module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 352,
"column": 22
} | {
"line": 352,
"column": 67
} | {
"line": 354,
"column": 0
} | [
{
"pp": "case mpr.cons\nA : Type u_1\ninst✝ : CStarAlgebra A\nx : ↥(selfAdjoint A)\nxs : List ↥(selfAdjoint A)\nih : (List.map expUnitary xs).prod ∈ pathComponent 1\n⊢ (List.map expUnitary (x :: xs)).prod ∈ pathComponent 1",
"ppTerm": "?mpr.cons",
"assigned": true,
"usedConstants": [
"CStarAlg... | [] | simpa using! (joined_one_expUnitary x).mul ih | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 352,
"column": 22
} | {
"line": 352,
"column": 67
} | {
"line": 354,
"column": 0
} | [
{
"pp": "case mpr.cons\nA : Type u_1\ninst✝ : CStarAlgebra A\nx : ↥(selfAdjoint A)\nxs : List ↥(selfAdjoint A)\nih : (List.map expUnitary xs).prod ∈ pathComponent 1\n⊢ (List.map expUnitary (x :: xs)).prod ∈ pathComponent 1",
"ppTerm": "?mpr.cons",
"assigned": true,
"usedConstants": [
"CStarAlg... | [] | simpa using! (joined_one_expUnitary x).mul ih | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 77
} | {
"line": 146,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nA : E →L[𝕜] F\nx : E\n⊢ ‖A x‖ ^ 2 = re ⟪(adjoint A ∘SL A) x,... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nA : E →L[𝕜] F\nx : E\nh : ⟪(adjoint A ∘SL A) x, x⟫_𝕜 = ⟪A x, A x⟫_𝕜\n⊢... | have h : ⟪(A† ∘L A) x, x⟫ = ⟪A x, A x⟫ := by rw [← adjoint_inner_left]; rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 178,
"column": 2
} | {
"line": 178,
"column": 6
} | {
"line": 179,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : CompleteSpace E\nU : Submodule 𝕜 E\ninst✝ : CompleteSpace ↥U\n⊢ adjoint U.subtypeL = U.orthogonalProjectionOnto",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": ... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : CompleteSpace E\nU : Submodule 𝕜 E\ninst✝ : CompleteSpace ↥U\n⊢ U.orthogonalProjectionOnto = adjoint U.subtypeL"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 345,
"column": 2
} | {
"line": 345,
"column": 6
} | {
"line": 346,
"column": 2
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : CharZero k\ns : Simplex k P n\ni : Fin (n + 1)\n⊢ s.points i -ᵥ s.centroid = ↑n • (s.centroid -ᵥ s.faceOppositeCentroid i)",
"... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : CharZero k\ns : Simplex k P n\ni : Fin (n + 1)\n⊢ ↑n • (s.centroid -ᵥ s.faceOppositeCentroid i) = s.points i -ᵥ s.centroid"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 352,
"column": 2
} | {
"line": 352,
"column": 6
} | {
"line": 353,
"column": 2
} | [
{
"pp": "case e_a\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : CharZero k\ns : Simplex k P n\ni : Fin (n + 1)\n⊢ ↑n * (↑n + 1)⁻¹ = ↑n * (↑n)⁻¹ - (↑n + 1)⁻¹",
"ppTerm": "?e_a✝",
... | [
"case e_a\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : CharZero k\ns : Simplex k P n\ni : Fin (n + 1)\n⊢ ↑n * (↑n)⁻¹ - (↑n + 1)⁻¹ = ↑n * (↑n + 1)⁻¹"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Analysis.Calculus.BumpFunction.Basic | {
"line": 132,
"column": 75
} | {
"line": 133,
"column": 39
} | {
"line": 135,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : HasContDiffBump E\nf : ContDiffBump 0\nx : E\n⊢ ↑f (-x) = ↑f x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"_private.Mathlib.Analysis.... | [] | by
simp_rw [← zero_sub, f.sub, zero_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 97
} | {
"line": 236,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : Fintype ι\np : ι → P\nn : ℕ\nhc : Fintype.card ι = n + 2\n⊢ AffineIndependent k p ↔ ¬finrank k ↥(vectorSpan k (Set.range p)) ≤ n",
"ppTerm... | [] | rw [affineIndependent_iff_le_finrank_vectorSpan k p hc, ← Nat.lt_iff_add_one_le, lt_iff_not_ge] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 97
} | {
"line": 236,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : Fintype ι\np : ι → P\nn : ℕ\nhc : Fintype.card ι = n + 2\n⊢ AffineIndependent k p ↔ ¬finrank k ↥(vectorSpan k (Set.range p)) ≤ n",
"ppTerm... | [] | rw [affineIndependent_iff_le_finrank_vectorSpan k p hc, ← Nat.lt_iff_add_one_le, lt_iff_not_ge] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 97
} | {
"line": 236,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : Fintype ι\np : ι → P\nn : ℕ\nhc : Fintype.card ι = n + 2\n⊢ AffineIndependent k p ↔ ¬finrank k ↥(vectorSpan k (Set.range p)) ≤ n",
"ppTerm... | [] | rw [affineIndependent_iff_le_finrank_vectorSpan k p hc, ← Nat.lt_iff_add_one_le, lt_iff_not_ge] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Group.Integral | {
"line": 43,
"column": 2
} | {
"line": 44,
"column": 40
} | {
"line": 46,
"column": 0
} | [
{
"pp": "G : Type u_4\nE : Type u_5\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : Group G\ninst✝¹ : MeasurableInv G\nf : G → E\nμ : Measure G\ninst✝ : μ.IsInvInvariant\n⊢ ∫ (x : G), f x⁻¹ ∂μ = ∫ (x : G), f x ∂μ",
"ppTerm": "?m.25",
"assigned": true,
"... | [] | have h : MeasurableEmbedding fun x : G => x⁻¹ := (MeasurableEquiv.inv G).measurableEmbedding
rw [← h.integral_map, map_inv_eq_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Group.Integral | {
"line": 43,
"column": 2
} | {
"line": 44,
"column": 40
} | {
"line": 46,
"column": 0
} | [
{
"pp": "G : Type u_4\nE : Type u_5\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : Group G\ninst✝¹ : MeasurableInv G\nf : G → E\nμ : Measure G\ninst✝ : μ.IsInvInvariant\n⊢ ∫ (x : G), f x⁻¹ ∂μ = ∫ (x : G), f x ∂μ",
"ppTerm": "?m.25",
"assigned": true,
"... | [] | have h : MeasurableEmbedding fun x : G => x⁻¹ := (MeasurableEquiv.inv G).measurableEmbedding
rw [← h.integral_map, map_inv_eq_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 654,
"column": 2
} | {
"line": 657,
"column": 88
} | {
"line": 658,
"column": 2
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np₁ p₂ p₃ p : P\nha : AffineIndependent k ![p₁, p₂, p₃]\nhcol : Collinear k {p₂, p₃, p}\nhne : p₂ ≠ p\nh : Collinear k {p₁, p₂, p}\n⊢ False",
"ppTerm": "?m.63",
... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np₁ p₂ p₃ p : P\nha : AffineIndependent k ![p₁, p₂, p₃]\nhcol : Collinear k {p₂, p₃, p}\nhne : p₂ ≠ p\nh : Collinear k {p₁, p₂, p}\nh1 : Collinear k {p₁, p₃, p₂, p}\n⊢ False"
] | have h1 : Collinear k {p₁, p₃, p₂, p} := by
apply collinear_insert_insert_of_mem_affineSpan_pair
· apply Collinear.mem_affineSpan_of_mem_of_ne h (by simp) (by simp) (by simp) hne
· apply Collinear.mem_affineSpan_of_mem_of_ne hcol (by simp) (by simp) (by simp) hne | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.SmoothSeries | {
"line": 221,
"column": 6
} | {
"line": 221,
"column": 38
} | {
"line": 221,
"column": 38
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : IsRCLikeNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nv : ℕ → α → ℝ\nN : ℕ... | [
"α : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : IsRCLikeNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nv : ℕ → α → ℝ\nN : ℕ∞\nhf : ∀ (i... | iteratedFDeriv_tsum hf hv h'f hk | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.SmoothSeries | {
"line": 231,
"column": 4
} | {
"line": 231,
"column": 14
} | {
"line": 232,
"column": 4
} | [
{
"pp": "case left\nα : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : IsRCLikeNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nv : ℕ → α... | [
"case left\nα : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : IsRCLikeNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nv : ℕ → α → ℝ\nN : ℕ∞... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Calculus.SmoothSeries | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 14
} | {
