module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute | {
"line": 50,
"column": 2
} | {
"line": 65,
"column": 43
} | [
{
"pp": "𝕜 : 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 b\nhb₂ : Commute... | induction f using ContinuousMap.induction_on_of_compact with
| const r =>
conv =>
enter [1, 2]
equals algebraMap 𝕜 _ r => rfl
rw [AlgHomClass.commutes]
exact Algebra.commute_algebraMap_left r b
| id => rwa [cfcHom_id ha]
| star_id => rwa [map_star, cfcHom_id]
| add f g hf hg => rw [map_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Integral | {
"line": 192,
"column": 2
} | {
"line": 197,
"column": 83
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : MeasurableSpace X\nμ : Measure X\ninst✝⁸ : NormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : NormedAlgebra 𝕜 A\ninst✝⁵ : ContinuousFunctionalCalculus 𝕜 A p\ninst✝⁴ : CompleteSpace A\ninst✝³ : NormedSpace ℝ A\ninst✝² : To... | have : ∀ᵐ (x : X) ∂(μ.restrict s), ContinuousOn (f x) (spectrum 𝕜 a) :=
ae_restrict_of_forall_mem hs fun x hx ↦
hf.comp (Continuous.prodMk_right x).continuousOn fun _ hz ↦ ⟨hx, hz⟩
refine cfc_setIntegral' _ _ this ⟨?_, ?_⟩ ha
· exact aeStronglyMeasurable_restrict_mkD_restrict_of_uncurry hs _ _ hf
· exa... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Integral | {
"line": 192,
"column": 2
} | {
"line": 197,
"column": 83
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : MeasurableSpace X\nμ : Measure X\ninst✝⁸ : NormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : NormedAlgebra 𝕜 A\ninst✝⁵ : ContinuousFunctionalCalculus 𝕜 A p\ninst✝⁴ : CompleteSpace A\ninst✝³ : NormedSpace ℝ A\ninst✝² : To... | have : ∀ᵐ (x : X) ∂(μ.restrict s), ContinuousOn (f x) (spectrum 𝕜 a) :=
ae_restrict_of_forall_mem hs fun x hx ↦
hf.comp (Continuous.prodMk_right x).continuousOn fun _ hz ↦ ⟨hx, hz⟩
refine cfc_setIntegral' _ _ this ⟨?_, ?_⟩ ha
· exact aeStronglyMeasurable_restrict_mkD_restrict_of_uncurry hs _ _ hf
· exa... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unitary | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 25
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Ring A\ninst✝² : StarRing A\ninst✝¹ : Algebra ℂ A\ninst✝ : ContinuousFunctionalCalculus ℂ A IsStarNormal\nu : A\nhu : IsStarNormal u\n⊢ u ∈ unitary A ↔ spectrum ℂ u ⊆ ↑(unitary ℂ)",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNon... | nth_rw 1 [← cfc_id ℂ u] | Mathlib.Tactic._aux_Mathlib_Tactic_NthRewrite___macroRules_Mathlib_Tactic_tacticNth_rw______1 | Mathlib.Tactic.tacticNth_rw_____ |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 77
} | [
{
"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 z ↦ ↑z.re + I * ↑z.im) a = ↑(ℜ a) + I • ↑(ℑ a)",
"usedConstants": [
"No... | simp [mul_comm I, re_add_im, cfc_id' .., realPart_add_I_smul_imaginaryPart] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 782,
"column": 10
} | {
"line": 782,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : NonUnitalCStarAlgebra A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\nm : Type u_2\nn : Type u_3\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nM : CStarMatrix n n A\nv : C⋆ᵐᵒᵈ(A, n → A)\nh₁ : ‖⟪v, (toCLM Mᴴ) ((toCLM M) v)⟫_A‖ ≤ ‖M * star M‖ * ‖v‖ ^ 2\nh₂ : ‖v‖ = √(‖v‖ ^ 2)\n⊢ √... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Dual | {
"line": 233,
"column": 4
} | {
"line": 233,
"column": 35
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁴ : RCLike 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nx : E\ny : F\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ finrank 𝕜 ↥(𝕜 ∙ x) = 1",
"usedConstants": [
"NormedSpace.toModu... | exact finrank_span_singleton hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.InnerProductSpace.Dual | {
"line": 233,
"column": 4
} | {
"line": 233,
"column": 35
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁴ : RCLike 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nx : E\ny : F\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ finrank 𝕜 ↥(𝕜 ∙ x) = 1",
"usedConstants": [
"NormedSpace.toModu... | exact finrank_span_singleton hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Dual | {
"line": 233,
"column": 4
} | {
"line": 233,
"column": 35
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁴ : RCLike 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nx : E\ny : F\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ finrank 𝕜 ↥(𝕜 ∙ x) = 1",
"usedConstants": [
"NormedSpace.toModu... | exact finrank_span_singleton hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.LinearMapCompletion | {
"line": 46,
"column": 14
} | {
"line": 46,
"column": 42
} | [
{
"pp": "case ih\nα : Type u_1\nβ : Type u_2\nR₁ : Type u_3\nR₂ : Type u_4\ninst✝¹¹ : UniformSpace α\ninst✝¹⁰ : AddCommGroup α\ninst✝⁹ : IsUniformAddGroup α\ninst✝⁸ : Semiring R₁\ninst✝⁷ : Module R₁ α\ninst✝⁶ : UniformContinuousConstSMul R₁ α\ninst✝⁵ : Semiring R₂\ninst✝⁴ : UniformSpace β\ninst✝³ : AddCommGroup... | simp [← Completion.coe_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.LinearMapCompletion | {
"line": 46,
"column": 14
} | {
"line": 46,
"column": 42
} | [
{
"pp": "case ih\nα : Type u_1\nβ : Type u_2\nR₁ : Type u_3\nR₂ : Type u_4\ninst✝¹¹ : UniformSpace α\ninst✝¹⁰ : AddCommGroup α\ninst✝⁹ : IsUniformAddGroup α\ninst✝⁸ : Semiring R₁\ninst✝⁷ : Module R₁ α\ninst✝⁶ : UniformContinuousConstSMul R₁ α\ninst✝⁵ : Semiring R₂\ninst✝⁴ : UniformSpace β\ninst✝³ : AddCommGroup... | simp [← Completion.coe_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.LinearMapCompletion | {
