1 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd)\nh2 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)\n⊢ (i, j) = (i', j')", "state_before": "case hi.mk.mk.node\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\na✝¹ : (i, j) ∈ Nat.antidiagonal n\ni' j' : ℕ\na✝ : (i', j') ∈ Nat.antidiagonal n\nh : (i, j) ≠ (i', j')\na : Unit\nl r : Tree Unit\nh1 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd)\nh2 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)\n⊢ False", "tactic": "apply h" }, { "state_after": "n : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\na✝¹ : (i, j) ∈ Nat.antidiagonal n\ni' j' : ℕ\na✝ : (i', j') ∈ Nat.antidiagonal n\nh : (i, j) ≠ (i', j')\na : Unit\nl r : Tree Unit\nh1 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd)\nh2 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)\n⊢ (i, j) = (numNodes l, numNodes r)\n\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\na✝¹ : (i, j) ∈ Nat.antidiagonal n\ni' j' : ℕ\na✝ : (i', j') ∈ Nat.antidiagonal n\nh : (i, j) ≠ (i', j')\na : Unit\nl r : Tree Unit\nh1 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd)\nh2 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)\n⊢ (numNodes l, numNodes r) = (i', j')", "state_before": "case hi.mk.mk.node\nn : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\na✝¹ : (i, j) ∈ Nat.antidiagonal n\ni' j' : ℕ\na✝ : (i', j') ∈ Nat.antidiagonal n\nh : (i, j) ≠ (i', j')\na : Unit\nl r : Tree Unit\nh1 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd)\nh2 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)\n⊢ (i, j) = (i', j')", "tactic": "trans (numNodes l, numNodes r)" }, { "state_after": "n : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\na✝¹ : (i, j) ∈ Nat.antidiagonal n\ni' j' : ℕ\na✝ : (i', j') ∈ Nat.antidiagonal n\nh : (i, j) ≠ (i', j')\na : Unit\nl r : Tree Unit\nh2 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)\nh1 : numNodes l = i ∧ numNodes r = j\n⊢ (i, j) = (numNodes l, numNodes r)", "state_before": "n : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\na✝¹ : (i, j) ∈ Nat.antidiagonal n\ni' j' : ℕ\na✝ : (i', j') ∈ Nat.antidiagonal n\nh : (i, j) ≠ (i', j')\na : Unit\nl r : Tree Unit\nh1 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd)\nh2 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)\n⊢ (i, j) = (numNodes l, numNodes r)", "tactic": "simp at h1" }, { "state_after": "no goals", "state_before": "n : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\na✝¹ : (i, j) ∈ Nat.antidiagonal n\ni' j' : ℕ\na✝ : (i', j') ∈ Nat.antidiagonal n\nh : (i, j) ≠ (i', j')\na : Unit\nl r : Tree Unit\nh2 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)\nh1 : numNodes l = i ∧ numNodes r = j\n⊢ (i, j) = (numNodes l, numNodes r)", "tactic": "simp [h1]" }, { "state_after": "n : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\na✝¹ : (i, j) ∈ Nat.antidiagonal n\ni' j' : ℕ\na✝ : (i', j') ∈ Nat.antidiagonal n\nh : (i, j) ≠ (i', j')\na : Unit\nl r : Tree Unit\nh1 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd)\nh2 : numNodes l = i' ∧ numNodes r = j'\n⊢ (numNodes l, numNodes r) = (i', j')", "state_before": "n : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\na✝¹ : (i, j) ∈ Nat.antidiagonal n\ni' j' : ℕ\na✝ : (i', j') ∈ Nat.antidiagonal n\nh : (i, j) ≠ (i', j')\na : Unit\nl r : Tree Unit\nh1 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd)\nh2 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i', j').fst) (treesOfNumNodesEq (i', j').snd)\n⊢ (numNodes l, numNodes r) = (i', j')", "tactic": "simp at h2" }, { "state_after": "no goals", "state_before": "n : ℕ\nih : ∀ (m : ℕ), m ≤ n → card (treesOfNumNodesEq m) = catalan m\ni j : ℕ\na✝¹ : (i, j) ∈ Nat.antidiagonal n\ni' j' : ℕ\na✝ : (i', j') ∈ Nat.antidiagonal n\nh : (i, j) ≠ (i', j')\na : Unit\nl r : Tree Unit\nh1 : node a l r ∈ pairwiseNode (treesOfNumNodesEq (i, j).fst) (treesOfNumNodesEq (i, j).snd)\nh2 : numNodes l = i' ∧ numNodes r = j'\n⊢ (numNodes l, numNodes r) = (i', j')", "tactic": "simp [h2]" } ]

[ 229, 30 ]

[ 212, 1 ]

[ { "state_after": "M : Type ?u.2760785\nN : Type ?u.2760788\nG : Type ?u.2760791\nR : Type u_1\nS : Type ?u.2760797\nF : Type ?u.2760800\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\nζ : R\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ 1 ∈ primitiveRoots 1 R ∧ ∀ (x : R), x ∈ primitiveRoots 1 R → x = 1", "state_before": "M : Type ?u.2760785\nN : Type ?u.2760788\nG : Type ?u.2760791\nR : Type u_1\nS : Type ?u.2760797\nF : Type ?u.2760800\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\nζ : R\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ primitiveRoots 1 R = {1}", "tactic": "apply Finset.eq_singleton_iff_unique_mem.2" }, { "state_after": "case left\nM : Type ?u.2760785\nN : Type ?u.2760788\nG : Type ?u.2760791\nR : Type u_1\nS : Type ?u.2760797\nF : Type ?u.2760800\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\nζ : R\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ 1 ∈ primitiveRoots 1 R\n\ncase right\nM : Type ?u.2760785\nN : Type ?u.2760788\nG : Type ?u.2760791\nR : Type u_1\nS : Type ?u.2760797\nF : Type ?u.2760800\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\nζ : R\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ ∀ (x : R), x ∈ primitiveRoots 1 R → x = 1", "state_before": "M : Type ?u.2760785\nN : Type ?u.2760788\nG : Type ?u.2760791\nR : Type u_1\nS : Type ?u.2760797\nF : Type ?u.2760800\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\nζ : R\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ 1 ∈ primitiveRoots 1 R ∧ ∀ (x : R), x ∈ primitiveRoots 1 R → x = 1", "tactic": "constructor" }, { "state_after": "no goals", "state_before": "case left\nM : Type ?u.2760785\nN : Type ?u.2760788\nG : Type ?u.2760791\nR : Type u_1\nS : Type ?u.2760797\nF : Type ?u.2760800\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\nζ : R\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ 1 ∈ primitiveRoots 1 R", "tactic": "simp only [IsPrimitiveRoot.one_right_iff, mem_primitiveRoots zero_lt_one]" }, { "state_after": "case right\nM : Type ?u.2760785\nN : Type ?u.2760788\nG : Type ?u.2760791\nR : Type u_1\nS : Type ?u.2760797\nF : Type ?u.2760800\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\nζ : R\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : R\nhx : x ∈ primitiveRoots 1 R\n⊢ x = 1", "state_before": "case right\nM : Type ?u.2760785\nN : Type ?u.2760788\nG : Type ?u.2760791\nR : Type u_1\nS : Type ?u.2760797\nF : Type ?u.2760800\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\nζ : R\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ ∀ (x : R), x ∈ primitiveRoots 1 R → x = 1", "tactic": "intro x hx" }, { "state_after": "case right\nM : Type ?u.2760785\nN : Type ?u.2760788\nG : Type ?u.2760791\nR : Type u_1\nS : Type ?u.2760797\nF : Type ?u.2760800\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\nζ : R\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : R\nhx : x = 1\n⊢ x = 1", "state_before": "case right\nM : Type ?u.2760785\nN : Type ?u.2760788\nG : Type ?u.2760791\nR : Type u_1\nS : Type ?u.2760797\nF : Type ?u.2760800\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\nζ : R\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : R\nhx : x ∈ primitiveRoots 1 R\n⊢ x = 1", "tactic": "rw [mem_primitiveRoots zero_lt_one, IsPrimitiveRoot.one_right_iff] at hx" }, { "state_after": "no goals", "state_before": "case right\nM : Type ?u.2760785\nN : Type ?u.2760788\nG : Type ?u.2760791\nR : Type u_1\nS : Type ?u.2760797\nF : Type ?u.2760800\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\nζ : R\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : R\nhx : x = 1\n⊢ x = 1", "tactic": "exact hx" } ]

[ 613, 13 ]

[ 607, 1 ]

[ { "state_after": "C : Type u\ninst✝³ : Category C\nI : MultispanIndex C\ninst✝² : HasMulticoequalizer I\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nb : I.R\n⊢ Sigma.ι I.right b ≫ coequalizer.π (MultispanIndex.fstSigmaMap I) (MultispanIndex.sndSigmaMap I) =\n π I b ≫\n (colimit.isoColimitCocone\n {\n cocone :=\n (MultispanIndex.multicoforkEquivSigmaCofork I).inverse.obj\n (colimit.cocone (parallelPair (MultispanIndex.fstSigmaMap I) (MultispanIndex.sndSigmaMap I))),\n isColimit :=\n IsColimit.ofPreservesCoconeInitial (MultispanIndex.multicoforkEquivSigmaCofork I).inverse\n (colimit.isColimit (parallelPair (MultispanIndex.fstSigmaMap I) (MultispanIndex.sndSigmaMap I))) }).hom", "state_before": "C : Type u\ninst✝³ : Category C\nI : MultispanIndex C\ninst✝² : HasMulticoequalizer I\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nb : I.R\n⊢ Sigma.ι I.right b ≫ sigmaπ I = π I b", "tactic": "rw [sigmaπ, ← Category.assoc, Iso.comp_inv_eq, isoCoequalizer]" }, { "state_after": "C : Type u\ninst✝³ : Category C\nI : MultispanIndex C\ninst✝² : HasMulticoequalizer I\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nb : I.R\n⊢ Sigma.ι I.right b ≫ coequalizer.π (MultispanIndex.fstSigmaMap I) (MultispanIndex.sndSigmaMap I) =\n Multicofork.π\n (Multicofork.ofSigmaCofork I\n (colimit.cocone (parallelPair (MultispanIndex.fstSigmaMap I) (MultispanIndex.sndSigmaMap I))))\n b", "state_before": "C : Type u\ninst✝³ : Category C\nI : MultispanIndex C\ninst✝² : HasMulticoequalizer I\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nb : I.R\n⊢ Sigma.ι I.right b ≫ coequalizer.π (MultispanIndex.fstSigmaMap I) (MultispanIndex.sndSigmaMap I) =\n π I b ≫\n (colimit.isoColimitCocone\n {\n cocone :=\n (MultispanIndex.multicoforkEquivSigmaCofork I).inverse.obj\n (colimit.cocone (parallelPair (MultispanIndex.fstSigmaMap I) (MultispanIndex.sndSigmaMap I))),\n isColimit :=\n IsColimit.ofPreservesCoconeInitial (MultispanIndex.multicoforkEquivSigmaCofork I).inverse\n (colimit.isColimit (parallelPair (MultispanIndex.fstSigmaMap I) (MultispanIndex.sndSigmaMap I))) }).hom", "tactic": "simp only [MultispanIndex.multicoforkEquivSigmaCofork_inverse,\n MultispanIndex.ofSigmaCoforkFunctor_obj, colimit.isoColimitCocone_ι_hom,\n Multicofork.ofSigmaCofork_pt, colimit.cocone_x, Multicofork.π_eq_app_right]" }, { "state_after": "no goals", "state_before": "C : Type u\ninst✝³ : Category C\nI : MultispanIndex C\ninst✝² : HasMulticoequalizer I\ninst✝¹ : HasCoproduct I.left\ninst✝ : HasCoproduct I.right\nb : I.R\n⊢ Sigma.ι I.right b ≫ coequalizer.π (MultispanIndex.fstSigmaMap I) (MultispanIndex.sndSigmaMap I) =\n Multicofork.π\n (Multicofork.ofSigmaCofork I\n (colimit.cocone (parallelPair (MultispanIndex.fstSigmaMap I) (MultispanIndex.sndSigmaMap I))))\n b", "tactic": "rfl" } ]

