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tasksource

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leandojo

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[ { "state_after": "case a\nG : Type u\ninst✝¹ : Monoid G\ninst✝ : Subsingleton G\n⊢ exponent G ≤ 1\n\ncase a\nG : Type u\ninst✝¹ : Monoid G\ninst✝ : Subsingleton G\n⊢ 1 ≤ exponent G", "state_before": "G : Type u\ninst✝¹ : Monoid G\ninst✝ : Subsingleton G\n⊢ exponent G = 1", "tactic": "apply le_antisymm" }, { "state_after": "case a\nG : Type u\ninst✝¹ : Monoid G\ninst✝ : Subsingleton G\n⊢ ∀ (g : G), g ^ 1 = 1", "state_before": "case a\nG : Type u\ninst✝¹ : Monoid G\ninst✝ : Subsingleton G\n⊢ exponent G ≤ 1", "tactic": "apply exponent_min' _ Nat.one_pos" }, { "state_after": "no goals", "state_before": "case a\nG : Type u\ninst✝¹ : Monoid G\ninst✝ : Subsingleton G\n⊢ ∀ (g : G), g ^ 1 = 1", "tactic": "simp" }, { "state_after": "case a.h\nG : Type u\ninst✝¹ : Monoid G\ninst✝ : Subsingleton G\n⊢ 0 < exponent G", "state_before": "case a\nG : Type u\ninst✝¹ : Monoid G\ninst✝ : Subsingleton G\n⊢ 1 ≤ exponent G", "tactic": "apply Nat.succ_le_of_lt" }, { "state_after": "case a.h\nG : Type u\ninst✝¹ : Monoid G\ninst✝ : Subsingleton G\n⊢ ∀ (g : G), g ^ 1 = 1", "state_before": "case a.h\nG : Type u\ninst✝¹ : Monoid G\ninst✝ : Subsingleton G\n⊢ 0 < exponent G", "tactic": "apply exponent_pos_of_exists 1 Nat.one_pos" }, { "state_after": "no goals", "state_before": "case a.h\nG : Type u\ninst✝¹ : Monoid G\ninst✝ : Subsingleton G\n⊢ ∀ (g : G), g ^ 1 = 1", "tactic": "simp" } ]

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[ { "state_after": "no goals", "state_before": "R : Type u_2\nM : Type u_1\ninst✝² : CommSemiring R\nσ : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : SemilatticeSup M\nw : σ → M\n⊢ weightedTotalDegree' w 0 = ⊥", "tactic": "simp only [weightedTotalDegree', support_zero, Finset.sup_empty]" } ]

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[ { "state_after": "no goals", "state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.265379\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.265474\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nc : F\nh : DifferentiableAt 𝕜 (fun y => c + f y) x\n⊢ DifferentiableAt 𝕜 f x", "tactic": "simpa using h.const_add (-c)" } ]

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[ { "state_after": "no goals", "state_before": "α β : Type u\n⊢ succ 0 = 1", "tactic": "norm_cast" } ]

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[ { "state_after": "ι : Type u\ns : Finset ι\nw z : ι → ℝ\nx : ℝ\nhw : ∀ (i : ι), i ∈ s → 0 ≤ w i\nhw' : ∑ i in s, w i = 1\nhz : ∀ (i : ι), i ∈ s → 0 ≤ z i\nhx : ∀ (i : ι), i ∈ s → w i ≠ 0 → z i = x\ni : ι\nhi : i ∈ s\n⊢ z i ^ w i = x ^ w i", "state_before": "ι : Type u\ns : Finset ι\nw z : ι → ℝ\nx : ℝ\nhw : ∀ (i : ι), i ∈ s → 0 ≤ w i\nhw' : ∑ i in s, w i = 1\nhz : ∀ (i : ι), i ∈ s → 0 ≤ z i\nhx : ∀ (i : ι), i ∈ s → w i ≠ 0 → z i = x\n⊢ ∏ i in s, z i ^ w i = ∏ i in s, x ^ w i", "tactic": "refine' prod_congr rfl fun i hi => _" }, { "state_after": "case inl\nι : Type u\ns : Finset ι\nw z : ι → ℝ\nx : ℝ\nhw : ∀ (i : ι), i ∈ s → 0 ≤ w i\nhw' : ∑ i in s, w i = 1\nhz : ∀ (i : ι), i ∈ s → 0 ≤ z i\nhx : ∀ (i : ι), i ∈ s → w i ≠ 0 → z i = x\ni : ι\nhi : i ∈ s\nh₀ : w i = 0\n⊢ z i ^ w i = x ^ w i\n\ncase inr\nι : Type u\ns : Finset ι\nw z : ι → ℝ\nx : ℝ\nhw : ∀ (i : ι), i ∈ s → 0 ≤ w i\nhw' : ∑ i in s, w i = 1\nhz : ∀ (i : ι), i ∈ s → 0 ≤ z i\nhx : ∀ (i : ι), i ∈ s → w i ≠ 0 → z i = x\ni : ι\nhi : i ∈ s\nh₀ : w i ≠ 0\n⊢ z i ^ w i = x ^ w i", "state_before": "ι : Type u\ns : Finset ι\nw z : ι → ℝ\nx : ℝ\nhw : ∀ (i : ι), i ∈ s → 0 ≤ w i\nhw' : ∑ i in s, w i = 1\nhz : ∀ (i : ι), i ∈ s → 0 ≤ z i\nhx : ∀ (i : ι), i ∈ s → w i ≠ 0 → z i = x\ni : ι\nhi : i ∈ s\n⊢ z i ^ w i = x ^ w i", "tactic": "cases' eq_or_ne (w i) 0 with h₀ h₀" }, { "state_after": "no goals", "state_before": "case inl\nι : Type u\ns : Finset ι\nw z : ι → ℝ\nx : ℝ\nhw : ∀ (i : ι), i ∈ s → 0 ≤ w i\nhw' : ∑ i in s, w i = 1\nhz : ∀ (i : ι), i ∈ s → 0 ≤ z i\nhx : ∀ (i : ι), i ∈ s → w i ≠ 0 → z i = x\ni : ι\nhi : i ∈ s\nh₀ : w i = 0\n⊢ z i ^ w i = x ^ w i", "tactic": "rw [h₀, rpow_zero, rpow_zero]" }, { "state_after": "no goals", "state_before": "case inr\nι : Type u\ns : Finset ι\nw z : ι → ℝ\nx : ℝ\nhw : ∀ (i : ι), i ∈ s → 0 ≤ w i\nhw' : ∑ i in s, w i = 1\nhz : ∀ (i : ι), i ∈ s → 0 ≤ z i\nhx : ∀ (i : ι), i ∈ s → w i ≠ 0 → z i = x\ni : ι\nhi : i ∈ s\nh₀ : w i ≠ 0\n⊢ z i ^ w i = x ^ w i", "tactic": "rw [hx i hi h₀]" }, { "state_after": "ι : Type u\ns : Finset ι\nw z : ι → ℝ\nx : ℝ\nhw : ∀ (i : ι), i ∈ s → 0 ≤ w i\nhw' : ∑ i in s, w i = 1\nhz : ∀ (i : ι), i ∈ s → 0 ≤ z i\nhx : ∀ (i : ι), i ∈ s → w i ≠ 0 → z i = x\n⊢ 0 ≤ x", "state_before": "ι : Type u\ns : Finset ι\nw z : ι → ℝ\nx : ℝ\nhw : ∀ (i : ι), i ∈ s → 0 ≤ w i\nhw' : ∑ i in s, w i = 1\nhz : ∀ (i : ι), i ∈ s → 0 ≤ z i\nhx : ∀ (i : ι), i ∈ s → w i ≠ 0 → z i = x\n⊢ ∏ i in s, x ^ w i = x", "tactic": "rw [← rpow_sum_of_nonneg _ hw, hw', rpow_one]" }, { "state_after": "ι : Type u\ns : Finset ι\nw z : ι → ℝ\nx : ℝ\nhw : ∀ (i : ι), i ∈ s → 0 ≤ w i\nhw' : ∑ i in s, w i = 1\nhz : ∀ (i : ι), i ∈ s → 0 ≤ z i\nhx : ∀ (i : ι), i ∈ s → w i ≠ 0 → z i = x\nthis : ∑ i in s, w i ≠ 0\n⊢ 0 ≤ x", "state_before": "ι : Type u\ns : Finset ι\nw z : ι → ℝ\nx : ℝ\nhw : ∀ (i : ι), i ∈ s → 0 ≤ w i\nhw' : ∑ i in s, w i = 1\nhz : ∀ (i : ι), i ∈ s → 0 ≤ z i\nhx : ∀ (i : ι), i ∈ s → w i ≠ 0 → z i = x\n⊢ 0 ≤ x", "tactic": "have : (∑ i in s, w i) ≠ 0 := by\n rw [hw']\n exact one_ne_zero" }, { "state_after": "case intro.intro\nι : Type u\ns : Finset ι\nw z : ι → ℝ\nx : ℝ\nhw : ∀ (i : ι), i ∈ s → 0 ≤ w i\nhw' : ∑ i in s, w i = 1\nhz : ∀ (i : ι), i ∈ s → 0 ≤ z i\nhx : ∀ (i : ι), i ∈ s → w i ≠ 0 → z i = x\nthis : ∑ i in s, w i ≠ 0\ni : ι\nhis : i ∈ s\nhi : w i ≠ 0\n⊢ 0 ≤ x", "state_before": "ι : Type u\ns : Finset ι\nw z : ι → ℝ\nx : ℝ\nhw : ∀ (i : ι), i ∈ s → 0 ≤ w i\nhw' : ∑ i in s, w i = 1\nhz : ∀ (i : ι), i ∈ s → 0 ≤ z i\nhx : ∀ (i : ι), i ∈ s → w i ≠ 0 → z i = x\nthis : ∑ i in s, w i ≠ 0\n⊢ 0 ≤ x", "tactic": "obtain ⟨i, his, hi⟩ := exists_ne_zero_of_sum_ne_zero this" }, { "state_after": "case intro.intro\nι : Type u\ns : Finset ι\nw z : ι → ℝ\nx : ℝ\nhw : ∀ (i : ι), i ∈ s → 0 ≤ w i\nhw' : ∑ i in s, w i = 1\nhz : ∀ (i : ι), i ∈ s → 0 ≤ z i\nhx : ∀ (i : ι), i ∈ s → w i ≠ 0 → z i = x\nthis : ∑ i in s, w i ≠ 0\ni : ι\nhis : i ∈ s\nhi : w i ≠ 0\n⊢ 0 ≤ z i", "state_before": "case intro.intro\nι : Type u\ns : Finset ι\nw z : ι → ℝ\nx : ℝ\nhw : ∀ (i : ι), i ∈ s → 0 ≤ w i\nhw' : ∑ i in s, w i = 1\nhz : ∀ (i : ι), i ∈ s → 0 ≤ z i\nhx : ∀ (i : ι), i ∈ s → w i ≠ 0 → z i = x\nthis : ∑ i in s, w i ≠ 0\ni : ι\nhis : i ∈ s\nhi : w i ≠ 0\n⊢ 0 ≤ x", "tactic": "rw [← hx i his hi]" }, { "state_after": "no goals", "state_before": "case intro.intro\nι : Type u\ns : Finset ι\nw z : ι → ℝ\nx : ℝ\nhw : ∀ (i : ι), i ∈ s → 0 ≤ w i\nhw' : ∑ i in s, w i = 1\nhz : ∀ (i : ι), i ∈ s → 0 ≤ z i\nhx : ∀ (i : ι), i ∈ s → w i ≠ 0 → z i = x\nthis : ∑ i in s, w i ≠ 0\ni : ι\nhis : i ∈ s\nhi : w i ≠ 0\n⊢ 0 ≤ z i", "tactic": "exact hz i his" }, { "state_after": "ι : Type u\ns : Finset ι\nw z : ι → ℝ\nx : ℝ\nhw : ∀ (i : ι), i ∈ s → 0 ≤ w i\nhw' : ∑ i in s, w i = 1\nhz : ∀ (i : ι), i ∈ s → 0 ≤ z i\nhx : ∀ (i : ι), i ∈ s → w i ≠ 0 → z i = x\n⊢ 1 ≠ 0", "state_before": "ι : Type u\ns : Finset ι\nw z : ι → ℝ\nx : ℝ\nhw : ∀ (i : ι), i ∈ s → 0 ≤ w i\nhw' : ∑ i in s, w i = 1\nhz : ∀ (i : ι), i ∈ s → 0 ≤ z i\nhx : ∀ (i : ι), i ∈ s → w i ≠ 0 → z i = x\n⊢ ∑ i in s, w i ≠ 0", "tactic": "rw [hw']" }, { "state_after": "no goals", "state_before": "ι : Type u\ns : Finset ι\nw z : ι → ℝ\nx : ℝ\nhw : ∀ (i : ι), i ∈ s → 0 ≤ w i\nhw' : ∑ i in s, w i = 1\nhz : ∀ (i : ι), i ∈ s → 0 ≤ z i\nhx : ∀ (i : ι), i ∈ s → w i ≠ 0 → z i = x\n⊢ 1 ≠ 0", "tactic": "exact one_ne_zero" } ]

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[ { "state_after": "case inl\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F\nhf : ker f = ⊥\nh✝ : Subsingleton E\n⊢ ∃ K, K > 0 ∧ AntilipschitzWith K ↑f\n\ncase inr\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F\nhf : ker f = ⊥\nh✝ : Nontrivial E\n⊢ ∃ K, K > 0 ∧ AntilipschitzWith K ↑f", "state_before": "𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F\nhf : ker f = ⊥\n⊢ ∃ K, K > 0 ∧ AntilipschitzWith K ↑f", "tactic": "cases subsingleton_or_nontrivial E" }, { "state_after": "no goals", "state_before": "case inl\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F\nhf : ker f = ⊥\nh✝ : Subsingleton E\n⊢ ∃ K, K > 0 ∧ AntilipschitzWith K ↑f", "tactic": "exact ⟨1, zero_lt_one, AntilipschitzWith.of_subsingleton⟩" }, { "state_after": "case inr\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F\nhf : Function.Injective ↑f\nh✝ : Nontrivial E\n⊢ ∃ K, K > 0 ∧ AntilipschitzWith K ↑f", "state_before": "case inr\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F\nhf : ker f = ⊥\nh✝ : Nontrivial E\n⊢ ∃ K, K > 0 ∧ AntilipschitzWith K ↑f", "tactic": "rw [LinearMap.ker_eq_bot] at hf" }, { "state_after": "case inr\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F\nhf : Function.Injective ↑f\nh✝ : Nontrivial E\ne : E ≃L[𝕜] { x // x ∈ range f } := LinearEquiv.toContinuousLinearEquiv (LinearEquiv.ofInjective f hf)\n⊢ ∃ K, K > 0 ∧ AntilipschitzWith K ↑f", "state_before": "case inr\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F\nhf : Function.Injective ↑f\nh✝ : Nontrivial E\n⊢ ∃ K, K > 0 ∧ AntilipschitzWith K ↑f", "tactic": "let e : E ≃L[𝕜] LinearMap.range f := (LinearEquiv.ofInjective f hf).toContinuousLinearEquiv" }, { "state_after": "no goals", "state_before": "case inr\n𝕜 : Type u\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁶ : AddCommGroup F'\ninst✝⁵ : Module 𝕜 F'\ninst✝⁴ : TopologicalSpace F'\ninst✝³ : TopologicalAddGroup F'\ninst✝² : ContinuousSMul 𝕜 F'\ninst✝¹ : CompleteSpace 𝕜\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F\nhf : Function.Injective ↑f\nh✝ : Nontrivial E\ne : E ≃L[𝕜] { x // x ∈ range f } := LinearEquiv.toContinuousLinearEquiv (LinearEquiv.ofInjective f hf)\n⊢ ∃ K, K > 0 ∧ AntilipschitzWith K ↑f", "tactic": "exact ⟨_, e.nnnorm_symm_pos, e.antilipschitz⟩" } ]

[ 232, 50 ]

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[ { "state_after": "α : Type u_1\nβ : Type ?u.660001\nm : MeasurableSpace α\nμ✝ ν : Measure α\nc✝ c : ComplexMeasure α\nμ : Measure α\n⊢ Memℒp (fun x => ↑IsROrC.re (rnDeriv c μ x)) 1 ∧ Memℒp (fun x => ↑IsROrC.im (rnDeriv c μ x)) 1", "state_before": "α : Type u_1\nβ : Type ?u.660001\nm : MeasurableSpace α\nμ✝ ν : Measure α\nc✝ c : ComplexMeasure α\nμ : Measure α\n⊢ Integrable (rnDeriv c μ)", "tactic": "rw [← memℒp_one_iff_integrable, ← memℒp_re_im_iff]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.660001\nm : MeasurableSpace α\nμ✝ ν : Measure α\nc✝ c : ComplexMeasure α\nμ : Measure α\n⊢ Memℒp (fun x => ↑IsROrC.re (rnDeriv c μ x)) 1 ∧ Memℒp (fun x => ↑IsROrC.im (rnDeriv c μ x)) 1", "tactic": "exact\n ⟨memℒp_one_iff_integrable.2 (SignedMeasure.integrable_rnDeriv _ _),\n memℒp_one_iff_integrable.2 (SignedMeasure.integrable_rnDeriv _ _)⟩" } ]