"line": 46,
"column": 14
} | {
"line": 46,
"column": 42
} | [
{
"pp": "case ih\nα : Type u_1\nβ : Type u_2\nR₁ : Type u_3\nR₂ : Type u_4\ninst✝¹¹ : UniformSpace α\ninst✝¹⁰ : AddCommGroup α\ninst✝⁹ : IsUniformAddGroup α\ninst✝⁸ : Semiring R₁\ninst✝⁷ : Module R₁ α\ninst✝⁶ : UniformContinuousConstSMul R₁ α\ninst✝⁵ : Semiring R₂\ninst✝⁴ : UniformSpace β\ninst✝³ : AddCommGroup... | simp [← Completion.coe_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.Extreme | {
"line": 48,
"column": 45
} | {
"line": 54,
"column": 67
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\ne : A\nhe : IsStarProjection e\na : A\nha : 0 ≤ a\nha1 : ‖a‖ ≤ 1\nb : A\nhb : 0 ≤ b\nhb1 : ‖b‖ ≤ 1\nx✝ : e ∈ openSegment ℝ a b\nt s : ℝ\nh0t : 0 < t\nh0s : 0 < s\nhts : t + s = 1\nhlin : t • a + s • b = ... | by
rw [← h0t.ne'.isUnit.smul_left_cancel, ← he.conjugate_of_nonneg_of_le
(smul_nonneg h0t.le ha) ?hae]
case hae => simpa [hlin] using le_add_of_nonneg_right (a := t • a) (by positivity : 0 ≤ s • b)
apply inr_injective (R := ℂ) <| Eq.symm ?_
simpa only [mul_one_sub_mul, he.inr.isIdempot... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Circle | {
"line": 161,
"column": 4
} | {
"line": 161,
"column": 53
} | [
{
"pp": "case refine_1\ne : AddChar ℝ Circle\nx : ℝ\n⊢ ↑(e x) ∈ Submonoid.unitSphere ℂ",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"Real.instAddMonoid",
"Subsemigroup.mk",
"Complex.instNormedField",
"SeminormedRing.toRing",
"AddChar",
"... | simp only [Submonoid.unitSphere, SetLike.coe_mem] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.Circle | {
"line": 161,
"column": 4
} | {
"line": 161,
"column": 53
} | [
{
"pp": "case refine_1\ne : AddChar ℝ Circle\nx : ℝ\n⊢ ↑(e x) ∈ Submonoid.unitSphere ℂ",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"Real.instAddMonoid",
"Subsemigroup.mk",
"Complex.instNormedField",
"SeminormedRing.toRing",
"AddChar",
"... | simp only [Submonoid.unitSphere, SetLike.coe_mem] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Circle | {
"line": 161,
"column": 4
} | {
"line": 161,
"column": 53
} | [
{
"pp": "case refine_1\ne : AddChar ℝ Circle\nx : ℝ\n⊢ ↑(e x) ∈ Submonoid.unitSphere ℂ",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"Real.instAddMonoid",
"Subsemigroup.mk",
"Complex.instNormedField",
"SeminormedRing.toRing",
"AddChar",
"... | simp only [Submonoid.unitSphere, SetLike.coe_mem] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.OpenPartialHomeomorph.Constructions | {
"line": 368,
"column": 18
} | {
"line": 368,
"column": 26
} | [
{
"pp": "case neg\nX✝ : Type u_1\nX'✝ : Type u_2\nY : Type u_3\nY' : Type u_4\nZ✝ : Type u_5\nZ' : Type u_6\ninst✝⁹ : TopologicalSpace X✝\ninst✝⁸ : TopologicalSpace X'✝\ninst✝⁷ : TopologicalSpace Y\ninst✝⁶ : TopologicalSpace Y'\ninst✝⁵ : TopologicalSpace Z✝\ninst✝⁴ : TopologicalSpace Z'\ne✝ : OpenPartialHomeomo... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.OpenPartialHomeomorph.Constructions | {
"line": 367,
"column": 4
} | {
"line": 381,
"column": 37
} | [
{
"pp": "X✝ : Type u_1\nX'✝ : Type u_2\nY : Type u_3\nY' : Type u_4\nZ✝ : Type u_5\nZ' : Type u_6\ninst✝⁹ : TopologicalSpace X✝\ninst✝⁸ : TopologicalSpace X'✝\ninst✝⁷ : TopologicalSpace Y\ninst✝⁶ : TopologicalSpace Y'\ninst✝⁵ : TopologicalSpace Z✝\ninst✝⁴ : TopologicalSpace Z'\ne✝ : OpenPartialHomeomorph X✝ Y\n... | by_cases Nonempty X; swap
· intro x hx; simp_all
set F := (extend f e (fun _ ↦ (Classical.arbitrary Z))) with F_eq
have heq : EqOn F (e ∘ (hf.toOpenPartialHomeomorph).symm) (f '' e.source) := by
intro x ⟨x₀, hx₀, hxx₀⟩
rw [← hxx₀, F_eq, hf.injective.extend_apply e, comp_apply,
hf.toOpenP... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.OpenPartialHomeomorph.Constructions | {
"line": 367,
"column": 4
} | {
"line": 381,
"column": 37
} | [
{
"pp": "X✝ : Type u_1\nX'✝ : Type u_2\nY : Type u_3\nY' : Type u_4\nZ✝ : Type u_5\nZ' : Type u_6\ninst✝⁹ : TopologicalSpace X✝\ninst✝⁸ : TopologicalSpace X'✝\ninst✝⁷ : TopologicalSpace Y\ninst✝⁶ : TopologicalSpace Y'\ninst✝⁵ : TopologicalSpace Z✝\ninst✝⁴ : TopologicalSpace Z'\ne✝ : OpenPartialHomeomorph X✝ Y\n... | by_cases Nonempty X; swap
· intro x hx; simp_all
set F := (extend f e (fun _ ↦ (Classical.arbitrary Z))) with F_eq
have heq : EqOn F (e ∘ (hf.toOpenPartialHomeomorph).symm) (f '' e.source) := by
intro x ⟨x₀, hx₀, hxx₀⟩
rw [← hxx₀, F_eq, hf.injective.extend_apply e, comp_apply,
hf.toOpenP... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 137,
"column": 67
} | {
"line": 138,
"column": 43
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nZ : Type u_4\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nproj : Z → B\ne : Pretrivialization F proj\nx : B × F\n⊢ x ∈ e.target ↔ x.1 ∈ e.baseSet",
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
"SProd.sprod",
"congrArg",
"Partia... | by
rw [e.target_eq, prod_univ, mem_preimage] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 348,
"column": 23
} | {
"line": 348,
"column": 41
} | [
{
"pp": "case h\nB : 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\nhs : IsOpen[inst✝²] s\nproj : Z → ↑s\ne : Pretrivializ... | simp [e.source_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 939,
