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tasksource 14

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[ { "state_after": "no goals", "state_before": "L : Language\nT : Theory L\nα : Type w\nn : ℕ\nφ✝ ψ : BoundedFormula L α n\nφ : BoundedFormula L α (n + 1)\nM : Theory.ModelType T\nv : α → ↑M\nxs : Fin n → ↑M\n⊢ Realize (∃'φ ⇔ ∼(∀'∼φ)) v xs", "tactic": "simp" } ]

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[ { "state_after": "no goals", "state_before": "α : Type u_2\nβ : Type u_3\nι : Type ?u.400065\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : β → E\ng : α → β\na : α\nhg : Measurable g\nhf : StronglyMeasurable f\n⊢ (∫ (x : β), f x ∂↑(deterministic g hg) a) = f (g a)", "tactic": "rw [kernel.deterministic_apply, integral_dirac' _ _ hf]" } ]

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[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : RegularSpace α\na✝ : α\ns : Set α\na b : α\n⊢ Disjoint (𝓝 a) (𝓝 b) ↔ ¬a ⤳ b", "tactic": "rw [← nhdsSet_singleton, disjoint_nhdsSet_nhds, specializes_iff_mem_closure]" } ]

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[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nι : Type ?u.30840\nπ : ι → Type ?u.30845\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\na : α\nf : Filter α\nhf : NeBot f\nhfa : f ≤ 𝓟 {a}\n⊢ 𝓟 {a} ≤ 𝓝 a", "tactic": "simpa only [principal_singleton] using pure_le_nhds a" } ]

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[ { "state_after": "ι : Type u_1\nV : Type u\ninst✝² : Category V\ninst✝¹ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ninst✝ : HasKernels V\nC₁ C₂ C₃ : HomologicalComplex V c\nf : C₁ ⟶ C₂\ni : ι\n⊢ Subobject.factorThru (cycles C₁ i) (Subobject.arrow (cycles C₁ i) ≫ Hom.f (𝟙 C₁) i)\n (_ : Subobject.Factors (kernelSubobject (dFrom C₁ i)) (Subobject.arrow (cycles C₁ i) ≫ Hom.f (𝟙 C₁) i)) =\n 𝟙 (Subobject.underlying.obj (cycles C₁ i))", "state_before": "ι : Type u_1\nV : Type u\ninst✝² : Category V\ninst✝¹ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ninst✝ : HasKernels V\nC₁ C₂ C₃ : HomologicalComplex V c\nf : C₁ ⟶ C₂\ni : ι\n⊢ cyclesMap (𝟙 C₁) i = 𝟙 (Subobject.underlying.obj (cycles C₁ i))", "tactic": "dsimp only [cyclesMap]" }, { "state_after": "no goals", "state_before": "ι : Type u_1\nV : Type u\ninst✝² : Category V\ninst✝¹ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ninst✝ : HasKernels V\nC₁ C₂ C₃ : HomologicalComplex V c\nf : C₁ ⟶ C₂\ni : ι\n⊢ Subobject.factorThru (cycles C₁ i) (Subobject.arrow (cycles C₁ i) ≫ Hom.f (𝟙 C₁) i)\n (_ : Subobject.Factors (kernelSubobject (dFrom C₁ i)) (Subobject.arrow (cycles C₁ i) ≫ Hom.f (𝟙 C₁) i)) =\n 𝟙 (Subobject.underlying.obj (cycles C₁ i))", "tactic": "simp" } ]

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[ { "state_after": "F : Type ?u.24201\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.24210\ninst✝³ : LinearOrderedField α\ninst✝² : ConditionallyCompleteLinearOrderedField β\ninst✝¹ : ConditionallyCompleteLinearOrderedField γ\ninst✝ : Archimedean α\na : α\nb : β\nq : ℚ\nx y : α\n⊢ sSup (cutMap β x + cutMap β y) = inducedMap α β x + inducedMap α β y", "state_before": "F : Type ?u.24201\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.24210\ninst✝³ : LinearOrderedField α\ninst✝² : ConditionallyCompleteLinearOrderedField β\ninst✝¹ : ConditionallyCompleteLinearOrderedField γ\ninst✝ : Archimedean α\na : α\nb : β\nq : ℚ\nx y : α\n⊢ inducedMap α β (x + y) = inducedMap α β x + inducedMap α β y", "tactic": "rw [inducedMap, cutMap_add]" }, { "state_after": "no goals", "state_before": "F : Type ?u.24201\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.24210\ninst✝³ : LinearOrderedField α\ninst✝² : ConditionallyCompleteLinearOrderedField β\ninst✝¹ : ConditionallyCompleteLinearOrderedField γ\ninst✝ : Archimedean α\na : α\nb : β\nq : ℚ\nx y : α\n⊢ sSup (cutMap β x + cutMap β y) = inducedMap α β x + inducedMap α β y", "tactic": "exact csSup_add (cutMap_nonempty β x) (cutMap_bddAbove β x) (cutMap_nonempty β y)\n (cutMap_bddAbove β y)" } ]

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[ { "state_after": "ι : Type ?u.15491641\n𝕜 : Type ?u.15491644\nE : Type u_1\nF : Type ?u.15491650\nA : Type ?u.15491653\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\n⊢ (∀ (k : ℕ), k ∈ Ico m n → IntervalIntegrable f μ (a k) (a (k + 1))) →\n (∑ k in Finset.Ico m n, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a n, f x ∂μ", "state_before": "ι : Type ?u.15491641\n𝕜 : Type ?u.15491644\nE : Type u_1\nF : Type ?u.15491650\nA : Type ?u.15491653\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\nhint : ∀ (k : ℕ), k ∈ Ico m n → IntervalIntegrable f μ (a k) (a (k + 1))\n⊢ (∑ k in Finset.Ico m n, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a n, f x ∂μ", "tactic": "revert hint" }, { "state_after": "case refine'_1\nι : Type ?u.15491641\n𝕜 : Type ?u.15491644\nE : Type u_1\nF : Type ?u.15491650\nA : Type ?u.15491653\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\n⊢ (∀ (k : ℕ), k ∈ Ico m m → IntervalIntegrable f μ (a k) (a (k + 1))) →\n (∑ k in Finset.Ico m m, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a m, f x ∂μ\n\ncase refine'_2\nι : Type ?u.15491641\n𝕜 : Type ?u.15491644\nE : Type u_1\nF : Type ?u.15491650\nA : Type ?u.15491653\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\n⊢ ∀ (n : ℕ),\n m ≤ n →\n ((∀ (k : ℕ), k ∈ Ico m n → IntervalIntegrable f μ (a k) (a (k + 1))) →\n (∑ k in Finset.Ico m n, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a n, f x ∂μ) →\n (∀ (k : ℕ), k ∈ Ico m (n + 1) → IntervalIntegrable f μ (a k) (a (k + 1))) →\n (∑ k in Finset.Ico m (n + 1), ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a (n + 1), f x ∂μ", "state_before": "ι : Type ?u.15491641\n𝕜 : Type ?u.15491644\nE : Type u_1\nF : Type ?u.15491650\nA : Type ?u.15491653\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\n⊢ (∀ (k : ℕ), k ∈ Ico m n → IntervalIntegrable f μ (a k) (a (k + 1))) →\n (∑ k in Finset.Ico m n, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a n, f x ∂μ", "tactic": "refine' Nat.le_induction _ _ n hmn" }, { "state_after": "no goals", "state_before": "case refine'_1\nι : Type ?u.15491641\n𝕜 : Type ?u.15491644\nE : Type u_1\nF : Type ?u.15491650\nA : Type ?u.15491653\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\n⊢ (∀ (k : ℕ), k ∈ Ico m m → IntervalIntegrable f μ (a k) (a (k + 1))) →\n (∑ k in Finset.Ico m m, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a m, f x ∂μ", "tactic": "simp" }, { "state_after": "case refine'_2\nι : Type ?u.15491641\n𝕜 : Type ?u.15491644\nE : Type u_1\nF : Type ?u.15491650\nA : Type ?u.15491653\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\np : ℕ\nhmp : m ≤ p\nIH :\n (∀ (k : ℕ), k ∈ Ico m p → IntervalIntegrable f μ (a k) (a (k + 1))) →\n (∑ k in Finset.Ico m p, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a p, f x ∂μ\nh : ∀ (k : ℕ), k ∈ Ico m (p + 1) → IntervalIntegrable f μ (a k) (a (k + 1))\n⊢ (∑ k in Finset.Ico m (p + 1), ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a (p + 1), f x ∂μ", "state_before": "case refine'_2\nι : Type ?u.15491641\n𝕜 : Type ?u.15491644\nE : Type u_1\nF : Type ?u.15491650\nA : Type ?u.15491653\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\n⊢ ∀ (n : ℕ),\n m ≤ n →\n ((∀ (k : ℕ), k ∈ Ico m n → IntervalIntegrable f μ (a k) (a (k + 1))) →\n (∑ k in Finset.Ico m n, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a n, f x ∂μ) →\n (∀ (k : ℕ), k ∈ Ico m (n + 1) → IntervalIntegrable f μ (a k) (a (k + 1))) →\n (∑ k in Finset.Ico m (n + 1), ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a (n + 1), f x ∂μ", "tactic": "intro p hmp IH h" }, { "state_after": "case refine'_2.hab\nι : Type ?u.15491641\n𝕜 : Type ?u.15491644\nE : Type u_1\nF : Type ?u.15491650\nA : Type ?u.15491653\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\np : ℕ\nhmp : m ≤ p\nIH :\n (∀ (k : ℕ), k ∈ Ico m p → IntervalIntegrable f μ (a k) (a (k + 1))) →\n (∑ k in Finset.Ico m p, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a p, f x ∂μ\nh : ∀ (k : ℕ), k ∈ Ico m (p + 1) → IntervalIntegrable f μ (a k) (a (k + 1))\n⊢ IntervalIntegrable (fun x => f x) μ (a m) (a p)\n\ncase refine'_2.hbc\nι : Type ?u.15491641\n𝕜 : Type ?u.15491644\nE : Type u_1\nF : Type ?u.15491650\nA : Type ?u.15491653\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\np : ℕ\nhmp : m ≤ p\nIH :\n (∀ (k : ℕ), k ∈ Ico m p → IntervalIntegrable f μ (a k) (a (k + 1))) →\n (∑ k in Finset.Ico m p, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a p, f x ∂μ\nh : ∀ (k : ℕ), k ∈ Ico m (p + 1) → IntervalIntegrable f μ (a k) (a (k + 1))\n⊢ IntervalIntegrable (fun x => f x) μ (a p) (a (p + 1))\n\ncase refine'_2\nι : Type ?u.15491641\n𝕜 : Type ?u.15491644\nE : Type u_1\nF : Type ?u.15491650\nA : Type ?u.15491653\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\np : ℕ\nhmp : m ≤ p\nIH :\n (∀ (k : ℕ), k ∈ Ico m p → IntervalIntegrable f μ (a k) (a (k + 1))) →\n (∑ k in Finset.Ico m p, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a p, f x ∂μ\nh : ∀ (k : ℕ), k ∈ Ico m (p + 1) → IntervalIntegrable f μ (a k) (a (k + 1))\n⊢ ∀ (k : ℕ), k ∈ Ico m p → IntervalIntegrable f μ (a k) (a (k + 1))", "state_before": "case refine'_2\nι : Type ?u.15491641\n𝕜 : Type ?u.15491644\nE : Type u_1\nF : Type ?u.15491650\nA : Type ?u.15491653\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\np : ℕ\nhmp : m ≤ p\nIH :\n (∀ (k : ℕ), k ∈ Ico m p → IntervalIntegrable f μ (a k) (a (k + 1))) →\n (∑ k in Finset.Ico m p, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a p, f x ∂μ\nh : ∀ (k : ℕ), k ∈ Ico m (p + 1) → IntervalIntegrable f μ (a k) (a (k + 1))\n⊢ (∑ k in Finset.Ico m (p + 1), ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a (p + 1), f x ∂μ", "tactic": "rw [Finset.sum_Ico_succ_top hmp, IH, integral_add_adjacent_intervals]" }, { "state_after": "case refine'_2.hab\nι : Type ?u.15491641\n𝕜 : Type ?u.15491644\nE : Type u_1\nF : Type ?u.15491650\nA : Type ?u.15491653\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\np : ℕ\nhmp : m ≤ p\nIH :\n (∀ (k : ℕ), k ∈ Ico m p → IntervalIntegrable f μ (a k) (a (k + 1))) →\n (∑ k in Finset.Ico m p, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a p, f x ∂μ\nh : ∀ (k : ℕ), k ∈ Ico m (p + 1) → IntervalIntegrable f μ (a k) (a (k + 1))\nk : ℕ\nhk : k ∈ Ico m p\n⊢ k ∈ Ico m (p + 1)", "state_before": "case refine'_2.hab\nι : Type ?u.15491641\n𝕜 : Type ?u.15491644\nE : Type u_1\nF : Type ?u.15491650\nA : Type ?u.15491653\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\np : ℕ\nhmp : m ≤ p\nIH :\n (∀ (k : ℕ), k ∈ Ico m p → IntervalIntegrable f μ (a k) (a (k + 1))) →\n (∑ k in Finset.Ico m p, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a p, f x ∂μ\nh : ∀ (k : ℕ), k ∈ Ico m (p + 1) → IntervalIntegrable f μ (a k) (a (k + 1))\n⊢ IntervalIntegrable (fun x => f x) μ (a m) (a p)", "tactic": "refine IntervalIntegrable.trans_iterate_Ico hmp fun k hk => h k ?_" }, { "state_after": "no goals", "state_before": "case refine'_2.hab\nι : Type ?u.15491641\n𝕜 : Type ?u.15491644\nE : Type u_1\nF : Type ?u.15491650\nA : Type ?u.15491653\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\np : ℕ\nhmp : m ≤ p\nIH :\n (∀ (k : ℕ), k ∈ Ico m p → IntervalIntegrable f μ (a k) (a (k + 1))) →\n (∑ k in Finset.Ico m p, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a p, f x ∂μ\nh : ∀ (k : ℕ), k ∈ Ico m (p + 1) → IntervalIntegrable f μ (a k) (a (k + 1))\nk : ℕ\nhk : k ∈ Ico m p\n⊢ k ∈ Ico m (p + 1)", "tactic": "exact (Ico_subset_Ico le_rfl (Nat.le_succ _)) hk" }, { "state_after": "case refine'_2.hbc.a\nι : Type ?u.15491641\n𝕜 : Type ?u.15491644\nE : Type u_1\nF : Type ?u.15491650\nA : Type ?u.15491653\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\np : ℕ\nhmp : m ≤ p\nIH :\n (∀ (k : ℕ), k ∈ Ico m p → IntervalIntegrable f μ (a k) (a (k + 1))) →\n (∑ k in Finset.Ico m p, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a p, f x ∂μ\nh : ∀ (k : ℕ), k ∈ Ico m (p + 1) → IntervalIntegrable f μ (a k) (a (k + 1))\n⊢ p ∈ Ico m (p + 1)", "state_before": "case refine'_2.hbc\nι : Type ?u.15491641\n𝕜 : Type ?u.15491644\nE : Type u_1\nF : Type ?u.15491650\nA : Type ?u.15491653\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\np : ℕ\nhmp : m ≤ p\nIH :\n (∀ (k : ℕ), k ∈ Ico m p → IntervalIntegrable f μ (a k) (a (k + 1))) →\n (∑ k in Finset.Ico m p, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a p, f x ∂μ\nh : ∀ (k : ℕ), k ∈ Ico m (p + 1) → IntervalIntegrable f μ (a k) (a (k + 1))\n⊢ IntervalIntegrable (fun x => f x) μ (a p) (a (p + 1))", "tactic": "apply h" }, { "state_after": "no goals", "state_before": "case refine'_2.hbc.a\nι : Type ?u.15491641\n𝕜 : Type ?u.15491644\nE : Type u_1\nF : Type ?u.15491650\nA : Type ?u.15491653\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\np : ℕ\nhmp : m ≤ p\nIH :\n (∀ (k : ℕ), k ∈ Ico m p → IntervalIntegrable f μ (a k) (a (k + 1))) →\n (∑ k in Finset.Ico m p, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a p, f x ∂μ\nh : ∀ (k : ℕ), k ∈ Ico m (p + 1) → IntervalIntegrable f μ (a k) (a (k + 1))\n⊢ p ∈ Ico m (p + 1)", "tactic": "simp [hmp]" }, { "state_after": "case refine'_2\nι : Type ?u.15491641\n𝕜 : Type ?u.15491644\nE : Type u_1\nF : Type ?u.15491650\nA : Type ?u.15491653\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\np : ℕ\nhmp : m ≤ p\nIH :\n (∀ (k : ℕ), k ∈ Ico m p → IntervalIntegrable f μ (a k) (a (k + 1))) →\n (∑ k in Finset.Ico m p, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a p, f x ∂μ\nh : ∀ (k : ℕ), k ∈ Ico m (p + 1) → IntervalIntegrable f μ (a k) (a (k + 1))\nk : ℕ\nhk : k ∈ Ico m p\n⊢ IntervalIntegrable f μ (a k) (a (k + 1))", "state_before": "case refine'_2\nι : Type ?u.15491641\n𝕜 : Type ?u.15491644\nE : Type u_1\nF : Type ?u.15491650\nA : Type ?u.15491653\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\np : ℕ\nhmp : m ≤ p\nIH :\n (∀ (k : ℕ), k ∈ Ico m p → IntervalIntegrable f μ (a k) (a (k + 1))) →\n (∑ k in Finset.Ico m p, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a p, f x ∂μ\nh : ∀ (k : ℕ), k ∈ Ico m (p + 1) → IntervalIntegrable f μ (a k) (a (k + 1))\n⊢ ∀ (k : ℕ), k ∈ Ico m p → IntervalIntegrable f μ (a k) (a (k + 1))", "tactic": "intro k hk" }, { "state_after": "no goals", "state_before": "case refine'_2\nι : Type ?u.15491641\n𝕜 : Type ?u.15491644\nE : Type u_1\nF : Type ?u.15491650\nA : Type ?u.15491653\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na✝ b c d : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\na : ℕ → ℝ\nm n : ℕ\nhmn : m ≤ n\np : ℕ\nhmp : m ≤ p\nIH :\n (∀ (k : ℕ), k ∈ Ico m p → IntervalIntegrable f μ (a k) (a (k + 1))) →\n (∑ k in Finset.Ico m p, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ) = ∫ (x : ℝ) in a m..a p, f x ∂μ\nh : ∀ (k : ℕ), k ∈ Ico m (p + 1) → IntervalIntegrable f μ (a k) (a (k + 1))\nk : ℕ\nhk : k ∈ Ico m p\n⊢ IntervalIntegrable f μ (a k) (a (k + 1))", "tactic": "exact h _ (Ico_subset_Ico_right p.le_succ hk)" } ]

[ 921, 52 ]

[ 908, 1 ]

