module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute | {
"line": 62,
"column": 4
} | {
"line": 65,
"column": 43
} | {
"line": 67,
"column": 0
} | [
{
"pp": "case frequently\n𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : Ring A\ninst✝⁵ : StarRing A\ninst✝⁴ : Algebra 𝕜 A\ninst✝³ : TopologicalSpace A\ninst✝² : ContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : IsSemitopologicalRing A\ninst✝ : T2Space A\na b : A\nha : p a\nhb₁ : Commute a... | [] | rw [commute_iff_eq, ← Set.mem_setOf (p := fun x => x * b = b * x),
← (isClosed_eq (by fun_prop) (by fun_prop)).closure_eq]
apply mem_closure_of_frequently_of_tendsto hf
exact cfcHom_continuous ha |>.tendsto _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute | {
"line": 145,
"column": 21
} | {
"line": 145,
"column": 33
} | {
"line": 145,
"column": 33
} | [
{
"pp": "case mul\n𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : Module 𝕜 A\ninst✝⁵ : IsScalarTower 𝕜 A A\ninst✝⁴ : SMulCommClass 𝕜 A A\ninst✝³ : TopologicalSpace A\ninst✝² : NonUnitalContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : IsTopo... | [
"case mul\n𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : Module 𝕜 A\ninst✝⁵ : IsScalarTower 𝕜 A A\ninst✝⁴ : SMulCommClass 𝕜 A A\ninst✝³ : TopologicalSpace A\ninst✝² : NonUnitalContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : IsTopologicalRing ... | rw [map_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 804,
"column": 2
} | {
"line": 804,
"column": 75
} | {
"line": 805,
"column": 2
} | [
{
"pp": "𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NonUnitalNormedRing A\ninst✝⁶ : StarRing A\ninst✝⁵ : NormedSpace 𝕜 A\ninst✝⁴ : IsScalarTower 𝕜 A A\ninst✝³ : SMulCommClass 𝕜 A A\ninst✝² : ContinuousStar A\ninst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p\ninst✝ :... | [
"𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NonUnitalNormedRing A\ninst✝⁶ : StarRing A\ninst✝⁵ : NormedSpace 𝕜 A\ninst✝⁴ : IsScalarTower 𝕜 A A\ninst✝³ : SMulCommClass 𝕜 A A\ninst✝² : ContinuousStar A\ninst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p\ninst✝ : CompleteSpa... | refine continuousOn_of_locally_continuousOn fun (f, a) ⟨hf, ha, has⟩ ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 496,
"column": 4
} | {
"line": 496,
"column": 42
} | {
"line": 498,
"column": 0
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\nS : Type u_4\nA : Type u_5\nB : Type u_6\ninst✝¹ : Fintype m\ninst✝ : NonUnitalCStarAlgebra A\nc : ℂ\nM : CStarMatrix m n A\nx : C⋆ᵐᵒᵈ(A, m → A)\ni : n\n⊢ ((WithCStarModule.equiv A (m → A)) x ᵥ* (c • M)) i = c • ((WithCStarModule.equiv A (m → A)) x ᵥ* M) i",
... | [] | rw [Matrix.vecMul_smul, Pi.smul_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 999,
"column": 2
} | {
"line": 999,
"column": 75
} | {
"line": 1000,
"column": 2
} | [
{
"pp": "A : Type u_2\ninst✝¹² : NonUnitalNormedRing A\ninst✝¹¹ : StarRing A\ninst✝¹⁰ : NormedSpace ℝ A\ninst✝⁹ : IsScalarTower ℝ A A\ninst✝⁸ : SMulCommClass ℝ A A\ninst✝⁷ : ContinuousStar A\ninst✝⁶ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁵ : PartialOrder A\ninst✝⁴ : StarOrdered... | [
"A : Type u_2\ninst✝¹² : NonUnitalNormedRing A\ninst✝¹¹ : StarRing A\ninst✝¹⁰ : NormedSpace ℝ A\ninst✝⁹ : IsScalarTower ℝ A A\ninst✝⁸ : SMulCommClass ℝ A A\ninst✝⁷ : ContinuousStar A\ninst✝⁶ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁵ : PartialOrder A\ninst✝⁴ : StarOrderedRing A\ninst... | refine continuousOn_of_locally_continuousOn fun (f, a) ⟨hf, ha, has⟩ ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 689,
"column": 2
} | {
"line": 694,
"column": 5
} | {
"line": 696,
"column": 0
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nA : Type u_3\ninst✝⁴ : Fintype m\ninst✝³ : Fintype n\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\n⊢ 𝓤 (CStarMatrix m n A) = 𝓤 (CStarMatrix m n A)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Matrix.i... | [] | have :
(fun x : CStarMatrix m n A × CStarMatrix m n A => ⟨ofMatrix.symm x.1, ofMatrix.symm x.2⟩)
= id := by
ext i <;> rfl
rw [← uniformInducing_toMatrixAux.comap_uniformity, this, Filter.comap_id]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 689,
"column": 2
} | {
"line": 694,
"column": 5
} | {
"line": 696,
"column": 0
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nA : Type u_3\ninst✝⁴ : Fintype m\ninst✝³ : Fintype n\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\n⊢ 𝓤 (CStarMatrix m n A) = 𝓤 (CStarMatrix m n A)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Matrix.i... | [] | have :
(fun x : CStarMatrix m n A × CStarMatrix m n A => ⟨ofMatrix.symm x.1, ofMatrix.symm x.2⟩)
= id := by
ext i <;> rfl
rw [← uniformInducing_toMatrixAux.comap_uniformity, this, Filter.comap_id]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Integral | {
"line": 148,
"column": 11
} | {
"line": 148,
"column": 31
} | {
"line": 148,
"column": 32
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : MeasurableSpace X\nμ : Measure X\ninst✝⁵ : NormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : ContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : CompleteSpace A\ninst✝ : NormedSpace ℝ A\nf : X → 𝕜 → ... | [
"X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : MeasurableSpace X\nμ : Measure X\ninst✝⁵ : NormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : ContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : CompleteSpace A\ninst✝ : NormedSpace ℝ A\nf : X → 𝕜 → 𝕜\na : A\nh... | cfc_eq_cfcL_mkD _ a, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart | {
"line": 113,
"column": 2
} | {
"line": 115,
"column": 9
} | {
"line": 116,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : Algebra ℂ A\ninst✝¹ : StarModule ℂ A\ninst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal\na : A\nhp : IsStarNormal a\n⊢ cfc (fun x ↦ ↑x.im) a = ↑(ℑ a)",
"ppTerm": "?m.43",
"assigned": true,
"usedCon... | [
"A : Type u_1\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : Algebra ℂ A\ninst✝¹ : StarModule ℂ A\ninst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal\na : A\nhp : IsStarNormal a\n⊢ cfc (fun z ↦ ↑z.re + I * ↑z.im) a = ↑(ℜ a) + I • ↑(ℑ a)"
] | suffices cfc (fun z : ℂ ↦ re z + I * im z) a = ℜ a + I • ℑ a by
rw [cfc_add .., cfc_const_mul .., cfc_re_id a] at this
simpa | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.InnerProductSpace.Dual | {
"line": 160,
"column": 20
} | {
"line": 160,
"column": 47
} | {
"line": 160,
"column": 47
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nℓ : StrongDual 𝕜 E\nY : Submodule 𝕜 E := (↑ℓ).ker\nz : E\nhz : z ∈ Yᗮ\nz_ne_0 : z ≠ 0\nx : E\nh₁ : ℓ z • x - ℓ x • z ∈ Y\n⊢ failed to pretty print expression (use ... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nℓ : StrongDual 𝕜 E\nY : Submodule 𝕜 E := (↑ℓ).ker\nz : E\nhz : z ∈ Yᗮ\nz_ne_0 : z ≠ 0\nx : E\nh₁ : ℓ z • x - ℓ x • z ∈ Y\n⊢ 0 = 0",
"case a\n𝕜 : Type u_1\nE : Type u_2\nins... | (Y.mem_orthogonal' z).mp hz | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.OpenPartialHomeomorph.Constructions | {
"line": 303,
"column": 2
} | {
"line": 303,