[ 923, 6 ]

[ 918, 1 ]

[ { "state_after": "case h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.224777\nι : Sort u_3\nι' : Sort ?u.224783\nι₂ : Sort ?u.224786\nκ : ι → Sort ?u.224791\nκ₁ : ι → Sort ?u.224796\nκ₂ : ι → Sort ?u.224801\nκ' : ι' → Sort ?u.224806\nf : α → β\ns : ι → Set β\nx✝ : α\n⊢ (x✝ ∈ f ⁻¹' ⋂ (i : ι), s i) ↔ x✝ ∈ ⋂ (i : ι), f ⁻¹' s i", "state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.224777\nι : Sort u_3\nι' : Sort ?u.224783\nι₂ : Sort ?u.224786\nκ : ι → Sort ?u.224791\nκ₁ : ι → Sort ?u.224796\nκ₂ : ι → Sort ?u.224801\nκ' : ι' → Sort ?u.224806\nf : α → β\ns : ι → Set β\n⊢ (f ⁻¹' ⋂ (i : ι), s i) = ⋂ (i : ι), f ⁻¹' s i", "tactic": "ext" }, { "state_after": "no goals", "state_before": "case h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.224777\nι : Sort u_3\nι' : Sort ?u.224783\nι₂ : Sort ?u.224786\nκ : ι → Sort ?u.224791\nκ₁ : ι → Sort ?u.224796\nκ₂ : ι → Sort ?u.224801\nκ' : ι' → Sort ?u.224806\nf : α → β\ns : ι → Set β\nx✝ : α\n⊢ (x✝ ∈ f ⁻¹' ⋂ (i : ι), s i) ↔ x✝ ∈ ⋂ (i : ι), f ⁻¹' s i", "tactic": "simp" } ]

[ 1731, 12 ]

[ 1730, 1 ]

[]

[ 70, 6 ]

[ 69, 11 ]

[]

[ 77, 50 ]

[ 75, 1 ]

[ { "state_after": "no goals", "state_before": "F : Type ?u.124190\nα : Type u_1\nβ : Type ?u.124196\nγ : Type ?u.124199\ninst✝ : Group α\ns t : Set α\na b : α\n⊢ (fun b => a⁻¹ * b) '' t = (fun b => a * b) ⁻¹' t", "tactic": "simp" } ]

[ 1211, 86 ]

[ 1211, 1 ]

[ { "state_after": "⊢ 0 * 0 = 0", "state_before": "⊢ Principal (fun x x_1 => x * x_1) 1", "tactic": "rw [principal_one_iff]" }, { "state_after": "no goals", "state_before": "⊢ 0 * 0 = 0", "tactic": "exact zero_mul _" } ]

[ 283, 19 ]

[ 281, 1 ]

[]

[ 400, 21 ]

[ 399, 1 ]

[]

[ 147, 20 ]

[ 143, 9 ]

[]

[ 159, 4 ]

[ 158, 1 ]

[]

[ 209, 34 ]

[ 208, 1 ]

[ { "state_after": "no goals", "state_before": "a b : Int\ne : a = ↑(natAbs a)\n⊢ ↑(natAbs a) ∣ b ↔ a ∣ b", "tactic": "rw [← e]" }, { "state_after": "no goals", "state_before": "a b : Int\ne : a = -↑(natAbs a)\n⊢ ↑(natAbs a) ∣ b ↔ a ∣ b", "tactic": "rw [← Int.neg_dvd, ← e]" } ]

[ 654, 41 ]

[ 651, 1 ]

[]

[ 771, 30 ]

[ 770, 1 ]

[]

[ 1077, 19 ]

[ 1075, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type ?u.7567\nf g : Ultrafilter α\ns t : Set α\np q : α → Prop\n⊢ sᶜ ∈ f ↔ ¬s ∈ f", "tactic": "rw [← compl_not_mem_iff, compl_compl]" } ]

[ 137, 91 ]

[ 137, 1 ]

[ { "state_after": "M : Type u_1\ninst✝ : Monoid M\nu : Mˣ\nx : M\n⊢ ↑u * x = ↑u * x * ↑u⁻¹ * ↑u", "state_before": "M : Type u_1\ninst✝ : Monoid M\nu : Mˣ\nx : M\n⊢ SemiconjBy (↑u) x (↑u * x * ↑u⁻¹)", "tactic": "unfold SemiconjBy" }, { "state_after": "no goals", "state_before": "M : Type u_1\ninst✝ : Monoid M\nu : Mˣ\nx : M\n⊢ ↑u * x = ↑u * x * ↑u⁻¹ * ↑u", "tactic": "rw [Units.inv_mul_cancel_right]" } ]

[ 249, 53 ]

[ 248, 1 ]

[]

[ 1480, 38 ]

[ 1479, 1 ]

[ { "state_after": "no goals", "state_before": "u : Lean.Level\nR : Type u_1\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝ : CommSemiring R\na : R\n⊢ a ^ 1 = a", "tactic": "simp" } ]

[ 659, 55 ]

[ 659, 1 ]

[]

[ 163, 6 ]

[ 162, 1 ]

[]

[ 122, 6 ]

[ 120, 1 ]