[ { "state_after": "α : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ninst✝ : DecidableRel Disjoint\ns s₁ s₂ t t₁ t₂ u : Finset α\na b c : α\n⊢ (∃ x a, a ∈ s ∧ ∃ b, b ∈ t ∧ Disjoint a b ∧ a ⊔ b = x) → ∃ x, x ∈ s", "state_before": "α : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ninst✝ : DecidableRel Disjoint\ns s₁ s₂ t t₁ t₂ u : Finset α\na b c : α\n⊢ Finset.Nonempty (s ○ t) → Finset.Nonempty s", "tactic": "simp_rw [Finset.Nonempty, mem_disjSups]" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝³ : DecidableEq α\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ninst✝ : DecidableRel Disjoint\ns s₁ s₂ t t₁ t₂ u : Finset α\na b c : α\n⊢ (∃ x a, a ∈ s ∧ ∃ b, b ∈ t ∧ Disjoint a b ∧ a ⊔ b = x) → ∃ x, x ∈ s", "tactic": "exact fun ⟨_, a, ha, _⟩ => ⟨a, ha⟩" } ]

[ 481, 37 ]

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[]

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[]

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[]

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[]

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[]

[ 107, 46 ]

[ 106, 1 ]

[ { "state_after": "no goals", "state_before": "a b c : ℝ≥0\nh : a ≤ b * c\nh0 : c = 0\n⊢ a / c ≤ b", "tactic": "simp [h0]" } ]

[ 817, 59 ]

[ 816, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type ?u.286751\ninst✝ : Preorder α\na b c d : ℕ\nlr₁ : 3 * a ≤ b + c + 1 + d\nmr₂ : b + c + 1 ≤ 3 * d\nmm₁ : b ≤ 3 * c\n⊢ a + b + 1 ≤ 3 * (c + d + 1)", "tactic": "linarith" } ]

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[]

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[ { "state_after": "case h\nα : Type ?u.48928\nβ : Type ?u.48931\nA : Type u_2\nF : Type u_1\ninst✝¹ : MulZeroOneClass A\ninst✝ : MonoidWithZeroHomClass F ℕ A\nf g : F\nh_pos : ∀ {n : ℕ}, 0 < n → ↑f n = ↑g n\n⊢ ∀ (x : ℕ), ↑f x = ↑g x", "state_before": "α : Type ?u.48928\nβ : Type ?u.48931\nA : Type u_2\nF : Type u_1\ninst✝¹ : MulZeroOneClass A\ninst✝ : MonoidWithZeroHomClass F ℕ A\nf g : F\nh_pos : ∀ {n : ℕ}, 0 < n → ↑f n = ↑g n\n⊢ f = g", "tactic": "apply FunLike.ext" }, { "state_after": "case h.zero\nα : Type ?u.48928\nβ : Type ?u.48931\nA : Type u_2\nF : Type u_1\ninst✝¹ : MulZeroOneClass A\ninst✝ : MonoidWithZeroHomClass F ℕ A\nf g : F\nh_pos : ∀ {n : ℕ}, 0 < n → ↑f n = ↑g n\n⊢ ↑f Nat.zero = ↑g Nat.zero\n\ncase h.succ\nα : Type ?u.48928\nβ : Type ?u.48931\nA : Type u_2\nF : Type u_1\ninst✝¹ : MulZeroOneClass A\ninst✝ : MonoidWithZeroHomClass F ℕ A\nf g : F\nh_pos : ∀ {n : ℕ}, 0 < n → ↑f n = ↑g n\nn : ℕ\n⊢ ↑f (Nat.succ n) = ↑g (Nat.succ n)", "state_before": "case h\nα : Type ?u.48928\nβ : Type ?u.48931\nA : Type u_2\nF : Type u_1\ninst✝¹ : MulZeroOneClass A\ninst✝ : MonoidWithZeroHomClass F ℕ A\nf g : F\nh_pos : ∀ {n : ℕ}, 0 < n → ↑f n = ↑g n\n⊢ ∀ (x : ℕ), ↑f x = ↑g x", "tactic": "rintro (_ | n)" }, { "state_after": "no goals", "state_before": "case h.zero\nα : Type ?u.48928\nβ : Type ?u.48931\nA : Type u_2\nF : Type u_1\ninst✝¹ : MulZeroOneClass A\ninst✝ : MonoidWithZeroHomClass F ℕ A\nf g : F\nh_pos : ∀ {n : ℕ}, 0 < n → ↑f n = ↑g n\n⊢ ↑f Nat.zero = ↑g Nat.zero", "tactic": "simp [map_zero f, map_zero g]" }, { "state_after": "no goals", "state_before": "case h.succ\nα : Type ?u.48928\nβ : Type ?u.48931\nA : Type u_2\nF : Type u_1\ninst✝¹ : MulZeroOneClass A\ninst✝ : MonoidWithZeroHomClass F ℕ A\nf g : F\nh_pos : ∀ {n : ℕ}, 0 < n → ↑f n = ↑g n\nn : ℕ\n⊢ ↑f (Nat.succ n) = ↑g (Nat.succ n)", "tactic": "exact h_pos n.succ_pos" } ]

[ 243, 27 ]

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[ { "state_after": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.532921\nE : Type u_1\nEₗ : Type ?u.532927\nF : Type u_2\nFₗ : Type ?u.532933\nG : Type ?u.532936\nGₗ : Type ?u.532939\n𝓕 : Type ?u.532942\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf✝ g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx✝ : E\na b : ℝ\nha : 0 < a\nhb : 0 < b\nf : E →SL[σ₁₂] F\nhf : f ∈ closedBall 0 (a / b)\nx : E\nhx : x ∈ closedBall 0 b\n⊢ ↑f x ∈ closedBall 0 a", "state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.532921\nE : Type u_1\nEₗ : Type ?u.532927\nF : Type u_2\nFₗ : Type ?u.532933\nG : Type ?u.532936\nGₗ : Type ?u.532939\n𝓕 : Type ?u.532942\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx : E\na b : ℝ\nha : 0 < a\nhb : 0 < b\n⊢ closedBall 0 (a / b) ⊆ {f | ∀ (x : E), x ∈ closedBall 0 b → ↑f x ∈ closedBall 0 a}", "tactic": "intro f hf x hx" }, { "state_after": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.532921\nE : Type u_1\nEₗ : Type ?u.532927\nF : Type u_2\nFₗ : Type ?u.532933\nG : Type ?u.532936\nGₗ : Type ?u.532939\n𝓕 : Type ?u.532942\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf✝ g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx✝ : E\na b : ℝ\nha : 0 < a\nhb : 0 < b\nf : E →SL[σ₁₂] F\nhf : ‖f‖ ≤ a / b\nx : E\nhx : ‖x‖ ≤ b\n⊢ ‖↑f x‖ ≤ a", "state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.532921\nE : Type u_1\nEₗ : Type ?u.532927\nF : Type u_2\nFₗ : Type ?u.532933\nG : Type ?u.532936\nGₗ : Type ?u.532939\n𝓕 : Type ?u.532942\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf✝ g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx✝ : E\na b : ℝ\nha : 0 < a\nhb : 0 < b\nf : E →SL[σ₁₂] F\nhf : f ∈ closedBall 0 (a / b)\nx : E\nhx : x ∈ closedBall 0 b\n⊢ ↑f x ∈ closedBall 0 a", "tactic": "rw [mem_closedBall_zero_iff] at hf hx⊢" }, { "state_after": "no goals", "state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.532921\nE : Type u_1\nEₗ : Type ?u.532927\nF : Type u_2\nFₗ : Type ?u.532933\nG : Type ?u.532936\nGₗ : Type ?u.532939\n𝓕 : Type ?u.532942\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf✝ g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx✝ : E\na b : ℝ\nha : 0 < a\nhb : 0 < b\nf : E →SL[σ₁₂] F\nhf : ‖f‖ ≤ a / b\nx : E\nhx : ‖x‖ ≤ b\n⊢ ‖↑f x‖ ≤ a", "tactic": "calc\n ‖f x‖ ≤ ‖f‖ * ‖x‖ := le_op_norm _ _\n _ ≤ a / b * b := by gcongr\n _ = a := div_mul_cancel a hb.ne.symm" }, { "state_after": "no goals", "state_before": "𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type ?u.532921\nE : Type u_1\nEₗ : Type ?u.532927\nF : Type u_2\nFₗ : Type ?u.532933\nG : Type ?u.532936\nGₗ : Type ?u.532939\n𝓕 : Type ?u.532942\ninst✝¹⁷ : SeminormedAddCommGroup E\ninst✝¹⁶ : SeminormedAddCommGroup Eₗ\ninst✝¹⁵ : SeminormedAddCommGroup F\ninst✝¹⁴ : SeminormedAddCommGroup Fₗ\ninst✝¹³ : SeminormedAddCommGroup G\ninst✝¹² : SeminormedAddCommGroup Gₗ\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜₂\ninst✝⁹ : NontriviallyNormedField 𝕜₃\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 Eₗ\ninst✝⁶ : NormedSpace 𝕜₂ F\ninst✝⁵ : NormedSpace 𝕜 Fₗ\ninst✝⁴ : NormedSpace 𝕜₃ G\ninst✝³ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝¹ : RingHomIsometric σ₁₂\ninst✝ : RingHomIsometric σ₂₃\nf✝ g : E →SL[σ₁₂] F\nh : F →SL[σ₂₃] G\nx✝ : E\na b : ℝ\nha : 0 < a\nhb : 0 < b\nf : E →SL[σ₁₂] F\nhf : ‖f‖ ≤ a / b\nx : E\nhx : ‖x‖ ≤ b\n⊢ ‖f‖ * ‖x‖ ≤ a / b * b", "tactic": "gcongr" } ]

[ 359, 41 ]

[ 351, 11 ]

[]

[ 1130, 6 ]

[ 1129, 1 ]

[]

[ 1773, 79 ]

[ 1772, 1 ]

[ { "state_after": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\np a : α\nh : p ∈ factors a\nha : a = 0\n⊢ False", "state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\np a : α\nh : p ∈ factors a\n⊢ a ≠ 0", "tactic": "intro ha" }, { "state_after": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\np a : α\nh : p ∈ 0\nha : a = 0\n⊢ False", "state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\np a : α\nh : p ∈ factors a\nha : a = 0\n⊢ False", "tactic": "rw [factors, dif_pos ha] at h" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : DecidableEq α\ninst✝ : UniqueFactorizationMonoid α\np a : α\nh : p ∈ 0\nha : a = 0\n⊢ False", "tactic": "exact Multiset.not_mem_zero _ h" } ]

[ 456, 34 ]

[ 453, 1 ]

[ { "state_after": "X : Type u_1\nY : Type ?u.48324\nZ : Type ?u.48327\nα : Type ?u.48330\nι : Type ?u.48333\nπ : ι → Type ?u.48338\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : (i : ι) → TopologicalSpace (π i)\nx y z : X\ns : Set X\nf : X → Y\nt : Set (SeparationQuotient X)\nhs : IsOpen s\n⊢ mk ⁻¹' (mk '' s) ⊆ s", "state_before": "X : Type u_1\nY : Type ?u.48324\nZ : Type ?u.48327\nα : Type ?u.48330\nι : Type ?u.48333\nπ : ι → Type ?u.48338\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : (i : ι) → TopologicalSpace (π i)\nx y z : X\ns : Set X\nf : X → Y\nt : Set (SeparationQuotient X)\nhs : IsOpen s\n⊢ mk ⁻¹' (mk '' s) = s", "tactic": "refine' Subset.antisymm _ (subset_preimage_image _ _)" }, { "state_after": "case intro.intro\nX : Type u_1\nY : Type ?u.48324\nZ : Type ?u.48327\nα : Type ?u.48330\nι : Type ?u.48333\nπ : ι → Type ?u.48338\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : (i : ι) → TopologicalSpace (π i)\nx✝ y✝ z : X\ns : Set X\nf : X → Y\nt : Set (SeparationQuotient X)\nhs : IsOpen s\nx y : X\nhys : y ∈ s\nhxy : mk y = mk x\n⊢ x ∈ s", "state_before": "X : Type u_1\nY : Type ?u.48324\nZ : Type ?u.48327\nα : Type ?u.48330\nι : Type ?u.48333\nπ : ι → Type ?u.48338\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : (i : ι) → TopologicalSpace (π i)\nx y z : X\ns : Set X\nf : X → Y\nt : Set (SeparationQuotient X)\nhs : IsOpen s\n⊢ mk ⁻¹' (mk '' s) ⊆ s", "tactic": "rintro x ⟨y, hys, hxy⟩" }, { "state_after": "no goals", "state_before": "case intro.intro\nX : Type u_1\nY : Type ?u.48324\nZ : Type ?u.48327\nα : Type ?u.48330\nι : Type ?u.48333\nπ : ι → Type ?u.48338\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : (i : ι) → TopologicalSpace (π i)\nx✝ y✝ z : X\ns : Set X\nf : X → Y\nt : Set (SeparationQuotient X)\nhs : IsOpen s\nx y : X\nhys : y ∈ s\nhxy : mk y = mk x\n⊢ x ∈ s", "tactic": "exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys" } ]

[ 454, 49 ]

[ 451, 1 ]

[]

[ 1159, 6 ]

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[ { "state_after": "no goals", "state_before": "i j : Nat\nh : i = 0 ∧ j = 0\n⊢ gcd i j = 0", "tactic": "simp [h]" } ]

[ 178, 25 ]

[ 176, 1 ]

[ { "state_after": "α : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns : Set α\nhs : s ∈ insert ∅ S\nt : Set α\nht : t ∈ insert ∅ S\nhst : Set.Nonempty (s ∩ t)\n⊢ s ∩ t ∈ insert ∅ S", "state_before": "α : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\n⊢ IsPiSystem (insert ∅ S)", "tactic": "intro s hs t ht hst" }, { "state_after": "case inl\nα : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns t : Set α\nht : t ∈ insert ∅ S\nhst : Set.Nonempty (s ∩ t)\nhs : s = ∅\n⊢ s ∩ t ∈ insert ∅ S\n\ncase inr\nα : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns t : Set α\nht : t ∈ insert ∅ S\nhst : Set.Nonempty (s ∩ t)\nhs : s ∈ S\n⊢ s ∩ t ∈ insert ∅ S", "state_before": "α : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns : Set α\nhs : s ∈ insert ∅ S\nt : Set α\nht : t ∈ insert ∅ S\nhst : Set.Nonempty (s ∩ t)\n⊢ s ∩ t ∈ insert ∅ S", "tactic": "cases' hs with hs hs" }, { "state_after": "no goals", "state_before": "case inl\nα : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns t : Set α\nht : t ∈ insert ∅ S\nhst : Set.Nonempty (s ∩ t)\nhs : s = ∅\n⊢ s ∩ t ∈ insert ∅ S", "tactic": "simp [hs]" }, { "state_after": "case inr.inl\nα : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns t : Set α\nhst : Set.Nonempty (s ∩ t)\nhs : s ∈ S\nht : t = ∅\n⊢ s ∩ t ∈ insert ∅ S\n\ncase inr.inr\nα : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns t : Set α\nhst : Set.Nonempty (s ∩ t)\nhs : s ∈ S\nht : t ∈ S\n⊢ s ∩ t ∈ insert ∅ S", "state_before": "case inr\nα : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns t : Set α\nht : t ∈ insert ∅ S\nhst : Set.Nonempty (s ∩ t)\nhs : s ∈ S\n⊢ s ∩ t ∈ insert ∅ S", "tactic": "cases' ht with ht ht" }, { "state_after": "no goals", "state_before": "case inr.inl\nα : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns t : Set α\nhst : Set.Nonempty (s ∩ t)\nhs : s ∈ S\nht : t = ∅\n⊢ s ∩ t ∈ insert ∅ S", "tactic": "simp [ht]" }, { "state_after": "no goals", "state_before": "case inr.inr\nα : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns t : Set α\nhst : Set.Nonempty (s ∩ t)\nhs : s ∈ S\nht : t ∈ S\n⊢ s ∩ t ∈ insert ∅ S", "tactic": "exact Set.mem_insert_of_mem _ (h_pi s hs t ht hst)" } ]

[ 94, 57 ]

[ 87, 1 ]

[]

[ 350, 33 ]

[ 348, 1 ]

[ { "state_after": "α : Type u_3\nE : Type u_2\nF : Type ?u.5042165\nG : Type ?u.5042168\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nι : Type u_1\ns : Finset ι\nf : ι → α → E\nhf : ∀ (i : ι), i ∈ s → Memℒp (f i) p\nthis : DecidableEq ι\n⊢ Memℒp (fun a => ∑ i in s, f i a) p", "state_before": "α : Type u_3\nE : Type u_2\nF : Type ?u.5042165\nG : Type ?u.5042168\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nι : Type u_1\ns : Finset ι\nf : ι → α → E\nhf : ∀ (i : ι), i ∈ s → Memℒp (f i) p\n⊢ Memℒp (fun a => ∑ i in s, f i a) p", "tactic": "haveI : DecidableEq ι := Classical.decEq _" }, { "state_after": "α : Type u_3\nE : Type u_2\nF : Type ?u.5042165\nG : Type ?u.5042168\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nι : Type u_1\ns : Finset ι\nf : ι → α → E\nthis : DecidableEq ι\n⊢ (∀ (i : ι), i ∈ s → Memℒp (f i) p) → Memℒp (fun a => ∑ i in s, f i a) p", "state_before": "α : Type u_3\nE : Type u_2\nF : Type ?u.5042165\nG : Type ?u.5042168\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nι : Type u_1\ns : Finset ι\nf : ι → α → E\nhf : ∀ (i : ι), i ∈ s → Memℒp (f i) p\nthis : DecidableEq ι\n⊢ Memℒp (fun a => ∑ i in s, f i a) p", "tactic": "revert hf" }, { "state_after": "case refine'_1\nα : Type u_3\nE : Type u_2\nF : Type ?u.5042165\nG : Type ?u.5042168\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nι : Type u_1\ns : Finset ι\nf : ι → α → E\nthis : DecidableEq ι\n⊢ (∀ (i : ι), i ∈ ∅ → Memℒp (f i) p) → Memℒp (fun a => ∑ i in ∅, f i a) p\n\ncase refine'_2\nα : Type u_3\nE : Type u_2\nF : Type ?u.5042165\nG : Type ?u.5042168\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nι : Type u_1\ns : Finset ι\nf : ι → α → E\nthis : DecidableEq ι\n⊢ ∀ ⦃a : ι⦄ {s : Finset ι},\n ¬a ∈ s →\n ((∀ (i : ι), i ∈ s → Memℒp (f i) p) → Memℒp (fun a => ∑ i in s, f i a) p) →\n (∀ (i : ι), i ∈ insert a s → Memℒp (f i) p) → Memℒp (fun a_3 => ∑ i in insert a s, f i a_3) p", "state_before": "α : Type u_3\nE : Type u_2\nF : Type ?u.5042165\nG : Type ?u.5042168\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nι : Type u_1\ns : Finset ι\nf : ι → α → E\nthis : DecidableEq ι\n⊢ (∀ (i : ι), i ∈ s → Memℒp (f i) p) → Memℒp (fun a => ∑ i in s, f i a) p", "tactic": "refine' Finset.induction_on s _ _" }, { "state_after": "no goals", "state_before": "case refine'_1\nα : Type u_3\nE : Type u_2\nF : Type ?u.5042165\nG : Type ?u.5042168\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nι : Type u_1\ns : Finset ι\nf : ι → α → E\nthis : DecidableEq ι\n⊢ (∀ (i : ι), i ∈ ∅ → Memℒp (f i) p) → Memℒp (fun a => ∑ i in ∅, f i a) p", "tactic": "simp only [zero_mem_ℒp', Finset.sum_empty, imp_true_iff]" }, { "state_after": "case refine'_2\nα : Type u_3\nE : Type u_2\nF : Type ?u.5042165\nG : Type ?u.5042168\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nι : Type u_1\ns✝ : Finset ι\nf : ι → α → E\nthis : DecidableEq ι\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nih : (∀ (i : ι), i ∈ s → Memℒp (f i) p) → Memℒp (fun a => ∑ i in s, f i a) p\nhf : ∀ (i_1 : ι), i_1 ∈ insert i s → Memℒp (f i_1) p\n⊢ Memℒp (fun a => ∑ i in insert i s, f i a) p", "state_before": "case refine'_2\nα : Type u_3\nE : Type u_2\nF : Type ?u.5042165\nG : Type ?u.5042168\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nι : Type u_1\ns : Finset ι\nf : ι → α → E\nthis : DecidableEq ι\n⊢ ∀ ⦃a : ι⦄ {s : Finset ι},\n ¬a ∈ s →\n ((∀ (i : ι), i ∈ s → Memℒp (f i) p) → Memℒp (fun a => ∑ i in s, f i a) p) →\n (∀ (i : ι), i ∈ insert a s → Memℒp (f i) p) → Memℒp (fun a_3 => ∑ i in insert a s, f i a_3) p", "tactic": "intro i s his ih hf" }, { "state_after": "case refine'_2\nα : Type u_3\nE : Type u_2\nF : Type ?u.5042165\nG : Type ?u.5042168\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nι : Type u_1\ns✝ : Finset ι\nf : ι → α → E\nthis : DecidableEq ι\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nih : (∀ (i : ι), i ∈ s → Memℒp (f i) p) → Memℒp (fun a => ∑ i in s, f i a) p\nhf : ∀ (i_1 : ι), i_1 ∈ insert i s → Memℒp (f i_1) p\n⊢ Memℒp (fun a => f i a + ∑ i in s, f i a) p", "state_before": "case refine'_2\nα : Type u_3\nE : Type u_2\nF : Type ?u.5042165\nG : Type ?u.5042168\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nι : Type u_1\ns✝ : Finset ι\nf : ι → α → E\nthis : DecidableEq ι\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nih : (∀ (i : ι), i ∈ s → Memℒp (f i) p) → Memℒp (fun a => ∑ i in s, f i a) p\nhf : ∀ (i_1 : ι), i_1 ∈ insert i s → Memℒp (f i_1) p\n⊢ Memℒp (fun a => ∑ i in insert i s, f i a) p", "tactic": "simp only [his, Finset.sum_insert, not_false_iff]" }, { "state_after": "no goals", "state_before": "case refine'_2\nα : Type u_3\nE : Type u_2\nF : Type ?u.5042165\nG : Type ?u.5042168\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nι : Type u_1\ns✝ : Finset ι\nf : ι → α → E\nthis : DecidableEq ι\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nih : (∀ (i : ι), i ∈ s → Memℒp (f i) p) → Memℒp (fun a => ∑ i in s, f i a) p\nhf : ∀ (i_1 : ι), i_1 ∈ insert i s → Memℒp (f i_1) p\n⊢ Memℒp (fun a => f i a + ∑ i in s, f i a) p", "tactic": "exact (hf i (s.mem_insert_self i)).add (ih fun j hj => hf j (Finset.mem_insert_of_mem hj))" } ]