[ 1223, 73 ]

[ 1219, 1 ]

[ { "state_after": "no goals", "state_before": "b₀ b₁ : Bool\nh : b₀ ≤ b₁\n⊢ toNat b₀ ≤ toNat b₁", "tactic": "cases h with\n| inl h => subst h; exact Nat.zero_le _\n| inr h => subst h; cases b₀ <;> simp;" }, { "state_after": "case inl\nb₁ : Bool\n⊢ toNat false ≤ toNat b₁", "state_before": "case inl\nb₀ b₁ : Bool\nh : b₀ = false\n⊢ toNat b₀ ≤ toNat b₁", "tactic": "subst h" }, { "state_after": "no goals", "state_before": "case inl\nb₁ : Bool\n⊢ toNat false ≤ toNat b₁", "tactic": "exact Nat.zero_le _" }, { "state_after": "case inr\nb₀ : Bool\n⊢ toNat b₀ ≤ toNat true", "state_before": "case inr\nb₀ b₁ : Bool\nh : b₁ = true\n⊢ toNat b₀ ≤ toNat b₁", "tactic": "subst h" }, { "state_after": "no goals", "state_before": "case inr\nb₀ : Bool\n⊢ toNat b₀ ≤ toNat true", "tactic": "cases b₀ <;> simp" } ]

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[]

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[]

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[ { "state_after": "case zero\nα : Type u_2\nβ : Type u_1\nf : β → α → β\nb : β\nv : Vector α Nat.zero\ni : Fin Nat.zero\n⊢ get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (↑Fin.castSucc i)) (get v i)\n\ncase succ\nα : Type u_2\nβ : Type u_1\nf : β → α → β\nb : β\nn : ℕ\nv : Vector α (Nat.succ n)\ni : Fin (Nat.succ n)\n⊢ get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (↑Fin.castSucc i)) (get v i)", "state_before": "n : ℕ\nα : Type u_2\nβ : Type u_1\nf : β → α → β\nb : β\nv : Vector α n\ni : Fin n\n⊢ get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (↑Fin.castSucc i)) (get v i)", "tactic": "cases' n with n" }, { "state_after": "case succ.zero\nα : Type u_2\nβ : Type u_1\nf : β → α → β\nb✝ : β\nn : ℕ\nv✝ : Vector α (Nat.succ n)\ni✝ : Fin (Nat.succ n)\nb : β\nv : Vector α (Nat.succ Nat.zero)\ni : Fin (Nat.succ Nat.zero)\n⊢ get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (↑Fin.castSucc i)) (get v i)\n\ncase succ.succ\nα : Type u_2\nβ : Type u_1\nf : β → α → β\nb✝ : β\nn✝ : ℕ\nv✝ : Vector α (Nat.succ n✝)\ni✝ : Fin (Nat.succ n✝)\nn : ℕ\nhn :\n ∀ (b : β) (v : Vector α (Nat.succ n)) (i : Fin (Nat.succ n)),\n get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (↑Fin.castSucc i)) (get v i)\nb : β\nv : Vector α (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\n⊢ get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (↑Fin.castSucc i)) (get v i)", "state_before": "case succ\nα : Type u_2\nβ : Type u_1\nf : β → α → β\nb : β\nn : ℕ\nv : Vector α (Nat.succ n)\ni : Fin (Nat.succ n)\n⊢ get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (↑Fin.castSucc i)) (get v i)", "tactic": "induction' n with n hn generalizing b" }, { "state_after": "no goals", "state_before": "case zero\nα : Type u_2\nβ : Type u_1\nf : β → α → β\nb : β\nv : Vector α Nat.zero\ni : Fin Nat.zero\n⊢ get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (↑Fin.castSucc i)) (get v i)", "tactic": "exact i.elim0" }, { "state_after": "case succ.zero\nα : Type u_2\nβ : Type u_1\nf : β → α → β\nb✝ : β\nn : ℕ\nv✝ : Vector α (Nat.succ n)\ni✝ : Fin (Nat.succ n)\nb : β\nv : Vector α (Nat.succ Nat.zero)\ni : Fin (Nat.succ Nat.zero)\ni0 : i = 0\n⊢ get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (↑Fin.castSucc i)) (get v i)", "state_before": "case succ.zero\nα : Type u_2\nβ : Type u_1\nf : β → α → β\nb✝ : β\nn : ℕ\nv✝ : Vector α (Nat.succ n)\ni✝ : Fin (Nat.succ n)\nb : β\nv : Vector α (Nat.succ Nat.zero)\ni : Fin (Nat.succ Nat.zero)\n⊢ get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (↑Fin.castSucc i)) (get v i)", "tactic": "have i0 : i = 0 := Fin.eq_zero _" }, { "state_after": "case succ.zero\nα : Type u_2\nβ : Type u_1\nf : β → α → β\nb✝ : β\nn : ℕ\nv✝ : Vector α (Nat.succ n)\ni✝ : Fin (Nat.succ n)\nb : β\nv : Vector α (Nat.succ Nat.zero)\ni : Fin (Nat.succ Nat.zero)\ni0 : i = 0\n⊢ get (b ::ᵥ f b (head v) ::ᵥ nil) 1 = f b (head v)", "state_before": "case succ.zero\nα : Type u_2\nβ : Type u_1\nf : β → α → β\nb✝ : β\nn : ℕ\nv✝ : Vector α (Nat.succ n)\ni✝ : Fin (Nat.succ n)\nb : β\nv : Vector α (Nat.succ Nat.zero)\ni : Fin (Nat.succ Nat.zero)\ni0 : i = 0\n⊢ get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (↑Fin.castSucc i)) (get v i)", "tactic": "simp [scanl_singleton, i0, get_zero]" }, { "state_after": "no goals", "state_before": "case succ.zero\nα : Type u_2\nβ : Type u_1\nf : β → α → β\nb✝ : β\nn : ℕ\nv✝ : Vector α (Nat.succ n)\ni✝ : Fin (Nat.succ n)\nb : β\nv : Vector α (Nat.succ Nat.zero)\ni : Fin (Nat.succ Nat.zero)\ni0 : i = 0\n⊢ get (b ::ᵥ f b (head v) ::ᵥ nil) 1 = f b (head v)", "tactic": "simp [get_eq_get]" }, { "state_after": "case succ.succ\nα : Type u_2\nβ : Type u_1\nf : β → α → β\nb✝ : β\nn✝ : ℕ\nv✝ : Vector α (Nat.succ n✝)\ni✝ : Fin (Nat.succ n✝)\nn : ℕ\nhn :\n ∀ (b : β) (v : Vector α (Nat.succ n)) (i : Fin (Nat.succ n)),\n get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (↑Fin.castSucc i)) (get v i)\nb : β\nv : Vector α (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\n⊢ get (scanl f (f b (head v)) (tail v)) i =\n f (get (b ::ᵥ scanl f (f b (head v)) (tail v)) (↑Fin.castSucc i)) (get (head v ::ᵥ tail v) i)", "state_before": "case succ.succ\nα : Type u_2\nβ : Type u_1\nf : β → α → β\nb✝ : β\nn✝ : ℕ\nv✝ : Vector α (Nat.succ n✝)\ni✝ : Fin (Nat.succ n✝)\nn : ℕ\nhn :\n ∀ (b : β) (v : Vector α (Nat.succ n)) (i : Fin (Nat.succ n)),\n get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (↑Fin.castSucc i)) (get v i)\nb : β\nv : Vector α (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\n⊢ get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (↑Fin.castSucc i)) (get v i)", "tactic": "rw [← cons_head_tail v, scanl_cons, get_cons_succ]" }, { "state_after": "case succ.succ.refine'_1\nα : Type u_2\nβ : Type u_1\nf : β → α → β\nb✝ : β\nn✝ : ℕ\nv✝ : Vector α (Nat.succ n✝)\ni✝ : Fin (Nat.succ n✝)\nn : ℕ\nhn :\n ∀ (b : β) (v : Vector α (Nat.succ n)) (i : Fin (Nat.succ n)),\n get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (↑Fin.castSucc i)) (get v i)\nb : β\nv : Vector α (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\n⊢ get (scanl f (f b (head v)) (tail v)) 0 =\n f (get (b ::ᵥ scanl f (f b (head v)) (tail v)) (↑Fin.castSucc 0)) (get (head v ::ᵥ tail v) 0)\n\ncase succ.succ.refine'_2\nα : Type u_2\nβ : Type u_1\nf : β → α → β\nb✝ : β\nn✝ : ℕ\nv✝ : Vector α (Nat.succ n✝)\ni✝ : Fin (Nat.succ n✝)\nn : ℕ\nhn :\n ∀ (b : β) (v : Vector α (Nat.succ n)) (i : Fin (Nat.succ n)),\n get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (↑Fin.castSucc i)) (get v i)\nb : β\nv : Vector α (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\n⊢ ∀ (i : Fin (n + 1)),\n get (scanl f (f b (head v)) (tail v)) (Fin.succ i) =\n f (get (b ::ᵥ scanl f (f b (head v)) (tail v)) (↑Fin.castSucc (Fin.succ i)))\n (get (head v ::ᵥ tail v) (Fin.succ i))", "state_before": "case succ.succ\nα : Type u_2\nβ : Type u_1\nf : β → α → β\nb✝ : β\nn✝ : ℕ\nv✝ : Vector α (Nat.succ n✝)\ni✝ : Fin (Nat.succ n✝)\nn : ℕ\nhn :\n ∀ (b : β) (v : Vector α (Nat.succ n)) (i : Fin (Nat.succ n)),\n get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (↑Fin.castSucc i)) (get v i)\nb : β\nv : Vector α (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\n⊢ get (scanl f (f b (head v)) (tail v)) i =\n f (get (b ::ᵥ scanl f (f b (head v)) (tail v)) (↑Fin.castSucc i)) (get (head v ::ᵥ tail v) i)", "tactic": "refine' Fin.cases _ _ i" }, { "state_after": "no goals", "state_before": "case succ.succ.refine'_1\nα : Type u_2\nβ : Type u_1\nf : β → α → β\nb✝ : β\nn✝ : ℕ\nv✝ : Vector α (Nat.succ n✝)\ni✝ : Fin (Nat.succ n✝)\nn : ℕ\nhn :\n ∀ (b : β) (v : Vector α (Nat.succ n)) (i : Fin (Nat.succ n)),\n get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (↑Fin.castSucc i)) (get v i)\nb : β\nv : Vector α (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\n⊢ get (scanl f (f b (head v)) (tail v)) 0 =\n f (get (b ::ᵥ scanl f (f b (head v)) (tail v)) (↑Fin.castSucc 0)) (get (head v ::ᵥ tail v) 0)", "tactic": "simp only [get_zero, scanl_head, Fin.castSucc_zero, head_cons]" }, { "state_after": "case succ.succ.refine'_2\nα : Type u_2\nβ : Type u_1\nf : β → α → β\nb✝ : β\nn✝ : ℕ\nv✝ : Vector α (Nat.succ n✝)\ni✝ : Fin (Nat.succ n✝)\nn : ℕ\nhn :\n ∀ (b : β) (v : Vector α (Nat.succ n)) (i : Fin (Nat.succ n)),\n get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (↑Fin.castSucc i)) (get v i)\nb : β\nv : Vector α (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\ni' : Fin (n + 1)\n⊢ get (scanl f (f b (head v)) (tail v)) (Fin.succ i') =\n f (get (b ::ᵥ scanl f (f b (head v)) (tail v)) (↑Fin.castSucc (Fin.succ i')))\n (get (head v ::ᵥ tail v) (Fin.succ i'))", "state_before": "case succ.succ.refine'_2\nα : Type u_2\nβ : Type u_1\nf : β → α → β\nb✝ : β\nn✝ : ℕ\nv✝ : Vector α (Nat.succ n✝)\ni✝ : Fin (Nat.succ n✝)\nn : ℕ\nhn :\n ∀ (b : β) (v : Vector α (Nat.succ n)) (i : Fin (Nat.succ n)),\n get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (↑Fin.castSucc i)) (get v i)\nb : β\nv : Vector α (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\n⊢ ∀ (i : Fin (n + 1)),\n get (scanl f (f b (head v)) (tail v)) (Fin.succ i) =\n f (get (b ::ᵥ scanl f (f b (head v)) (tail v)) (↑Fin.castSucc (Fin.succ i)))\n (get (head v ::ᵥ tail v) (Fin.succ i))", "tactic": "intro i'" }, { "state_after": "no goals", "state_before": "case succ.succ.refine'_2\nα : Type u_2\nβ : Type u_1\nf : β → α → β\nb✝ : β\nn✝ : ℕ\nv✝ : Vector α (Nat.succ n✝)\ni✝ : Fin (Nat.succ n✝)\nn : ℕ\nhn :\n ∀ (b : β) (v : Vector α (Nat.succ n)) (i : Fin (Nat.succ n)),\n get (scanl f b v) (Fin.succ i) = f (get (scanl f b v) (↑Fin.castSucc i)) (get v i)\nb : β\nv : Vector α (Nat.succ (Nat.succ n))\ni : Fin (Nat.succ (Nat.succ n))\ni' : Fin (n + 1)\n⊢ get (scanl f (f b (head v)) (tail v)) (Fin.succ i') =\n f (get (b ::ᵥ scanl f (f b (head v)) (tail v)) (↑Fin.castSucc (Fin.succ i')))\n (get (head v ::ᵥ tail v) (Fin.succ i'))", "tactic": "simp only [hn, Fin.castSucc_fin_succ, get_cons_succ]" } ]

[ 393, 59 ]

[ 382, 1 ]

[]

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[]

[ 361, 63 ]

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[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na b : α\nm : ℕ\nha : (fun b => some a = b) (Stream'.nth (↑s) m)\nn : ℕ\nhb : (fun b_1 => some b = b_1) (Stream'.nth (↑s) n)\n⊢ a = b", "tactic": "injection\n (le_stable s (le_max_left m n) ha.symm).symm.trans (le_stable s (le_max_right m n) hb.symm)" } ]

[ 341, 98 ]

[ 338, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_2\nβ : Type ?u.16749\nγ : Type ?u.16752\nδ : Type ?u.16755\ninst✝¹ : TopologicalSpace α\nι : Sort u_1\ninst✝ : Finite ι\ns : ι → Set α\na : α\n⊢ 𝓝[⋃ (i : ι), s i] a = ⨆ (i : ι), 𝓝[s i] a", "tactic": "rw [← sUnion_range, nhdsWithin_sUnion (finite_range s), iSup_range]" } ]

[ 258, 70 ]

[ 256, 1 ]

[ { "state_after": "L : Language\nM : Type w\nN : Type ?u.34066\nP : Type ?u.34069\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\ns : Set M\n⊢ lift (#↑(range (Term.realize Subtype.val))) ≤ (#Term L ↑s)", "state_before": "L : Language\nM : Type w\nN : Type ?u.34066\nP : Type ?u.34069\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\ns : Set M\n⊢ lift (#{ x // x ∈ LowerAdjoint.toFun (closure L) s }) ≤ (#Term L ↑s)", "tactic": "rw [← SetLike.coe_sort_coe, coe_closure_eq_range_term_realize]" }, { "state_after": "L : Language\nM : Type w\nN : Type ?u.34066\nP : Type ?u.34069\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\ns : Set M\n⊢ lift (#↑(range (Term.realize Subtype.val))) ≤ lift (#Term L ↑s)", "state_before": "L : Language\nM : Type w\nN : Type ?u.34066\nP : Type ?u.34069\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\ns : Set M\n⊢ lift (#↑(range (Term.realize Subtype.val))) ≤ (#Term L ↑s)", "tactic": "rw [← Cardinal.lift_id'.{w, max u w} (#L.Term s)]" }, { "state_after": "no goals", "state_before": "L : Language\nM : Type w\nN : Type ?u.34066\nP : Type ?u.34069\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\ns : Set M\n⊢ lift (#↑(range (Term.realize Subtype.val))) ≤ lift (#Term L ↑s)", "tactic": "exact Cardinal.mk_range_le_lift" } ]

[ 326, 34 ]

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[]