"line": 239,
"column": 4
} | [
{
"pp": "case right\nα : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : IsRCLikeNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nv : ℕ → ... | [
"case right\nα : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : IsRCLikeNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nv : ℕ → α → ℝ\nN : ℕ... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Calculus.FDeriv.WithLp | {
"line": 60,
"column": 77
} | {
"line": 63,
"column": 5
} | {
"line": 65,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nι : Type u_2\nE : ι → Type u_3\nH : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup H\ninst✝⁴ : (i : ι) → NormedAddCommGroup (E i)\ninst✝³ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝² : NormedSpace 𝕜 H\ninst✝¹ : Finite ι\np : ENNReal\ninst✝ : Fact (1 ≤ p)\nf : H →... | [] | by
have := Fintype.ofFinite ι
rw [← (PiLp.continuousLinearEquiv p 𝕜 E).comp_hasFDerivWithinAt_iff, hasFDerivWithinAt_pi']
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 718,
"column": 11
} | {
"line": 718,
"column": 37
} | {
"line": 718,
"column": 37
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_3\nE : α → Type u_4\np : ℝ≥0∞\ninst✝³ : (i : α) → NormedAddCommGroup (E i)\ninst✝² : NormedRing 𝕜\ninst✝¹ : (i : α) → Module 𝕜 (E i)\ninst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : ↥(lp E p)\nhp : 0 < p.toReal\ninst : NNNorm ↥(lp E p) := { nnnorm := fun ... | [
"𝕜 : Type u_1\nα : Type u_3\nE : α → Type u_4\np : ℝ≥0∞\ninst✝³ : (i : α) → NormedAddCommGroup (E i)\ninst✝² : NormedRing 𝕜\ninst✝¹ : (i : α) → Module 𝕜 (E i)\ninst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : ↥(lp E p)\nhp : 0 < p.toReal\ninst : NNNorm ↥(lp E p) := { nnnorm := fun f ↦ ⟨‖f‖, ⋯⟩... | NNReal.rpow_le_rpow_iff hp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ParametricIntegral | {
"line": 124,
"column": 8
} | {
"line": 124,
"column": 25
} | {
"line": 124,
"column": 26
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\ns : ... | [
"α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\ns : Set H\nF' : ... | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convolution | {
"line": 448,
"column": 56
} | {
"line": 448,
"column": 69
} | {
"line": 448,
"column": 69
} | [
{
"pp": "𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : NormedAddCommGroup F\ng : G → E'\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜 E'\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[... | [] | integral_zero | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Convolution | {
"line": 453,
"column": 59
} | {
"line": 453,
"column": 72
} | {
"line": 453,
"column": 72
} | [
{
"pp": "𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : NormedAddCommGroup F\nf : G → E\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜 E'\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[�... | [] | integral_zero | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Convolution | {
"line": 797,
"column": 2
} | {
"line": 797,
"column": 39
} | {
"line": 798,
"column": 2
} | [
{
"pp": "G : Type uG\nE' : Type uE'\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : MeasurableSpace G\nμ : Measure G\ninst✝⁶ : SeminormedAddCommGroup G\ninst✝⁵ : BorelSpace G\ninst✝⁴ : SecondCountableTopology G\ninst✝³ : μ.IsAddLeftInvariant\ninst✝² : SFinite μ\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : CompleteSpace E'\nι ... | [
"G : Type uG\nE' : Type uE'\ninst✝⁸ : NormedAddCommGroup E'\ninst✝⁷ : MeasurableSpace G\nμ : Measure G\ninst✝⁶ : SeminormedAddCommGroup G\ninst✝⁵ : BorelSpace G\ninst✝⁴ : SecondCountableTopology G\ninst✝³ : μ.IsAddLeftInvariant\ninst✝² : SFinite μ\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : CompleteSpace E'\nι : Type u_1\n... | simp_rw [tendsto_smallSets_iff] at hφ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Calculus.ContDiff.Convolution | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 59
} | {
"line": 223,
"column": 2
} | [
{
"pp": "𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\nP : Type uP\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedAddCommGroup F\nf : G → E\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : No... | [
"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\nP : Type uP\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedAddCommGroup F\nf : G → E\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedSpace 𝕜... | have hK' : IsCompact K' := hk.neg.add isCompact_singleton | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convolution | {
"line": 960,
"column": 8
} | {
"line": 961,
"column": 60
} | {
"line": 962,
"column": 4
} | [
{
"pp": "case inr\nE : Type uE\nE' : Type uE'\nF : Type uF\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup E'\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace ℝ E'\ninst✝¹ : NormedSpace ℝ F\nf : ℝ → E\ng : ℝ → E'\nL : E →L[ℝ] E' →L[ℝ] F\nν : Measure ℝ\ninst✝ : NullSingl... | [] | rcases lt_or_ge t x with (h' | h')
exacts [Or.inr (Or.inl ⟨h, h'⟩), Or.inr (Or.inr h')] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convolution | {
"line": 960,
"column": 8
} | {
"line": 961,
"column": 60
} | {
"line": 962,
"column": 4
} | [
{
"pp": "case inr\nE : Type uE\nE' : Type uE'\nF : Type uF\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup E'\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace ℝ E'\ninst✝¹ : NormedSpace ℝ F\nf : ℝ → E\ng : ℝ → E'\nL : E →L[ℝ] E' →L[ℝ] F\nν : Measure ℝ\ninst✝ : NullSingl... | [] | rcases lt_or_ge t x with (h' | h')
exacts [Or.inr (Or.inl ⟨h, h'⟩), Or.inr (Or.inr h')] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.BumpFunction.Normed | {
"line": 64,
"column": 2
} | {
"line": 66,
"column": 39
} | {
"line": 68,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : HasContDiffBump E\ninst✝⁴ : MeasurableSpace E\nc : E\nf : ContDiffBump c\nμ : Measure E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : μ.IsOpenPosMeasure\n⊢ 0 < ∫ (x : E),... | [] | refine (integral_pos_iff_support_of_nonneg f.nonneg' f.integrable).mpr ?_
rw [f.support_eq]
exact measure_ball_pos μ c f.rOut_pos | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.BumpFunction.Normed | {
"line": 64,
"column": 2
} | {
"line": 66,
"column": 39
} | {
"line": 68,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : HasContDiffBump E\ninst✝⁴ : MeasurableSpace E\nc : E\nf : ContDiffBump c\nμ : Measure E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : μ.IsOpenPosMeasure\n⊢ 0 < ∫ (x : E),... | [] | refine (integral_pos_iff_support_of_nonneg f.nonneg' f.integrable).mpr ?_
rw [f.support_eq]
exact measure_ball_pos μ c f.rOut_pos | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.BumpFunction.Normed | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 17
} | {
"line": 111,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : HasContDiffBump E\ninst✝⁴ : MeasurableSpace E\nc : E\nf : ContDiffBump c\nμ : Measure E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : μ.IsOpenPosMeasure\nx : E\n⊢ f.norme... | [
"E : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : HasContDiffBump E\ninst✝⁴ : MeasurableSpace E\nc : E\nf : ContDiffBump c\nμ : Measure E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : μ.IsOpenPosMeasure\nx : E\n⊢ ↑f x / ∫ (x : E), ↑... | rw [normed_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 251,
"column": 2
} | {
"line": 252,
"column": 100
} | {
"line": 253,
"column": 2
} | [
{
"pp": "case pos\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ' μ : Measure G\ninst✝² : μ.IsHaarMeasure\ninst✝¹ : IsFiniteMeasureOnCompacts μ'\ninst✝ : μ'.IsMulLeftInvariant\nH : LocallyCompactSpace G\ng : G → ℝ... | [