[ 1241, 95 ]

[ 1233, 1 ]

[ { "state_after": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nthis : 2 * card s ≤ Fintype.card α\n⊢ ∃ t, s ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t", "state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\n⊢ ∃ t, s ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t", "tactic": "have := hs.card_le" }, { "state_after": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nthis : card s ≤ Fintype.card α / 2\n⊢ ∃ t, s ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t", "state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nthis : 2 * card s ≤ Fintype.card α\n⊢ ∃ t, s ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t", "tactic": "rw [mul_comm, ← Nat.le_div_iff_mul_le' two_pos] at this" }, { "state_after": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nthis : card s ≤ Fintype.card α / 2\n⊢ Intersecting ↑s → ∃ t, s ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t", "state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nthis : card s ≤ Fintype.card α / 2\n⊢ ∃ t, s ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t", "tactic": "revert hs" }, { "state_after": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nthis : card s ≤ Fintype.card α / 2\n⊢ ∀ (t₁ : Finset α),\n (∀ {t₂ : Finset α},\n card t₂ ≤ Fintype.card α / 2 →\n t₁ ⊂ t₂ → Intersecting ↑t₂ → ∃ t, t₂ ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t) →\n card t₁ ≤ Fintype.card α / 2 → Intersecting ↑t₁ → ∃ t, t₁ ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t", "state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nthis : card s ≤ Fintype.card α / 2\n⊢ Intersecting ↑s → ∃ t, s ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t", "tactic": "refine' s.strongDownwardInductionOn _ this" }, { "state_after": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns✝ : Finset α\nthis : card s✝ ≤ Fintype.card α / 2\ns : Finset α\nih :\n ∀ {t₂ : Finset α},\n card t₂ ≤ Fintype.card α / 2 →\n s ⊂ t₂ → Intersecting ↑t₂ → ∃ t, t₂ ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t\n_hcard : card s ≤ Fintype.card α / 2\nhs : Intersecting ↑s\n⊢ ∃ t, s ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t", "state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nthis : card s ≤ Fintype.card α / 2\n⊢ ∀ (t₁ : Finset α),\n (∀ {t₂ : Finset α},\n card t₂ ≤ Fintype.card α / 2 →\n t₁ ⊂ t₂ → Intersecting ↑t₂ → ∃ t, t₂ ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t) →\n card t₁ ≤ Fintype.card α / 2 → Intersecting ↑t₁ → ∃ t, t₁ ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t", "tactic": "rintro s ih _hcard hs" }, { "state_after": "case pos\nα : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns✝ : Finset α\nthis : card s✝ ≤ Fintype.card α / 2\ns : Finset α\nih :\n ∀ {t₂ : Finset α},\n card t₂ ≤ Fintype.card α / 2 →\n s ⊂ t₂ → Intersecting ↑t₂ → ∃ t, t₂ ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t\n_hcard : card s ≤ Fintype.card α / 2\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\n⊢ ∃ t, s ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t\n\ncase neg\nα : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns✝ : Finset α\nthis : card s✝ ≤ Fintype.card α / 2\ns : Finset α\nih :\n ∀ {t₂ : Finset α},\n card t₂ ≤ Fintype.card α / 2 →\n s ⊂ t₂ → Intersecting ↑t₂ → ∃ t, t₂ ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t\n_hcard : card s ≤ Fintype.card α / 2\nhs : Intersecting ↑s\nh : ¬∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\n⊢ ∃ t, s ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t", "state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns✝ : Finset α\nthis : card s✝ ≤ Fintype.card α / 2\ns : Finset α\nih :\n ∀ {t₂ : Finset α},\n card t₂ ≤ Fintype.card α / 2 →\n s ⊂ t₂ → Intersecting ↑t₂ → ∃ t, t₂ ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t\n_hcard : card s ≤ Fintype.card α / 2\nhs : Intersecting ↑s\n⊢ ∃ t, s ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t", "tactic": "by_cases h : ∀ t : Finset α, (t : Set α).Intersecting → s ⊆ t → s = t" }, { "state_after": "case neg\nα : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns✝ : Finset α\nthis : card s✝ ≤ Fintype.card α / 2\ns : Finset α\nih :\n ∀ {t₂ : Finset α},\n card t₂ ≤ Fintype.card α / 2 →\n s ⊂ t₂ → Intersecting ↑t₂ → ∃ t, t₂ ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t\n_hcard : card s ≤ Fintype.card α / 2\nhs : Intersecting ↑s\nh : ∃ t, Intersecting ↑t ∧ s ⊆ t ∧ s ≠ t\n⊢ ∃ t, s ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t", "state_before": "case neg\nα : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns✝ : Finset α\nthis : card s✝ ≤ Fintype.card α / 2\ns : Finset α\nih :\n ∀ {t₂ : Finset α},\n card t₂ ≤ Fintype.card α / 2 →\n s ⊂ t₂ → Intersecting ↑t₂ → ∃ t, t₂ ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t\n_hcard : card s ≤ Fintype.card α / 2\nhs : Intersecting ↑s\nh : ¬∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\n⊢ ∃ t, s ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t", "tactic": "push_neg at h" }, { "state_after": "case neg.intro.intro\nα : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns✝ : Finset α\nthis : card s✝ ≤ Fintype.card α / 2\ns : Finset α\nih :\n ∀ {t₂ : Finset α},\n card t₂ ≤ Fintype.card α / 2 →\n s ⊂ t₂ → Intersecting ↑t₂ → ∃ t, t₂ ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t\n_hcard : card s ≤ Fintype.card α / 2\nhs : Intersecting ↑s\nt : Finset α\nht : Intersecting ↑t\nhst : s ⊆ t ∧ s ≠ t\n⊢ ∃ t, s ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t", "state_before": "case neg\nα : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns✝ : Finset α\nthis : card s✝ ≤ Fintype.card α / 2\ns : Finset α\nih :\n ∀ {t₂ : Finset α},\n card t₂ ≤ Fintype.card α / 2 →\n s ⊂ t₂ → Intersecting ↑t₂ → ∃ t, t₂ ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t\n_hcard : card s ≤ Fintype.card α / 2\nhs : Intersecting ↑s\nh : ∃ t, Intersecting ↑t ∧ s ⊆ t ∧ s ≠ t\n⊢ ∃ t, s ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t", "tactic": "obtain ⟨t, ht, hst⟩ := h" }, { "state_after": "case neg.intro.intro\nα : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns✝ : Finset α\nthis : card s✝ ≤ Fintype.card α / 2\ns : Finset α\nih :\n ∀ {t₂ : Finset α},\n card t₂ ≤ Fintype.card α / 2 →\n s ⊂ t₂ → Intersecting ↑t₂ → ∃ t, t₂ ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t\n_hcard : card s ≤ Fintype.card α / 2\nhs : Intersecting ↑s\nt : Finset α\nht : Intersecting ↑t\nhst : s ⊆ t ∧ s ≠ t\n⊢ card t ≤ Fintype.card α / 2", "state_before": "case neg.intro.intro\nα : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns✝ : Finset α\nthis : card s✝ ≤ Fintype.card α / 2\ns : Finset α\nih :\n ∀ {t₂ : Finset α},\n card t₂ ≤ Fintype.card α / 2 →\n s ⊂ t₂ → Intersecting ↑t₂ → ∃ t, t₂ ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t\n_hcard : card s ≤ Fintype.card α / 2\nhs : Intersecting ↑s\nt : Finset α\nht : Intersecting ↑t\nhst : s ⊆ t ∧ s ≠ t\n⊢ ∃ t, s ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t", "tactic": "refine' (ih _ (_root_.ssubset_iff_subset_ne.2 hst) ht).imp fun u => And.imp_left hst.1.trans" }, { "state_after": "case neg.intro.intro\nα : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns✝ : Finset α\nthis : card s✝ ≤ Fintype.card α / 2\ns : Finset α\nih :\n ∀ {t₂ : Finset α},\n card t₂ ≤ Fintype.card α / 2 →\n s ⊂ t₂ → Intersecting ↑t₂ → ∃ t, t₂ ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t\n_hcard : card s ≤ Fintype.card α / 2\nhs : Intersecting ↑s\nt : Finset α\nht : Intersecting ↑t\nhst : s ⊆ t ∧ s ≠ t\n⊢ 2 * card t ≤ Fintype.card α", "state_before": "case neg.intro.intro\nα : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns✝ : Finset α\nthis : card s✝ ≤ Fintype.card α / 2\ns : Finset α\nih :\n ∀ {t₂ : Finset α},\n card t₂ ≤ Fintype.card α / 2 →\n s ⊂ t₂ → Intersecting ↑t₂ → ∃ t, t₂ ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t\n_hcard : card s ≤ Fintype.card α / 2\nhs : Intersecting ↑s\nt : Finset α\nht : Intersecting ↑t\nhst : s ⊆ t ∧ s ≠ t\n⊢ card t ≤ Fintype.card α / 2", "tactic": "rw [Nat.le_div_iff_mul_le' two_pos, mul_comm]" }, { "state_after": "no goals", "state_before": "case neg.intro.intro\nα : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns✝ : Finset α\nthis : card s✝ ≤ Fintype.card α / 2\ns : Finset α\nih :\n ∀ {t₂ : Finset α},\n card t₂ ≤ Fintype.card α / 2 →\n s ⊂ t₂ → Intersecting ↑t₂ → ∃ t, t₂ ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t\n_hcard : card s ≤ Fintype.card α / 2\nhs : Intersecting ↑s\nt : Finset α\nht : Intersecting ↑t\nhst : s ⊆ t ∧ s ≠ t\n⊢ 2 * card t ≤ Fintype.card α", "tactic": "exact ht.card_le" }, { "state_after": "no goals", "state_before": "case pos\nα : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns✝ : Finset α\nthis : card s✝ ≤ Fintype.card α / 2\ns : Finset α\nih :\n ∀ {t₂ : Finset α},\n card t₂ ≤ Fintype.card α / 2 →\n s ⊂ t₂ → Intersecting ↑t₂ → ∃ t, t₂ ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t\n_hcard : card s ≤ Fintype.card α / 2\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\n⊢ ∃ t, s ⊆ t ∧ 2 * card t = Fintype.card α ∧ Intersecting ↑t", "tactic": "exact ⟨s, Subset.rfl, hs.is_max_iff_card_eq.1 h, hs⟩" } ]

[ 213, 19 ]

[ 200, 1 ]

[ { "state_after": "no goals", "state_before": "𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.267674\nE : Type u_3\nEₗ : Type ?u.267680\nF : Type u_4\nFₗ : Type ?u.267686\nG : Type ?u.267689\nGₗ : Type ?u.267692\n𝓕 : Type ?u.267695\ninst✝¹⁵ : SeminormedAddCommGroup E\ninst✝¹⁴ : SeminormedAddCommGroup Eₗ\ninst✝¹³ : SeminormedAddCommGroup F\ninst✝¹² : SeminormedAddCommGroup Fₗ\ninst✝¹¹ : SeminormedAddCommGroup G\ninst✝¹⁰ : SeminormedAddCommGroup Gₗ\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NontriviallyNormedField 𝕜₂\ninst✝⁷ : NontriviallyNormedField 𝕜₃\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 Eₗ\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜 Fₗ\ninst✝² : NormedSpace 𝕜₃ G\ninst✝¹ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\nf : E →SL[σ₁₂] F\nM : ℝ\nhMp : 0 ≤ M\nhM : ∀ (x : E), ‖x‖ ≠ 0 → ‖↑f x‖ ≤ M * ‖x‖\nx : E\nh : ‖x‖ = 0\n⊢ ‖↑f x‖ ≤ M * ‖x‖", "tactic": "simp only [h, MulZeroClass.mul_zero, norm_image_of_norm_zero f f.2 h, le_refl]" } ]

[ 171, 85 ]

[ 167, 1 ]

[ { "state_after": "case h\nα : Type u_1\ninst✝ : DecidableEq α\ns s' : Cycle α\na✝ : List α\nh : Nodup (Quot.mk Setoid.r a✝)\nhn : Nontrivial (Quot.mk Setoid.r a✝)\n⊢ IsCycle (formPerm (Quot.mk Setoid.r a✝) h)", "state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ s' s : Cycle α\nh : Nodup s\nhn : Nontrivial s\n⊢ IsCycle (formPerm s h)", "tactic": "induction s using Quot.inductionOn" }, { "state_after": "no goals", "state_before": "case h\nα : Type u_1\ninst✝ : DecidableEq α\ns s' : Cycle α\na✝ : List α\nh : Nodup (Quot.mk Setoid.r a✝)\nhn : Nontrivial (Quot.mk Setoid.r a✝)\n⊢ IsCycle (formPerm (Quot.mk Setoid.r a✝) h)", "tactic": "exact List.isCycle_formPerm h (length_nontrivial hn)" } ]

[ 166, 55 ]

[ 163, 1 ]

[ { "state_after": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf : Perm (Fin n)\nx✝ : (_ : Fin n) × Fin n\na₁ a₂ : Fin n\nha : { fst := a₁, snd := a₂ } ∈ finPairsLT n\nhxa : ↑f⁻¹ a₂ < ↑f⁻¹ a₁\n⊢ { fst := a₁, snd := a₂ } =\n if ↑f (↑f⁻¹ a₂) < ↑f (↑f⁻¹ a₁) then { fst := ↑f (↑f⁻¹ a₁), snd := ↑f (↑f⁻¹ a₂) }\n else { fst := ↑f (↑f⁻¹ a₂), snd := ↑f (↑f⁻¹ a₁) }", "state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf : Perm (Fin n)\nx✝ : (_ : Fin n) × Fin n\na₁ a₂ : Fin n\nha : { fst := a₁, snd := a₂ } ∈ finPairsLT n\nhxa : ↑f⁻¹ a₂ < ↑f⁻¹ a₁\n⊢ { fst := a₁, snd := a₂ } = signBijAux f { fst := ↑f⁻¹ a₁, snd := ↑f⁻¹ a₂ }", "tactic": "dsimp [signBijAux]" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf : Perm (Fin n)\nx✝ : (_ : Fin n) × Fin n\na₁ a₂ : Fin n\nha : { fst := a₁, snd := a₂ } ∈ finPairsLT n\nhxa : ↑f⁻¹ a₂ < ↑f⁻¹ a₁\n⊢ { fst := a₁, snd := a₂ } =\n if ↑f (↑f⁻¹ a₂) < ↑f (↑f⁻¹ a₁) then { fst := ↑f (↑f⁻¹ a₁), snd := ↑f (↑f⁻¹ a₂) }\n else { fst := ↑f (↑f⁻¹ a₂), snd := ↑f (↑f⁻¹ a₁) }", "tactic": "rw [apply_inv_self, apply_inv_self, if_pos (mem_finPairsLT.1 ha)]" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf : Perm (Fin n)\nx✝ : (_ : Fin n) × Fin n\na₁ a₂ : Fin n\nha : { fst := a₁, snd := a₂ } ∈ finPairsLT n\nhxa : ¬↑f⁻¹ a₂ < ↑f⁻¹ a₁\nh : ↑f⁻¹ a₁ = ↑f⁻¹ a₂\n⊢ False", "tactic": "simp [mem_finPairsLT, f⁻¹.injective h, lt_irrefl] at ha" }, { "state_after": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf : Perm (Fin n)\nx✝ : (_ : Fin n) × Fin n\na₁ a₂ : Fin n\nha : { fst := a₁, snd := a₂ } ∈ finPairsLT n\nhxa : ¬↑f⁻¹ a₂ < ↑f⁻¹ a₁\n⊢ { fst := a₁, snd := a₂ } =\n if ↑f (↑f⁻¹ a₁) < ↑f (↑f⁻¹ a₂) then { fst := ↑f (↑f⁻¹ a₂), snd := ↑f (↑f⁻¹ a₁) }\n else { fst := ↑f (↑f⁻¹ a₁), snd := ↑f (↑f⁻¹ a₂) }", "state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf : Perm (Fin n)\nx✝ : (_ : Fin n) × Fin n\na₁ a₂ : Fin n\nha : { fst := a₁, snd := a₂ } ∈ finPairsLT n\nhxa : ¬↑f⁻¹ a₂ < ↑f⁻¹ a₁\n⊢ { fst := a₁, snd := a₂ } = signBijAux f { fst := ↑f⁻¹ a₂, snd := ↑f⁻¹ a₁ }", "tactic": "dsimp [signBijAux]" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\ninst✝ : DecidableEq α\nn : ℕ\nf : Perm (Fin n)\nx✝ : (_ : Fin n) × Fin n\na₁ a₂ : Fin n\nha : { fst := a₁, snd := a₂ } ∈ finPairsLT n\nhxa : ¬↑f⁻¹ a₂ < ↑f⁻¹ a₁\n⊢ { fst := a₁, snd := a₂ } =\n if ↑f (↑f⁻¹ a₁) < ↑f (↑f⁻¹ a₂) then { fst := ↑f (↑f⁻¹ a₂), snd := ↑f (↑f⁻¹ a₁) }\n else { fst := ↑f (↑f⁻¹ a₁), snd := ↑f (↑f⁻¹ a₂) }", "tactic": "rw [apply_inv_self, apply_inv_self, if_neg (mem_finPairsLT.1 ha).le.not_lt]" } ]