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[ { "state_after": "case refine'_1\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type ?u.130650\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : TopologicalSpace Y\ninst✝² : NormedAddCommGroup E\nf✝ : X → E\nμ✝ : Measure X\ns : Set X\ninst✝¹ : BorelSpace X\ninst✝ : BorelSpace Y\ne : X ≃ₜ Y\nf : Y → E\nμ : Measure X\nh : LocallyIntegrable f\nx : X\n⊢ IntegrableAtFilter (f ∘ ↑e) (𝓝 x)\n\ncase refine'_2\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type ?u.130650\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : TopologicalSpace Y\ninst✝² : NormedAddCommGroup E\nf✝ : X → E\nμ✝ : Measure X\ns : Set X\ninst✝¹ : BorelSpace X\ninst✝ : BorelSpace Y\ne : X ≃ₜ Y\nf : Y → E\nμ : Measure X\nh : LocallyIntegrable (f ∘ ↑e)\nx : Y\n⊢ IntegrableAtFilter f (𝓝 x)", "state_before": "X : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type ?u.130650\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : TopologicalSpace Y\ninst✝² : NormedAddCommGroup E\nf✝ : X → E\nμ✝ : Measure X\ns : Set X\ninst✝¹ : BorelSpace X\ninst✝ : BorelSpace Y\ne : X ≃ₜ Y\nf : Y → E\nμ : Measure X\n⊢ LocallyIntegrable f ↔ LocallyIntegrable (f ∘ ↑e)", "tactic": "refine' ⟨fun h x => _, fun h x => _⟩" }, { "state_after": "case refine'_1.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type ?u.130650\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : TopologicalSpace Y\ninst✝² : NormedAddCommGroup E\nf✝ : X → E\nμ✝ : Measure X\ns : Set X\ninst✝¹ : BorelSpace X\ninst✝ : BorelSpace Y\ne : X ≃ₜ Y\nf : Y → E\nμ : Measure X\nh : LocallyIntegrable f\nx : X\nU : Set Y\nhU : U ∈ 𝓝 (↑e x)\nh'U : IntegrableOn f U\n⊢ IntegrableAtFilter (f ∘ ↑e) (𝓝 x)", "state_before": "case refine'_1\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type ?u.130650\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : TopologicalSpace Y\ninst✝² : NormedAddCommGroup E\nf✝ : X → E\nμ✝ : Measure X\ns : Set X\ninst✝¹ : BorelSpace X\ninst✝ : BorelSpace Y\ne : X ≃ₜ Y\nf : Y → E\nμ : Measure X\nh : LocallyIntegrable f\nx : X\n⊢ IntegrableAtFilter (f ∘ ↑e) (𝓝 x)", "tactic": "rcases h (e x) with ⟨U, hU, h'U⟩" }, { "state_after": "case refine'_1.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type ?u.130650\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : TopologicalSpace Y\ninst✝² : NormedAddCommGroup E\nf✝ : X → E\nμ✝ : Measure X\ns : Set X\ninst✝¹ : BorelSpace X\ninst✝ : BorelSpace Y\ne : X ≃ₜ Y\nf : Y → E\nμ : Measure X\nh : LocallyIntegrable f\nx : X\nU : Set Y\nhU : U ∈ 𝓝 (↑e x)\nh'U : IntegrableOn f U\n⊢ IntegrableOn (f ∘ ↑e) (↑e ⁻¹' U)", "state_before": "case refine'_1.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type ?u.130650\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : TopologicalSpace Y\ninst✝² : NormedAddCommGroup E\nf✝ : X → E\nμ✝ : Measure X\ns : Set X\ninst✝¹ : BorelSpace X\ninst✝ : BorelSpace Y\ne : X ≃ₜ Y\nf : Y → E\nμ : Measure X\nh : LocallyIntegrable f\nx : X\nU : Set Y\nhU : U ∈ 𝓝 (↑e x)\nh'U : IntegrableOn f U\n⊢ IntegrableAtFilter (f ∘ ↑e) (𝓝 x)", "tactic": "refine' ⟨e ⁻¹' U, e.continuous.continuousAt.preimage_mem_nhds hU, _⟩" }, { "state_after": "no goals", "state_before": "case refine'_1.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type ?u.130650\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : TopologicalSpace Y\ninst✝² : NormedAddCommGroup E\nf✝ : X → E\nμ✝ : Measure X\ns : Set X\ninst✝¹ : BorelSpace X\ninst✝ : BorelSpace Y\ne : X ≃ₜ Y\nf : Y → E\nμ : Measure X\nh : LocallyIntegrable f\nx : X\nU : Set Y\nhU : U ∈ 𝓝 (↑e x)\nh'U : IntegrableOn f U\n⊢ IntegrableOn (f ∘ ↑e) (↑e ⁻¹' U)", "tactic": "exact (integrableOn_map_equiv e.toMeasurableEquiv).1 h'U" }, { "state_after": 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"state_after": "case refine'_2.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type ?u.130650\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : TopologicalSpace Y\ninst✝² : NormedAddCommGroup E\nf✝ : X → E\nμ✝ : Measure X\ns : Set X\ninst✝¹ : BorelSpace X\ninst✝ : BorelSpace Y\ne : X ≃ₜ Y\nf : Y → E\nμ : Measure X\nh : LocallyIntegrable (f ∘ ↑e)\nx : Y\nU : Set X\nhU : U ∈ 𝓝 (↑(Homeomorph.symm e) x)\nh'U : IntegrableOn (f ∘ ↑e) U\n⊢ IntegrableOn f (↑(Homeomorph.symm e) ⁻¹' U)", "state_before": "case refine'_2.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type ?u.130650\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : TopologicalSpace Y\ninst✝² : NormedAddCommGroup E\nf✝ : X → E\nμ✝ : Measure X\ns : Set X\ninst✝¹ : BorelSpace X\ninst✝ : BorelSpace Y\ne : X ≃ₜ Y\nf : Y → E\nμ : Measure X\nh : LocallyIntegrable (f ∘ ↑e)\nx : Y\nU : Set X\nhU : U ∈ 𝓝 (↑(Homeomorph.symm e) 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Y\ninst✝² : NormedAddCommGroup E\nf✝ : X → E\nμ✝ : Measure X\ns : Set X\ninst✝¹ : BorelSpace X\ninst✝ : BorelSpace Y\ne : X ≃ₜ Y\nf : Y → E\nμ : Measure X\nh : LocallyIntegrable (f ∘ ↑e)\nx : Y\nU : Set X\nhU : U ∈ 𝓝 (↑(Homeomorph.symm e) x)\nh'U : IntegrableOn (f ∘ ↑e) U\n⊢ IntegrableOn f (↑(Homeomorph.symm e) ⁻¹' U)", "tactic": "apply (integrableOn_map_equiv e.toMeasurableEquiv).2" }, { "state_after": "case refine'_2.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type ?u.130650\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : TopologicalSpace Y\ninst✝² : NormedAddCommGroup E\nf✝ : X → E\nμ✝ : Measure X\ns : Set X\ninst✝¹ : BorelSpace X\ninst✝ : BorelSpace Y\ne : X ≃ₜ Y\nf : Y → E\nμ : Measure X\nh : LocallyIntegrable (f ∘ ↑e)\nx : Y\nU : Set X\nhU : U ∈ 𝓝 (↑(Homeomorph.symm e) x)\nh'U : IntegrableOn (f ∘ ↑e) U\n⊢ IntegrableOn (f ∘ ↑e) (↑e ⁻¹' (↑(Homeomorph.symm e) ⁻¹' U))", "state_before": "case refine'_2.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type ?u.130650\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : TopologicalSpace Y\ninst✝² : NormedAddCommGroup E\nf✝ : X → E\nμ✝ : Measure X\ns : Set X\ninst✝¹ : BorelSpace X\ninst✝ : BorelSpace Y\ne : X ≃ₜ Y\nf : Y → E\nμ : Measure X\nh : LocallyIntegrable (f ∘ ↑e)\nx : Y\nU : Set X\nhU : U ∈ 𝓝 (↑(Homeomorph.symm e) x)\nh'U : IntegrableOn (f ∘ ↑e) U\n⊢ IntegrableOn (f ∘ ↑(Homeomorph.toMeasurableEquiv e))\n (↑(Homeomorph.toMeasurableEquiv e) ⁻¹' (↑(Homeomorph.symm e) ⁻¹' U))", "tactic": "simp only [Homeomorph.toMeasurableEquiv_coe]" }, { "state_after": "case h.e'_6\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type ?u.130650\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : TopologicalSpace Y\ninst✝² : NormedAddCommGroup E\nf✝ : X → E\nμ✝ : Measure X\ns : Set X\ninst✝¹ : BorelSpace X\ninst✝ : BorelSpace Y\ne : X ≃ₜ Y\nf : Y → E\nμ : Measure X\nh : LocallyIntegrable (f ∘ ↑e)\nx : Y\nU : Set X\nhU : U ∈ 𝓝 (↑(Homeomorph.symm e) x)\nh'U : IntegrableOn (f ∘ ↑e) U\n⊢ ↑e ⁻¹' (↑(Homeomorph.symm e) ⁻¹' U) = U", "state_before": "case refine'_2.intro.intro\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type ?u.130650\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : TopologicalSpace Y\ninst✝² : NormedAddCommGroup E\nf✝ : X → E\nμ✝ : Measure X\ns : Set X\ninst✝¹ : BorelSpace X\ninst✝ : BorelSpace Y\ne : X ≃ₜ Y\nf : Y → E\nμ : Measure X\nh : LocallyIntegrable (f ∘ ↑e)\nx : Y\nU : Set X\nhU : U ∈ 𝓝 (↑(Homeomorph.symm e) x)\nh'U : IntegrableOn (f ∘ ↑e) U\n⊢ IntegrableOn (f ∘ ↑e) (↑e ⁻¹' (↑(Homeomorph.symm e) ⁻¹' U))", "tactic": "convert h'U" }, { "state_after": "case h.e'_6.h\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type ?u.130650\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : TopologicalSpace Y\ninst✝² : NormedAddCommGroup E\nf✝ : X → E\nμ✝ : Measure X\ns : Set X\ninst✝¹ : BorelSpace X\ninst✝ : BorelSpace Y\ne : X ≃ₜ Y\nf : Y → E\nμ : Measure X\nh : LocallyIntegrable (f ∘ ↑e)\nx✝ : Y\nU : Set X\nhU : U ∈ 𝓝 (↑(Homeomorph.symm e) x✝)\nh'U : IntegrableOn (f ∘ ↑e) U\nx : X\n⊢ x ∈ ↑e ⁻¹' (↑(Homeomorph.symm e) ⁻¹' U) ↔ x ∈ U", "state_before": "case h.e'_6\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type ?u.130650\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : TopologicalSpace Y\ninst✝² : NormedAddCommGroup E\nf✝ : X → E\nμ✝ : Measure X\ns : Set X\ninst✝¹ : BorelSpace X\ninst✝ : BorelSpace Y\ne : X ≃ₜ Y\nf : Y → E\nμ : Measure X\nh : LocallyIntegrable (f ∘ ↑e)\nx : Y\nU : Set X\nhU : U ∈ 𝓝 (↑(Homeomorph.symm e) x)\nh'U : IntegrableOn (f ∘ ↑e) U\n⊢ ↑e ⁻¹' (↑(Homeomorph.symm e) ⁻¹' U) = U", "tactic": "ext x" }, { "state_after": "no goals", "state_before": "case h.e'_6.h\nX : Type u_1\nY : Type u_2\nE : Type u_3\nR : Type ?u.130650\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : TopologicalSpace Y\ninst✝² : NormedAddCommGroup E\nf✝ : X → E\nμ✝ : Measure X\ns : Set X\ninst✝¹ : BorelSpace X\ninst✝ : BorelSpace Y\ne : X ≃ₜ Y\nf : Y → E\nμ : Measure X\nh : LocallyIntegrable (f ∘ ↑e)\nx✝ : Y\nU : Set X\nhU : U ∈ 𝓝 (↑(Homeomorph.symm e) x✝)\nh'U : IntegrableOn (f ∘ ↑e) U\nx : X\n⊢ x ∈ ↑e ⁻¹' (↑(Homeomorph.symm e) ⁻¹' U) ↔ x ∈ U", "tactic": "simp only [mem_preimage, Homeomorph.symm_apply_apply]" } ]

[ 237, 58 ]

[ 225, 1 ]

[ { "state_after": "α : Type u_2\nβ : Type ?u.42764\nι : Type ?u.42767\nM : Type u_1\nN : Type ?u.42773\ninst✝¹ : One M\ninst✝ : One N\ns t : Set α\nf g : α → M\na : α\nr : M → M → Prop\nh1 : r 1 1\nha : a ∈ s → r (f a) (g a)\n⊢ r (if a ∈ s then f a else 1) (if a ∈ s then g a else 1)", "state_before": "α : Type u_2\nβ : Type ?u.42764\nι : Type ?u.42767\nM : Type u_1\nN : Type ?u.42773\ninst✝¹ : One M\ninst✝ : One N\ns t : Set α\nf g : α → M\na : α\nr : M → M → Prop\nh1 : r 1 1\nha : a ∈ s → r (f a) (g a)\n⊢ r (mulIndicator s f a) (mulIndicator s g a)", "tactic": "simp only [mulIndicator]" }, { "state_after": "case inl\nα : Type u_2\nβ : Type ?u.42764\nι : Type ?u.42767\nM : Type u_1\nN : Type ?u.42773\ninst✝¹ : One M\ninst✝ : One N\ns t : Set α\nf g : α → M\na : α\nr : M → M → Prop\nh1 : r 1 1\nha : a ∈ s → r (f a) (g a)\nhas : a ∈ s\n⊢ r (f a) (g a)\n\ncase inr\nα : Type u_2\nβ : Type ?u.42764\nι : Type ?u.42767\nM : Type u_1\nN : Type ?u.42773\ninst✝¹ : One M\ninst✝ : One N\ns t : Set α\nf g : α → M\na : α\nr : M → M → Prop\nh1 : r 1 1\nha : a ∈ s → r (f a) (g a)\nhas : ¬a ∈ s\n⊢ r 1 1", "state_before": "α : Type u_2\nβ : Type ?u.42764\nι : Type ?u.42767\nM : Type u_1\nN : Type ?u.42773\ninst✝¹ : One M\ninst✝ : One N\ns t : Set α\nf g : α → M\na : α\nr : M → M → Prop\nh1 : r 1 1\nha : a ∈ s → r (f a) (g a)\n⊢ r (if a ∈ s then f a else 1) (if a ∈ s then g a else 1)", "tactic": "split_ifs with has" }, { "state_after": "no goals", "state_before": "case inl\nα : Type u_2\nβ : Type ?u.42764\nι : Type ?u.42767\nM : Type u_1\nN : Type ?u.42773\ninst✝¹ : One M\ninst✝ : One N\ns t : Set α\nf g : α → M\na : α\nr : M → M → Prop\nh1 : r 1 1\nha : a ∈ s → r (f a) (g a)\nhas : a ∈ s\n⊢ r (f a) (g a)\n\ncase inr\nα : Type u_2\nβ : Type ?u.42764\nι : Type ?u.42767\nM : Type u_1\nN : Type ?u.42773\ninst✝¹ : One M\ninst✝ : One N\ns t : Set α\nf g : α → M\na : α\nr : M → M → Prop\nh1 : r 1 1\nha : a ∈ s → r (f a) (g a)\nhas : ¬a ∈ s\n⊢ r 1 1", "tactic": "exacts [ha has, h1]" } ]

[ 360, 22 ]

[ 356, 1 ]

[]

[ 90, 31 ]

[ 89, 1 ]

