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Mathlib/Analysis/Calculus/ContDiff.lean | norm_iteratedFDerivWithin_comp_le | [
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"tactic": "let Fu : Type max uF uG := ULift.{uG, uF} F"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"tactic": "let Gu : Type max uF uG := ULift.{uF, uG} G"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"tactic": "have isoF : Fu ≃ₗᵢ[𝕜] F := LinearIsometryEquiv.ulift 𝕜 F"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"tactic": "have isoG : Gu ≃ₗᵢ[𝕜] G := LinearIsometryEquiv.ulift 𝕜 G"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"tactic": "let fu : E → Fu := isoF.symm ∘ f"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"tactic": "let gu : Fu → Gu := isoG.symm ∘ g ∘ isoF"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"tactic": "let tu := isoF ⁻¹' t"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"tactic": "have htu : UniqueDiffOn 𝕜 tu := isoF.toContinuousLinearEquiv.uniqueDiffOn_preimage_iff.2 ht"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"tactic": "have hstu : MapsTo fu s tu := fun y hy ↦ by\n simpa only [mem_preimage, comp_apply, LinearIsometryEquiv.apply_symm_apply] using hst hy"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x) = f x\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"tactic": "have Ffu : isoF (fu x) = f x := by simp only [comp_apply, LinearIsometryEquiv.apply_symm_apply]"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x) = f x\nhfu : ContDiffOn 𝕜 (↑n) fu s\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x) = f x\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"tactic": "have hfu : ContDiffOn 𝕜 n fu s := isoF.symm.contDiff.comp_contDiffOn (hf.of_le hn)"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x) = f x\nhfu : ContDiffOn 𝕜 (↑n) fu s\nhgu : ContDiffOn 𝕜 (↑n) gu tu\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x) = f x\nhfu : ContDiffOn 𝕜 (↑n) fu s\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"tactic": "have hgu : ContDiffOn 𝕜 n gu tu :=\n isoG.symm.contDiff.comp_contDiffOn\n ((hg.of_le hn).comp_continuousLinearMap (isoF : Fu →L[𝕜] F))"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x) = f x\nhfu : ContDiffOn 𝕜 (↑n) fu s\nhgu : ContDiffOn 𝕜 (↑n) gu tu\nNfu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i fu s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x) = f x\nhfu : ContDiffOn 𝕜 (↑n) fu s\nhgu : ContDiffOn 𝕜 (↑n) gu tu\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"tactic": "have Nfu : ∀ i, ‖iteratedFDerivWithin 𝕜 i fu s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖ := fun i ↦ by\n rw [LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_left _ _ hs hx]"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x) = f x\nhfu : ContDiffOn 𝕜 (↑n) fu s\nhgu : ContDiffOn 𝕜 (↑n) gu tu\nNfu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i fu s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoF) ∘ f) s x‖ ≤ D ^ i\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x) = f x\nhfu : ContDiffOn 𝕜 (↑n) fu s\nhgu : ContDiffOn 𝕜 (↑n) gu tu\nNfu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i fu s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"tactic": "simp_rw [← Nfu] at hD"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x) = f x\nhfu : ContDiffOn 𝕜 (↑n) fu s\nhgu : ContDiffOn 𝕜 (↑n) gu tu\nNfu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i fu s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoF) ∘ f) s x‖ ≤ D ^ i\nNgu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i gu tu (fu x)‖ = ‖iteratedFDerivWithin 𝕜 i g t (f x)‖\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x) = f x\nhfu : ContDiffOn 𝕜 (↑n) fu s\nhgu : ContDiffOn 𝕜 (↑n) gu tu\nNfu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i fu s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoF) ∘ f) s x‖ ≤ D ^ i\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"tactic": "have Ngu : ∀ i,\n ‖iteratedFDerivWithin 𝕜 i gu tu (fu x)‖ = ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ := fun i ↦ by\n rw [LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_left _ _ htu (hstu hx)]\n rw [LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_right _ _ ht, Ffu]\n rw [Ffu]\n exact hst hx"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x) = f x\nhfu : ContDiffOn 𝕜 (↑n) fu s\nhgu : ContDiffOn 𝕜 (↑n) gu tu\nNfu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i fu s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoF) ∘ f) s x‖ ≤ D ^ i\nNgu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i gu tu (fu x)‖ = ‖iteratedFDerivWithin 𝕜 i g t (f x)‖\nhC :\n ∀ (i : ℕ),\n i ≤ n →\n ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF) (↑isoF ⁻¹' t)\n ((↑(LinearIsometryEquiv.symm isoF) ∘ f) x)‖ ≤\n C\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x) = f x\nhfu : ContDiffOn 𝕜 (↑n) fu s\nhgu : ContDiffOn 𝕜 (↑n) gu tu\nNfu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i fu s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoF) ∘ f) s x‖ ≤ D ^ i\nNgu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i gu tu (fu x)‖ = ‖iteratedFDerivWithin 𝕜 i g t (f x)‖\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"tactic": "simp_rw [← Ngu] at hC"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x) = f x\nhfu : ContDiffOn 𝕜 (↑n) fu s\nhgu : ContDiffOn 𝕜 (↑n) gu tu\nNfu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i fu s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoF) ∘ f) s x‖ ≤ D ^ i\nNgu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i gu tu (fu x)‖ = ‖iteratedFDerivWithin 𝕜 i g t (f x)‖\nhC :\n ∀ (i : ℕ),\n i ≤ n →\n ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF) (↑isoF ⁻¹' t)\n ((↑(LinearIsometryEquiv.symm isoF) ∘ f) x)‖ ≤\n C\nNfgu : ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ = ‖iteratedFDerivWithin 𝕜 n (gu ∘ fu) s x‖\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x) = f x\nhfu : ContDiffOn 𝕜 (↑n) fu s\nhgu : ContDiffOn 𝕜 (↑n) gu tu\nNfu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i fu s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoF) ∘ f) s x‖ ≤ D ^ i\nNgu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i gu tu (fu x)‖ = ‖iteratedFDerivWithin 𝕜 i g t (f x)‖\nhC :\n ∀ (i : ℕ),\n i ≤ n →\n ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF) (↑isoF ⁻¹' t)\n ((↑(LinearIsometryEquiv.symm isoF) ∘ f) x)‖ ≤\n C\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"tactic": "have Nfgu :\n ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ = ‖iteratedFDerivWithin 𝕜 n (gu ∘ fu) s x‖ := by\n have : gu ∘ fu = isoG.symm ∘ g ∘ f := by\n ext x\n simp only [comp_apply, LinearIsometryEquiv.map_eq_iff, LinearIsometryEquiv.apply_symm_apply]\n rw [this, LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_left _ _ hs hx]"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x) = f x\nhfu : ContDiffOn 𝕜 (↑n) fu s\nhgu : ContDiffOn 𝕜 (↑n) gu tu\nNfu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i fu s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoF) ∘ f) s x‖ ≤ D ^ i\nNgu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i gu tu (fu x)‖ = ‖iteratedFDerivWithin 𝕜 i g t (f x)‖\nhC :\n ∀ (i : ℕ),\n i ≤ n →\n ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF) (↑isoF ⁻¹' t)\n ((↑(LinearIsometryEquiv.symm isoF) ∘ f) x)‖ ≤\n C\nNfgu : ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ = ‖iteratedFDerivWithin 𝕜 n (gu ∘ fu) s x‖\n⊢ ‖iteratedFDerivWithin 𝕜 n (gu ∘ fu) s x‖ ≤ ↑n ! * C * D ^ n",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x) = f x\nhfu : ContDiffOn 𝕜 (↑n) fu s\nhgu : ContDiffOn 𝕜 (↑n) gu tu\nNfu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i fu s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoF) ∘ f) s x‖ ≤ D ^ i\nNgu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i gu tu (fu x)‖ = ‖iteratedFDerivWithin 𝕜 i g t (f x)‖\nhC :\n ∀ (i : ℕ),\n i ≤ n →\n ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF) (↑isoF ⁻¹' t)\n ((↑(LinearIsometryEquiv.symm isoF) ∘ f) x)‖ ≤\n C\nNfgu : ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ = ‖iteratedFDerivWithin 𝕜 n (gu ∘ fu) s x‖\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n",
"tactic": "rw [Nfgu]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x) = f x\nhfu : ContDiffOn 𝕜 (↑n) fu s\nhgu : ContDiffOn 𝕜 (↑n) gu tu\nNfu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i fu s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoF) ∘ f) s x‖ ≤ D ^ i\nNgu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i gu tu (fu x)‖ = ‖iteratedFDerivWithin 𝕜 i g t (f x)‖\nhC :\n ∀ (i : ℕ),\n i ≤ n →\n ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF) (↑isoF ⁻¹' t)\n ((↑(LinearIsometryEquiv.symm isoF) ∘ f) x)‖ ≤\n C\nNfgu : ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ = ‖iteratedFDerivWithin 𝕜 n (gu ∘ fu) s x‖\n⊢ ‖iteratedFDerivWithin 𝕜 n (gu ∘ fu) s x‖ ≤ ↑n ! * C * D ^ n",
"tactic": "exact norm_iteratedFDerivWithin_comp_le_aux hgu hfu htu hs hstu hx hC hD"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nhC : ∀ (i : ℕ), i ≤ n → ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\ny : E\nhy : y ∈ s\n⊢ fu y ∈ tu",
"tactic": "simpa only [mem_preimage, comp_apply, LinearIsometryEquiv.apply_symm_apply] using hst hy"
},
{
"state_after": "no goals",
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"tactic": "rw [LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_left _ _ hs hx]"
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"tactic": "rw [LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_left _ _ htu (hstu hx)]"
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"tactic": "exact hst hx"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x) = f x\nhfu : ContDiffOn 𝕜 (↑n) fu s\nhgu : ContDiffOn 𝕜 (↑n) gu tu\nNfu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i fu s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoF) ∘ f) s x‖ ≤ D ^ i\nNgu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i gu tu (fu x)‖ = ‖iteratedFDerivWithin 𝕜 i g t (f x)‖\nhC :\n ∀ (i : ℕ),\n i ≤ n →\n ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF) (↑isoF ⁻¹' t)\n ((↑(LinearIsometryEquiv.symm isoF) ∘ f) x)‖ ≤\n C\nthis : gu ∘ fu = ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ f\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ = ‖iteratedFDerivWithin 𝕜 n (gu ∘ fu) s x‖",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x) = f x\nhfu : ContDiffOn 𝕜 (↑n) fu s\nhgu : ContDiffOn 𝕜 (↑n) gu tu\nNfu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i fu s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoF) ∘ f) s x‖ ≤ D ^ i\nNgu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i gu tu (fu x)‖ = ‖iteratedFDerivWithin 𝕜 i g t (f x)‖\nhC :\n ∀ (i : ℕ),\n i ≤ n →\n ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF) (↑isoF ⁻¹' t)\n ((↑(LinearIsometryEquiv.symm isoF) ∘ f) x)‖ ≤\n C\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ = ‖iteratedFDerivWithin 𝕜 n (gu ∘ fu) s x‖",