[ 359, 91 ]

[ 346, 1 ]

[ { "state_after": "no goals", "state_before": "p q r : ℕ+\n⊢ sumInv {p, q, r} = (↑↑p)⁻¹ + (↑↑q)⁻¹ + (↑↑r)⁻¹", "tactic": "simp only [sumInv, add_zero, insert_eq_cons, add_assoc, map_cons, sum_cons,\n map_singleton, sum_singleton]" } ]

[ 121, 34 ]

[ 119, 1 ]

[]

[ 147, 6 ]

[ 146, 1 ]

[ { "state_after": "case intro\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NeZero 2\na b c x : K\nha : a ≠ 0\ns : K\nhs : discrim a b c = s * s\n⊢ ∃ x, a * x * x + b * x + c = 0", "state_before": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NeZero 2\na b c x : K\nha : a ≠ 0\nh : ∃ s, discrim a b c = s * s\n⊢ ∃ x, a * x * x + b * x + c = 0", "tactic": "rcases h with ⟨s, hs⟩" }, { "state_after": "case intro\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NeZero 2\na b c x : K\nha : a ≠ 0\ns : K\nhs : discrim a b c = s * s\n⊢ a * ((-b + s) / (2 * a)) * ((-b + s) / (2 * a)) + b * ((-b + s) / (2 * a)) + c = 0", "state_before": "case intro\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NeZero 2\na b c x : K\nha : a ≠ 0\ns : K\nhs : discrim a b c = s * s\n⊢ ∃ x, a * x * x + b * x + c = 0", "tactic": "use (-b + s) / (2 * a)" }, { "state_after": "case intro\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NeZero 2\na b c x : K\nha : a ≠ 0\ns : K\nhs : discrim a b c = s * s\n⊢ (-b + s) / (2 * a) = (-b + s) / (2 * a) ∨ (-b + s) / (2 * a) = (-b - s) / (2 * a)", "state_before": "case intro\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NeZero 2\na b c x : K\nha : a ≠ 0\ns : K\nhs : discrim a b c = s * s\n⊢ a * ((-b + s) / (2 * a)) * ((-b + s) / (2 * a)) + b * ((-b + s) / (2 * a)) + c = 0", "tactic": "rw [quadratic_eq_zero_iff ha hs]" }, { "state_after": "no goals", "state_before": "case intro\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NeZero 2\na b c x : K\nha : a ≠ 0\ns : K\nhs : discrim a b c = s * s\n⊢ (-b + s) / (2 * a) = (-b + s) / (2 * a) ∨ (-b + s) / (2 * a) = (-b - s) / (2 * a)", "tactic": "simp" } ]

[ 104, 7 ]

[ 99, 1 ]

[]

[ 231, 21 ]

[ 229, 1 ]

[ { "state_after": "C : Type u\ninst✝² : Category C\nX Y : C\nf : X ⟶ Y\nZ : C\ng : Y ⟶ Z\ninst✝¹ : HasImage g\ninst✝ : HasImage (f ≫ g)\n⊢ lift (MonoFactorisation.mk (image g) (ι g) (f ≫ factorThruImage g)) ≫ ι g = ι (f ≫ g)", "state_before": "C : Type u\ninst✝² : Category C\nX Y : C\nf : X ⟶ Y\nZ : C\ng : Y ⟶ Z\ninst✝¹ : HasImage g\ninst✝ : HasImage (f ≫ g)\n⊢ preComp f g ≫ ι g = ι (f ≫ g)", "tactic": "dsimp [image.preComp]" }, { "state_after": "no goals", "state_before": "C : Type u\ninst✝² : Category C\nX Y : C\nf : X ⟶ Y\nZ : C\ng : Y ⟶ Z\ninst✝¹ : HasImage g\ninst✝ : HasImage (f ≫ g)\n⊢ lift (MonoFactorisation.mk (image g) (ι g) (f ≫ factorThruImage g)) ≫ ι g = ι (f ≫ g)", "tactic": "rw [image.lift_fac]" } ]

[ 553, 26 ]

[ 550, 1 ]

[]

[ 1076, 24 ]

[ 1074, 8 ]

[ { "state_after": "case mk.intro.mk.intro\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nf : G ⧸ H → G\nhf : ∀ (q : G ⧸ H), ↑(f q) = q\nq₁ q₂ : G ⧸ H\nh :\n Set.restrict (Set.range f) Quotient.mk'' { val := f q₁, property := (_ : ∃ y, f y = f q₁) } =\n Set.restrict (Set.range f) Quotient.mk'' { val := f q₂, property := (_ : ∃ y, f y = f q₂) }\n⊢ { val := f q₁, property := (_ : ∃ y, f y = f q₁) } = { val := f q₂, property := (_ : ∃ y, f y = f q₂) }", "state_before": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nf : G ⧸ H → G\nhf : ∀ (q : G ⧸ H), ↑(f q) = q\n⊢ Function.Injective (Set.restrict (Set.range f) Quotient.mk'')", "tactic": "rintro ⟨-, q₁, rfl⟩ ⟨-, q₂, rfl⟩ h" }, { "state_after": "no goals", "state_before": "case mk.intro.mk.intro\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nf : G ⧸ H → G\nhf : ∀ (q : G ⧸ H), ↑(f q) = q\nq₁ q₂ : G ⧸ H\nh :\n Set.restrict (Set.range f) Quotient.mk'' { val := f q₁, property := (_ : ∃ y, f y = f q₁) } =\n Set.restrict (Set.range f) Quotient.mk'' { val := f q₂, property := (_ : ∃ y, f y = f q₂) }\n⊢ { val := f q₁, property := (_ : ∃ y, f y = f q₁) } = { val := f q₂, property := (_ : ∃ y, f y = f q₂) }", "tactic": "exact Subtype.ext $ congr_arg f $ ((hf q₁).symm.trans h).trans (hf q₂)" } ]

[ 300, 38 ]

[ 295, 1 ]

[ { "state_after": "x : ℂ\nhx : cos x = 0\n⊢ Tendsto (fun x => ‖sin x‖ / ‖cos x‖) (𝓝[{x}ᶜ] x) atTop", "state_before": "x : ℂ\nhx : cos x = 0\n⊢ Tendsto (fun x => ↑abs (tan x)) (𝓝[{x}ᶜ] x) atTop", "tactic": "simp only [tan_eq_sin_div_cos, ← norm_eq_abs, norm_div]" }, { "state_after": "x : ℂ\nhx : cos x = 0\nA : sin x ≠ 0\n⊢ Tendsto (fun x => ‖sin x‖ / ‖cos x‖) (𝓝[{x}ᶜ] x) atTop", "state_before": "x : ℂ\nhx : cos x = 0\n⊢ Tendsto (fun x => ‖sin x‖ / ‖cos x‖) (𝓝[{x}ᶜ] x) atTop", "tactic": "have A : sin x ≠ 0 := fun h => by simpa [*, sq] using sin_sq_add_cos_sq x" }, { "state_after": "x : ℂ\nhx : cos x = 0\nA : sin x ≠ 0\nB : Tendsto cos (𝓝[{x}ᶜ] x) (𝓝[{0}ᶜ] 0)\n⊢ Tendsto (fun x => ‖sin x‖ / ‖cos x‖) (𝓝[{x}ᶜ] x) atTop", "state_before": "x : ℂ\nhx : cos x = 0\nA : sin x ≠ 0\n⊢ Tendsto (fun x => ‖sin x‖ / ‖cos x‖) (𝓝[{x}ᶜ] x) atTop", "tactic": "have B : Tendsto cos (𝓝[≠] x) (𝓝[≠] 0) :=\n hx ▸ (hasDerivAt_cos x).tendsto_punctured_nhds (neg_ne_zero.2 A)" }, { "state_after": "no goals", "state_before": "x : ℂ\nhx : cos x = 0\nA : sin x ≠ 0\nB : Tendsto cos (𝓝[{x}ᶜ] x) (𝓝[{0}ᶜ] 0)\n⊢ Tendsto (fun x => ‖sin x‖ / ‖cos x‖) (𝓝[{x}ᶜ] x) atTop", "tactic": "exact continuous_sin.continuousWithinAt.norm.mul_atTop (norm_pos_iff.2 A)\n (tendsto_norm_nhdsWithin_zero.comp B).inv_tendsto_zero" }, { "state_after": "no goals", "state_before": "x : ℂ\nhx : cos x = 0\nh : sin x = 0\n⊢ False", "tactic": "simpa [*, sq] using sin_sq_add_cos_sq x" } ]

[ 49, 59 ]

[ 42, 1 ]

[]

[ 3056, 28 ]

[ 3055, 1 ]

[ { "state_after": "case inl\nα : Type u_1\nβ : Type ?u.97957\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhcd : c ≤ d\nh : c ≤ b\n⊢ Iio b ∪ Ico c d = Iio (max b d)\n\ncase inr\nα : Type u_1\nβ : Type ?u.97957\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhcd : d ≤ c\nh : d ≤ b\n⊢ Iio b ∪ Ico c d = Iio (max b d)", "state_before": "α : Type u_1\nβ : Type ?u.97957\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh : min c d ≤ b\n⊢ Iio b ∪ Ico c d = Iio (max b d)", "tactic": "cases' le_total c d with hcd hcd <;> simp [hcd] at h" }, { "state_after": "no goals", "state_before": "case inl\nα : Type u_1\nβ : Type ?u.97957\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhcd : c ≤ d\nh : c ≤ b\n⊢ Iio b ∪ Ico c d = Iio (max b d)", "tactic": "exact Iio_union_Ico' h" }, { "state_after": "no goals", "state_before": "case inr\nα : Type u_1\nβ : Type ?u.97957\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhcd : d ≤ c\nh : d ≤ b\n⊢ Iio b ∪ Ico c d = Iio (max b d)", "tactic": "simp [*]" } ]

[ 1395, 13 ]

[ 1392, 1 ]

[]

[ 100, 73 ]

[ 99, 1 ]

[ { "state_after": "G : Type u\ninst✝⁴ : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst✝³ : Group H\nI : Type u_1\ninst✝² : Group I\nM : Subgroup H\nO : Subgroup I\ninst✝¹ : Subgroup.Normal M\ninst✝ : Subgroup.Normal O\nf : G →* H\ng : H →* I\nhf : N ≤ Subgroup.comap f M\nhg : M ≤ Subgroup.comap g O\nhgf : optParam (N ≤ Subgroup.comap (MonoidHom.comp g f) O) (_ : N ≤ Subgroup.comap (MonoidHom.comp g f) O)\nx✝ : G ⧸ N\nx : G\n⊢ ↑(map M O g hg) (↑(map N M f hf) ↑x) = ↑(map N O (MonoidHom.comp g f) hgf) ↑x", "state_before": "G : Type u\ninst✝⁴ : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst✝³ : Group H\nI : Type u_1\ninst✝² : Group I\nM : Subgroup H\nO : Subgroup I\ninst✝¹ : Subgroup.Normal M\ninst✝ : Subgroup.Normal O\nf : G →* H\ng : H →* I\nhf : N ≤ Subgroup.comap f M\nhg : M ≤ Subgroup.comap g O\nhgf : optParam (N ≤ Subgroup.comap (MonoidHom.comp g f) O) (_ : N ≤ Subgroup.comap (MonoidHom.comp g f) O)\nx : G ⧸ N\n⊢ ↑(map M O g hg) (↑(map N M f hf) x) = ↑(map N O (MonoidHom.comp g f) hgf) x", "tactic": "refine' induction_on' x fun x => _" }, { "state_after": "no goals", "state_before": "G : Type u\ninst✝⁴ : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst✝³ : Group H\nI : Type u_1\ninst✝² : Group I\nM : Subgroup H\nO : Subgroup I\ninst✝¹ : Subgroup.Normal M\ninst✝ : Subgroup.Normal O\nf : G →* H\ng : H →* I\nhf : N ≤ Subgroup.comap f M\nhg : M ≤ Subgroup.comap g O\nhgf : optParam (N ≤ Subgroup.comap (MonoidHom.comp g f) O) (_ : N ≤ Subgroup.comap (MonoidHom.comp g f) O)\nx✝ : G ⧸ N\nx : G\n⊢ ↑(map M O g hg) (↑(map N M f hf) ↑x) = ↑(map N O (MonoidHom.comp g f) hgf) ↑x", "tactic": "simp only [map_mk, MonoidHom.comp_apply]" } ]

[ 271, 43 ]

[ 265, 1 ]

[ { "state_after": "case refl\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ : σ₂\n⊢ ∃ c₁ c₂, Reaches f₂ a₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁\n\ncase tail\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\nIH : ∃ c₁ c₂_1, Reaches f₂ c₂ c₂_1 ∧ tr c₁ c₂_1 ∧ Reaches f₁ a₁ c₁\n⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "state_before": "σ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ : σ₂\nab : Reaches f₂ a₂ b₂\n⊢ ∃ c₁ c₂, Reaches f₂ b₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "tactic": "induction' ab with c₂ d₂ _ cd IH" }, { "state_after": "no goals", "state_before": "case refl\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ : σ₂\n⊢ ∃ c₁ c₂, Reaches f₂ a₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "tactic": "exact ⟨_, _, ReflTransGen.refl, aa, ReflTransGen.refl⟩" }, { "state_after": "case tail.intro.intro.intro.intro\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\ne₂ : σ₂\nce : Reaches f₂ c₂ e₂\nee : tr e₁ e₂\nae : Reaches f₁ a₁ e₁\n⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "state_before": "case tail\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\nIH : ∃ c₁ c₂_1, Reaches f₂ c₂ c₂_1 ∧ tr c₁ c₂_1 ∧ Reaches f₁ a₁ c₁\n⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "tactic": "rcases IH with ⟨e₁, e₂, ce, ee, ae⟩" }, { "state_after": "case tail.intro.intro.intro.intro.inl\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\nae : Reaches f₁ a₁ e₁\nce : Reaches f₂ c₂ c₂\nee : tr e₁ c₂\n⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁\n\ncase tail.intro.intro.intro.intro.inr.intro.intro\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\ne₂ : σ₂\nce : Reaches f₂ c₂ e₂\nee : tr e₁ e₂\nae : Reaches f₁ a₁ e₁\nd' : σ₂\ncd' : d' ∈ f₂ c₂\nde : ReflTransGen (fun a b => b ∈ f₂ a) d' e₂\n⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "state_before": "case tail.intro.intro.intro.intro\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\ne₂ : σ₂\nce : Reaches f₂ c₂ e₂\nee : tr e₁ e₂\nae : Reaches f₁ a₁ e₁\n⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "tactic": "rcases ReflTransGen.cases_head ce with (rfl | ⟨d', cd', de⟩)" }, { "state_after": "case tail.intro.intro.intro.intro.inl\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\nae : Reaches f₁ a₁ e₁\nce : Reaches f₂ c₂ c₂\nee : tr e₁ c₂\nthis :\n match f₁ e₁ with\n | some b₁ => ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ c₂ b₂\n | none => f₂ c₂ = none\n⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "state_before": "case tail.intro.intro.intro.intro.inl\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\nae : Reaches f₁ a₁ e₁\nce : Reaches f₂ c₂ c₂\nee : tr e₁ c₂\n⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "tactic": "have := H ee" }, { "state_after": "case tail.intro.intro.intro.intro.inl\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\nae : Reaches f₁ a₁ e₁\nce : Reaches f₂ c₂ c₂\nee : tr e₁ c₂\n⊢ (match f₁ e₁ with\n | some b₁ => ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ c₂ b₂\n | none => f₂ c₂ = none) →\n ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "state_before": "case tail.intro.intro.intro.intro.inl\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\nae : Reaches f₁ a₁ e₁\nce : Reaches f₂ c₂ c₂\nee : tr e₁ c₂\nthis :\n match f₁ e₁ with\n | some b₁ => ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ c₂ b₂\n | none => f₂ c₂ = none\n⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "tactic": "revert this" }, { "state_after": "case tail.intro.intro.intro.intro.inl.none\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\nae : Reaches f₁ a₁ e₁\nce : Reaches f₂ c₂ c₂\nee : tr e₁ c₂\neg : f₁ e₁ = none\n⊢ f₂ c₂ = none → ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁\n\ncase tail.intro.intro.intro.intro.inl.some\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\nae : Reaches f₁ a₁ e₁\nce : Reaches f₂ c₂ c₂\nee : tr e₁ c₂\ng₁ : σ₁\neg : f₁ e₁ = some g₁\n⊢ ∀ (x : σ₂), tr g₁ x → Reaches₁ f₂ c₂ x → ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "state_before": "case tail.intro.intro.intro.intro.inl\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\nae : Reaches f₁ a₁ e₁\nce : Reaches f₂ c₂ c₂\nee : tr e₁ c₂\n⊢ (match f₁ e₁ with\n | some b₁ => ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ c₂ b₂\n | none => f₂ c₂ = none) →\n ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "tactic": "cases' eg : f₁ e₁ with g₁ <;> simp only [Respects, and_imp, exists_imp]" }, { "state_after": "case tail.intro.intro.intro.intro.inl.none\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\nae : Reaches f₁ a₁ e₁\nce : Reaches f₂ c₂ c₂\nee : tr e₁ c₂\neg : f₁ e₁ = none\nc0 : f₂ c₂ = none\n⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "state_before": "case tail.intro.intro.intro.intro.inl.none\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\nae : Reaches f₁ a₁ e₁\nce : Reaches f₂ c₂ c₂\nee : tr e₁ c₂\neg : f₁ e₁ = none\n⊢ f₂ c₂ = none → ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "tactic": "intro c0" }, { "state_after": "no goals", "state_before": "case tail.intro.intro.intro.intro.inl.none\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\nae : Reaches f₁ a₁ e₁\nce : Reaches f₂ c₂ c₂\nee : tr e₁ c₂\neg : f₁ e₁ = none\nc0 : f₂ c₂ = none\n⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "tactic": "cases cd.symm.trans c0" }, { "state_after": "case tail.intro.intro.intro.intro.inl.some\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\nae : Reaches f₁ a₁ e₁\nce : Reaches f₂ c₂ c₂\nee : tr e₁ c₂\ng₁ : σ₁\neg : f₁ e₁ = some g₁\ng₂ : σ₂\ngg : tr g₁ g₂\ncg : Reaches₁ f₂ c₂ g₂\n⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "state_before": "case tail.intro.intro.intro.intro.inl.some\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\nae : Reaches f₁ a₁ e₁\nce : Reaches f₂ c₂ c₂\nee : tr e₁ c₂\ng₁ : σ₁\neg : f₁ e₁ = some g₁\n⊢ ∀ (x : σ₂), tr g₁ x → Reaches₁ f₂ c₂ x → ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "tactic": "intro g₂ gg cg" }, { "state_after": "case tail.intro.intro.intro.intro.inl.some.intro.intro\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\nae : Reaches f₁ a₁ e₁\nce : Reaches f₂ c₂ c₂\nee : tr e₁ c₂\ng₁ : σ₁\neg : f₁ e₁ = some g₁\ng₂ : σ₂\ngg : tr g₁ g₂\ncg : Reaches₁ f₂ c₂ g₂\nd' : σ₂\ncd' : d' ∈ f₂ c₂\ndg : ReflTransGen (fun a b => b ∈ f₂ a) d' g₂\n⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "state_before": "case tail.intro.intro.intro.intro.inl.some\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\nae : Reaches f₁ a₁ e₁\nce : Reaches f₂ c₂ c₂\nee : tr e₁ c₂\ng₁ : σ₁\neg : f₁ e₁ = some g₁\ng₂ : σ₂\ngg : tr g₁ g₂\ncg : Reaches₁ f₂ c₂ g₂\n⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "tactic": "rcases TransGen.head'_iff.1 cg with ⟨d', cd', dg⟩" }, { "state_after": "case tail.intro.intro.intro.intro.inl.some.intro.intro.refl\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\nae : Reaches f₁ a₁ e₁\nce : Reaches f₂ c₂ c₂\nee : tr e₁ c₂\ng₁ : σ₁\neg : f₁ e₁ = some g₁\ng₂ : σ₂\ngg : tr g₁ g₂\ncg : Reaches₁ f₂ c₂ g₂\ncd' : d₂ ∈ f₂ c₂\ndg : ReflTransGen (fun a b => b ∈ f₂ a) d₂ g₂\n⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "state_before": "case tail.intro.intro.intro.intro.inl.some.intro.intro\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\nae : Reaches f₁ a₁ e₁\nce : Reaches f₂ c₂ c₂\nee : tr e₁ c₂\ng₁ : σ₁\neg : f₁ e₁ = some g₁\ng₂ : σ₂\ngg : tr g₁ g₂\ncg : Reaches₁ f₂ c₂ g₂\nd' : σ₂\ncd' : d' ∈ f₂ c₂\ndg : ReflTransGen (fun a b => b ∈ f₂ a) d' g₂\n⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "tactic": "cases Option.mem_unique cd cd'" }, { "state_after": "no goals", "state_before": "case tail.intro.intro.intro.intro.inl.some.intro.intro.refl\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\nae : Reaches f₁ a₁ e₁\nce : Reaches f₂ c₂ c₂\nee : tr e₁ c₂\ng₁ : σ₁\neg : f₁ e₁ = some g₁\ng₂ : σ₂\ngg : tr g₁ g₂\ncg : Reaches₁ f₂ c₂ g₂\ncd' : d₂ ∈ f₂ c₂\ndg : ReflTransGen (fun a b => b ∈ f₂ a) d₂ g₂\n⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "tactic": "exact ⟨_, _, dg, gg, ae.tail eg⟩" }, { "state_after": "case tail.intro.intro.intro.intro.inr.intro.intro.refl\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\ne₂ : σ₂\nce : Reaches f₂ c₂ e₂\nee : tr e₁ e₂\nae : Reaches f₁ a₁ e₁\ncd' : d₂ ∈ f₂ c₂\nde : ReflTransGen (fun a b => b ∈ f₂ a) d₂ e₂\n⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "state_before": "case tail.intro.intro.intro.intro.inr.intro.intro\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\ne₂ : σ₂\nce : Reaches f₂ c₂ e₂\nee : tr e₁ e₂\nae : Reaches f₁ a₁ e₁\nd' : σ₂\ncd' : d' ∈ f₂ c₂\nde : ReflTransGen (fun a b => b ∈ f₂ a) d' e₂\n⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "tactic": "cases Option.mem_unique cd cd'" }, { "state_after": "no goals", "state_before": "case tail.intro.intro.intro.intro.inr.intro.intro.refl\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₂ c₂ d₂ : σ₂\na✝ : ReflTransGen (fun a b => b ∈ f₂ a) a₂ c₂\ncd : d₂ ∈ f₂ c₂\ne₁ : σ₁\ne₂ : σ₂\nce : Reaches f₂ c₂ e₂\nee : tr e₁ e₂\nae : Reaches f₁ a₁ e₁\ncd' : d₂ ∈ f₂ c₂\nde : ReflTransGen (fun a b => b ∈ f₂ a) d₂ e₂\n⊢ ∃ c₁ c₂, Reaches f₂ d₂ c₂ ∧ tr c₁ c₂ ∧ Reaches f₁ a₁ c₁", "tactic": "exact ⟨_, _, de, ee, ae⟩" } ]