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CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type ?u.321532\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u_1\ninst✝ : Finite ι\nb : Basis ι R M\nval✝ : Fintype ι\n⊢ ∀ (i : ι × ι),\n ↑(comp (trace R M) (dualTensorHom R M M)) (↑(Basis.tensorProduct (Basis.dualBasis b) b) i) =\n ↑(contractLeft R M) (↑(Basis.tensorProduct (Basis.dualBasis b) b) i)", "tactic": "rintro ⟨i, j⟩" }, { "state_after": "case intro.mk\nR : Type u_2\ninst✝⁵ : CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type ?u.321532\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u_1\ninst✝ : Finite ι\nb : Basis ι R M\nval✝ : Fintype ι\ni j : ι\n⊢ ↑(trace R M) (↑(dualTensorHom R M M) (Basis.coord b i ⊗ₜ[R] ↑b j)) = ↑(contractLeft R M) (Basis.coord b i ⊗ₜ[R] ↑b j)", "state_before": "case intro.mk\nR : Type u_2\ninst✝⁵ : CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type ?u.321532\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u_1\ninst✝ : Finite ι\nb : Basis ι R M\nval✝ : Fintype ι\ni j : ι\n⊢ ↑(comp (trace R M) (dualTensorHom R M M)) (↑(Basis.tensorProduct (Basis.dualBasis b) b) (i, j)) =\n ↑(contractLeft R M) (↑(Basis.tensorProduct (Basis.dualBasis b) b) (i, j))", "tactic": "simp only [Function.comp_apply, Basis.tensorProduct_apply, Basis.coe_dualBasis, coe_comp]" }, { "state_after": "case intro.mk\nR : Type u_2\ninst✝⁵ : CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type ?u.321532\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u_1\ninst✝ : Finite ι\nb : Basis ι R M\nval✝ : Fintype ι\ni j : ι\n⊢ Matrix.trace (stdBasisMatrix j i 1) = ↑(contractLeft R M) (Basis.coord b i ⊗ₜ[R] ↑b j)", "state_before": "case intro.mk\nR : Type u_2\ninst✝⁵ : CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type ?u.321532\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u_1\ninst✝ : Finite ι\nb : Basis ι R M\nval✝ : Fintype ι\ni j : ι\n⊢ ↑(trace R M) (↑(dualTensorHom R M M) (Basis.coord b i ⊗ₜ[R] ↑b j)) = ↑(contractLeft R M) (Basis.coord b i ⊗ₜ[R] ↑b j)", "tactic": "rw [trace_eq_matrix_trace R b, toMatrix_dualTensorHom]" }, { "state_after": "case pos\nR : Type u_2\ninst✝⁵ : CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type ?u.321532\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u_1\ninst✝ : Finite ι\nb : Basis ι R M\nval✝ : Fintype ι\ni j : ι\nhij : i = j\n⊢ Matrix.trace (stdBasisMatrix j i 1) = ↑(contractLeft R M) (Basis.coord b i ⊗ₜ[R] ↑b j)\n\ncase neg\nR : Type u_2\ninst✝⁵ : CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type ?u.321532\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u_1\ninst✝ : Finite ι\nb : Basis ι R M\nval✝ : Fintype ι\ni j : ι\nhij : ¬i = j\n⊢ Matrix.trace (stdBasisMatrix j i 1) = ↑(contractLeft R M) (Basis.coord b i ⊗ₜ[R] ↑b j)", "state_before": "case intro.mk\nR : Type u_2\ninst✝⁵ : CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type ?u.321532\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u_1\ninst✝ : Finite ι\nb : Basis ι R M\nval✝ : Fintype ι\ni j : ι\n⊢ Matrix.trace (stdBasisMatrix j i 1) = ↑(contractLeft R M) (Basis.coord b i ⊗ₜ[R] ↑b j)", "tactic": "by_cases hij : i = j" }, { "state_after": "case neg\nR : Type u_2\ninst✝⁵ : CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type ?u.321532\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u_1\ninst✝ : Finite ι\nb : Basis ι R M\nval✝ : Fintype ι\ni j : ι\nhij : ¬i = j\n⊢ 0 = ↑(contractLeft R M) (Basis.coord b i ⊗ₜ[R] ↑b j)", "state_before": "case neg\nR : Type u_2\ninst✝⁵ : CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type ?u.321532\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u_1\ninst✝ : Finite ι\nb : Basis ι R M\nval✝ : Fintype ι\ni j : ι\nhij : ¬i = j\n⊢ Matrix.trace (stdBasisMatrix j i 1) = ↑(contractLeft R M) (Basis.coord b i ⊗ₜ[R] ↑b j)", "tactic": "rw [Matrix.StdBasisMatrix.trace_zero j i (1 : R) hij]" }, { "state_after": "no goals", "state_before": "case neg\nR : Type u_2\ninst✝⁵ : CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type ?u.321532\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u_1\ninst✝ : Finite ι\nb : Basis ι R M\nval✝ : Fintype ι\ni j : ι\nhij : ¬i = j\n⊢ 0 = ↑(contractLeft R M) (Basis.coord b i ⊗ₜ[R] ↑b j)", "tactic": "simp [Finsupp.single_eq_pi_single, hij]" }, { "state_after": "case pos\nR : Type u_2\ninst✝⁵ : CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type ?u.321532\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u_1\ninst✝ : Finite ι\nb : Basis ι R M\nval✝ : Fintype ι\ni j : ι\nhij : i = j\n⊢ Matrix.trace (stdBasisMatrix j j 1) = ↑(contractLeft R M) (Basis.coord b j ⊗ₜ[R] ↑b j)", "state_before": "case pos\nR : Type u_2\ninst✝⁵ : CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type ?u.321532\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u_1\ninst✝ : Finite ι\nb : Basis ι R M\nval✝ : Fintype ι\ni j : ι\nhij : i = j\n⊢ Matrix.trace (stdBasisMatrix j i 1) = ↑(contractLeft R M) (Basis.coord b i ⊗ₜ[R] ↑b j)", "tactic": "rw [hij]" }, { "state_after": "no goals", "state_before": "case pos\nR : Type u_2\ninst✝⁵ : CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type ?u.321532\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u_1\ninst✝ : Finite ι\nb : Basis ι R M\nval✝ : Fintype ι\ni j : ι\nhij : i = j\n⊢ Matrix.trace (stdBasisMatrix j j 1) = ↑(contractLeft R M) (Basis.coord b j ⊗ₜ[R] ↑b j)", "tactic": "simp" } ]

[ 151, 44 ]

[ 139, 1 ]

[]

[ 736, 43 ]

[ 734, 1 ]

[]

[ 659, 23 ]

[ 657, 1 ]

[]

[ 167, 18 ]

[ 166, 1 ]

[ { "state_after": "α : Type u_2\nβ✝ : Type ?u.5720\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\nβ : Type u_1\nf : List α → β\nb : Filter β\na : α\nl : List α\nthis : 𝓝 (a :: l) = Filter.map (fun p => p.fst :: p.snd) (𝓝 a ×ˢ 𝓝 l)\n⊢ Tendsto (f ∘ fun p => p.fst :: p.snd) (𝓝 a ×ˢ 𝓝 l) b ↔ Tendsto (fun p => f (p.fst :: p.snd)) (𝓝 a ×ˢ 𝓝 l) b", "state_before": "α : Type u_2\nβ✝ : Type ?u.5720\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\nβ : Type u_1\nf : List α → β\nb : Filter β\na : α\nl : List α\nthis : 𝓝 (a :: l) = Filter.map (fun p => p.fst :: p.snd) (𝓝 a ×ˢ 𝓝 l)\n⊢ Tendsto f (𝓝 (a :: l)) b ↔ Tendsto (fun p => f (p.fst :: p.snd)) (𝓝 a ×ˢ 𝓝 l) b", "tactic": "rw [this, Filter.tendsto_map'_iff]" }, { "state_after": "α : Type u_2\nβ✝ : Type ?u.5720\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\nβ : Type u_1\nf : List α → β\nb : Filter β\na : α\nl : List α\nthis : 𝓝 (a :: l) = Filter.map (fun p => p.fst :: p.snd) (𝓝 a ×ˢ 𝓝 l)\n⊢ Tendsto (f ∘ fun p => p.fst :: p.snd) (𝓝 a ×ˢ 𝓝 l) b ↔ Tendsto (fun p => f (p.fst :: p.snd)) (𝓝 a ×ˢ 𝓝 l) b", "state_before": "α : Type u_2\nβ✝ : Type ?u.5720\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\nβ : Type u_1\nf : List α → β\nb : Filter β\na : α\nl : List α\nthis : 𝓝 (a :: l) = Filter.map (fun p => p.fst :: p.snd) (𝓝 a ×ˢ 𝓝 l)\n⊢ Tendsto (f ∘ fun p => p.fst :: p.snd) (𝓝 a ×ˢ 𝓝 l) b ↔ Tendsto (fun p => f (p.fst :: p.snd)) (𝓝 a ×ˢ 𝓝 l) b", "tactic": "dsimp" }, { "state_after": "no goals", "state_before": "α : Type u_2\nβ✝ : Type ?u.5720\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\nβ : Type u_1\nf : List α → β\nb : Filter β\na : α\nl : List α\nthis : 𝓝 (a :: l) = Filter.map (fun p => p.fst :: p.snd) (𝓝 a ×ˢ 𝓝 l)\n⊢ Tendsto (f ∘ fun p => p.fst :: p.snd) (𝓝 a ×ˢ 𝓝 l) b ↔ Tendsto (fun p => f (p.fst :: p.snd)) (𝓝 a ×ˢ 𝓝 l) b", "tactic": "rfl" }, { "state_after": "α : Type u_2\nβ✝ : Type ?u.5720\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\nβ : Type u_1\nf : List α → β\nb : Filter β\na : α\nl : List α\n⊢ (Seq.seq (cons <$> 𝓝 a) fun x => 𝓝 l) = (fun p => p.fst :: p.snd) <$> Seq.seq (Prod.mk <$> 𝓝 a) fun x => 𝓝 l", "state_before": "α : Type u_2\nβ✝ : Type ?u.5720\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β✝\nβ : Type u_1\nf : List α → β\nb : Filter β\na : α\nl : List α\n⊢ 𝓝 (a :: l) = Filter.map (fun p => p.fst :: p.snd) (𝓝 a ×ˢ 𝓝 l)", "tactic": "simp only [nhds_cons, Filter.prod_eq, (Filter.map_def _ _).symm,\n (Filter.seq_eq_filter_seq _ _).symm]" } ]

[ 96, 50 ]

[ 90, 1 ]

[ { "state_after": "R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nhsur : Function.Surjective ↑(algebraMap R A)\nX : Set M\nm : M\nhm : m ∈ restrictScalars R (span A X)\n⊢ m ∈ span R X", "state_before": "R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nhsur : Function.Surjective ↑(algebraMap R A)\nX : Set M\n⊢ restrictScalars R (span A X) = span R X", "tactic": "refine' ((span_le_restrictScalars R A X).antisymm fun m hm => _).symm" }, { "state_after": "R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nhsur : Function.Surjective ↑(algebraMap R A)\nX : Set M\nm✝ : M\nhm✝ : m✝ ∈ restrictScalars R (span A X)\na : A\nm : M\nhm : m ∈ span R X\n⊢ a • m ∈ span R X", "state_before": "R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nhsur : Function.Surjective ↑(algebraMap R A)\nX : Set M\nm : M\nhm : m ∈ restrictScalars R (span A X)\n⊢ m ∈ span R X", "tactic": "refine' span_induction hm subset_span (zero_mem _) (fun _ _ => add_mem) fun a m hm => _" }, { "state_after": "case intro\nR : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nhsur : Function.Surjective ↑(algebraMap R A)\nX : Set M\nm✝ : M\nhm✝ : m✝ ∈ restrictScalars R (span A X)\nm : M\nhm : m ∈ span R X\nr : R\n⊢ ↑(algebraMap R A) r • m ∈ span R X", "state_before": "R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nhsur : Function.Surjective ↑(algebraMap R A)\nX : Set M\nm✝ : M\nhm✝ : m✝ ∈ restrictScalars R (span A X)\na : A\nm : M\nhm : m ∈ span R X\n⊢ a • m ∈ span R X", "tactic": "obtain ⟨r, rfl⟩ := hsur a" }, { "state_after": "no goals", "state_before": "case intro\nR : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nhsur : Function.Surjective ↑(algebraMap R A)\nX : Set M\nm✝ : M\nhm✝ : m✝ ∈ restrictScalars R (span A X)\nm : M\nhm : m ∈ span R X\nr : R\n⊢ ↑(algebraMap R A) r • m ∈ span R X", "tactic": "simpa [algebraMap_smul] using smul_mem _ r hm" } ]

[ 257, 48 ]

[ 252, 1 ]

[]

[ 266, 6 ]

[ 265, 1 ]

[ { "state_after": "α : Type u\nβ : Type ?u.43077\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nh : y < z \\ x\nhxz : x ≤ z\n⊢ x ⊔ y < x ⊔ z \\ x", "state_before": "α : Type u\nβ : Type ?u.43077\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nh : y < z \\ x\nhxz : x ≤ z\n⊢ x ⊔ y < z", "tactic": "rw [← sup_sdiff_cancel_right hxz]" }, { "state_after": "α : Type u\nβ : Type ?u.43077\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nh : y < z \\ x\nhxz : x ≤ z\nh' : x ⊔ z \\ x ≤ x ⊔ y\n⊢ z \\ x ≤ y", "state_before": "α : Type u\nβ : Type ?u.43077\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nh : y < z \\ x\nhxz : x ≤ z\n⊢ x ⊔ y < x ⊔ z \\ x", "tactic": "refine' (sup_le_sup_left h.le _).lt_of_not_le fun h' => h.not_le _" }, { "state_after": "α : Type u\nβ : Type ?u.43077\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nh : y < z \\ x\nhxz : x ≤ z\nh' : x ⊔ z \\ x ≤ x ⊔ y\n⊢ (z \\ x) \\ x ≤ y", "state_before": "α : Type u\nβ : Type ?u.43077\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nh : y < z \\ x\nhxz : x ≤ z\nh' : x ⊔ z \\ x ≤ x ⊔ y\n⊢ z \\ x ≤ y", "tactic": "rw [← sdiff_idem]" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type ?u.43077\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nh : y < z \\ x\nhxz : x ≤ z\nh' : x ⊔ z \\ x ≤ x ⊔ y\n⊢ (z \\ x) \\ x ≤ y", "tactic": "exact (sdiff_le_sdiff_of_sup_le_sup_left h').trans sdiff_le" } ]

[ 478, 62 ]

[ 474, 1 ]

[]

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[ 67, 1 ]

[]

[ 74, 29 ]

[ 73, 1 ]

[ { "state_after": "case hC.a\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.833445\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\ni : R\n⊢ ↑(RingHom.comp (↑(finSuccEquiv R n)) C) i =\n ↑(RingHom.comp\n (eval₂Hom (RingHom.comp Polynomial.C C) fun i => Fin.cases Polynomial.X (fun k => ↑Polynomial.C (X k)) i) C)\n i\n\ncase hX.a\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.833445\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\ni : Fin (n + 1)\nn✝ : ℕ\n⊢ Polynomial.coeff (↑↑(finSuccEquiv R n) (X i)) n✝ =\n Polynomial.coeff\n (↑(eval₂Hom (RingHom.comp Polynomial.C C) fun i => Fin.cases Polynomial.X (fun k => ↑Polynomial.C (X k)) i) (X i))\n n✝", "state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.833445\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\n⊢ ↑(finSuccEquiv R n) =\n eval₂Hom (RingHom.comp Polynomial.C C) fun i => Fin.cases Polynomial.X (fun k => ↑Polynomial.C (X k)) i", "tactic": "ext i : 2" }, { "state_after": "case hC.a\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.833445\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\ni : R\n⊢ ↑(algebraMap R (MvPolynomial (Fin n) R)[X]) i = ↑Polynomial.C (↑C i)", "state_before": "case hC.a\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.833445\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\ni : R\n⊢ ↑(RingHom.comp (↑(finSuccEquiv R n)) C) i =\n ↑(RingHom.comp\n (eval₂Hom (RingHom.comp Polynomial.C C) fun i => Fin.cases Polynomial.X (fun k => ↑Polynomial.C (X k)) i) C)\n i", "tactic": "simp only [finSuccEquiv, optionEquivLeft_apply, aeval_C, AlgEquiv.coe_trans, RingHom.coe_coe,\n coe_eval₂Hom, comp_apply, renameEquiv_apply, eval₂_C, RingHom.coe_comp, rename_C]" }, { "state_after": "no goals", "state_before": "case hC.a\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.833445\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\ni : R\n⊢ ↑(algebraMap R (MvPolynomial (Fin n) R)[X]) i = ↑Polynomial.C (↑C i)", "tactic": "rfl" }, { "state_after": "no goals", "state_before": "case hX.a\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.833445\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\ni : Fin (n + 1)\nn✝ : ℕ\n⊢ Polynomial.coeff (↑↑(finSuccEquiv R n) (X i)) n✝ =\n Polynomial.coeff\n (↑(eval₂Hom (RingHom.comp Polynomial.C C) fun i => Fin.cases Polynomial.X (fun k => ↑Polynomial.C (X k)) i) (X i))\n n✝", "tactic": "refine' Fin.cases _ _ i <;> simp [finSuccEquiv]" } ]

[ 331, 52 ]

[ 323, 1 ]

[ { "state_after": "G : Type u_1\ninst✝ : DivInvMonoid G\na✝ b a : G\nn : ℕ\n⊢ a ^ Int.negSucc n = (a ^ ↑(n + 1))⁻¹", "state_before": "G : Type u_1\ninst✝ : DivInvMonoid G\na✝ b a : G\nn : ℕ\n⊢ a ^ Int.negSucc n = (a ^ (n + 1))⁻¹", "tactic": "rw [← zpow_ofNat]" }, { "state_after": "no goals", "state_before": "G : Type u_1\ninst✝ : DivInvMonoid G\na✝ b a : G\nn : ℕ\n⊢ a ^ Int.negSucc n = (a ^ ↑(n + 1))⁻¹", "tactic": "exact DivInvMonoid.zpow_neg' n a" } ]

[ 937, 35 ]

[ 935, 1 ]

[]

[ 121, 6 ]

[ 120, 1 ]