"tactic": "have : gu ∘ fu = isoG.symm ∘ g ∘ f := by\n ext x\n simp only [comp_apply, LinearIsometryEquiv.map_eq_iff, LinearIsometryEquiv.apply_symm_apply]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x) = f x\nhfu : ContDiffOn 𝕜 (↑n) fu s\nhgu : ContDiffOn 𝕜 (↑n) gu tu\nNfu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i fu s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoF) ∘ f) s x‖ ≤ D ^ i\nNgu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i gu tu (fu x)‖ = ‖iteratedFDerivWithin 𝕜 i g t (f x)‖\nhC :\n ∀ (i : ℕ),\n i ≤ n →\n ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF) (↑isoF ⁻¹' t)\n ((↑(LinearIsometryEquiv.symm isoF) ∘ f) x)‖ ≤\n C\nthis : gu ∘ fu = ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ f\n⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ = ‖iteratedFDerivWithin 𝕜 n (gu ∘ fu) s x‖",
"tactic": "rw [this, LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_left _ _ hs hx]"
},
{
"state_after": "case h.h\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝¹ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx✝ : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x✝ ∈ s\nC D : ℝ\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x✝) = f x✝\nhfu : ContDiffOn 𝕜 (↑n) fu s\nhgu : ContDiffOn 𝕜 (↑n) gu tu\nNfu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i fu s x✝‖ = ‖iteratedFDerivWithin 𝕜 i f s x✝‖\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoF) ∘ f) s x✝‖ ≤ D ^ i\nNgu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i gu tu (fu x✝)‖ = ‖iteratedFDerivWithin 𝕜 i g t (f x✝)‖\nhC :\n ∀ (i : ℕ),\n i ≤ n →\n ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF) (↑isoF ⁻¹' t)\n ((↑(LinearIsometryEquiv.symm isoF) ∘ f) x✝)‖ ≤\n C\nx : E\n⊢ ((gu ∘ fu) x).down = ((↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ f) x).down",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x ∈ s\nC D : ℝ\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x) = f x\nhfu : ContDiffOn 𝕜 (↑n) fu s\nhgu : ContDiffOn 𝕜 (↑n) gu tu\nNfu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i fu s x‖ = ‖iteratedFDerivWithin 𝕜 i f s x‖\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoF) ∘ f) s x‖ ≤ D ^ i\nNgu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i gu tu (fu x)‖ = ‖iteratedFDerivWithin 𝕜 i g t (f x)‖\nhC :\n ∀ (i : ℕ),\n i ≤ n →\n ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF) (↑isoF ⁻¹' t)\n ((↑(LinearIsometryEquiv.symm isoF) ∘ f) x)‖ ≤\n C\n⊢ gu ∘ fu = ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ f",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h.h\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD✝ : Type uD\ninst✝⁹ : NormedAddCommGroup D✝\ninst✝⁸ : NormedSpace 𝕜 D✝\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.4322453\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns✝ s₁ t✝ u : Set E\nf✝ f₁ : E → F\ng✝ : F → G\nx✝¹ x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\nt : Set F\nx✝ : E\nN : ℕ∞\nhg : ContDiffOn 𝕜 N g t\nhf : ContDiffOn 𝕜 N f s\nhn : ↑n ≤ N\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\nhx : x✝ ∈ s\nC D : ℝ\nFu : Type (max uF uG) := ULift F\nGu : Type (max uF uG) := ULift G\nisoF : Fu ≃ₗᵢ[𝕜] F\nisoG : Gu ≃ₗᵢ[𝕜] G\nfu : E → Fu := ↑(LinearIsometryEquiv.symm isoF) ∘ f\ngu : Fu → Gu := ↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF\ntu : Set Fu := ↑isoF ⁻¹' t\nhtu : UniqueDiffOn 𝕜 tu\nhstu : MapsTo fu s tu\nFfu : ↑isoF (fu x✝) = f x✝\nhfu : ContDiffOn 𝕜 (↑n) fu s\nhgu : ContDiffOn 𝕜 (↑n) gu tu\nNfu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i fu s x✝‖ = ‖iteratedFDerivWithin 𝕜 i f s x✝‖\nhD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoF) ∘ f) s x✝‖ ≤ D ^ i\nNgu : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 i gu tu (fu x✝)‖ = ‖iteratedFDerivWithin 𝕜 i g t (f x✝)‖\nhC :\n ∀ (i : ℕ),\n i ≤ n →\n ‖iteratedFDerivWithin 𝕜 i (↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ ↑isoF) (↑isoF ⁻¹' t)\n ((↑(LinearIsometryEquiv.symm isoF) ∘ f) x✝)‖ ≤\n C\nx : E\n⊢ ((gu ∘ fu) x).down = ((↑(LinearIsometryEquiv.symm isoG) ∘ g ∘ f) x).down",
"tactic": "simp only [comp_apply, LinearIsometryEquiv.map_eq_iff, LinearIsometryEquiv.apply_symm_apply]"
}
] | [
2696,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2652,
1
] |
Mathlib/Analysis/InnerProductSpace/Projection.lean | eq_orthogonalProjection_of_mem_orthogonal' | [
{
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"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.799854\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nu v z : E\nhv : v ∈ K\nhz : z ∈ Kᗮ\nhu : u = v + z\n⊢ u - v ∈ Kᗮ",
"tactic": "simpa [hu]"
}
] | [
816,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
814,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | AffineSubspace.coe_direction_eq_vsub_set_right | [
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p ∈ s\n⊢ ↑s -ᵥ ↑s = (fun x => x -ᵥ p) '' ↑s",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p ∈ s\n⊢ ↑(direction s) = (fun x => x -ᵥ p) '' ↑s",
"tactic": "rw [coe_direction_eq_vsub_set ⟨p, hp⟩]"
},
{
"state_after": "case refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p ∈ s\n⊢ ↑s -ᵥ ↑s ≤ (fun x => x -ᵥ p) '' ↑s\n\ncase refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p ∈ s\n⊢ (fun x => x -ᵥ p) '' ↑s ≤ ↑s -ᵥ ↑s",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p ∈ s\n⊢ ↑s -ᵥ ↑s = (fun x => x -ᵥ p) '' ↑s",
"tactic": "refine' le_antisymm _ _"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p ∈ s\np1 p2 : P\nhp1 : p1 ∈ ↑s\nhp2 : p2 ∈ ↑s\n⊢ (fun x x_1 => x -ᵥ x_1) p1 p2 ∈ (fun x => x -ᵥ p) '' ↑s",
"state_before": "case refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p ∈ s\n⊢ ↑s -ᵥ ↑s ≤ (fun x => x -ᵥ p) '' ↑s",
"tactic": "rintro v ⟨p1, p2, hp1, hp2, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p ∈ s\np1 p2 : P\nhp1 : p1 ∈ ↑s\nhp2 : p2 ∈ ↑s\n⊢ (fun x x_1 => x -ᵥ x_1) p1 p2 ∈ (fun x => x -ᵥ p) '' ↑s",
"tactic": "exact ⟨p1 -ᵥ p2 +ᵥ p, vadd_mem_of_mem_direction (vsub_mem_direction hp1 hp2) hp, vadd_vsub _ _⟩"
},
{
"state_after": "case refine'_2.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p ∈ s\np2 : P\nhp2 : p2 ∈ ↑s\n⊢ (fun x => x -ᵥ p) p2 ∈ ↑s -ᵥ ↑s",
"state_before": "case refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p ∈ s\n⊢ (fun x => x -ᵥ p) '' ↑s ≤ ↑s -ᵥ ↑s",
"tactic": "rintro v ⟨p2, hp2, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p ∈ s\np2 : P\nhp2 : p2 ∈ ↑s\n⊢ (fun x => x -ᵥ p) p2 ∈ ↑s -ᵥ ↑s",
"tactic": "exact ⟨p2, p, hp2, hp, rfl⟩"
}
] | [
302,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
295,
1
] |
Mathlib/Algebra/Associated.lean | irreducible_mul_isUnit | [] | [
279,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
277,
1
] |
Mathlib/GroupTheory/Submonoid/Operations.lean | Submonoid.prod_top | [
{
"state_after": "no goals",
"state_before": "M : Type u_1\nN : Type u_2\nP : Type ?u.98294\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.98315\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ns : Submonoid M\nx : M × N\n⊢ x ∈ prod s ⊤ ↔ x ∈ comap (MonoidHom.fst M N) s",
"tactic": "simp [mem_prod, MonoidHom.coe_fst]"
}
] | [
873,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
872,
1
] |
Mathlib/Data/Set/Image.lean | Set.image_univ_of_surjective | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.32858\nβ : Type u_2\nγ : Type ?u.32864\nι✝ : Sort ?u.32867\nι' : Sort ?u.32870\nf✝ : α → β\ns t : Set α\nι : Type u_1\nf : ι → β\nH : Surjective f\n⊢ ∀ (x : β), x ∈ f '' univ",
"tactic": "simpa [image]"
}
] | [
338,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
337,
1
] |
Mathlib/Analysis/NormedSpace/PiLp.lean | PiLp.lipschitzWith_equiv | [] | [
518,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
517,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Uniform.lean | Pmf.toMeasure_ofMultiset_apply | [] | [
214,
96
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
212,
1
] |
Mathlib/Order/Filter/SmallSets.lean | Filter.HasAntitoneBasis.tendsto_smallSets | [] | [
89,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
87,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean | MeasureTheory.AEEqFun.quot_mk_eq_mk | [] | [
157,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
155,
1
] |
Mathlib/Logic/Basic.lean | eq_ite_iff | [] | [
1136,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1135,
1
] |
Mathlib/Data/Finset/Fold.lean | Finset.fold_singleton | [] | [
63,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
62,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean | fderivWithin_congr_set | [] | [
811,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
810,
1
] |
Mathlib/Topology/Algebra/ConstMulAction.lean | IsClosed.smul | [] | [
268,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
267,
1
] |
Mathlib/Data/Real/CauSeq.lean | CauSeq.cauchy₂ | [] | [
175,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
173,
1
] |
Mathlib/Computability/Language.lean | Language.one_def | [] | [
74,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
73,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean | dist_div_norm_sq_smul | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.2668535\nE : Type ?u.2668538\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : F\nhx : x ≠ 0\nhy : y ≠ 0\nR : ℝ\nhx' : ‖x‖ ≠ 0\nhy' : ‖y‖ ≠ 0\n⊢ dist ((R / ‖x‖) ^ 2 • x) ((R / ‖y‖) ^ 2 • y) = sqrt (‖(R / ‖x‖) ^ 2 • x - (R / ‖y‖) ^ 2 • y‖ ^ 2)",
"tactic": "rw [dist_eq_norm, sqrt_sq (norm_nonneg _)]"
},
{
"state_after": "𝕜 : Type ?u.2668535\nE : Type ?u.2668538\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : F\nhx : x ≠ 0\nhy : y ≠ 0\nR : ℝ\nhx' : ‖x‖ ≠ 0\nhy' : ‖y‖ ≠ 0\n⊢ ((R * R * ‖x‖ * (R * R * ‖x‖) * (‖y‖ * ‖y‖ * (‖x‖ * ‖x‖)) -\n ‖x‖ * ‖x‖ * (‖x‖ * ‖x‖) * (2 * (R * R * (R * R * inner x y)))) *\n (‖y‖ * ‖y‖ * (‖y‖ * ‖y‖)) +\n R * R * ‖y‖ * (R * R * ‖y‖) * (‖x‖ * ‖x‖ * (‖x‖ * ‖x‖) * (‖y‖ * ‖y‖ * (‖x‖ * ‖x‖)))) *\n (‖x‖ * ‖y‖ * (‖x‖ * ‖y‖)) =\n R * R * (R * R) * (‖x‖ * ‖x‖ - 2 * inner x y + ‖y‖ * ‖y‖) *\n (‖x‖ * ‖x‖ * (‖x‖ * ‖x‖) * (‖y‖ * ‖y‖ * (‖x‖ * ‖x‖)) * (‖y‖ * ‖y‖ * (‖y‖ * ‖y‖)))",
"state_before": "𝕜 : Type ?u.2668535\nE : Type ?u.2668538\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : F\nhx : x ≠ 0\nhy : y ≠ 0\nR : ℝ\nhx' : ‖x‖ ≠ 0\nhy' : ‖y‖ ≠ 0\n⊢ ‖(R / ‖x‖) ^ 2 • x - (R / ‖y‖) ^ 2 • y‖ ^ 2 = (R ^ 2 / (‖x‖ * ‖y‖)) ^ 2 * ‖x - y‖ ^ 2",
"tactic": "field_simp [sq, norm_sub_mul_self_real, norm_smul, real_inner_smul_left, inner_smul_right,\n Real.norm_of_nonneg (mul_self_nonneg _)]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.2668535\nE : Type ?u.2668538\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : F\nhx : x ≠ 0\nhy : y ≠ 0\nR : ℝ\nhx' : ‖x‖ ≠ 0\nhy' : ‖y‖ ≠ 0\n⊢ ((R * R * ‖x‖ * (R * R * ‖x‖) * (‖y‖ * ‖y‖ * (‖x‖ * ‖x‖)) -\n ‖x‖ * ‖x‖ * (‖x‖ * ‖x‖) * (2 * (R * R * (R * R * inner x y)))) *\n (‖y‖ * ‖y‖ * (‖y‖ * ‖y‖)) +\n R * R * ‖y‖ * (R * R * ‖y‖) * (‖x‖ * ‖x‖ * (‖x‖ * ‖x‖) * (‖y‖ * ‖y‖ * (‖x‖ * ‖x‖)))) *\n (‖x‖ * ‖y‖ * (‖x‖ * ‖y‖)) =\n R * R * (R * R) * (‖x‖ * ‖x‖ - 2 * inner x y + ‖y‖ * ‖y‖) *\n (‖x‖ * ‖x‖ * (‖x‖ * ‖x‖) * (‖y‖ * ‖y‖ * (‖x‖ * ‖x‖)) * (‖y‖ * ‖y‖ * (‖y‖ * ‖y‖)))",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.2668535\nE : Type ?u.2668538\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : F\nhx : x ≠ 0\nhy : y ≠ 0\nR : ℝ\nhx' : ‖x‖ ≠ 0\nhy' : ‖y‖ ≠ 0\n⊢ sqrt ((R ^ 2 / (‖x‖ * ‖y‖)) ^ 2 * ‖x - y‖ ^ 2) = R ^ 2 / (‖x‖ * ‖y‖) * dist x y",
"tactic": "rw [sqrt_mul (sq_nonneg _), sqrt_sq (norm_nonneg _),\n sqrt_sq (div_nonneg (sq_nonneg _) (mul_nonneg (norm_nonneg _) (norm_nonneg _))),\n dist_eq_norm]"
}
] | [