"column": 10
} | {
"line": 939,
"column": 30
} | [
{
"pp": "case inl\nB : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace F\nproj : Z → B\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace (TotalSpace F E)\ne✝ : Trivialization F proj\nx : Z\ne'✝ : Trivialization F TotalSpace.proj\nb : B\ny : E b... | piecewise_eq_of_mem, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Covering.Quotient | {
"line": 106,
"column": 4
} | {
"line": 107,
"column": 73
} | [
{
"pp": "case refine_1\nE : Type u_1\nX : Type u_2\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : TopologicalSpace X\nf : E → X\nG : Type u_3\ninst✝⁴ : Group G\ninst✝³ : MulAction G E\nhf : IsQuotientMap f\ninst✝² : ContinuousConstSMul G E\nhfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂\ninst✝¹ : Topologi... | rw [← hf.isOpen_preimage, preim_im]
exact isOpen_iUnion fun g ↦ open_U.preimage (continuous_const_smul g) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Covering.Quotient | {
"line": 106,
"column": 4
} | {
"line": 107,
"column": 73
} | [
{
"pp": "case refine_1\nE : Type u_1\nX : Type u_2\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : TopologicalSpace X\nf : E → X\nG : Type u_3\ninst✝⁴ : Group G\ninst✝³ : MulAction G E\nhf : IsQuotientMap f\ninst✝² : ContinuousConstSMul G E\nhfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂\ninst✝¹ : Topologi... | rw [← hf.isOpen_preimage, preim_im]
exact isOpen_iUnion fun g ↦ open_U.preimage (continuous_const_smul g) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.Multiplier | {
"line": 521,
"column": 2
} | {
"line": 521,
"column": 27
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NonUnitalNormedRing A\ninst✝³ : NormedSpace 𝕜 A\ninst✝² : SMulCommClass 𝕜 A A\ninst✝¹ : IsScalarTower 𝕜 A A\ninst✝ : CompleteSpace A\n⊢ IsComplete (Set.range toProdMulOpposite)",
"usedConstants": [
"NormedCommRing.t... | apply IsClosed.isComplete | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Covering.Basic | {
"line": 544,
"column": 23
} | {
"line": 544,
"column": 25
} | [
{
"pp": "case h.e'_3\nE : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\nf : E → X\ninst✝ : T2Space E\nx : X\nhf : IsClosedMap f\nfin : (f ⁻¹' {x}).Finite\nh✝¹ : ∀ e ∈ f ⁻¹' {x}, ∃ φ, e ∈ φ.source ∧ ↑φ = f\nthis✝² : DiscreteTopology ↑(f ⁻¹' {x})\nφ : ↑(f ⁻¹' {x}) → OpenPartial... | V' | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Covering.Quotient | {
"line": 242,
"column": 92
} | {
"line": 249,
"column": 38
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G E\nh :\n IsCoveringMap f ∧\n Function.Surjective f ∧\n ContinuousConstSMul G E ∧ IsCancelSMul G E ∧ ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.o... | by
rintro ⟨_, ⟨_, ⟨x, hx, rfl⟩, rfl⟩, y, hy, eq⟩
have := h.2.2.2.1
apply IsCancelSMul.right_cancel _ _ x.1
simp_rw [← eq, one_smul]
refine congr($(H.injective <| Prod.ext (Subtype.ext ?_) <| hy.trans hx.symm))
simp_rw [hH]
exact h.2.2.2.2.mpr ⟨_, eq.symm⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Basic | {
"line": 146,
"column": 57
} | {
"line": 147,
"column": 18
} | [
{
"pp": "A : Type u_1\ninst✝⁶ : NormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : NormedAlgebra ℝ A\ninst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : NonnegSpectrumClass ℝ A\nr : ℝ\na : A\nhr : 0 < r\nha : IsStrictlyPositive a\n⊢ log (r • a) = (alge... | by
grind [log_smul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.Unitary.Span | {
"line": 61,
"column": 85
} | {
"line": 62,
"column": 97
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : ↥(selfAdjoint A)\nha_norm : ‖a‖ ≤ 1\n⊢ star ↑(unitarySelfAddISMul a ha_norm) = ↑a - I • CFC.sqrt (1 - ↑a ^ 2)",
"usedConstants": [
"NegZeroClass.toNeg",
"CStarAlgebra.toNonUnitalCStarAlgebra",... | by
simp [IsSelfAdjoint.star_eq, ← sub_eq_add_neg, (CFC.sqrt_nonneg (1 - a ^ 2 : A)).isSelfAdjoint] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.Unitary.Span | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 39
} | [
{
"pp": "case inl\nA : Type u_1\ninst✝ : CStarAlgebra A\nx✝¹ : (A : Type ?u.39122.6) → [NonUnitalCStarAlgebra A] → PartialOrder A := spectralOrder\nx✝ : ∀ (A : Type ?u.39122.17) [inst : NonUnitalCStarAlgebra A], StarOrderedRing A := spectralOrderedRing\n⊢ ∃ u c, 0 = ∑ i, c i • ↑(u i) ∧ ∀ (i : Fin 4), ‖c i‖ ≤ ‖0... | · exact ⟨![1, -1, 1, -1], 0, by simp⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 367,
"column": 4
} | {
"line": 369,
"column": 47
} | [
{
"pp": "case inl\nk : 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\ns : AffineSubspace k P\ninst✝ : FiniteDimensional k ↥s.direction\np : P\nhs : ↑s = ∅\n⊢ FiniteDimensional k ↥(affineSpan k (insert p ↑(affin... | rw [coe_eq_bot_iff] at hs
rw [hs, bot_coe, span_empty, bot_coe, direction_affineSpan]
convert! finiteDimensional_bot k V <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 367,
"column": 4
} | {
"line": 369,
"column": 47
} | [
{
"pp": "case inl\nk : 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\ns : AffineSubspace k P\ninst✝ : FiniteDimensional k ↥s.direction\np : P\nhs : ↑s = ∅\n⊢ FiniteDimensional k ↥(affineSpan k (insert p ↑(affin... | rw [coe_eq_bot_iff] at hs
rw [hs, bot_coe, span_empty, bot_coe, direction_affineSpan]
convert! finiteDimensional_bot k V <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 183,