"column": 81
} | {
"line": 304,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ns : Opens X\nhs : Nonempty ↥s\nf f' : OpenPartialHomeomorph X Y\nopenness₁ : IsOpen[inst✝] (f.target ∩ ↑f.symm ⁻¹' ↑s)\nset_identity : f.symm.source ∩ (f.target ∩ ↑f.symm ⁻¹' ↑s) = f.symm.source ∩ ↑f.symm ⁻¹' ↑s\nopenn... | [
"X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ns : Opens X\nhs : Nonempty ↥s\nf f' : OpenPartialHomeomorph X Y\nopenness₁ : IsOpen[inst✝] (f.target ∩ ↑f.symm ⁻¹' ↑s)\nset_identity : f.symm.source ∩ (f.target ∩ ↑f.symm ⁻¹' ↑s) = f.symm.source ∩ ↑f.symm ⁻¹' ↑s\nopenness₂ : IsOpe... | refine Setoid.trans (symm_trans_self (s.openPartialHomeomorphSubtypeCoe hs)) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 215,
"column": 60
} | {
"line": 216,
"column": 11
} | {
"line": 218,
"column": 0
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\ne' : Pretrivialization F TotalSpace.proj\nb : B\ny : E b\nhb : b ∈ e'.baseSet\n⊢ (↑e' { proj := b, snd := y }).1 = b",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"congrA... | [] | by
simp [hb] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 329,
"column": 63
} | {
"line": 329,
"column": 88
} | {
"line": 331,
"column": 0
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ne✝ : Pretrivialization F proj\nx : Z\ne' : Pretrivialization F TotalSpace.proj\nb : B\ny : E b\ns : Set B\ne : Pretrivialization F fun z ↦ proj ↑z\ninst✝ : Nonempty (Z → F... | [] | simp [hzp, e.coe_fst hze] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 747,
"column": 6
} | {
"line": 747,
"column": 18
} | {
"line": 747,
"column": 19
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ninst✝ : TopologicalSpace Z\ne₁ e₂ : Trivialization F proj\nb : B\nh₁ : b ∈ e₁.baseSet\nh₂ : b ∈ e₂.baseSet\nx : F\n⊢ (b, e₁.coordChange e₂ b x).1 = (↑e₂ (↑e₁.symm (b, x))).1",
"ppTerm":... | [
"B : Type u_1\nF : Type u_2\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ninst✝ : TopologicalSpace Z\ne₁ e₂ : Trivialization F proj\nb : B\nh₁ : b ∈ e₁.baseSet\nh₂ : b ∈ e₂.baseSet\nx : F\n⊢ (b, e₁.coordChange e₂ b x).1 = proj (↑e₁.symm (b, x))",
"B : Type u_1\nF : Type u_... | e₂.coe_fst', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.SeparatedMap | {
"line": 89,
"column": 2
} | {
"line": 90,
"column": 28
} | {
"line": 91,
"column": 2
} | [
{
"pp": "case refine_1\nX : Type u_1\nY : Sort u_2\ninst✝ : TopologicalSpace X\nf : X → Y\nx₁ x₂ : X\nx✝¹ : f x₁ = f x₂\nx✝ : x₁ ≠ x₂\nh : ∃ s₁ ∈ 𝓝 x₁, ∃ s₂ ∈ 𝓝 x₂, Disjoint s₁ s₂\n⊢ (Function.pullbackDiagonal f)ᶜ ∈ comap Subtype.val (𝓝 x₁ ×ˢ 𝓝 x₂)",
"ppTerm": "?refine_1",
"assigned": true,
"use... | [
"case refine_2\nX : Type u_1\nY : Sort u_2\ninst✝ : TopologicalSpace X\nf : X → Y\nx₁ x₂ : X\nx✝² : f x₁ = f x₂\nx✝¹ : x₁ ≠ x₂\nx✝ : (Function.pullbackDiagonal f)ᶜ ∈ comap Subtype.val (𝓝 x₁ ×ˢ 𝓝 x₂)\nt : Set (X × X)\nht : t ∈ 𝓝 x₁ ×ˢ 𝓝 x₂\nt_sub : Subtype.val ⁻¹' t ⊆ (Function.pullbackDiagonal f)ᶜ\n⊢ ∃ s₁ ∈ 𝓝 ... | · simp_rw [← Filter.disjoint_iff, ← compl_diagonal_mem_prod] at h
exact ⟨_, h, subset_rfl⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.CStarAlgebra.Multiplier | {
"line": 187,
"column": 6
} | {
"line": 188,
"column": 23
} | {
"line": 188,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NonUnitalNormedRing A\ninst✝² : NormedSpace 𝕜 A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : IsScalarTower 𝕜 A A\nn : ℕ\nx y : A\n⊢ ↑n x * y = x * ↑n y",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"No... | [] | simp only [← Nat.smul_one_eq_cast, smul_apply, one_apply_eq_self, mul_smul_comm,
smul_mul_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.IsLocalHomeomorph | {
"line": 270,
"column": 4
} | {
"line": 270,
"column": 32
} | {
"line": 271,
"column": 4
} | [
{
"pp": "case refine_1\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nhf : IsLocalHomeomorph f\n⊢ ∀ u ∈ {U | ∃ V, IsOpen[inst✝] V ∧ ∃ s, f ∘ ⇑s = Subtype.val ∧ Set.range ⇑s = U}, IsOpen[inst✝¹] u",
"ppTerm": "?refine_1",
"assigned": true,
"usedConsta... | [
"case refine_1\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nhf : IsLocalHomeomorph f\nU : Set Y\nhU : IsOpen[inst✝] U\ns : C(↑U, X)\nhs : f ∘ ⇑s = Subtype.val\n⊢ IsOpen[inst✝¹] (Set.range ⇑s)"
] | rintro _ ⟨U, hU, s, hs, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 34
} | {
"line": 145,
"column": 0
} | [
{
"pp": "r x : ℝ\nhx : x ∈ {x | |x| < r}\n⊢ 0 < r",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real",
"Real.lattice",
"abs",
"covariant_swap_add_of_covariant_add",
"PartialOrder.toPreorder",
"Preorder.toLE",
... | [] | exact (abs_nonneg x).trans_lt hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 23
} | {
"line": 284,
"column": 2
} | [
{
"pp": "x y : Circle\nh : x ≠ y\n⊢ ⇑(y.path x) '' Ioo 0 1 ⊆ {x}ᶜ",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Real.partialOrder",
"Real",
"Set.Icc.instZero",
"Compl.compl",
"PartialOrder.toPreorder",
"Membership.mem",... | [
"x y : Circle\nh : x ≠ y\nt : ↑unitInterval\nht : t ∈ Ioo 0 1\n⊢ (y.path x) t ∈ {x}ᶜ"
] | rintro z ⟨t, ht, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 287,
"column": 2
} | {
"line": 293,
"column": 10
} | {
"line": 295,
"column": 0
} | [
{
"pp": "x y : Circle\n⊢ range ⇑(x.path y) ⊂ univ",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCom... | [] | rw [ssubset_univ_iff_nonempty_compl]
obtain rfl | hne := eq_or_ne x y
· use -x, by simp [neg_ne_self]
rw [compl_range_path hne]
use y.path x ⟨2⁻¹, by simp only [mem_Icc, inv_nonneg, Nat.ofNat_nonneg, true_and]; linarith⟩
refine mem_image_of_mem _ ⟨by simp [← unitInterval.coe_pos], unitInterval.coe_lt_one.mp ?... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 287,
"column": 2
} | {
"line": 293,
"column": 10
} | {
"line": 295,
"column": 0
} | [
{
"pp": "x y : Circle\n⊢ range ⇑(x.path y) ⊂ univ",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCom... | [] | rw [ssubset_univ_iff_nonempty_compl]
obtain rfl | hne := eq_or_ne x y
· use -x, by simp [neg_ne_self]
rw [compl_range_path hne]
use y.path x ⟨2⁻¹, by simp only [mem_Icc, inv_nonneg, Nat.ofNat_nonneg, true_and]; linarith⟩
refine mem_image_of_mem _ ⟨by simp [← unitInterval.coe_pos], unitInterval.coe_lt_one.mp ?... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 660,
"column": 14
} | {
"line": 661,
"column": 36
} | {
"line": 661,
"column": 36
} | [
{
"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 : FiniteDimensional 𝕜 E\ninst✝ : FiniteDimensional 𝕜 F\nA : E →ₗ[𝕜] F\n⊢ finrank 𝕜 F - finrank 𝕜 ↥(a... | [
"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 : FiniteDimensional 𝕜 E\ninst✝ : FiniteDimensional 𝕜 F\nA : E →ₗ[𝕜] F\n⊢ finrank 𝕜 ↥(adjoint A).ker + finrank 𝕜 ... | rw [← A.adjoint.ker.finrank_add_finrank_orthogonal,
orthogonal_ker, adjoint_adjoint] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Affine.AddTorsorBases | {
"line": 93,
"column": 86
} | {
"line": 93,
"column": 98
} | {
"line": 93,
"column": 98
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ :... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ : s ⊆ t\nht₂ ... | div_mul_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Affine.AddTorsorBases | {