[]

[ 216, 84 ]

[ 215, 1 ]

[]

[ 200, 6 ]

[ 199, 1 ]

[]

[ 1280, 89 ]

[ 1277, 1 ]

[ { "state_after": "ι : Type ?u.5056\nα : Type u_1\nβ : Type ?u.5062\nr r' : α → α → Prop\ns t : Set α\nx y : α\nh : WellFounded fun a b => r' a b ∧ a ∈ t ∧ b ∈ t\nhle : r ≤ r'\nhst : s ⊆ t\n⊢ WellFounded fun a b => r a b ∧ a ∈ s ∧ b ∈ s", "state_before": "ι : Type ?u.5056\nα : Type u_1\nβ : Type ?u.5062\nr r' : α → α → Prop\ns t : Set α\nx y : α\nh : WellFoundedOn t r'\nhle : r ≤ r'\nhst : s ⊆ t\n⊢ WellFoundedOn s r", "tactic": "rw [wellFoundedOn_iff] at *" }, { "state_after": "no goals", "state_before": "ι : Type ?u.5056\nα : Type u_1\nβ : Type ?u.5062\nr r' : α → α → Prop\ns t : Set α\nx y : α\nh : WellFounded fun a b => r' a b ∧ a ∈ t ∧ b ∈ t\nhle : r ≤ r'\nhst : s ⊆ t\n⊢ WellFounded fun a b => r a b ∧ a ∈ s ∧ b ∈ s", "tactic": "exact Subrelation.wf (fun xy => ⟨hle _ _ xy.1, hst xy.2.1, hst xy.2.2⟩) h" } ]

[ 104, 76 ]

[ 101, 11 ]

[ { "state_after": "no goals", "state_before": "a : ZMod 2\n⊢ -a = a", "tactic": "fin_cases a <;> apply Fin.ext <;> simp [Fin.coe_neg, Int.natMod]" } ]

[ 828, 67 ]

[ 827, 1 ]

[]

[ 287, 77 ]

[ 286, 1 ]

[]

[ 47, 71 ]

[ 46, 1 ]

[ { "state_after": "F : Type ?u.452327\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.452336\nι : Type ?u.452339\nκ : Type ?u.452342\ninst✝¹ : CompleteLattice β\ninst✝ : DecidableEq α\nx : α\nt : Finset α\nf : α → β\ns : β\nhx : ¬x ∈ t\n⊢ (s ⊔ ⨆ (x_1 : α) (_ : x_1 ∈ t), update f x s x_1) = s ⊔ ⨆ (i : α) (_ : i ∈ t), f i", "state_before": "F : Type ?u.452327\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.452336\nι : Type ?u.452339\nκ : Type ?u.452342\ninst✝¹ : CompleteLattice β\ninst✝ : DecidableEq α\nx : α\nt : Finset α\nf : α → β\ns : β\nhx : ¬x ∈ t\n⊢ (⨆ (i : α) (_ : i ∈ insert x t), update f x s i) = s ⊔ ⨆ (i : α) (_ : i ∈ t), f i", "tactic": "simp only [Finset.iSup_insert, update_same]" }, { "state_after": "case e_a.e_s.h.e_s.h\nF : Type ?u.452327\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.452336\nι : Type ?u.452339\nκ : Type ?u.452342\ninst✝¹ : CompleteLattice β\ninst✝ : DecidableEq α\nx : α\nt : Finset α\nf : α → β\ns : β\nhx : ¬x ∈ t\ni : α\nhi : i ∈ t\n⊢ update f x s i = f i", "state_before": "F : Type ?u.452327\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.452336\nι : Type ?u.452339\nκ : Type ?u.452342\ninst✝¹ : CompleteLattice β\ninst✝ : DecidableEq α\nx : α\nt : Finset α\nf : α → β\ns : β\nhx : ¬x ∈ t\n⊢ (s ⊔ ⨆ (x_1 : α) (_ : x_1 ∈ t), update f x s x_1) = s ⊔ ⨆ (i : α) (_ : i ∈ t), f i", "tactic": "rcongr (i hi)" }, { "state_after": "case e_a.e_s.h.e_s.h.h\nF : Type ?u.452327\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.452336\nι : Type ?u.452339\nκ : Type ?u.452342\ninst✝¹ : CompleteLattice β\ninst✝ : DecidableEq α\nx : α\nt : Finset α\nf : α → β\ns : β\nhx : ¬x ∈ t\ni : α\nhi : i ∈ t\n⊢ i ≠ x", "state_before": "case e_a.e_s.h.e_s.h\nF : Type ?u.452327\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.452336\nι : Type ?u.452339\nκ : Type ?u.452342\ninst✝¹ : CompleteLattice β\ninst✝ : DecidableEq α\nx : α\nt : Finset α\nf : α → β\ns : β\nhx : ¬x ∈ t\ni : α\nhi : i ∈ t\n⊢ update f x s i = f i", "tactic": "apply update_noteq" }, { "state_after": "case e_a.e_s.h.e_s.h.h\nF : Type ?u.452327\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.452336\nι : Type ?u.452339\nκ : Type ?u.452342\ninst✝¹ : CompleteLattice β\ninst✝ : DecidableEq α\nt : Finset α\nf : α → β\ns : β\ni : α\nhi : i ∈ t\nhx : ¬i ∈ t\n⊢ False", "state_before": "case e_a.e_s.h.e_s.h.h\nF : Type ?u.452327\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.452336\nι : Type ?u.452339\nκ : Type ?u.452342\ninst✝¹ : CompleteLattice β\ninst✝ : DecidableEq α\nx : α\nt : Finset α\nf : α → β\ns : β\nhx : ¬x ∈ t\ni : α\nhi : i ∈ t\n⊢ i ≠ x", "tactic": "rintro rfl" }, { "state_after": "no goals", "state_before": "case e_a.e_s.h.e_s.h.h\nF : Type ?u.452327\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.452336\nι : Type ?u.452339\nκ : Type ?u.452342\ninst✝¹ : CompleteLattice β\ninst✝ : DecidableEq α\nt : Finset α\nf : α → β\ns : β\ni : α\nhi : i ∈ t\nhx : ¬i ∈ t\n⊢ False", "tactic": "exact hx hi" } ]

[ 1979, 61 ]

[ 1976, 1 ]

[]

[ 284, 42 ]

[ 283, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedField β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nhf : SuperpolynomialDecay l k f\n⊢ SuperpolynomialDecay l k (k⁻¹ * f)", "tactic": "simpa using hf.param_zpow_mul hk (-1)" } ]

[ 274, 40 ]

[ 272, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type ?u.19001\nf✝ fa : α → α\nfb : β → β\nx✝ y : α\nm n : ℕ\nf : α → α\nx : α\n⊢ IsPeriodicPt f (if h : x ∈ periodicPts f then Nat.find h else 0) x", "state_before": "α : Type u_1\nβ : Type ?u.19001\nf✝ fa : α → α\nfb : β → β\nx✝ y : α\nm n : ℕ\nf : α → α\nx : α\n⊢ IsPeriodicPt f (minimalPeriod f x) x", "tactic": "delta minimalPeriod" }, { "state_after": "case inl\nα : Type u_1\nβ : Type ?u.19001\nf✝ fa : α → α\nfb : β → β\nx✝ y : α\nm n : ℕ\nf : α → α\nx : α\nhx : x ∈ periodicPts f\n⊢ IsPeriodicPt f (Nat.find hx) x\n\ncase inr\nα : Type u_1\nβ : Type ?u.19001\nf✝ fa : α → α\nfb : β → β\nx✝ y : α\nm n : ℕ\nf : α → α\nx : α\nhx : ¬x ∈ periodicPts f\n⊢ IsPeriodicPt f 0 x", "state_before": "α : Type u_1\nβ : Type ?u.19001\nf✝ fa : α → α\nfb : β → β\nx✝ y : α\nm n : ℕ\nf : α → α\nx : α\n⊢ IsPeriodicPt f (if h : x ∈ periodicPts f then Nat.find h else 0) x", "tactic": "split_ifs with hx" }, { "state_after": "no goals", "state_before": "case inl\nα : Type u_1\nβ : Type ?u.19001\nf✝ fa : α → α\nfb : β → β\nx✝ y : α\nm n : ℕ\nf : α → α\nx : α\nhx : x ∈ periodicPts f\n⊢ IsPeriodicPt f (Nat.find hx) x", "tactic": "exact (Nat.find_spec hx).2" }, { "state_after": "no goals", "state_before": "case inr\nα : Type u_1\nβ : Type ?u.19001\nf✝ fa : α → α\nfb : β → β\nx✝ y : α\nm n : ℕ\nf : α → α\nx : α\nhx : ¬x ∈ periodicPts f\n⊢ IsPeriodicPt f 0 x", "tactic": "exact isPeriodicPt_zero f x" } ]

[ 281, 32 ]

[ 277, 1 ]

[]

[ 318, 86 ]

[ 315, 1 ]

[]

[ 207, 33 ]

[ 205, 1 ]

[]

[ 167, 50 ]

[ 166, 1 ]

[]

[ 201, 46 ]

[ 199, 1 ]

[]

[ 986, 6 ]

[ 984, 1 ]

[]

[ 799, 99 ]

[ 799, 9 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\nl : List α\n⊢ length (sublists' (a :: l)) = 2 ^ length (a :: l)", "tactic": "simp_arith only [sublists'_cons, length_append, length_sublists' l,\n length_map, length, Nat.pow_succ', mul_succ, mul_zero, zero_add]" } ]

[ 103, 71 ]

[ 99, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type ?u.977234\nA : Type u_1\nK : Type ?u.977240\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Field K\ninst✝¹ : IsDomain A\ninst✝ : IsDedekindDomain A\nI : Ideal R\nJ : Ideal A\nX : ↑{p | p ∣ J}\n⊢ ↑(idealFactorsFunOfQuotHom (_ : Function.Surjective ↑(RingHom.id (A ⧸ J)))) X = ↑OrderHom.id X", "tactic": "simp only [idealFactorsFunOfQuotHom, map_id, OrderHom.coe_fun_mk, OrderHom.id_coe, id.def,\n comap_map_of_surjective (Ideal.Quotient.mk J) Quotient.mk_surjective, ←\n RingHom.ker_eq_comap_bot (Ideal.Quotient.mk J), mk_ker,\n sup_eq_left.mpr (dvd_iff_le.mp X.prop), Subtype.coe_eta]" } ]

[ 1081, 66 ]

[ 1074, 1 ]

[ { "state_after": "no goals", "state_before": "F : Type ?u.65500\nG : Type ?u.65503\nα : Type u_1\nM : Type ?u.65509\nN : Type ?u.65512\ninst✝ : DivisionCommMonoid α\na b c d : α\nhb : IsUnit b\nhd : IsUnit d\n⊢ a / b = c / d ↔ a * d = c * b", "tactic": "rw [← (hb.mul hd).mul_left_inj, ← mul_assoc, hb.div_mul_cancel, ← mul_assoc, mul_right_comm,\n hd.div_mul_cancel]" } ]

[ 499, 23 ]

[ 496, 11 ]

[ { "state_after": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.54964\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nμa : Measure α\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\ng : α → β\ne : β ≃ᵐ γ\nh : MeasurePreserving ↑e\nhg : MeasurePreserving (↑e ∘ g)\n⊢ MeasurePreserving g", "state_before": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.54964\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nμa : Measure α\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\ng : α → β\ne : β ≃ᵐ γ\nh : MeasurePreserving ↑e\n⊢ MeasurePreserving (↑e ∘ g) ↔ MeasurePreserving g", "tactic": "refine' ⟨fun hg => _, fun hg => h.comp hg⟩" }, { "state_after": "case h.e'_5\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.54964\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nμa : Measure α\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\ng : α → β\ne : β ≃ᵐ γ\nh : MeasurePreserving ↑e\nhg : MeasurePreserving (↑e ∘ g)\n⊢ g = ↑(MeasurableEquiv.symm e) ∘ ↑e ∘ g", "state_before": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.54964\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nμa : Measure α\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\ng : α → β\ne : β ≃ᵐ γ\nh : MeasurePreserving ↑e\nhg : MeasurePreserving (↑e ∘ g)\n⊢ MeasurePreserving g", "tactic": "convert (MeasurePreserving.symm e h).comp hg" }, { "state_after": "no goals", "state_before": "case h.e'_5\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.54964\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nμa : Measure α\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\ng : α → β\ne : β ≃ᵐ γ\nh : MeasurePreserving ↑e\nhg : MeasurePreserving (↑e ∘ g)\n⊢ g = ↑(MeasurableEquiv.symm e) ∘ ↑e ∘ g", "tactic": "simp [← Function.comp.assoc e.symm e g]" } ]

[ 109, 42 ]

[ 105, 11 ]

[ { "state_after": "no goals", "state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf : G →g G'\nf' : G' →g G''\nu v u' v' : V\np✝ : Walk G u v\np : Walk G u u\n⊢ Walk.map f p = nil ↔ p = nil", "tactic": "cases p <;> simp" } ]

[ 1495, 89 ]

[ 1495, 1 ]

[ { "state_after": "no goals", "state_before": "C : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\nf : X ⟶ Y\ninst✝ : Mono f\n⊢ 𝟙 X ≫ MonoOver.arrow (MonoOver.mk' f) = f", "tactic": "simp" } ]

[ 84, 17 ]

[ 83, 1 ]

[]

[ 294, 47 ]

[ 292, 1 ]

[ { "state_after": "ι : Type ?u.788537\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝¹ : CommMonoid β\nR : Setoid α\ninst✝ : DecidableRel Setoid.r\nh : ∀ (x : α), x ∈ s → ∏ a in filter (fun y => y ≈ x) s, f a = 1\n⊢ ∀ (x : Quotient R), x ∈ image Quotient.mk'' s → ∏ y in filter (fun x_1 => Quotient.mk R x_1 = x) s, f y = 1", "state_before": "ι : Type ?u.788537\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝¹ : CommMonoid β\nR : Setoid α\ninst✝ : DecidableRel Setoid.r\nh : ∀ (x : α), x ∈ s → ∏ a in filter (fun y => y ≈ x) s, f a = 1\n⊢ ∏ x in s, f x = 1", "tactic": "rw [prod_partition R, ← Finset.prod_eq_one]" }, { "state_after": "ι : Type ?u.788537\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝¹ : CommMonoid β\nR : Setoid α\ninst✝ : DecidableRel Setoid.r\nh : ∀ (x : α), x ∈ s → ∏ a in filter (fun y => y ≈ x) s, f a = 1\nxbar : Quotient R\nxbar_in_s : xbar ∈ image Quotient.mk'' s\n⊢ ∏ y in filter (fun x => Quotient.mk R x = xbar) s, f y = 1", "state_before": "ι : Type ?u.788537\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝¹ : CommMonoid β\nR : Setoid α\ninst✝ : DecidableRel Setoid.r\nh : ∀ (x : α), x ∈ s → ∏ a in filter (fun y => y ≈ x) s, f a = 1\n⊢ ∀ (x : Quotient R), x ∈ image Quotient.mk'' s → ∏ y in filter (fun x_1 => Quotient.mk R x_1 = x) s, f y = 1", "tactic": "intro xbar xbar_in_s" }, { "state_after": "case intro.intro\nι : Type ?u.788537\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝¹ : CommMonoid β\nR : Setoid α\ninst✝ : DecidableRel Setoid.r\nh : ∀ (x : α), x ∈ s → ∏ a in filter (fun y => y ≈ x) s, f a = 1\nx : α\nx_in_s : x ∈ s\nxbar_in_s : Quotient.mk'' x ∈ image Quotient.mk'' s\n⊢ ∏ y in filter (fun x_1 => Quotient.mk R x_1 = Quotient.mk'' x) s, f y = 1", "state_before": "ι : Type ?u.788537\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝¹ : CommMonoid β\nR : Setoid α\ninst✝ : DecidableRel Setoid.r\nh : ∀ (x : α), x ∈ s → ∏ a in filter (fun y => y ≈ x) s, f a = 1\nxbar : Quotient R\nxbar_in_s : xbar ∈ image Quotient.mk'' s\n⊢ ∏ y in filter (fun x => Quotient.mk R x = xbar) s, f y = 1", "tactic": "obtain ⟨x, x_in_s, rfl⟩ := mem_image.mp xbar_in_s" }, { "state_after": "case intro.intro\nι : Type ?u.788537\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝¹ : CommMonoid β\nR : Setoid α\ninst✝ : DecidableRel Setoid.r\nx : α\nx_in_s : x ∈ s\nxbar_in_s : Quotient.mk'' x ∈ image Quotient.mk'' s\nh : ∀ (x : α), x ∈ s → ∏ x in filter (fun y => Quotient.mk R y = Quotient.mk R x) s, f x = 1\n⊢ ∏ y in filter (fun x_1 => Quotient.mk R x_1 = Quotient.mk'' x) s, f y = 1", "state_before": "case intro.intro\nι : Type ?u.788537\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝¹ : CommMonoid β\nR : Setoid α\ninst✝ : DecidableRel Setoid.r\nh : ∀ (x : α), x ∈ s → ∏ a in filter (fun y => y ≈ x) s, f a = 1\nx : α\nx_in_s : x ∈ s\nxbar_in_s : Quotient.mk'' x ∈ image Quotient.mk'' s\n⊢ ∏ y in filter (fun x_1 => Quotient.mk R x_1 = Quotient.mk'' x) s, f y = 1", "tactic": "simp only [← Quotient.eq] at h" }, { "state_after": "no goals", "state_before": "case intro.intro\nι : Type ?u.788537\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝¹ : CommMonoid β\nR : Setoid α\ninst✝ : DecidableRel Setoid.r\nx : α\nx_in_s : x ∈ s\nxbar_in_s : Quotient.mk'' x ∈ image Quotient.mk'' s\nh : ∀ (x : α), x ∈ s → ∏ x in filter (fun y => Quotient.mk R y = Quotient.mk R x) s, f x = 1\n⊢ ∏ y in filter (fun x_1 => Quotient.mk R x_1 = Quotient.mk'' x) s, f y = 1", "tactic": "exact h x x_in_s" } ]