[ { "state_after": "X : Type u_1\nY : Type ?u.15129\ninst✝ : UniformSpace X\ns : Set X\nh : IsSeqCompact s\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\n⊢ ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : X) (_ : y ∈ t), {x | (x, y) ∈ V}", "state_before": "X : Type u_1\nY : Type ?u.15129\ninst✝ : UniformSpace X\ns : Set X\nh : IsSeqCompact s\n⊢ TotallyBounded s", "tactic": "intro V V_in" }, { "state_after": "X : Type u_1\nY : Type ?u.15129\ninst✝ : UniformSpace X\ns : Set X\nh : ∀ ⦃x : ℕ → X⦄, (∀ (n : ℕ), x n ∈ s) → ∃ a, a ∈ s ∧ ∃ φ, StrictMono φ ∧ Tendsto (x ∘ φ) atTop (𝓝 a)\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\n⊢ ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : X) (_ : y ∈ t), {x | (x, y) ∈ V}", "state_before": "X : Type u_1\nY : Type ?u.15129\ninst✝ : UniformSpace X\ns : Set X\nh : IsSeqCompact s\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\n⊢ ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : X) (_ : y ∈ t), {x | (x, y) ∈ V}", "tactic": "unfold IsSeqCompact at h" }, { "state_after": "X : Type u_1\nY : Type ?u.15129\ninst✝ : UniformSpace X\ns : Set X\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\nh : ∀ (t : Set X), Set.Finite t → ¬s ⊆ ⋃ (y : X) (_ : y ∈ t), {x | (x, y) ∈ V}\n⊢ Exists fun ⦃x⦄ => (∀ (n : ℕ), x n ∈ s) ∧ ∀ (a : X), a ∈ s → ∀ (φ : ℕ → ℕ), StrictMono φ → ¬Tendsto (x ∘ φ) atTop (𝓝 a)", "state_before": "X : Type u_1\nY : Type ?u.15129\ninst✝ : UniformSpace X\ns : Set X\nh : ∀ ⦃x : ℕ → X⦄, (∀ (n : ℕ), x n ∈ s) → ∃ a, a ∈ s ∧ ∃ φ, StrictMono φ ∧ Tendsto (x ∘ φ) atTop (𝓝 a)\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\n⊢ ∃ t, Set.Finite t ∧ s ⊆ ⋃ (y : X) (_ : y ∈ t), {x | (x, y) ∈ V}", "tactic": "contrapose! h" }, { "state_after": "case intro.intro\nX : Type u_1\nY : Type ?u.15129\ninst✝ : UniformSpace X\ns : Set X\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\nh : ∀ (t : Set X), Set.Finite t → ¬s ⊆ ⋃ (y : X) (_ : y ∈ t), {x | (x, y) ∈ V}\nu : ℕ → X\nu_in : ∀ (n : ℕ), u n ∈ s\nhu : ∀ (n m : ℕ), m < n → ¬u m ∈ ball (u n) V\n⊢ Exists fun ⦃x⦄ => (∀ (n : ℕ), x n ∈ s) ∧ ∀ (a : X), a ∈ s → ∀ (φ : ℕ → ℕ), StrictMono φ → ¬Tendsto (x ∘ φ) atTop (𝓝 a)", "state_before": "X : Type u_1\nY : Type ?u.15129\ninst✝ : UniformSpace X\ns : Set X\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\nh : ∀ (t : Set X), Set.Finite t → ¬s ⊆ ⋃ (y : X) (_ : y ∈ t), {x | (x, y) ∈ V}\n⊢ Exists fun ⦃x⦄ => (∀ (n : ℕ), x n ∈ s) ∧ ∀ (a : X), a ∈ s → ∀ (φ : ℕ → ℕ), StrictMono φ → ¬Tendsto (x ∘ φ) atTop (𝓝 a)", "tactic": "obtain ⟨u, u_in, hu⟩ : ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ ∀ n m, m < n → u m ∉ ball (u n) V := by\n simp only [not_subset, mem_iUnion₂, not_exists, exists_prop] at h\n simpa only [forall_and, ball_image_iff, not_and] using seq_of_forall_finite_exists h" }, { "state_after": "case intro.intro\nX : Type u_1\nY : Type ?u.15129\ninst✝ : UniformSpace X\ns : Set X\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\nh : ∀ (t : Set X), Set.Finite t → ¬s ⊆ ⋃ (y : X) (_ : y ∈ t), {x | (x, y) ∈ V}\nu : ℕ → X\nu_in : ∀ (n : ℕ), u n ∈ s\nhu : ∀ (n m : ℕ), m < n → ¬u m ∈ ball (u n) V\nx : X\nx✝ : x ∈ s\nφ : ℕ → ℕ\nhφ : StrictMono φ\nhuφ : Tendsto (u ∘ φ) atTop (𝓝 x)\n⊢ False", "state_before": "case intro.intro\nX : Type u_1\nY : Type ?u.15129\ninst✝ : UniformSpace X\ns : Set X\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\nh : ∀ (t : Set X), Set.Finite t → ¬s ⊆ ⋃ (y : X) (_ : y ∈ t), {x | (x, y) ∈ V}\nu : ℕ → X\nu_in : ∀ (n : ℕ), u n ∈ s\nhu : ∀ (n m : ℕ), m < n → ¬u m ∈ ball (u n) V\n⊢ Exists fun ⦃x⦄ => (∀ (n : ℕ), x n ∈ s) ∧ ∀ (a : X), a ∈ s → ∀ (φ : ℕ → ℕ), StrictMono φ → ¬Tendsto (x ∘ φ) atTop (𝓝 a)", "tactic": "refine' ⟨u, u_in, fun x _ φ hφ huφ => _⟩" }, { "state_after": "X : Type u_1\nY : Type ?u.15129\ninst✝ : UniformSpace X\ns : Set X\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\nh : ∀ (t : Set X), Set.Finite t → ¬s ⊆ ⋃ (y : X) (_ : y ∈ t), {x | (x, y) ∈ V}\nu : ℕ → X\nu_in : ∀ (n : ℕ), u n ∈ s\nhu : ∀ (n m : ℕ), m < n → ¬u m ∈ ball (u n) V\nx : X\nx✝ : x ∈ s\nφ : ℕ → ℕ\nhφ : StrictMono φ\nhuφ : Tendsto (u ∘ φ) atTop (𝓝 x)\n⊢ ∃ N, ∀ (p q : ℕ), p ≥ N → q ≥ N → (u (φ p), u (φ q)) ∈ V\n\ncase intro.intro.intro\nX : Type u_1\nY : Type ?u.15129\ninst✝ : UniformSpace X\ns : Set X\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\nh : ∀ (t : Set X), Set.Finite t → ¬s ⊆ ⋃ (y : X) (_ : y ∈ t), {x | (x, y) ∈ V}\nu : ℕ → X\nu_in : ∀ (n : ℕ), u n ∈ s\nhu : ∀ (n m : ℕ), m < n → ¬u m ∈ ball (u n) V\nx : X\nx✝ : x ∈ s\nφ : ℕ → ℕ\nhφ : StrictMono φ\nhuφ : Tendsto (u ∘ φ) atTop (𝓝 x)\nN : ℕ\nhN : ∀ (p q : ℕ), p ≥ N → q ≥ N → (u (φ p), u (φ q)) ∈ V\n⊢ False", "state_before": "case intro.intro\nX : Type u_1\nY : Type ?u.15129\ninst✝ : UniformSpace X\ns : Set X\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\nh : ∀ (t : Set X), Set.Finite t → ¬s ⊆ ⋃ (y : X) (_ : y ∈ t), {x | (x, y) ∈ V}\nu : ℕ → X\nu_in : ∀ (n : ℕ), u n ∈ s\nhu : ∀ (n m : ℕ), m < n → ¬u m ∈ ball (u n) V\nx : X\nx✝ : x ∈ s\nφ : ℕ → ℕ\nhφ : StrictMono φ\nhuφ : Tendsto (u ∘ φ) atTop (𝓝 x)\n⊢ False", "tactic": "obtain ⟨N, hN⟩ : ∃ N, ∀ p q, p ≥ N → q ≥ N → (u (φ p), u (φ q)) ∈ V" }, { "state_after": "case intro.intro.intro\nX : Type u_1\nY : Type ?u.15129\ninst✝ : UniformSpace X\ns : Set X\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\nh : ∀ (t : Set X), Set.Finite t → ¬s ⊆ ⋃ (y : X) (_ : y ∈ t), {x | (x, y) ∈ V}\nu : ℕ → X\nu_in : ∀ (n : ℕ), u n ∈ s\nhu : ∀ (n m : ℕ), m < n → ¬u m ∈ ball (u n) V\nx : X\nx✝ : x ∈ s\nφ : ℕ → ℕ\nhφ : StrictMono φ\nhuφ : Tendsto (u ∘ φ) atTop (𝓝 x)\nN : ℕ\nhN : ∀ (p q : ℕ), p ≥ N → q ≥ N → (u (φ p), u (φ q)) ∈ V\n⊢ False", "state_before": "X : Type u_1\nY : Type ?u.15129\ninst✝ : UniformSpace X\ns : Set X\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\nh : ∀ (t : Set X), Set.Finite t → ¬s ⊆ ⋃ (y : X) (_ : y ∈ t), {x | (x, y) ∈ V}\nu : ℕ → X\nu_in : ∀ (n : ℕ), u n ∈ s\nhu : ∀ (n m : ℕ), m < n → ¬u m ∈ ball (u n) V\nx : X\nx✝ : x ∈ s\nφ : ℕ → ℕ\nhφ : StrictMono φ\nhuφ : Tendsto (u ∘ φ) atTop (𝓝 x)\n⊢ ∃ N, ∀ (p q : ℕ), p ≥ N → q ≥ N → (u (φ p), u (φ q)) ∈ V\n\ncase intro.intro.intro\nX : Type u_1\nY : Type ?u.15129\ninst✝ : UniformSpace X\ns : Set X\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\nh : ∀ (t : Set X), Set.Finite t → ¬s ⊆ ⋃ (y : X) (_ : y ∈ t), {x | (x, y) ∈ V}\nu : ℕ → X\nu_in : ∀ (n : ℕ), u n ∈ s\nhu : ∀ (n m : ℕ), m < n → ¬u m ∈ ball (u n) V\nx : X\nx✝ : x ∈ s\nφ : ℕ → ℕ\nhφ : StrictMono φ\nhuφ : Tendsto (u ∘ φ) atTop (𝓝 x)\nN : ℕ\nhN : ∀ (p q : ℕ), p ≥ N → q ≥ N → (u (φ p), u (φ q)) ∈ V\n⊢ False", "tactic": "exact huφ.cauchySeq.mem_entourage V_in" }, { "state_after": "no goals", "state_before": "case intro.intro.intro\nX : Type u_1\nY : Type ?u.15129\ninst✝ : UniformSpace X\ns : Set X\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\nh : ∀ (t : Set X), Set.Finite t → ¬s ⊆ ⋃ (y : X) (_ : y ∈ t), {x | (x, y) ∈ V}\nu : ℕ → X\nu_in : ∀ (n : ℕ), u n ∈ s\nhu : ∀ (n m : ℕ), m < n → ¬u m ∈ ball (u n) V\nx : X\nx✝ : x ∈ s\nφ : ℕ → ℕ\nhφ : StrictMono φ\nhuφ : Tendsto (u ∘ φ) atTop (𝓝 x)\nN : ℕ\nhN : ∀ (p q : ℕ), p ≥ N → q ≥ N → (u (φ p), u (φ q)) ∈ V\n⊢ False", "tactic": "exact hu (φ <| N + 1) (φ N) (hφ <| lt_add_one N) (hN (N + 1) N N.le_succ le_rfl)" }, { "state_after": "X : Type u_1\nY : Type ?u.15129\ninst✝ : UniformSpace X\ns : Set X\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\nh : ∀ (t : Set X), Set.Finite t → ∃ a, a ∈ s ∧ ∀ (x : X), ¬(x ∈ t ∧ a ∈ {x_1 | (x_1, x) ∈ V})\n⊢ ∃ u, (∀ (n : ℕ), u n ∈ s) ∧ ∀ (n m : ℕ), m < n → ¬u m ∈ ball (u n) V", "state_before": "X : Type u_1\nY : Type ?u.15129\ninst✝ : UniformSpace X\ns : Set X\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\nh : ∀ (t : Set X), Set.Finite t → ¬s ⊆ ⋃ (y : X) (_ : y ∈ t), {x | (x, y) ∈ V}\n⊢ ∃ u, (∀ (n : ℕ), u n ∈ s) ∧ ∀ (n m : ℕ), m < n → ¬u m ∈ ball (u n) V", "tactic": "simp only [not_subset, mem_iUnion₂, not_exists, exists_prop] at h" }, { "state_after": "no goals", "state_before": "X : Type u_1\nY : Type ?u.15129\ninst✝ : UniformSpace X\ns : Set X\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\nh : ∀ (t : Set X), Set.Finite t → ∃ a, a ∈ s ∧ ∀ (x : X), ¬(x ∈ t ∧ a ∈ {x_1 | (x_1, x) ∈ V})\n⊢ ∃ u, (∀ (n : ℕ), u n ∈ s) ∧ ∀ (n m : ℕ), m < n → ¬u m ∈ ball (u n) V", "tactic": "simpa only [forall_and, ball_image_iff, not_and] using seq_of_forall_finite_exists h" } ]

[ 350, 83 ]

[ 340, 11 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.300349\nγ : Type ?u.300352\np : α → Prop\ninst✝ : DecidablePred p\ns : Multiset α\nn : ℕ\n⊢ countp p (n • s) = n * countp p s", "tactic": "induction n <;> simp [*, succ_nsmul', succ_mul, zero_nsmul]" } ]

[ 2240, 62 ]

[ 2239, 1 ]

[]

[ 239, 11 ]

[ 237, 1 ]

[]

[ 1260, 44 ]

[ 1259, 1 ]

[]

[ 592, 10 ]

[ 591, 11 ]

[]

[ 319, 6 ]

[ 318, 1 ]

[ { "state_after": "G : Type u_1\nH : Type u_2\ninst✝¹ : Mul G\ninst✝ : Mul H\nA✝ B✝ : Finset G\na0✝ b0✝ : G\nf : G →ₙ* H\nhf : Function.Injective ↑f\na0 b0 : G\nA B : Finset H\nu : UniqueMul A B (↑f a0) (↑f b0)\na b : G\nha : a ∈ Finset.preimage A ↑f (_ : Set.InjOn (↑f) (↑f ⁻¹' ↑A))\nhb : b ∈ Finset.preimage B ↑f (_ : Set.InjOn (↑f) (↑f ⁻¹' ↑B))\nab : a * b = a0 * b0\n⊢ a = a0 ∧ b = b0", "state_before": "G : Type u_1\nH : Type u_2\ninst✝¹ : Mul G\ninst✝ : Mul H\nA✝ B✝ : Finset G\na0✝ b0✝ : G\nf : G →ₙ* H\nhf : Function.Injective ↑f\na0 b0 : G\nA B : Finset H\nu : UniqueMul A B (↑f a0) (↑f b0)\n⊢ UniqueMul (Finset.preimage A ↑f (_ : Set.InjOn (↑f) (↑f ⁻¹' ↑A)))\n (Finset.preimage B ↑f (_ : Set.InjOn (↑f) (↑f ⁻¹' ↑B))) a0 b0", "tactic": "intro a b ha hb ab" }, { "state_after": "G : Type u_1\nH : Type u_2\ninst✝¹ : Mul G\ninst✝ : Mul H\nA✝ B✝ : Finset G\na0✝ b0✝ : G\nf : G →ₙ* H\nhf : Function.Injective ↑f\na0 b0 : G\nA B : Finset H\nu : UniqueMul A B (↑f a0) (↑f b0)\na b : G\nha : a ∈ Finset.preimage A ↑f (_ : Set.InjOn (↑f) (↑f ⁻¹' ↑A))\nhb : b ∈ Finset.preimage B ↑f (_ : Set.InjOn (↑f) (↑f ⁻¹' ↑B))\nab : a * b = a0 * b0\n⊢ ↑f a = ↑f a0 ∧ ↑f b = ↑f b0", "state_before": "G : Type u_1\nH : Type u_2\ninst✝¹ : Mul G\ninst✝ : Mul H\nA✝ B✝ : Finset G\na0✝ b0✝ : G\nf : G →ₙ* H\nhf : Function.Injective ↑f\na0 b0 : G\nA B : Finset H\nu : UniqueMul A B (↑f a0) (↑f b0)\na b : G\nha : a ∈ Finset.preimage A ↑f (_ : Set.InjOn (↑f) (↑f ⁻¹' ↑A))\nhb : b ∈ Finset.preimage B ↑f (_ : Set.InjOn (↑f) (↑f ⁻¹' ↑B))\nab : a * b = a0 * b0\n⊢ a = a0 ∧ b = b0", "tactic": "rw [← hf.eq_iff, ← hf.eq_iff]" }, { "state_after": "G : Type u_1\nH : Type u_2\ninst✝¹ : Mul G\ninst✝ : Mul H\nA✝ B✝ : Finset G\na0✝ b0✝ : G\nf : G →ₙ* H\nhf : Function.Injective ↑f\na0 b0 : G\nA B : Finset H\nu : UniqueMul A B (↑f a0) (↑f b0)\na b : G\nha : a ∈ Finset.preimage A ↑f (_ : Set.InjOn (↑f) (↑f ⁻¹' ↑A))\nhb : b ∈ Finset.preimage B ↑f (_ : Set.InjOn (↑f) (↑f ⁻¹' ↑B))\nab : ↑f a * ↑f b = ↑f a0 * ↑f b0\n⊢ ↑f a = ↑f a0 ∧ ↑f b = ↑f b0", "state_before": "G : Type u_1\nH : Type u_2\ninst✝¹ : Mul G\ninst✝ : Mul H\nA✝ B✝ : Finset G\na0✝ b0✝ : G\nf : G →ₙ* H\nhf : Function.Injective ↑f\na0 b0 : G\nA B : Finset H\nu : UniqueMul A B (↑f a0) (↑f b0)\na b : G\nha : a ∈ Finset.preimage A ↑f (_ : Set.InjOn (↑f) (↑f ⁻¹' ↑A))\nhb : b ∈ Finset.preimage B ↑f (_ : Set.InjOn (↑f) (↑f ⁻¹' ↑B))\nab : a * b = a0 * b0\n⊢ ↑f a = ↑f a0 ∧ ↑f b = ↑f b0", "tactic": "rw [← hf.eq_iff, map_mul, map_mul] at ab" }, { "state_after": "no goals", "state_before": "G : Type u_1\nH : Type u_2\ninst✝¹ : Mul G\ninst✝ : Mul H\nA✝ B✝ : Finset G\na0✝ b0✝ : G\nf : G →ₙ* H\nhf : Function.Injective ↑f\na0 b0 : G\nA B : Finset H\nu : UniqueMul A B (↑f a0) (↑f b0)\na b : G\nha : a ∈ Finset.preimage A ↑f (_ : Set.InjOn (↑f) (↑f ⁻¹' ↑A))\nhb : b ∈ Finset.preimage B ↑f (_ : Set.InjOn (↑f) (↑f ⁻¹' ↑B))\nab : ↑f a * ↑f b = ↑f a0 * ↑f b0\n⊢ ↑f a = ↑f a0 ∧ ↑f b = ↑f b0", "tactic": "exact u (Finset.mem_preimage.mp ha) (Finset.mem_preimage.mp hb) ab" } ]