1193,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1177,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Products.lean | CategoryTheory.Limits.has_smallest_products_of_hasProducts | [] | [
299,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
298,
1
] |
Mathlib/Analysis/SpecialFunctions/Sqrt.lean | ContDiffAt.sqrt | [] | [
165,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
163,
1
] |
Mathlib/Analysis/Normed/Group/Pointwise.lean | mul_ball | [
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx y : E\n⊢ s * ball x δ = x • thickening δ s",
"tactic": "rw [← smul_ball_one, mul_smul_comm, mul_ball_one]"
}
] | [
233,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
232,
1
] |
Std/Data/List/Init/Lemmas.lean | List.map_map | [
{
"state_after": "no goals",
"state_before": "β : Type u_1\nγ : Type u_2\nα : Type u_3\ng : β → γ\nf : α → β\nl : List α\n⊢ map g (map f l) = map (g ∘ f) l",
"tactic": "induction l <;> simp_all"
}
] | [
96,
65
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
95,
9
] |
Mathlib/Data/MvPolynomial/Rename.lean | MvPolynomial.rename_X | [] | [
71,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
70,
1
] |
Mathlib/CategoryTheory/Limits/Final.lean | CategoryTheory.Functor.initial_of_adjunction | [
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nL : C ⥤ D\nR : D ⥤ C\nadj : L ⊣ R\nd : D\nu : CostructuredArrow L d := CostructuredArrow.mk (adj.counit.app d)\nf g : CostructuredArrow L d\n⊢ L.map (↑(Adjunction.homEquiv adj f.left d) f.hom) ≫ u.hom = f.hom",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nL : C ⥤ D\nR : D ⥤ C\nadj : L ⊣ R\nd : D\nu : CostructuredArrow L d := CostructuredArrow.mk (adj.counit.app d)\nf g : CostructuredArrow L d\n⊢ L.map (↑(Adjunction.homEquiv adj g.left d) g.hom) ≫ u.hom = g.hom",
"tactic": "simp"
}
] | [
152,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
141,
1
] |
Mathlib/MeasureTheory/Measure/GiryMonad.lean | MeasureTheory.Measure.join_map_dirac | [] | [
231,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
230,
1
] |
Mathlib/Combinatorics/Pigeonhole.lean | Finset.exists_le_card_fiber_of_nsmul_le_card_of_maps_to | [
{
"state_after": "α : Type u\nβ : Type v\nM : Type w\ninst✝¹ : DecidableEq β\ns : Finset α\nt : Finset β\nf : α → β\nw : α → M\nb : M\nn : ℕ\ninst✝ : LinearOrderedCommSemiring M\nhf : ∀ (a : α), a ∈ s → f a ∈ t\nht : Finset.Nonempty t\nhb : card t • b ≤ ∑ a in s, 1\n⊢ ∃ y, y ∈ t ∧ b ≤ ∑ a in filter (fun x => f x = y) s, 1",
"state_before": "α : Type u\nβ : Type v\nM : Type w\ninst✝¹ : DecidableEq β\ns : Finset α\nt : Finset β\nf : α → β\nw : α → M\nb : M\nn : ℕ\ninst✝ : LinearOrderedCommSemiring M\nhf : ∀ (a : α), a ∈ s → f a ∈ t\nht : Finset.Nonempty t\nhb : card t • b ≤ ↑(card s)\n⊢ ∃ y, y ∈ t ∧ b ≤ ↑(card (filter (fun x => f x = y) s))",
"tactic": "simp_rw [cast_card] at hb⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nM : Type w\ninst✝¹ : DecidableEq β\ns : Finset α\nt : Finset β\nf : α → β\nw : α → M\nb : M\nn : ℕ\ninst✝ : LinearOrderedCommSemiring M\nhf : ∀ (a : α), a ∈ s → f a ∈ t\nht : Finset.Nonempty t\nhb : card t • b ≤ ∑ a in s, 1\n⊢ ∃ y, y ∈ t ∧ b ≤ ∑ a in filter (fun x => f x = y) s, 1",
"tactic": "exact exists_le_sum_fiber_of_maps_to_of_nsmul_le_sum hf ht hb"
}
] | [
280,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
277,
1
] |
Mathlib/Data/Setoid/Basic.lean | Quotient.eq_rel | [] | [
54,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
52,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean | Cardinal.toPartENat_cast | [
{
"state_after": "no goals",
"state_before": "α β : Type u\nn : ℕ\n⊢ ↑toPartENat ↑n = ↑n",
"tactic": "rw [toPartENat_apply_of_lt_aleph0 (nat_lt_aleph0 n), toNat_cast]"
}
] | [
1866,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1865,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.lean | Real.hasDerivWithinAt_arcsin_Ici | [
{
"state_after": "case inl\nh : 1 ≠ -1\n⊢ HasDerivWithinAt arcsin (1 / sqrt (1 - 1 ^ 2)) (Ici 1) 1\n\ncase inr\nx : ℝ\nh : x ≠ -1\nh' : x ≠ 1\n⊢ HasDerivWithinAt arcsin (1 / sqrt (1 - x ^ 2)) (Ici x) x",
"state_before": "x : ℝ\nh : x ≠ -1\n⊢ HasDerivWithinAt arcsin (1 / sqrt (1 - x ^ 2)) (Ici x) x",
"tactic": "rcases eq_or_ne x 1 with (rfl | h')"
},
{
"state_after": "no goals",
"state_before": "case inl\nh : 1 ≠ -1\n⊢ HasDerivWithinAt arcsin (1 / sqrt (1 - 1 ^ 2)) (Ici 1) 1",
"tactic": "convert (hasDerivWithinAt_const (1 : ℝ) _ (π / 2)).congr _ _ <;>\n simp (config := { contextual := true }) [arcsin_of_one_le]"
},
{
"state_after": "no goals",
"state_before": "case inr\nx : ℝ\nh : x ≠ -1\nh' : x ≠ 1\n⊢ HasDerivWithinAt arcsin (1 / sqrt (1 - x ^ 2)) (Ici x) x",
"tactic": "exact (hasDerivAt_arcsin h h').hasDerivWithinAt"
}
] | [
74,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
69,
1
] |
Mathlib/LinearAlgebra/Lagrange.lean | Lagrange.eval_basis_of_ne | [
{
"state_after": "F : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\ni j : ι\nhij : i ≠ j\nhj : j ∈ s\n⊢ ∃ a, a ∈ Finset.erase s i ∧ eval (v j) (basisDivisor (v i) (v a)) = 0",
"state_before": "F : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\ni j : ι\nhij : i ≠ j\nhj : j ∈ s\n⊢ eval (v j) (Lagrange.basis s v i) = 0",
"tactic": "simp_rw [Lagrange.basis, eval_prod, prod_eq_zero_iff]"
},
{
"state_after": "no goals",
"state_before": "F : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns : Finset ι\nv : ι → F\ni j : ι\nhij : i ≠ j\nhj : j ∈ s\n⊢ ∃ a, a ∈ Finset.erase s i ∧ eval (v j) (basisDivisor (v i) (v a)) = 0",
"tactic": "exact ⟨j, ⟨mem_erase.mpr ⟨hij.symm, hj⟩, eval_basisDivisor_right⟩⟩"
}
] | [
236,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
234,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.inter_singleton_of_mem | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.188015\nγ : Type ?u.188018\ninst✝ : DecidableEq α\ns✝ s₁ s₂ t t₁ t₂ u v : Finset α\na✝ b a : α\ns : Finset α\nh : a ∈ s\n⊢ s ∩ {a} = {a}",
"tactic": "rw [inter_comm, singleton_inter_of_mem h]"
}
] | [
1687,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1686,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean | HasDerivWithinAt.congr | [] | [
580,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
578,
1
] |
Mathlib/Dynamics/PeriodicPts.lean | Function.Semiconj.mapsTo_ptsOfPeriod | [] | [
206,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
204,
1
] |
Std/Data/String/Lemmas.lean | String.Iterator.ValidFor.remainingToString | [
{
"state_after": "case refl\nl r : List Char\nh : ValidFor l r { s := { data := List.reverseAux l r }, i := { byteIdx := utf8Len l } }\n⊢ Iterator.remainingToString { s := { data := List.reverseAux l r }, i := { byteIdx := utf8Len l } } = { data := r }",
"state_before": "l r : List Char\nit : Iterator\nh : ValidFor l r it\n⊢ Iterator.remainingToString it = { data := r }",
"tactic": "cases h.out"
},
{
"state_after": "no goals",
"state_before": "case refl\nl r : List Char\nh : ValidFor l r { s := { data := List.reverseAux l r }, i := { byteIdx := utf8Len l } }\n⊢ Iterator.remainingToString { s := { data := List.reverseAux l r }, i := { byteIdx := utf8Len l } } = { data := r }",
"tactic": "simpa [Iterator.remainingToString, List.reverseAux_eq] using extract_of_valid l.reverse r []"
}
] | [
603,
95
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
601,
1
] |
Mathlib/CategoryTheory/Limits/ColimitLimit.lean | CategoryTheory.Limits.ι_colimitLimitToLimitColimit_π | [
{
"state_after": "J K : Type v\ninst✝⁴ : SmallCategory J\ninst✝³ : SmallCategory K\nC : Type u\ninst✝² : Category C\nF : J × K ⥤ C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : HasColimitsOfShape K C\nj : J\nk : K\n⊢ colimit.ι (curry.obj (Prod.swap K J ⋙ F) ⋙ lim) k ≫\n limit.lift (curry.obj F ⋙ colim)\n { pt := colimit (curry.obj (Prod.swap K J ⋙ F) ⋙ lim),\n π :=\n NatTrans.mk fun j =>\n colimit.desc (curry.obj (Prod.swap K J ⋙ F) ⋙ lim)\n { pt := colimit ((curry.obj F).obj j),\n ι :=\n NatTrans.mk fun k =>\n limit.π ((curry.obj (Prod.swap K J ⋙ F)).obj k) j ≫ colimit.ι ((curry.obj F).obj j) k } } ≫\n limit.π (curry.obj F ⋙ colim) j =\n limit.π ((curry.obj (Prod.swap K J ⋙ F)).obj k) j ≫ colimit.ι ((curry.obj F).obj j) k",
"state_before": "J K : Type v\ninst✝⁴ : SmallCategory J\ninst✝³ : SmallCategory K\nC : Type u\ninst✝² : Category C\nF : J × K ⥤ C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : HasColimitsOfShape K C\nj : J\nk : K\n⊢ colimit.ι (curry.obj (Prod.swap K J ⋙ F) ⋙ lim) k ≫ colimitLimitToLimitColimit F ≫ limit.π (curry.obj F ⋙ colim) j =\n limit.π ((curry.obj (Prod.swap K J ⋙ F)).obj k) j ≫ colimit.ι ((curry.obj F).obj j) k",
"tactic": "dsimp [colimitLimitToLimitColimit]"
},
{
"state_after": "no goals",
"state_before": "J K : Type v\ninst✝⁴ : SmallCategory J\ninst✝³ : SmallCategory K\nC : Type u\ninst✝² : Category C\nF : J × K ⥤ C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : HasColimitsOfShape K C\nj : J\nk : K\n⊢ colimit.ι (curry.obj (Prod.swap K J ⋙ F) ⋙ lim) k ≫\n limit.lift (curry.obj F ⋙ colim)\n { pt := colimit (curry.obj (Prod.swap K J ⋙ F) ⋙ lim),\n π :=\n NatTrans.mk fun j =>\n colimit.desc (curry.obj (Prod.swap K J ⋙ F) ⋙ lim)\n { pt := colimit ((curry.obj F).obj j),\n ι :=\n NatTrans.mk fun k =>\n limit.π ((curry.obj (Prod.swap K J ⋙ F)).obj k) j ≫ colimit.ι ((curry.obj F).obj j) k } } ≫\n limit.π (curry.obj F ⋙ colim) j =\n limit.π ((curry.obj (Prod.swap K J ⋙ F)).obj k) j ≫ colimit.ι ((curry.obj F).obj j) k",
"tactic": "simp"
}
] | [
99,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
95,
1
] |
Mathlib/CategoryTheory/Limits/Preserves/Limits.lean | CategoryTheory.preserves_lift_mapCone | [
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝³ : Category C\nD : Type u₂\ninst✝² : Category D\nG : C ⥤ D\nJ : Type w\ninst✝¹ : Category J\nF : J ⥤ C\ninst✝ : PreservesLimit F G\nc₁ c₂ : Cone F\nt : IsLimit c₁\n⊢ ∀ (j : J), G.map (IsLimit.lift t c₂) ≫ (G.mapCone c₁).π.app j = (G.mapCone c₂).π.app j",
"tactic": "simp [← G.map_comp]"
}
] | [
51,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
49,
1
] |
Mathlib/RingTheory/Localization/Basic.lean | Localization.add_mk_self | [
{
"state_after": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type ?u.2768138\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type ?u.2768305\ninst✝ : CommSemiring P\na : R\nb : { x // x ∈ M }\nc : R\n⊢ ↑(r' M) (↑b * c + ↑b * a, b * b) (a + c, b)",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type ?u.2768138\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type ?u.2768305\ninst✝ : CommSemiring P\na : R\nb : { x // x ∈ M }\nc : R\n⊢ mk a b + mk c b = mk (a + c) b",
"tactic": "rw [add_mk, mk_eq_mk_iff, r_eq_r']"
},
{
"state_after": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type ?u.2768138\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type ?u.2768305\ninst✝ : CommSemiring P\na : R\nb : { x // x ∈ M }\nc : R\n⊢ ↑1 * (↑(↑b * c + ↑b * a, b * b).snd * (a + c, b).fst) = ↑1 * (↑(a + c, b).snd * (↑b * c + ↑b * a, b * b).fst)",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type ?u.2768138\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type ?u.2768305\ninst✝ : CommSemiring P\na : R\nb : { x // x ∈ M }\nc : R\n⊢ ↑(r' M) (↑b * c + ↑b * a, b * b) (a + c, b)",
"tactic": "refine' (r' M).symm ⟨1, _⟩"
},
{