[ 923, 31 ]

[ 906, 1 ]

[ { "state_after": "V : Type u\ninst✝⁴ : Category V\ninst✝³ : HasImages V\nA✝ B✝ C D : V\nf✝ : A✝ ⟶ B✝\ng : B✝ ⟶ C\nh✝ : C ⟶ D\ninst✝² : HasZeroObject V\ninst✝¹ : Preadditive V\ninst✝ : HasEqualizers V\nA B : V\nf : A ⟶ B\nh : Exact f 0\ne₁ : Epi (imageToKernel f 0 (_ : f ≫ 0 = 0))\n⊢ Epi f", "state_before": "V : Type u\ninst✝⁴ : Category V\ninst✝³ : HasImages V\nA✝ B✝ C D : V\nf✝ : A✝ ⟶ B✝\ng : B✝ ⟶ C\nh✝ : C ⟶ D\ninst✝² : HasZeroObject V\ninst✝¹ : Preadditive V\ninst✝ : HasEqualizers V\nA B : V\nf : A ⟶ B\nh : Exact f 0\n⊢ Epi f", "tactic": "have e₁ := h.epi" }, { "state_after": "V : Type u\ninst✝⁴ : Category V\ninst✝³ : HasImages V\nA✝ B✝ C D : V\nf✝ : A✝ ⟶ B✝\ng : B✝ ⟶ C\nh✝ : C ⟶ D\ninst✝² : HasZeroObject V\ninst✝¹ : Preadditive V\ninst✝ : HasEqualizers V\nA B : V\nf : A ⟶ B\nh : Exact f 0\ne₁ : Epi (Subobject.arrow (imageSubobject f) ≫ inv (Subobject.arrow (kernelSubobject 0)))\n⊢ Epi f", "state_before": "V : Type u\ninst✝⁴ : Category V\ninst✝³ : HasImages V\nA✝ B✝ C D : V\nf✝ : A✝ ⟶ B✝\ng : B✝ ⟶ C\nh✝ : C ⟶ D\ninst✝² : HasZeroObject V\ninst✝¹ : Preadditive V\ninst✝ : HasEqualizers V\nA B : V\nf : A ⟶ B\nh : Exact f 0\ne₁ : Epi (imageToKernel f 0 (_ : f ≫ 0 = 0))\n⊢ Epi f", "tactic": "rw [imageToKernel_zero_right] at e₁" }, { "state_after": "V : Type u\ninst✝⁴ : Category V\ninst✝³ : HasImages V\nA✝ B✝ C D : V\nf✝ : A✝ ⟶ B✝\ng : B✝ ⟶ C\nh✝ : C ⟶ D\ninst✝² : HasZeroObject V\ninst✝¹ : Preadditive V\ninst✝ : HasEqualizers V\nA B : V\nf : A ⟶ B\nh : Exact f 0\ne₁ : Epi (Subobject.arrow (imageSubobject f) ≫ inv (Subobject.arrow (kernelSubobject 0)))\ne₂ :\n Epi\n ((Subobject.arrow (imageSubobject f) ≫ inv (Subobject.arrow (kernelSubobject 0))) ≫\n Subobject.arrow (kernelSubobject 0))\n⊢ Epi f", "state_before": "V : Type u\ninst✝⁴ : Category V\ninst✝³ : HasImages V\nA✝ B✝ C D : V\nf✝ : A✝ ⟶ B✝\ng : B✝ ⟶ C\nh✝ : C ⟶ D\ninst✝² : HasZeroObject V\ninst✝¹ : Preadditive V\ninst✝ : HasEqualizers V\nA B : V\nf : A ⟶ B\nh : Exact f 0\ne₁ : Epi (Subobject.arrow (imageSubobject f) ≫ inv (Subobject.arrow (kernelSubobject 0)))\n⊢ Epi f", "tactic": "have e₂ : Epi (((imageSubobject f).arrow ≫ inv (kernelSubobject 0).arrow) ≫\n (kernelSubobject 0).arrow) := @epi_comp _ _ _ _ _ _ e₁ _ _" }, { "state_after": "V : Type u\ninst✝⁴ : Category V\ninst✝³ : HasImages V\nA✝ B✝ C D : V\nf✝ : A✝ ⟶ B✝\ng : B✝ ⟶ C\nh✝ : C ⟶ D\ninst✝² : HasZeroObject V\ninst✝¹ : Preadditive V\ninst✝ : HasEqualizers V\nA B : V\nf : A ⟶ B\nh : Exact f 0\ne₁ : Epi (Subobject.arrow (imageSubobject f) ≫ inv (Subobject.arrow (kernelSubobject 0)))\ne₂ : Epi (Subobject.arrow (imageSubobject f))\n⊢ Epi f", "state_before": "V : Type u\ninst✝⁴ : Category V\ninst✝³ : HasImages V\nA✝ B✝ C D : V\nf✝ : A✝ ⟶ B✝\ng : B✝ ⟶ C\nh✝ : C ⟶ D\ninst✝² : HasZeroObject V\ninst✝¹ : Preadditive V\ninst✝ : HasEqualizers V\nA B : V\nf : A ⟶ B\nh : Exact f 0\ne₁ : Epi (Subobject.arrow (imageSubobject f) ≫ inv (Subobject.arrow (kernelSubobject 0)))\ne₂ :\n Epi\n ((Subobject.arrow (imageSubobject f) ≫ inv (Subobject.arrow (kernelSubobject 0))) ≫\n Subobject.arrow (kernelSubobject 0))\n⊢ Epi f", "tactic": "rw [Category.assoc, IsIso.inv_hom_id, Category.comp_id] at e₂" }, { "state_after": "V : Type u\ninst✝⁴ : Category V\ninst✝³ : HasImages V\nA✝ B✝ C D : V\nf✝ : A✝ ⟶ B✝\ng : B✝ ⟶ C\nh✝ : C ⟶ D\ninst✝² : HasZeroObject V\ninst✝¹ : Preadditive V\ninst✝ : HasEqualizers V\nA B : V\nf : A ⟶ B\nh : Exact f 0\ne₁ : Epi (Subobject.arrow (imageSubobject f) ≫ inv (Subobject.arrow (kernelSubobject 0)))\ne₂ : Epi ((imageSubobjectIso f).hom ≫ image.ι f)\n⊢ Epi f", "state_before": "V : Type u\ninst✝⁴ : Category V\ninst✝³ : HasImages V\nA✝ B✝ C D : V\nf✝ : A✝ ⟶ B✝\ng : B✝ ⟶ C\nh✝ : C ⟶ D\ninst✝² : HasZeroObject V\ninst✝¹ : Preadditive V\ninst✝ : HasEqualizers V\nA B : V\nf : A ⟶ B\nh : Exact f 0\ne₁ : Epi (Subobject.arrow (imageSubobject f) ≫ inv (Subobject.arrow (kernelSubobject 0)))\ne₂ : Epi (Subobject.arrow (imageSubobject f))\n⊢ Epi f", "tactic": "rw [← imageSubobject_arrow] at e₂" }, { "state_after": "V : Type u\ninst✝⁴ : Category V\ninst✝³ : HasImages V\nA✝ B✝ C D : V\nf✝ : A✝ ⟶ B✝\ng : B✝ ⟶ C\nh✝ : C ⟶ D\ninst✝² : HasZeroObject V\ninst✝¹ : Preadditive V\ninst✝ : HasEqualizers V\nA B : V\nf : A ⟶ B\nh : Exact f 0\ne₁ : Epi (Subobject.arrow (imageSubobject f) ≫ inv (Subobject.arrow (kernelSubobject 0)))\ne₂ : Epi ((imageSubobjectIso f).hom ≫ image.ι f)\nthis : Epi (image.ι f)\n⊢ Epi f", "state_before": "V : Type u\ninst✝⁴ : Category V\ninst✝³ : HasImages V\nA✝ B✝ C D : V\nf✝ : A✝ ⟶ B✝\ng : B✝ ⟶ C\nh✝ : C ⟶ D\ninst✝² : HasZeroObject V\ninst✝¹ : Preadditive V\ninst✝ : HasEqualizers V\nA B : V\nf : A ⟶ B\nh : Exact f 0\ne₁ : Epi (Subobject.arrow (imageSubobject f) ≫ inv (Subobject.arrow (kernelSubobject 0)))\ne₂ : Epi ((imageSubobjectIso f).hom ≫ image.ι f)\n⊢ Epi f", "tactic": "haveI : Epi (image.ι f) := epi_of_epi (imageSubobjectIso f).hom (image.ι f)" }, { "state_after": "no goals", "state_before": "V : Type u\ninst✝⁴ : Category V\ninst✝³ : HasImages V\nA✝ B✝ C D : V\nf✝ : A✝ ⟶ B✝\ng : B✝ ⟶ C\nh✝ : C ⟶ D\ninst✝² : HasZeroObject V\ninst✝¹ : Preadditive V\ninst✝ : HasEqualizers V\nA B : V\nf : A ⟶ B\nh : Exact f 0\ne₁ : Epi (Subobject.arrow (imageSubobject f) ≫ inv (Subobject.arrow (kernelSubobject 0)))\ne₂ : Epi ((imageSubobjectIso f).hom ≫ image.ι f)\nthis : Epi (image.ι f)\n⊢ Epi f", "tactic": "apply epi_of_epi_image" } ]

[ 346, 28 ]

[ 336, 1 ]

[ { "state_after": "case intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nL : E →L[𝕜] F\nr ε δ : ℝ\nh : ε ≤ δ\nx : E\nr' : ℝ\nr'r : r' ∈ Ioc (r / 2) r\nhr' : ∀ (y : E), y ∈ ball x r' → ∀ (z : E), z ∈ ball x r' → ‖f z - f y - ↑L (z - y)‖ ≤ ε * r\n⊢ x ∈ A f L r δ", "state_before": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nL : E →L[𝕜] F\nr ε δ : ℝ\nh : ε ≤ δ\n⊢ A f L r ε ⊆ A f L r δ", "tactic": "rintro x ⟨r', r'r, hr'⟩" }, { "state_after": "case intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nL : E →L[𝕜] F\nr ε δ : ℝ\nh : ε ≤ δ\nx : E\nr' : ℝ\nr'r : r' ∈ Ioc (r / 2) r\nhr' : ∀ (y : E), y ∈ ball x r' → ∀ (z : E), z ∈ ball x r' → ‖f z - f y - ↑L (z - y)‖ ≤ ε * r\ny : E\nhy : y ∈ ball x r'\nz : E\nhz : z ∈ ball x r'\n⊢ 0 ≤ r", "state_before": "case intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nL : E →L[𝕜] F\nr ε δ : ℝ\nh : ε ≤ δ\nx : E\nr' : ℝ\nr'r : r' ∈ Ioc (r / 2) r\nhr' : ∀ (y : E), y ∈ ball x r' → ∀ (z : E), z ∈ ball x r' → ‖f z - f y - ↑L (z - y)‖ ≤ ε * r\n⊢ x ∈ A f L r δ", "tactic": "refine' ⟨r', r'r, fun y hy z hz => (hr' y hy z hz).trans (mul_le_mul_of_nonneg_right h _)⟩" }, { "state_after": "no goals", "state_before": "case intro.intro\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nL : E →L[𝕜] F\nr ε δ : ℝ\nh : ε ≤ δ\nx : E\nr' : ℝ\nr'r : r' ∈ Ioc (r / 2) r\nhr' : ∀ (y : E), y ∈ ball x r' → ∀ (z : E), z ∈ ball x r' → ‖f z - f y - ↑L (z - y)‖ ≤ ε * r\ny : E\nhy : y ∈ ball x r'\nz : E\nhz : z ∈ ball x r'\n⊢ 0 ≤ r", "tactic": "linarith [mem_ball.1 hy, r'r.2, @dist_nonneg _ _ y x]" } ]

[ 156, 56 ]

[ 153, 1 ]

[]

[ 96, 6 ]

[ 94, 1 ]

[]

[ 614, 48 ]

[ 612, 1 ]

[ { "state_after": "case h\nR : Type u_2\nR₂ : Type ?u.109627\nK : Type ?u.109630\nM : Type u_1\nM₂ : Type ?u.109636\nV : Type ?u.109639\nS : Type ?u.109642\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nx : M\n⊢ x ∈ (p ⊔ p').toAddSubmonoid ↔ x ∈ p.toAddSubmonoid ⊔ p'.toAddSubmonoid", "state_before": "R : Type u_2\nR₂ : Type ?u.109627\nK : Type ?u.109630\nM : Type u_1\nM₂ : Type ?u.109636\nV : Type ?u.109639\nS : Type ?u.109642\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\n⊢ (p ⊔ p').toAddSubmonoid = p.toAddSubmonoid ⊔ p'.toAddSubmonoid", "tactic": "ext x" }, { "state_after": "case h\nR : Type u_2\nR₂ : Type ?u.109627\nK : Type ?u.109630\nM : Type u_1\nM₂ : Type ?u.109636\nV : Type ?u.109639\nS : Type ?u.109642\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nx : M\n⊢ (∃ y, y ∈ p ∧ ∃ z, z ∈ p' ∧ y + z = x) ↔ ∃ y, y ∈ p.toAddSubmonoid ∧ ∃ z, z ∈ p'.toAddSubmonoid ∧ y + z = x", "state_before": "case h\nR : Type u_2\nR₂ : Type ?u.109627\nK : Type ?u.109630\nM : Type u_1\nM₂ : Type ?u.109636\nV : Type ?u.109639\nS : Type ?u.109642\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nx : M\n⊢ x ∈ (p ⊔ p').toAddSubmonoid ↔ x ∈ p.toAddSubmonoid ⊔ p'.toAddSubmonoid", "tactic": "rw [mem_toAddSubmonoid, mem_sup, AddSubmonoid.mem_sup]" }, { "state_after": "no goals", "state_before": "case h\nR : Type u_2\nR₂ : Type ?u.109627\nK : Type ?u.109630\nM : Type u_1\nM₂ : Type ?u.109636\nV : Type ?u.109639\nS : Type ?u.109642\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nx : M\n⊢ (∃ y, y ∈ p ∧ ∃ z, z ∈ p' ∧ y + z = x) ↔ ∃ y, y ∈ p.toAddSubmonoid ∧ ∃ z, z ∈ p'.toAddSubmonoid ∧ y + z = x", "tactic": "rfl" } ]