[ 120, 69 ]

[ 113, 1 ]

[]

[ 89, 6 ]

[ 88, 1 ]

[ { "state_after": "no goals", "state_before": "ι✝ : Type ?u.46874\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι✝ → Type ?u.46885\nr : α✝ → α✝ → Prop\nι : Type u\nα : ι → Type v\ninst✝ : (i : ι) → Preorder (α i)\nx y : (i : ι) → α i\n⊢ x < y ↔ x ≤ y ∧ ∃ i, x i < y i", "tactic": "simp (config := { contextual := true }) [lt_iff_le_not_le, Pi.le_def]" } ]

[ 809, 72 ]

[ 807, 1 ]

[ { "state_after": "case pos\nA : Type u_3\nB : Type u_1\nB' : Type ?u.27580\ninst✝⁷ : CommRing A\ninst✝⁶ : Ring B\ninst✝⁵ : Ring B'\ninst✝⁴ : Algebra A B\ninst✝³ : Algebra A B'\nx : B\ninst✝² : Nontrivial B\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\nf : A →+* R\nhx : IsIntegral A x\n⊢ map f (minpoly A x) ≠ 1\n\ncase neg\nA : Type u_3\nB : Type u_1\nB' : Type ?u.27580\ninst✝⁷ : CommRing A\ninst✝⁶ : Ring B\ninst✝⁵ : Ring B'\ninst✝⁴ : Algebra A B\ninst✝³ : Algebra A B'\nx : B\ninst✝² : Nontrivial B\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\nf : A →+* R\nhx : ¬IsIntegral A x\n⊢ map f (minpoly A x) ≠ 1", "state_before": "A : Type u_3\nB : Type u_1\nB' : Type ?u.27580\ninst✝⁷ : CommRing A\ninst✝⁶ : Ring B\ninst✝⁵ : Ring B'\ninst✝⁴ : Algebra A B\ninst✝³ : Algebra A B'\nx : B\ninst✝² : Nontrivial B\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\nf : A →+* R\n⊢ map f (minpoly A x) ≠ 1", "tactic": "by_cases hx : IsIntegral A x" }, { "state_after": "no goals", "state_before": "case pos\nA : Type u_3\nB : Type u_1\nB' : Type ?u.27580\ninst✝⁷ : CommRing A\ninst✝⁶ : Ring B\ninst✝⁵ : Ring B'\ninst✝⁴ : Algebra A B\ninst✝³ : Algebra A B'\nx : B\ninst✝² : Nontrivial B\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\nf : A →+* R\nhx : IsIntegral A x\n⊢ map f (minpoly A x) ≠ 1", "tactic": "exact mt ((monic hx).eq_one_of_map_eq_one f) (ne_one A x)" }, { "state_after": "case neg\nA : Type u_3\nB : Type u_1\nB' : Type ?u.27580\ninst✝⁷ : CommRing A\ninst✝⁶ : Ring B\ninst✝⁵ : Ring B'\ninst✝⁴ : Algebra A B\ninst✝³ : Algebra A B'\nx : B\ninst✝² : Nontrivial B\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\nf : A →+* R\nhx : ¬IsIntegral A x\n⊢ 0 ≠ 1", "state_before": "case neg\nA : Type u_3\nB : Type u_1\nB' : Type ?u.27580\ninst✝⁷ : CommRing A\ninst✝⁶ : Ring B\ninst✝⁵ : Ring B'\ninst✝⁴ : Algebra A B\ninst✝³ : Algebra A B'\nx : B\ninst✝² : Nontrivial B\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\nf : A →+* R\nhx : ¬IsIntegral A x\n⊢ map f (minpoly A x) ≠ 1", "tactic": "rw [eq_zero hx, Polynomial.map_zero]" }, { "state_after": "no goals", "state_before": "case neg\nA : Type u_3\nB : Type u_1\nB' : Type ?u.27580\ninst✝⁷ : CommRing A\ninst✝⁶ : Ring B\ninst✝⁵ : Ring B'\ninst✝⁴ : Algebra A B\ninst✝³ : Algebra A B'\nx : B\ninst✝² : Nontrivial B\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\nf : A →+* R\nhx : ¬IsIntegral A x\n⊢ 0 ≠ 1", "tactic": "exact zero_ne_one" } ]

[ 106, 22 ]

[ 101, 1 ]

[ { "state_after": "no goals", "state_before": "⊢ SurjOn sin (Icc (-(π / 2)) (π / 2)) (Icc (-1) 1)", "tactic": "simpa only [sin_neg, sin_pi_div_two] using\n intermediate_value_Icc (neg_le_self pi_div_two_pos.le) continuous_sin.continuousOn" } ]

[ 620, 87 ]

[ 618, 1 ]

[ { "state_after": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.59708\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : Subsingleton F\n⊢ ContDiffWithinAt 𝕜 n (fun x => 0) s x", "state_before": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.59708\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : Subsingleton F\n⊢ ContDiffWithinAt 𝕜 n f s x", "tactic": "rw [Subsingleton.elim f fun _ => 0]" }, { "state_after": "no goals", "state_before": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹⁰ : NormedAddCommGroup D\ninst✝⁹ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nX : Type ?u.59708\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ninst✝ : Subsingleton F\n⊢ ContDiffWithinAt 𝕜 n (fun x => 0) s x", "tactic": "exact contDiffWithinAt_const" } ]

[ 121, 68 ]

[ 120, 1 ]

[ { "state_after": "case w\nC : Type u\ninst✝ : Category C\nX : SimplicialObject C\nn : ℕ\nh : n = n\n⊢ (eqToIso X h).hom = (Iso.refl (X.obj [n].op)).hom", "state_before": "C : Type u\ninst✝ : Category C\nX : SimplicialObject C\nn : ℕ\nh : n = n\n⊢ eqToIso X h = Iso.refl (X.obj [n].op)", "tactic": "ext" }, { "state_after": "no goals", "state_before": "case w\nC : Type u\ninst✝ : Category C\nX : SimplicialObject C\nn : ℕ\nh : n = n\n⊢ (eqToIso X h).hom = (Iso.refl (X.obj [n].op)).hom", "tactic": "simp [eqToIso]" } ]

[ 106, 17 ]

[ 104, 1 ]

[]

[ 91, 14 ]

[ 90, 1 ]