[ 1603, 19 ]

[ 1597, 1 ]

[ { "state_after": "R : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\n⊢ ∑ i in Finite.toFinset (_ : Set.Finite (Function.support fun m => ↑(weightedHomogeneousComponent w m) φ)),\n ↑(weightedHomogeneousComponent w i) φ =\n φ", "state_before": "R : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\n⊢ (∑ᶠ (m : M), ↑(weightedHomogeneousComponent w m) φ) = φ", "tactic": "rw [finsum_eq_sum _ (weightedHomogeneousComponent_finsupp φ)]" }, { "state_after": "case a\nR : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\n⊢ coeff d\n (∑ i in Finite.toFinset (_ : Set.Finite (Function.support fun m => ↑(weightedHomogeneousComponent w m) φ)),\n ↑(weightedHomogeneousComponent w i) φ) =\n coeff d φ", "state_before": "R : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\n⊢ ∑ i in Finite.toFinset (_ : Set.Finite (Function.support fun m => ↑(weightedHomogeneousComponent w m) φ)),\n ↑(weightedHomogeneousComponent w i) φ =\n φ", "tactic": "ext1 d" }, { "state_after": "case a\nR : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\n⊢ (∑ x in Finite.toFinset (_ : Set.Finite (Function.support fun m => ↑(weightedHomogeneousComponent w m) φ)),\n if ↑(weightedDegree' w) d = x then coeff d φ else 0) =\n coeff d φ", "state_before": "case a\nR : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\n⊢ coeff d\n (∑ i in Finite.toFinset (_ : Set.Finite (Function.support fun m => ↑(weightedHomogeneousComponent w m) φ)),\n ↑(weightedHomogeneousComponent w i) φ) =\n coeff d φ", "tactic": "simp only [coeff_sum, coeff_weightedHomogeneousComponent]" }, { "state_after": "case a\nR : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\n⊢ (if ↑(weightedDegree' w) d = ↑(weightedDegree' w) d then coeff d φ else 0) = coeff d φ\n\ncase a.h₀\nR : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\n⊢ ∀ (b : (fun x => M) d),\n b ∈ Finite.toFinset (_ : Set.Finite (Function.support fun m => ↑(weightedHomogeneousComponent w m) φ)) →\n b ≠ ↑(weightedDegree' w) d → (if ↑(weightedDegree' w) d = b then coeff d φ else 0) = 0\n\ncase a.h₁\nR : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\n⊢ ¬↑(weightedDegree' w) d ∈\n Finite.toFinset (_ : Set.Finite (Function.support fun m => ↑(weightedHomogeneousComponent w m) φ)) →\n (if ↑(weightedDegree' w) d = ↑(weightedDegree' w) d then coeff d φ else 0) = 0", "state_before": "case a\nR : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\n⊢ (∑ x in Finite.toFinset (_ : Set.Finite (Function.support fun m => ↑(weightedHomogeneousComponent w m) φ)),\n if ↑(weightedDegree' w) d = x then coeff d φ else 0) =\n coeff d φ", "tactic": "rw [Finset.sum_eq_single (weightedDegree' w d)]" }, { "state_after": "no goals", "state_before": "case a\nR : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\n⊢ (if ↑(weightedDegree' w) d = ↑(weightedDegree' w) d then coeff d φ else 0) = coeff d φ", "tactic": "rw [if_pos rfl]" }, { "state_after": "case a.h₀\nR : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\nm : M\na✝ : m ∈ Finite.toFinset (_ : Set.Finite (Function.support fun m => ↑(weightedHomogeneousComponent w m) φ))\nhm' : m ≠ ↑(weightedDegree' w) d\n⊢ (if ↑(weightedDegree' w) d = m then coeff d φ else 0) = 0", "state_before": "case a.h₀\nR : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\n⊢ ∀ (b : (fun x => M) d),\n b ∈ Finite.toFinset (_ : Set.Finite (Function.support fun m => ↑(weightedHomogeneousComponent w m) φ)) →\n b ≠ ↑(weightedDegree' w) d → (if ↑(weightedDegree' w) d = b then coeff d φ else 0) = 0", "tactic": "intro m _ hm'" }, { "state_after": "no goals", "state_before": "case a.h₀\nR : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\nm : M\na✝ : m ∈ Finite.toFinset (_ : Set.Finite (Function.support fun m => ↑(weightedHomogeneousComponent w m) φ))\nhm' : m ≠ ↑(weightedDegree' w) d\n⊢ (if ↑(weightedDegree' w) d = m then coeff d φ else 0) = 0", "tactic": "rw [if_neg hm'.symm]" }, { "state_after": "case a.h₁\nR : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\nhm :\n ¬↑(weightedDegree' w) d ∈\n Finite.toFinset (_ : Set.Finite (Function.support fun m => ↑(weightedHomogeneousComponent w m) φ))\n⊢ (if ↑(weightedDegree' w) d = ↑(weightedDegree' w) d then coeff d φ else 0) = 0", "state_before": "case a.h₁\nR : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\n⊢ ¬↑(weightedDegree' w) d ∈\n Finite.toFinset (_ : Set.Finite (Function.support fun m => ↑(weightedHomogeneousComponent w m) φ)) →\n (if ↑(weightedDegree' w) d = ↑(weightedDegree' w) d then coeff d φ else 0) = 0", "tactic": "intro hm" }, { "state_after": "case a.h₁\nR : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\nhm :\n ¬↑(weightedDegree' w) d ∈\n Finite.toFinset (_ : Set.Finite (Function.support fun m => ↑(weightedHomogeneousComponent w m) φ))\n⊢ coeff d φ = 0", "state_before": "case a.h₁\nR : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\nhm :\n ¬↑(weightedDegree' w) d ∈\n Finite.toFinset (_ : Set.Finite (Function.support fun m => ↑(weightedHomogeneousComponent w m) φ))\n⊢ (if ↑(weightedDegree' w) d = ↑(weightedDegree' w) d then coeff d φ else 0) = 0", "tactic": "rw [if_pos rfl]" }, { "state_after": "case a.h₁\nR : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\nhm : ↑(weightedHomogeneousComponent w (↑(weightedDegree' w) d)) φ = 0\n⊢ coeff d φ = 0", "state_before": "case a.h₁\nR : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\nhm :\n ¬↑(weightedDegree' w) d ∈\n Finite.toFinset (_ : Set.Finite (Function.support fun m => ↑(weightedHomogeneousComponent w m) φ))\n⊢ coeff d φ = 0", "tactic": "simp only [Finite.mem_toFinset, mem_support, Ne.def, Classical.not_not] at hm" }, { "state_after": "case a.h₁\nR : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\nhm : ↑(weightedHomogeneousComponent w (↑(weightedDegree' w) d)) φ = 0\nthis :\n coeff d (↑(weightedHomogeneousComponent w (↑(weightedDegree' w) d)) φ) =\n if ↑(weightedDegree' w) d = ↑(weightedDegree' w) d then coeff d φ else 0\n⊢ coeff d φ = 0", "state_before": "case a.h₁\nR : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\nhm : ↑(weightedHomogeneousComponent w (↑(weightedDegree' w) d)) φ = 0\n⊢ coeff d φ = 0", "tactic": "have := coeff_weightedHomogeneousComponent (w := w) (weightedDegree' w d) φ d" }, { "state_after": "case a.h₁\nR : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\nhm : ↑(weightedHomogeneousComponent w (↑(weightedDegree' w) d)) φ = 0\nthis : 0 = coeff d φ\n⊢ coeff d φ = 0", "state_before": "case a.h₁\nR : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\nhm : ↑(weightedHomogeneousComponent w (↑(weightedDegree' w) d)) φ = 0\nthis :\n coeff d (↑(weightedHomogeneousComponent w (↑(weightedDegree' w) d)) φ) =\n if ↑(weightedDegree' w) d = ↑(weightedDegree' w) d then coeff d φ else 0\n⊢ coeff d φ = 0", "tactic": "rw [hm, if_pos rfl, coeff_zero] at this" }, { "state_after": "no goals", "state_before": "case a.h₁\nR : Type u_1\nM : Type u_3\ninst✝¹ : CommSemiring R\nσ : Type u_2\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\nhm : ↑(weightedHomogeneousComponent w (↑(weightedDegree' w) d)) φ = 0\nthis : 0 = coeff d φ\n⊢ coeff d φ = 0", "tactic": "exact this.symm" } ]

[ 420, 20 ]

[ 405, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : PredOrder α\na b : α\ninst✝ : NoMinOrder α\n⊢ pred a = pred b ↔ a = b", "tactic": "simp_rw [eq_iff_le_not_lt, pred_le_pred_iff, pred_lt_pred_iff]" } ]

[ 827, 65 ]

[ 826, 1 ]

[ { "state_after": "𝕜 : Type ?u.299226\nE : Type ?u.299229\nF : Type u_1\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\ninst✝² : NormedSpace ℝ E\nG : Type ?u.299361\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf g : G → E\nf' g' : G →L[ℝ] E\ns : Set G\nx✝ : G\nn : ℕ∞\nx : F\n⊢ HasStrictFDerivAt (fun x => ↑re (inner x x)) (2 • ↑(innerSL ℝ) x) x", "state_before": "𝕜 : Type ?u.299226\nE : Type ?u.299229\nF : Type u_1\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\ninst✝² : NormedSpace ℝ E\nG : Type ?u.299361\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf g : G → E\nf' g' : G →L[ℝ] E\ns : Set G\nx✝ : G\nn : ℕ∞\nx : F\n⊢ HasStrictFDerivAt (fun x => ‖x‖ ^ 2) (2 • ↑(innerSL ℝ) x) x", "tactic": "simp only [sq, ← @inner_self_eq_norm_mul_norm ℝ]" }, { "state_after": "case h.e'_10.h.h\n𝕜 : Type ?u.299226\nE : Type ?u.299229\nF : Type u_1\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\ninst✝² : NormedSpace ℝ E\nG : Type ?u.299361\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf g : G → E\nf' g' : G →L[ℝ] E\ns : Set G\nx✝ : G\nn : ℕ∞\nx : F\ne_7✝ : normedAddCommGroup = NonUnitalNormedRing.toNormedAddCommGroup\nhe✝ : InnerProductSpace.toNormedSpace = NormedAlgebra.toNormedSpace'\n⊢ 2 • ↑(innerSL ℝ) x =\n ContinuousLinearMap.comp (fderivInnerClm ℝ (id x, id x))\n (ContinuousLinearMap.prod (ContinuousLinearMap.id ℝ F) (ContinuousLinearMap.id ℝ F))", "state_before": "𝕜 : Type ?u.299226\nE : Type ?u.299229\nF : Type u_1\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\ninst✝² : NormedSpace ℝ E\nG : Type ?u.299361\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf g : G → E\nf' g' : G →L[ℝ] E\ns : Set G\nx✝ : G\nn : ℕ∞\nx : F\n⊢ HasStrictFDerivAt (fun x => ↑re (inner x x)) (2 • ↑(innerSL ℝ) x) x", "tactic": "convert (hasStrictFDerivAt_id x).inner ℝ (hasStrictFDerivAt_id x)" }, { "state_after": "case h.e'_10.h.h.h\n𝕜 : Type ?u.299226\nE : Type ?u.299229\nF : Type u_1\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\ninst✝² : NormedSpace ℝ E\nG : Type ?u.299361\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf g : G → E\nf' g' : G →L[ℝ] E\ns : Set G\nx✝ : G\nn : ℕ∞\nx : F\ne_7✝ : normedAddCommGroup = NonUnitalNormedRing.toNormedAddCommGroup\nhe✝ : InnerProductSpace.toNormedSpace = NormedAlgebra.toNormedSpace'\ny : F\n⊢ ↑(2 • ↑(innerSL ℝ) x) y =\n ↑(ContinuousLinearMap.comp (fderivInnerClm ℝ (id x, id x))\n (ContinuousLinearMap.prod (ContinuousLinearMap.id ℝ F) (ContinuousLinearMap.id ℝ F)))\n y", "state_before": "case h.e'_10.h.h\n𝕜 : Type ?u.299226\nE : Type ?u.299229\nF : Type u_1\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\ninst✝² : NormedSpace ℝ E\nG : Type ?u.299361\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf g : G → E\nf' g' : G →L[ℝ] E\ns : Set G\nx✝ : G\nn : ℕ∞\nx : F\ne_7✝ : normedAddCommGroup = NonUnitalNormedRing.toNormedAddCommGroup\nhe✝ : InnerProductSpace.toNormedSpace = NormedAlgebra.toNormedSpace'\n⊢ 2 • ↑(innerSL ℝ) x =\n ContinuousLinearMap.comp (fderivInnerClm ℝ (id x, id x))\n (ContinuousLinearMap.prod (ContinuousLinearMap.id ℝ F) (ContinuousLinearMap.id ℝ F))", "tactic": "ext y" }, { "state_after": "no goals", "state_before": "case h.e'_10.h.h.h\n𝕜 : Type ?u.299226\nE : Type ?u.299229\nF : Type u_1\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\ninst✝² : NormedSpace ℝ E\nG : Type ?u.299361\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf g : G → E\nf' g' : G →L[ℝ] E\ns : Set G\nx✝ : G\nn : ℕ∞\nx : F\ne_7✝ : normedAddCommGroup = NonUnitalNormedRing.toNormedAddCommGroup\nhe✝ : InnerProductSpace.toNormedSpace = NormedAlgebra.toNormedSpace'\ny : F\n⊢ ↑(2 • ↑(innerSL ℝ) x) y =\n ↑(ContinuousLinearMap.comp (fderivInnerClm ℝ (id x, id x))\n (ContinuousLinearMap.prod (ContinuousLinearMap.id ℝ F) (ContinuousLinearMap.id ℝ F)))\n y", "tactic": "simp [two_smul, real_inner_comm]" } ]

[ 229, 35 ]

[ 224, 1 ]

[]

[ 248, 89 ]

[ 246, 1 ]