[ 377, 6 ]

[ 374, 1 ]

[]

[ 130, 6 ]

[ 129, 1 ]

[]

[ 272, 33 ]

[ 271, 1 ]

[ { "state_after": "α : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nl : List α\nH : Pairwise R l\nthis : Pairwise (Lex (swap fun b a => R a b)) (sublists' (reverse l))\n⊢ Pairwise (fun l₁ l₂ => Lex R (reverse l₁) (reverse l₂)) (sublists l)", "state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nl : List α\nH : Pairwise R l\n⊢ Pairwise (fun l₁ l₂ => Lex R (reverse l₁) (reverse l₂)) (sublists l)", "tactic": "have := (pairwise_reverse.2 H).sublists'" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → α → Prop\nl : List α\nH : Pairwise R l\nthis : Pairwise (Lex (swap fun b a => R a b)) (sublists' (reverse l))\n⊢ Pairwise (fun l₁ l₂ => Lex R (reverse l₁) (reverse l₂)) (sublists l)", "tactic": "rwa [sublists'_reverse, pairwise_map] at this" } ]

[ 369, 48 ]

[ 366, 1 ]

[ { "state_after": "no goals", "state_before": "n : ℕ\n⊢ antidiagonal (n + 1) = (n + 1, 0) ::ₘ map (Prod.map id Nat.succ) (antidiagonal n)", "tactic": "rw [antidiagonal, List.Nat.antidiagonal_succ', ← coe_add, add_comm, antidiagonal, coe_map,\n coe_add, List.singleton_append, cons_coe]" } ]

[ 70, 46 ]

[ 67, 1 ]

[]

[ 389, 23 ]

[ 387, 1 ]

[]

[ 375, 23 ]

[ 374, 1 ]

[]

[ 1347, 44 ]

[ 1346, 1 ]

[]

[ 844, 78 ]

[ 842, 1 ]

[]

[ 286, 79 ]

[ 284, 1 ]

[ { "state_after": "α : Type u_1\nf : Char → α → α\nl m r : List Char\na : α\n⊢ foldrAux f a { data := l ++ List.reverse (List.reverse m) ++ r }\n { byteIdx := utf8Len l + utf8Len (List.reverse (List.reverse m)) } { byteIdx := utf8Len l } =\n List.foldr f a (List.reverse (List.reverse m))", "state_before": "α : Type u_1\nf : Char → α → α\nl m r : List Char\na : α\n⊢ foldrAux f a { data := l ++ m ++ r } { byteIdx := utf8Len l + utf8Len m } { byteIdx := utf8Len l } = List.foldr f a m", "tactic": "rw [← m.reverse_reverse]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nf : Char → α → α\nl m r : List Char\na : α\n⊢ foldrAux f a { data := l ++ List.reverse (List.reverse m) ++ r }\n { byteIdx := utf8Len l + utf8Len (List.reverse (List.reverse m)) } { byteIdx := utf8Len l } =\n List.foldr f a (List.reverse (List.reverse m))", "tactic": "induction m.reverse generalizing r a with (unfold foldrAux; simp)\n| cons c m IH =>\n rw [dif_pos (by exact Nat.lt_add_of_pos_right add_csize_pos)]\n simp [← Nat.add_assoc, by simpa using prev_of_valid (l++m.reverse) c r]\n simp [by simpa using get_of_valid (l++m.reverse) (c::r)]\n simpa using IH (c::r) (f c a)" }, { "state_after": "case nil\nα : Type u_1\nf : Char → α → α\nl m r : List Char\na : α\n⊢ (if h : { byteIdx := utf8Len l } < { byteIdx := utf8Len l + utf8Len (List.reverse []) } then\n let_fun this :=\n (_ :\n (prev { data := l ++ List.reverse [] ++ r } { byteIdx := utf8Len l + utf8Len (List.reverse []) }).byteIdx <\n { byteIdx := utf8Len l + utf8Len (List.reverse []) }.byteIdx);\n let i := prev { data := l ++ List.reverse [] ++ r } { byteIdx := utf8Len l + utf8Len (List.reverse []) };\n let a := f (get { data := l ++ List.reverse [] ++ r } i) a;\n foldrAux f a { data := l ++ List.reverse [] ++ r } i { byteIdx := utf8Len l }\n else a) =\n List.foldr f a (List.reverse [])", "state_before": "case nil\nα : Type u_1\nf : Char → α → α\nl m r : List Char\na : α\n⊢ foldrAux f a { data := l ++ List.reverse [] ++ r } { byteIdx := utf8Len l + utf8Len (List.reverse []) }\n { byteIdx := utf8Len l } =\n List.foldr f a (List.reverse [])", "tactic": "unfold foldrAux" }, { "state_after": "no goals", "state_before": "case nil\nα : Type u_1\nf : Char → α → α\nl m r : List Char\na : α\n⊢ (if h : { byteIdx := utf8Len l } < { byteIdx := utf8Len l + utf8Len (List.reverse []) } then\n let_fun this :=\n (_ :\n (prev { data := l ++ List.reverse [] ++ r } { byteIdx := utf8Len l + utf8Len (List.reverse []) }).byteIdx <\n { byteIdx := utf8Len l + utf8Len (List.reverse []) }.byteIdx);\n let i := prev { data := l ++ List.reverse [] ++ r } { byteIdx := utf8Len l + utf8Len (List.reverse []) };\n let a := f (get { data := l ++ List.reverse [] ++ r } i) a;\n foldrAux f a { data := l ++ List.reverse [] ++ r } i { byteIdx := utf8Len l }\n else a) =\n List.foldr f a (List.reverse [])", "tactic": "simp" }, { "state_after": "case cons\nα : Type u_1\nf : Char → α → α\nl m✝ : List Char\nc : Char\nm : List Char\nIH :\n ∀ (r : List Char) (a : α),\n foldrAux f a { data := l ++ List.reverse m ++ r } { byteIdx := utf8Len l + utf8Len (List.reverse m) }\n { byteIdx := utf8Len l } =\n List.foldr f a (List.reverse m)\nr : List Char\na : α\n⊢ foldrAux f\n (f\n (get { data := l ++ (List.reverse m ++ c :: r) }\n (prev { data := l ++ (List.reverse m ++ c :: r) } { byteIdx := utf8Len l + (utf8Len m + csize c) }))\n a)\n { data := l ++ (List.reverse m ++ c :: r) }\n (prev { data := l ++ (List.reverse m ++ c :: r) } { byteIdx := utf8Len l + (utf8Len m + csize c) })\n { byteIdx := utf8Len l } =\n List.foldl (fun x y => f y x) (f c a) m", "state_before": "case cons\nα : Type u_1\nf : Char → α → α\nl m✝ : List Char\nc : Char\nm : List Char\nIH :\n ∀ (r : List Char) (a : α),\n foldrAux f a { data := l ++ List.reverse m ++ r } { byteIdx := utf8Len l + utf8Len (List.reverse m) }\n { byteIdx := utf8Len l } =\n List.foldr f a (List.reverse m)\nr : List Char\na : α\n⊢ (if h : utf8Len l < utf8Len l + (utf8Len m + csize c) then\n foldrAux f\n (f\n (get { data := l ++ (List.reverse m ++ c :: r) }\n (prev { data := l ++ (List.reverse m ++ c :: r) } { byteIdx := utf8Len l + (utf8Len m + csize c) }))\n a)\n { data := l ++ (List.reverse m ++ c :: r) }\n (prev { data := l ++ (List.reverse m ++ c :: r) } { byteIdx := utf8Len l + (utf8Len m + csize c) })\n { byteIdx := utf8Len l }\n else a) =\n List.foldl (fun x y => f y x) (f c a) m", "tactic": "rw [dif_pos (by exact Nat.lt_add_of_pos_right add_csize_pos)]" }, { "state_after": "case cons\nα : Type u_1\nf : Char → α → α\nl m✝ : List Char\nc : Char\nm : List Char\nIH :\n ∀ (r : List Char) (a : α),\n foldrAux f a { data := l ++ List.reverse m ++ r } { byteIdx := utf8Len l + utf8Len (List.reverse m) }\n { byteIdx := utf8Len l } =\n List.foldr f a (List.reverse m)\nr : List Char\na : α\n⊢ foldrAux f (f (get { data := l ++ (List.reverse m ++ c :: r) } { byteIdx := utf8Len l + utf8Len m }) a)\n { data := l ++ (List.reverse m ++ c :: r) } { byteIdx := utf8Len l + utf8Len m } { byteIdx := utf8Len l } =\n List.foldl (fun x y => f y x) (f c a) m", "state_before": "case cons\nα : Type u_1\nf : Char → α → α\nl m✝ : List Char\nc : Char\nm : List Char\nIH :\n ∀ (r : List Char) (a : α),\n foldrAux f a { data := l ++ List.reverse m ++ r } { byteIdx := utf8Len l + utf8Len (List.reverse m) }\n { byteIdx := utf8Len l } =\n List.foldr f a (List.reverse m)\nr : List Char\na : α\n⊢ foldrAux f\n (f\n (get { data := l ++ (List.reverse m ++ c :: r) }\n (prev { data := l ++ (List.reverse m ++ c :: r) } { byteIdx := utf8Len l + (utf8Len m + csize c) }))\n a)\n { data := l ++ (List.reverse m ++ c :: r) }\n (prev { data := l ++ (List.reverse m ++ c :: r) } { byteIdx := utf8Len l + (utf8Len m + csize c) })\n { byteIdx := utf8Len l } =\n List.foldl (fun x y => f y x) (f c a) m", "tactic": "simp [← Nat.add_assoc, by simpa using prev_of_valid (l++m.reverse) c r]" }, { "state_after": "case cons\nα : Type u_1\nf : Char → α → α\nl m✝ : List Char\nc : Char\nm : List Char\nIH :\n ∀ (r : List Char) (a : α),\n foldrAux f a { data := l ++ List.reverse m ++ r } { byteIdx := utf8Len l + utf8Len (List.reverse m) }\n { byteIdx := utf8Len l } =\n List.foldr f a (List.reverse m)\nr : List Char\na : α\n⊢ foldrAux f (f c a) { data := l ++ (List.reverse m ++ c :: r) } { byteIdx := utf8Len l + utf8Len m }\n { byteIdx := utf8Len l } =\n List.foldl (fun x y => f y x) (f c a) m", "state_before": "case cons\nα : Type u_1\nf : Char → α → α\nl m✝ : List Char\nc : Char\nm : List Char\nIH :\n ∀ (r : List Char) (a : α),\n foldrAux f a { data := l ++ List.reverse m ++ r } { byteIdx := utf8Len l + utf8Len (List.reverse m) }\n { byteIdx := utf8Len l } =\n List.foldr f a (List.reverse m)\nr : List Char\na : α\n⊢ foldrAux f (f (get { data := l ++ (List.reverse m ++ c :: r) } { byteIdx := utf8Len l + utf8Len m }) a)\n { data := l ++ (List.reverse m ++ c :: r) } { byteIdx := utf8Len l + utf8Len m } { byteIdx := utf8Len l } =\n List.foldl (fun x y => f y x) (f c a) m", "tactic": "simp [by simpa using get_of_valid (l++m.reverse) (c::r)]" }, { "state_after": "no goals", "state_before": "case cons\nα : Type u_1\nf : Char → α → α\nl m✝ : List Char\nc : Char\nm : List Char\nIH :\n ∀ (r : List Char) (a : α),\n foldrAux f a { data := l ++ List.reverse m ++ r } { byteIdx := utf8Len l + utf8Len (List.reverse m) }\n { byteIdx := utf8Len l } =\n List.foldr f a (List.reverse m)\nr : List Char\na : α\n⊢ foldrAux f (f c a) { data := l ++ (List.reverse m ++ c :: r) } { byteIdx := utf8Len l + utf8Len m }\n { byteIdx := utf8Len l } =\n List.foldl (fun x y => f y x) (f c a) m", "tactic": "simpa using IH (c::r) (f c a)" }, { "state_after": "no goals", "state_before": "α : Type u_1\nf : Char → α → α\nl m✝ : List Char\nc : Char\nm : List Char\nIH :\n ∀ (r : List Char) (a : α),\n foldrAux f a { data := l ++ List.reverse m ++ r } { byteIdx := utf8Len l + utf8Len (List.reverse m) }\n { byteIdx := utf8Len l } =\n List.foldr f a (List.reverse m)\nr : List Char\na : α\n⊢ utf8Len l < utf8Len l + (utf8Len m + csize c)", "tactic": "exact Nat.lt_add_of_pos_right add_csize_pos" }, { "state_after": "no goals", "state_before": "α : Type u_1\nf : Char → α → α\nl m✝ : List Char\nc : Char\nm : List Char\nIH :\n ∀ (r : List Char) (a : α),\n foldrAux f a { data := l ++ List.reverse m ++ r } { byteIdx := utf8Len l + utf8Len (List.reverse m) }\n { byteIdx := utf8Len l } =\n List.foldr f a (List.reverse m)\nr : List Char\na : α\n⊢ ?m.168183", "tactic": "simpa using prev_of_valid (l++m.reverse) c r" }, { "state_after": "no goals", "state_before": "α : Type u_1\nf : Char → α → α\nl m✝ : List Char\nc : Char\nm : List Char\nIH :\n ∀ (r : List Char) (a : α),\n foldrAux f a { data := l ++ List.reverse m ++ r } { byteIdx := utf8Len l + utf8Len (List.reverse m) }\n { byteIdx := utf8Len l } =\n List.foldr f a (List.reverse m)\nr : List Char\na : α\n⊢ ?m.168355", "tactic": "simpa using get_of_valid (l++m.reverse) (c::r)" } ]

[ 701, 34 ]

[ 693, 1 ]

[]

[ 41, 6 ]

[ 39, 1 ]

[]

[ 275, 96 ]

[ 273, 1 ]

[ { "state_after": "case inl\nR : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n b : ℕ\nhn : IsPrimePow n\nhab : coprime 0 b\n⊢ n ∣ 0 * b ↔ n ∣ 0 ∨ n ∣ b\n\ncase inr\nR : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a b : ℕ\nhab : coprime a b\nhn : IsPrimePow n\nha : a ≠ 0\n⊢ n ∣ a * b ↔ n ∣ a ∨ n ∣ b", "state_before": "R : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a b : ℕ\nhab : coprime a b\nhn : IsPrimePow n\n⊢ n ∣ a * b ↔ n ∣ a ∨ n ∣ b", "tactic": "rcases eq_or_ne a 0 with (rfl | ha)" }, { "state_after": "case inr.inl\nR : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a : ℕ\nhn : IsPrimePow n\nha : a ≠ 0\nhab : coprime a 0\n⊢ n ∣ a * 0 ↔ n ∣ a ∨ n ∣ 0\n\ncase inr.inr\nR : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a b : ℕ\nhab : coprime a b\nhn : IsPrimePow n\nha : a ≠ 0\nhb : b ≠ 0\n⊢ n ∣ a * b ↔ n ∣ a ∨ n ∣ b", "state_before": "case inr\nR : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a b : ℕ\nhab : coprime a b\nhn : IsPrimePow n\nha : a ≠ 0\n⊢ n ∣ a * b ↔ n ∣ a ∨ n ∣ b", "tactic": "rcases eq_or_ne b 0 with (rfl | hb)" }, { "state_after": "case inr.inr\nR : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a b : ℕ\nhab : coprime a b\nhn : IsPrimePow n\nha : a ≠ 0\nhb : b ≠ 0\n⊢ n ∣ a * b → n ∣ a ∨ n ∣ b", "state_before": "case inr.inr\nR : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a b : ℕ\nhab : coprime a b\nhn : IsPrimePow n\nha : a ≠ 0\nhb : b ≠ 0\n⊢ n ∣ a * b ↔ n ∣ a ∨ n ∣ b", "tactic": "refine'\n ⟨_, fun h =>\n Or.elim h (fun i => i.trans ((@dvd_mul_right a b a hab).mpr (dvd_refl a)))\n fun i => i.trans ((@dvd_mul_left a b b hab.symm).mpr (dvd_refl b))⟩" }, { "state_after": "case inr.inr.intro.intro.intro.intro\nR : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn p✝ : R\nk✝ a b : ℕ\nhab : coprime a b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\n⊢ p ^ k ∣ a * b → p ^ k ∣ a ∨ p ^ k ∣ b", "state_before": "case inr.inr\nR : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a b : ℕ\nhab : coprime a b\nhn : IsPrimePow n\nha : a ≠ 0\nhb : b ≠ 0\n⊢ n ∣ a * b → n ∣ a ∨ n ∣ b", "tactic": "obtain ⟨p, k, hp, _, rfl⟩ := (isPrimePow_nat_iff _).1 hn" }, { "state_after": "case inr.inr.intro.intro.intro.intro\nR : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn p✝ : R\nk✝ a b : ℕ\nhab : coprime a b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\n⊢ k ≤ ↑(factorization a) p + ↑(factorization b) p → k ≤ ↑(factorization a) p ∨ k ≤ ↑(factorization b) p", "state_before": "case inr.inr.intro.intro.intro.intro\nR : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn p✝ : R\nk✝ a b : ℕ\nhab : coprime a b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\n⊢ p ^ k ∣ a * b → p ^ k ∣ a ∨ p ^ k ∣ b", "tactic": "simp only [hp.pow_dvd_iff_le_factorization (mul_ne_zero ha hb), Nat.factorization_mul ha hb,\n hp.pow_dvd_iff_le_factorization ha, hp.pow_dvd_iff_le_factorization hb, Pi.add_apply,\n Finsupp.coe_add]" }, { "state_after": "case inr.inr.intro.intro.intro.intro\nR : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn p✝ : R\nk✝ a b : ℕ\nhab : coprime a b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\nthis : ↑(factorization a) p = 0 ∨ ↑(factorization b) p = 0\n⊢ k ≤ ↑(factorization a) p + ↑(factorization b) p → k ≤ ↑(factorization a) p ∨ k ≤ ↑(factorization b) p", "state_before": "case inr.inr.intro.intro.intro.intro\nR : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn p✝ : R\nk✝ a b : ℕ\nhab : coprime a b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\n⊢ k ≤ ↑(factorization a) p + ↑(factorization b) p → k ≤ ↑(factorization a) p ∨ k ≤ ↑(factorization b) p", "tactic": "have : a.factorization p = 0 ∨ b.factorization p = 0 := by\n rw [← Finsupp.not_mem_support_iff, ← Finsupp.not_mem_support_iff, ← not_and_or, ←\n Finset.mem_inter]\n intro t simpa using (Nat.factorization_disjoint_of_coprime hab).le_bot t" }, { "state_after": "no goals", "state_before": "case inr.inr.intro.intro.intro.intro\nR : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn p✝ : R\nk✝ a b : ℕ\nhab : coprime a b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\nthis : ↑(factorization a) p = 0 ∨ ↑(factorization b) p = 0\n⊢ k ≤ ↑(factorization a) p + ↑(factorization b) p → k ≤ ↑(factorization a) p ∨ k ≤ ↑(factorization b) p", "tactic": "cases' this with h h <;> simp [h, imp_or]" }, { "state_after": "case inl\nR : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n b : ℕ\nhn : IsPrimePow n\nhab : b = 1\n⊢ n ∣ 0 * b ↔ n ∣ 0 ∨ n ∣ b", "state_before": "case inl\nR : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n b : ℕ\nhn : IsPrimePow n\nhab : coprime 0 b\n⊢ n ∣ 0 * b ↔ n ∣ 0 ∨ n ∣ b", "tactic": "simp only [Nat.coprime_zero_left] at hab" }, { "state_after": "no goals", "state_before": "case inl\nR : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n b : ℕ\nhn : IsPrimePow n\nhab : b = 1\n⊢ n ∣ 0 * b ↔ n ∣ 0 ∨ n ∣ b", "tactic": "simp [hab, Finset.filter_singleton, not_isPrimePow_one]" }, { "state_after": "case inr.inl\nR : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a : ℕ\nhn : IsPrimePow n\nha : a ≠ 0\nhab : a = 1\n⊢ n ∣ a * 0 ↔ n ∣ a ∨ n ∣ 0", "state_before": "case inr.inl\nR : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a : ℕ\nhn : IsPrimePow n\nha : a ≠ 0\nhab : coprime a 0\n⊢ n ∣ a * 0 ↔ n ∣ a ∨ n ∣ 0", "tactic": "simp only [Nat.coprime_zero_right] at hab" }, { "state_after": "no goals", "state_before": "case inr.inl\nR : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a : ℕ\nhn : IsPrimePow n\nha : a ≠ 0\nhab : a = 1\n⊢ n ∣ a * 0 ↔ n ∣ a ∨ n ∣ 0", "tactic": "simp [hab, Finset.filter_singleton, not_isPrimePow_one]" }, { "state_after": "R : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn p✝ : R\nk✝ a b : ℕ\nhab : coprime a b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\n⊢ ¬p ∈ (factorization a).support ∩ (factorization b).support", "state_before": "R : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn p✝ : R\nk✝ a b : ℕ\nhab : coprime a b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\n⊢ ↑(factorization a) p = 0 ∨ ↑(factorization b) p = 0", "tactic": "rw [← Finsupp.not_mem_support_iff, ← Finsupp.not_mem_support_iff, ← not_and_or, ←\n Finset.mem_inter]" }, { "state_after": "R : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn p✝ : R\nk✝ a b : ℕ\nhab : coprime a b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\nt : p ∈ (factorization a).support ∩ (factorization b).support\n⊢ False", "state_before": "R : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn p✝ : R\nk✝ a b : ℕ\nhab : coprime a b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\n⊢ ¬p ∈ (factorization a).support ∩ (factorization b).support", "tactic": "intro t" }, { "state_after": "no goals", "state_before": "R : Type ?u.12477\ninst✝ : CommMonoidWithZero R\nn p✝ : R\nk✝ a b : ℕ\nhab : coprime a b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\nt : p ∈ (factorization a).support ∩ (factorization b).support\n⊢ False", "tactic": "simpa using (Nat.factorization_disjoint_of_coprime hab).le_bot t" } ]