"state_after": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type ?u.2768138\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type ?u.2768305\ninst✝ : CommSemiring P\na : R\nb : { x // x ∈ M }\nc : R\n⊢ 1 * (↑b * ↑b * (a + c)) = 1 * (↑b * (↑b * c + ↑b * a))",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type ?u.2768138\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type ?u.2768305\ninst✝ : CommSemiring P\na : R\nb : { x // x ∈ M }\nc : R\n⊢ ↑1 * (↑(↑b * c + ↑b * a, b * b).snd * (a + c, b).fst) = ↑1 * (↑(a + c, b).snd * (↑b * c + ↑b * a, b * b).fst)",
"tactic": "simp only [Submonoid.coe_one, Submonoid.coe_mul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type ?u.2768138\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type ?u.2768305\ninst✝ : CommSemiring P\na : R\nb : { x // x ∈ M }\nc : R\n⊢ 1 * (↑b * ↑b * (a + c)) = 1 * (↑b * (↑b * c + ↑b * a))",
"tactic": "ring"
}
] | [
893,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
889,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean | zpow_bit0 | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : DivisionMonoid α\na : α\nn : ℕ\n⊢ a ^ bit0 ↑n = a ^ ↑n * a ^ ↑n",
"tactic": "simp only [zpow_ofNat, ← Int.ofNat_bit0, pow_bit0]"
},
{
"state_after": "α : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : DivisionMonoid α\na : α\nn : ℕ\n⊢ a ^ bit0 -[n+1] = (a ^ bit0 (n + 1))⁻¹",
"state_before": "α : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : DivisionMonoid α\na : α\nn : ℕ\n⊢ a ^ bit0 -[n+1] = a ^ -[n+1] * a ^ -[n+1]",
"tactic": "simp [← mul_inv_rev, ← pow_bit0]"
},
{
"state_after": "α : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : DivisionMonoid α\na : α\nn : ℕ\n⊢ (a ^ bit0 (↑n + 1))⁻¹ = (a ^ bit0 (n + 1))⁻¹",
"state_before": "α : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : DivisionMonoid α\na : α\nn : ℕ\n⊢ a ^ bit0 -[n+1] = (a ^ bit0 (n + 1))⁻¹",
"tactic": "rw [negSucc_eq, bit0_neg, zpow_neg]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : DivisionMonoid α\na : α\nn : ℕ\n⊢ (a ^ bit0 (↑n + 1))⁻¹ = (a ^ bit0 (n + 1))⁻¹",
"tactic": "norm_cast"
}
] | [
173,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
168,
1
] |
Mathlib/Data/Seq/Seq.lean | Stream'.Seq.tail_cons | [
{
"state_after": "case mk\nα : Type u\nβ : Type v\nγ : Type w\na : α\nf : Stream' (Option α)\nal : IsSeq f\n⊢ tail (cons a { val := f, property := al }) = { val := f, property := al }",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\ns : Seq α\n⊢ tail (cons a s) = s",
"tactic": "cases' s with f al"
},
{
"state_after": "case mk.a\nα : Type u\nβ : Type v\nγ : Type w\na : α\nf : Stream' (Option α)\nal : IsSeq f\n⊢ ↑(tail (cons a { val := f, property := al })) = ↑{ val := f, property := al }",
"state_before": "case mk\nα : Type u\nβ : Type v\nγ : Type w\na : α\nf : Stream' (Option α)\nal : IsSeq f\n⊢ tail (cons a { val := f, property := al }) = { val := f, property := al }",
"tactic": "apply Subtype.eq"
},
{
"state_after": "no goals",
"state_before": "case mk.a\nα : Type u\nβ : Type v\nγ : Type w\na : α\nf : Stream' (Option α)\nal : IsSeq f\n⊢ ↑(tail (cons a { val := f, property := al })) = ↑{ val := f, property := al }",
"tactic": "dsimp [tail, cons]"
}
] | [
274,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
271,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean | abs_dist_sub_le_dist_mul_mul | [
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.649048\n𝕜 : Type ?u.649051\nα : Type ?u.649054\nι : Type ?u.649057\nκ : Type ?u.649060\nE : Type u_1\nF : Type ?u.649066\nG : Type ?u.649069\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na a₁✝ a₂✝ b b₁✝ b₂✝ : E\nr r₁ r₂ : ℝ\na₁ a₂ b₁ b₂ : E\n⊢ abs (dist a₁ b₁ - dist a₂ b₂) ≤ dist (a₁ * a₂) (b₁ * b₂)",
"tactic": "simpa only [dist_mul_left, dist_mul_right, dist_comm b₂] using\n abs_dist_sub_le (a₁ * a₂) (b₁ * b₂) (b₁ * a₂)"
}
] | [
1457,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1454,
1
] |
Mathlib/Algebra/Ring/Prod.lean | RingHom.snd_comp_prod | [] | [
237,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
236,
1
] |
Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean | Differentiable.exp | [] | [
309,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
308,
1
] |
Mathlib/Data/Set/Basic.lean | Set.compl_univ | [] | [
1674,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1673,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean | Polynomial.natDegree_eq_zero_iff_degree_le_zero | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.701471\n⊢ natDegree p = 0 ↔ degree p ≤ 0",
"tactic": "rw [← nonpos_iff_eq_zero, natDegree_le_iff_degree_le,\n Nat.cast_withBot, WithBot.coe_zero]"
}
] | [
1075,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1072,
1
] |
Mathlib/Data/List/Basic.lean | List.nthLe_insertNth_add_succ | [
{
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"state_before": "ι : Type ?u.128455\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn k : ℕ\nhk' : n + k < length l\n⊢ n + k + 1 < length (insertNth n x l)",
"tactic": "rw [length_insertNth _ _ (le_self_add.trans hk'.le)]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.128455\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn k : ℕ\nhk' : n + k < length l\n⊢ n + k + 1 < length l + 1",
"tactic": "exact Nat.succ_lt_succ_iff.2 hk'"
}
] | [
1753,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1747,
1
] |
Mathlib/Algebra/CubicDiscriminant.lean | Cubic.c_eq_three_roots | [
{
"state_after": "no goals",
"state_before": "R : Type ?u.1067442\nS : Type ?u.1067445\nF : Type u_1\nK : Type u_2\nP : Cubic F\ninst✝¹ : Field F\ninst✝ : Field K\nφ : F →+* K\nx y z : K\nha : P.a ≠ 0\nh3 : roots (map φ P) = {x, y, z}\n⊢ ↑φ P.c = ↑φ P.a * (x * y + x * z + y * z)",
"tactic": "injection eq_sum_three_roots ha h3"
}
] | [
549,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
547,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean | toIocDiv_sub' | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIocDiv hp (a - p) b = toIocDiv hp a b + 1",
"tactic": "simpa only [one_zsmul] using toIocDiv_sub_zsmul' hp a b 1"
}
] | [
355,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
354,
1
] |
Mathlib/Data/Pi/Interval.lean | Pi.card_Icc | [] | [
46,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
45,
1
] |
Mathlib/Data/Set/Lattice.lean | Set.prod_eq_seq | [
{
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"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.274480\nι : Sort ?u.274483\nι' : Sort ?u.274486\nι₂ : Sort ?u.274489\nκ : ι → Sort ?u.274494\nκ₁ : ι → Sort ?u.274499\nκ₂ : ι → Sort ?u.274504\nκ' : ι' → Sort ?u.274509\ns : Set α\nt : Set β\n⊢ s ×ˢ t = seq (Prod.mk '' s) t",
"tactic": "ext ⟨a, b⟩"
},
{
"state_after": "case h.mk.mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.274480\nι : Sort ?u.274483\nι' : Sort ?u.274486\nι₂ : Sort ?u.274489\nκ : ι → Sort ?u.274494\nκ₁ : ι → Sort ?u.274499\nκ₂ : ι → Sort ?u.274504\nκ' : ι' → Sort ?u.274509\ns : Set α\nt : Set β\na : α\nb : β\n⊢ (a, b) ∈ s ×ˢ t → (a, b) ∈ seq (Prod.mk '' s) t\n\ncase h.mk.mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.274480\nι : Sort ?u.274483\nι' : Sort ?u.274486\nι₂ : Sort ?u.274489\nκ : ι → Sort ?u.274494\nκ₁ : ι → Sort ?u.274499\nκ₂ : ι → Sort ?u.274504\nκ' : ι' → Sort ?u.274509\ns : Set α\nt : Set β\na : α\nb : β\n⊢ (a, b) ∈ seq (Prod.mk '' s) t → (a, b) ∈ s ×ˢ t",
"state_before": "case h.mk\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.274480\nι : Sort ?u.274483\nι' : Sort ?u.274486\nι₂ : Sort ?u.274489\nκ : ι → Sort ?u.274494\nκ₁ : ι → Sort ?u.274499\nκ₂ : ι → Sort ?u.274504\nκ' : ι' → Sort ?u.274509\ns : Set α\nt : Set β\na : α\nb : β\n⊢ (a, b) ∈ s ×ˢ t ↔ (a, b) ∈ seq (Prod.mk '' s) t",
"tactic": "constructor"
},
{
"state_after": "case h.mk.mp.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.274480\nι : Sort ?u.274483\nι' : Sort ?u.274486\nι₂ : Sort ?u.274489\nκ : ι → Sort ?u.274494\nκ₁ : ι → Sort ?u.274499\nκ₂ : ι → Sort ?u.274504\nκ' : ι' → Sort ?u.274509\ns : Set α\nt : Set β\na : α\nb : β\nha : (a, b).fst ∈ s\nhb : (a, b).snd ∈ t\n⊢ (a, b) ∈ seq (Prod.mk '' s) t",
"state_before": "case h.mk.mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.274480\nι : Sort ?u.274483\nι' : Sort ?u.274486\nι₂ : Sort ?u.274489\nκ : ι → Sort ?u.274494\nκ₁ : ι → Sort ?u.274499\nκ₂ : ι → Sort ?u.274504\nκ' : ι' → Sort ?u.274509\ns : Set α\nt : Set β\na : α\nb : β\n⊢ (a, b) ∈ s ×ˢ t → (a, b) ∈ seq (Prod.mk '' s) t",
"tactic": "rintro ⟨ha, hb⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mk.mp.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.274480\nι : Sort ?u.274483\nι' : Sort ?u.274486\nι₂ : Sort ?u.274489\nκ : ι → Sort ?u.274494\nκ₁ : ι → Sort ?u.274499\nκ₂ : ι → Sort ?u.274504\nκ' : ι' → Sort ?u.274509\ns : Set α\nt : Set β\na : α\nb : β\nha : (a, b).fst ∈ s\nhb : (a, b).snd ∈ t\n⊢ (a, b) ∈ seq (Prod.mk '' s) t",
"tactic": "exact ⟨Prod.mk a, ⟨a, ha, rfl⟩, b, hb, rfl⟩"
},
{
"state_after": "case h.mk.mpr.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.274480\nι : Sort ?u.274483\nι' : Sort ?u.274486\nι₂ : Sort ?u.274489\nκ : ι → Sort ?u.274494\nκ₁ : ι → Sort ?u.274499\nκ₂ : ι → Sort ?u.274504\nκ' : ι' → Sort ?u.274509\ns : Set α\nt : Set β\na : α\nb : β\nx : α\nhx : x ∈ s\ny : β\nhy : y ∈ t\neq : (x, y) = (a, b)\n⊢ (a, b) ∈ s ×ˢ t",
"state_before": "case h.mk.mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.274480\nι : Sort ?u.274483\nι' : Sort ?u.274486\nι₂ : Sort ?u.274489\nκ : ι → Sort ?u.274494\nκ₁ : ι → Sort ?u.274499\nκ₂ : ι → Sort ?u.274504\nκ' : ι' → Sort ?u.274509\ns : Set α\nt : Set β\na : α\nb : β\n⊢ (a, b) ∈ seq (Prod.mk '' s) t → (a, b) ∈ s ×ˢ t",
"tactic": "rintro ⟨f, ⟨x, hx, rfl⟩, y, hy, eq⟩"
},
{
"state_after": "case h.mk.mpr.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.274480\nι : Sort ?u.274483\nι' : Sort ?u.274486\nι₂ : Sort ?u.274489\nκ : ι → Sort ?u.274494\nκ₁ : ι → Sort ?u.274499\nκ₂ : ι → Sort ?u.274504\nκ' : ι' → Sort ?u.274509\ns : Set α\nt : Set β\na : α\nb : β\nx : α\nhx : x ∈ s\ny : β\nhy : y ∈ t\neq : (x, y) = (a, b)\n⊢ (x, y) ∈ s ×ˢ t",
"state_before": "case h.mk.mpr.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.274480\nι : Sort ?u.274483\nι' : Sort ?u.274486\nι₂ : Sort ?u.274489\nκ : ι → Sort ?u.274494\nκ₁ : ι → Sort ?u.274499\nκ₂ : ι → Sort ?u.274504\nκ' : ι' → Sort ?u.274509\ns : Set α\nt : Set β\na : α\nb : β\nx : α\nhx : x ∈ s\ny : β\nhy : y ∈ t\neq : (x, y) = (a, b)\n⊢ (a, b) ∈ s ×ˢ t",
"tactic": "rw [← eq]"
},
{
"state_after": "no goals",
"state_before": "case h.mk.mpr.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.274480\nι : Sort ?u.274483\nι' : Sort ?u.274486\nι₂ : Sort ?u.274489\nκ : ι → Sort ?u.274494\nκ₁ : ι → Sort ?u.274499\nκ₂ : ι → Sort ?u.274504\nκ' : ι' → Sort ?u.274509\ns : Set α\nt : Set β\na : α\nb : β\nx : α\nhx : x ∈ s\ny : β\nhy : y ∈ t\neq : (x, y) = (a, b)\n⊢ (x, y) ∈ s ×ˢ t",
"tactic": "exact ⟨hx, hy⟩"
}
] | [
2000,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1993,
1
] |
Mathlib/LinearAlgebra/Basic.lean | LinearMap.coe_smulRight | [] | [
303,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
302,
1
] |
Mathlib/Algebra/Order/Floor.lean | Int.le_floor_add_floor | [
{
"state_after": "F : Type ?u.132012\nα : Type u_1\nβ : Type ?u.132018\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\n⊢ ↑⌊a + b⌋ - 1 - a ≤ ↑⌊b⌋",