"line": 138,
"column": 12
} | {
"line": 138,
"column": 38
} | {
"line": 138,
"column": 39
} | [
{
"pp": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\ninst✝ : FiniteDimensional ℝ V\ns : Set V\nh : affineSpan ℝ s = ⊤\nt : Set V\nhts : t ⊆ s\nb : AffineBasis ↑t ℝ V\nhb : ⇑b = Subtype.val\nthis : (interior ((convexHull ℝ) (range Subtype.val))).Nonempty\n⊢ (interior ((convexHull ℝ) s)... | [
"V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\ninst✝ : FiniteDimensional ℝ V\ns : Set V\nh : affineSpan ℝ s = ⊤\nt : Set V\nhts : t ⊆ s\nb : AffineBasis ↑t ℝ V\nhb : ⇑b = Subtype.val\nthis : (interior ((convexHull ℝ) {x | x ∈ t})).Nonempty\n⊢ (interior ((convexHull ℝ) s)).Nonempty"
] | Subtype.range_coe_subtype, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 448,
"column": 2
} | {
"line": 448,
"column": 54
} | {
"line": 449,
"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\np : P\n⊢ Collinear k {p}",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Collinear",
"vectorSpa... | [
"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\np : P\n⊢ Module.rank k ↥⊥ ≤ 1"
] | rw [collinear_iff_rank_le_one, vectorSpan_singleton] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.UniformLimitsDeriv | {
"line": 221,
"column": 4
} | {
"line": 221,
"column": 63
} | {
"line": 222,
"column": 4
} | [
{
"pp": "case inr.right\nι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\nf' : ι → E → E →... | [
"case inr.right\nι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\nf' : ι → E → E →L[𝕜] G\nx :... | refine ⟨fun n => f n x ∈ t, ht, fun n => f n x ∈ t, ht, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Calculus.UniformLimitsDeriv | {
"line": 292,
"column": 35
} | {
"line": 292,
"column": 47
} | {
"line": 292,
"column": 47
} | [
{
"pp": "ι : Type u_1\nl : Filter ι\nE : Type u_5\ninst✝⁴ : NormedAddCommGroup E\n𝕜 : Type u_6\ninst✝³ : RCLike 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_7\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\nhf' : TendstoUnifo... | [
"ι : Type u_1\nl : Filter ι\nE : Type u_5\ninst✝⁴ : NormedAddCommGroup E\n𝕜 : Type u_6\ninst✝³ : RCLike 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_7\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\nhf' : TendstoUniformlyOnFilter... | norm_pos_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 1110,
"column": 4
} | {
"line": 1115,
"column": 93
} | {
"line": 1117,
"column": 0
} | [
{
"pp": "case coe\nα : Type u_3\nE : α → Type u_4\np✝ : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\ninst✝ : DecidableEq α\ni : α\nx : E i\nthis : Nonempty α\np : ℝ≥0\nhp : 0 < ↑p\n⊢ ‖lp.single (↑p) i x‖ = ‖x‖",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtracti... | [] | have : 0 < (p : ℝ≥0∞).toReal := by simpa using hp
rw [norm_eq_tsum_rpow this, tsum_eq_single i, lp.coeFn_single, one_div,
Real.rpow_rpow_inv _ this.ne', Pi.single_eq_same]
· exact norm_nonneg _
· intro j hji
rw [lp.coeFn_single, Pi.single_eq_of_ne hji, _root_.norm_zero, Real.zero_rpow this.ne'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 1110,
"column": 4
} | {
"line": 1115,
"column": 93
} | {
"line": 1117,
"column": 0
} | [
{
"pp": "case coe\nα : Type u_3\nE : α → Type u_4\np✝ : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\ninst✝ : DecidableEq α\ni : α\nx : E i\nthis : Nonempty α\np : ℝ≥0\nhp : 0 < ↑p\n⊢ ‖lp.single (↑p) i x‖ = ‖x‖",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtracti... | [] | have : 0 < (p : ℝ≥0∞).toReal := by simpa using hp
rw [norm_eq_tsum_rpow this, tsum_eq_single i, lp.coeFn_single, one_div,
Real.rpow_rpow_inv _ this.ne', Pi.single_eq_same]
· exact norm_nonneg _
· intro j hji
rw [lp.coeFn_single, Pi.single_eq_of_ne hji, _root_.norm_zero, Real.zero_rpow this.ne'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 1300,
"column": 2
} | {
"line": 1300,
"column": 55
} | {
"line": 1301,
"column": 2
} | [
{
"pp": "α : Type u_3\nE : α → Type u_4\np : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_5\nl : Filter ι\ninst✝ : l.NeBot\n_i : Fact (1 ≤ p)\nhp : p ≠ ∞\nC : ℝ\nF : ι → ↥(lp E p)\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i ↦ ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' ... | [
"α : Type u_3\nE : α → Type u_4\np : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\nι : Type u_5\nl : Filter ι\ninst✝ : l.NeBot\n_i : Fact (1 ≤ p)\nhp : p ≠ ∞\nC : ℝ\nF : ι → ↥(lp E p)\nhCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C\nf : (a : α) → E a\nhf : Tendsto (id fun i ↦ ↑(F i)) l (𝓝 f)\ns : Finset α\nhp' : p ≠ 0\nhp'... | have hp'' : 0 < p.toReal := ENNReal.toReal_pos hp' hp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.Calculus | {
"line": 106,
"column": 74
} | {
"line": 107,
"column": 82
} | {
"line": 109,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nf g : ℝ → E\nf' g' : E\ns : Set ℝ\nx : ℝ\nhf : HasDerivWithinAt f f' s x\nhg : HasDerivWithinAt g g' s x\n⊢ HasDerivWithinAt (fun t ↦ ⟪f t, g t⟫) (⟪f x, g'⟫ + ⟪f', g... | [] | by
simpa using (hf.hasFDerivWithinAt.inner 𝕜 hg.hasFDerivWithinAt).hasDerivWithinAt | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.UniformLimitsDeriv | {
"line": 391,
"column": 10
} | {
"line": 391,
"column": 22
} | {
"line": 391,
"column": 22
} | [
{
"pp": "ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : IsRCLikeNormedField 𝕜\ninst✝³ : NormedSpace 𝕜 E\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜... | [
"ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : IsRCLikeNormedField 𝕜\ninst✝³ : NormedSpace 𝕜 E\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E ... | norm_pos_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.ProperSpace.Lemmas | {
"line": 50,
"column": 2
} | {
"line": 52,
"column": 63
} | {
"line": 53,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝¹ : PseudoMetricSpace α\ninst✝ : ProperSpace α\nx : α\nr : ℝ\ns : Set α\nhs : IsClosed s\nh : s ⊆ ball x r\nhr : r ≤ 0\n⊢ ∃ r' < r, s ⊆ ball x r'",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"Rea... | [
"case inr\nα : Type u_1\ninst✝¹ : PseudoMetricSpace α\ninst✝ : ProperSpace α\nx : α\nr : ℝ\ns : Set α\nhs : IsClosed s\nh : s ⊆ ball x r\nhr : 0 < r\n⊢ ∃ r' < r, s ⊆ ball x r'"
] | · rw [ball_eq_empty.2 hr, subset_empty_iff] at h
subst s
exact (exists_lt r).imp fun r' hr' => ⟨hr', empty_subset _⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convolution | {
"line": 512,
"column": 4
} | {
"line": 512,
"column": 26
} | {
"line": 513,
"column": 2
} | [
{
"pp": "case e'_7.inl\n𝕜 : 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\ng : G → E'\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜 E'\ninst✝³ : N... | [] | rw [h, (L _).map_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convolution | {
"line": 512,
"column": 4
} | {
"line": 512,
"column": 26
} | {
"line": 513,
"column": 2
} | [
{
"pp": "case e'_7.inl\n𝕜 : 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\ng : G → E'\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜 E'\ninst✝³ : N... | [] | rw [h, (L _).map_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convolution | {
