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.Normed.Operator.Extend | {
"line": 354,
"column": 15
} | {
"line": 354,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_3\nEₗ : Type u_4\nF : Type u_5\nFₗ : Type u_6\ninst✝¹³ : NormedField 𝕜\ninst✝¹² : NormedField 𝕜₂\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : Module 𝕜 E\ninst✝⁹ : AddCommGroup F\ninst✝⁸ : Module 𝕜₂ F\ninst✝⁷ : NormedAddCommGroup Eₗ\ninst✝⁶ : NormedSpace 𝕜 Eₗ\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 285,
"column": 4
} | {
"line": 285,
"column": 54
} | [
{
"pp": "case pos\nα : Type u_1\nG : Type u_5\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace ℝ G\nm : MeasurableSpace α\nμ : Measure α\nR : Type u_6\ninst✝³ : NormedRing R\ninst✝² : Module R G\ninst✝¹ : IsBoundedSMul R G\ninst✝ : SMulCommClass ℝ R G\nc : R\nf : α → G\nhf : Integrable f μ\nhG : CompleteSpa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 297,
"column": 2
} | {
"line": 297,
"column": 37
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nL : Type u_6\ninst✝ : RCLike L\nr : L\nf : α → L\n⊢ ∫ (a : α), f a / r ∂μ = (∫ (a : α), f a ∂μ) / r",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 673,
"column": 2
} | {
"line": 673,
"column": 47
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : PartialOrder E\ninst✝⁴ : IsOrderedAddMonoid E\ninst✝³ : IsOrderedModule ℝ E\ninst✝² : ClosedIciTopology E\nβ : Type u_6\ninst✝¹ : AddCommMonoid β\ninst✝ : Module ℝ β\nf : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 685,
"column": 2
} | {
"line": 685,
"column": 81
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0\nhfi : Integrable (fun x ↦ ↑(f x)) μ\n⊢ ∫⁻ (a : α), ↑(f a) ∂μ ≠ ∞",
"usedConstants": [
"Eq.mpr",
"ENNReal.ofNNReal",
"Preorder.toLT",
"PartialOrder.toPreorder",
"Preorder.toLE",
"_private.Mathlib.Mea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 702,
"column": 60
} | {
"line": 704,
"column": 32
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0\nhfi : Integrable (fun x ↦ ↑(f x)) μ\nb : ℝ≥0\n⊢ ∫⁻ (a : α), ↑(f a) ∂μ ≤ ↑b ↔ ∫ (a : α), ↑(f a) ∂μ ≤ ↑b",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"ENNReal.ofNNReal",
"ENNReal.ofReal",
"co... | by
rw [lintegral_coe_eq_integral f hfi, ENNReal.ofReal, ENNReal.coe_le_coe,
Real.toNNReal_le_iff_le_coe] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 715,
"column": 26
} | {
"line": 715,
"column": 60
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : 0 ≤ᶠ[ae μ] f\nhfi : Integrable f μ\n⊢ ∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ = 0 ∨ ¬∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ < ∞ ↔ f =ᶠ[ae μ] 0",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"NormedCommRing.toSeminorm... | ← hasFiniteIntegral_iff_ofReal hf, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 890,
"column": 6
} | {
"line": 893,
"column": 23
} | [
{
"pp": "case inr.const.inr\nα : Type u_1\nE : Type u_4\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ∞\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤... | have : μ s < ∞ := SimpleFunc.measure_lt_top_of_memLp_indicator hp_pos hp_ne_top hc hs Hs
rcases h0P c hs this εpos with ⟨g, hg, Pg⟩
rw [← eLpNorm_neg, neg_sub] at hg
exact ⟨g, hg, Pg⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 890,
"column": 6
} | {
"line": 893,
"column": 23
} | [
{
"pp": "case inr.const.inr\nα : Type u_1\nE : Type u_4\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ∞\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤... | have : μ s < ∞ := SimpleFunc.measure_lt_top_of_memLp_indicator hp_pos hp_ne_top hc hs Hs
rcases h0P c hs this εpos with ⟨g, hg, Pg⟩
rw [← eLpNorm_neg, neg_sub] at hg
exact ⟨g, hg, Pg⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 897,
"column": 6
} | {
"line": 897,
"column": 75
} | [
{