"column": 58
} | {
"line": 184,
"column": 44
} | [
{
"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.orthogonalProjection = U.subtypeL",
"usedConstants": [
"LinearIsometryEquiv.instEquivLike",
... | by
rw [← U.adjoint_subtypeL, adjoint_adjoint] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 210,
"column": 49
} | {
"line": 210,
"column": 57
} | [
{
"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\nT : E →L[𝕜] F\nx✝¹ : E\nx✝ : x✝¹ ∈ (↑T).ker\n⊢ x✝¹ ∈ (↑(adjo... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 210,
"column": 49
} | {
"line": 210,
"column": 57
} | [
{
"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\nT : E →L[𝕜] F\nx✝¹ : E\nx✝ : x✝¹ ∈ (↑T).ker\n⊢ x✝¹ ∈ (↑(adjo... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 210,
"column": 49
} | {
"line": 210,
"column": 57
} | [
{
"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\nT : E →L[𝕜] F\nx✝¹ : E\nx✝ : x✝¹ ∈ (↑T).ker\n⊢ x✝¹ ∈ (↑(adjo... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 10
} | [
{
"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\nT : E →L[𝕜] F\nx✝¹ : E\nx✝ : x✝¹ ∈ (↑(adjoint T ∘SL T)).ker\... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 789,
"column": 2
} | {
"line": 789,
"column": 30
} | [
{
"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\ns : Set P\nh : finrank k V = 2\nthis : FiniteDimensional k V\n⊢ finrank k ↥(vectorSpan k s) ≤ finrank k V",
"usedConstants": [
"IsNoetherianRing.strongRank... | exact Submodule.finrank_le _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Group.Integral | {
"line": 48,
"column": 2
} | {
"line": 51,
"column": 16
} | [
{
"pp": "G : Type u_4\nF : Type u_6\ninst✝⁴ : MeasurableSpace G\ninst✝³ : NormedAddCommGroup F\nμ : Measure G\ninst✝² : Group G\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsInvInvariant\nf : G → F\ns : Set G\nhf : IntegrableOn f s μ\n⊢ IntegrableOn (fun x ↦ f x⁻¹) s⁻¹ μ",
"usedConstants": [
"MeasurableEquiv... | apply (integrable_map_equiv (MeasurableEquiv.inv G) f).mp
have : s⁻¹ = MeasurableEquiv.inv G ⁻¹' s := by simp
rw [this, ← MeasurableEquiv.restrict_map]
simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Group.Integral | {
"line": 48,
"column": 2
} | {
"line": 51,
"column": 16
} | [
{
"pp": "G : Type u_4\nF : Type u_6\ninst✝⁴ : MeasurableSpace G\ninst✝³ : NormedAddCommGroup F\nμ : Measure G\ninst✝² : Group G\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsInvInvariant\nf : G → F\ns : Set G\nhf : IntegrableOn f s μ\n⊢ IntegrableOn (fun x ↦ f x⁻¹) s⁻¹ μ",
"usedConstants": [
"MeasurableEquiv... | apply (integrable_map_equiv (MeasurableEquiv.inv G) f).mp
have : s⁻¹ = MeasurableEquiv.inv G ⁻¹' s := by simp
rw [this, ← MeasurableEquiv.restrict_map]
simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.UniformLimitsDeriv | {
"line": 203,
"column": 6
} | {
"line": 203,
"column": 24
} | [
{
"pp": "ι : 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 : E\n... | simp_rw [mul_comm] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.SpecialFunctions.Sqrt | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 94
} | [
{
"pp": "case inr.left\nx : ℝ\nhx✝ : x ≠ 0\nhx : 0 < x\nthis : 2 * √x ^ (2 - 1) ≠ 0\n⊢ HasStrictDerivAt (fun x ↦ √x) (1 / (2 * √x)) x",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"Real",
"DivInvMonoid... | · simpa using sqPartialHomeomorph.hasStrictDerivAt_symm hx this (hasStrictDerivAt_pow 2 _) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 200,
"column": 4
} | {
"line": 201,
"column": 27
} | [
{
"pp": "case inr.inl\nα : Type u_3\nE : α → Type u_4\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhf : Memℓp f ∞\n⊢ Memℓp (-f) ∞",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NegZeroClass.toNeg",
... | apply memℓp_infty
simpa using hf.bddAbove | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 200,
"column": 4
} | {
"line": 201,
"column": 27
} | [
{
"pp": "case inr.inl\nα : Type u_3\nE : α → Type u_4\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhf : Memℓp f ∞\n⊢ Memℓp (-f) ∞",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NegZeroClass.toNeg",
... | apply memℓp_infty
simpa using hf.bddAbove | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 197,
"column": 2
} | {
"line": 203,
"column": 30
} | [
{
"pp": "α : Type u_3\nE : α → Type u_4\np : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhf : Memℓp f p\n⊢ Memℓp (-f) p",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NegZeroClass.toNeg",
... | rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
simp [hf.finite_dsupport]
· apply memℓp_infty
simpa using hf.bddAbove
· apply memℓp_gen
simpa using hf.summable hp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 197,
"column": 2
} | {
"line": 203,
"column": 30
} | [
{
"pp": "α : Type u_3\nE : α → Type u_4\np : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : (i : α) → E i\nhf : Memℓp f p\n⊢ Memℓp (-f) p",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NegZeroClass.toNeg",
... | rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
simp [hf.finite_dsupport]
· apply memℓp_infty
simpa using hf.bddAbove
· apply memℓp_gen
simpa using hf.summable hp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Ball.Homeomorph | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 29