"line": 512,
"column": 4
} | {
"line": 512,
"column": 26
} | {
"line": 513,
"column": 2
} | [
{
"pp": "case e'_7.inl\n𝕜 : 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\ng : G → E'\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜 E'\ninst✝³ : N... | [] | rw [h, (L _).map_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.EverywherePos | {
"line": 255,
"column": 63
} | {
"line": 255,
"column": 68
} | {
"line": 255,
"column": 68
} | [
{
"pp": "G : Type u_2\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : LocallyCompactSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : μ.InnerRegularCompactLTTop\nk : Set ... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.MeasureTheory.Measure.EverywherePos | {
"line": 255,
"column": 63
} | {
"line": 255,
"column": 68
} | {
"line": 255,
"column": 68
} | [
{
"pp": "G : Type u_2\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : LocallyCompactSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : μ.InnerRegularCompactLTTop\nk : Set ... | [] | group | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.EverywherePos | {
"line": 255,
"column": 63
} | {
"line": 255,
"column": 68
} | {
"line": 255,
"column": 68
} | [
{
"pp": "G : Type u_2\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : LocallyCompactSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : μ.InnerRegularCompactLTTop\nk : Set ... | [] | group | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.EverywherePos | {
"line": 264,
"column": 4
} | {
"line": 264,
"column": 9
} | {
"line": 265,
"column": 4
} | [
{
"pp": "G : Type u_2\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : LocallyCompactSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : μ.InnerRegularCompactLTTop\nk : Set ... | [
"G : Type u_2\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : LocallyCompactSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : μ.InnerRegularCompactLTTop\nk : Set G\nh : μ.IsE... | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.Analysis.Convolution | {
"line": 711,
"column": 26
} | {
"line": 711,
"column": 48
} | {
"line": 711,
"column": 49
} | [
{
"pp": "case pos\n𝕜 : 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\ng : G → E'\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜 E'\ninst✝³ : Normed... | [
"case pos\n𝕜 : 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\ng : G → E'\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜 E'\ninst✝³ : NormedSpace 𝕜 F\n... | add_mem_ball_iff_norm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convolution | {
"line": 743,
"column": 26
} | {
"line": 743,
"column": 48
} | {
"line": 743,
"column": 49
} | [
{
"pp": "case pos\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedAddCommGroup F\nf : G → E\ng : G → E'\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : No... | [
"case pos\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedAddCommGroup F\nf : G → E\ng : G → E'\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace 𝕜... | add_mem_ball_iff_norm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension | {
"line": 69,
"column": 4
} | {
"line": 70,
"column": 42
} | {
"line": 71,
"column": 2
} | [
{
"pp": "case refine_2\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nx : E\nn : ℕ∞\nhs : s ∈ 𝓝 x\nd : ℝ\nd_pos : 0 < d\nhd : Euclidean.closedBall x d ⊆ s\nc : ContDiffBump (toEuclidean x) := { rIn := d / 2, rOut := d, rIn_pos := ⋯, rIn_lt_rOut... | [] | apply c.contDiff.comp
exact ContinuousLinearEquiv.contDiff _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension | {
"line": 69,
"column": 4
} | {
"line": 70,
"column": 42
} | {
"line": 71,
"column": 2
} | [
{
"pp": "case refine_2\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nx : E\nn : ℕ∞\nhs : s ∈ 𝓝 x\nd : ℝ\nd_pos : 0 < d\nhd : Euclidean.closedBall x d ⊆ s\nc : ContDiffBump (toEuclidean x) := { rIn := d / 2, rOut := d, rIn_pos := ⋯, rIn_lt_rOut... | [] | apply c.contDiff.comp
exact ContinuousLinearEquiv.contDiff _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 175,
"column": 44
} | {
"line": 175,
"column": 49
} | {
"line": 175,
"column": 49
} | [
{
"pp": "G : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : IsTopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\nμ ν : Measure G\ninst✝⁴ : IsFiniteMeasureOnCompacts μ\ninst✝³ : IsFiniteMeasureOnCompacts ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : ν.IsMulRightInvariant\ninst... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 175,
"column": 44
} | {
"line": 175,
"column": 49
} | {
"line": 175,
"column": 49
} | [
{
"pp": "G : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : IsTopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\nμ ν : Measure G\ninst✝⁴ : IsFiniteMeasureOnCompacts μ\ninst✝³ : IsFiniteMeasureOnCompacts ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : ν.IsMulRightInvariant\ninst... | [] | group | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 175,
"column": 44
} | {
"line": 175,
"column": 49
} | {
"line": 175,
"column": 49
} | [
{
"pp": "G : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : IsTopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\nμ ν : Measure G\ninst✝⁴ : IsFiniteMeasureOnCompacts μ\ninst✝³ : IsFiniteMeasureOnCompacts ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : ν.IsMulRightInvariant\ninst... | [] | group | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 184,
"column": 6
} | {
"line": 184,
"column": 66
} | {
"line": 185,
"column": 6
} | [
{
"pp": "G : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : IsTopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\nμ ν : Measure G\ninst✝⁴ : IsFiniteMeasureOnCompacts μ\ninst✝³ : IsFiniteMeasureOnCompacts ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : ν.IsMulRightInvariant\ninst... | [
"G : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : IsTopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\nμ ν : Measure G\ninst✝⁴ : IsFiniteMeasureOnCompacts μ\ninst✝³ : IsFiniteMeasureOnCompacts ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : ν.IsMulRightInvariant\ninst✝ : ν.IsOpen... | apply (integral_integral_swap_of_hasCompactSupport _ _).symm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 207,
"column": 42
} | {
"line": 207,
"column": 47
} | {
"line": 207,
"column": 47
} | [
{
"pp": "G : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : IsTopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\nμ ν : Measure G\ninst✝⁴ : IsFiniteMeasureOnCompacts μ\ninst✝³ : IsFiniteMeasureOnCompacts ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : ν.IsMulRightInvariant\ninst... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 207,
"column": 42
} | {
"line": 207,
"column": 47
} | {
"line": 207,
"column": 47
} | [
{
"pp": "G : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : IsTopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\nμ ν : Measure G\ninst✝⁴ : IsFiniteMeasureOnCompacts μ\ninst✝³ : IsFiniteMeasureOnCompacts ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : ν.IsMulRightInvariant\ninst... | [] | group | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 207,