"pp": "case inr.add\nα : Type u_1\nE : Type u_4\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ∞\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧ P... | memLp_add_of_disjoint hff' f.stronglyMeasurable f'.stronglyMeasurable | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 15
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝ : NormedAddCommGroup β\nu v : α → β\nl : Filter α\nhuv : u ~[l] v\nhu : u =o[l] fun _x ↦ 1\n⊢ v =o[l] fun _x ↦ 1",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"Real",
"SeminormedA... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 53
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : NormedAddCommGroup β\nu v w : α → β\nl : Filter α\nhuv : u ~[l] v\nhwv : w =o[l] v\n⊢ u + w ~[l] v",
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"AddMonoid.toAddZeroClass",
"HSub.hSub",
"AddCommGrou... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 35
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : NormedAddCommGroup β\nu v w : α → β\nl : Filter α\nhuv : u ~[l] v\nhwv : w =o[l] v\n⊢ u - w ~[l] v",
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddMonoid.toAddZeroClass",
"sub_eq_add_neg",
"HSub.hSub",
"AddCommGroup.toAddGroup",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 797,
"column": 6
} | {
"line": 797,
"column": 27
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf : ∀ (n : ℕ), Integrable (f n) μ\nhF : Integrable F μ\nh_mono : ∀ᵐ (x : α) ∂μ, Antitone fun n ↦ f n x\nh_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))\nthis✝ : Tendsto (fun n ↦ ∫ (x : α), -f n x ∂μ) atTo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : NormedField β\nu v : α → β\nl : Filter α\nhuv : Tendsto (u / v) l (𝓝 1)\nh : ∃ᶠ (t : α) in l, (u / v) t = 0\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 348,
"column": 2
} | {
"line": 348,
"column": 25
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : PartialOrder F\ninst✝¹ : IsOrderedAddMonoid F\ninst✝ : IsOrderedModule ℝ F\nν : Measure α\nf : α →ₛ F\nhf : 0 ≤ᶠ[ae ν] ⇑f\nhμν : μ ≤ ν\nhfν : Integrable (⇑f) ν\n⊢ ∑ x ∈ f.... | apply Finset.sum_le_sum | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 358,
"column": 41
} | {
"line": 358,
"column": 52
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : PartialOrder F\ninst✝¹ : IsOrderedAddMonoid F\ninst✝ : IsOrderedModule ℝ F\nν : Measure α\nf : α →ₛ F\nhf : 0 ≤ᶠ[ae ν] ⇑f\nhμν : μ ≤ ν\nhfν : Integrable (⇑f) ν\nx : α\nhx ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 35
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\ninst✝ : NormedField β\nt u v w : α → β\nl : Filter α\nhtu : t ~[l] u\nhvw : v ~[l] w\n⊢ t / v ~[l] u / w",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 299,
"column": 12
} | {
"line": 299,
"column": 23
} | [
{
"pp": "case zero\nα : Type u_1\nβ : Type u_3\ninst✝ : NormedField β\nt u : α → β\nl : Filter α\nh : t ~[l] u\n⊢ t ^ 0 ~[l] u ^ 0",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"NormedField.toFiel... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 300,
"column": 17
} | {
"line": 300,
"column": 39
} | [
{
"pp": "case succ\nα : Type u_1\nβ : Type u_3\ninst✝ : NormedField β\nt u : α → β\nl : Filter α\nh : t ~[l] u\nn✝ : ℕ\nih : t ^ n✝ ~[l] u ^ n✝\n⊢ t ^ (n✝ + 1) ~[l] u ^ (n✝ + 1)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"HMul.hMul",
"Monoid.toMulOneClass",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 304,
"column": 19
} | {
"line": 304,
"column": 30
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\ninst✝ : NormedField β\nt u : α → β\nl : Filter α\nh : t ~[l] u\nz : ℤ\na✝ : ℕ\n⊢ t ^ Int.ofNat a✝ ~[l] u ^ Int.ofNat a✝",
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"congrArg",
"DivInvMonoid.toZPow",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 305,
"column": 21
} | {
"line": 305,
"column": 32