"case pos\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ' μ : Measure G\ninst✝² : μ.IsHaarMeasure\ninst✝¹ : IsFiniteMeasureOnCompacts μ'\ninst✝ : μ'.IsMulLeftInvariant\nH : LocallyCompactSpace G\ng : G → ℝ\ng_cont : C... | have A : ∫ x, f x ∂μ = (∫ y, f y * (∫ z, g (z⁻¹ * y) ∂ν)⁻¹ ∂ν) * ∫ x, g x ∂μ :=
integral_isMulLeftInvariant_isMulRightInvariant_combo f_cont f_comp g_cont g_comp g_nonneg g_one | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 412,
"column": 4
} | {
"line": 412,
"column": 44
} | {
"line": 412,
"column": 44
} | [
{
"pp": "case pos\nG : Type u_1\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : Group G\ninst✝⁴ : IsTopologicalGroup G\ninst✝³ : MeasurableSpace G\ninst✝² : BorelSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\nφ : G ≃ₜ* G\nhG : LocallyCompactSpace G\nf : G → ℝ\nf_cont : Continuous[inst✝⁶, ... | [
"case pos\nG : Type u_1\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : Group G\ninst✝⁴ : IsTopologicalGroup G\ninst✝³ : MeasurableSpace G\ninst✝² : BorelSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\nφ : G ≃ₜ* G\nhG : LocallyCompactSpace G\nf : G → ℝ\nf_cont : Continuous[inst✝⁶, _] f\nhf :\n... | integral_map (by fun_prop) (by fun_prop) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 377,
"column": 2
} | {
"line": 381,
"column": 71
} | {
"line": 382,
"column": 2
} | [
{
"pp": "case ind\nα : Type u_1\ninst✝¹ : MetricSpace α\nβ : Type u\ninst✝ : Nonempty β\np : TauPackage β α\nN : ℕ\nhN : IsEmpty (SatelliteConfig α N p.τ)\ni : Ordinal.{u}\nIH : ∀ y < i, y < p.lastStep → p.color y < N\nhi : i < p.lastStep\nA : Set ℕ :=\n ⋃ j,\n ⋃ (_ :\n (closedBall (p.c (p.index ↑j)) (... | [
"case ind\nα : Type u_1\ninst✝¹ : MetricSpace α\nβ : Type u\ninst✝ : Nonempty β\np : TauPackage β α\nN : ℕ\nhN : IsEmpty (SatelliteConfig α N p.τ)\ni : Ordinal.{u}\nIH : ∀ y < i, y < p.lastStep → p.color y < N\nhi : i < p.lastStep\nA : Set ℕ :=\n ⋃ j,\n ⋃ (_ :\n (closedBall (p.c (p.index ↑j)) (p.r (p.index... | have G_lt_last : ∀ n, n ≤ N → G n < p.lastStep := by
intro n hn
rcases hn.eq_or_lt with (rfl | H)
· simp only [G]; simp only [hi, if_true]
· simp only [G]; simp only [H.ne, (hg n H).left.trans hi, if_false] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.ContDiff.Convolution | {
"line": 295,
"column": 2
} | {
"line": 328,
"column": 35
} | {
"line": 330,
"column": 0
} | [
{
"pp": "𝕜 : Type u𝕜\nE : Type uE\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : RCLike 𝕜\ninst✝¹¹ : NormedSpace 𝕜 E\nG E' F P : Type uP\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedSpace 𝕜 F\ninst✝⁵ : MeasurableSpace G\nμ... | [] | induction n using ENat.nat_induction generalizing g E' F with
| zero =>
rw [WithTop.coe_zero, contDiffOn_zero] at hg ⊢
exact continuousOn_convolution_right_with_param L hk hgs hf hg
| succ n ih =>
simp only [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one, WithTop.coe_add,
WithTop.coe_natCast, Wit... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.Calculus.ContDiff.Convolution | {
"line": 295,
"column": 2
} | {
"line": 328,
"column": 35
} | {
"line": 330,
"column": 0
} | [
{
"pp": "𝕜 : Type u𝕜\nE : Type uE\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : RCLike 𝕜\ninst✝¹¹ : NormedSpace 𝕜 E\nG E' F P : Type uP\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedSpace 𝕜 F\ninst✝⁵ : MeasurableSpace G\nμ... | [] | induction n using ENat.nat_induction generalizing g E' F with
| zero =>
rw [WithTop.coe_zero, contDiffOn_zero] at hg ⊢
exact continuousOn_convolution_right_with_param L hk hgs hf hg
| succ n ih =>
simp only [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one, WithTop.coe_add,
WithTop.coe_natCast, Wit... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Convolution | {
"line": 295,
"column": 2
} | {
"line": 328,
"column": 35
} | {
"line": 330,
"column": 0
} | [
{
"pp": "𝕜 : Type u𝕜\nE : Type uE\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : RCLike 𝕜\ninst✝¹¹ : NormedSpace 𝕜 E\nG E' F P : Type uP\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedSpace 𝕜 F\ninst✝⁵ : MeasurableSpace G\nμ... | [] | induction n using ENat.nat_induction generalizing g E' F with
| zero =>
rw [WithTop.coe_zero, contDiffOn_zero] at hg ⊢
exact continuousOn_convolution_right_with_param L hk hgs hf hg
| succ n ih =>
simp only [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one, WithTop.coe_add,
WithTop.coe_natCast, Wit... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace | {
"line": 286,
"column": 4
} | {
"line": 286,
"column": 17
} | {
"line": 287,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nn : ℕ\nf : Fin n → E\nh : ∀ (i : Fin n), ‖f i‖ ≤ 2\nh' : Pairwise fun i j ↦ 1 - goodδ E ≤ ‖f i - f j‖\nfinj : Function.Injective f\ns : Finset E := Finset.image f Finset.univ\ns_card : s.card = n\nhs :... | [] | exact h' this | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.BumpFunction.SmoothApprox | {
"line": 67,
"column": 16
} | {
"line": 67,
"column": 26
} | {
"line": 67,
"column": 26
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : C(E, F)\nK : Set E\nε : ℝ\nhK : IsCompact K\nhε : 0 < ε\nthis : UniformContinuousOn (⇑f) (cthickenin... | [
"E : Type u_1\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : C(E, F)\nK : Set E\nε : ℝ\nhK : IsCompact K\nhε : 0 < ε\nthis : UniformContinuousOn (⇑f) (cthickening 1 K)\nδ : ... | lt_min_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 633,
"column": 8
} | {
"line": 633,
"column": 38
} | {
"line": 634,
"column": 8
} | [
{
"pp": "case h₁\nα : Type u_1\ninst✝⁴ : MetricSpace α\ninst✝³ : SecondCountableTopology α\ninst✝² : MeasurableSpace α\ninst✝¹ : OpensMeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nN : ℕ\nτ : ℝ\nhτ : 1 < τ\nhN : IsEmpty (SatelliteConfig α N τ)\ns : Set α\nr : α → ℝ\nrpos : ∀ x ∈ s, 0 < r x\nrle : ... | [
"α : Type u_1\ninst✝⁴ : MetricSpace α\ninst✝³ : SecondCountableTopology α\ninst✝² : MeasurableSpace α\ninst✝¹ : OpensMeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nN : ℕ\nτ : ℝ\nhτ : 1 < τ\nhN : IsEmpty (SatelliteConfig α N τ)\ns : Set α\nr : α → ℝ\nrpos : ∀ x ∈ s, 0 < r x\nrle : ∀ x ∈ s, r x ≤ 1\nhμs... | apply hw.le.trans (le_of_eq _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Calculus.ContDiff.RestrictScalars | {
"line": 66,
"column": 6
} | {
"line": 66,
"column": 39
} | {
"line": 66,
"column": 39
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NormedAlgebra 𝕜 𝕜'\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜' E\ninst✝⁴ : IsScalarTower 𝕜 𝕜' E\nF : Type u_4\ninst✝³ : NormedAdd... | [
"𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NormedAlgebra 𝕜 𝕜'\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜' E\ninst✝⁴ : IsScalarTower 𝕜 𝕜' E\nF : Type u_4\ninst✝³ : NormedAddCommGroup F\... | fderivWithin_restrictScalars_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ContDiff.RestrictScalars | {
"line": 50,
"column": 2
} | {
"line": 73,
"column": 18
} | {
"line": 75,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NormedAlgebra 𝕜 𝕜'\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜' E\ninst✝⁴ : IsScalarTower 𝕜 𝕜' E\nF : Type u_4\ninst✝³ : NormedAdd... | [] | induction n with
| zero =>
filter_upwards with a
ext m
simp
| succ n hn =>
have t₀ := h.of_le (Nat.cast_le.mpr (n.le_add_right 1))
have t₁ : ∀ᶠ (y : E) in 𝓝[s] x, ContDiffWithinAt 𝕜' (↑(n + 1)) f s y := by
nth_rw 2 [← s.insert_eq_of_mem hx]
apply h.eventually (by simp)