[ { "state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\na b : α\ns : Set α\nhs : IsClosed (s ∩ Icc a b)\nha : a ∈ s\nhgt : ∀ (x : α), x ∈ s ∩ Ico a b → ∀ (y : α), y ∈ Ioi x → Set.Nonempty (s ∩ Ioc x y)\ny : α\nhy : y ∈ Icc a b\n⊢ y ∈ s", "state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\na b : α\ns : Set α\nhs : IsClosed (s ∩ Icc a b)\nha : a ∈ s\nhgt : ∀ (x : α), x ∈ s ∩ Ico a b → ∀ (y : α), y ∈ Ioi x → Set.Nonempty (s ∩ Ioc x y)\n⊢ Icc a b ⊆ s", "tactic": "intro y hy" }, { "state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\na b : α\ns : Set α\nhs : IsClosed (s ∩ Icc a b)\nha : a ∈ s\nhgt : ∀ (x : α), x ∈ s ∩ Ico a b → ∀ (y : α), y ∈ Ioi x → Set.Nonempty (s ∩ Ioc x y)\ny : α\nhy : y ∈ Icc a b\nthis : IsClosed (s ∩ Icc a y)\n⊢ y ∈ s", "state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\na b : α\ns : Set α\nhs : IsClosed (s ∩ Icc a b)\nha : a ∈ s\nhgt : ∀ (x : α), x ∈ s ∩ Ico a b → ∀ (y : α), y ∈ Ioi x → Set.Nonempty (s ∩ Ioc x y)\ny : α\nhy : y ∈ Icc a b\n⊢ y ∈ s", "tactic": "have : IsClosed (s ∩ Icc a y) := by\n suffices s ∩ Icc a y = s ∩ Icc a b ∩ Icc a y by\n rw [this]\n exact IsClosed.inter hs isClosed_Icc\n rw [inter_assoc]\n congr\n exact (inter_eq_self_of_subset_right <| Icc_subset_Icc_right hy.2).symm" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\na b : α\ns : Set α\nhs : IsClosed (s ∩ Icc a b)\nha : a ∈ s\nhgt : ∀ (x : α), x ∈ s ∩ Ico a b → ∀ (y : α), y ∈ Ioi x → Set.Nonempty (s ∩ Ioc x y)\ny : α\nhy : y ∈ Icc a b\nthis : IsClosed (s ∩ Icc a y)\n⊢ y ∈ s", "tactic": "exact\n IsClosed.mem_of_ge_of_forall_exists_gt this ha hy.1 fun x hx =>\n hgt x ⟨hx.1, Ico_subset_Ico_right hy.2 hx.2⟩ y hx.2.2" }, { "state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\na b : α\ns : Set α\nhs : IsClosed (s ∩ Icc a b)\nha : a ∈ s\nhgt : ∀ (x : α), x ∈ s ∩ Ico a b → ∀ (y : α), y ∈ Ioi x → Set.Nonempty (s ∩ Ioc x y)\ny : α\nhy : y ∈ Icc a b\n⊢ s ∩ Icc a y = s ∩ Icc a b ∩ Icc a y", "state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\na b : α\ns : Set α\nhs : IsClosed (s ∩ Icc a b)\nha : a ∈ s\nhgt : ∀ (x : α), x ∈ s ∩ Ico a b → ∀ (y : α), y ∈ Ioi x → Set.Nonempty (s ∩ Ioc x y)\ny : α\nhy : y ∈ Icc a b\n⊢ IsClosed (s ∩ Icc a y)", "tactic": "suffices s ∩ Icc a y = s ∩ Icc a b ∩ Icc a y by\n rw [this]\n exact IsClosed.inter hs isClosed_Icc" }, { "state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\na b : α\ns : Set α\nhs : IsClosed (s ∩ Icc a b)\nha : a ∈ s\nhgt : ∀ (x : α), x ∈ s ∩ Ico a b → ∀ (y : α), y ∈ Ioi x → Set.Nonempty (s ∩ Ioc x y)\ny : α\nhy : y ∈ Icc a b\n⊢ s ∩ Icc a y = s ∩ (Icc a b ∩ Icc a y)", "state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\na b : α\ns : Set α\nhs : IsClosed (s ∩ Icc a b)\nha : a ∈ s\nhgt : ∀ (x : α), x ∈ s ∩ Ico a b → ∀ (y : α), y ∈ Ioi x → Set.Nonempty (s ∩ Ioc x y)\ny : α\nhy : y ∈ Icc a b\n⊢ s ∩ Icc a y = s ∩ Icc a b ∩ Icc a y", "tactic": "rw [inter_assoc]" }, { "state_after": "case e_a\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\na b : α\ns : Set α\nhs : IsClosed (s ∩ Icc a b)\nha : a ∈ s\nhgt : ∀ (x : α), x ∈ s ∩ Ico a b → ∀ (y : α), y ∈ Ioi x → Set.Nonempty (s ∩ Ioc x y)\ny : α\nhy : y ∈ Icc a b\n⊢ Icc a y = Icc a b ∩ Icc a y", "state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\na b : α\ns : Set α\nhs : IsClosed (s ∩ Icc a b)\nha : a ∈ s\nhgt : ∀ (x : α), x ∈ s ∩ Ico a b → ∀ (y : α), y ∈ Ioi x → Set.Nonempty (s ∩ Ioc x y)\ny : α\nhy : y ∈ Icc a b\n⊢ s ∩ Icc a y = s ∩ (Icc a b ∩ Icc a y)", "tactic": "congr" }, { "state_after": "no goals", "state_before": "case e_a\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\na b : α\ns : Set α\nhs : IsClosed (s ∩ Icc a b)\nha : a ∈ s\nhgt : ∀ (x : α), x ∈ s ∩ Ico a b → ∀ (y : α), y ∈ Ioi x → Set.Nonempty (s ∩ Ioc x y)\ny : α\nhy : y ∈ Icc a b\n⊢ Icc a y = Icc a b ∩ Icc a y", "tactic": "exact (inter_eq_self_of_subset_right <| Icc_subset_Icc_right hy.2).symm" }, { "state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\na b : α\ns : Set α\nhs : IsClosed (s ∩ Icc a b)\nha : a ∈ s\nhgt : ∀ (x : α), x ∈ s ∩ Ico a b → ∀ (y : α), y ∈ Ioi x → Set.Nonempty (s ∩ Ioc x y)\ny : α\nhy : y ∈ Icc a b\nthis : s ∩ Icc a y = s ∩ Icc a b ∩ Icc a y\n⊢ IsClosed (s ∩ Icc a b ∩ Icc a y)", "state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\na b : α\ns : Set α\nhs : IsClosed (s ∩ Icc a b)\nha : a ∈ s\nhgt : ∀ (x : α), x ∈ s ∩ Ico a b → ∀ (y : α), y ∈ Ioi x → Set.Nonempty (s ∩ Ioc x y)\ny : α\nhy : y ∈ Icc a b\nthis : s ∩ Icc a y = s ∩ Icc a b ∩ Icc a y\n⊢ IsClosed (s ∩ Icc a y)", "tactic": "rw [this]" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\na b : α\ns : Set α\nhs : IsClosed (s ∩ Icc a b)\nha : a ∈ s\nhgt : ∀ (x : α), x ∈ s ∩ Ico a b → ∀ (y : α), y ∈ Ioi x → Set.Nonempty (s ∩ Ioc x y)\ny : α\nhy : y ∈ Icc a b\nthis : s ∩ Icc a y = s ∩ Icc a b ∩ Icc a y\n⊢ IsClosed (s ∩ Icc a b ∩ Icc a y)", "tactic": "exact IsClosed.inter hs isClosed_Icc" } ]

[ 375, 60 ]

[ 363, 1 ]

[ { "state_after": "no goals", "state_before": "X : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx : X\nf g : GenLoop 0 x\n⊢ Joined (↑genLoopZeroEquiv f) (↑genLoopZeroEquiv g) ↔ GenLoop.Homotopic f g", "tactic": "rw [← homotopic_genLoopZeroEquiv_symm_iff, Equiv.symm_apply_apply, Equiv.symm_apply_apply]" } ]

[ 240, 93 ]

[ 238, 1 ]

[]

[ 2900, 31 ]

[ 2898, 1 ]

[]

[ 899, 6 ]

[ 898, 1 ]

[]

[ 767, 48 ]

[ 766, 11 ]

[ { "state_after": "𝕜 : Type u_1\nE : Type u_2\ninst✝¹ : NormedField 𝕜\ninst✝ : Norm E\na : 𝕜\nf : 𝕜 → E\nh : IsBoundedUnder (fun x x_1 => x ≤ x_1) (𝓝[{a}ᶜ] a) (norm ∘ f)\n⊢ Tendsto (norm ∘ fun x => (x - a)⁻¹) (𝓝[{a}ᶜ] a) atTop", "state_before": "𝕜 : Type u_1\nE : Type u_2\ninst✝¹ : NormedField 𝕜\ninst✝ : Norm E\na : 𝕜\nf : 𝕜 → E\nh : IsBoundedUnder (fun x x_1 => x ≤ x_1) (𝓝[{a}ᶜ] a) (norm ∘ f)\n⊢ f =o[𝓝[{a}ᶜ] a] fun x => (x - a)⁻¹", "tactic": "refine' (h.isBigO_const (one_ne_zero' ℝ)).trans_isLittleO (isLittleO_const_left.2 <| Or.inr _)" }, { "state_after": "no goals", "state_before": "𝕜 : Type u_1\nE : Type u_2\ninst✝¹ : NormedField 𝕜\ninst✝ : Norm E\na : 𝕜\nf : 𝕜 → E\nh : IsBoundedUnder (fun x x_1 => x ≤ x_1) (𝓝[{a}ᶜ] a) (norm ∘ f)\n⊢ Tendsto (fun x => ‖x - a‖⁻¹) (𝓝[{a}ᶜ] a) atTop", "tactic": "exact (tendsto_norm_sub_self_punctured_nhds a).inv_tendsto_zero" } ]

[ 35, 66 ]

[ 30, 1 ]

[]

[ 120, 50 ]

[ 119, 1 ]

[ { "state_after": "case pos\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np✝ q✝ p q : R[X]\nhp : p = 0\n⊢ leadingCoeff (p * q) = leadingCoeff p * leadingCoeff q\n\ncase neg\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np✝ q✝ p q : R[X]\nhp : ¬p = 0\n⊢ leadingCoeff (p * q) = leadingCoeff p * leadingCoeff q", "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np✝ q✝ p q : R[X]\n⊢ leadingCoeff (p * q) = leadingCoeff p * leadingCoeff q", "tactic": "by_cases hp : p = 0" }, { "state_after": "no goals", "state_before": "case pos\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np✝ q✝ p q : R[X]\nhp : p = 0\n⊢ leadingCoeff (p * q) = leadingCoeff p * leadingCoeff q", "tactic": "simp only [hp, zero_mul, leadingCoeff_zero]" }, { "state_after": "case pos\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np✝ q✝ p q : R[X]\nhp : ¬p = 0\nhq : q = 0\n⊢ leadingCoeff (p * q) = leadingCoeff p * leadingCoeff q\n\ncase neg\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np✝ q✝ p q : R[X]\nhp : ¬p = 0\nhq : ¬q = 0\n⊢ leadingCoeff (p * q) = leadingCoeff p * leadingCoeff q", "state_before": "case neg\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np✝ q✝ p q : R[X]\nhp : ¬p = 0\n⊢ leadingCoeff (p * q) = leadingCoeff p * leadingCoeff q", "tactic": "by_cases hq : q = 0" }, { "state_after": "no goals", "state_before": "case pos\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np✝ q✝ p q : R[X]\nhp : ¬p = 0\nhq : q = 0\n⊢ leadingCoeff (p * q) = leadingCoeff p * leadingCoeff q", "tactic": "simp only [hq, mul_zero, leadingCoeff_zero]" }, { "state_after": "case neg\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np✝ q✝ p q : R[X]\nhp : ¬p = 0\nhq : ¬q = 0\n⊢ leadingCoeff p * leadingCoeff q ≠ 0", "state_before": "case neg\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np✝ q✝ p q : R[X]\nhp : ¬p = 0\nhq : ¬q = 0\n⊢ leadingCoeff (p * q) = leadingCoeff p * leadingCoeff q", "tactic": "rw [leadingCoeff_mul']" }, { "state_after": "no goals", "state_before": "case neg\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\np✝ q✝ p q : R[X]\nhp : ¬p = 0\nhq : ¬q = 0\n⊢ leadingCoeff p * leadingCoeff q ≠ 0", "tactic": "exact mul_ne_zero (mt leadingCoeff_eq_zero.1 hp) (mt leadingCoeff_eq_zero.1 hq)" } ]

[ 1544, 86 ]

[ 1538, 1 ]

[ { "state_after": "α : Type u_2\nι : Type ?u.581469\nγ : Type ?u.581472\nA : Type ?u.581475\nB : Type ?u.581478\nC : Type ?u.581481\ninst✝⁴ : AddCommMonoid A\ninst✝³ : AddCommMonoid B\ninst✝² : 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.584611\nM : Type u_1\nM' : Type ?u.584617\nN : Type ?u.584620\nP : Type ?u.584623\nG : Type ?u.584626\nH : Type ?u.584629\nR : Type ?u.584632\nS : Type ?u.584635\ninst✝¹ : AddCommMonoid M\nf1 f2 : α →₀ M\nhd : Disjoint f1.support f2.support\nβ : Type u_3\ninst✝ : CommMonoid β\ng : α → M → β\nthis : ∀ {f1 f2 : α →₀ M}, Disjoint f1.support f2.support → ∏ x in f1.support, g x (↑f1 x + ↑f2 x) = prod f1 g\n⊢ prod (f1 + f2) g = prod f1 g * prod f2 g", "state_before": "α : Type u_2\nι : Type ?u.581469\nγ : Type ?u.581472\nA : Type ?u.581475\nB : Type ?u.581478\nC : Type ?u.581481\ninst✝⁴ : AddCommMonoid A\ninst✝³ : AddCommMonoid B\ninst✝² : 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.584611\nM : Type u_1\nM' : Type ?u.584617\nN : Type ?u.584620\nP : Type ?u.584623\nG : Type ?u.584626\nH : Type ?u.584629\nR : Type ?u.584632\nS : Type ?u.584635\ninst✝¹ : AddCommMonoid M\nf1 f2 : α →₀ M\nhd : Disjoint f1.support f2.support\nβ : Type u_3\ninst✝ : CommMonoid β\ng : α → M → β\n⊢ prod (f1 + f2) g = prod f1 g * prod f2 g", "tactic": "have :\n ∀ {f1 f2 : α →₀ M},\n Disjoint f1.support f2.support → (∏ x in f1.support, g x (f1 x + f2 x)) = f1.prod g :=\n fun hd =>\n Finset.prod_congr rfl fun x hx => by\n simp only [not_mem_support_iff.mp (disjoint_left.mp hd hx), add_zero]" }, { "state_after": "no goals", "state_before": "α : Type u_2\nι : Type ?u.581469\nγ : Type ?u.581472\nA : Type ?u.581475\nB : Type ?u.581478\nC : Type ?u.581481\ninst✝⁴ : AddCommMonoid A\ninst✝³ : AddCommMonoid B\ninst✝² : 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.584611\nM : Type u_1\nM' : Type ?u.584617\nN : Type ?u.584620\nP : Type ?u.584623\nG : Type ?u.584626\nH : Type ?u.584629\nR : Type ?u.584632\nS : Type ?u.584635\ninst✝¹ : AddCommMonoid M\nf1 f2 : α →₀ M\nhd : Disjoint f1.support f2.support\nβ : Type u_3\ninst✝ : CommMonoid β\ng : α → M → β\nthis : ∀ {f1 f2 : α →₀ M}, Disjoint f1.support f2.support → ∏ x in f1.support, g x (↑f1 x + ↑f2 x) = prod f1 g\n⊢ prod (f1 + f2) g = prod f1 g * prod f2 g", "tactic": "classical simp_rw [← this hd, ← this hd.symm, add_comm (f2 _), Finsupp.prod, support_add_eq hd,\n prod_union hd, add_apply]" }, { "state_after": "no goals", "state_before": "α : Type u_2\nι : Type ?u.581469\nγ : Type ?u.581472\nA : Type ?u.581475\nB : Type ?u.581478\nC : Type ?u.581481\ninst✝⁴ : AddCommMonoid A\ninst✝³ : AddCommMonoid B\ninst✝² : 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.584611\nM : Type u_1\nM' : Type ?u.584617\nN : Type ?u.584620\nP : Type ?u.584623\nG : Type ?u.584626\nH : Type ?u.584629\nR : Type ?u.584632\nS : Type ?u.584635\ninst✝¹ : AddCommMonoid M\nf1 f2 : α →₀ M\nhd✝ : Disjoint f1.support f2.support\nβ : Type u_3\ninst✝ : CommMonoid β\ng : α → M → β\nf1✝ f2✝ : α →₀ M\nhd : Disjoint f1✝.support f2✝.support\nx : α\nhx : x ∈ f1✝.support\n⊢ g x (↑f1✝ x + ↑f2✝ x) = g x (↑f1✝ x)", "tactic": "simp only [not_mem_support_iff.mp (disjoint_left.mp hd hx), add_zero]" }, { "state_after": "no goals", "state_before": "α : Type u_2\nι : Type ?u.581469\nγ : Type ?u.581472\nA : Type ?u.581475\nB : Type ?u.581478\nC : Type ?u.581481\ninst✝⁴ : AddCommMonoid A\ninst✝³ : AddCommMonoid B\ninst✝² : 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.584611\nM : Type u_1\nM' : Type ?u.584617\nN : Type ?u.584620\nP : Type ?u.584623\nG : Type ?u.584626\nH : Type ?u.584629\nR : Type ?u.584632\nS : Type ?u.584635\ninst✝¹ : AddCommMonoid M\nf1 f2 : α →₀ M\nhd : Disjoint f1.support f2.support\nβ : Type u_3\ninst✝ : CommMonoid β\ng : α → M → β\nthis : ∀ {f1 f2 : α →₀ M}, Disjoint f1.support f2.support → ∏ x in f1.support, g x (↑f1 x + ↑f2 x) = prod f1 g\n⊢ prod (f1 + f2) g = prod f1 g * prod f2 g", "tactic": "simp_rw [← this hd, ← this hd.symm, add_comm (f2 _), Finsupp.prod, support_add_eq hd,\nprod_union hd, add_apply]" } ]

[ 589, 32 ]