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\ninst✝ : NormedField β\nt u : α → β\nl : Filter α\nh : t ~[l] u\nz : ℤ\na✝ : ℕ\n⊢ t ^ Int.negSucc a✝ ~[l] u ^ Int.negSucc a✝",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"DivInvMonoid.toInv",
"congrArg",
"zpow_negSucc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 374,
"column": 24
} | {
"line": 374,
"column": 79
} | [
{
"pp": "case h\nα : Type u_1\nu v t w : α → ℝ\nl : Filter α\nhu : 0 ≤ v\nhw : 0 ≤ w\nhtu : u ~[l] v\nhvw : t ~[l] w\nx : α\n| v x + w x",
"usedConstants": [
"Iff.mpr",
"Real",
"abs",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"AddCommGroup.toAddGroup",... | rw [← abs_eq_self.mpr (hu x), ← abs_eq_self.mpr (hw x)] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 375,
"column": 2
} | {
"line": 375,
"column": 34
} | [
{
"pp": "α : Type u_1\nu v t w : α → ℝ\nl : Filter α\nhu : 0 ≤ v\nhw : 0 ≤ w\nhtu : u ~[l] v\nhvw : t ~[l] w\n⊢ (fun x ↦ (u - v) x + (t - w) x) =o[l] fun x ↦ |v x| + |w x|",
"usedConstants": [
"Real",
"abs",
"Real.instSub",
"HSub.hSub",
"AddCommGroup.toAddGroup",
"Distrib... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.AddTorsor | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 13
} | [
{
"pp": "V : Type u_2\nP : Type u_3\ninst✝² : SeminormedAddCommGroup V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\nv v' : V\np p' : P\n⊢ dist (v +ᵥ p) (v' +ᵥ p') ≤ dist v v' + dist p p'",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 461,
"column": 4
} | {
"line": 461,
"column": 25
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedSpace ℝ E\nf : ↥(simpleFunc E 1 μ)\n⊢ ‖{ toFun := integral, map_add' := ⋯, map_smul' := ⋯ } f‖ ≤ 1 * ‖f‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"MeasureTheory.Lp.s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.AddTorsor | {
"line": 201,
"column": 29
} | {
"line": 201,
"column": 40
} | [
{
"pp": "α : Type u_1\nV✝ : Type u_2\nP✝ : Type u_3\nW : Type u_4\nQ : Type u_5\ninst✝⁷ : SeminormedAddCommGroup V✝\ninst✝⁶ : PseudoMetricSpace P✝\ninst✝⁵ : NormedAddTorsor V✝ P✝\ninst✝⁴ : SeminormedAddCommGroup W\ninst✝³ : PseudoMetricSpace Q\ninst✝² : NormedAddTorsor W Q\nV : Type u_6\nP : Type u_7\ninst✝¹ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 472,
"column": 4
} | {
"line": 472,
"column": 11
} | [
{
"pp": "case h.e'_3\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ↥(simpleFunc ℝ 1 μ)\neq : ∀ (a : α), (toSimpleFunc f).posPart a = max ((toSimpleFunc f) a) 0\na✝¹ : α\na✝ : (toSimpleFunc (posPart f)) a✝¹ = ↑↑↑(posPart f) a✝¹\nh₂ : ↑↑(Lp.posPart ↑f) a✝¹ = max (↑↑↑f a✝¹) 0\nh₃ : (toSimpleFunc f) a✝¹ ... | rw [h₃] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1118,
"column": 4
} | {
"line": 1118,
"column": 37
} | [
{
"pp": "case inl\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf_nonneg : 0 ≤ᶠ[ae μ] f\nhf_int : Integrable f μ\nε : ℝ\nhμ : μ {x | ε ≤ f x} = ∞\n⊢ ε * μ.real {x | ε ≤ f x} ≤ ∫ (x : α), f x ∂μ",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"MeasureTheory.Me... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1117,
"column": 2
} | {
"line": 1125,
"column": 72
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf_nonneg : 0 ≤ᶠ[ae μ] f\nhf_int : Integrable f μ\nε : ℝ\n⊢ ε * μ.real {x | ε ≤ f x} ≤ ∫ (x : α), f x ∂μ",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"MeasureTheory.Integrable.mono_measur... | rcases eq_top_or_lt_top (μ {x | ε ≤ f x}) with hμ | hμ
· simpa [measureReal_def, hμ] using integral_nonneg_of_ae hf_nonneg
· have := Fact.mk hμ
calc
ε * μ.real { x | ε ≤ f x } = ∫ _ in {x | ε ≤ f x}, ε ∂μ := by simp [mul_comm]
_ ≤ ∫ x in {x | ε ≤ f x}, f x ∂μ :=
integral_mono_ae (integrable_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1117,
"column": 2
} | {
"line": 1125,
"column": 72
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf_nonneg : 0 ≤ᶠ[ae μ] f\nhf_int : Integrable f μ\nε : ℝ\n⊢ ε * μ.real {x | ε ≤ f x} ≤ ∫ (x : α), f x ∂μ",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"MeasureTheory.Integrable.mono_measur... | rcases eq_top_or_lt_top (μ {x | ε ≤ f x}) with hμ | hμ