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nP : Type u_2\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor E P\nc : P\nr : ℝ\nhr : 0 < r\n⊢ ball c r ⊆ (univBall c r).target",
"usedConstants": [
"OpenPartialHomeomorph.univBall",
"Eq.mpr",
... | rw [univBall_target c hr] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Module.Ball.Homeomorph | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 29
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nP : Type u_2\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor E P\nc : P\nr : ℝ\nhr : 0 < r\n⊢ ball c r ⊆ (univBall c r).target",
"usedConstants": [
"OpenPartialHomeomorph.univBall",
"Eq.mpr",
... | rw [univBall_target c hr] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.Ball.Homeomorph | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 29
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nP : Type u_2\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor E P\nc : P\nr : ℝ\nhr : 0 < r\n⊢ ball c r ⊆ (univBall c r).target",
"usedConstants": [
"OpenPartialHomeomorph.univBall",
"Eq.mpr",
... | rw [univBall_target c hr] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 776,
"column": 4
} | {
"line": 777,
"column": 27
} | [
{
"pp": "case inr.inl\nα : Type u_3\nE : α → Type u_4\ninst✝² : (i : α) → NormedAddCommGroup (E i)\ninst✝¹ : (i : α) → StarAddMonoid (E i)\ninst✝ : ∀ (i : α), NormedStarGroup (E i)\nf : (i : α) → E i\nhf : Memℓp f ∞\n⊢ Memℓp (star f) ∞",
"usedConstants": [
"norm_star",
"Norm.norm",
"Eq.mpr... | apply memℓp_infty
simpa using hf.bddAbove | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 776,
"column": 4
} | {
"line": 777,
"column": 27
} | [
{
"pp": "case inr.inl\nα : Type u_3\nE : α → Type u_4\ninst✝² : (i : α) → NormedAddCommGroup (E i)\ninst✝¹ : (i : α) → StarAddMonoid (E i)\ninst✝ : ∀ (i : α), NormedStarGroup (E i)\nf : (i : α) → E i\nhf : Memℓp f ∞\n⊢ Memℓp (star f) ∞",
"usedConstants": [
"norm_star",
"Norm.norm",
"Eq.mpr... | apply memℓp_infty
simpa using hf.bddAbove | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 773,
"column": 2
} | {
"line": 779,
"column": 30
} | [
{
"pp": "α : Type u_3\nE : α → Type u_4\np : ℝ≥0∞\ninst✝² : (i : α) → NormedAddCommGroup (E i)\ninst✝¹ : (i : α) → StarAddMonoid (E i)\ninst✝ : ∀ (i : α), NormedStarGroup (E i)\nf : (i : α) → E i\nhf : Memℓp f p\n⊢ Memℓp (star f) p",
"usedConstants": [
"norm_star",
"Norm.norm",
"Eq.mpr",
... | rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
simp [hf.finite_dsupport]
· apply memℓp_infty
simpa using hf.bddAbove
· apply memℓp_gen
simpa using hf.summable hp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 773,
"column": 2
} | {
"line": 779,
"column": 30
} | [
{
"pp": "α : Type u_3\nE : α → Type u_4\np : ℝ≥0∞\ninst✝² : (i : α) → NormedAddCommGroup (E i)\ninst✝¹ : (i : α) → StarAddMonoid (E i)\ninst✝ : ∀ (i : α), NormedStarGroup (E i)\nf : (i : α) → E i\nhf : Memℓp f p\n⊢ Memℓp (star f) p",
"usedConstants": [
"norm_star",
"Norm.norm",
"Eq.mpr",
... | rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
simp [hf.finite_dsupport]
· apply memℓp_infty
simpa using hf.bddAbove
· apply memℓp_gen
simpa using hf.summable hp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.UniformLimitsDeriv | {
"line": 371,
"column": 8
} | {
"line": 371,
"column": 32
} | [
{
"pp": "case refine_2\nι : 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' : ... | hasFDerivAt_iff_tendsto, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 1250,
"column": 39
} | {
"line": 1252,
"column": 80
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\nα : Type u_3\nE : α → Type u_4\np✝ q✝ : ℝ≥0∞\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\ninst✝³ : NormedRing 𝕜\ninst✝² : (i : α) → Module 𝕜 (E i)\ninst✝¹ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)\np q r : ℝ≥0∞\ninst✝ : Fact (1 ≤ p)\ni : α\nx : ↥(lp E p)\n⊢ ‖(evalₗ E p i) x‖... | by
have hp : p ≠ 0 := zero_lt_one.trans_le Fact.out |>.ne'
simpa only [evalₗ_apply, one_mul, ge_iff_le] using norm_apply_le_norm hp x i | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.UniformLimitsDeriv | {
"line": 519,
"column": 2
} | {
"line": 520,
"column": 46
} | [
{
"pp": "ι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nG : Type u_3\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\ninst✝¹ : IsRCLikeNormedField 𝕜\ninst✝ : l.NeBot\nhf' : TendstoUniformly f' g' l\nhf : ∀ᶠ (n ... | have hf : ∀ᶠ n in l, ∀ x : 𝕜, x ∈ Set.univ → HasDerivAt (f n) (f' n x) x := by
filter_upwards [hf] with n h x _ using h x | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 1371,
"column": 10
} | {
"line": 1371,
"column": 59
} | [
{
"pp": "case h₁\nα : Type u_3\nι : Type u_5\ninst✝ : PseudoMetricSpace α\ng : α → ι → ℝ\nK : ℝ≥0\nhg : ∀ (i : ι), LipschitzWith K fun x ↦ g x i\na₀ a : α\nM : ℝ\nhM : M ∈ upperBounds (Set.range fun i ↦ ‖g a₀ i‖)\ni : ι\n⊢ |g a i - g a₀ i| ≤ ↑K * dist a a₀",
"usedConstants": [
"Real.instLE",
"Re... | exact lipschitzWith_iff_dist_le_mul.1 (hg i) a a₀ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 1371,
"column": 10
} | {
"line": 1371,
"column": 59
} | [
{
"pp": "case h₁\nα : Type u_3\nι : Type u_5\ninst✝ : PseudoMetricSpace α\ng : α → ι → ℝ\nK : ℝ≥0\nhg : ∀ (i : ι), LipschitzWith K fun x ↦ g x i\na₀ a : α\nM : ℝ\nhM : M ∈ upperBounds (Set.range fun i ↦ ‖g a₀ i‖)\ni : ι\n⊢ |g a i - g a₀ i| ≤ ↑K * dist a a₀",
"usedConstants": [
"Real.instLE",