"state_before": "F : Type ?u.132012\nα : Type u_1\nβ : Type ?u.132018\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\n⊢ ⌊a + b⌋ - 1 ≤ ⌊a⌋ + ⌊b⌋",
"tactic": "rw [← sub_le_iff_le_add, le_floor, Int.cast_sub, sub_le_comm, Int.cast_sub, Int.cast_one]"
},
{
"state_after": "F : Type ?u.132012\nα : Type u_1\nβ : Type ?u.132018\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\n⊢ ↑⌊a + b⌋ - 1 - a ≤ b - 1",
"state_before": "F : Type ?u.132012\nα : Type u_1\nβ : Type ?u.132018\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\n⊢ ↑⌊a + b⌋ - 1 - a ≤ ↑⌊b⌋",
"tactic": "refine' le_trans _ (sub_one_lt_floor _).le"
},
{
"state_after": "F : Type ?u.132012\nα : Type u_1\nβ : Type ?u.132018\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\n⊢ ↑⌊a + b⌋ ≤ a + b",
"state_before": "F : Type ?u.132012\nα : Type u_1\nβ : Type ?u.132018\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\n⊢ ↑⌊a + b⌋ - 1 - a ≤ b - 1",
"tactic": "rw [sub_le_iff_le_add', ← add_sub_assoc, sub_le_sub_iff_right]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.132012\nα : Type u_1\nβ : Type ?u.132018\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\n⊢ ↑⌊a + b⌋ ≤ a + b",
"tactic": "exact floor_le _"
}
] | [
743,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
739,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.disjoint_right | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.42944\nγ : Type ?u.42947\nf : α → β\ns t u : Finset α\na b : α\n⊢ _root_.Disjoint s t ↔ ∀ ⦃a : α⦄, a ∈ t → ¬a ∈ s",
"tactic": "rw [_root_.disjoint_comm, disjoint_left]"
}
] | [
919,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
918,
1
] |
Mathlib/Data/List/Count.lean | List.countp_eq_zero | [
{
"state_after": "α : Type u_1\nl : List α\np q : α → Bool\n⊢ (∃ a, a ∈ l ∧ p a = true) ↔ ¬∀ (a : α), a ∈ l → ¬p a = true",
"state_before": "α : Type u_1\nl : List α\np q : α → Bool\n⊢ countp p l = 0 ↔ ∀ (a : α), a ∈ l → ¬p a = true",
"tactic": "rw [← not_iff_not, ← Ne.def, ← pos_iff_ne_zero, countp_pos]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : List α\np q : α → Bool\n⊢ (∃ a, a ∈ l ∧ p a = true) ↔ ¬∀ (a : α), a ∈ l → ¬p a = true",
"tactic": "simp"
}
] | [
108,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
106,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean | Cardinal.sup_lt_ord_lift_of_isRegular | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.158147\nr : α → α → Prop\nι : Type u\nf : ι → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : lift (#ι) < c\n⊢ lift (#ι) < Ordinal.cof (ord c)",
"tactic": "rwa [hc.cof_eq]"
}
] | [
1073,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1071,
1
] |
Mathlib/LinearAlgebra/Orientation.lean | Orientation.eq_or_eq_neg_of_isEmpty | [
{
"state_after": "case h\nR : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nx : AlternatingMap R M R ι\nhx : x ≠ 0\n⊢ rayOfNeZero R x hx = positiveOrientation ∨ rayOfNeZero R x hx = -positiveOrientation",
"state_before": "R : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\no : Orientation R M ι\n⊢ o = positiveOrientation ∨ o = -positiveOrientation",
"tactic": "induction' o using Module.Ray.ind with x hx"
},
{
"state_after": "case h\nR : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nx : AlternatingMap R M R ι\nhx : x ≠ 0\n⊢ rayOfNeZero R x hx =\n rayOfNeZero R (AlternatingMap.constOfIsEmpty R M ι 1)\n (_ :\n ↑AlternatingMap.constLinearEquivOfIsEmpty 1 ≠ ↑AlternatingMap.constLinearEquivOfIsEmpty AddMonoid.toZero.1) ∨\n rayOfNeZero R x hx =\n rayOfNeZero R (-AlternatingMap.constOfIsEmpty R M ι 1) (_ : ¬-AlternatingMap.constOfIsEmpty R M ι 1 = 0)",
"state_before": "case h\nR : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nx : AlternatingMap R M R ι\nhx : x ≠ 0\n⊢ rayOfNeZero R x hx = positiveOrientation ∨ rayOfNeZero R x hx = -positiveOrientation",
"tactic": "dsimp [positiveOrientation]"
},
{
"state_after": "case h\nR : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nx : AlternatingMap R M R ι\nhx : x ≠ 0\n⊢ SameRay R x (AlternatingMap.constOfIsEmpty R M ι 1) ∨ SameRay R x (-AlternatingMap.constOfIsEmpty R M ι 1)",
"state_before": "case h\nR : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nx : AlternatingMap R M R ι\nhx : x ≠ 0\n⊢ rayOfNeZero R x hx =\n rayOfNeZero R (AlternatingMap.constOfIsEmpty R M ι 1)\n (_ :\n ↑AlternatingMap.constLinearEquivOfIsEmpty 1 ≠ ↑AlternatingMap.constLinearEquivOfIsEmpty AddMonoid.toZero.1) ∨\n rayOfNeZero R x hx =\n rayOfNeZero R (-AlternatingMap.constOfIsEmpty R M ι 1) (_ : ¬-AlternatingMap.constOfIsEmpty R M ι 1 = 0)",
"tactic": "simp only [ray_eq_iff, sameRay_neg_swap]"
},
{
"state_after": "case h\nR : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nx : AlternatingMap R M R ι\nhx : x ≠ 0\n⊢ ¬LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]",
"state_before": "case h\nR : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nx : AlternatingMap R M R ι\nhx : x ≠ 0\n⊢ SameRay R x (AlternatingMap.constOfIsEmpty R M ι 1) ∨ SameRay R x (-AlternatingMap.constOfIsEmpty R M ι 1)",
"tactic": "rw [sameRay_or_sameRay_neg_iff_not_linearIndependent]"
},
{
"state_after": "case h\nR : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nx : AlternatingMap R M R ι\nhx : x ≠ 0\nh : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]\n⊢ False",
"state_before": "case h\nR : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nx : AlternatingMap R M R ι\nhx : x ≠ 0\n⊢ ¬LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]",
"tactic": "intro h"
},
{
"state_after": "case h\nR : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nx : AlternatingMap R M R ι\nhx : x ≠ 0\nh : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]\nf : AlternatingMap R M R ι ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty\n⊢ False",
"state_before": "case h\nR : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nx : AlternatingMap R M R ι\nhx : x ≠ 0\nh : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]\n⊢ False",
"tactic": "set f : AlternatingMap R M R ι ≃ₗ[R] R := AlternatingMap.constLinearEquivOfIsEmpty.symm"
},
{
"state_after": "case h\nR : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nx : AlternatingMap R M R ι\nhx : x ≠ 0\nh : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]\nf : AlternatingMap R M R ι ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty\nH : LinearIndependent R ![↑f x, 1]\n⊢ False",
"state_before": "case h\nR : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nx : AlternatingMap R M R ι\nhx : x ≠ 0\nh : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]\nf : AlternatingMap R M R ι ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty\n⊢ False",
"tactic": "have H : LinearIndependent R ![f x, 1] := by\n convert h.map' f.toLinearMap f.ker\n ext i\n fin_cases i <;> simp"
},
{
"state_after": "case h\nR : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nx : AlternatingMap R M R ι\nhx : x ≠ 0\nh : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]\nf : AlternatingMap R M R ι ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty\nH :\n ∀ (s : Finset (Fin (Nat.succ (Nat.succ 0)))) (g : Fin (Nat.succ (Nat.succ 0)) → R),\n ∑ i in s, g i • Matrix.vecCons (↑f x) ![1] i = 0 → ∀ (i : Fin (Nat.succ (Nat.succ 0))), i ∈ s → g i = 0\n⊢ False",
"state_before": "case h\nR : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nx : AlternatingMap R M R ι\nhx : x ≠ 0\nh : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]\nf : AlternatingMap R M R ι ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty\nH : LinearIndependent R ![↑f x, 1]\n⊢ False",
"tactic": "rw [linearIndependent_iff'] at H"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nx : AlternatingMap R M R ι\nhx : x ≠ 0\nh : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]\nf : AlternatingMap R M R ι ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty\nH :\n ∀ (s : Finset (Fin (Nat.succ (Nat.succ 0)))) (g : Fin (Nat.succ (Nat.succ 0)) → R),\n ∑ i in s, g i • Matrix.vecCons (↑f x) ![1] i = 0 → ∀ (i : Fin (Nat.succ (Nat.succ 0))), i ∈ s → g i = 0\n⊢ False",
"tactic": "simpa using H Finset.univ ![1, -f x] (by simp [Fin.sum_univ_succ]) 0 (by simp)"
},
{
"state_after": "case h.e'_4\nR : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nx : AlternatingMap R M R ι\nhx : x ≠ 0\nh : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]\nf : AlternatingMap R M R ι ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty\n⊢ ![↑f x, 1] = ↑↑f ∘ ![x, AlternatingMap.constOfIsEmpty R M ι 1]",
"state_before": "R : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nx : AlternatingMap R M R ι\nhx : x ≠ 0\nh : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]\nf : AlternatingMap R M R ι ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty\n⊢ LinearIndependent R ![↑f x, 1]",
"tactic": "convert h.map' f.toLinearMap f.ker"
},
{
"state_after": "case h.e'_4.h\nR : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nx : AlternatingMap R M R ι\nhx : x ≠ 0\nh : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]\nf : AlternatingMap R M R ι ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty\ni : Fin (Nat.succ (Nat.succ 0))\n⊢ Matrix.vecCons (↑f x) ![1] i = (↑↑f ∘ ![x, AlternatingMap.constOfIsEmpty R M ι 1]) i",
"state_before": "case h.e'_4\nR : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nx : AlternatingMap R M R ι\nhx : x ≠ 0\nh : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]\nf : AlternatingMap R M R ι ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty\n⊢ ![↑f x, 1] = ↑↑f ∘ ![x, AlternatingMap.constOfIsEmpty R M ι 1]",
"tactic": "ext i"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4.h\nR : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nx : AlternatingMap R M R ι\nhx : x ≠ 0\nh : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]\nf : AlternatingMap R M R ι ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty\ni : Fin (Nat.succ (Nat.succ 0))\n⊢ Matrix.vecCons (↑f x) ![1] i = (↑↑f ∘ ![x, AlternatingMap.constOfIsEmpty R M ι 1]) i",
"tactic": "fin_cases i <;> simp"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁴ : LinearOrderedCommRing R\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_2\ninst✝¹ : Nontrivial R\ninst✝ : IsEmpty ι\nx : AlternatingMap R M R ι\nhx : x ≠ 0\nh : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]\nf : AlternatingMap R M R ι ≃ₗ[R] R := LinearEquiv.symm AlternatingMap.constLinearEquivOfIsEmpty\nH :\n ∀ (s : Finset (Fin (Nat.succ (Nat.succ 0)))) (g : Fin (Nat.succ (Nat.succ 0)) → R),\n ∑ i in s, g i • Matrix.vecCons (↑f x) ![1] i = 0 → ∀ (i : Fin (Nat.succ (Nat.succ 0))), i ∈ s → g i = 0\n⊢ ∑ i : Fin (Nat.succ (Nat.succ 0)), Matrix.vecCons 1 ![-↑f x] i • Matrix.vecCons (↑f x) ![1] i = 0",
"tactic": "simp [Fin.sum_univ_succ]"
},
{
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Mathlib/Data/Num/Lemmas.lean | ZNum.ofInt'_neg | [
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Mathlib/Order/UpperLower/Basic.lean | UpperSet.coe_sup | [] | [
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Mathlib/RingTheory/OreLocalization/Basic.lean | OreLocalization.expand | [
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Mathlib/Algebra/Star/SelfAdjoint.lean | IsSelfAdjoint.inv | [
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Mathlib/Algebra/GroupPower/Lemmas.lean | zpow_mul_comm | [] | [