· simpa [measureReal_def, hμ] using integral_nonneg_of_ae hf_nonneg
· have := Fact.mk hμ
calc
ε * μ.real { x | ε ≤ f x } = ∫ _ in {x | ε ≤ f x}, ε ∂μ := by simp [mul_comm]
_ ≤ ∫ x in {x | ε ≤ f x}, f x ∂μ :=
integral_mono_ae (integrable_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1151,
"column": 23
} | {
"line": 1151,
"column": 79
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_6\ninst✝ : NormedAddCommGroup E\nf g : α → E\np q : ℝ\nhpq : p.HolderConjugate q\nhf : MemLp f (ENNReal.ofReal p) μ\nhg : MemLp g (ENNReal.ofReal q) μ\nh_left : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a : α), ((fun x ↦ ‖f x‖ₑ) *... | ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) hpq.nonneg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.RCLike.Basic | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 34
} | [
{
"pp": "case inr\n𝕜 : Type u_1\ninst✝⁷ : RCLike 𝕜\n𝓕 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : FunLike 𝓕 E F\ninst✝¹ : AddMonoidHomClass 𝓕 E F\ninst✝ : MulActionHomClass 𝓕 𝕜 E F\nf ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.RieszLemma | {
"line": 98,
"column": 4
} | {
"line": 98,
"column": 15
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nR : ℝ\nhR : ‖c‖ < R\nF : Subspace 𝕜 E\nhFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F\nhF : ∃ x, x ∉ F\nRpos : 0 < R\n⊢ ‖c‖ < 1 * R",
"usedCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1238,
"column": 41
} | {
"line": 1246,
"column": 83
} | [
{
"pp": "F : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nβ : Type u_6\nm m0 : MeasurableSpace β\nμ : Measure β\nhm : m ≤ m0\nf : β →ₛ F\nhf_int : Integrable (⇑f) μ\n⊢ ∫ (x : β), f x ∂μ = ∫ (x : β), f x ∂μ.trim hm",
"usedConstants": [
"Eq.mpr",
"le_... | by
have hf : StronglyMeasurable[m] f := @SimpleFunc.stronglyMeasurable β F m _ f
have hf_int_m := hf_int.trim hm hf
rw [integral_simpleFunc_larger_space (le_refl m) f hf_int_m,
integral_simpleFunc_larger_space hm f hf_int]
congr with x
simp only [measureReal_def]
congr 2
exact (trim_measurableSet_eq h... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.RieszLemma | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 13
} | [
{
"pp": "case refine_2.ha\n𝕜 : Type u_4\ninst✝² : RCLike 𝕜\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nF : Subspace 𝕜 E\nhFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F\nhF : ∃ x, x ∉ F\nr : ℝ\nhr : r < 1\nx₀ : E\nhx₀ : x₀ ∉ F\nh : ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.LinearMap | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 37
} | [
{
"pp": "K : Type u\nV : Type v\nV' V'₁ : Type v'\ninst✝⁶ : Ring K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V'\ninst✝² : Module K V'\ninst✝¹ : AddCommGroup V'₁\ninst✝ : Module K V'₁\ng : V →ₗ[K] V'\nf : V' →ₗ[K] V'₁\n⊢ (f ∘ₗ g).rank ≤ g.rank",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.LinearMap | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 37
} | [
{
"pp": "K : Type u\nV : Type v\nV' V'₁ : Type v'\ninst✝⁶ : Ring K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup V'\ninst✝² : Module K V'\ninst✝¹ : AddCommGroup V'₁\ninst✝ : Module K V'₁\ng : V →ₗ[K] V'\nf : V' →ₗ[K] V'₁\n⊢ (f ∘ₗ g).rank ≤ min f.rank g.rank",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.LinearMap | {
"line": 115,
"column": 6
} | {
"line": 115,
"column": 27
} | [
{
"pp": "K : Type u\nV : Type v\nV' : Type v'\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : AddCommGroup V'\ninst✝ : Module K V'\nf : V →ₗ[K] V'\ng : ↥f.range →ₗ[K] V\nhg : f.rangeRestrict ∘ₗ g = id\ns : Set ↥f.range\nsi : LinearIndepOn K _root_.id s\nh : #↑s ≤ f.rank\nfg : ∀ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Affine.Isometry | {
"line": 323,
"column": 4