"Re... | exact lipschitzWith_iff_dist_le_mul.1 (hg i) a a₀ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 1371,
"column": 10
} | {
"line": 1371,
"column": 59
} | [
{
"pp": "case h₁\nα : Type u_3\nι : Type u_5\ninst✝ : PseudoMetricSpace α\ng : α → ι → ℝ\nK : ℝ≥0\nhg : ∀ (i : ι), LipschitzWith K fun x ↦ g x i\na₀ a : α\nM : ℝ\nhM : M ∈ upperBounds (Set.range fun i ↦ ‖g a₀ i‖)\ni : ι\n⊢ |g a i - g a₀ i| ≤ ↑K * dist a a₀",
"usedConstants": [
"Real.instLE",
"Re... | exact lipschitzWith_iff_dist_le_mul.1 (hg i) a a₀ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.SmoothTransition | {
"line": 129,
"column": 2
} | {
"line": 135,
"column": 56
} | [
{
"pp": "n : ℕ∞\np : ℝ[X]\nm : ℕ\n⊢ ContDiff ℝ ↑m fun x ↦ Polynomial.eval x⁻¹ p * expNegInvGlue x",
"usedConstants": [
"Differentiable",
"Iff.mpr",
"NormedCommRing.toNormedRing",
"Polynomial.derivative",
"Eq.mpr",
"Polynomial.eval",
"NormedCommRing.toSeminormedCommR... | induction m generalizing p with
| zero => exact contDiff_zero.2 <| continuous_polynomial_eval_inv_mul _
| succ m ihm =>
rw [show ((m + 1 : ℕ) : WithTop ℕ∞) = m + 1 from rfl]
refine contDiff_succ_iff_deriv.2 ⟨differentiable_polynomial_eval_inv_mul _, by simp, ?_⟩
convert! ihm (X ^ 2 * (p - derivative (R ... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.Calculus.ParametricIntegral | {
"line": 128,
"column": 6
} | {
"line": 128,
"column": 30
} | [
{
"pp": "case pos\nα : 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... | hasFDerivAt_iff_tendsto, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ParametricIntegral | {
"line": 160,
"column": 9
} | {
"line": 160,
"column": 33
} | [
{
"pp": "case pos.h_lim\nα : 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\nbou... | hasFDerivAt_iff_tendsto, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Haar.NormedSpace | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 83
} | [
{
"pp": "F : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ng : ℝ → F\na : ℝ\n⊢ ∫ (x : ℝ), g (a * x) = |a⁻¹| • ∫ (y : ℝ), g y",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"instHSMul",
"instIsAddHaarMeasureVolume",
"instSMulO... | simp_rw [← smul_eq_mul, Measure.integral_comp_smul, Module.finrank_self, pow_one] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Measure.Haar.NormedSpace | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 83
} | [
{
"pp": "F : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ng : ℝ → F\na : ℝ\n⊢ ∫ (x : ℝ), g (a * x) = |a⁻¹| • ∫ (y : ℝ), g y",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"instHSMul",
"instIsAddHaarMeasureVolume",
"instSMulO... | simp_rw [← smul_eq_mul, Measure.integral_comp_smul, Module.finrank_self, pow_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.NormedSpace | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 83
} | [
{
"pp": "F : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ng : ℝ → F\na : ℝ\n⊢ ∫ (x : ℝ), g (a * x) = |a⁻¹| • ∫ (y : ℝ), g y",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"instHSMul",
"instIsAddHaarMeasureVolume",
"instSMulO... | simp_rw [← smul_eq_mul, Measure.integral_comp_smul, Module.finrank_self, pow_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.EverywherePos | {
"line": 297,
"column": 4
} | {
"line": 297,
"column": 33
} | [
{
"pp": "case refine_1\nG : 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.InnerRegularCompact... | exact ⟨f, f_cont, f_comp, Lf⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension | {
"line": 490,
"column": 10
} | {
"line": 492,
"column": 74
} | [
{
"pp": "E✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E✝\ninst✝⁴ : NormedSpace ℝ E✝\ninst✝³ : FiniteDimensional ℝ E✝\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nthis✝¹ : MeasurableSpace E := borel E\nthis✝ : BorelSpace E\nIR : ∀ (R : ℝ), 1 < R → 0 < (R - ... | apply this.congr
rintro ⟨R, x⟩ ⟨hR : 1 < R, _⟩
simp only [hR, uncurry_apply_pair, if_true, Function.comp_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension | {
"line": 490,
"column": 10
} | {
"line": 492,
"column": 74
} | [
{
"pp": "E✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E✝\ninst✝⁴ : NormedSpace ℝ E✝\ninst✝³ : FiniteDimensional ℝ E✝\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nthis✝¹ : MeasurableSpace E := borel E\nthis✝ : BorelSpace E\nIR : ∀ (R : ℝ), 1 < R → 0 < (R - ... | apply this.congr
rintro ⟨R, x⟩ ⟨hR : 1 < R, _⟩
simp only [hR, uncurry_apply_pair, if_true, Function.comp_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.BumpFunction.Normed | {
"line": 117,
"column": 54
} | {
"line": 120,
"column": 18
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : HasContDiffBump E\ninst✝³ : MeasurableSpace E\nc : E\nf : ContDiffBump c\nμ : Measure E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : IsLocallyFiniteMeasure μ\n⊢ ∫ (x : E), ↑f x ∂μ = ∫ (x : E) in closedBal... | by
apply (setIntegral_eq_integral_of_forall_compl_eq_zero (fun x hx ↦ ?_)).symm
apply f.zero_of_le_dist (le_of_lt _)
simpa using hx | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 349,
"column": 4
} | {
"line": 349,
"column": 16
} | [
{
"pp": "α : 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.ind... | intro j ji _ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 402,
"column": 2
} | {
"line": 403,
"column": 89
} | [
{