[ { "state_after": "m n : ℕ∞\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_1\ninst✝¹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type ?u.103687\ninst✝ : TopologicalSpace M\ns : Set H\nhs : IsOpen s\n⊢ Pregroupoid.property\n {\n property := fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I),\n comp :=\n (_ :\n ∀ {f g : H → H} {u v : Set H},\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I) →\n ContDiffOn 𝕜 n (↑I ∘ g ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' v ∩ range ↑I) →\n IsOpen u →\n IsOpen v →\n IsOpen (u ∩ f ⁻¹' v) →\n (fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I))\n (g ∘ f) (u ∩ f ⁻¹' v)),\n id_mem :=\n (_ : ContDiffOn 𝕜 n (↑I ∘ id ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' univ ∩ range ↑I)),\n locality :=\n (_ :\n ∀ {f : H → H} {u : Set H},\n IsOpen u →\n (∀ (x : H),\n x ∈ u →\n ∃ v,\n IsOpen v ∧\n x ∈ v ∧\n (fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I))\n f (u ∩ v)) →\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I)),\n congr :=\n (_ :\n ∀ {f g : H → H} {u : Set H},\n IsOpen u →\n (∀ (x : H), x ∈ u → g x = f x) →\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I) →\n ContDiffOn 𝕜 n (↑I ∘ g ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I)) }\n (↑(LocalHomeomorph.ofSet s hs)) (LocalHomeomorph.ofSet s hs).toLocalEquiv.source ∧\n Pregroupoid.property\n {\n property := fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I),\n comp :=\n (_ :\n ∀ {f g : H → H} {u v : Set H},\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I) →\n ContDiffOn 𝕜 n (↑I ∘ g ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' v ∩ range ↑I) →\n IsOpen u →\n IsOpen v →\n IsOpen (u ∩ f ⁻¹' v) →\n (fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I))\n (g ∘ f) (u ∩ f ⁻¹' v)),\n id_mem :=\n (_ : ContDiffOn 𝕜 n (↑I ∘ id ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' univ ∩ range ↑I)),\n locality :=\n (_ :\n ∀ {f : H → H} {u : Set H},\n IsOpen u →\n (∀ (x : H),\n x ∈ u →\n ∃ v,\n IsOpen v ∧\n x ∈ v ∧\n (fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I))\n f (u ∩ v)) →\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I)),\n congr :=\n (_ :\n ∀ {f g : H → H} {u : Set H},\n IsOpen u →\n (∀ (x : H), x ∈ u → g x = f x) →\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I) →\n ContDiffOn 𝕜 n (↑I ∘ g ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I)) }\n (↑(LocalHomeomorph.symm (LocalHomeomorph.ofSet s hs))) (LocalHomeomorph.ofSet s hs).toLocalEquiv.target", "state_before": "m n : ℕ∞\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_1\ninst✝¹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type ?u.103687\ninst✝ : TopologicalSpace M\ns : Set H\nhs : IsOpen s\n⊢ LocalHomeomorph.ofSet s hs ∈ contDiffGroupoid n I", "tactic": "rw [contDiffGroupoid, mem_groupoid_of_pregroupoid]" }, { "state_after": "m n : ℕ∞\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_1\ninst✝¹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type ?u.103687\ninst✝ : TopologicalSpace M\ns : Set H\nhs : IsOpen s\nh : ContDiffOn 𝕜 n (↑I ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I)\n⊢ Pregroupoid.property\n {\n property := fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I),\n comp :=\n (_ :\n ∀ {f g : H → H} {u v : Set H},\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I) →\n ContDiffOn 𝕜 n (↑I ∘ g ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' v ∩ range ↑I) →\n IsOpen u →\n IsOpen v →\n IsOpen (u ∩ f ⁻¹' v) →\n (fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I))\n (g ∘ f) (u ∩ f ⁻¹' v)),\n id_mem :=\n (_ : ContDiffOn 𝕜 n (↑I ∘ id ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' univ ∩ range ↑I)),\n locality :=\n (_ :\n ∀ {f : H → H} {u : Set H},\n IsOpen u →\n (∀ (x : H),\n x ∈ u →\n ∃ v,\n IsOpen v ∧\n x ∈ v ∧\n (fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I))\n f (u ∩ v)) →\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I)),\n congr :=\n (_ :\n ∀ {f g : H → H} {u : Set H},\n IsOpen u →\n (∀ (x : H), x ∈ u → g x = f x) →\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I) →\n ContDiffOn 𝕜 n (↑I ∘ g ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I)) }\n (↑(LocalHomeomorph.ofSet s hs)) (LocalHomeomorph.ofSet s hs).toLocalEquiv.source ∧\n Pregroupoid.property\n {\n property := fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I),\n comp :=\n (_ :\n ∀ {f g : H → H} {u v : Set H},\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I) →\n ContDiffOn 𝕜 n (↑I ∘ g ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' v ∩ range ↑I) →\n IsOpen u →\n IsOpen v →\n IsOpen (u ∩ f ⁻¹' v) →\n (fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I))\n (g ∘ f) (u ∩ f ⁻¹' v)),\n id_mem :=\n (_ : ContDiffOn 𝕜 n (↑I ∘ id ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' univ ∩ range ↑I)),\n locality :=\n (_ :\n ∀ {f : H → H} {u : Set H},\n IsOpen u →\n (∀ (x : H),\n x ∈ u →\n ∃ v,\n IsOpen v ∧\n x ∈ v ∧\n (fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I))\n f (u ∩ v)) →\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I)),\n congr :=\n (_ :\n ∀ {f g : H → H} {u : Set H},\n IsOpen u →\n (∀ (x : H), x ∈ u → g x = f x) →\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I) →\n ContDiffOn 𝕜 n (↑I ∘ g ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I)) }\n (↑(LocalHomeomorph.symm (LocalHomeomorph.ofSet s hs))) (LocalHomeomorph.ofSet s hs).toLocalEquiv.target\n\ncase h\nm n : ℕ∞\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_1\ninst✝¹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type ?u.103687\ninst✝ : TopologicalSpace M\ns : Set H\nhs : IsOpen s\n⊢ ContDiffOn 𝕜 n (↑I ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I)", "state_before": "m n : ℕ∞\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_1\ninst✝¹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type ?u.103687\ninst✝ : TopologicalSpace M\ns : Set H\nhs : IsOpen s\n⊢ Pregroupoid.property\n {\n property := fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I),\n comp :=\n (_ :\n ∀ {f g : H → H} {u v : Set H},\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I) →\n ContDiffOn 𝕜 n (↑I ∘ g ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' v ∩ range ↑I) →\n IsOpen u →\n IsOpen v →\n IsOpen (u ∩ f ⁻¹' v) →\n (fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I))\n (g ∘ f) (u ∩ f ⁻¹' v)),\n id_mem :=\n (_ : ContDiffOn 𝕜 n (↑I ∘ id ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' univ ∩ range ↑I)),\n locality :=\n (_ :\n ∀ {f : H → H} {u : Set H},\n IsOpen u →\n (∀ (x : H),\n x ∈ u →\n ∃ v,\n IsOpen v ∧\n x ∈ v ∧\n (fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I))\n f (u ∩ v)) →\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I)),\n congr :=\n (_ :\n ∀ {f g : H → H} {u : Set H},\n IsOpen u →\n (∀ (x : H), x ∈ u → g x = f x) →\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I) →\n ContDiffOn 𝕜 n (↑I ∘ g ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I)) }\n (↑(LocalHomeomorph.ofSet s hs)) (LocalHomeomorph.ofSet s hs).toLocalEquiv.source ∧\n Pregroupoid.property\n {\n property := fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I),\n comp :=\n (_ :\n ∀ {f g : H → H} {u v : Set H},\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I) →\n ContDiffOn 𝕜 n (↑I ∘ g ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' v ∩ range ↑I) →\n IsOpen u →\n IsOpen v →\n IsOpen (u ∩ f ⁻¹' v) →\n (fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I))\n (g ∘ f) (u ∩ f ⁻¹' v)),\n id_mem :=\n (_ : ContDiffOn 𝕜 n (↑I ∘ id ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' univ ∩ range ↑I)),\n locality :=\n (_ :\n ∀ {f : H → H} {u : Set H},\n IsOpen u →\n (∀ (x : H),\n x ∈ u →\n ∃ v,\n IsOpen v ∧\n x ∈ v ∧\n (fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I))\n f (u ∩ v)) →\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I)),\n congr :=\n (_ :\n ∀ {f g : H → H} {u : Set H},\n IsOpen u →\n (∀ (x : H), x ∈ u → g x = f x) →\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I) →\n ContDiffOn 𝕜 n (↑I ∘ g ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I)) }\n (↑(LocalHomeomorph.symm (LocalHomeomorph.ofSet s hs))) (LocalHomeomorph.ofSet s hs).toLocalEquiv.target", "tactic": "suffices h : ContDiffOn 𝕜 n (I ∘ I.symm) (I.symm ⁻¹' s ∩ range I)" }, { "state_after": "case h\nm n : ℕ∞\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_1\ninst✝¹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type ?u.103687\ninst✝ : TopologicalSpace M\ns : Set H\nhs : IsOpen s\nthis : ContDiffOn 𝕜 n id univ\n⊢ ContDiffOn 𝕜 n (↑I ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I)", "state_before": "case h\nm n : ℕ∞\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_1\ninst✝¹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type ?u.103687\ninst✝ : TopologicalSpace M\ns : Set H\nhs : IsOpen s\n⊢ ContDiffOn 𝕜 n (↑I ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I)", "tactic": "have : ContDiffOn 𝕜 n id (univ : Set E) := contDiff_id.contDiffOn" }, { "state_after": "no goals", "state_before": "case h\nm n : ℕ∞\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_1\ninst✝¹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type ?u.103687\ninst✝ : TopologicalSpace M\ns : Set H\nhs : IsOpen s\nthis : ContDiffOn 𝕜 n id univ\n⊢ ContDiffOn 𝕜 n (↑I ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I)", "tactic": "exact this.congr_mono (fun x hx => I.right_inv hx.2) (subset_univ _)" }, { "state_after": "no goals", "state_before": "m n : ℕ∞\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_1\ninst✝¹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type ?u.103687\ninst✝ : TopologicalSpace M\ns : Set H\nhs : IsOpen s\nh : ContDiffOn 𝕜 n (↑I ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I)\n⊢ Pregroupoid.property\n {\n property := fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I),\n comp :=\n (_ :\n ∀ {f g : H → H} {u v : Set H},\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I) →\n ContDiffOn 𝕜 n (↑I ∘ g ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' v ∩ range ↑I) →\n IsOpen u →\n IsOpen v →\n IsOpen (u ∩ f ⁻¹' v) →\n (fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I))\n (g ∘ f) (u ∩ f ⁻¹' v)),\n id_mem :=\n (_ : ContDiffOn 𝕜 n (↑I ∘ id ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' univ ∩ range ↑I)),\n locality :=\n (_ :\n ∀ {f : H → H} {u : Set H},\n IsOpen u →\n (∀ (x : H),\n x ∈ u →\n ∃ v,\n IsOpen v ∧\n x ∈ v ∧\n (fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I))\n f (u ∩ v)) →\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I)),\n congr :=\n (_ :\n ∀ {f g : H → H} {u : Set H},\n IsOpen u →\n (∀ (x : H), x ∈ u → g x = f x) →\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I) →\n ContDiffOn 𝕜 n (↑I ∘ g ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I)) }\n (↑(LocalHomeomorph.ofSet s hs)) (LocalHomeomorph.ofSet s hs).toLocalEquiv.source ∧\n Pregroupoid.property\n {\n property := fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I),\n comp :=\n (_ :\n ∀ {f g : H → H} {u v : Set H},\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I) →\n ContDiffOn 𝕜 n (↑I ∘ g ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' v ∩ range ↑I) →\n IsOpen u →\n IsOpen v →\n IsOpen (u ∩ f ⁻¹' v) →\n (fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I))\n (g ∘ f) (u ∩ f ⁻¹' v)),\n id_mem :=\n (_ : ContDiffOn 𝕜 n (↑I ∘ id ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' univ ∩ range ↑I)),\n locality :=\n (_ :\n ∀ {f : H → H} {u : Set H},\n IsOpen u →\n (∀ (x : H),\n x ∈ u →\n ∃ v,\n IsOpen v ∧\n x ∈ v ∧\n (fun f s =>\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I))\n f (u ∩ v)) →\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I)),\n congr :=\n (_ :\n ∀ {f g : H → H} {u : Set H},\n IsOpen u →\n (∀ (x : H), x ∈ u → g x = f x) →\n ContDiffOn 𝕜 n (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I) →\n ContDiffOn 𝕜 n (↑I ∘ g ∘ ↑(ModelWithCorners.symm I))\n (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I)) }\n (↑(LocalHomeomorph.symm (LocalHomeomorph.ofSet s hs))) (LocalHomeomorph.ofSet s hs).toLocalEquiv.target", "tactic": "simp [h]" } ]

[ 595, 71 ]

[ 589, 1 ]

[ { "state_after": "no goals", "state_before": "n k : ℕ\n⊢ 1 * (k / 2 ^ n) + k % 2 ^ n = k / 2 ^ n + k % 2 ^ n", "tactic": "rw [one_mul]" } ]

[ 601, 49 ]

[ 595, 1 ]

[]

[ 344, 38 ]

[ 343, 1 ]

[]

[ 2193, 30 ]

[ 2192, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nas : List α\nbs : Array α\n⊢ Array.size (toArrayAux as bs) = length as + Array.size bs", "tactic": "induction as generalizing bs with\n| nil => simp [toArrayAux]\n| cons a as ih => simp_arith [toArrayAux, *]" }, { "state_after": "no goals", "state_before": "case nil\nα : Type u_1\nbs : Array α\n⊢ Array.size (toArrayAux [] bs) = length [] + Array.size bs", "tactic": "simp [toArrayAux]" }, { "state_after": "no goals", "state_before": "case cons\nα : Type u_1\na : α\nas : List α\nih : ∀ (bs : Array α), Array.size (toArrayAux as bs) = length as + Array.size bs\nbs : Array α\n⊢ Array.size (toArrayAux (a :: as) bs) = length (a :: as) + Array.size bs", "tactic": "simp_arith [toArrayAux, *]" } ]

[ 20, 47 ]

[ 17, 9 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.109653\nγ : Type ?u.109656\ninst✝ : DecidableEq α\ns✝ t u v : Finset α\na✝ b✝ : α\ns : Finset α\na b : α\n⊢ ↑s = {a, b} ↔ s = {a, b}", "tactic": "rw [← coe_pair, coe_inj]" } ]

[ 1129, 27 ]

[ 1128, 1 ]

[ { "state_after": "no goals", "state_before": "V : Type u_2\nα : Type u_1\nβ : Type ?u.16625\ninst✝³ : DecidableEq α\ninst✝² : DecidableEq V\nA : Matrix V V α\ninst✝¹ : Zero α\ninst✝ : One α\nh : IsSymm A\n⊢ IsSymm (compl A)", "tactic": "simp [h]" } ]