} | {
"line": 323,
"column": 47
} | [
{
"pp": "case mk.mk.e_toAffineEquiv\n𝕜 : Type u_1\nV : Type u_2\nV₁ : Type u_3\nV₁' : Type u_4\nV₂ : Type u_5\nV₃ : Type u_6\nV₄ : Type u_7\nP₁ : Type u_8\nP₁' : Type u_9\nP : Type u_10\nP₂ : Type u_11\nP₃ : Type u_12\nP₄ : Type u_13\ninst✝²⁴ : NormedField 𝕜\ninst✝²³ : SeminormedAddCommGroup V\ninst✝²² : Norm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.SumMeasure | {
"line": 91,
"column": 8
} | {
"line": 91,
"column": 19
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nE : Type u_3\ninst✝² : Countable ι\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\nf : X → E\ninst✝ : MeasurableSingletonClass X\nx : ι → X\nc : ι → ℝ≥0∞\nhc : ∀ (i : ι), c i ≠ ∞\nh : Summable fun i ↦ (c i).toReal * ‖f (x i)‖\n⊢ Summable fun i ↦ ∫ (x : X), ‖f x‖ ∂c i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.SumMeasure | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 13
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nE : Type u_3\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\nf : X → E\ninst✝ : MeasurableSingletonClass X\nx : ι → X\nc : ι → ℝ≥0∞\nhf : Integrable f (Measure.sum fun i ↦ c i • Measure.dirac (x i))\n⊢ Summable fun i ↦ (c i).toReal * ‖f (x i)‖",
"usedConstants": ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.SumMeasure | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 47
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nE : Type u_3\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\nμ : ι → Measure X\nf : X → E\ninst✝ : NormedSpace ℝ E\nhf : Integrable f (Measure.sum μ)\nhfi : ∀ (i : ι), Integrable f (μ i)\nε : ℝ≥0\nε0 : 0 < ↑ε\nhf_lt : ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < ∞\nhmem : ∀ᶠ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Bornology.BoundedOperation | {
"line": 118,
"column": 6
} | {
"line": 118,
"column": 36
} | [
{
"pp": "R : Type u_2\ninst✝² : Bornology R\ninst✝¹ : Monoid R\ninst✝ : BoundedMul R\ns : Set R\ns_bdd : Bornology.IsBounded s\nn : ℕ\nhn : Bornology.IsBounded ((fun x ↦ x ^ n) '' s)\nx y : R\ny_in_s : y ∈ s\nypow_eq_x : y ^ (n + 1) = x\n⊢ x ∈ (fun x ↦ x ^ n) '' s * s",
"usedConstants": [
"Eq.mpr",
... | rw [← ypow_eq_x, pow_succ y n] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Bornology.BoundedOperation | {
"line": 147,
"column": 6
} | {
"line": 147,
"column": 26
} | [
{
"pp": "case h.e'_3.h.mpr\nR : Type u_1\ninst✝² : PseudoMetricSpace R\ninst✝¹ : Monoid R\ninst✝ : LipschitzMul R\ns t : Set R\ns_bdd : Bornology.IsBounded s\nt_bdd : Bornology.IsBounded t\nbdd : Bornology.IsBounded (s ×ˢ t)\nC : ℝ≥0\nmul_lip : LipschitzWith C fun p ↦ p.1 * p.2\np a b : R\na_in_s : a ∈ s\nb_in_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.SumMeasure | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 13
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nE : Type u_3\ninst✝⁴ : Countable ι\nmX : MeasurableSpace X\ninst✝³ : NormedAddCommGroup E\nf : X → E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : MeasurableSingletonClass X\nx : ι → X\nc : ι → ℝ≥0∞\ninst✝ : CompleteSpace E\nhc : ∀ (i : ι), c i ≠ ∞\nhf : Summable fun i ↦ (c i).toReal ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Bornology.BoundedOperation | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝ : SeminormedAddCommGroup R\nx : R\n⊢ Tendsto (fun x_1 ↦ x_1 - x) (cobounded R) (cobounded R)",
"usedConstants": [
"Eq.mpr",
"PseudoMetricSpace.toBornology",
"congrArg",
"AddMonoid.toAddZeroClass",
"sub_eq_add_neg",
"HSub.hSub",
"AddCommG... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Bornology.BoundedOperation | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝ : SeminormedAddCommGroup R\nx : R\n⊢ Tendsto (fun x_1 ↦ x - x_1) (cobounded R) (cobounded R)",
"usedConstants": [
"Eq.mpr",
"PseudoMetricSpace.toBornology",
"congrArg",
"AddMonoid.toAddZeroClass",