"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, ... | have int_f_ne_zero : ∫ (x : G), f x ∂(map φ μ) ≠ 0 :=
ne_of_gt (f_cont.integral_pos_of_hasCompactSupport_nonneg_nonzero hf.1 hf.2.1 hf.2.2) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convolution | {
"line": 963,
"column": 4
} | {
"line": 964,
"column": 49
} | [
{
"pp": "case pos.e_f.h.inl\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✝ :... | · rw [indicator_of_notMem (notMem_Ioo_of_le ht), indicator_of_notMem (notMem_Ioi.mpr ht),
map_zero, ContinuousLinearMap.zero_apply] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 507,
"column": 4
} | {
"line": 514,
"column": 55
} | [
{
"pp": "case inr.refine_2\nα : 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), ⋃ (... | refine range_subset_iff.2 fun b => ?_
obtain ⟨a, ha⟩ :
∃ a : Ordinal, a < p.lastStep ∧ dist (p.c b) (p.c (p.index a)) < p.r (p.index a) := by
simpa only [iUnionUpTo, exists_prop, mem_iUnion, mem_ball, Subtype.exists,
Subtype.coe_mk] using p.mem_iUnionUpTo_lastStep b
simp only [s, exists_prop... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 507,
"column": 4
} | {
"line": 514,
"column": 55
} | [
{
"pp": "case inr.refine_2\nα : 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), ⋃ (... | refine range_subset_iff.2 fun b => ?_
obtain ⟨a, ha⟩ :
∃ a : Ordinal, a < p.lastStep ∧ dist (p.c b) (p.c (p.index a)) < p.r (p.index a) := by
simpa only [iUnionUpTo, exists_prop, mem_iUnion, mem_ball, Subtype.exists,
Subtype.coe_mk] using p.mem_iUnionUpTo_lastStep b
simp only [s, exists_prop... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 571,
"column": 4
} | {
"line": 574,
"column": 53
} | [
{
"pp": "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, ... | obtain ⟨i, y, hxy, h'⟩ :
∃ (i : Fin N) (i_1 : ↥s), i_1 ∈ u i ∧ x ∈ ball (↑i_1) (r ↑i_1) := by
have : x ∈ range a.c := by simpa only [a, Subtype.range_coe_subtype, setOf_mem_eq]
simpa only [mem_iUnion, bex_def] using hu' this | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 595,
"column": 4
} | {
"line": 595,
"column": 27
} | [
{
"pp": "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, ... | rw [ENNReal.inv_lt_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 678,
"column": 2
} | {
"line": 679,
"column": 16
} | [
{
"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 : CompactSpace G\nμ : Measure G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : IsFiniteMeasureOnCompacts μ\n⊢ μ.InnerRegular",
"usedConstants": [
"Eq.... | rw [isMulInvariant_eq_smul_of_compactSpace μ haar]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 678,
"column": 2
} | {
"line": 679,
"column": 16
} | [
{
"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 : CompactSpace G\nμ : Measure G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : IsFiniteMeasureOnCompacts μ\n⊢ μ.InnerRegular",
"usedConstants": [
"Eq.... | rw [isMulInvariant_eq_smul_of_compactSpace μ haar]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 685,
"column": 2
} | {
"line": 686,
"column": 16
} | [
{
"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 : CompactSpace G\nμ : Measure G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : IsFiniteMeasureOnCompacts μ\n⊢ μ.Regular",
"usedConstants": [
"Eq.mpr",... | rw [isMulInvariant_eq_smul_of_compactSpace μ haar]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 685,
"column": 2
} | {
"line": 686,
"column": 16
} | [
{
"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 : CompactSpace G\nμ : Measure G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : IsFiniteMeasureOnCompacts μ\n⊢ μ.Regular",
"usedConstants": [
"Eq.mpr",... | rw [isMulInvariant_eq_smul_of_compactSpace μ haar]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 801,
"column": 6
} | {
"line": 801,
"column": 51
} | [
{
"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... | have : Countable m := countable_univ_iff.mp C | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace | {
"line": 399,
"column": 63
} | {
"line": 450,
"column": 11
} | [
{
"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 < ‖a.c i‖\nhij : ‖a.c i‖ ≤ ‖a.c j‖\n⊢ 1 - δ ≤ ‖(2 / ‖a.c i‖) ... | by
have ah :
Pairwise fun i j => a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨
a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j := by
simpa only [dist_eq_norm] using a.h
have δnonneg : 0 ≤ δ := by linarith only [hτ, hδ1]
have hcrj : ‖a.c j‖ ≤ a.r j + 1 := by simpa only [lastc, lastr, dist_zero_right]... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace | {
"line": 467,
"column": 35
} | {
"line": 467,
"column": 42
} | [
{
"pp": "case inl\nE : 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\nhδ2 : δ ≤ 1\nc' : Fin N.succ → E := fun i ↦ if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) •... | if_true | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 930,
"column": 4
} | {
"line": 930,
"column": 30
} | [
{
"pp": "α : 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 : ∀ x ∈ s, ∀ δ > 0, (f ... | simp only [r, if_neg this] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 363,
"column": 4
} | {
"line": 364,
"column": 54
} | [
{
"pp": "case hz\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... | simpa [norm_iteratedFDerivWithin_zero, Nat.factorial_zero, algebraMap.coe_one, one_mul,
pow_zero, mul_one, comp_apply] using hC 0 le_rfl | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 363,