"column": 42
} | {
"line": 207,
"column": 47
} | {
"line": 207,
"column": 47
} | [
{
"pp": "G : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : IsTopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\nμ ν : Measure G\ninst✝⁴ : IsFiniteMeasureOnCompacts μ\ninst✝³ : IsFiniteMeasureOnCompacts ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : ν.IsMulRightInvariant\ninst... | [] | group | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 147,
"column": 2
} | {
"line": 215,
"column": 69
} | {
"line": 217,
"column": 0
} | [
{
"pp": "case inr\nG : Type u_1\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : Group G\ninst✝⁷ : IsTopologicalGroup G\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : BorelSpace G\nμ ν : Measure G\ninst✝⁴ : IsFiniteMeasureOnCompacts μ\ninst✝³ : IsFiniteMeasureOnCompacts ν\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : ν.IsMulRightInvar... | [] | calc
∫ x, f x ∂μ = ∫ x, f x * (D x)⁻¹ * D x ∂μ := by
congr with x; rw [mul_assoc, inv_mul_cancel₀ (D_pos x).ne', mul_one]
_ = ∫ x, (∫ y, f x * (D x)⁻¹ * g (y⁻¹ * x) ∂ν) ∂μ := by simp_rw [D, integral_const_mul]
_ = ∫ y, (∫ x, f x * (D x)⁻¹ * g (y⁻¹ * x) ∂μ) ∂ν := by
apply integral_integral_swap_of_hasCom... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.Calculus.BumpFunction.Normed | {
"line": 84,
"column": 87
} | {
"line": 90,
"column": 30
} | {
"line": 92,
"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\nμ : Measure E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : μ.IsOpenPosMeasure\nι : Type u_2\nφ : ι → ContDiffBump c... | [] | by
simp_rw [NormedAddGroup.tendsto_nhds_zero, Real.norm_eq_abs,
abs_eq_self.mpr (φ _).rOut_pos.le] at hφ
rw [nhds_basis_ball.smallSets.tendsto_right_iff]
refine fun ε hε ↦ (hφ ε hε).mono fun i hi ↦ ?_
rw [(φ i).support_normed_eq]
exact ball_subset_ball hi.le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 50
} | {
"line": 182,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : MetricSpace α\nN : ℕ\nτ : ℝ\na : SatelliteConfig α N τ\ni : Fin N.succ\nH : last N ≤ i\nI : i = last N\nthis : 0 ≤ a.r (last N)\n⊢ dist (a.c i) (a.c (last N)) ≤ a.r i + a.r (last N)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Real.instLE",... | [] | simp only [I, add_nonneg this this, dist_self] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace | {
"line": 239,
"column": 29
} | {
"line": 239,
"column": 60
} | {
"line": 240,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nh✝ :\n ∀ (δ : ℝ),\n 0 < δ → δ < 1 → ∃ s, (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card\nN : ℕ := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : ℝ ... | [] | by rw [h] at hij; exact hij rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace | {
"line": 286,
"column": 29
} | {
"line": 286,
"column": 60
} | {
"line": 287,
"column": 4
} | [
{
"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 ... | [] | by rw [h] at hij; exact hij rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 481,
"column": 6
} | {
"line": 481,
"column": 73
} | {
"line": 482,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : MetricSpace α\nβ : Type u\nN : ℕ\nτ : ℝ\nhτ : 1 < τ\nhN : IsEmpty (SatelliteConfig α N τ)\nq : BallPackage β α\nh✝ : Nonempty β\np : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }\ns : Fin N → Set β := fun i ↦ ⋃ k, ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = ↑i)... | [] | simpa only [s, exists_prop, mem_iUnion, mem_singleton_iff] using hy | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 481,
"column": 6
} | {
"line": 481,
"column": 73
} | {
"line": 482,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : MetricSpace α\nβ : Type u\nN : ℕ\nτ : ℝ\nhτ : 1 < τ\nhN : IsEmpty (SatelliteConfig α N τ)\nq : BallPackage β α\nh✝ : Nonempty β\np : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }\ns : Fin N → Set β := fun i ↦ ⋃ k, ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = ↑i)... | [] | simpa only [s, exists_prop, mem_iUnion, mem_singleton_iff] using hy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 481,
"column": 6
} | {
"line": 481,
"column": 73
} | {
"line": 482,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : MetricSpace α\nβ : Type u\nN : ℕ\nτ : ℝ\nhτ : 1 < τ\nhN : IsEmpty (SatelliteConfig α N τ)\nq : BallPackage β α\nh✝ : Nonempty β\np : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }\ns : Fin N → Set β := fun i ↦ ⋃ k, ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = ↑i)... | [] | simpa only [s, exists_prop, mem_iUnion, mem_singleton_iff] using hy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 758,
"column": 54
} | {
"line": 758,
"column": 59
} | {
"line": 758,
"column": 59
} | [
{
"pp": "G : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 758,
"column": 54
} | {
"line": 758,
"column": 59
} | {
"line": 758,
"column": 59
} | [
{
"pp": "G : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s... | [] | group | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 758,
"column": 54
} | {
"line": 758,
"column": 59
} | {
"line": 758,
"column": 59
} | [
{
"pp": "G : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s... | [] | group | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace | {
"line": 412,
"column": 6
} | {
"line": 412,
"column": 49
} | {
"line": 413,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : a.c (last N) = 0\nlastr : a.r (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\ni j : Fin N.succ\ninej : i ≠ j\nhi : 2 < 0\nhij : ‖a.c i‖ ≤ ‖a.c j‖\nah : Pairwise fun i j ↦ a.r i ≤... | [] | exact lt_irrefl _ (zero_le_two.trans_lt hi) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 845,
"column": 22
} | {
"line": 845,
"column": 25
} | {
"line": 846,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : Group G\ninst✝⁸ : IsTopologicalGroup G\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : BorelSpace G\ninst✝⁵ : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝⁴ : μ.IsHaarMeasure\ninst✝³ : IsFiniteMeasureOnCompacts μ'\ninst✝² : μ'.IsMulLeftInvariant\ninst✝¹ : μ.I... | [
"G : Type u_1\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : Group G\ninst✝⁸ : IsTopologicalGroup G\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : BorelSpace G\ninst✝⁵ : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝⁴ : μ.IsHaarMeasure\ninst✝³ : IsFiniteMeasureOnCompacts μ'\ninst✝² : μ'.IsMulLeftInvariant\ninst✝¹ : μ.InnerRegularC... | h's | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Calculus.ContDiff.FiniteDimension | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 22
} | {
"line": 45,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁷ : NormedAddCommGroup D\ninst✝⁶ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type uF\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nn : ℕ∞ω\ninst✝¹ : CompleteSpace 𝕜\nf ... | [
"𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁷ : NormedAddCommGroup D\ninst✝⁶ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type uF\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nn : ℕ∞ω\ninst✝¹ : CompleteSpace 𝕜\nf : D → E →L[�... | let d := finrank 𝕜 E | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Calculus.ContDiffHolder.Pointwise | {
"line": 243,
"column": 4
} | {
"line": 244,
"column": 77
} | {
"line": 246,
"column": 0
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nk l : ℕ\nα : ↑I\nf : E → F\na : E\nhf : ContDiffPointwiseHolderAt k α f a\nhl : l < k\n⊢ (fun x ↦ ‖iteratedFDeriv ℝ l (fderiv ℝ f) x - iteratedFDeriv ℝ l (fderiv ... | [] | simpa [iteratedFDeriv_succ_eq_comp_right, Function.comp_def, ← dist_eq_norm_sub]
using hf.of_order_le (Nat.add_one_le_iff.mpr hl) |>.isBigO |>.norm_left | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.ContDiffHolder.Pointwise | {
"line": 243,
"column": 4
} | {
"line": 244,
"column": 77
} | {
"line": 246,
"column": 0
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nk l : ℕ\nα : ↑I\nf : E → F\na : E\nhf : ContDiffPointwiseHolderAt k α f a\nhl : l < k\n⊢ (fun x ↦ ‖iteratedFDeriv ℝ l (fderiv ℝ f) x - iteratedFDeriv ℝ l (fderiv ... | [] | simpa [iteratedFDeriv_succ_eq_comp_right, Function.comp_def, ← dist_eq_norm_sub]
using hf.of_order_le (Nat.add_one_le_iff.mpr hl) |>.isBigO |>.norm_left | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiffHolder.Pointwise | {
"line": 243,
"column": 4
} | {
"line": 244,
"column": 77
} | {
"line": 246,
"column": 0
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nk l : ℕ\nα : ↑I\nf : E → F\na : E\nhf : ContDiffPointwiseHolderAt k α f a\nhl : l < k\n⊢ (fun x ↦ ‖iteratedFDeriv ℝ l (fderiv ℝ f) x - iteratedFDeriv ℝ l (fderiv ... | [] | simpa [iteratedFDeriv_succ_eq_comp_right, Function.comp_def, ← dist_eq_norm_sub]
using hf.of_order_le (Nat.add_one_le_iff.mpr hl) |>.isBigO |>.norm_left | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 78
} | {
"line": 179,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁷ : NormedAddCommGroup D\ninst✝⁶ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type uF\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type uG\ninst✝¹ : NormedAddCommGro... | [
"case hx\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁷ : NormedAddCommGroup D\ninst✝⁶ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type uF\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type uG\ninst✝¹ : NormedAddCommGroup ... | rw [LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_right _ _ hs, xu_x] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 90
} | {
"line": 77,
"column": 2
} | [
{
"pp": "case pos\nf : ℝ → ℝ\na b : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\ng₀ : a ≤ b\n⊢ ContinuousOn f (Icc a b)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
"R... | [
"case neg\nf : ℝ → ℝ\na b : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\ng₀ : b < a\n⊢ ContinuousOn f (Icc a b)"
] | · exact Ioo_union_both g₀ ▸ hd₀.continuousOn.union_continuousAt isOpen_Ioo (by simp_all) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 78
} | {
"line": 184,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁷ : NormedAddCommGroup D\ninst✝⁶ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type uF\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type uG\ninst✝¹ : NormedAddCommGro... | [
"case hx\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁷ : NormedAddCommGroup D\ninst✝⁶ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type uF\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type uG\ninst✝¹ : NormedAddCommGroup ... | rw [LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_right _ _ hs, xu_x] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 190,
"column": 4
} | {
"line": 190,
"column": 78
} | {
"line": 191,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁷ : NormedAddCommGroup D\ninst✝⁶ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type uF\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type uG\ninst✝¹ : NormedAddCommGro... | [
"case hx\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁷ : NormedAddCommGroup D\ninst✝⁶ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type uF\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type uG\ninst✝¹ : NormedAddCommGroup ... | rw [LinearIsometryEquiv.norm_iteratedFDerivWithin_comp_right _ _ hs, xu_x] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 843,
"column": 76
} | {
"line": 846,
"column": 51
} | {
"line": 847,
"column": 2
} | [
{
"pp": "α : 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✝ : SFinite μ\nf : α → Set ℝ\ns : Set α\nhf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty\nR : α → ℝ\nhR : ∀ x ∈ s,... | [] | by
rcases hf x hx (min δ (R x)) (lt_min δpos (hR x hx)) with ⟨r, hr⟩
exact ⟨r, ⟨⟨hr.1, hr.2.1, hr.2.2.trans_le (min_le_right _ _)⟩,
⟨hr.2.1, hr.2.2.trans_le (min_le_left _ _)⟩⟩⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.FDeriv.Symmetric | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 56
} | {
"line": 170,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nhs : UniqueDiffOn 𝕜 s\nhx : x ∈ s\n⊢ (∀ (v w : E), ((fderivWithin 𝕜 (fderivWi... | [] | simp [iteratedFDerivWithin_two_apply f hs hx, eq_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 426,
"column": 6
} | {
"line": 428,
"column": 10
} | {
"line": 429,
"column": 4
} | [
{
"pp": "𝕜 : 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 ... | [] | congr! 1 with i hi
simp only [Nat.cast_choose ℝ (Finset.mem_range_succ_iff.1 hi), div_eq_inv_mul, mul_inv]
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 426,
"column": 6
} | {
"line": 428,
"column": 10
} | {
"line": 429,
"column": 4
} | [
{
"pp": "𝕜 : 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 ... | [] | congr! 1 with i hi
simp only [Nat.cast_choose ℝ (Finset.mem_range_succ_iff.1 hi), div_eq_inv_mul, mul_inv]
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.VectorField | {
"line": 451,
"column": 2
} | {
"line": 456,
"column": 49
} | {
"line": 458,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nhn : minSmoothness 𝕜 2 ≤ n\nU V W : E → E\ns : Set E\nx : E\nhs : UniqueDiffOn 𝕜 s\nh'x : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (interior s)\n... | [] | apply leibniz_identity_lieBracketWithin_of_isSymmSndFDerivWithinAt hs hx
(hU.of_le (le_minSmoothness.trans hn)) (hV.of_le (le_minSmoothness.trans hn))
(hW.of_le (le_minSmoothness.trans hn))
· exact hU.isSymmSndFDerivWithinAt hn hs h'x hx
· exact hV.isSymmSndFDerivWithinAt hn hs h'x hx
· exact hW.isSymmSnd... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.VectorField | {
"line": 451,
"column": 2
} | {
"line": 456,
"column": 49
} | {
"line": 458,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nhn : minSmoothness 𝕜 2 ≤ n\nU V W : E → E\ns : Set E\nx : E\nhs : UniqueDiffOn 𝕜 s\nh'x : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (interior s)\n... | [] | apply leibniz_identity_lieBracketWithin_of_isSymmSndFDerivWithinAt hs hx
(hU.of_le (le_minSmoothness.trans hn)) (hV.of_le (le_minSmoothness.trans hn))
(hW.of_le (le_minSmoothness.trans hn))