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Mathlib/CategoryTheory/Preadditive/Biproducts.lean | CategoryTheory.Limits.HasBiproduct.of_hasProduct | [
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Mathlib/Topology/Algebra/Module/Basic.lean | ContinuousLinearMap.projKerOfRightInverse_comp_inv | [
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Mathlib/Topology/Algebra/Group/Basic.lean | isOpenMap_mul_left | [] | [
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Mathlib/Dynamics/PeriodicPts.lean | Function.IsPeriodicPt.gcd | [
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Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean | CategoryTheory.Limits.Multiequalizer.multifork_π_app_left | [] | [
780,
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Mathlib/Analysis/Normed/Group/HomCompletion.lean | NormedAddGroupHom.extension_unique | [
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Mathlib/Data/List/Destutter.lean | List.destutter_is_chain' | [] | [
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Mathlib/Analysis/Normed/Group/Basic.lean | uniformContinuous_nnnorm' | [] | [
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Mathlib/FieldTheory/IntermediateField.lean | IntermediateField.coe_nat_mem | [
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Mathlib/MeasureTheory/Function/LpSpace.lean | MeasureTheory.Memℒp.coeFn_toLp | [] | [
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Mathlib/Analysis/Complex/LocallyUniformLimit.lean | Complex.norm_cderiv_sub_lt | [] | [
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Mathlib/Data/Real/Hyperreal.lean | Hyperreal.IsSt.mul | [] | [
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Mathlib/MeasureTheory/Covering/Differentiation.lean | VitaliFamily.withDensity_limRatioMeas_eq | [
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{
"state_after": "case h\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.5186182\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nthis : Tendsto (fun t => ↑t ^ 2 * ↑↑ρ s) (𝓝[Ioi 1] 1) (𝓝 (↑↑ρ s))\na✝ : ℝ≥0\nht : a✝ ∈ Ioi 1\n⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑a✝ ^ 2 * ↑↑ρ s",
"state_before": "case h.refine'_1\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.5186182\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nthis : Tendsto (fun t => ↑t ^ 2 * ↑↑ρ s) (𝓝[Ioi 1] 1) (𝓝 (↑↑ρ s))\n⊢ ∀ᶠ (c : ℝ≥0) in 𝓝[Ioi 1] 1, ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑c ^ 2 * ↑↑ρ s",
"tactic": "filter_upwards [self_mem_nhdsWithin] with _ ht"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.5186182\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nthis : Tendsto (fun t => ↑t ^ 2 * ↑↑ρ s) (𝓝[Ioi 1] 1) (𝓝 (↑↑ρ s))\na✝ : ℝ≥0\nht : a✝ ∈ Ioi 1\n⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑a✝ ^ 2 * ↑↑ρ s",
"tactic": "exact v.withDensity_le_mul hρ hs ht"
},
{
"state_after": "case refine'_1\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.5186182\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\n⊢ Tendsto (fun t => ↑t ^ 2) (𝓝[Ioi 1] 1) (𝓝 (1 ^ 2))\n\ncase refine'_2\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.5186182\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\n⊢ 1 ^ 2 ≠ 0 ∨ ↑↑ρ s ≠ ⊤\n\ncase refine'_3\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.5186182\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑ρ s ≠ 0 ∨ 1 ^ 2 ≠ ⊤",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.5186182\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\n⊢ Tendsto (fun t => ↑t ^ 2 * ↑↑ρ s) (𝓝[Ioi 1] 1) (𝓝 (1 ^ 2 * ↑↑ρ s))",
"tactic": "refine' ENNReal.Tendsto.mul _ _ tendsto_const_nhds _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.5186182\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\n⊢ Tendsto (fun t => ↑t ^ 2) (𝓝[Ioi 1] 1) (𝓝 (1 ^ 2))",
"tactic": "exact ENNReal.Tendsto.pow (ENNReal.tendsto_coe.2 nhdsWithin_le_nhds)"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.5186182\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\n⊢ 1 ^ 2 ≠ 0 ∨ ↑↑ρ s ≠ ⊤",
"tactic": "simp only [one_pow, ENNReal.coe_one, true_or_iff, Ne.def, not_false_iff, one_ne_zero]"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.5186182\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑ρ s ≠ 0 ∨ 1 ^ 2 ≠ ⊤",
"tactic": "simp only [one_pow, ENNReal.coe_one, Ne.def, or_true_iff, ENNReal.one_ne_top, not_false_iff]"
},
{
"state_after": "case h.refine'_2\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.5186182\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nthis :\n Tendsto (fun t => ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s) (𝓝[Ioi 1] 1)\n (𝓝 (1 * ↑↑(withDensity μ (limRatioMeas v hρ)) s))\n⊢ ↑↑ρ s ≤ ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"state_before": "case h.refine'_2\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.5186182\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑ρ s ≤ ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"tactic": "have :\n Tendsto (fun t : ℝ≥0 => (t : ℝ≥0∞) * μ.withDensity (v.limRatioMeas hρ) s) (𝓝[>] 1)\n (𝓝 ((1 : ℝ≥0∞) * μ.withDensity (v.limRatioMeas hρ) s)) := by\n refine' ENNReal.Tendsto.mul_const (ENNReal.tendsto_coe.2 nhdsWithin_le_nhds) _\n simp only [ENNReal.coe_one, true_or_iff, Ne.def, not_false_iff, one_ne_zero]"
},
{
"state_after": "case h.refine'_2\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.5186182\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nthis :\n Tendsto (fun t => ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s) (𝓝[Ioi 1] 1)\n (𝓝 (↑↑(withDensity μ (limRatioMeas v hρ)) s))\n⊢ ↑↑ρ s ≤ ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"state_before": "case h.refine'_2\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.5186182\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nthis :\n Tendsto (fun t => ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s) (𝓝[Ioi 1] 1)\n (𝓝 (1 * ↑↑(withDensity μ (limRatioMeas v hρ)) s))\n⊢ ↑↑ρ s ≤ ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"tactic": "simp only [one_mul, ENNReal.coe_one] at this"
},
{
"state_after": "case h.refine'_2\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.5186182\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nthis :\n Tendsto (fun t => ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s) (𝓝[Ioi 1] 1)\n (𝓝 (↑↑(withDensity μ (limRatioMeas v hρ)) s))\n⊢ ∀ᶠ (c : ℝ≥0) in 𝓝[Ioi 1] 1, ↑↑ρ s ≤ ↑c * ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"state_before": "case h.refine'_2\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.5186182\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nthis :\n Tendsto (fun t => ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s) (𝓝[Ioi 1] 1)\n (𝓝 (↑↑(withDensity μ (limRatioMeas v hρ)) s))\n⊢ ↑↑ρ s ≤ ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"tactic": "refine' ge_of_tendsto this _"
},
{
"state_after": "case h\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.5186182\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nthis :\n Tendsto (fun t => ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s) (𝓝[Ioi 1] 1)\n (𝓝 (↑↑(withDensity μ (limRatioMeas v hρ)) s))\na✝ : ℝ≥0\nht : a✝ ∈ Ioi 1\n⊢ ↑↑ρ s ≤ ↑a✝ * ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"state_before": "case h.refine'_2\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.5186182\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nthis :\n Tendsto (fun t => ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s) (𝓝[Ioi 1] 1)\n (𝓝 (↑↑(withDensity μ (limRatioMeas v hρ)) s))\n⊢ ∀ᶠ (c : ℝ≥0) in 𝓝[Ioi 1] 1, ↑↑ρ s ≤ ↑c * ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"tactic": "filter_upwards [self_mem_nhdsWithin] with _ ht"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.5186182\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nthis :\n Tendsto (fun t => ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s) (𝓝[Ioi 1] 1)\n (𝓝 (↑↑(withDensity μ (limRatioMeas v hρ)) s))\na✝ : ℝ≥0\nht : a✝ ∈ Ioi 1\n⊢ ↑↑ρ s ≤ ↑a✝ * ↑↑(withDensity μ (limRatioMeas v hρ)) s",
"tactic": "exact v.le_mul_withDensity hρ hs ht"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.5186182\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\n⊢ 1 ≠ 0 ∨ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≠ ⊤",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.5186182\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\n⊢ Tendsto (fun t => ↑t * ↑↑(withDensity μ (limRatioMeas v hρ)) s) (𝓝[Ioi 1] 1)\n (𝓝 (1 * ↑↑(withDensity μ (limRatioMeas v hρ)) s))",
"tactic": "refine' ENNReal.Tendsto.mul_const (ENNReal.tendsto_coe.2 nhdsWithin_le_nhds) _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.5186182\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\n⊢ 1 ≠ 0 ∨ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≠ ⊤",
"tactic": "simp only [ENNReal.coe_one, true_or_iff, Ne.def, not_false_iff, one_ne_zero]"
}
] | [
689,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
668,
1
] |
Mathlib/MeasureTheory/Function/SimpleFuncDense.lean | MeasureTheory.SimpleFunc.nearestPtInd_succ | [
{
"state_after": "α : Type u_1\nβ : Type ?u.11058\nι : Type ?u.11061\nE : Type ?u.11064\nF : Type ?u.11067\n𝕜 : Type ?u.11070\ninst✝² : MeasurableSpace α\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : OpensMeasurableSpace α\ne : ℕ → α\nN : ℕ\nx : α\n⊢ (if x ∈ ⋂ (k : ℕ) (_ : k ≤ Nat.add N 0), {x | edist (e (Nat.add N 0 + 1)) x < edist (e k) x} then\n ↑(const α (Nat.add N 0 + 1)) x\n else ↑(nearestPtInd e (Nat.add N 0)) x) =\n if ∀ (k : ℕ), k ≤ N → edist (e (N + 1)) x < edist (e k) x then N + 1 else ↑(nearestPtInd e N) x",
"state_before": "α : Type u_1\nβ : Type ?u.11058\nι : Type ?u.11061\nE : Type ?u.11064\nF : Type ?u.11067\n𝕜 : Type ?u.11070\ninst✝² : MeasurableSpace α\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : OpensMeasurableSpace α\ne : ℕ → α\nN : ℕ\nx : α\n⊢ ↑(nearestPtInd e (N + 1)) x =\n if ∀ (k : ℕ), k ≤ N → edist (e (N + 1)) x < edist (e k) x then N + 1 else ↑(nearestPtInd e N) x",
"tactic": "simp only [nearestPtInd, coe_piecewise, Set.piecewise]"
},
{
"state_after": "case e_c\nα : Type u_1\nβ : Type ?u.11058\nι : Type ?u.11061\nE : Type ?u.11064\nF : Type ?u.11067\n𝕜 : Type ?u.11070\ninst✝² : MeasurableSpace α\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : OpensMeasurableSpace α\ne : ℕ → α\nN : ℕ\nx : α\n⊢ (x ∈ ⋂ (k : ℕ) (_ : k ≤ Nat.add N 0), {x | edist (e (Nat.add N 0 + 1)) x < edist (e k) x}) =\n ∀ (k : ℕ), k ≤ N → edist (e (N + 1)) x < edist (e k) x",
"state_before": "α : Type u_1\nβ : Type ?u.11058\nι : Type ?u.11061\nE : Type ?u.11064\nF : Type ?u.11067\n𝕜 : Type ?u.11070\ninst✝² : MeasurableSpace α\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : OpensMeasurableSpace α\ne : ℕ → α\nN : ℕ\nx : α\n⊢ (if x ∈ ⋂ (k : ℕ) (_ : k ≤ Nat.add N 0), {x | edist (e (Nat.add N 0 + 1)) x < edist (e k) x} then\n ↑(const α (Nat.add N 0 + 1)) x\n else ↑(nearestPtInd e (Nat.add N 0)) x) =\n if ∀ (k : ℕ), k ≤ N → edist (e (N + 1)) x < edist (e k) x then N + 1 else ↑(nearestPtInd e N) x",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_c\nα : Type u_1\nβ : Type ?u.11058\nι : Type ?u.11061\nE : Type ?u.11064\nF : Type ?u.11067\n𝕜 : Type ?u.11070\ninst✝² : MeasurableSpace α\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : OpensMeasurableSpace α\ne : ℕ → α\nN : ℕ\nx : α\n⊢ (x ∈ ⋂ (k : ℕ) (_ : k ≤ Nat.add N 0), {x | edist (e (Nat.add N 0 + 1)) x < edist (e k) x}) =\n ∀ (k : ℕ), k ≤ N → edist (e (N + 1)) x < edist (e k) x",