"sub_eq_add_neg",
"HSub.hSub",
"AddCommG... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Bornology.BoundedOperation | {
"line": 254,
"column": 22
} | {
"line": 254,
"column": 53
} | [
{
"pp": "s t : Set ℝ≥0\nhs : Bornology.IsBounded s\nht : Bornology.IsBounded t\nAf : ℝ\nhAf : s ⊆ closedBall 0 Af\nAg : ℝ\nhAg : t ⊆ closedBall 0 Ag\nkey : IsCompact (closedBall 0 Af ×ˢ closedBall 0 Ag)\na✝ x : ℝ≥0\nx_in_s : x ∈ s\ny : ℝ≥0\ny_in_t : y ∈ t\nxy_eq : (fun x1 x2 ↦ x1 * x2) x y = a✝\n⊢ (x, y) ∈ clos... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Bounded.Basic | {
"line": 223,
"column": 8
} | {
"line": 223,
"column": 23
} | [
{
"pp": "case inl\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : PseudoMetricSpace β\nf g : α →ᵇ β\nh✝ : IsEmpty α\n⊢ dist f g = ⨆ x, dist (f x) (g x)",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"iSup",
"Real.instSupSet",
"BoundedContinuousFunction.i... | iSup_of_empty', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.ContinuousMap.Bounded.Basic | {
"line": 231,
"column": 25
} | {
"line": 231,
"column": 39
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : PseudoMetricSpace β\nf g : α →ᵇ β\n⊢ ↑(nndist f g) = ⨆ x, ↑(nndist (f x) (g x))",
"usedConstants": [
"Eq.mpr",
"NNDist.nndist",
"ENNReal.ofNNReal",
"congrArg",
"iSup",
"PseudoMetricSpace.toNNDist",
... | nndist_eq_iSup | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 175,
"column": 4
} | {
"line": 175,
"column": 69
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : CompleteSpace 𝕜\nh : ¬∃ s, Nonempty (Basis (↥s) 𝕜 E)\n⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun f ↦\n (if H : ∃ s, Nonempty (Basi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Bounded.Normed | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 13
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : SeminormedAddCommGroup β\nf : α →ᵇ β\nC : ℝ\nC0 : 0 ≤ C\n⊢ ‖f‖ ≤ C ↔ ∀ (x : α), ‖f x‖ ≤ C",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Bounded.Basic | {
"line": 631,
"column": 6
} | {
"line": 631,
"column": 40
} | [
{
"pp": "F : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : AddMonoid β\ninst✝ : LipschitzAdd β\nf g : α →ᵇ β\nx : α\nC : ℝ\nC_nonneg : 0 ≤ ↑(LipschitzAdd.C β)\n⊢ LipschitzWith (LipschitzAdd.C β) fun p ↦ p.1 + p.2",
"usedConstants": [
... | rw [lipschitzWith_iff_dist_le_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.ContinuousMap.Bounded.Normed | {
"line": 190,
"column": 25
} | {
"line": 190,
"column": 36
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : TopologicalSpace α\ninst✝ : SeminormedAddCommGroup β\nf✝ g : α →ᵇ β\nx : α\nC : ℝ\nn : ℤ\nf : α →ᵇ β\n⊢ ∃ C, ∀ (x y : α), dist ((n • f.toContinuousMap).toFun x) ((n • f.toContinuousMap).toFun y) ≤ C",
"usedConstants": [
"ContinuousMap.instZSMul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Bounded.Normed | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 39
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : SeminormedAddCommGroup β\nf : α →ᵇ β\n⊢ ‖f‖ₑ = ⨆ x, ‖f x‖ₑ",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"SeminormedAddGroup.toAddGroup",
"congr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Bounded.Normed | {
"line": 311,
"column": 42
} | {
"line": 311,
"column": 85
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\nR : Type u_1\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nf g : α →ᵇ R\nh : f * g = 0\n⊢ ∀ (x : α), f x = 0 ∨ g x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Bounded.Normed | {
"line": 325,
"column": 2
} | {
"line": 325,
"column": 30
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\nR : Type u_1\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nf g : α →ᵇ R\nh : f * g = 0\n⊢ ‖f - g‖ = max ‖f‖ ‖g‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"congrArg",
"AddMonoid.toAddZeroClass",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Bounded.Normed | {