· exact hU.isSymmSndFDerivWithinAt hn hs h'x hx
· exact hV.isSymmSndFDerivWithinAt hn hs h'x hx
· exact hW.isSymmSnd... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Norm | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 92
} | {
"line": 63,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nn : WithTop ℕ∞\nx : E\nt : ℝ\nht : t ≠ 0\nh : ContDiffAt ℝ n (fun x ↦ ‖x‖) x\n⊢ ContDiffAt ℝ n (fun x ↦ ‖x‖) (t • x)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"ContDiffAt",
"Real",
"instH... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nn : WithTop ℕ∞\nx : E\nt : ℝ\nht : t ≠ 0\nh : ContDiffAt ℝ n (fun x ↦ ‖x‖) x\nh1 : ContDiffAt ℝ n (fun y ↦ t⁻¹ • y) (t • x)\n⊢ ContDiffAt ℝ n (fun x ↦ ‖x‖) (t • x)"
] | have h1 : ContDiffAt ℝ n (fun y ↦ t⁻¹ • y) (t • x) := (contDiff_const_smul t⁻¹).contDiffAt | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.FDeriv.Norm | {
"line": 148,
"column": 4
} | {
"line": 148,
"column": 83
} | {
"line": 149,
"column": 4
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx : E\nh : DifferentiableAt ℝ (fun x ↦ ‖x‖) (0 • x)\n⊢ DifferentiableAt ℝ (fun x ↦ ‖x‖) x",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
"Semiring.toModule",
... | [
"case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx : E\nh : DifferentiableAt ℝ (fun x ↦ ‖x‖) (0 • x)\n⊢ Subsingleton E"
] | suffices Subsingleton E from (hasFDerivAt_of_subsingleton _ _).differentiableAt | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.Calculus.Gradient.Basic | {
"line": 298,
"column": 17
} | {
"line": 298,
"column": 30
} | {
"line": 298,
"column": 30
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : CompleteSpace F\nx : F\nL : Filter F\nf₀ f₁ : F → 𝕜\nf₀' f₁' : F\nh₀ : f₀ =ᶠ[L] f₁\nhx : f₀ x = f₁ x\nh₁ : f₀' = f₁'\n⊢ f₀ =ᶠ[pure x] f₁",
"ppTerm": "?m.30",
"assigned": tru... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno | {
"line": 211,
"column": 2
} | {
"line": 212,
"column": 35
} | {
"line": 214,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\ng f : 𝕜 → 𝕜\nx : 𝕜\nn : ℕ∞ω\ni : ℕ\nhg : ContDiffAt 𝕜 n g (f x)\nhf : ContDiffAt 𝕜 n f x\nhi : ↑i ≤ n\n⊢ iteratedDeriv i (g ∘ f) x = ∑ c, iteratedDeriv c.length g (f x) * ∏ j, iteratedDeriv (c.partSize j) f x",
"ppTerm": "?m.55",
"assigned... | [] | rw [iteratedDeriv_scomp_eq_sum_orderedFinpartition hg hf hi]
simp only [smul_eq_mul, mul_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno | {
"line": 211,
"column": 2
} | {
"line": 212,
"column": 35
} | {
"line": 214,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\ng f : 𝕜 → 𝕜\nx : 𝕜\nn : ℕ∞ω\ni : ℕ\nhg : ContDiffAt 𝕜 n g (f x)\nhf : ContDiffAt 𝕜 n f x\nhi : ↑i ≤ n\n⊢ iteratedDeriv i (g ∘ f) x = ∑ c, iteratedDeriv c.length g (f x) * ∏ j, iteratedDeriv (c.partSize j) f x",
"ppTerm": "?m.55",
"assigned... | [] | rw [iteratedDeriv_scomp_eq_sum_orderedFinpartition hg hf hi]
simp only [smul_eq_mul, mul_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.LHopital | {
"line": 68,
"column": 2
} | {
"line": 74,
"column": 75
} | {
"line": 75,
"column": 2
} | [
{
"pp": "a b : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhab : a < b\nhff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x\nhg' : ∀ x ∈ Ioo a b, 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\nsub : ∀ x... | [
"a b : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhab : a < b\nhff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x\nhg' : ∀ x ∈ Ioo a b, 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\nsub : ∀ x ∈ Ioo a b, ... | have : ∀ x ∈ Ioo a b, ∃ c ∈ Ioo a x, f x * g' c = g x * f' c := by
intro x hx
rw [← sub_zero (f x), ← sub_zero (g x)]
exact exists_ratio_hasDerivAt_eq_ratio_slope' g g' hx.1 f f' (fun y hy => hgg' y <| sub x hx hy)
(fun y hy => hff' y <| sub x hx hy) hga hfa
(tendsto_nhdsWithin_of_tendsto_nhds (... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 187,
"column": 2
} | {
"line": 194,
"column": 51
} | {
"line": 195,
"column": 2
} | [
{
"pp": "s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nhf : MonotoneOn f s\nu : ℝ → ℝ≥0∞\na b c : Set ℝ\nh_union : a ∪ (b ∪ c) = s\nha : MeasurableSet a\nhb : MeasurableSet b\nhc : MeasurableSet c\nh_disj : Disjoint a (b ∪ c)\nh_disj' : Disjoint b c\na_count : a.Co... | [
"s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nhf : MonotoneOn f s\nu : ℝ → ℝ≥0∞\na b c : Set ℝ\nh_union : a ∪ (b ∪ c) = s\nha : MeasurableSet a\nhb : MeasurableSet b\nhc : MeasurableSet c\nh_disj : Disjoint a (b ∪ c)\nh_disj' : Disjoint b c\na_count : a.Countable\nfb_... | have I : ∫⁻ x in s, ENNReal.ofReal (f' x) * u (f x)
= ∫⁻ x in c, ENNReal.ofReal (f' x) * u (f x) := by
have : ∫⁻ x in a, ENNReal.ofReal (f' x) * u (f x) = 0 :=
setLIntegral_measure_zero a _ (a_count.measure_zero volume)
rw [← h_union, lintegral_union (hb.union hc) h_disj, this, zero_add]
have : ... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 220,
"column": 4
} | {
"line": 220,
"column": 66
} | {
"line": 221,
"column": 4
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝⁴ : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝³ : LinearOrder α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderClosedTopology α\ninst✝ : OpensMeasurableSpace α\nb c : ι → α\nB : α\nhb : Tendsto b l (𝓝 B)\nhc : Tendsto c l atBot\n⊢ ∀ᵐ (x : α) ∂μ.restrict (Iio B)... | [
"α : Type u_1\nι : Type u_2\ninst✝⁴ : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝³ : LinearOrder α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderClosedTopology α\ninst✝ : OpensMeasurableSpace α\nb c : ι → α\nB : α\nhb : Tendsto b l (𝓝 B)\nhc : Tendsto c l atBot\n_x : α\nhx : _x ∈ Iio B\n⊢ ∀ᶠ (i : ι) in l, ... | refine (ae_restrict_mem measurableSet_Iio).mono fun _x hx ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 693,
"column": 2
} | {
"line": 696,
"column": 69
} | {
"line": 697,
"column": 2
} | [
{
"pp": "case neg\nι : Type u_1\nE : Type u_2\nμ : Measure ℝ\nl : Filter ι\ninst✝² : l.IsCountablyGenerated\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na : ι → ℝ\nf : ℝ → E\nha : Tendsto a l atTop\nb : ℝ\nhb : IntegrableOn f (Ici b) μ\n⊢ Tendsto (fun i ↦ ∫ (x : ℝ) in Ici (a i), f x ∂μ) l (𝓝 0)",
... | [
"case neg\nι : Type u_1\nE : Type u_2\nμ : Measure ℝ\nl : Filter ι\ninst✝² : l.IsCountablyGenerated\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na : ι → ℝ\nf : ℝ → E\nha : Tendsto a l atTop\nb : ℝ\nhb : IntegrableOn f (Ici b) μ\nthis : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in Ici b, f x ∂μ - ∫ (x : ℝ) in Ico b (a ... | have : ∀ᶠ i in l, ∫ x in Ici b, f x ∂μ - ∫ x in Ico b (a i), f x ∂μ =
∫ x in Ici (a i), f x ∂μ := by
filter_upwards [ha.eventually_mem (Ici_mem_atTop b)] with i hi
rw [sub_eq_iff_comm, intervalIntegral.integral_Ici_sub_Ici hb hi] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 511,
"column": 62
} | {
"line": 512,
"column": 66
} | {
"line": 514,
"column": 0
} | [
{
"pp": "a b : ℝ\nf f' g : ℝ → ℝ\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : ContinuousOn f' [[a, b]]\nhg : ContinuousOn g (f '' [[a, b]])\n⊢ ∫ (x : ℝ) in a..b, (g ∘ f) x * f' x = ∫ (u : ℝ) in f a..f b, g u",