"tactic": "simp"
}
] | [
94,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
89,
1
] |
Mathlib/Data/Set/Image.lean | Disjoint.of_preimage | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.120582\nf : α → β\ns✝ t✝ : Set α\nhf : Surjective f\ns t : Set β\nh : Disjoint (f ⁻¹' s) (f ⁻¹' t)\n⊢ Disjoint s t",
"tactic": "rw [disjoint_iff_inter_eq_empty, ← image_preimage_eq (_ ∩ _) hf, preimage_inter, h.inter_eq,\n image_empty]"
}
] | [
1608,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1605,
1
] |
Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean | bot_eq_one' | [] | [
392,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
391,
1
] |
Mathlib/Analysis/Normed/Group/Quotient.lean | QuotientAddGroup.norm_eq_infDist | [
{
"state_after": "no goals",
"state_before": "M : Type u_1\nN : Type ?u.25731\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nx : M ⧸ S\n⊢ ‖x‖ = infDist 0 {m | ↑m = x}",
"tactic": "simp only [AddSubgroup.quotient_norm_eq, infDist_eq_iInf, sInf_image', dist_zero_left]"
}
] | [
117,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
115,
1
] |
Mathlib/Data/Ordmap/Ordset.lean | Ordnode.findMax_dual | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nt : Ordnode α\n⊢ findMax (dual t) = findMin t",
"tactic": "rw [← findMin_dual, dual_dual]"
}
] | [
595,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
594,
1
] |
Mathlib/Topology/MetricSpace/Gluing.lean | Metric.isometry_inl | [] | [
283,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
282,
1
] |
Std/Data/List/Lemmas.lean | List.self_mem_range_succ | [
{
"state_after": "no goals",
"state_before": "n : Nat\n⊢ n ∈ range (n + 1)",
"tactic": "simp"
}
] | [
1918,
69
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1918,
1
] |
Mathlib/Data/Finsupp/Basic.lean | Finsupp.some_zero | [
{
"state_after": "case h\nα : Type u_2\nβ : Type ?u.319084\nγ : Type ?u.319087\nι : Type ?u.319090\nM : Type u_1\nM' : Type ?u.319096\nN : Type ?u.319099\nP : Type ?u.319102\nG : Type ?u.319105\nH : Type ?u.319108\nR : Type ?u.319111\nS : Type ?u.319114\ninst✝ : Zero M\na✝ : α\n⊢ ↑(some 0) a✝ = ↑0 a✝",
"state_before": "α : Type u_2\nβ : Type ?u.319084\nγ : Type ?u.319087\nι : Type ?u.319090\nM : Type u_1\nM' : Type ?u.319096\nN : Type ?u.319099\nP : Type ?u.319102\nG : Type ?u.319105\nH : Type ?u.319108\nR : Type ?u.319111\nS : Type ?u.319114\ninst✝ : Zero M\n⊢ some 0 = 0",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_2\nβ : Type ?u.319084\nγ : Type ?u.319087\nι : Type ?u.319090\nM : Type u_1\nM' : Type ?u.319096\nN : Type ?u.319099\nP : Type ?u.319102\nG : Type ?u.319105\nH : Type ?u.319108\nR : Type ?u.319111\nS : Type ?u.319114\ninst✝ : Zero M\na✝ : α\n⊢ ↑(some 0) a✝ = ↑0 a✝",
"tactic": "simp"
}
] | [
819,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
817,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean | EMetric.controlled_of_uniformEmbedding | [] | [
313,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
310,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean | MeasureTheory.snorm_indicator_const' | [
{
"state_after": "case pos\nα : Type u_1\nE : Type ?u.645570\nF : Type ?u.645573\nG : Type u_2\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc✝ : E\nf : α → E\nhf : AEStronglyMeasurable f μ\nc : G\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ 0\nhp : p ≠ 0\nhp_top : p = ⊤\n⊢ snorm (Set.indicator s fun x => c) p μ = ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal p)\n\ncase neg\nα : Type u_1\nE : Type ?u.645570\nF : Type ?u.645573\nG : Type u_2\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc✝ : E\nf : α → E\nhf : AEStronglyMeasurable f μ\nc : G\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ 0\nhp : p ≠ 0\nhp_top : ¬p = ⊤\n⊢ snorm (Set.indicator s fun x => c) p μ = ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal p)",
"state_before": "α : Type u_1\nE : Type ?u.645570\nF : Type ?u.645573\nG : Type u_2\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc✝ : E\nf : α → E\nhf : AEStronglyMeasurable f μ\nc : G\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ 0\nhp : p ≠ 0\n⊢ snorm (Set.indicator s fun x => c) p μ = ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal p)",
"tactic": "by_cases hp_top : p = ∞"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type ?u.645570\nF : Type ?u.645573\nG : Type u_2\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc✝ : E\nf : α → E\nhf : AEStronglyMeasurable f μ\nc : G\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ 0\nhp : p ≠ 0\nhp_top : p = ⊤\n⊢ snorm (Set.indicator s fun x => c) p μ = ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal p)",
"tactic": "simp [hp_top, snormEssSup_indicator_const_eq s c hμs]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.645570\nF : Type ?u.645573\nG : Type u_2\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ns : Set α\nhs✝ : MeasurableSet s\nc✝ : E\nf : α → E\nhf : AEStronglyMeasurable f μ\nc : G\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ 0\nhp : p ≠ 0\nhp_top : ¬p = ⊤\n⊢ snorm (Set.indicator s fun x => c) p μ = ↑‖c‖₊ * ↑↑μ s ^ (1 / ENNReal.toReal p)",
"tactic": "exact snorm_indicator_const hs hp hp_top"
}
] | [
613,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
609,
1
] |
Mathlib/Data/Option/Basic.lean | Option.bind_eq_some' | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1609\nδ : Type ?u.1612\nx : Option α\nf : α → Option β\nb : β\n⊢ Option.bind x f = some b ↔ ∃ a, x = some a ∧ f a = some b",
"tactic": "cases x <;> simp"
}
] | [
91,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
89,
1
] |
Mathlib/Data/List/Nodup.lean | List.Nodup.ne_singleton_iff | [
{
"state_after": "case nil\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\nl : List α\nh✝ : Nodup l\nx : α\nh : Nodup []\n⊢ [] ≠ [x] ↔ [] = [] ∨ ∃ y, y ∈ [] ∧ y ≠ x\n\ncase cons\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\nl : List α\nh✝ : Nodup l\nx hd : α\ntl : List α\nhl : Nodup tl → (tl ≠ [x] ↔ tl = [] ∨ ∃ y, y ∈ tl ∧ y ≠ x)\nh : Nodup (hd :: tl)\n⊢ hd :: tl ≠ [x] ↔ hd :: tl = [] ∨ ∃ y, y ∈ hd :: tl ∧ y ≠ x",
"state_before": "α : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\nl : List α\nh : Nodup l\nx : α\n⊢ l ≠ [x] ↔ l = [] ∨ ∃ y, y ∈ l ∧ y ≠ x",
"tactic": "induction' l with hd tl hl"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\nl : List α\nh✝ : Nodup l\nx : α\nh : Nodup []\n⊢ [] ≠ [x] ↔ [] = [] ∨ ∃ y, y ∈ [] ∧ y ≠ x",
"tactic": "simp"
},
{
"state_after": "case cons\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\nl : List α\nh✝ : Nodup l\nx hd : α\ntl : List α\nh : Nodup (hd :: tl)\nhl : tl ≠ [x] ↔ tl = [] ∨ ∃ y, y ∈ tl ∧ y ≠ x\n⊢ hd :: tl ≠ [x] ↔ hd :: tl = [] ∨ ∃ y, y ∈ hd :: tl ∧ y ≠ x",
"state_before": "case cons\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\nl : List α\nh✝ : Nodup l\nx hd : α\ntl : List α\nhl : Nodup tl → (tl ≠ [x] ↔ tl = [] ∨ ∃ y, y ∈ tl ∧ y ≠ x)\nh : Nodup (hd :: tl)\n⊢ hd :: tl ≠ [x] ↔ hd :: tl = [] ∨ ∃ y, y ∈ hd :: tl ∧ y ≠ x",
"tactic": "specialize hl h.of_cons"
},
{
"state_after": "case pos\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\nl : List α\nh✝ : Nodup l\nx hd : α\ntl : List α\nh : Nodup (hd :: tl)\nhl : tl ≠ [x] ↔ tl = [] ∨ ∃ y, y ∈ tl ∧ y ≠ x\nhx : tl = [x]\n⊢ hd :: tl ≠ [x] ↔ hd :: tl = [] ∨ ∃ y, y ∈ hd :: tl ∧ y ≠ x\n\ncase neg\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\nl : List α\nh✝ : Nodup l\nx hd : α\ntl : List α\nh : Nodup (hd :: tl)\nhl : tl ≠ [x] ↔ tl = [] ∨ ∃ y, y ∈ tl ∧ y ≠ x\nhx : ¬tl = [x]\n⊢ hd :: tl ≠ [x] ↔ hd :: tl = [] ∨ ∃ y, y ∈ hd :: tl ∧ y ≠ x",
"state_before": "case cons\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\nl : List α\nh✝ : Nodup l\nx hd : α\ntl : List α\nh : Nodup (hd :: tl)\nhl : tl ≠ [x] ↔ tl = [] ∨ ∃ y, y ∈ tl ∧ y ≠ x\n⊢ hd :: tl ≠ [x] ↔ hd :: tl = [] ∨ ∃ y, y ∈ hd :: tl ∧ y ≠ x",
"tactic": "by_cases hx : tl = [x]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\nl : List α\nh✝ : Nodup l\nx hd : α\ntl : List α\nh : Nodup (hd :: tl)\nhl : tl ≠ [x] ↔ tl = [] ∨ ∃ y, y ∈ tl ∧ y ≠ x\nhx : tl = [x]\n⊢ hd :: tl ≠ [x] ↔ hd :: tl = [] ∨ ∃ y, y ∈ hd :: tl ∧ y ≠ x",
"tactic": "simpa [hx, and_comm, and_or_left] using h"
},
{
"state_after": "case neg\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\nl : List α\nh✝ : Nodup l\nx hd : α\ntl : List α\nh : Nodup (hd :: tl)\nhl : tl ≠ [x] ↔ tl = [] ∨ ∃ y, y ∈ tl ∧ y ≠ x\nhx : tl = [] ∨ ∃ y, y ∈ tl ∧ y ≠ x\n⊢ hd :: tl ≠ [x] ↔ hd :: tl = [] ∨ ∃ y, y ∈ hd :: tl ∧ y ≠ x",
"state_before": "case neg\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\nl : List α\nh✝ : Nodup l\nx hd : α\ntl : List α\nh : Nodup (hd :: tl)\nhl : tl ≠ [x] ↔ tl = [] ∨ ∃ y, y ∈ tl ∧ y ≠ x\nhx : ¬tl = [x]\n⊢ hd :: tl ≠ [x] ↔ hd :: tl = [] ∨ ∃ y, y ∈ hd :: tl ∧ y ≠ x",
"tactic": "rw [← Ne.def, hl] at hx"
},
{
"state_after": "case neg.inl\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\nl : List α\nh✝ : Nodup l\nx hd : α\nh : Nodup [hd]\nhl : [] ≠ [x] ↔ [] = [] ∨ ∃ y, y ∈ [] ∧ y ≠ x\n⊢ [hd] ≠ [x] ↔ [hd] = [] ∨ ∃ y, y ∈ [hd] ∧ y ≠ x\n\ncase neg.inr.intro.intro\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\nl : List α\nh✝ : Nodup l\nx hd : α\ntl : List α\nh : Nodup (hd :: tl)\nhl : tl ≠ [x] ↔ tl = [] ∨ ∃ y, y ∈ tl ∧ y ≠ x\ny : α\nhy : y ∈ tl\nhx : y ≠ x\n⊢ hd :: tl ≠ [x] ↔ hd :: tl = [] ∨ ∃ y, y ∈ hd :: tl ∧ y ≠ x",
"state_before": "case neg\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\nl : List α\nh✝ : Nodup l\nx hd : α\ntl : List α\nh : Nodup (hd :: tl)\nhl : tl ≠ [x] ↔ tl = [] ∨ ∃ y, y ∈ tl ∧ y ≠ x\nhx : tl = [] ∨ ∃ y, y ∈ tl ∧ y ≠ x\n⊢ hd :: tl ≠ [x] ↔ hd :: tl = [] ∨ ∃ y, y ∈ hd :: tl ∧ y ≠ x",
"tactic": "rcases hx with (rfl | ⟨y, hy, hx⟩)"
},
{
"state_after": "no goals",
"state_before": "case neg.inl\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\nl : List α\nh✝ : Nodup l\nx hd : α\nh : Nodup [hd]\nhl : [] ≠ [x] ↔ [] = [] ∨ ∃ y, y ∈ [] ∧ y ≠ x\n⊢ [hd] ≠ [x] ↔ [hd] = [] ∨ ∃ y, y ∈ [hd] ∧ y ≠ x",
"tactic": "simp"
},
{
"state_after": "case neg.inr.intro.intro\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\nl : List α\nh✝ : Nodup l\nx hd : α\ntl : List α\nh : Nodup (hd :: tl)\nhl : tl ≠ [x] ↔ tl = [] ∨ ∃ y, y ∈ tl ∧ y ≠ x\ny : α\nhy : y ∈ tl\nhx : y ≠ x\n⊢ ∃ y x_1, y ≠ x",
"state_before": "case neg.inr.intro.intro\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\nl : List α\nh✝ : Nodup l\nx hd : α\ntl : List α\nh : Nodup (hd :: tl)\nhl : tl ≠ [x] ↔ tl = [] ∨ ∃ y, y ∈ tl ∧ y ≠ x\ny : α\nhy : y ∈ tl\nhx : y ≠ x\n⊢ hd :: tl ≠ [x] ↔ hd :: tl = [] ∨ ∃ y, y ∈ hd :: tl ∧ y ≠ x",
"tactic": "suffices ∃ (y : α) (_ : y ∈ hd :: tl), y ≠ x by simpa [ne_nil_of_mem hy]"
},
{
"state_after": "no goals",
"state_before": "case neg.inr.intro.intro\nα : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\nl : List α\nh✝ : Nodup l\nx hd : α\ntl : List α\nh : Nodup (hd :: tl)\nhl : tl ≠ [x] ↔ tl = [] ∨ ∃ y, y ∈ tl ∧ y ≠ x\ny : α\nhy : y ∈ tl\nhx : y ≠ x\n⊢ ∃ y x_1, y ≠ x",
"tactic": "exact ⟨y, mem_cons_of_mem _ hy, hx⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nl✝ l₁ l₂ : List α\nr : α → α → Prop\na b : α\nl : List α\nh✝ : Nodup l\nx hd : α\ntl : List α\nh : Nodup (hd :: tl)\nhl : tl ≠ [x] ↔ tl = [] ∨ ∃ y, y ∈ tl ∧ y ≠ x\ny : α\nhy : y ∈ tl\nhx : y ≠ x\nthis : ∃ y x_1, y ≠ x\n⊢ hd :: tl ≠ [x] ↔ hd :: tl = [] ∨ ∃ y, y ∈ hd :: tl ∧ y ≠ x",