"line": 341,
"column": 39
} | {
"line": 341,
"column": 61
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\nR : Type u_1\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nι : Type u_2\nf : ι → α →ᵇ R\nh : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)\nj : ι\ns : Finset ι\nhj : j ∉ s\nih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊\nthis : f j * ∑ i ∈ s, f i = 0\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Bounded.Normed | {
"line": 342,
"column": 4
} | {
"line": 342,
"column": 32
} | [
{
"pp": "case insert\nα : Type u\ninst✝² : TopologicalSpace α\nR : Type u_1\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nι : Type u_2\nf : ι → α →ᵇ R\nh : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)\nj : ι\ns : Finset ι\nhj : j ∉ s\nih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊\n⊢ f j * ∑ i ∈ s, ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Bounded.Normed | {
"line": 372,
"column": 25
} | {
"line": 372,
"column": 50
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : TopologicalSpace α\nR : Type u_1\ninst✝ : SeminormedRing R\nf : α →ᵇ R\nn : ℕ\n⊢ ∃ C, ∀ (x y : α), dist ((f.toContinuousMap ^ n).toFun x) ((f.toContinuousMap ^ n).toFun y) ≤ C",
"usedConstants": [
"Real.instLE",
"Real",
"Ring.toNonAssoc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Bounded.Normed | {
"line": 587,
"column": 35
} | {
"line": 587,
"column": 46
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : NormedAddCommGroup β\ninst✝² : Lattice β\ninst✝¹ : HasSolidNorm β\ninst✝ : IsOrderedAddMonoid β\nf g : α →ᵇ β\nh₁ : f ≤ g\nh : α →ᵇ β\nt : α\n⊢ (fun f ↦ f.toFun) (f + h) t ≤ (fun f ↦ f.toFun) (g + h) t",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.ThickenedIndicator | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 25
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEDist x E\nkey : 1 - infEDist x E / ENNReal.ofReal δ ≤ 1 - 1\n⊢ 1 - infEDist x E / ENNReal.ofReal δ ≤ ⊥",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Compact | {
"line": 150,
"column": 4
} | {
"line": 150,
"column": 41
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nE : Type u_3\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : CompactSpace α\ninst✝⁶ : PseudoMetricSpace β\ninst✝⁵ : SeminormedAddCommGroup E\nf✝ g✝ : C(α, β)\nC : ℝ\nR : Type u_4\ninst✝⁴ : Zero R\ninst✝³ : Zero β\ninst✝² : PseudoMetricSpace R\ninst✝¹ : SMul R β\ninst✝ : IsBoundedSMul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Compact | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 41
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nE : Type u_3\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : CompactSpace α\ninst✝⁶ : PseudoMetricSpace β\ninst✝⁵ : SeminormedAddCommGroup E\nf✝ g : C(α, β)\nC : ℝ\nR : Type u_4\ninst✝⁴ : Zero R\ninst✝³ : Zero β\ninst✝² : PseudoMetricSpace R\ninst✝¹ : SMul R β\ninst✝ : IsBoundedSMul ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Compact | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nE✝ : Type u_3\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : CompactSpace α\ninst✝⁴ : PseudoMetricSpace β\ninst✝³ : SeminormedAddCommGroup E✝\ninst✝² : Nonempty α\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : Nontrivial E\n⊢ NontrivialTopology C(α, E)",
"usedConstants": ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.ThickenedIndicator | {
"line": 151,
"column": 40
} | {
"line": 151,
"column": 91
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδseq : ℕ → ℝ\nE : Set α\nx : α\nx_mem_closure : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E\nε : ℝ\nε_pos : 0 < ε\nε_lt : ENNReal.ofReal ε < infEDist x E\nN : ℕ\nhN : ∀ b ≥ N, |δseq b| < ε\nn : ℕ\nn_large : n ≥ N\n⊢ x ∉ thickening ε E"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Compact | {
"line": 361,
"column": 61