filter_u... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.Calculus.ContDiff.RestrictScalars | {
"line": 50,
"column": 2
} | {
"line": 73,
"column": 18
} | {
"line": 75,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NormedAlgebra 𝕜 𝕜'\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜' E\ninst✝⁴ : IsScalarTower 𝕜 𝕜' E\nF : Type u_4\ninst✝³ : NormedAdd... | [] | induction n with
| zero =>
filter_upwards with a
ext m
simp
| succ n hn =>
have t₀ := h.of_le (Nat.cast_le.mpr (n.le_add_right 1))
have t₁ : ∀ᶠ (y : E) in 𝓝[s] x, ContDiffWithinAt 𝕜' (↑(n + 1)) f s y := by
nth_rw 2 [← s.insert_eq_of_mem hx]
apply h.eventually (by simp)
filter_u... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.RestrictScalars | {
"line": 50,
"column": 2
} | {
"line": 73,
"column": 18
} | {
"line": 75,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NormedAlgebra 𝕜 𝕜'\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜' E\ninst✝⁴ : IsScalarTower 𝕜 𝕜' E\nF : Type u_4\ninst✝³ : NormedAdd... | [] | induction n with
| zero =>
filter_upwards with a
ext m
simp
| succ n hn =>
have t₀ := h.of_le (Nat.cast_le.mpr (n.le_add_right 1))
have t₁ : ∀ᶠ (y : E) in 𝓝[s] x, ContDiffWithinAt 𝕜' (↑(n + 1)) f s y := by
nth_rw 2 [← s.insert_eq_of_mem hx]
apply h.eventually (by simp)
filter_u... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 79,
"column": 10
} | {
"line": 80,
"column": 59
} | {
"line": 81,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nDu : Type u\ninst✝⁷ : NormedAddCommGroup Du\ninst✝⁶ : NormedSpace 𝕜 Du\ns : Set Du\nx : Du\nhs : UniqueDiffOn 𝕜 s\nhx : x ∈ s\nn : ℕ\nIH :\n ∀ {Eu Fu Gu : Type u} [inst : NormedAddCommGroup Eu] [inst_1 : NormedSpace 𝕜 Eu] [inst_2 : NormedAddCommGr... | [] | rw [Nat.succ_sub (Nat.lt_succ_iff.1 (Finset.mem_range.1 hi)),
← norm_iteratedFDerivWithin_fderivWithin hs hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.ContDiffHolder.Pointwise | {
"line": 188,
"column": 8
} | {
"line": 188,
"column": 94
} | {
"line": 189,
"column": 6
} | [
{
"pp": "case hq₁_bdd\nE : Type u_1\nF : Type u_2\nG : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nk : ℕ\nα : ↑I\nf : E → F\na : E\ng : F → G\nhg : ContDiffPointwiseHolderAt k ... | [] | exact (hf.contDiffAt.continuousAt_iteratedFDeriv (mod_cast hi)).norm.isBoundedUnder_le | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.ContDiffHolder.Pointwise | {
"line": 188,
"column": 8
} | {
"line": 188,
"column": 94
} | {
"line": 189,
"column": 6
} | [
{
"pp": "case hq₁_bdd\nE : Type u_1\nF : Type u_2\nG : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nk : ℕ\nα : ↑I\nf : E → F\na : E\ng : F → G\nhg : ContDiffPointwiseHolderAt k ... | [] | exact (hf.contDiffAt.continuousAt_iteratedFDeriv (mod_cast hi)).norm.isBoundedUnder_le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiffHolder.Pointwise | {
"line": 188,
"column": 8
} | {
"line": 188,
"column": 94
} | {
"line": 189,
"column": 6
} | [
{
"pp": "case hq₁_bdd\nE : Type u_1\nF : Type u_2\nG : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nk : ℕ\nα : ↑I\nf : E → F\na : E\ng : F → G\nhg : ContDiffPointwiseHolderAt k ... | [] | exact (hf.contDiffAt.continuousAt_iteratedFDeriv (mod_cast hi)).norm.isBoundedUnder_le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 716,
"column": 8
} | {
"line": 716,
"column": 57
} | {
"line": 717,
"column": 8
} | [
{
"pp": "case refine_1.left\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : HasBesicovitchCovering α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nf : α → Set ℝ\ns : Set α\nhf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonemp... | [
"case refine_1.left\nα : Type u_1\ninst✝⁵ : MetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : HasBesicovitchCovering α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nf : α → Set ℝ\ns : Set α\nhf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty\nN : ℕ\nτ... | rcases (mem_image _ _ _).1 hp with ⟨p', p'v, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Calculus.Deriv.Star | {
"line": 54,
"column": 72
} | {
"line": 57,
"column": 57
} | {
"line": 59,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : StarRing 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : StarAddMonoid F\ninst✝² : StarModule 𝕜 F\ninst✝¹ : ContinuousStar F\nf : 𝕜 → F\nx : 𝕜\ninst✝ : TrivialStar 𝕜\ns : Set 𝕜\n⊢ derivWithin (fun y ↦ sta... | [] | by
by_cases hxs : UniqueDiffWithinAt 𝕜 s x
· exact DFunLike.congr_fun (fderivWithin_star hxs) _
· simp [derivWithin_zero_of_not_uniqueDiffWithinAt hxs] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 307,
"column": 17
} | {
"line": 307,
"column": 94
} | {
"line": 307,
"column": 94
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ns : Set E\nι : Type u_2\nA' : Type u_4\ninst✝³ : NormedCommRing A'\ninst✝² : NormedAlgebra 𝕜 A'\ninst✝¹ : DecidableEq ι\ninst✝ : NormOneClass A'\nf : ι → E → A'\nN : ℕ∞ω\nhs : Uni... | [] | by simp only [← comp_apply (g := Finset.symInsertEquiv hi), comp_assoc]; simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 341,
"column": 2
} | {
"line": 342,
"column": 65
} | {
"line": 343,
"column": 2
} | [
{
"pp": "f : ℝ → ℝ\nx₀ : ℝ\nhf : deriv f x₀ > 0\nhx : f x₀ = 0\n⊢ ∀ᶠ (x : ℝ) in 𝓝[≠] x₀, sign (f x) = sign (x - x₀)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"NormedSpace.toIsBoundedSMul",
"DifferentiableAt.hasDe... | [
"f : ℝ → ℝ\nx₀ : ℝ\nhf : deriv f x₀ > 0\nhx : f x₀ = 0\nh_tendsto : Tendsto (slope f x₀) (𝓝[≠] x₀) (𝓝 (deriv f x₀))\n⊢ ∀ᶠ (x : ℝ) in 𝓝[≠] x₀, sign (f x) = sign (x - x₀)"
] | have h_tendsto := hasDerivAt_iff_tendsto_slope.mp
(differentiableAt_of_deriv_ne_zero <| ne_of_gt hf).hasDerivAt | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 390,
"column": 2
} | {
"line": 394,
"column": 57
} | {
"line": 396,
"column": 2
} | [
{
"pp": "case hi\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nFu : Type u\ninst✝³ : NormedAddCommGroup Fu\ninst✝² : NormedSpace 𝕜 Fu\nf : E → Fu\ns : Set E\nt : Set Fu\nx : E\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : Ma... | [
"case hi\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nFu : Type u\ninst✝³ : NormedAddCommGroup Fu\ninst✝² : NormedSpace 𝕜 Fu\nf : E → Fu\ns : Set E\nt : Set Fu\nx : E\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : MapsTo f s t\n... | have J : ∀ i, ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 f s) s x‖ ≤ D ^ (n - i + 1) := by
intro i
have : ‖iteratedFDerivWithin 𝕜 (n - i + 1) f s x‖ ≤ D ^ (n - i + 1) :=
hD (n - i + 1) (by simp) (Nat.succ_le_succ tsub_le_self)
simpa [iteratedFDerivWithin_succ_eq_comp_right hs hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 938,
"column": 6
} | {
"line": 942,
"column": 48
} | {
"line": 943,
"column": 2
} | [
{
"pp": "case refine_3.inr\nα : Type u_1\ninst✝⁶ : MetricSpace α\ninst✝⁵ : SecondCountableTopology α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : OpensMeasurableSpace α\ninst✝² : HasBesicovitchCovering α\nμ : Measure α\ninst✝¹ : SFinite μ\ninst✝ : μ.OuterRegular\nε : ℝ≥0∞\nhε : ε ≠ 0\nf : α → Set ℝ\ns : Set α\nhf : ∀ ... | [] | have h'x : x ∈ s' := by
simp only [mem_iUnion, mem_image] at hx
rcases hx with ⟨i, y, _, rfl⟩
exact y.2
simp only [r, if_pos h'x, (hr1 x h'x).1.1] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 938,
"column": 6
} | {
"line": 942,
"column": 48
} | {
"line": 943,
"column": 2
} | [
{
"pp": "case refine_3.inr\nα : Type u_1\ninst✝⁶ : MetricSpace α\ninst✝⁵ : SecondCountableTopology α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : OpensMeasurableSpace α\ninst✝² : HasBesicovitchCovering α\nμ : Measure α\ninst✝¹ : SFinite μ\ninst✝ : μ.OuterRegular\nε : ℝ≥0∞\nhε : ε ≠ 0\nf : α → Set ℝ\ns : Set α\nhf : ∀ ... | [] | have h'x : x ∈ s' := by