[ 125, 26 ]

[ 124, 1 ]

[]

[ 188, 33 ]

[ 185, 11 ]

[]

[ 173, 6 ]

[ 172, 1 ]

[ { "state_after": "α : Sort u\nβ : Sort v\nγ : Sort w\nx : α\ny : (fun x => β) x\nf : α ≃ β\n⊢ ↑f x = ↑f (↑f.symm y) ↔ x = ↑f.symm y", "state_before": "α : Sort u\nβ : Sort v\nγ : Sort w\nx : α\ny : (fun x => β) x\nf : α ≃ β\n⊢ ↑f x = y ↔ x = ↑f.symm y", "tactic": "conv_lhs => rw [← apply_symm_apply f y]" }, { "state_after": "no goals", "state_before": "α : Sort u\nβ : Sort v\nγ : Sort w\nx : α\ny : (fun x => β) x\nf : α ≃ β\n⊢ ↑f x = ↑f (↑f.symm y) ↔ x = ↑f.symm y", "tactic": "rw [apply_eq_iff_eq]" } ]

[ 309, 23 ]

[ 307, 1 ]

[]

[ 763, 6 ]

[ 762, 1 ]

[]

[ 669, 6 ]

[ 668, 1 ]

[ { "state_after": "m n : ℕ\nα : Fin (n + 1) → Type u\nβ : Type v\nx : α 0\np : (j : Fin n) → α (↑(succAbove 0) j)\n⊢ p = fun j => cons x (fun j => _root_.cast (_ : α (↑(succAbove 0) j) = α (succ j)) (p j)) (↑(succAbove 0) j)", "state_before": "m n : ℕ\nα : Fin (n + 1) → Type u\nβ : Type v\nx : α 0\np : (j : Fin n) → α (↑(succAbove 0) j)\n⊢ insertNth 0 x p = cons x fun j => _root_.cast (_ : α (↑(succAbove 0) j) = α (succ j)) (p j)", "tactic": "refine' insertNth_eq_iff.2 ⟨by simp, _⟩" }, { "state_after": "case h\nm n : ℕ\nα : Fin (n + 1) → Type u\nβ : Type v\nx : α 0\np : (j : Fin n) → α (↑(succAbove 0) j)\nj : Fin n\n⊢ p j = cons x (fun j => _root_.cast (_ : α (↑(succAbove 0) j) = α (succ j)) (p j)) (↑(succAbove 0) j)", "state_before": "m n : ℕ\nα : Fin (n + 1) → Type u\nβ : Type v\nx : α 0\np : (j : Fin n) → α (↑(succAbove 0) j)\n⊢ p = fun j => cons x (fun j => _root_.cast (_ : α (↑(succAbove 0) j) = α (succ j)) (p j)) (↑(succAbove 0) j)", "tactic": "ext j" }, { "state_after": "no goals", "state_before": "case h\nm n : ℕ\nα : Fin (n + 1) → Type u\nβ : Type v\nx : α 0\np : (j : Fin n) → α (↑(succAbove 0) j)\nj : Fin n\n⊢ p j = cons x (fun j => _root_.cast (_ : α (↑(succAbove 0) j) = α (succ j)) (p j)) (↑(succAbove 0) j)", "tactic": "convert (cons_succ x p j).symm" }, { "state_after": "no goals", "state_before": "m n : ℕ\nα : Fin (n + 1) → Type u\nβ : Type v\nx : α 0\np : (j : Fin n) → α (↑(succAbove 0) j)\n⊢ cons x (fun j => _root_.cast (_ : α (↑(succAbove 0) j) = α (succ j)) (p j)) 0 = x", "tactic": "simp" } ]

[ 727, 33 ]

[ 722, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.21194\nγ : Type ?u.21197\ninst✝¹ : Fintype α\ns✝ t : Finset α\ninst✝ : DecidableEq α\ns : Finset α\na : α\n⊢ a ∈ univ ∩ s ↔ a ∈ s", "tactic": "simp" } ]

[ 275, 23 ]

[ 274, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nhead✝ : α\nl : List α\n⊢ (head✝ :: l) ×ˢ [] = []", "tactic": "simp [product_cons, product_nil]" } ]

[ 43, 50 ]

[ 41, 1 ]

[ { "state_after": "U : Type u_1\ninst✝ : Quiver U\nu v v' w : U\np : Path u v\np' : Path u v'\ne : v ⟶ w\ne' : v' ⟶ w\nh : cons p e = cons p' e'\n⊢ HEq e e'", "state_before": "U : Type u_1\ninst✝ : Quiver U\nu v v' w : U\np : Path u v\np' : Path u v'\ne : v ⟶ w\ne' : v' ⟶ w\nh : cons p e = cons p' e'\n⊢ Hom.cast (_ : v = v') (_ : w = w) e = e'", "tactic": "rw [Hom.cast_eq_iff_heq]" }, { "state_after": "no goals", "state_before": "U : Type u_1\ninst✝ : Quiver U\nu v v' w : U\np : Path u v\np' : Path u v'\ne : v ⟶ w\ne' : v' ⟶ w\nh : cons p e = cons p' e'\n⊢ HEq e e'", "tactic": "exact hom_heq_of_cons_eq_cons h" } ]

[ 148, 34 ]

[ 145, 1 ]

[ { "state_after": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime (y + x * z) x\n⊢ IsCoprime y x", "state_before": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime x (y + x * z)\n⊢ IsCoprime x y", "tactic": "rw [isCoprime_comm] at h⊢" }, { "state_after": "no goals", "state_before": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nh : IsCoprime (y + x * z) x\n⊢ IsCoprime y x", "tactic": "exact h.of_add_mul_left_left" } ]

[ 196, 31 ]

[ 194, 1 ]

[]

[ 92, 6 ]

[ 91, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.27075\nγ : Type ?u.27078\nι : Type ?u.27081\ninst✝¹ : TopologicalSpace α\ninst✝ : BaireSpace α\nT : Set (Set α)\ns : Set α\nhs : IsGδ s\nhd : Dense s\nhc : Set.Countable T\nhc' : ∀ (t : Set α), t ∈ T → IsClosed t\nhU : s ⊆ ⋃₀ T\n⊢ s ⊆ ⋃ (i : Set α) (_ : i ∈ T), i", "tactic": "rwa [← sUnion_eq_biUnion]" } ]

[ 327, 80 ]

[ 324, 1 ]

[]

[ 780, 23 ]

[ 778, 1 ]

[]

[ 179, 97 ]

[ 177, 1 ]

[]

[ 1279, 45 ]

[ 1277, 1 ]

[ { "state_after": "no goals", "state_before": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi✝ : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁹ : Fintype ι\ninst✝¹⁸ : Fintype ι'\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁵ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹⁴ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹³ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝¹² : (i : ι') → NormedAddCommGroup (E' i)\ninst✝¹¹ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝¹⁰ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei✝ i)\ninst✝⁹ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei✝ i)\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedSpace 𝕜 G\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : NormedSpace 𝕜 G'\nc : 𝕜\nf✝ g : ContinuousMultilinearMap 𝕜 E G\nm✝ : (i : ι) → E i\n𝕜' : Type ?u.527868\ninst✝⁴ : NormedField 𝕜'\ninst✝³ : NormedSpace 𝕜' G\ninst✝² : SMulCommClass 𝕜 𝕜' G\nEi : Fin n → Type u_1\ninst✝¹ : (i : Fin n) → NormedAddCommGroup (Ei i)\ninst✝ : (i : Fin n) → NormedSpace 𝕜 (Ei i)\nf : ContinuousMultilinearMap 𝕜 Ei G\nm : (i : Fin n) → Ei i\nb : ℝ\nhm : ‖m‖ ≤ b\n⊢ ‖↑f m‖ ≤ ‖f‖ * b ^ n", "tactic": "simpa only [Fintype.card_fin] using\n f.le_op_norm_mul_pow_card_of_le m fun i => (norm_le_pi_norm m i).trans hm" } ]

[ 461, 78 ]

[ 457, 1 ]

[ { "state_after": "α : Type u_4\nβ : Type u_3\nγ : Type u_1\nδ : Type u_2\ninst✝¹ : Nonempty γ\ninst✝ : Nonempty δ\nf : α → γ\ng : β → δ\nh : Surjective (map f g)\nc : γ\ninhabited_h : Inhabited δ\n⊢ ∃ a, f a = c", "state_before": "α : Type u_4\nβ : Type u_3\nγ : Type u_1\nδ : Type u_2\ninst✝¹ : Nonempty γ\ninst✝ : Nonempty δ\nf : α → γ\ng : β → δ\nh : Surjective (map f g)\nc : γ\n⊢ ∃ a, f a = c", "tactic": "inhabit δ" }, { "state_after": "case intro.mk\nα : Type u_4\nβ : Type u_3\nγ : Type u_1\nδ : Type u_2\ninst✝¹ : Nonempty γ\ninst✝ : Nonempty δ\nf : α → γ\ng : β → δ\nh✝ : Surjective (map f g)\nc : γ\ninhabited_h : Inhabited δ\na : α\nb : β\nh : map f g (a, b) = (c, default)\n⊢ ∃ a, f a = c", "state_before": "α : Type u_4\nβ : Type u_3\nγ : Type u_1\nδ : Type u_2\ninst✝¹ : Nonempty γ\ninst✝ : Nonempty δ\nf : α → γ\ng : β → δ\nh : Surjective (map f g)\nc : γ\ninhabited_h : Inhabited δ\n⊢ ∃ a, f a = c", "tactic": "obtain ⟨⟨a, b⟩, h⟩ := h (c, default)" }, { "state_after": "no goals", "state_before": "case intro.mk\nα : Type u_4\nβ : Type u_3\nγ : Type u_1\nδ : Type u_2\ninst✝¹ : Nonempty γ\ninst✝ : Nonempty δ\nf : α → γ\ng : β → δ\nh✝ : Surjective (map f g)\nc : γ\ninhabited_h : Inhabited δ\na : α\nb : β\nh : map f g (a, b) = (c, default)\n⊢ ∃ a, f a = c", "tactic": "exact ⟨a, congr_arg Prod.fst h⟩" }, { "state_after": "α : Type u_4\nβ : Type u_3\nγ : Type u_1\nδ : Type u_2\ninst✝¹ : Nonempty γ\ninst✝ : Nonempty δ\nf : α → γ\ng : β → δ\nh : Surjective (map f g)\nd : δ\ninhabited_h : Inhabited γ\n⊢ ∃ a, g a = d", "state_before": "α : Type u_4\nβ : Type u_3\nγ : Type u_1\nδ : Type u_2\ninst✝¹ : Nonempty γ\ninst✝ : Nonempty δ\nf : α → γ\ng : β → δ\nh : Surjective (map f g)\nd : δ\n⊢ ∃ a, g a = d", "tactic": "inhabit γ" }, { "state_after": "case intro.mk\nα : Type u_4\nβ : Type u_3\nγ : Type u_1\nδ : Type u_2\ninst✝¹ : Nonempty γ\ninst✝ : Nonempty δ\nf : α → γ\ng : β → δ\nh✝ : Surjective (map f g)\nd : δ\ninhabited_h : Inhabited γ\na : α\nb : β\nh : map f g (a, b) = (default, d)\n⊢ ∃ a, g a = d", "state_before": "α : Type u_4\nβ : Type u_3\nγ : Type u_1\nδ : Type u_2\ninst✝¹ : Nonempty γ\ninst✝ : Nonempty δ\nf : α → γ\ng : β → δ\nh : Surjective (map f g)\nd : δ\ninhabited_h : Inhabited γ\n⊢ ∃ a, g a = d", "tactic": "obtain ⟨⟨a, b⟩, h⟩ := h (default, d)" }, { "state_after": "no goals", "state_before": "case intro.mk\nα : Type u_4\nβ : Type u_3\nγ : Type u_1\nδ : Type u_2\ninst✝¹ : Nonempty γ\ninst✝ : Nonempty δ\nf : α → γ\ng : β → δ\nh✝ : Surjective (map f g)\nd : δ\ninhabited_h : Inhabited γ\na : α\nb : β\nh : map f g (a, b) = (default, d)\n⊢ ∃ a, g a = d", "tactic": "exact ⟨b, congr_arg Prod.snd h⟩" } ]