[ 144, 44 ]

[ 123, 1 ]

[ { "state_after": "ι : Type ?u.1119536\nα : Type u_1\nβ : Type ?u.1119542\nf g : Perm α\nx y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : IsCycle f\nhg : IsCycle g\nh : support f ⊆ support g\nh' : ∀ (x : α), x ∈ support f → ↑f x = ↑g x\nthis : support f = support g\n⊢ f = g", "state_before": "ι : Type ?u.1119536\nα : Type u_1\nβ : Type ?u.1119542\nf g : Perm α\nx y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : IsCycle f\nhg : IsCycle g\nh : support f ⊆ support g\nh' : ∀ (x : α), x ∈ support f → ↑f x = ↑g x\n⊢ f = g", "tactic": "have : f.support = g.support := by\n refine' le_antisymm h _\n intro z hz\n obtain ⟨x, hx, _⟩ := id hf\n have hx' : g x ≠ x := by rwa [← h' x (mem_support.mpr hx)]\n obtain ⟨m, hm⟩ := hg.exists_pow_eq hx' (mem_support.mp hz)\n have h'' : ∀ x ∈ f.support ∩ g.support, f x = g x := by\n intro x hx\n exact h' x (mem_of_mem_inter_left hx)\n rwa [← hm, ←\n pow_eq_on_of_mem_support h'' _ x\n (mem_inter_of_mem (mem_support.mpr hx) (mem_support.mpr hx')),\n pow_apply_mem_support, mem_support]" }, { "state_after": "ι : Type ?u.1119536\nα : Type u_1\nβ : Type ?u.1119542\nf g : Perm α\nx y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : IsCycle f\nhg : IsCycle g\nh : support f ⊆ support g\nh' : ∀ (x : α), x ∈ support f → ↑f x = ↑g x\nthis : support f = support g\n⊢ ∀ (x : α), x ∈ support g → ↑f x = ↑g x", "state_before": "ι : Type ?u.1119536\nα : Type u_1\nβ : Type ?u.1119542\nf g : Perm α\nx y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : IsCycle f\nhg : IsCycle g\nh : support f ⊆ support g\nh' : ∀ (x : α), x ∈ support f → ↑f x = ↑g x\nthis : support f = support g\n⊢ f = g", "tactic": "refine' Equiv.Perm.support_congr h _" }, { "state_after": "no goals", "state_before": "ι : Type ?u.1119536\nα : Type u_1\nβ : Type ?u.1119542\nf g : Perm α\nx y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : IsCycle f\nhg : IsCycle g\nh : support f ⊆ support g\nh' : ∀ (x : α), x ∈ support f → ↑f x = ↑g x\nthis : support f = support g\n⊢ ∀ (x : α), x ∈ support g → ↑f x = ↑g x", "tactic": "simpa [← this] using h'" }, { "state_after": "ι : Type ?u.1119536\nα : Type u_1\nβ : Type ?u.1119542\nf g : Perm α\nx y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : IsCycle f\nhg : IsCycle g\nh : support f ⊆ support g\nh' : ∀ (x : α), x ∈ support f → ↑f x = ↑g x\n⊢ support g ≤ support f", "state_before": "ι : Type ?u.1119536\nα : Type u_1\nβ : Type ?u.1119542\nf g : Perm α\nx y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : IsCycle f\nhg : IsCycle g\nh : support f ⊆ support g\nh' : ∀ (x : α), x ∈ support f → ↑f x = ↑g x\n⊢ support f = support g", "tactic": "refine' le_antisymm h _" }, { "state_after": "ι : Type ?u.1119536\nα : Type u_1\nβ : Type ?u.1119542\nf g : Perm α\nx y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : IsCycle f\nhg : IsCycle g\nh : support f ⊆ support g\nh' : ∀ (x : α), x ∈ support f → ↑f x = ↑g x\nz : α\nhz : z ∈ support g\n⊢ z ∈ support f", "state_before": "ι : Type ?u.1119536\nα : Type u_1\nβ : Type ?u.1119542\nf g : Perm α\nx y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : IsCycle f\nhg : IsCycle g\nh : support f ⊆ support g\nh' : ∀ (x : α), x ∈ support f → ↑f x = ↑g x\n⊢ support g ≤ support f", "tactic": "intro z hz" }, { "state_after": "case intro.intro\nι : Type ?u.1119536\nα : Type u_1\nβ : Type ?u.1119542\nf g : Perm α\nx✝ y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : IsCycle f\nhg : IsCycle g\nh : support f ⊆ support g\nh' : ∀ (x : α), x ∈ support f → ↑f x = ↑g x\nz : α\nhz : z ∈ support g\nx : α\nhx : ↑f x ≠ x\nright✝ : ∀ ⦃y : α⦄, ↑f y ≠ y → SameCycle f x y\n⊢ z ∈ support f", "state_before": "ι : Type ?u.1119536\nα : Type u_1\nβ : Type ?u.1119542\nf g : Perm α\nx y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : IsCycle f\nhg : IsCycle g\nh : support f ⊆ support g\nh' : ∀ (x : α), x ∈ support f → ↑f x = ↑g x\nz : α\nhz : z ∈ support g\n⊢ z ∈ support f", "tactic": "obtain ⟨x, hx, _⟩ := id hf" }, { "state_after": "case intro.intro\nι : Type ?u.1119536\nα : Type u_1\nβ : Type ?u.1119542\nf g : Perm α\nx✝ y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : IsCycle f\nhg : IsCycle g\nh : support f ⊆ support g\nh' : ∀ (x : α), x ∈ support f → ↑f x = ↑g x\nz : α\nhz : z ∈ support g\nx : α\nhx : ↑f x ≠ x\nright✝ : ∀ ⦃y : α⦄, ↑f y ≠ y → SameCycle f x y\nhx' : ↑g x ≠ x\n⊢ z ∈ support f", "state_before": "case intro.intro\nι : Type ?u.1119536\nα : Type u_1\nβ : Type ?u.1119542\nf g : Perm α\nx✝ y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : IsCycle f\nhg : IsCycle g\nh : support f ⊆ support g\nh' : ∀ (x : α), x ∈ support f → ↑f x = ↑g x\nz : α\nhz : z ∈ support g\nx : α\nhx : ↑f x ≠ x\nright✝ : ∀ ⦃y : α⦄, ↑f y ≠ y → SameCycle f x y\n⊢ z ∈ support f", "tactic": "have hx' : g x ≠ x := by rwa [← h' x (mem_support.mpr hx)]" }, { "state_after": "case intro.intro.intro\nι : Type ?u.1119536\nα : Type u_1\nβ : Type ?u.1119542\nf g : Perm α\nx✝ y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : IsCycle f\nhg : IsCycle g\nh : support f ⊆ support g\nh' : ∀ (x : α), x ∈ support f → ↑f x = ↑g x\nz : α\nhz : z ∈ support g\nx : α\nhx : ↑f x ≠ x\nright✝ : ∀ ⦃y : α⦄, ↑f y ≠ y → SameCycle f x y\nhx' : ↑g x ≠ x\nm : ℕ\nhm : ↑(g ^ m) x = z\n⊢ z ∈ support f", "state_before": "case intro.intro\nι : Type ?u.1119536\nα : Type u_1\nβ : Type ?u.1119542\nf g : Perm α\nx✝ y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : IsCycle f\nhg : IsCycle g\nh : support f ⊆ support g\nh' : ∀ (x : α), x ∈ support f → ↑f x = ↑g x\nz : α\nhz : z ∈ support g\nx : α\nhx : ↑f x ≠ x\nright✝ : ∀ ⦃y : α⦄, ↑f y ≠ y → SameCycle f x y\nhx' : ↑g x ≠ x\n⊢ z ∈ support f", "tactic": "obtain ⟨m, hm⟩ := hg.exists_pow_eq hx' (mem_support.mp hz)" }, { "state_after": "case intro.intro.intro\nι : Type ?u.1119536\nα : Type u_1\nβ : Type ?u.1119542\nf g : Perm α\nx✝ y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : IsCycle f\nhg : IsCycle g\nh : support f ⊆ support g\nh' : ∀ (x : α), x ∈ support f → ↑f x = ↑g x\nz : α\nhz : z ∈ support g\nx : α\nhx : ↑f x ≠ x\nright✝ : ∀ ⦃y : α⦄, ↑f y ≠ y → SameCycle f x y\nhx' : ↑g x ≠ x\nm : ℕ\nhm : ↑(g ^ m) x = z\nh'' : ∀ (x : α), x ∈ support f ∩ support g → ↑f x = ↑g x\n⊢ z ∈ support f", "state_before": "case intro.intro.intro\nι : Type ?u.1119536\nα : Type u_1\nβ : Type ?u.1119542\nf g : Perm α\nx✝ y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : IsCycle f\nhg : IsCycle g\nh : support f ⊆ support g\nh' : ∀ (x : α), x ∈ support f → ↑f x = ↑g x\nz : α\nhz : z ∈ support g\nx : α\nhx : ↑f x ≠ x\nright✝ : ∀ ⦃y : α⦄, ↑f y ≠ y → SameCycle f x y\nhx' : ↑g x ≠ x\nm : ℕ\nhm : ↑(g ^ m) x = z\n⊢ z ∈ support f", "tactic": "have h'' : ∀ x ∈ f.support ∩ g.support, f x = g x := by\n intro x hx\n exact h' x (mem_of_mem_inter_left hx)" }, { "state_after": "no goals", "state_before": "case intro.intro.intro\nι : Type ?u.1119536\nα : Type u_1\nβ : Type ?u.1119542\nf g : Perm α\nx✝ y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : IsCycle f\nhg : IsCycle g\nh : support f ⊆ support g\nh' : ∀ (x : α), x ∈ support f → ↑f x = ↑g x\nz : α\nhz : z ∈ support g\nx : α\nhx : ↑f x ≠ x\nright✝ : ∀ ⦃y : α⦄, ↑f y ≠ y → SameCycle f x y\nhx' : ↑g x ≠ x\nm : ℕ\nhm : ↑(g ^ m) x = z\nh'' : ∀ (x : α), x ∈ support f ∩ support g → ↑f x = ↑g x\n⊢ z ∈ support f", "tactic": "rwa [← hm, ←\n pow_eq_on_of_mem_support h'' _ x\n (mem_inter_of_mem (mem_support.mpr hx) (mem_support.mpr hx')),\n pow_apply_mem_support, mem_support]" }, { "state_after": "no goals", "state_before": "ι : Type ?u.1119536\nα : Type u_1\nβ : Type ?u.1119542\nf g : Perm α\nx✝ y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : IsCycle f\nhg : IsCycle g\nh : support f ⊆ support g\nh' : ∀ (x : α), x ∈ support f → ↑f x = ↑g x\nz : α\nhz : z ∈ support g\nx : α\nhx : ↑f x ≠ x\nright✝ : ∀ ⦃y : α⦄, ↑f y ≠ y → SameCycle f x y\n⊢ ↑g x ≠ x", "tactic": "rwa [← h' x (mem_support.mpr hx)]" }, { "state_after": "ι : Type ?u.1119536\nα : Type u_1\nβ : Type ?u.1119542\nf g : Perm α\nx✝¹ y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : IsCycle f\nhg : IsCycle g\nh : support f ⊆ support g\nh' : ∀ (x : α), x ∈ support f → ↑f x = ↑g x\nz : α\nhz : z ∈ support g\nx✝ : α\nhx✝ : ↑f x✝ ≠ x✝\nright✝ : ∀ ⦃y : α⦄, ↑f y ≠ y → SameCycle f x✝ y\nhx' : ↑g x✝ ≠ x✝\nm : ℕ\nhm : ↑(g ^ m) x✝ = z\nx : α\nhx : x ∈ support f ∩ support g\n⊢ ↑f x = ↑g x", "state_before": "ι : Type ?u.1119536\nα : Type u_1\nβ : Type ?u.1119542\nf g : Perm α\nx✝ y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : IsCycle f\nhg : IsCycle g\nh : support f ⊆ support g\nh' : ∀ (x : α), x ∈ support f → ↑f x = ↑g x\nz : α\nhz : z ∈ support g\nx : α\nhx : ↑f x ≠ x\nright✝ : ∀ ⦃y : α⦄, ↑f y ≠ y → SameCycle f x y\nhx' : ↑g x ≠ x\nm : ℕ\nhm : ↑(g ^ m) x = z\n⊢ ∀ (x : α), x ∈ support f ∩ support g → ↑f x = ↑g x", "tactic": "intro x hx" }, { "state_after": "no goals", "state_before": "ι : Type ?u.1119536\nα : Type u_1\nβ : Type ?u.1119542\nf g : Perm α\nx✝¹ y : α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhf : IsCycle f\nhg : IsCycle g\nh : support f ⊆ support g\nh' : ∀ (x : α), x ∈ support f → ↑f x = ↑g x\nz : α\nhz : z ∈ support g\nx✝ : α\nhx✝ : ↑f x✝ ≠ x✝\nright✝ : ∀ ⦃y : α⦄, ↑f y ≠ y → SameCycle f x✝ y\nhx' : ↑g x✝ ≠ x✝\nm : ℕ\nhm : ↑(g ^ m) x✝ = z\nx : α\nhx : x ∈ support f ∩ support g\n⊢ ↑f x = ↑g x", "tactic": "exact h' x (mem_of_mem_inter_left hx)" } ]

[ 604, 26 ]

[ 588, 1 ]

[]

[ 807, 17 ]

[ 806, 1 ]