"tactic": "simpa [ne_nil_of_mem hy]"
}
] | [
150,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
139,
1
] |
Std/Data/Nat/Gcd.lean | Nat.gcd_ne_zero_right | [] | [
114,
83
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
114,
1
] |
Mathlib/RingTheory/Polynomial/Quotient.lean | MvPolynomial.quotient_map_C_eq_zero | [
{
"state_after": "R : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nr : R\nI : Ideal R\ni : R\nhi : i ∈ I\n⊢ ↑C i ∈ Ideal.map C I",
"state_before": "R : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nr : R\nI : Ideal R\ni : R\nhi : i ∈ I\n⊢ ↑(RingHom.comp (Ideal.Quotient.mk (Ideal.map C I)) C) i = 0",
"tactic": "simp only [Function.comp_apply, RingHom.coe_comp, Ideal.Quotient.eq_zero_iff_mem]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nr : R\nI : Ideal R\ni : R\nhi : i ∈ I\n⊢ ↑C i ∈ Ideal.map C I",
"tactic": "exact Ideal.mem_map_of_mem _ hi"
}
] | [
201,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
197,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean | HasDerivAt.deriv | [] | [
453,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
452,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean | Set.Iic_union_Ici | [] | [
1220,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1219,
1
] |
Mathlib/Order/Filter/FilterProduct.lean | Filter.Germ.lt_def | [
{
"state_after": "case h.mk.h.mk.a\nα : Type u\nβ : Type v\nφ : Ultrafilter α\ninst✝ : Preorder β\nx✝¹ : Germ (↑φ) β\nf : α → β\nx✝ : Germ (↑φ) β\ng : α → β\n⊢ Quot.mk Setoid.r f < Quot.mk Setoid.r g ↔ LiftRel (fun x x_1 => x < x_1) (Quot.mk Setoid.r f) (Quot.mk Setoid.r g)",
"state_before": "α : Type u\nβ : Type v\nφ : Ultrafilter α\ninst✝ : Preorder β\n⊢ (fun x x_1 => x < x_1) = LiftRel fun x x_1 => x < x_1",
"tactic": "ext ⟨f⟩ ⟨g⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mk.h.mk.a\nα : Type u\nβ : Type v\nφ : Ultrafilter α\ninst✝ : Preorder β\nx✝¹ : Germ (↑φ) β\nf : α → β\nx✝ : Germ (↑φ) β\ng : α → β\n⊢ Quot.mk Setoid.r f < Quot.mk Setoid.r g ↔ LiftRel (fun x x_1 => x < x_1) (Quot.mk Setoid.r f) (Quot.mk Setoid.r g)",
"tactic": "exact coe_lt"
}
] | [
83,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
81,
1
] |
Mathlib/Algebra/CubicDiscriminant.lean | Cubic.degree_of_a_ne_zero | [] | [
305,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
304,
1
] |
Mathlib/Order/CompleteLattice.lean | iSup_congr_Prop | [
{
"state_after": "α : Type u_1\nβ : Type ?u.63159\nβ₂ : Type ?u.63162\nγ : Type ?u.63165\nι : Sort ?u.63168\nι' : Sort ?u.63171\nκ : ι → Sort ?u.63176\nκ' : ι' → Sort ?u.63181\ninst✝ : SupSet α\nf✝ g : ι → α\np : Prop\nf₁ f₂ : p → α\npq : p ↔ p\nf : ∀ (x : p), f₁ (_ : p) = f₂ x\n⊢ iSup f₁ = iSup f₂",
"state_before": "α : Type u_1\nβ : Type ?u.63159\nβ₂ : Type ?u.63162\nγ : Type ?u.63165\nι : Sort ?u.63168\nι' : Sort ?u.63171\nκ : ι → Sort ?u.63176\nκ' : ι' → Sort ?u.63181\ninst✝ : SupSet α\nf✝ g : ι → α\np q : Prop\nf₁ : p → α\nf₂ : q → α\npq : p ↔ q\nf : ∀ (x : q), f₁ (_ : p) = f₂ x\n⊢ iSup f₁ = iSup f₂",
"tactic": "obtain rfl := propext pq"
},
{
"state_after": "case e_s.h\nα : Type u_1\nβ : Type ?u.63159\nβ₂ : Type ?u.63162\nγ : Type ?u.63165\nι : Sort ?u.63168\nι' : Sort ?u.63171\nκ : ι → Sort ?u.63176\nκ' : ι' → Sort ?u.63181\ninst✝ : SupSet α\nf✝ g : ι → α\np : Prop\nf₁ f₂ : p → α\npq : p ↔ p\nf : ∀ (x : p), f₁ (_ : p) = f₂ x\nx : p\n⊢ f₁ x = f₂ x",
"state_before": "α : Type u_1\nβ : Type ?u.63159\nβ₂ : Type ?u.63162\nγ : Type ?u.63165\nι : Sort ?u.63168\nι' : Sort ?u.63171\nκ : ι → Sort ?u.63176\nκ' : ι' → Sort ?u.63181\ninst✝ : SupSet α\nf✝ g : ι → α\np : Prop\nf₁ f₂ : p → α\npq : p ↔ p\nf : ∀ (x : p), f₁ (_ : p) = f₂ x\n⊢ iSup f₁ = iSup f₂",
"tactic": "congr with x"
},
{
"state_after": "no goals",
"state_before": "case e_s.h\nα : Type u_1\nβ : Type ?u.63159\nβ₂ : Type ?u.63162\nγ : Type ?u.63165\nι : Sort ?u.63168\nι' : Sort ?u.63171\nκ : ι → Sort ?u.63176\nκ' : ι' → Sort ?u.63181\ninst✝ : SupSet α\nf✝ g : ι → α\np : Prop\nf₁ f₂ : p → α\npq : p ↔ p\nf : ∀ (x : p), f₁ (_ : p) = f₂ x\nx : p\n⊢ f₁ x = f₂ x",
"tactic": "apply f"
}
] | [
664,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
660,
1
] |
Mathlib/RingTheory/Algebraic.lean | Subalgebra.inv_mem_of_root_of_coeff_zero_ne_zero | [
{
"state_after": "R : Type ?u.788449\nS : Type ?u.788452\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\naeval_eq : ↑(aeval x) p = 0\ncoeff_zero_ne : coeff p 0 ≠ 0\n⊢ (↑x)⁻¹ = ↑((-coeff p 0)⁻¹ • ↑(aeval x) (divX p))",
"state_before": "R : Type ?u.788449\nS : Type ?u.788452\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\naeval_eq : ↑(aeval x) p = 0\ncoeff_zero_ne : coeff p 0 ≠ 0\n⊢ (↑x)⁻¹ ∈ A",
"tactic": "suffices (x⁻¹ : L) = (-p.coeff 0)⁻¹ • aeval x (divX p) by\n rw [this]\n exact A.smul_mem (aeval x _).2 _"
},
{
"state_after": "R : Type ?u.788449\nS : Type ?u.788452\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\naeval_eq : ↑(aeval x) p = 0\ncoeff_zero_ne : coeff p 0 ≠ 0\nthis : ↑(aeval ↑x) p = 0\n⊢ (↑x)⁻¹ = ↑((-coeff p 0)⁻¹ • ↑(aeval x) (divX p))",
"state_before": "R : Type ?u.788449\nS : Type ?u.788452\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\naeval_eq : ↑(aeval x) p = 0\ncoeff_zero_ne : coeff p 0 ≠ 0\n⊢ (↑x)⁻¹ = ↑((-coeff p 0)⁻¹ • ↑(aeval x) (divX p))",
"tactic": "have : aeval (x : L) p = 0 := by rw [Subalgebra.aeval_coe, aeval_eq, Subalgebra.coe_zero]"
},
{
"state_after": "R : Type ?u.788449\nS : Type ?u.788452\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\naeval_eq : ↑(aeval x) p = 0\ncoeff_zero_ne : coeff p 0 ≠ 0\nthis : ↑(aeval ↑x) p = 0\n⊢ -((↑(algebraMap K L) (coeff p 0))⁻¹ * ↑(aeval ↑x) (divX p)) =\n ↑(↑(algebraMap K ((fun x => { x // x ∈ A }) (divX p))) (-coeff p 0)⁻¹ * ↑(aeval x) (divX p))",
"state_before": "R : Type ?u.788449\nS : Type ?u.788452\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\naeval_eq : ↑(aeval x) p = 0\ncoeff_zero_ne : coeff p 0 ≠ 0\nthis : ↑(aeval ↑x) p = 0\n⊢ (↑x)⁻¹ = ↑((-coeff p 0)⁻¹ • ↑(aeval x) (divX p))",
"tactic": "rw [inv_eq_of_root_of_coeff_zero_ne_zero this coeff_zero_ne, div_eq_inv_mul, Algebra.smul_def]"
},
{
"state_after": "R : Type ?u.788449\nS : Type ?u.788452\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\naeval_eq : ↑(aeval x) p = 0\ncoeff_zero_ne : coeff p 0 ≠ 0\nthis : ↑(aeval ↑x) p = 0\n⊢ -((↑(algebraMap K L) (coeff p 0))⁻¹ * ↑(↑(aeval x) (divX p))) =\n ↑(algebraMap K L) (-coeff p 0)⁻¹ * ↑(↑(aeval x) (divX p))",
"state_before": "R : Type ?u.788449\nS : Type ?u.788452\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\naeval_eq : ↑(aeval x) p = 0\ncoeff_zero_ne : coeff p 0 ≠ 0\nthis : ↑(aeval ↑x) p = 0\n⊢ -((↑(algebraMap K L) (coeff p 0))⁻¹ * ↑(aeval ↑x) (divX p)) =\n ↑(↑(algebraMap K ((fun x => { x // x ∈ A }) (divX p))) (-coeff p 0)⁻¹ * ↑(aeval x) (divX p))",
"tactic": "simp only [aeval_coe, Submonoid.coe_mul, Subsemiring.coe_toSubmonoid, coe_toSubsemiring,\n coe_algebraMap]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.788449\nS : Type ?u.788452\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\naeval_eq : ↑(aeval x) p = 0\ncoeff_zero_ne : coeff p 0 ≠ 0\nthis : ↑(aeval ↑x) p = 0\n⊢ -((↑(algebraMap K L) (coeff p 0))⁻¹ * ↑(↑(aeval x) (divX p))) =\n ↑(algebraMap K L) (-coeff p 0)⁻¹ * ↑(↑(aeval x) (divX p))",
"tactic": "rw [map_inv₀, map_neg, inv_neg, neg_mul]"
},
{
"state_after": "R : Type ?u.788449\nS : Type ?u.788452\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\naeval_eq : ↑(aeval x) p = 0\ncoeff_zero_ne : coeff p 0 ≠ 0\nthis : (↑x)⁻¹ = ↑((-coeff p 0)⁻¹ • ↑(aeval x) (divX p))\n⊢ ↑((-coeff p 0)⁻¹ • ↑(aeval x) (divX p)) ∈ A",
"state_before": "R : Type ?u.788449\nS : Type ?u.788452\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\naeval_eq : ↑(aeval x) p = 0\ncoeff_zero_ne : coeff p 0 ≠ 0\nthis : (↑x)⁻¹ = ↑((-coeff p 0)⁻¹ • ↑(aeval x) (divX p))\n⊢ (↑x)⁻¹ ∈ A",
"tactic": "rw [this]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.788449\nS : Type ?u.788452\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\naeval_eq : ↑(aeval x) p = 0\ncoeff_zero_ne : coeff p 0 ≠ 0\nthis : (↑x)⁻¹ = ↑((-coeff p 0)⁻¹ • ↑(aeval x) (divX p))\n⊢ ↑((-coeff p 0)⁻¹ • ↑(aeval x) (divX p)) ∈ A",
"tactic": "exact A.smul_mem (aeval x _).2 _"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.788449\nS : Type ?u.788452\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_2\nL : Type u_1\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : { x // x ∈ A }\np : K[X]\naeval_eq : ↑(aeval x) p = 0\ncoeff_zero_ne : coeff p 0 ≠ 0\n⊢ ↑(aeval ↑x) p = 0",
"tactic": "rw [Subalgebra.aeval_coe, aeval_eq, Subalgebra.coe_zero]"
}
] | [
372,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
362,
1
] |
Mathlib/CategoryTheory/Abelian/NonPreadditive.lean | CategoryTheory.NonPreadditiveAbelian.lift_σ | [
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : NonPreadditiveAbelian C\nX : C\n⊢ prod.lift (𝟙 X) 0 ≫ σ = 𝟙 X",
"tactic": "rw [← Category.assoc, IsIso.hom_inv_id]"
}
] | [
292,
99
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
292,
1
] |
Mathlib/NumberTheory/LegendreSymbol/MulCharacter.lean | MulChar.one_mul | [
{
"state_after": "case h\nR : Type u\ninst✝¹ : CommMonoid R\nR' : Type v\ninst✝ : CommMonoidWithZero R'\nχ : MulChar R R'\na✝ : Rˣ\n⊢ ↑(1 * χ) ↑a✝ = ↑χ ↑a✝",
"state_before": "R : Type u\ninst✝¹ : CommMonoid R\nR' : Type v\ninst✝ : CommMonoidWithZero R'\nχ : MulChar R R'\n⊢ 1 * χ = χ",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\ninst✝¹ : CommMonoid R\nR' : Type v\ninst✝ : CommMonoidWithZero R'\nχ : MulChar R R'\na✝ : Rˣ\n⊢ ↑(1 * χ) ↑a✝ = ↑χ ↑a✝",
"tactic": "simp only [one_mul, Pi.mul_apply, MulChar.coeToFun_mul, MulChar.one_apply_coe]"
}
] | [
318,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
316,
11
] |
Mathlib/MeasureTheory/Integral/CircleIntegral.lean | range_circleMap | [
{
"state_after": "E : Type ?u.12721\ninst✝ : NormedAddCommGroup E\nc : ℂ\nR : ℝ\n⊢ c +ᵥ sphere (R • 0) (‖R‖ * 1) = sphere c (Abs.abs R)",
"state_before": "E : Type ?u.12721\ninst✝ : NormedAddCommGroup E\nc : ℂ\nR : ℝ\n⊢ (c +ᵥ R • range fun θ => exp (↑θ * I)) = sphere c (Abs.abs R)",
"tactic": "rw [Complex.range_exp_mul_I, smul_sphere R 0 zero_le_one]"
},
{
"state_after": "no goals",
"state_before": "E : Type ?u.12721\ninst✝ : NormedAddCommGroup E\nc : ℂ\nR : ℝ\n⊢ c +ᵥ sphere (R • 0) (‖R‖ * 1) = sphere c (Abs.abs R)",
"tactic": "simp"
}
] | [
152,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
145,
1
] |
Mathlib/Topology/LocalHomeomorph.lean | LocalHomeomorph.mapsTo | [] | [
171,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
171,
11
] |
Mathlib/Algebra/Field/Opposite.lean | MulOpposite.op_ratCast | [] | [
31,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
30,
1
] |
Mathlib/Order/OmegaCompletePartialOrder.lean | OmegaCompletePartialOrder.ContinuousHom.coe_inj | [] | [
717,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
716,
11
] |
Mathlib/GroupTheory/FreeGroup.lean | FreeGroup.Red.sublist | [] | [
395,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
390,
11
] |
Mathlib/Data/Dfinsupp/Basic.lean | Dfinsupp.coe_mk' | [] | [
128,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
127,
1
] |
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