simp only [mem_iUnion, mem_image] at hx
rcases hx with ⟨i, y, _, rfl⟩
exact y.2
simp only [r, if_pos h'x, (hr1 x h'x).1.1] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.DifferentialForm.Basic | {
"line": 255,
"column": 4
} | {
"line": 256,
"column": 90
} | {
"line": 257,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nn : ℕ\nr : WithTop ℕ∞\ns : Set E\nx ... | [] | ext v
simp +unfoldPartialApp [alternatizeUncurryFin_apply, Fin.removeNth, Function.comp_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.DifferentialForm.Basic | {
"line": 255,
"column": 4
} | {
"line": 256,
"column": 90
} | {
"line": 257,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nn : ℕ\nr : WithTop ℕ∞\ns : Set E\nx ... | [] | ext v
simp +unfoldPartialApp [alternatizeUncurryFin_apply, Fin.removeNth, Function.comp_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.DifferentialForm.VectorField | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 6
} | {
"line": 95,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\ns : Set E\nx : E\nω : E → E [⋀^Fin (n + 1)]→L[𝕜] F\nV : Fin (n + 2) → E → E\nhω : DifferentiableWith... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\ns : Set E\nx : E\nω : E → E [⋀^Fin (n + 1)]→L[𝕜] F\nV : Fin (n + 2) → E → E\nhω : DifferentiableWithinAt 𝕜 ω s ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Analysis.Calculus.VectorField | {
"line": 227,
"column": 34
} | {
"line": 227,
"column": 40
} | {
"line": 227,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nV W : E → E\ns : Set E\nx : E\nf : E → F\nhf : ContDiffWithinAt 𝕜 2 f s x\nhsymm : IsSymmSndFDerivWithinAt ... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Analysis.Calculus.VectorField | {
"line": 227,
"column": 34
} | {
"line": 227,
"column": 40
} | {
"line": 227,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nV W : E → E\ns : Set E\nx : E\nf : E → F\nhf : ContDiffWithinAt 𝕜 2 f s x\nhsymm : IsSymmSndFDerivWithinAt ... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.VectorField | {
"line": 227,
"column": 34
} | {
"line": 227,
"column": 40
} | {
"line": 227,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nV W : E → E\ns : Set E\nx : E\nf : E → F\nhf : ContDiffWithinAt 𝕜 2 f s x\nhsymm : IsSymmSndFDerivWithinAt ... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.VectorField | {
"line": 265,
"column": 2
} | {
"line": 265,
"column": 71
} | {
"line": 266,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nV W : E → E\nx : E\nf : E → F\nn : ℕ∞ω\nhf : ContDiffAt 𝕜 n f x\nhn : minSmoothness 𝕜 2 ≤ n\nhW : Differen... | [
"case hf\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nV W : E → E\nx : E\nf : E → F\nn : ℕ∞ω\nhf : ContDiffAt 𝕜 n f x\nhn : minSmoothness 𝕜 2 ≤ n\nhW : Differentia... | apply fderiv_apply_lieBracket_of_isSymmSndFDerivAt <;> try assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.Calculus.VectorField | {
"line": 596,
"column": 4
} | {
"line": 597,
"column": 16
} | {
"line": 599,
"column": 0
} | [
{
"pp": "case hx\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace E\nf : E → F\nV : F → F\ns : Set E\nt : Set F\nx : E\nhV : DifferentiableWit... | [] | have hMx : M x = fderivWithin 𝕜 f s x := by apply mem_of_mem_nhdsWithin hx hM
simp [← hMx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.VectorField | {
"line": 596,
"column": 4
} | {
"line": 597,
"column": 16
} | {
"line": 599,
"column": 0
} | [
{
"pp": "case hx\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace E\nf : E → F\nV : F → F\ns : Set E\nt : Set F\nx : E\nhV : DifferentiableWit... | [] | have hMx : M x = fderivWithin 𝕜 f s x := by apply mem_of_mem_nhdsWithin hx hM
simp [← hMx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ImplicitFunction.ProdDomain | {
"line": 66,
"column": 6
} | {
"line": 70,
"column": 11
} | {
"line": 72,
"column": 0
} | [
{
"pp": "case codisjoint\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedA... | [] | rw [Submodule.codisjoint_iff_exists_add_eq]
intro v
have ⟨y, hy⟩ := if₂u.surjective (f'u v)
use v - (0, y), (0, y)
aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ImplicitFunction.ProdDomain | {
"line": 66,
"column": 6
} | {
"line": 70,
"column": 11
} | {
"line": 72,
"column": 0
} | [
{
"pp": "case codisjoint\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedA... | [] | rw [Submodule.codisjoint_iff_exists_add_eq]
intro v
have ⟨y, hy⟩ := if₂u.surjective (f'u v)
use v - (0, y), (0, y)
aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.IteratedDeriv.Analytic | {
"line": 42,
"column": 73
} | {
"line": 43,
"column": 46
} | {
"line": 44,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : CharZero 𝕜\ninst✝ : CompleteSpace 𝕜\nz₀ : 𝕜\nR R₁ : 𝕜 → 𝕜\nhf1 : ∀ (z : 𝕜), AnalyticAt 𝕜 R₁ z\nk : ℕ\nIH :\n ∀ {t : ℕ},\n (∀ (z : 𝕜), R z = (z - z₀) ^ (k + t) * R₁ z) →\n ∃ R₂,\n (∀ (z : 𝕜), AnalyticAt 𝕜 R₂ z) ∧\n ... | [] | by
rw [iteratedDeriv_succ, funext hR₂_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 60
} | {
"line": 166,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ng : 𝕜 → E\nf : 𝕜 → 𝕜\nx : 𝕜\nhg : ContDiffAt 𝕜 2 g (f x)\nhf : ContDiffAt 𝕜 2 f x\n⊢ iteratedDeriv 2 (g ∘ f) x = deriv f x ^ 2 • iteratedDeriv 2 g (f x) + iteratedDeriv 2 f x... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ng : 𝕜 → E\nf : 𝕜 → 𝕜\nx : 𝕜\nhg : ContDiffAt 𝕜 2 g (f x)\nhf : ContDiffAt 𝕜 2 f x\n⊢ iteratedDerivWithin 2 (g ∘ f) univ x =\n derivWithin f univ x ^ 2 • iteratedDerivWithin 2 g univ (... | simp only [← iteratedDerivWithin_univ, ← derivWithin_univ] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno | {
"line": 186,
"column": 2
} | {
"line": 186,
"column": 60
} | {
"line": 187,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ng : 𝕜 → E\nf : 𝕜 → 𝕜\nx : 𝕜\nhg : ContDiffAt 𝕜 3 g (f x)\nhf : ContDiffAt 𝕜 3 f x\n⊢ iteratedDeriv 3 (g ∘ f) x =\n deriv f x ^ 3 • iteratedDeriv 3 g (f x) + 3 • iteratedDe... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ng : 𝕜 → E\nf : 𝕜 → 𝕜\nx : 𝕜\nhg : ContDiffAt 𝕜 3 g (f x)\nhf : ContDiffAt 𝕜 3 f x\n⊢ iteratedDerivWithin 3 (g ∘ f) univ x =\n derivWithin f univ x ^ 3 • iteratedDerivWithin 3 g univ (... | simp only [← iteratedDerivWithin_univ, ← derivWithin_univ] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 60
} | {
"line": 227,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\ng f : 𝕜 → 𝕜\nx : 𝕜\nhg : ContDiffAt 𝕜 2 g (f x)\nhf : ContDiffAt 𝕜 2 f x\n⊢ iteratedDeriv 2 (g ∘ f) x = iteratedDeriv 2 g (f x) * deriv f x ^ 2 + deriv g (f x) * iteratedDeriv 2 f x",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants... | [
"𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\ng f : 𝕜 → 𝕜\nx : 𝕜\nhg : ContDiffAt 𝕜 2 g (f x)\nhf : ContDiffAt 𝕜 2 f x\n⊢ iteratedDerivWithin 2 (g ∘ f) univ x =\n iteratedDerivWithin 2 g univ (f x) * derivWithin f univ x ^ 2 +\n derivWithin g univ (f x) * iteratedDerivWithin 2 f univ x"
] | simp only [← iteratedDerivWithin_univ, ← derivWithin_univ] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 60
} | {
"line": 247,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\ng f : 𝕜 → 𝕜\nx : 𝕜\nhg : ContDiffAt 𝕜 3 g (f x)\nhf : ContDiffAt 𝕜 3 f x\n⊢ iteratedDeriv 3 (g ∘ f) x =\n iteratedDeriv 3 g (f x) * deriv f x ^ 3 + 3 * iteratedDeriv 2 g (f x) * iteratedDeriv 2 f x * deriv f x +\n deriv g (f x) * iteratedD... | [
"𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\ng f : 𝕜 → 𝕜\nx : 𝕜\nhg : ContDiffAt 𝕜 3 g (f x)\nhf : ContDiffAt 𝕜 3 f x\n⊢ iteratedDerivWithin 3 (g ∘ f) univ x =\n iteratedDerivWithin 3 g univ (f x) * derivWithin f univ x ^ 3 +\n 3 * iteratedDerivWithin 2 g univ (f x) * iteratedDerivWithin 2 f u... | simp only [← iteratedDerivWithin_univ, ← derivWithin_univ] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.LHopital | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 82
} | {
"line": 141,
"column": 2
} | [
{
"pp": "a : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x\nhgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x\nhg' : ∀ x ∈ Ioi a, g' x ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) atTop l\na' : ℝ\nhaa' : a < a'\nha' : 0 < a'\n⊢... | [
"a : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x\nhgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x\nhg' : ∀ x ∈ Ioi a, g' x ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) atTop l\na' : ℝ\nhaa' : a < a'\nha' : 0 < a'\nfact1 : ∀ x ∈... | have fact1 : ∀ x : ℝ, x ∈ Ioo 0 a'⁻¹ → x ≠ 0 := fun _ hx => (ne_of_lt hx.1).symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 164,
"column": 8
} | {
"line": 164,
"column": 38
} | {
"line": 164,
"column": 38
} | [
{
"pp": "s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ... | [
"s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ∈ s₁ ∧ x < y... | accPt_principal_iff_nhdsWithin | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.LHopital | {
"line": 324,
"column": 4
} | {
"line": 325,
"column": 47
} | {
"line": 325,
"column": 48
} | [
{
"pp": "case hff'\na : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt f (f' x) x\nhgg' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt g (g' x) x\nhg' : ∀ᶠ (x : ℝ) in 𝓝 a, g' x ≠ 0\nhfa : Tendsto f (𝓝 a) (𝓝 0)\nhga : Tendsto g (𝓝 a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝 a) l\n⊢ ∀ᶠ (x... | [
"case hff'\na : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt f (f' x) x\nhgg' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt g (g' x) x\nhg' : ∀ᶠ (x : ℝ) in 𝓝 a, g' x ≠ 0\nhfa : Tendsto f (𝓝 a) (𝓝 0)\nhga : Tendsto g (𝓝 a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝 a) l\n⊢ ∀ᶠ (x : ℝ) in 𝓝 ... | (first | apply eventually_nhdsWithin_of_eventually_nhds |
apply tendsto_nhdsWithin_of_tendsto_nhds) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.Analysis.Calculus.LHopital | {
"line": 324,
"column": 4
} | {
"line": 325,
"column": 47
} | {
"line": 325,
"column": 48
} | [
{
"pp": "case hgg'\na : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt f (f' x) x\nhgg' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt g (g' x) x\nhg' : ∀ᶠ (x : ℝ) in 𝓝 a, g' x ≠ 0\nhfa : Tendsto f (𝓝 a) (𝓝 0)\nhga : Tendsto g (𝓝 a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝 a) l\n⊢ ∀ᶠ (x... | [
"case hgg'\na : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt f (f' x) x\nhgg' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt g (g' x) x\nhg' : ∀ᶠ (x : ℝ) in 𝓝 a, g' x ≠ 0\nhfa : Tendsto f (𝓝 a) (𝓝 0)\nhga : Tendsto g (𝓝 a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝 a) l\n⊢ ∀ᶠ (x : ℝ) in 𝓝 ... | (first | apply eventually_nhdsWithin_of_eventually_nhds |
apply tendsto_nhdsWithin_of_tendsto_nhds) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.Analysis.Calculus.LHopital | {
"line": 324,
"column": 4
} | {
"line": 325,
"column": 47
} | {
"line": 325,
"column": 48
} | [
{
"pp": "case hg'\na : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt f (f' x) x\nhgg' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt g (g' x) x\nhg' : ∀ᶠ (x : ℝ) in 𝓝 a, g' x ≠ 0\nhfa : Tendsto f (𝓝 a) (𝓝 0)\nhga : Tendsto g (𝓝 a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝 a) l\n⊢ ∀ᶠ (x ... | [
"case hg'\na : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt f (f' x) x\nhgg' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt g (g' x) x\nhg' : ∀ᶠ (x : ℝ) in 𝓝 a, g' x ≠ 0\nhfa : Tendsto f (𝓝 a) (𝓝 0)\nhga : Tendsto g (𝓝 a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝 a) l\n⊢ ∀ᶠ (x : ℝ) in 𝓝 a... | (first | apply eventually_nhdsWithin_of_eventually_nhds |
apply tendsto_nhdsWithin_of_tendsto_nhds) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.Analysis.Calculus.LHopital | {
"line": 324,
"column": 4
} | {
"line": 325,
"column": 47
} | {
"line": 325,
"column": 48
} | [
{
"pp": "case hfa\na : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt f (f' x) x\nhgg' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt g (g' x) x\nhg' : ∀ᶠ (x : ℝ) in 𝓝 a, g' x ≠ 0\nhfa : Tendsto f (𝓝 a) (𝓝 0)\nhga : Tendsto g (𝓝 a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝 a) l\n⊢ Tendst... | [
"case hfa\na : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt f (f' x) x\nhgg' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt g (g' x) x\nhg' : ∀ᶠ (x : ℝ) in 𝓝 a, g' x ≠ 0\nhfa : Tendsto f (𝓝 a) (𝓝 0)\nhga : Tendsto g (𝓝 a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝 a) l\n⊢ Tendsto f (𝓝 a) (... | (first | apply eventually_nhdsWithin_of_eventually_nhds |
apply tendsto_nhdsWithin_of_tendsto_nhds) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.Analysis.Calculus.LHopital | {
"line": 324,
"column": 4
} | {
"line": 325,
"column": 47
} | {
"line": 325,
"column": 48
} | [
{
"pp": "case hga\na : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt f (f' x) x\nhgg' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt g (g' x) x\nhg' : ∀ᶠ (x : ℝ) in 𝓝 a, g' x ≠ 0\nhfa : Tendsto f (𝓝 a) (𝓝 0)\nhga : Tendsto g (𝓝 a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝 a) l\n⊢ Tendst... | [
"case hga\na : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt f (f' x) x\nhgg' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt g (g' x) x\nhg' : ∀ᶠ (x : ℝ) in 𝓝 a, g' x ≠ 0\nhfa : Tendsto f (𝓝 a) (𝓝 0)\nhga : Tendsto g (𝓝 a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝 a) l\n⊢ Tendsto g (𝓝 a) (... | (first | apply eventually_nhdsWithin_of_eventually_nhds |
apply tendsto_nhdsWithin_of_tendsto_nhds) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.Analysis.Calculus.LHopital | {
"line": 324,
"column": 4
} | {
"line": 325,
"column": 47
} | {
"line": 325,
"column": 48
} | [
{
"pp": "case hdiv\na : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt f (f' x) x\nhgg' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt g (g' x) x\nhg' : ∀ᶠ (x : ℝ) in 𝓝 a, g' x ≠ 0\nhfa : Tendsto f (𝓝 a) (𝓝 0)\nhga : Tendsto g (𝓝 a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝 a) l\n⊢ Tends... | [
"case hdiv\na : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt f (f' x) x\nhgg' : ∀ᶠ (x : ℝ) in 𝓝 a, HasDerivAt g (g' x) x\nhg' : ∀ᶠ (x : ℝ) in 𝓝 a, g' x ≠ 0\nhfa : Tendsto f (𝓝 a) (𝓝 0)\nhga : Tendsto g (𝓝 a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝 a) l\n⊢ Tendsto (fun x ↦ ... | (first | apply eventually_nhdsWithin_of_eventually_nhds |
apply tendsto_nhdsWithin_of_tendsto_nhds) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.Analysis.Calculus.LHopital | {
"line": 434,
"column": 4
} | {
"line": 435,
"column": 47
} | {
"line": 435,
"column": 48
} | [
{
"pp": "case hdf\na : ℝ\nl : Filter ℝ\nf g : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0\nhfa : Tendsto f (𝓝 a) (𝓝 0)\nhga : Tendsto g (𝓝 a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l\n⊢ ∀ᶠ (x : ℝ) in 𝓝[≠] a, DifferentiableAt ℝ f x",... | [
"case hdf\na : ℝ\nl : Filter ℝ\nf g : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0\nhfa : Tendsto f (𝓝 a) (𝓝 0)\nhga : Tendsto g (𝓝 a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l\n⊢ ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x"