[ 376, 31 ]

[ 365, 1 ]

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[]

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: Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E → F\n⊢ Integrable (g ∘ f) ↔ IntegrableOn (fun x => abs (ContinuousLinearMap.det (f' x)) • g (f x)) s", "tactic": "rw [restrict_withDensity hs, integrable_withDensity_iff_integrable_coe_smul₀, IntegrableOn]" }, { "state_after": "E : Type u_1\nF : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E → F\n⊢ (Integrable fun x => ↑(Real.toNNReal (abs (ContinuousLinearMap.det (f' x)))) • (g ∘ f) x) =\n Integrable fun x => abs 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x)) • g (f x)", "tactic": "rw [Real.coe_toNNReal _ (abs_nonneg _)]" }, { "state_after": "no goals", "state_before": "case e_f.h\nE : Type u_1\nF : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E → F\nx : E\n⊢ abs (ContinuousLinearMap.det (f' x)) • (g ∘ f) x = abs (ContinuousLinearMap.det (f' x)) • g (f x)", "tactic": "rfl" }, { "state_after": "no goals", "state_before": "case hf\nE : Type u_1\nF : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsAddHaarMeasure μ\nhs : MeasurableSet s\nhf' : ∀ (x : E), x ∈ s → HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : E → F\n⊢ AEMeasurable fun x => Real.toNNReal (abs (ContinuousLinearMap.det (f' x)))", "tactic": "exact aemeasurable_toNNReal_abs_det_fderivWithin μ hs hf'" } ]

[ 1204, 62 ]

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[]

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[]

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[]

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[ 143, 19 ]

[ 140, 1 ]

[ { "state_after": "ι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nc : Set (PartialRefinement u s)\ni : ι\nne : Set.Nonempty c\n⊢ i ∈ (if hi : ∃ v, v ∈ c ∧ i ∈ v.carrier then Exists.choose hi else Set.Nonempty.some ne).carrier ↔\n i ∈ chainSupCarrier c", "state_before": "ι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nc : Set (PartialRefinement u s)\ni : ι\nne : Set.Nonempty c\n⊢ i ∈ (find c ne i).carrier ↔ i ∈ chainSupCarrier c", "tactic": "rw [find]" }, { "state_after": "case inl\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nc : Set (PartialRefinement u s)\ni : ι\nne : Set.Nonempty c\nh : ∃ v, v ∈ c ∧ i ∈ v.carrier\n⊢ i ∈ (Exists.choose h).carrier ↔ i ∈ chainSupCarrier c\n\ncase inr\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nc : Set (PartialRefinement u s)\ni : ι\nne : Set.Nonempty c\nh : ¬∃ v, v ∈ c ∧ i ∈ v.carrier\n⊢ i ∈ (Set.Nonempty.some ne).carrier ↔ i ∈ chainSupCarrier c", "state_before": "ι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nc : Set (PartialRefinement u s)\ni : ι\nne : Set.Nonempty c\n⊢ i ∈ (if hi : ∃ v, v ∈ c ∧ i ∈ v.carrier then Exists.choose hi else Set.Nonempty.some ne).carrier ↔\n i ∈ chainSupCarrier c", "tactic": "split_ifs with h" }, { "state_after": "case inl\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nc : Set (PartialRefinement u s)\ni : ι\nne : Set.Nonempty c\nh : ∃ v, v ∈ c ∧ i ∈ v.carrier\nthis : Exists.choose h ∈ c ∧ i ∈ (Exists.choose h).carrier\n⊢ i ∈ (Exists.choose h).carrier ↔ i ∈ chainSupCarrier c", "state_before": "case inl\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nc : Set (PartialRefinement u s)\ni : ι\nne : Set.Nonempty c\nh : ∃ v, v ∈ c ∧ i ∈ v.carrier\n⊢ i ∈ (Exists.choose h).carrier ↔ i ∈ chainSupCarrier c", "tactic": "have := h.choose_spec" }, { "state_after": "no goals", "state_before": "case inl\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nc : Set (PartialRefinement u s)\ni : ι\nne : Set.Nonempty c\nh : ∃ v, v ∈ c ∧ i ∈ v.carrier\nthis : Exists.choose h ∈ c ∧ i ∈ (Exists.choose h).carrier\n⊢ i ∈ (Exists.choose h).carrier ↔ i ∈ chainSupCarrier c", "tactic": "exact iff_of_true this.2 (mem_iUnion₂.2 ⟨_, this.1, this.2⟩)" }, { "state_after": "case inr\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nc : Set (PartialRefinement u s)\ni : ι\nne : Set.Nonempty c\nh : ∀ (v : PartialRefinement u s), v ∈ c → ¬i ∈ v.carrier\n⊢ i ∈ (Set.Nonempty.some ne).carrier ↔ i ∈ chainSupCarrier c", "state_before": "case inr\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nc : Set (PartialRefinement u s)\ni : ι\nne : Set.Nonempty c\nh : ¬∃ v, v ∈ c ∧ i ∈ v.carrier\n⊢ i ∈ (Set.Nonempty.some ne).carrier ↔ i ∈ chainSupCarrier c", "tactic": "push_neg at h" }, { "state_after": "case inr\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nc : Set (PartialRefinement u s)\ni : ι\nne : Set.Nonempty c\nh : ∀ (v : PartialRefinement u s), v ∈ c → ¬i ∈ v.carrier\n⊢ ¬i ∈ chainSupCarrier c", "state_before": "case inr\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nc : Set (PartialRefinement u s)\ni : ι\nne : Set.Nonempty c\nh : ∀ (v : PartialRefinement u s), v ∈ c → ¬i ∈ v.carrier\n⊢ i ∈ (Set.Nonempty.some ne).carrier ↔ i ∈ chainSupCarrier c", "tactic": "refine iff_of_false (h _ ne.some_mem) ?_" }, { "state_after": "no goals", "state_before": "case inr\nι : Type u_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nc : Set (PartialRefinement u s)\ni : ι\nne : Set.Nonempty c\nh : ∀ (v : PartialRefinement u s), v ∈ c → ¬i ∈ v.carrier\n⊢ ¬i ∈ chainSupCarrier c", "tactic": "simpa only [chainSupCarrier, mem_iUnion₂, not_exists]" } ]

[ 135, 58 ]

[ 127, 1 ]

[]

[ 295, 6 ]

[ 294, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\n⊢ dual (node' l x r) = node' (dual r) x (dual l)", "tactic": "simp [node', add_comm]" } ]

[ 312, 80 ]

[ 311, 1 ]

[]

[ 834, 10 ]

[ 833, 1 ]

[]

[ 400, 46 ]

[ 399, 1 ]

[]

[ 2188, 46 ]

[ 2187, 1 ]

[]

[ 102, 6 ]

[ 100, 1 ]

[]

[ 1109, 7 ]

[ 1108, 1 ]

[]

[ 688, 17 ]

[ 687, 11 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.196535\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\na a₁ a₂ b b₁ b₂ c d : α\nh : b₁ ≤ b₂\n⊢ Ioo a₁ b₁ ∩ Ioc a₂ b₂ = Ioo (max a₁ a₂) b₁", "tactic": "rw [inter_comm, Ioc_inter_Ioo_of_right_le h, max_comm]" } ]

[ 1807, 57 ]

[ 1806, 1 ]

[ { "state_after": "case inl\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝¹ : Semiring R\np q r✝ : R[X]\ninst✝ : DecidableEq R\ni : ℕ\nr : R\nhr : r = 0\n⊢ natDegree (↑(monomial i) r) = 0\n\ncase inr\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝¹ : Semiring R\np q r✝ : R[X]\ninst✝ : DecidableEq R\ni : ℕ\nr : R\nhr : ¬r = 0\n⊢ natDegree (↑(monomial i) r) = i", "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝¹ : Semiring R\np q r✝ : R[X]\ninst✝ : DecidableEq R\ni : ℕ\nr : R\n⊢ natDegree (↑(monomial i) r) = if r = 0 then 0 else i", "tactic": "split_ifs with hr" }, { "state_after": "no goals", "state_before": "case inl\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝¹ : Semiring R\np q r✝ : R[X]\ninst✝ : DecidableEq R\ni : ℕ\nr : R\nhr : r = 0\n⊢ natDegree (↑(monomial i) r) = 0", "tactic": "simp [hr]" }, { "state_after": "no goals", "state_before": "case inr\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝¹ : Semiring R\np q r✝ : R[X]\ninst✝ : DecidableEq R\ni : ℕ\nr : R\nhr : ¬r = 0\n⊢ natDegree (↑(monomial i) r) = i", "tactic": "rw [← C_mul_X_pow_eq_monomial, natDegree_C_mul_X_pow i r hr]" } ]

[ 331, 65 ]

[ 327, 1 ]

[ { "state_after": "a b c : Int\nh : a + b = a + c\nh₁ : -a + (a + b) = -a + (a + c)\n⊢ b = c", "state_before": "a b c : Int\nh : a + b = a + c\n⊢ b = c", "tactic": "have h₁ : -a + (a + b) = -a + (a + c) := by rw [h]" }, { "state_after": "a b c : Int\nh : a + b = a + c\nh₁ : b = c\n⊢ b = c", "state_before": "a b c : Int\nh : a + b = a + c\nh₁ : -a + (a + b) = -a + (a + c)\n⊢ b = c", "tactic": "simp [← Int.add_assoc, Int.add_left_neg, Int.zero_add] at h₁" }, { "state_after": "no goals", "state_before": "a b c : Int\nh : a + b = a + c\nh₁ : b = c\n⊢ b = c", "tactic": "exact h₁" }, { "state_after": "no goals", "state_before": "a b c : Int\nh : a + b = a + c\n⊢ -a + (a + b) = -a + (a + c)", "tactic": "rw [h]" } ]

[ 331, 73 ]