"column": 4
} | {
"line": 364,
"column": 54
} | [
{
"pp": "case hz\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... | simpa [norm_iteratedFDerivWithin_zero, Nat.factorial_zero, algebraMap.coe_one, one_mul,
pow_zero, mul_one, comp_apply] using hC 0 le_rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 363,
"column": 4
} | {
"line": 364,
"column": 54
} | [
{
"pp": "case hz\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... | simpa [norm_iteratedFDerivWithin_zero, Nat.factorial_zero, algebraMap.coe_one, one_mul,
pow_zero, mul_one, comp_apply] using hC 0 le_rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Alternating.Curry | {
"line": 100,
"column": 47
} | {
"line": 100,
"column": 70
} | [
{
"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\ng : F [⋀^Fin (n + 1)]→L[𝕜] G... | cases i using Fin.cases | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 60,
"column": 82
} | {
"line": 60,
"column": 90
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nh : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\ng₀ : a < b\n⊢ ∀ x ∈ {b}, ContinuousAt f x",
"usedConstants": [
"Real",
"congrArg",
"ContinuousAt",
"PseudoMetricSpace.toUniformSpace",
"Membership.mem",
"Set.instSingletonSet",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 60,
"column": 82
} | {
"line": 60,
"column": 90
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nh : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\ng₀ : a < b\n⊢ ∀ x ∈ {b}, ContinuousAt f x",
"usedConstants": [
"Real",
"congrArg",
"ContinuousAt",
"PseudoMetricSpace.toUniformSpace",
"Membership.mem",
"Set.instSingletonSet",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 60,
"column": 82
} | {
"line": 60,
"column": 90
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nh : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\ng₀ : a < b\n⊢ ∀ x ∈ {b}, ContinuousAt f x",
"usedConstants": [
"Real",
"congrArg",
"ContinuousAt",
"PseudoMetricSpace.toUniformSpace",
"Membership.mem",
"Set.instSingletonSet",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 68,
"column": 81
} | {
"line": 68,
"column": 89
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nh : ContinuousAt f a\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\ng₀ : a < b\n⊢ ∀ x ∈ {a}, ContinuousAt f x",
"usedConstants": [
"Real",
"congrArg",
"ContinuousAt",
"PseudoMetricSpace.toUniformSpace",
"Membership.mem",
"Set.instSingletonSet",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 68,
"column": 81
} | {
"line": 68,
"column": 89
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nh : ContinuousAt f a\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\ng₀ : a < b\n⊢ ∀ x ∈ {a}, ContinuousAt f x",
"usedConstants": [
"Real",
"congrArg",
"ContinuousAt",
"PseudoMetricSpace.toUniformSpace",
"Membership.mem",
"Set.instSingletonSet",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 68,
"column": 81
} | {
"line": 68,
"column": 89
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nh : ContinuousAt f a\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\ng₀ : a < b\n⊢ ∀ x ∈ {a}, ContinuousAt f x",
"usedConstants": [
"Real",
"congrArg",
"ContinuousAt",
"PseudoMetricSpace.toUniformSpace",
"Membership.mem",
"Set.instSingletonSet",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 76,
"column": 81
} | {
"line": 76,
"column": 89
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\ng₀ : a ≤ b\n⊢ ∀ x ∈ {a, b}, ContinuousAt f x",
"usedConstants": [
"Real",
"congrArg",
"and_self",
"ContinuousAt",
"forall_eq_or_imp._simp_1",
"PseudoMetricSpac... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 76,
"column": 81
} | {
"line": 76,
"column": 89
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\ng₀ : a ≤ b\n⊢ ∀ x ∈ {a, b}, ContinuousAt f x",
"usedConstants": [
"Real",
"congrArg",
"and_self",
"ContinuousAt",
"forall_eq_or_imp._simp_1",
"PseudoMetricSpac... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 76,
"column": 81
} | {
"line": 76,
"column": 89
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\ng₀ : a ≤ b\n⊢ ∀ x ∈ {a, b}, ContinuousAt f x",
"usedConstants": [
"Real",
"congrArg",
"and_self",
"ContinuousAt",
"forall_eq_or_imp._simp_1",
"PseudoMetricSpac... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 100,
"column": 83
} | {
"line": 100,
"column": 91
} | [
{
"pp": "case refine_1.hf'\nf : ℝ → ℝ\na b c : ℝ\nh : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ DifferentiableOn ℝ f (interior (Ioc a b))",
"usedConstants": [
"Real.instIsOrdered... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 100,
"column": 83
} | {
"line": 100,
"column": 91
} | [
{
"pp": "case refine_1.hf'_nonneg\nf : ℝ → ℝ\na b c : ℝ\nh : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ ∀ x ∈ interior (Ioc a b), 0 ≤ deriv f x",
"usedConstants": [
"Real.instIsOr... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 101,
"column": 83
} | {
"line": 101,
"column": 91
} | [
{
"pp": "case refine_2.hf'\nf : ℝ → ℝ\na b c : ℝ\nh : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x\nh₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0\n⊢ DifferentiableOn ℝ f (interior (Ico b c))",
"usedConstants": [
"Real.instIsOrdered... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
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