[ 579, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_4\nβ : Type u_3\nγ : Type u_2\nf✝ : α → β → γ\na : Option α\nb : Option β\nc : Option γ\nδ : Type u_1\nf : γ → β → δ\ng : α → γ\n⊢ map₂ f (Option.map g a) b = map₂ (fun a b => f (g a) b) a b", "tactic": "cases a <;> rfl" } ]

[ 98, 79 ]

[ 97, 1 ]

[ { "state_after": "no goals", "state_before": "S : Type u_2\nR : Type u_3\nR₁ : Type ?u.15005\nM : Type u_4\ninst✝⁴ : Ring R\ninst✝³ : CommRing R₁\ninst✝² : AddCommGroup M\nF : Type u_1\ninst✝¹ : Ring S\ninst✝ : AddMonoidHomClass F R S\nf : M → R\ng : F\nx y : M\n⊢ polar (↑g ∘ f) x y = ↑g (polar f x y)", "tactic": "simp only [polar, Pi.smul_apply, Function.comp_apply, map_sub]" } ]

[ 124, 65 ]

[ 122, 1 ]

[]

[ 699, 22 ]

[ 698, 1 ]

[ { "state_after": "no goals", "state_before": "n✝ n : ℕ\n⊢ 10 % 9 = 1", "tactic": "norm_num" } ]

[ 556, 40 ]

[ 555, 1 ]

[ { "state_after": "X : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx : X\nf g : Path x x\n⊢ Path.Homotopic f g → GenLoop.Homotopic (↑genLoopOneEquivPathSelf.symm f) (↑genLoopOneEquivPathSelf.symm g)", "state_before": "X : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx : X\nf g : Path x x\n⊢ GenLoop.Homotopic (↑genLoopOneEquivPathSelf.symm f) (↑genLoopOneEquivPathSelf.symm g) ↔ Path.Homotopic f g", "tactic": "refine ⟨GenLoop.Homotopic.diagonal, ?_⟩" }, { "state_after": "case intro\nX : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx : X\nf g : Path x x\nH : Path.Homotopy f g\n⊢ GenLoop.Homotopic (↑genLoopOneEquivPathSelf.symm f) (↑genLoopOneEquivPathSelf.symm g)", "state_before": "X : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx : X\nf g : Path x x\n⊢ Path.Homotopic f g → GenLoop.Homotopic (↑genLoopOneEquivPathSelf.symm f) (↑genLoopOneEquivPathSelf.symm g)", "tactic": "rintro ⟨H⟩" }, { "state_after": "case intro\nX : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx : X\nf g : Path x x\nH : Path.Homotopy f g\n⊢ ∀ (t : ↑I) (x_1 : I^ 1),\n x_1 ∈ Cube.boundary 1 →\n ↑(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n (ContinuousMap.Homotopy.compContinuousMap H.toHomotopy (ContinuousMap.eval 0)).toContinuousMap\n (t, x_2))\n x_1 =\n ↑(↑genLoopOneEquivPathSelf.symm f).toContinuousMap x_1 ∧\n ↑(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n (ContinuousMap.Homotopy.compContinuousMap H.toHomotopy (ContinuousMap.eval 0)).toContinuousMap\n (t, x_2))\n x_1 =\n ↑(↑genLoopOneEquivPathSelf.symm g).toContinuousMap x_1", "state_before": "case intro\nX : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx : X\nf g : Path x x\nH : Path.Homotopy f g\n⊢ GenLoop.Homotopic (↑genLoopOneEquivPathSelf.symm f) (↑genLoopOneEquivPathSelf.symm g)", "tactic": "refine ⟨H.1.compContinuousMap _, ?_⟩" }, { "state_after": "case intro\nX : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx : X\nf g : Path x x\nH : Path.Homotopy f g\n⊢ ∀ (t : ↑I) (x_1 : I^ 1),\n x_1 ∈ {0, 1} →\n ↑(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n (ContinuousMap.Homotopy.compContinuousMap H.toHomotopy (ContinuousMap.eval 0)).toContinuousMap\n (t, x_2))\n x_1 =\n ↑(↑genLoopOneEquivPathSelf.symm f).toContinuousMap x_1 ∧\n ↑(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n (ContinuousMap.Homotopy.compContinuousMap H.toHomotopy (ContinuousMap.eval 0)).toContinuousMap\n (t, x_2))\n x_1 =\n ↑(↑genLoopOneEquivPathSelf.symm g).toContinuousMap x_1", "state_before": "case intro\nX : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx : X\nf g : Path x x\nH : Path.Homotopy f g\n⊢ ∀ (t : ↑I) (x_1 : I^ 1),\n x_1 ∈ Cube.boundary 1 →\n ↑(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n (ContinuousMap.Homotopy.compContinuousMap H.toHomotopy (ContinuousMap.eval 0)).toContinuousMap\n (t, x_2))\n x_1 =\n ↑(↑genLoopOneEquivPathSelf.symm f).toContinuousMap x_1 ∧\n ↑(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n (ContinuousMap.Homotopy.compContinuousMap H.toHomotopy (ContinuousMap.eval 0)).toContinuousMap\n (t, x_2))\n x_1 =\n ↑(↑genLoopOneEquivPathSelf.symm g).toContinuousMap x_1", "tactic": "rw [Cube.boundary_one]" }, { "state_after": "case intro.inl\nX : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx : X\nf g : Path x x\nH : Path.Homotopy f g\nt : ↑I\n⊢ ↑(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n (ContinuousMap.Homotopy.compContinuousMap H.toHomotopy (ContinuousMap.eval 0)).toContinuousMap (t, x_1))\n 0 =\n ↑(↑genLoopOneEquivPathSelf.symm f).toContinuousMap 0 ∧\n ↑(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n (ContinuousMap.Homotopy.compContinuousMap H.toHomotopy (ContinuousMap.eval 0)).toContinuousMap (t, x_1))\n 0 =\n ↑(↑genLoopOneEquivPathSelf.symm g).toContinuousMap 0\n\ncase intro.inr\nX : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx : X\nf g : Path x x\nH : Path.Homotopy f g\nt : ↑I\n⊢ ↑(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n (ContinuousMap.Homotopy.compContinuousMap H.toHomotopy (ContinuousMap.eval 0)).toContinuousMap (t, x_1))\n 1 =\n ↑(↑genLoopOneEquivPathSelf.symm f).toContinuousMap 1 ∧\n ↑(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n (ContinuousMap.Homotopy.compContinuousMap H.toHomotopy (ContinuousMap.eval 0)).toContinuousMap (t, x_1))\n 1 =\n ↑(↑genLoopOneEquivPathSelf.symm g).toContinuousMap 1", "state_before": "case intro\nX : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx : X\nf g : Path x x\nH : Path.Homotopy f g\n⊢ ∀ (t : ↑I) (x_1 : I^ 1),\n x_1 ∈ {0, 1} →\n ↑(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n (ContinuousMap.Homotopy.compContinuousMap H.toHomotopy (ContinuousMap.eval 0)).toContinuousMap\n (t, x_2))\n x_1 =\n ↑(↑genLoopOneEquivPathSelf.symm f).toContinuousMap x_1 ∧\n ↑(ContinuousMap.mk fun x_2 =>\n ContinuousMap.toFun\n (ContinuousMap.Homotopy.compContinuousMap H.toHomotopy (ContinuousMap.eval 0)).toContinuousMap\n (t, x_2))\n x_1 =\n ↑(↑genLoopOneEquivPathSelf.symm g).toContinuousMap x_1", "tactic": "rintro t _ (rfl | rfl)" }, { "state_after": "no goals", "state_before": "case intro.inl\nX : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx : X\nf g : Path x x\nH : Path.Homotopy f g\nt : ↑I\n⊢ ↑(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n (ContinuousMap.Homotopy.compContinuousMap H.toHomotopy (ContinuousMap.eval 0)).toContinuousMap (t, x_1))\n 0 =\n ↑(↑genLoopOneEquivPathSelf.symm f).toContinuousMap 0 ∧\n ↑(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n (ContinuousMap.Homotopy.compContinuousMap H.toHomotopy (ContinuousMap.eval 0)).toContinuousMap (t, x_1))\n 0 =\n ↑(↑genLoopOneEquivPathSelf.symm g).toContinuousMap 0", "tactic": "exact H.prop' _ _ (.inl rfl)" }, { "state_after": "no goals", "state_before": "case intro.inr\nX : Type u\ninst✝ : TopologicalSpace X\nn : ℕ\nx : X\nf g : Path x x\nH : Path.Homotopy f g\nt : ↑I\n⊢ ↑(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n (ContinuousMap.Homotopy.compContinuousMap H.toHomotopy (ContinuousMap.eval 0)).toContinuousMap (t, x_1))\n 1 =\n ↑(↑genLoopOneEquivPathSelf.symm f).toContinuousMap 1 ∧\n ↑(ContinuousMap.mk fun x_1 =>\n ContinuousMap.toFun\n (ContinuousMap.Homotopy.compContinuousMap H.toHomotopy (ContinuousMap.eval 0)).toContinuousMap (t, x_1))\n 1 =\n ↑(↑genLoopOneEquivPathSelf.symm g).toContinuousMap 1", "tactic": "exact H.prop' _ _ (.inr rfl)" } ]

[ 273, 33 ]

[ 264, 1 ]

[ { "state_after": "no goals", "state_before": "l : Type ?u.962514\nm : Type u_1\nn : Type u_2\no : Type ?u.962523\nm' : o → Type ?u.962528\nn' : o → Type ?u.962533\nR : Type ?u.962536\nS : Type ?u.962539\nα : Type v\nβ : Type w\nγ : Type ?u.962546\ninst✝¹ : AddMonoid α\ninst✝ : StarAddMonoid α\nM N : Matrix m n α\n⊢ ∀ (i : n) (j : m), (M + N)ᴴ i j = (Mᴴ + Nᴴ) i j", "tactic": "simp" } ]

[ 2151, 24 ]

[ 2149, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nf : R[X]\n⊢ ↑(RingEquiv.symm (polynomialQuotientEquivQuotientPolynomial I)) (↑(Quotient.mk (map C I)) f) =\n Polynomial.map (Quotient.mk I) f", "tactic": "rw [polynomialQuotientEquivQuotientPolynomial, RingEquiv.symm_mk, RingEquiv.coe_mk,\n Equiv.coe_fn_mk, Quotient.lift_mk, coe_eval₂RingHom, eval₂_eq_eval_map, ← Polynomial.map_map,\n ← eval₂_eq_eval_map, Polynomial.eval₂_C_X]" } ]

[ 146, 47 ]

[ 142, 1 ]

[]

[ 52, 4 ]

[ 51, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝ : Semiring R\np : ℕ\nx y : R\nhp : Nat.Prime p\nh : Commute x y\n⊢ (x + y) ^ p = x ^ p + y ^ p + ↑p * ∑ k in Ioo 0 p, x ^ k * y ^ (p - k) * ↑(Nat.choose p k / p)", "tactic": "simpa using h.add_pow_prime_pow_eq hp 1" } ]

[ 51, 45 ]

[ 48, 11 ]

[ { "state_after": "case h.e'_3\nα : Type u_1\ninst✝ : LinearOrderedField α\nf g h : CauSeq α abs\nfg : f < g\ngh : g ≈ h\n⊢ h - f = g - f + -(g - h)", "state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nf g h : CauSeq α abs\nfg : f < g\ngh : g ≈ h\n⊢ Pos (h - f)", "tactic": "convert pos_add_limZero fg (neg_limZero gh) using 1" }, { "state_after": "no goals", "state_before": "case h.e'_3\nα : Type u_1\ninst✝ : LinearOrderedField α\nf g h : CauSeq α abs\nfg : f < g\ngh : g ≈ h\n⊢ h - f = g - f + -(g - h)", "tactic": "simp" } ]

[ 730, 9 ]

[ 727, 1 ]

[]

[ 494, 6 ]

[ 493, 1 ]

[]

[ 1826, 8 ]

[ 1818, 1 ]

[]

[ 1049, 36 ]

[ 1048, 1 ]

[ { "state_after": "case nil\nα : Type u_1\nβ : Type ?u.49282\nl✝ l₁ l₂ l₃ : List α\na b : α\nm n✝¹ : ℕ\nl : List α\nn✝ : ℕ\nhn✝ : n✝ < length (tails l)\nn : ℕ\nhn : n < length (tails [])\n⊢ nthLe (tails []) n hn = drop n []\n\ncase cons\nα : Type u_1\nβ : Type ?u.49282\nl✝¹ l₁ l₂ l₃ : List α\na b : α\nm n✝¹ : ℕ\nl✝ : List α\nn✝ : ℕ\nhn✝ : n✝ < length (tails l✝)\nx : α\nl : List α\nIH : ∀ (n : ℕ) (hn : n < length (tails l)), nthLe (tails l) n hn = drop n l\nn : ℕ\nhn : n < length (tails (x :: l))\n⊢ nthLe (tails (x :: l)) n hn = drop n (x :: l)", "state_before": "α : Type u_1\nβ : Type ?u.49282\nl✝ l₁ l₂ l₃ : List α\na b : α\nm n✝ : ℕ\nl : List α\nn : ℕ\nhn : n < length (tails l)\n⊢ nthLe (tails l) n hn = drop n l", "tactic": "induction' l with x l IH generalizing n" }, { "state_after": "no goals", "state_before": "case nil\nα : Type u_1\nβ : Type ?u.49282\nl✝ l₁ l₂ l₃ : List α\na b : α\nm n✝¹ : ℕ\nl : List α\nn✝ : ℕ\nhn✝ : n✝ < length (tails l)\nn : ℕ\nhn : n < length (tails [])\n⊢ nthLe (tails []) n hn = drop n []", "tactic": "simp" }, { "state_after": "case cons.zero\nα : Type u_1\nβ : Type ?u.49282\nl✝¹ l₁ l₂ l₃ : List α\na b : α\nm n✝ : ℕ\nl✝ : List α\nn : ℕ\nhn✝ : n < length (tails l✝)\nx : α\nl : List α\nIH : ∀ (n : ℕ) (hn : n < length (tails l)), nthLe (tails l) n hn = drop n l\nhn : zero < length (tails (x :: l))\n⊢ nthLe (tails (x :: l)) zero hn = drop zero (x :: l)\n\ncase cons.succ\nα : Type u_1\nβ : Type ?u.49282\nl✝¹ l₁ l₂ l₃ : List α\na b : α\nm n✝¹ : ℕ\nl✝ : List α\nn : ℕ\nhn✝ : n < length (tails l✝)\nx : α\nl : List α\nIH : ∀ (n : ℕ) (hn : n < length (tails l)), nthLe (tails l) n hn = drop n l\nn✝ : ℕ\nhn : succ n✝ < length (tails (x :: l))\n⊢ nthLe (tails (x :: l)) (succ n✝) hn = drop (succ n✝) (x :: l)", "state_before": "case cons\nα : Type u_1\nβ : Type ?u.49282\nl✝¹ l₁ l₂ l₃ : List α\na b : α\nm n✝¹ : ℕ\nl✝ : List α\nn✝ : ℕ\nhn✝ : n✝ < length (tails l✝)\nx : α\nl : List α\nIH : ∀ (n : ℕ) (hn : n < length (tails l)), nthLe (tails l) n hn = drop n l\nn : ℕ\nhn : n < length (tails (x :: l))\n⊢ nthLe (tails (x :: l)) n hn = drop n (x :: l)", "tactic": "cases n" }, { "state_after": "no goals", "state_before": "case cons.zero\nα : Type u_1\nβ : Type ?u.49282\nl✝¹ l₁ l₂ l₃ : List α\na b : α\nm n✝ : ℕ\nl✝ : List α\nn : ℕ\nhn✝ : n < length (tails l✝)\nx : α\nl : List α\nIH : ∀ (n : ℕ) (hn : n < length (tails l)), nthLe (tails l) n hn = drop n l\nhn : zero < length (tails (x :: l))\n⊢ nthLe (tails (x :: l)) zero hn = drop zero (x :: l)", "tactic": "simp[nthLe_cons]" }, { "state_after": "no goals", "state_before": "case cons.succ\nα : Type u_1\nβ : Type ?u.49282\nl✝¹ l₁ l₂ l₃ : List α\na b : α\nm n✝¹ : ℕ\nl✝ : List α\nn : ℕ\nhn✝ : n < length (tails l✝)\nx : α\nl : List α\nIH : ∀ (n : ℕ) (hn : n < length (tails l)), nthLe (tails l) n hn = drop n l\nn✝ : ℕ\nhn : succ n✝ < length (tails (x :: l))\n⊢ nthLe (tails (x :: l)) (succ n✝) hn = drop (succ n✝) (x :: l)", "tactic": "simpa[nthLe_cons] using IH _ _" } ]

[ 446, 37 ]

[ 440, 1 ]

[]

[ 46, 23 ]

[ 44, 11 ]