[ { "state_after": "case left\nα : Type u\nβ : Type v\nδ : Type w\ns₁✝ s₂✝ : Stream' α\nhh : head s₁✝ = head s₂✝\nht₁ : s₁✝ = tail s₁✝\nht₂ : s₂✝ = tail s₂✝\ns₁ s₂ : Stream' α\nx✝ : (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) s₁ s₂\nh₁ : head s₁ = head s₂\nh₂ : s₁ = tail s₁\nh₃ : s₂ = tail s₂\n⊢ head s₁ = head s₂\n\ncase right\nα : Type u\nβ : Type v\nδ : Type w\ns₁✝ s₂✝ : Stream' α\nhh : head s₁✝ = head s₂✝\nht₁ : s₁✝ = tail s₁✝\nht₂ : s₂✝ = tail s₂✝\ns₁ s₂ : Stream' α\nx✝ : (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) s₁ s₂\nh₁ : head s₁ = head s₂\nh₂ : s₁ = tail s₁\nh₃ : s₂ = tail s₂\n⊢ (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) (tail s₁) (tail s₂)", "state_before": "α : Type u\nβ : Type v\nδ : Type w\ns₁✝ s₂✝ : Stream' α\nhh : head s₁✝ = head s₂✝\nht₁ : s₁✝ = tail s₁✝\nht₂ : s₂✝ = tail s₂✝\ns₁ s₂ : Stream' α\nx✝ : (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) s₁ s₂\nh₁ : head s₁ = head s₂\nh₂ : s₁ = tail s₁\nh₃ : s₂ = tail s₂\n⊢ head s₁ = head s₂ ∧ (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) (tail s₁) (tail s₂)", "tactic": "constructor" }, { "state_after": "case right\nα : Type u\nβ : Type v\nδ : Type w\ns₁✝ s₂✝ : Stream' α\nhh : head s₁✝ = head s₂✝\nht₁ : s₁✝ = tail s₁✝\nht₂ : s₂✝ = tail s₂✝\ns₁ s₂ : Stream' α\nx✝ : (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) s₁ s₂\nh₁ : head s₁ = head s₂\nh₂ : s₁ = tail s₁\nh₃ : s₂ = tail s₂\n⊢ (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) (tail s₁) (tail s₂)", "state_before": "case left\nα : Type u\nβ : Type v\nδ : Type w\ns₁✝ s₂✝ : Stream' α\nhh : head s₁✝ = head s₂✝\nht₁ : s₁✝ = tail s₁✝\nht₂ : s₂✝ = tail s₂✝\ns₁ s₂ : Stream' α\nx✝ : (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) s₁ s₂\nh₁ : head s₁ = head s₂\nh₂ : s₁ = tail s₁\nh₃ : s₂ = tail s₂\n⊢ head s₁ = head s₂\n\ncase right\nα : Type u\nβ : Type v\nδ : Type w\ns₁✝ s₂✝ : Stream' α\nhh : head s₁✝ = head s₂✝\nht₁ : s₁✝ = tail s₁✝\nht₂ : s₂✝ = tail s₂✝\ns₁ s₂ : Stream' α\nx✝ : (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) s₁ s₂\nh₁ : head s₁ = head s₂\nh₂ : s₁ = tail s₁\nh₃ : s₂ = tail s₂\n⊢ (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) (tail s₁) (tail s₂)", "tactic": "exact h₁" }, { "state_after": "case right\nα : Type u\nβ : Type v\nδ : Type w\ns₁✝ s₂✝ : Stream' α\nhh : head s₁✝ = head s₂✝\nht₁ : s₁✝ = tail s₁✝\nht₂ : s₂✝ = tail s₂✝\ns₁ s₂ : Stream' α\nx✝ : (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) s₁ s₂\nh₁ : head s₁ = head s₂\nh₂ : s₁ = tail s₁\nh₃ : s₂ = tail s₂\n⊢ (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) s₁ s₂", "state_before": "case right\nα : Type u\nβ : Type v\nδ : Type w\ns₁✝ s₂✝ : Stream' α\nhh : head s₁✝ = head s₂✝\nht₁ : s₁✝ = tail s₁✝\nht₂ : s₂✝ = tail s₂✝\ns₁ s₂ : Stream' α\nx✝ : (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) s₁ s₂\nh₁ : head s₁ = head s₂\nh₂ : s₁ = tail s₁\nh₃ : s₂ = tail s₂\n⊢ (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) (tail s₁) (tail s₂)", "tactic": "rw [← h₂, ← h₃]" }, { "state_after": "no goals", "state_before": "case right\nα : Type u\nβ : Type v\nδ : Type w\ns₁✝ s₂✝ : Stream' α\nhh : head s₁✝ = head s₂✝\nht₁ : s₁✝ = tail s₁✝\nht₂ : s₂✝ = tail s₂✝\ns₁ s₂ : Stream' α\nx✝ : (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) s₁ s₂\nh₁ : head s₁ = head s₂\nh₂ : s₁ = tail s₁\nh₃ : s₂ = tail s₂\n⊢ (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) s₁ s₂", "tactic": "(repeat' constructor) <;> assumption" }, { "state_after": "case right.left\nα : Type u\nβ : Type v\nδ : Type w\ns₁✝ s₂✝ : Stream' α\nhh : head s₁✝ = head s₂✝\nht₁ : s₁✝ = tail s₁✝\nht₂ : s₂✝ = tail s₂✝\ns₁ s₂ : Stream' α\nx✝ : (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) s₁ s₂\nh₁ : head s₁ = head s₂\nh₂ : s₁ = tail s₁\nh₃ : s₂ = tail s₂\n⊢ head s₁ = head s₂\n\ncase right.right.left\nα : Type u\nβ : Type v\nδ : Type w\ns₁✝ s₂✝ : Stream' α\nhh : head s₁✝ = head s₂✝\nht₁ : s₁✝ = tail s₁✝\nht₂ : s₂✝ = tail s₂✝\ns₁ s₂ : Stream' α\nx✝ : (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) s₁ s₂\nh₁ : head s₁ = head s₂\nh₂ : s₁ = tail s₁\nh₃ : s₂ = tail s₂\n⊢ s₁ = tail s₁\n\ncase right.right.right\nα : Type u\nβ : Type v\nδ : Type w\ns₁✝ s₂✝ : Stream' α\nhh : head s₁✝ = head s₂✝\nht₁ : s₁✝ = tail s₁✝\nht₂ : s₂✝ = tail s₂✝\ns₁ s₂ : Stream' α\nx✝ : (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) s₁ s₂\nh₁ : head s₁ = head s₂\nh₂ : s₁ = tail s₁\nh₃ : s₂ = tail s₂\n⊢ s₂ = tail s₂", "state_before": "case right\nα : Type u\nβ : Type v\nδ : Type w\ns₁✝ s₂✝ : Stream' α\nhh : head s₁✝ = head s₂✝\nht₁ : s₁✝ = tail s₁✝\nht₂ : s₂✝ = tail s₂✝\ns₁ s₂ : Stream' α\nx✝ : (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) s₁ s₂\nh₁ : head s₁ = head s₂\nh₂ : s₁ = tail s₁\nh₃ : s₂ = tail s₂\n⊢ (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) s₁ s₂", "tactic": "repeat' constructor" }, { "state_after": "case right.right.right\nα : Type u\nβ : Type v\nδ : Type w\ns₁✝ s₂✝ : Stream' α\nhh : head s₁✝ = head s₂✝\nht₁ : s₁✝ = tail s₁✝\nht₂ : s₂✝ = tail s₂✝\ns₁ s₂ : Stream' α\nx✝ : (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) s₁ s₂\nh₁ : head s₁ = head s₂\nh₂ : s₁ = tail s₁\nh₃ : s₂ = tail s₂\n⊢ s₂ = tail s₂", "state_before": "case right.right.right\nα : Type u\nβ : Type v\nδ : Type w\ns₁✝ s₂✝ : Stream' α\nhh : head s₁✝ = head s₂✝\nht₁ : s₁✝ = tail s₁✝\nht₂ : s₂✝ = tail s₂✝\ns₁ s₂ : Stream' α\nx✝ : (fun s₁ s₂ => head s₁ = head s₂ ∧ s₁ = tail s₁ ∧ s₂ = tail s₂) s₁ s₂\nh₁ : head s₁ = head s₂\nh₂ : s₁ = tail s₁\nh₃ : s₂ = tail s₂\n⊢ s₂ = tail s₂", "tactic": "constructor" } ]

[ 333, 39 ]

[ 327, 1 ]

[]

[ 508, 85 ]

[ 508, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_2\nβ : Type u_1\nR : Type ?u.3292\nR' : Type ?u.3295\nms : Set (OuterMeasure α)\nm✝ : OuterMeasure α\ninst✝ : Countable β\nm : OuterMeasure α\ns : β → Set α\nh : ∀ (i : β), ↑m (s i) = 0\n⊢ ↑m (⋃ (i : β), s i) = 0", "tactic": "simpa [h] using m.iUnion s" } ]

[ 119, 54 ]

[ 118, 1 ]

[ { "state_after": "α : Type ?u.268594\nβ : Type ?u.268597\nγ : Type ?u.268600\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b c : Ordinal\nh : c ∣ b\n⊢ a % b % c = (b * (a / b) + a % b) % c", "state_before": "α : Type ?u.268594\nβ : Type ?u.268597\nγ : Type ?u.268600\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b c : Ordinal\nh : c ∣ b\n⊢ a % b % c = a % c", "tactic": "nth_rw 2 [← div_add_mod a b]" }, { "state_after": "case intro\nα : Type ?u.268594\nβ : Type ?u.268597\nγ : Type ?u.268600\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na c d : Ordinal\n⊢ a % (c * d) % c = (c * d * (a / (c * d)) + a % (c * d)) % c", "state_before": "α : Type ?u.268594\nβ : Type ?u.268597\nγ : Type ?u.268600\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b c : Ordinal\nh : c ∣ b\n⊢ a % b % c = (b * (a / b) + a % b) % c", "tactic": "rcases h with ⟨d, rfl⟩" }, { "state_after": "no goals", "state_before": "case intro\nα : Type ?u.268594\nβ : Type ?u.268597\nγ : Type ?u.268600\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na c d : Ordinal\n⊢ a % (c * d) % c = (c * d * (a / (c * d)) + a % (c * d)) % c", "tactic": "rw [mul_assoc, mul_add_mod_self]" } ]

[ 1095, 35 ]

[ 1092, 1 ]

[]

[ 179, 26 ]

[ 178, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type ?u.113900\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : LocallyFiniteOrder α\na b : α\n⊢ Ioo (↑toDual a) (↑toDual b) = Ioo b a", "state_before": "α : Type u_1\nβ : Type ?u.113900\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : LocallyFiniteOrder α\na b : α\n⊢ Ioo (↑toDual a) (↑toDual b) = map (Equiv.toEmbedding toDual) (Ioo b a)", "tactic": "refine' Eq.trans _ map_refl.symm" }, { "state_after": "case a\nα : Type u_1\nβ : Type ?u.113900\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : LocallyFiniteOrder α\na b : α\nc : (fun x => αᵒᵈ) a\n⊢ c ∈ Ioo (↑toDual a) (↑toDual b) ↔ c ∈ Ioo b a", "state_before": "α : Type u_1\nβ : Type ?u.113900\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : LocallyFiniteOrder α\na b : α\n⊢ Ioo (↑toDual a) (↑toDual b) = Ioo b a", "tactic": "ext c" }, { "state_after": "case a\nα : Type u_1\nβ : Type ?u.113900\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : LocallyFiniteOrder α\na b : α\nc : (fun x => αᵒᵈ) a\n⊢ ↑toDual a < c ∧ c < ↑toDual b ↔ b < c ∧ c < a", "state_before": "case a\nα : Type u_1\nβ : Type ?u.113900\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : LocallyFiniteOrder α\na b : α\nc : (fun x => αᵒᵈ) a\n⊢ c ∈ Ioo (↑toDual a) (↑toDual b) ↔ c ∈ Ioo b a", "tactic": "rw [mem_Ioo, mem_Ioo (α := α)]" }, { "state_after": "no goals", "state_before": "case a\nα : Type u_1\nβ : Type ?u.113900\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : LocallyFiniteOrder α\na b : α\nc : (fun x => αᵒᵈ) a\n⊢ ↑toDual a < c ∧ c < ↑toDual b ↔ b < c ∧ c < a", "tactic": "exact and_comm" } ]

[ 852, 17 ]

[ 848, 1 ]

[]

[ 2730, 6 ]

[ 2728, 1 ]

[ { "state_after": "case inl\nd : ℕ\n⊢ d ∣ 0 ↔ ∀ (p k : ℕ), Prime p → p ^ k ∣ d → p ^ k ∣ 0\n\ncase inr\nn d : ℕ\nhn : n ≠ 0\n⊢ d ∣ n ↔ ∀ (p k : ℕ), Prime p → p ^ k ∣ d → p ^ k ∣ n", "state_before": "n d : ℕ\n⊢ d ∣ n ↔ ∀ (p k : ℕ), Prime p → p ^ k ∣ d → p ^ k ∣ n", "tactic": "rcases eq_or_ne n 0 with (rfl | hn)" }, { "state_after": "case inr.inl\nn : ℕ\nhn : n ≠ 0\n⊢ 0 ∣ n ↔ ∀ (p k : ℕ), Prime p → p ^ k ∣ 0 → p ^ k ∣ n\n\ncase inr.inr\nn d : ℕ\nhn : n ≠ 0\nhd : d ≠ 0\n⊢ d ∣ n ↔ ∀ (p k : ℕ), Prime p → p ^ k ∣ d → p ^ k ∣ n", "state_before": "case inr\nn d : ℕ\nhn : n ≠ 0\n⊢ d ∣ n ↔ ∀ (p k : ℕ), Prime p → p ^ k ∣ d → p ^ k ∣ n", "tactic": "rcases eq_or_ne d 0 with (rfl | hd)" }, { "state_after": "case inr.inr\nn d : ℕ\nhn : n ≠ 0\nhd : d ≠ 0\n⊢ (∀ (p k : ℕ), Prime p → p ^ k ∣ d → p ^ k ∣ n) → d ∣ n", "state_before": "case inr.inr\nn d : ℕ\nhn : n ≠ 0\nhd : d ≠ 0\n⊢ d ∣ n ↔ ∀ (p k : ℕ), Prime p → p ^ k ∣ d → p ^ k ∣ n", "tactic": "refine' ⟨fun h p k _ hpkd => dvd_trans hpkd h, _⟩" }, { "state_after": "case inr.inr\nn d : ℕ\nhn : n ≠ 0\nhd : d ≠ 0\n⊢ (∀ (p k : ℕ), Prime p → p ^ k ∣ d → p ^ k ∣ n) → ∀ (p : ℕ), Prime p → ↑(factorization d) p ≤ ↑(factorization n) p", "state_before": "case inr.inr\nn d : ℕ\nhn : n ≠ 0\nhd : d ≠ 0\n⊢ (∀ (p k : ℕ), Prime p → p ^ k ∣ d → p ^ k ∣ n) → d ∣ n", "tactic": "rw [← factorization_prime_le_iff_dvd hd hn]" }, { "state_after": "case inr.inr\nn d : ℕ\nhn : n ≠ 0\nhd : d ≠ 0\nh : ∀ (p k : ℕ), Prime p → p ^ k ∣ d → p ^ k ∣ n\np : ℕ\npp : Prime p\n⊢ ↑(factorization d) p ≤ ↑(factorization n) p", "state_before": "case inr.inr\nn d : ℕ\nhn : n ≠ 0\nhd : d ≠ 0\n⊢ (∀ (p k : ℕ), Prime p → p ^ k ∣ d → p ^ k ∣ n) → ∀ (p : ℕ), Prime p → ↑(factorization d) p ≤ ↑(factorization n) p", "tactic": "intro h p pp" }, { "state_after": "case inr.inr\nn d : ℕ\nhn : n ≠ 0\nhd : d ≠ 0\nh : ∀ (p k : ℕ), Prime p → p ^ k ∣ d → p ^ k ∣ n\np : ℕ\npp : Prime p\n⊢ p ^ ↑(factorization d) p ∣ n", "state_before": "case inr.inr\nn d : ℕ\nhn : n ≠ 0\nhd : d ≠ 0\nh : ∀ (p k : ℕ), Prime p → p ^ k ∣ d → p ^ k ∣ n\np : ℕ\npp : Prime p\n⊢ ↑(factorization d) p ≤ ↑(factorization n) p", "tactic": "simp_rw [← pp.pow_dvd_iff_le_factorization hn]" }, { "state_after": "no goals", "state_before": "case inr.inr\nn d : ℕ\nhn : n ≠ 0\nhd : d ≠ 0\nh : ∀ (p k : ℕ), Prime p → p ^ k ∣ d → p ^ k ∣ n\np : ℕ\npp : Prime p\n⊢ p ^ ↑(factorization d) p ∣ n", "tactic": "exact h p _ pp (ord_proj_dvd _ _)" }, { "state_after": "no goals", "state_before": "case inl\nd : ℕ\n⊢ d ∣ 0 ↔ ∀ (p k : ℕ), Prime p → p ^ k ∣ d → p ^ k ∣ 0", "tactic": "simp" }, { "state_after": "case inr.inl\nn : ℕ\nhn : n ≠ 0\n⊢ ∃ x x_1 h x_2, ¬x ^ x_1 ∣ n", "state_before": "case inr.inl\nn : ℕ\nhn : n ≠ 0\n⊢ 0 ∣ n ↔ ∀ (p k : ℕ), Prime p → p ^ k ∣ 0 → p ^ k ∣ n", "tactic": "simp only [zero_dvd_iff, hn, false_iff_iff, not_forall]" }, { "state_after": "no goals", "state_before": "case inr.inl\nn : ℕ\nhn : n ≠ 0\n⊢ ∃ x x_1 h x_2, ¬x ^ x_1 ∣ n", "tactic": "exact ⟨2, n, prime_two, dvd_zero _, mt (le_of_dvd hn.bot_lt) (lt_two_pow n).not_le⟩" } ]

[ 657, 36 ]

[ 646, 1 ]

[]

[ 250, 12 ]

[ 249, 11 ]

[]

[ 1111, 79 ]

[ 1110, 1 ]

[]

[ 269, 89 ]

[ 267, 1 ]

[]

[ 87, 11 ]

[ 86, 1 ]

[]

[ 620, 6 ]

[ 618, 1 ]

[ { "state_after": "no goals", "state_before": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx y : E\n⊢ {x} * closedBall 1 δ = closedBall x δ", "tactic": "simp" } ]

[ 181, 88 ]

[ 181, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nX : Type ?u.101350\nι : Type ?u.101353\ninst✝¹ : PseudoMetricSpace α\nx y z : α\nδ ε ε₁ ε₂ : ℝ\ns✝ : Set α\ninst✝ : PseudoMetricSpace β\nf : α → β\na : α\ns : Set α\n⊢ ContinuousWithinAt f s a ↔ ∀ (ε : ℝ), ε > 0 → ∃ δ, δ > 0 ∧ ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) (f a) < ε", "tactic": "rw [ContinuousWithinAt, tendsto_nhdsWithin_nhds]" } ]

[ 1060, 54 ]

[ 1057, 1 ]

[ { "state_after": "α : Type ?u.1359\nβ : Type ?u.1362\nι : Type ?u.1365\n⊢ Tendsto (fun a => ↑(↑a)⁻¹) atTop (𝓝 ↑0)", "state_before": "α : Type ?u.1359\nβ : Type ?u.1362\nι : Type ?u.1365\n⊢ Tendsto (fun n => (↑n)⁻¹) atTop (𝓝 0)", "tactic": "rw [← NNReal.tendsto_coe]" }, { "state_after": "no goals", "state_before": "α : Type ?u.1359\nβ : Type ?u.1362\nι : Type ?u.1365\n⊢ Tendsto (fun a => ↑(↑a)⁻¹) atTop (𝓝 ↑0)", "tactic": "exact _root_.tendsto_inverse_atTop_nhds_0_nat" } ]

[ 45, 48 ]

[ 42, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.1574917\nγ : Type ?u.1574920\nδ : Type ?u.1574923\ninst✝³ : MeasurableSpace α\nμ ν : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α →ₘ[μ] ℝ≥0∞\n⊢ (∫⁻ (a : α), ↑f a ∂μ) = lintegral f", "tactic": "rw [← lintegral_mk, mk_coeFn]" } ]

[ 854, 32 ]

[ 853, 1 ]

[]

[ 465, 38 ]

[ 464, 1 ]

[ { "state_after": "no goals", "state_before": "⊢ cos ↑π = -1", "tactic": "rw [cos_coe, Real.cos_pi]" } ]

[ 402, 74 ]

[ 402, 1 ]

[ { "state_after": "case h.e'_3\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : ContinuousMul G\nh : IsClosed {1}\nx : G\n⊢ {x} = (fun x_1 => x_1 * x) '' {1}", "state_before": "α : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : ContinuousMul G\nh : IsClosed {1}\nx : G\n⊢ IsClosed {x}", "tactic": "convert isClosedMap_mul_right x _ h" }, { "state_after": "no goals", "state_before": "case h.e'_3\nα : Type u\nβ : Type v\nG : Type w\nH : Type x\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : ContinuousMul G\nh : IsClosed {1}\nx : G\n⊢ {x} = (fun x_1 => x_1 * x) '' {1}", "tactic": "simp" } ]

[ 1419, 10 ]

[ 1416, 1 ]

[ { "state_after": "no goals", "state_before": "ι : Sort ?u.15646\nα : Type u_1\nβ : Type ?u.15652\nX : Type ?u.15655\nY : Type ?u.15658\ns : Set (Filter α)\nl : Filter α\n⊢ l ∈ interior s ↔ ∃ t, t ∈ l ∧ Iic (𝓟 t) ⊆ s", "tactic": "rw [mem_interior_iff_mem_nhds, mem_nhds_iff]" } ]

[ 163, 95 ]

[ 162, 11 ]

[]

[ 981, 37 ]

[ 980, 1 ]

[]

[ 820, 65 ]

[ 818, 9 ]

[]

[ 983, 23 ]

[ 982, 1 ]