] | (first | apply eventually_nhdsWithin_of_eventually_nhds |
apply tendsto_nhdsWithin_of_tendsto_nhds) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.Analysis.Calculus.LHopital | {
"line": 434,
"column": 4
} | {
"line": 435,
"column": 47
} | {
"line": 435,
"column": 48
} | [
{
"pp": "case hg'\na : ℝ\nl : Filter ℝ\nf g : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0\nhfa : Tendsto f (𝓝 a) (𝓝 0)\nhga : Tendsto g (𝓝 a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l\n⊢ ∀ᶠ (x : ℝ) in 𝓝[≠] a, deriv g x ≠ 0",
"ppT... | [
"case hg'\na : ℝ\nl : Filter ℝ\nf g : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0\nhfa : Tendsto f (𝓝 a) (𝓝 0)\nhga : Tendsto g (𝓝 a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l\n⊢ ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0"
] | (first | apply eventually_nhdsWithin_of_eventually_nhds |
apply tendsto_nhdsWithin_of_tendsto_nhds) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.Analysis.Calculus.LHopital | {
"line": 434,
"column": 4
} | {
"line": 435,
"column": 47
} | {
"line": 435,
"column": 48
} | [
{
"pp": "case hfa\na : ℝ\nl : Filter ℝ\nf g : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0\nhfa : Tendsto f (𝓝 a) (𝓝 0)\nhga : Tendsto g (𝓝 a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l\n⊢ Tendsto f (𝓝[≠] a) (𝓝 0)",
"ppTerm": "?hf... | [
"case hfa\na : ℝ\nl : Filter ℝ\nf g : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0\nhfa : Tendsto f (𝓝 a) (𝓝 0)\nhga : Tendsto g (𝓝 a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l\n⊢ Tendsto f (𝓝 a) (𝓝 0)"
] | (first | apply eventually_nhdsWithin_of_eventually_nhds |
apply tendsto_nhdsWithin_of_tendsto_nhds) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.Analysis.Calculus.LHopital | {
"line": 434,
"column": 4
} | {
"line": 435,
"column": 47
} | {
"line": 435,
"column": 48
} | [
{
"pp": "case hga\na : ℝ\nl : Filter ℝ\nf g : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0\nhfa : Tendsto f (𝓝 a) (𝓝 0)\nhga : Tendsto g (𝓝 a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l\n⊢ Tendsto g (𝓝[≠] a) (𝓝 0)",
"ppTerm": "?hg... | [
"case hga\na : ℝ\nl : Filter ℝ\nf g : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0\nhfa : Tendsto f (𝓝 a) (𝓝 0)\nhga : Tendsto g (𝓝 a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l\n⊢ Tendsto g (𝓝 a) (𝓝 0)"
] | (first | apply eventually_nhdsWithin_of_eventually_nhds |
apply tendsto_nhdsWithin_of_tendsto_nhds) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.Analysis.Calculus.LHopital | {
"line": 434,
"column": 4
} | {
"line": 435,
"column": 47
} | {
"line": 435,
"column": 48
} | [
{
"pp": "case hdiv\na : ℝ\nl : Filter ℝ\nf g : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0\nhfa : Tendsto f (𝓝 a) (𝓝 0)\nhga : Tendsto g (𝓝 a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l\n⊢ Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝[≠]... | [
"case hdiv\na : ℝ\nl : Filter ℝ\nf g : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in 𝓝 a, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in 𝓝 a, deriv g x ≠ 0\nhfa : Tendsto f (𝓝 a) (𝓝 0)\nhga : Tendsto g (𝓝 a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l\n⊢ Tendsto (fun x ↦ deriv f x / deriv g x) (𝓝 a) l"
] | (first | apply eventually_nhdsWithin_of_eventually_nhds |
apply tendsto_nhdsWithin_of_tendsto_nhds) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 451,
"column": 2
} | {
"line": 451,
"column": 37
} | {
"line": 452,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nE : Type u_3\ninst✝³ : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : l.NeBot\ninst✝ : l.IsCountablyGenerated\nφ : ι → Set α\nhφ : AECover μ l φ\nf : α → E\nI : ℝ\nhfi : ∀ (i : ι), IntegrableOn f (φ i) μ\nhbounded : ∀ᶠ (i : ι) in l, (... | [
"α : Type u_1\nι : Type u_2\nE : Type u_3\ninst✝³ : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : l.NeBot\ninst✝ : l.IsCountablyGenerated\nφ : ι → Set α\nhφ : AECover μ l φ\nf : α → E\nI : ℝ\nhfi : ∀ (i : ι), IntegrableOn f (φ i) μ\nhbounded : ∀ᶠ (i : ι) in l, (∫⁻ (a : α) i... | refine hbounded.mono fun i hi => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 799,
"column": 2
} | {
"line": 799,
"column": 6
} | {
"line": 800,
"column": 2
} | [
{
"pp": "E : Type u_1\nf f' : ℝ → E\na : ℝ\nm : E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhcont✝ : ContinuousWithinAt f (Ici a) a\nhderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Ioi a) volume\nhf : Tendsto f atTop (𝓝 m)\nhcont : ContinuousOn f (... | [
"E : Type u_1\nf f' : ℝ → E\na : ℝ\nm : E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhcont✝ : ContinuousWithinAt f (Ici a) a\nhderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Ioi a) volume\nhf : Tendsto f atTop (𝓝 m)\nhcont : ContinuousOn f (Ici a)\nx : ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 825,
"column": 83
} | {
"line": 825,
"column": 91
} | {
"line": 825,
"column": 91
} | [
{
"pp": "E : Type u_1\nf : ℝ → E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhf : ContDiff ℝ 1 f\nh2f : HasCompactSupport f\nb : ℝ\nthis : ∀ x ∈ Ioi b, HasDerivAt f (deriv f x) x\n⊢ ?m.74 - f b = -f b",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
... | [
"E : Type u_1\nf : ℝ → E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhf : ContDiff ℝ 1 f\nh2f : HasCompactSupport f\nb : ℝ\nthis : ∀ x ∈ Ioi b, HasDerivAt f (deriv f x) x\n⊢ -f b = -f b",
"case f'int\nE : Type u_1\nf : ℝ → E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpa... | zero_sub | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 849,
"column": 6
} | {
"line": 849,
"column": 10
} | {
"line": 850,
"column": 6
} | [
{
"pp": "g g' : ℝ → ℝ\na l : ℝ\nhcont✝ : ContinuousWithinAt g (Ici a) a\nhderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x\ng'pos : ∀ x ∈ Ioi a, 0 ≤ g' x\nhg : Tendsto g atTop (𝓝 l)\nhcont : ContinuousOn g (Ici a)\nx : ℝ\nhx : x ∈ Ioi a\nh'x : a ≤ id x\n⊢ g x - g a = ∫ (y : ℝ) in a..id x, g' y",
"ppTerm": "?m.20... | [
"g g' : ℝ → ℝ\na l : ℝ\nhcont✝ : ContinuousWithinAt g (Ici a) a\nhderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x\ng'pos : ∀ x ∈ Ioi a, 0 ≤ g' x\nhg : Tendsto g atTop (𝓝 l)\nhcont : ContinuousOn g (Ici a)\nx : ℝ\nhx : x ∈ Ioi a\nh'x : a ≤ id x\n⊢ ∫ (y : ℝ) in a..id x, g' y = g x - g a"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 995,
"column": 2
} | {
"line": 995,
"column": 6
} | {
"line": 996,
"column": 2
} | [
{
"pp": "E : Type u_1\nf f' : ℝ → E\na : ℝ\nm : E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhcont✝ : ContinuousWithinAt f (Iic a) a\nhderiv : ∀ x ∈ Iio a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nhf : Tendsto f atBot (𝓝 m)\nhcont : ContinuousOn f (... | [
"E : Type u_1\nf f' : ℝ → E\na : ℝ\nm : E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhcont✝ : ContinuousWithinAt f (Iic a) a\nhderiv : ∀ x ∈ Iio a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nhf : Tendsto f atBot (𝓝 m)\nhcont : ContinuousOn f (Iic a)\nx : ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Analysis.Complex.RemovableSingularity | {
"line": 150,
"column": 62
} | {
"line": 150,
"column": 68
} | {
"line": 150,
"column": 69
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Analysis.Complex.RemovableSingularity | {
"line": 150,
"column": 62
} | {
"line": 150,
"column": 68
} | {
"line": 150,
"column": 69
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.RemovableSingularity | {
"line": 150,
"column": 62
} | {
"line": 150,
"column": 68
} | {
"line": 150,
"column": 69