"ppTerm": "?m.60",
"assigne... | [] | by
simpa [mul_comm] using integral_deriv_smul_comp'' hf hff' hf' hg | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1128,
"column": 64
} | {
"line": 1130,
"column": 82
} | {
"line": 1132,
"column": 0
} | [
{
"pp": "f f' g : ℝ → ℝ\na : ℝ\nhf : ContinuousOn f (Ici a)\nhft : Tendsto f atTop atTop\nhff' : ∀ x ∈ Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x ↦ (g ∘ f) x * f' x) (Ici a) volume\n⊢ ∫ (x : ℝ) in Ioi a, ... | [] | by
have hg2' : IntegrableOn (fun x => f' x • (g ∘ f) x) (Ici a) := by simpa [mul_comm] using hg2
simpa [mul_comm] using integral_deriv_smul_comp_Ioi hf hft hff' hg_cont hg1 hg2' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.InfiniteSum.UniformOn | {
"line": 193,
"column": 2
} | {
"line": 193,
"column": 35
} | {
"line": 195,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝² : CommMonoid α\nf : ι → β → α\ng : β → α\ns : Set β\ninst✝¹ : UniformSpace α\ninst✝ : TopologicalSpace β\nh : TendstoUniformlyOn (fun x1 x2 ↦ ∏ i ∈ x1, f i x2) g atTop s\n⊢ TendstoLocallyUniformlyOn (fun x1 x2 ↦ ∏ i ∈ x1, f i x2) g atTop s",
"ppTerm"... | [] | exact h.tendstoLocallyUniformlyOn | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1302,
"column": 2
} | {
"line": 1302,
"column": 71
} | {
"line": 1303,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℝ A\na' b' : A\nu v u' v' : ℝ → A\ninst✝ : CompleteSpace A\nhu : ∀ x ∈ tsupport v, HasDerivAt u (u' x) x\nhv : ∀ x ∈ tsupport u, HasDerivAt v (v' x) x\nhuv : Integrable (u' * v + u * v') volume\nh_bot : Tendsto (u * v) atBot (𝓝 a')\nh_top : T... | [
"A : Type u_1\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℝ A\na' b' : A\nu v u' v' : ℝ → A\ninst✝ : CompleteSpace A\nhu : ∀ x ∈ tsupport v, HasDerivAt u (u' x) x\nhv : ∀ x ∈ tsupport u, HasDerivAt v (v' x) x\nhuv : Integrable (u' * v + u * v') volume\nh_bot : Tendsto (u * v) atBot (𝓝 a')\nh_top : Tendsto (u * ... | refine integral_of_hasDerivAt_of_tendsto (fun x ↦ ?_) huv h_bot h_top | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 782,
"column": 2
} | {
"line": 782,
"column": 39
} | {
"line": 783,
"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\nhs : NullMeasurableSet s μ\nhf' : ∀ x ∈ s, HasFDerivWithinAt f (... | [
"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\nhs : NullMeasurableSet s μ\nhf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x\nh... | apply MeasurableSet.nullMeasurableSet | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 158,
"column": 20
} | {
"line": 158,
"column": 24
} | {
"line": 159,
"column": 6
} | [
{
"pp": "H : Type u_1\ninst✝ : TopologicalSpace H\nS : Set (StructureGroupoid H)\ne e' : OpenPartialHomeomorph H H\nhe : ∀ i ∈ S, e ∈ i.members\n⊢ e' ≈ e → ∀ i ∈ S, e' ∈ i.members",
"ppTerm": "?m.111",
"assigned": true,
"usedConstants": [
"OpenPartialHomeomorph.eqOnSourceSetoid",
"HasEqu... | [
"H : Type u_1\ninst✝ : TopologicalSpace H\nS : Set (StructureGroupoid H)\ne e' : OpenPartialHomeomorph H H\nhe : ∀ i ∈ S, e ∈ i.members\nhe'e : e' ≈ e\n⊢ ∀ i ∈ S, e' ∈ i.members"
] | he'e | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 217,
"column": 4
} | {
"line": 237,
"column": 23
} | {
"line": 238,
"column": 2
} | [
{
"pp": "case inr\nH✝ : 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 = ∅}\nh : e.source.Nonempty\n⊢ e ∈ {OpenPartialHomeomorph.refl 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 hs : OpenPartialHomeomorph.restr e s = OpenPartialHomeomorph.refl H := by
... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.HasGroupoid | {
"line": 245,
"column": 81
} | {
"line": 245,
"column": 94
} | {
"line": 245,
"column": 94
} | [
{
"pp": "H : Type u\ninst✝² : TopologicalSpace H\nα : Type u_5\ninst✝¹ : TopologicalSpace α\ne : OpenPartialHomeomorph α H\nh : e.source = univ\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\ne' e'' : OpenPartialHomeomorph α H\nhe' : e' ∈ atlas H α\nhe'' : e'' ∈ atlas H α\n⊢ ClosedUnderRestriction G... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.ChartedSpace | {
"line": 518,
"column": 10
} | {
"line": 518,
"column": 22
} | {
"line": 518,
"column": 22
} | [
{
"pp": "case inr\nH : Type u\nH' : Type u_1\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : TopologicalSpace M'\ncm : ChartedSpace H M\ncm' : ChartedSpace H M'\ninst✝ : Nonempty H\nx : M'\n⊢ Sum.elim (fun x ↦ (ChartedSpace.chartAt x).lift_openEmb... | [
"case inr\nH : Type u\nH' : Type u_1\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : TopologicalSpace M'\ncm : ChartedSpace H M\ncm' : ChartedSpace H M'\ninst✝ : Nonempty H\nx : M'\n⊢ (ChartedSpace.chartAt x).lift_openEmbedding ⋯ ∈\n (fun e ↦ e.lif... | Sum.elim_inr | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 306,
"column": 6
} | {
"line": 306,
"column": 27
} | {
"line": 307,
"column": 6
} | [
{
"pp": "case left\nH : Type u_1\ninst✝ : TopologicalSpace H\nPG : Pregroupoid H\ne : OpenPartialHomeomorph H H\nhe :\n ∀ x ∈ e.source,\n ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}\nx : H\nxu : x ∈ e.source\ns : Set H\ns_open : IsOpen[inst✝] s... | [
"case e'_5\nH : Type u_1\ninst✝ : TopologicalSpace H\nPG : Pregroupoid H\ne : OpenPartialHomeomorph H H\nhe :\n ∀ x ∈ e.source,\n ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}\nx : H\nxu : x ∈ e.source\ns : Set H\ns_open : IsOpen[inst✝] s\nxs : x ∈ s... | convert! hs.1 using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 98,
"column": 6
} | {
"line": 98,
"column": 23
} | {
"line": 98,
"column": 24
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : TopologicalSpace H\ninst✝ : TopologicalSpace M\nf : OpenPartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\n⊢ (f.extend I).target = ↑(f.extend... | [
"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : TopologicalSpace H\ninst✝ : TopologicalSpace M\nf : OpenPartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\n⊢ ↑I '' f.target = ↑(f.extend I) '' f.source"
... | f.extend_target', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 246,
"column": 25
} | {
"line": 246,
"column": 68
} | {
"line": 246,
"column": 69
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : TopologicalSpace H\ninst✝ : TopologicalSpace M\nf : OpenPartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ny : M\nhy : y ∈ f.source\n⊢ map (↑... | [
"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : TopologicalSpace H\ninst✝ : TopologicalSpace M\nf : OpenPartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ny : M\nhy : y ∈ f.source\n⊢ map (↑(f.extend I)... | ← map_extend_symm_nhdsWithin f (I := I) hy, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 267,
"column": 18
} | {
"line": 267,
"column": 40
} | {
"line": 267,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf : OpenPartialHomeomorph M H\nI : ModelWithCorn... | [
"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf : OpenPartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\n... | extend_left_inv _ hz.2 | Lean.Elab.Tactic.evalRewriteSeq | null |
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