[ 329, 11 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\na✝ : α\nas as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\na : α\nh : a ∈ []\n⊢ ∃ n, a = get n []", "tactic": "cases h" }, { "state_after": "α : Type u\nβ : Type v\nγ : Type w\na✝ : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\na b : α\nas : List α\nh : a = b ∨ a ∈ as\n⊢ ∃ n, a = get n (b :: as)", "state_before": "α : Type u\nβ : Type v\nγ : Type w\na✝ : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\na b : α\nas : List α\nh : a ∈ b :: as\n⊢ ∃ n, a = get n (b :: as)", "tactic": "rw [mem_cons] at h" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\na✝ : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\na b : α\nas : List α\nh : a = b ∨ a ∈ as\n⊢ ∃ n, a = get n (b :: as)", "tactic": "cases h with\n| inl h => exact ⟨0, h⟩\n| inr h =>\n rcases eq_get_of_mem h with ⟨n, h⟩\n exact ⟨n + 1, h⟩" }, { "state_after": "no goals", "state_before": "case inl\nα : Type u\nβ : Type v\nγ : Type w\na✝ : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\na b : α\nas : List α\nh : a = b\n⊢ ∃ n, a = get n (b :: as)", "tactic": "exact ⟨0, h⟩" }, { "state_after": "case inr.intro\nα : Type u\nβ : Type v\nγ : Type w\na✝ : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\na b : α\nas : List α\nh✝ : a ∈ as\nn : ℕ\nh : a = get n as\n⊢ ∃ n, a = get n (b :: as)", "state_before": "case inr\nα : Type u\nβ : Type v\nγ : Type w\na✝ : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\na b : α\nas : List α\nh : a ∈ as\n⊢ ∃ n, a = get n (b :: as)", "tactic": "rcases eq_get_of_mem h with ⟨n, h⟩" }, { "state_after": "no goals", "state_before": "case inr.intro\nα : Type u\nβ : Type v\nγ : Type w\na✝ : α\nas✝ as1 as2 as3 : List α\ninst✝¹ : Inhabited α\ninst✝ : Inhabited β\na b : α\nas : List α\nh✝ : a ∈ as\nn : ℕ\nh : a = get n as\n⊢ ∃ n, a = get n (b :: as)", "tactic": "exact ⟨n + 1, h⟩" } ]

[ 145, 23 ]

[ 137, 1 ]

[ { "state_after": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nhf : IsCycle f\nx : α\nhx : ↑f x ≠ x\nkey : Finset.univ.val = x ::ₘ Multiset.erase Finset.univ.val x\n⊢ toCycle f hf = ↑(toList f x)", "state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nhf : IsCycle f\nx : α\nhx : ↑f x ≠ x\n⊢ toCycle f hf = ↑(toList f x)", "tactic": "have key : (Finset.univ : Finset α).val = x ::ₘ Finset.univ.val.erase x := by simp" }, { "state_after": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nhf : IsCycle f\nx : α\nhx : ↑f x ≠ x\nkey : Finset.univ.val = x ::ₘ Multiset.erase Finset.univ.val x\n⊢ Multiset.recOn (x ::ₘ Multiset.erase Finset.univ.val x) (Quot.mk Setoid.r [])\n (fun x x_1 l => if ↑f x = x then l else ↑(toList f x))\n (_ :\n ∀ (x y : α),\n Multiset α →\n ∀ (s : Cycle α),\n HEq (if ↑f x = x then if ↑f y = y then s else ↑(toList f y) else ↑(toList f x))\n (if ↑f y = y then if ↑f x = x then s else ↑(toList f x) else ↑(toList f y))) =\n ↑(toList f x)", "state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nhf : IsCycle f\nx : α\nhx : ↑f x ≠ x\nkey : Finset.univ.val = x ::ₘ Multiset.erase Finset.univ.val x\n⊢ toCycle f hf = ↑(toList f x)", "tactic": "rw [toCycle, key]" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nhf : IsCycle f\nx : α\nhx : ↑f x ≠ x\nkey : Finset.univ.val = x ::ₘ Multiset.erase Finset.univ.val x\n⊢ Multiset.recOn (x ::ₘ Multiset.erase Finset.univ.val x) (Quot.mk Setoid.r [])\n (fun x x_1 l => if ↑f x = x then l else ↑(toList f x))\n (_ :\n ∀ (x y : α),\n Multiset α →\n ∀ (s : Cycle α),\n HEq (if ↑f x = x then if ↑f y = y then s else ↑(toList f y) else ↑(toList f x))\n (if ↑f y = y then if ↑f x = x then s else ↑(toList f x) else ↑(toList f y))) =\n ↑(toList f x)", "tactic": "simp [hx]" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nhf : IsCycle f\nx : α\nhx : ↑f x ≠ x\n⊢ Finset.univ.val = x ::ₘ Multiset.erase Finset.univ.val x", "tactic": "simp" } ]

[ 415, 12 ]

[ 411, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝² : CanonicallyOrderedAddMonoid α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nha : AddLECancellable a\nhb : AddLECancellable b\nhc : AddLECancellable c\nhba : b ≤ a\nhca : c ≤ a\n⊢ a - b = a - c ↔ b = c", "tactic": "simp_rw [le_antisymm_iff, ha.tsub_le_tsub_iff_left hb hba, ha.tsub_le_tsub_iff_left hc hca,\n and_comm]" } ]

[ 377, 14 ]

[ 374, 11 ]

[]

[ 405, 7 ]

[ 403, 1 ]

[]

[ 233, 41 ]

[ 232, 1 ]

[ { "state_after": "case refine'_1\nι : Type u_1\ninst✝ : Fintype ι\n⊢ ↑(parallelepiped (Pi.basisFun ℝ ι)) = uIcc (fun i => 0) fun i => 1\n\ncase refine'_2\nι : Type u_1\ninst✝ : Fintype ι\n⊢ (fun i => 0) ≤ fun i => 1", "state_before": "ι : Type u_1\ninst✝ : Fintype ι\n⊢ ↑(parallelepiped (Pi.basisFun ℝ ι)) = ↑(PositiveCompacts.piIcc01 ι)", "tactic": "refine' Eq.trans _ ((uIcc_of_le _).trans (Set.pi_univ_Icc _ _).symm)" }, { "state_after": "no goals", "state_before": "case refine'_1\nι : Type u_1\ninst✝ : Fintype ι\n⊢ ↑(parallelepiped (Pi.basisFun ℝ ι)) = uIcc (fun i => 0) fun i => 1", "tactic": "classical convert parallelepiped_single (ι := ι) 1" }, { "state_after": "no goals", "state_before": "case refine'_1\nι : Type u_1\ninst✝ : Fintype ι\n⊢ ↑(parallelepiped (Pi.basisFun ℝ ι)) = uIcc (fun i => 0) fun i => 1", "tactic": "convert parallelepiped_single (ι := ι) 1" }, { "state_after": "no goals", "state_before": "case refine'_2\nι : Type u_1\ninst✝ : Fintype ι\n⊢ (fun i => 0) ≤ fun i => 1", "tactic": "exact zero_le_one" } ]

[ 82, 24 ]

[ 77, 1 ]

[ { "state_after": "case inl\nα : Type u_1\nβ : Type ?u.124878\ns✝ t✝ : Set α\na b x y : α\nf : α → β\ns t : Set α\nht : autoParam (Set.Finite t) _auto✝\nh : Set.Finite s\n⊢ ncard s ≤ ncard (s \\ t) + ncard t\n\ncase inr\nα : Type u_1\nβ : Type ?u.124878\ns✝ t✝ : Set α\na b x y : α\nf : α → β\ns t : Set α\nht : autoParam (Set.Finite t) _auto✝\nh : Set.Infinite s\n⊢ ncard s ≤ ncard (s \\ t) + ncard t", "state_before": "α : Type u_1\nβ : Type ?u.124878\ns✝ t✝ : Set α\na b x y : α\nf : α → β\ns t : Set α\nht : autoParam (Set.Finite t) _auto✝\n⊢ ncard s ≤ ncard (s \\ t) + ncard t", "tactic": "cases' s.finite_or_infinite with h h" }, { "state_after": "case h.e'_3\nα : Type u_1\nβ : Type ?u.124878\ns✝ t✝ : Set α\na b x y : α\nf : α → β\ns t : Set α\nht : autoParam (Set.Finite t) _auto✝\nh : Set.Infinite s\n⊢ ncard s = 0", "state_before": "case inr\nα : Type u_1\nβ : Type ?u.124878\ns✝ t✝ : Set α\na b x y : α\nf : α → β\ns t : Set α\nht : autoParam (Set.Finite t) _auto✝\nh : Set.Infinite s\n⊢ ncard s ≤ ncard (s \\ t) + ncard t", "tactic": "convert Nat.zero_le _" }, { "state_after": "no goals", "state_before": "case h.e'_3\nα : Type u_1\nβ : Type ?u.124878\ns✝ t✝ : Set α\na b x y : α\nf : α → β\ns t : Set α\nht : autoParam (Set.Finite t) _auto✝\nh : Set.Infinite s\n⊢ ncard s = 0", "tactic": "rw [h.ncard]" }, { "state_after": "case inl\nα : Type u_1\nβ : Type ?u.124878\ns✝ t✝ : Set α\na b x y : α\nf : α → β\ns t : Set α\nht : autoParam (Set.Finite t) _auto✝\nh : Set.Finite s\n⊢ ncard (t ∩ s) ≤ ncard t", "state_before": "case inl\nα : Type u_1\nβ : Type ?u.124878\ns✝ t✝ : Set α\na b x y : α\nf : α → β\ns t : Set α\nht : autoParam (Set.Finite t) _auto✝\nh : Set.Finite s\n⊢ ncard s ≤ ncard (s \\ t) + ncard t", "tactic": "rw [← diff_inter_self_eq_diff, ← ncard_diff_add_ncard_eq_ncard (inter_subset_right t s) h,\n add_le_add_iff_left]" }, { "state_after": "no goals", "state_before": "case inl\nα : Type u_1\nβ : Type ?u.124878\ns✝ t✝ : Set α\na b x y : α\nf : α → β\ns t : Set α\nht : autoParam (Set.Finite t) _auto✝\nh : Set.Finite s\n⊢ ncard (t ∩ s) ≤ ncard t", "tactic": "apply ncard_inter_le_ncard_left _ _ ht" } ]

[ 515, 15 ]

[ 508, 1 ]

[]

[ 171, 6 ]

[ 170, 1 ]

[]

[ 338, 35 ]

[ 336, 11 ]

[ { "state_after": "no goals", "state_before": "C : Type u_2\ninst✝ : Category C\nX : C\n⊢ (mk X (𝟙 X)).p = 𝟙 X", "tactic": "rfl" } ]

[ 143, 58 ]

[ 143, 1 ]

[ { "state_after": "case intro.intro\nF : Type ?u.313391\nα : Type u_1\nβ : Type ?u.313397\nγ : Type ?u.313400\nι : Type ?u.313403\nκ : Type ?u.313406\ninst✝ : LinearOrder α\ns : Finset α\na : α\nh✝ : a ∈ s\nb : α\nh : b ∈ sup s some\nright✝ : a ≤ b\n⊢ ∃ b, Finset.max s = ↑b", "state_before": "F : Type ?u.313391\nα : Type u_1\nβ : Type ?u.313397\nγ : Type ?u.313400\nι : Type ?u.313403\nκ : Type ?u.313406\ninst✝ : LinearOrder α\ns : Finset α\na : α\nh : a ∈ s\n⊢ ∃ b, Finset.max s = ↑b", "tactic": "obtain ⟨b, h, _⟩ := le_sup (α := WithBot α) h _ rfl" }, { "state_after": "no goals", "state_before": "case intro.intro\nF : Type ?u.313391\nα : Type u_1\nβ : Type ?u.313397\nγ : Type ?u.313400\nι : Type ?u.313403\nκ : Type ?u.313406\ninst✝ : LinearOrder α\ns : Finset α\na : α\nh✝ : a ∈ s\nb : α\nh : b ∈ sup s some\nright✝ : a ≤ b\n⊢ ∃ b, Finset.max s = ↑b", "tactic": "exact ⟨b, h⟩" } ]

[ 1182, 15 ]

[ 1180, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type u_3\nV : Type u_1\nV' : Type ?u.53134\nP : Type u_2\nP' : Type ?u.53140\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y z p : P\n⊢ p -ᵥ z ∈ affineSegment R (p -ᵥ x) (p -ᵥ y) ↔ z ∈ affineSegment R x y", "tactic": "rw [← affineSegment_const_vsub_image, (vsub_right_injective p).mem_set_image]" } ]

[ 133, 80 ]

[ 131, 1 ]