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ParametricIntervalIntegral | {
"line": 89,
"column": 2
} | {
"line": 90,
"column": 27
} | {
"line": 91,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nμ : Measure ℝ\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace 𝕜 E\na b : ℝ\nbound : ℝ → ℝ\nF : 𝕜 → ℝ → E\nF' : ℝ → E\nx₀ : 𝕜\ns : Set 𝕜\nhs : s ∈ 𝓝 x₀\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restr... | [
"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nμ : Measure ℝ\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace 𝕜 E\na b : ℝ\nbound : ℝ → ℝ\nF : 𝕜 → ℝ → E\nF' : ℝ → E\nx₀ : 𝕜\ns : Set 𝕜\nhs : s ∈ 𝓝 x₀\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))... | have := hasDerivAt_integral_of_dominated_loc_of_lip hs hF_meas hF_int hF'_meas h_lipsch
bound_integrable h_diff | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 203,
"column": 6
} | {
"line": 206,
"column": 45
} | {
"line": 207,
"column": 2
} | [
{
"pp": "case inr\nH✝ : Type u_1\ninst✝¹ : TopologicalSpace H✝\nH : Type u_2\ninst✝ : TopologicalSpace H\ne e' : OpenPartialHomeomorph H H\nhe' : e' ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}\nhe : e ∈ {e | e.source = ∅}\n⊢ e ≫ₕ e' ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}",
"ppTerm":... | [] | have : (e ≫ₕ e').source ⊆ e.source := sep_subset _ _
rw [he] at this
have : e ≫ₕ e' ∈ { e : OpenPartialHomeomorph H H | e.source = ∅ } := eq_bot_iff.2 this
exact (mem_union _ _ _).2 (Or.inr this) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 203,
"column": 6
} | {
"line": 206,
"column": 45
} | {
"line": 207,
"column": 2
} | [
{
"pp": "case inr\nH✝ : Type u_1\ninst✝¹ : TopologicalSpace H✝\nH : Type u_2\ninst✝ : TopologicalSpace H\ne e' : OpenPartialHomeomorph H H\nhe' : e' ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}\nhe : e ∈ {e | e.source = ∅}\n⊢ e ≫ₕ e' ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}",
"ppTerm":... | [] | have : (e ≫ₕ e').source ⊆ e.source := sep_subset _ _
rw [he] at this
have : e ≫ₕ e' ∈ { e : OpenPartialHomeomorph H H | e.source = ∅ } := eq_bot_iff.2 this
exact (mem_union _ _ _).2 (Or.inr this) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 660,
"column": 2
} | {
"line": 667,
"column": 51
} | {
"line": 668,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nhf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x\nh'f' : ∀ x ∈ s, (... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nhf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x\nh'f' : ∀ x ∈ s, (f' x).det = ... | have B : Tendsto (fun ε : ℝ≥0 => (ε : ℝ≥0∞) * μ (closedBall 0 R)) (𝓝[>] 0) (𝓝 0) := by
have :
Tendsto (fun ε : ℝ≥0 => (ε : ℝ≥0∞) * μ (closedBall 0 R)) (𝓝 0)
(𝓝 (((0 : ℝ≥0) : ℝ≥0∞) * μ (closedBall 0 R))) :=
ENNReal.Tendsto.mul_const (ENNReal.tendsto_coe.2 tendsto_id)
(Or.inr measure_c... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 214,
"column": 4
} | {
"line": 237,
"column": 23
} | {
"line": 238,
"column": 2
} | [
{
"pp": "H✝ : Type u_1\ninst✝¹ : TopologicalSpace H✝\nH : Type u_2\ninst✝ : TopologicalSpace H\ne : OpenPartialHomeomorph H H\nhe : ∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}\n⊢ e ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}",
"ppT... | [] | rcases e.source.eq_empty_or_nonempty with h | h
· right
exact h
· left
rcases h with ⟨x, hx⟩
rcases he x hx with ⟨s, open_s, xs, hs⟩
have x's : x ∈ (e.restr s).source := by
rw [restr_source, open_s.interior_eq]
exact ⟨hx, xs⟩
rcases hs with hs | hs
· replace h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 214,
"column": 4
} | {
"line": 237,
"column": 23
} | {
"line": 238,
"column": 2
} | [
{
"pp": "H✝ : Type u_1\ninst✝¹ : TopologicalSpace H✝\nH : Type u_2\ninst✝ : TopologicalSpace H\ne : OpenPartialHomeomorph H H\nhe : ∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}\n⊢ e ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}",
"ppT... | [] | rcases e.source.eq_empty_or_nonempty with h | h
· right
exact h
· left
rcases h with ⟨x, hx⟩
rcases he x hx with ⟨s, open_s, xs, hs⟩
have x's : x ∈ (e.restr s).source := by
rw [restr_source, open_s.interior_eq]
exact ⟨hx, xs⟩
rcases hs with hs | hs
· replace h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 247,
"column": 11
} | {
"line": 247,
"column": 28
} | {
"line": 247,
"column": 29
} | [
{
"pp": "case inr\nH✝ : Type u_1\ninst✝¹ : TopologicalSpace H✝\nH : Type u_2\ninst✝ : TopologicalSpace H\ne e' : OpenPartialHomeomorph H H\nhe'e : e' ≈ e\nhe✝ : e ∈ {e | e.source = ∅}\nhe : e.source = ∅\n⊢ e' ∈ {e | e.source = ∅}",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"case inr\nH✝ : Type u_1\ninst✝¹ : TopologicalSpace H✝\nH : Type u_2\ninst✝ : TopologicalSpace H\ne e' : OpenPartialHomeomorph H H\nhe'e : e' ≈ e\nhe✝ : e ∈ {e | e.source = ∅}\nhe : e.source = ∅\n⊢ e'.source = ∅"
] | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 252,
"column": 12
} | {
"line": 264,
"column": 19
} | {
"line": 266,
"column": 0
} | [
{
"pp": "H : Type u_1\ninst✝ : TopologicalSpace H\n⊢ ∀ (a : StructureGroupoid H), idGroupoid H ≤ a",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"idGroupoid",
"congrArg",
"StructureGroupoid.locality",
"False.elim",
"PartialHomeo... | [] | by
intro u f hf
have hf :
f ∈ {OpenPartialHomeomorph.refl H} ∪ { e : OpenPartialHomeomorph H H | e.source = ∅ } :=
hf
simp only [singleton_union, mem_setOf_eq, mem_insert_iff] at hf
rcases hf with hf | hf
· rw [hf]
apply u.id_mem
· apply u.locality
intro x hx
rw [... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.HasGroupoid | {
"line": 451,
"column": 2
} | {
"line": 451,
"column": 79
} | {
"line": 452,
"column": 2
} | [
{
"pp": "H : Type u\nM : Type u_2\ninst✝⁴ : TopologicalSpace H\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nG : StructureGroupoid H\ne : OpenPartialHomeomorph M H\nhe : e ∈ atlas H M\nhs : Nonempty ↑e.source\ninst✝¹ : HasGroupoid M G\ninst✝ : ClosedUnderRestriction G\ns : Opens M := { carrier := e.s... | [
"H : Type u\nM : Type u_2\ninst✝⁴ : TopologicalSpace H\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nG : StructureGroupoid H\ne : OpenPartialHomeomorph M H\nhe : e ∈ atlas H M\nhs : Nonempty ↑e.source\ninst✝¹ : HasGroupoid M G\ninst✝ : ClosedUnderRestriction G\ns : Opens M := { carrier := e.source, is_op... | rw [OpenPartialHomeomorph.subtypeRestr_def, OpenPartialHomeomorph.trans_refl] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 313,
"column": 93
} | {
"line": 315,
"column": 57
} | {
"line": 317,
"column": 0
} | [
{
"pp": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : TopologicalSpace H'\ninst✝¹ : TopologicalSpace M'\ninst✝ : ChartedSpace H' M'\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : OpenPartialHo... | [] | by
rw [← liftPropWithinAt_indep_chart_aux' hG he' xe' hf' xf' hgs,
liftPropWithinAt_indep_chart_aux' hG he xe hf xf hgs] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 380,
"column": 2
} | {
"line": 380,
"column": 75
} | {
"line": 381,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : TopologicalSpace H\ninst✝¹ : TopologicalSpace M\nn : WithTop ℕ∞\nI : ModelWithCorners 𝕜 E H\ne e' : OpenPartialHomeomorph M H\ninst✝ : Charte... | [
"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : TopologicalSpace H\ninst✝¹ : TopologicalSpace M\nn : WithTop ℕ∞\nI : ModelWithCorners 𝕜 E H\ne e' : OpenPartialHomeomorph M H\ninst✝ : ChartedSpace H M\n... | apply (I.contDiffOn_extendCoordChange he he' x hx).mono_of_mem_nhdsWithin | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
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