} | {
"line": 361,
"column": 72
} | [
{
"pp": "case h\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\nR : Type u_4\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nf g : C(α, R)\nh : f * g = 0\nx : α\n⊢ (mkOfCompact f * mkOfCompact g) x = 0 x",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"IsTo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 717,
"column": 19
} | {
"line": 717,
"column": 57
} | [
{
"pp": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nhT : DominatedFinMeasAdditive μ T C\nhF : CompleteSpace F\n⊢ (setToFun... | setToFun_eq hT (integrable_zero _ _ _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.ContinuousMap.Compact | {
"line": 362,
"column": 2
} | {
"line": 362,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\nR : Type u_4\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nf g : C(α, R)\nh : mkOfCompact f * mkOfCompact g = 0\n⊢ ‖f + g‖ = max ‖f‖ ‖g‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Compact | {
"line": 372,
"column": 2
} | {
"line": 372,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\nR : Type u_4\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nf g : C(α, R)\nh : f * g = 0\n⊢ ‖f - g‖ = max ‖f‖ ‖g‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"ContinuousMap.instNorm"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Compact | {
"line": 388,
"column": 39
} | {
"line": 388,
"column": 61
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\nR : Type u_4\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nι : Type u_5\nf : ι → C(α, R)\nh : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)\nj : ι\ns : Finset ι\nhj : j ∉ s\nih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊\nth... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Compact | {
"line": 389,
"column": 4
} | {
"line": 389,
"column": 32
} | [
{
"pp": "case insert\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\nR : Type u_4\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nι : Type u_5\nf : ι → C(α, R)\nh : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)\nj : ι\ns : Finset ι\nhj : j ∉ s\nih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.ThickenedIndicator | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\n⊢ (ENNReal.toNNReal ∘ thickenedIndicatorAux δ E) x ≤ 1",
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
"Function.comp",
"id",
"NNReal",
"ENNReal.toNNReal",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Compact | {
"line": 464,
"column": 2
} | {
"line": 464,
"column": 30
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\ninst✝² : LocallyCompactSpace X\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nι : Type u_3\nF : ι → C(X, E)\nhF : ∀ (K : Compacts X), Summable fun i ↦ ‖restrict (↑K) (F i)‖\nK : Compacts X\nA : ∀ (s : Finset ι), restrict (↑K) (∑ i ∈ s, ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 358,
"column": 2
} | {
"line": 358,
"column": 13
} | [
{
"pp": "𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type w\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Fintype ι\nv : Basis ι 𝕜 E\nu : E →L[𝕜] F\nM : ℝ\nhM : 0 ≤ M\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 375,
"column": 2
} | {
"line": 375,
"column": 13
} | [
{
"pp": "𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type w\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\nv : Basis ι 𝕜 E\nC : ℝ≥0\nhC : C > 0\nh : ∀ {u : E →L... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 781,
"column": 2
} | {
"line": 781,
"column": 13
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nhT : DominatedFinMeasAdditive μ T C\nf : α → E\n⊢ setToFun μ (-T) ⋯ f = -setToFu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 415,
"column": 4
} | {
"line": 415,
"column": 15
} | [
{
"pp": "𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : CompleteSpace 𝕜\nc : 𝕜\nhc : 1 < ‖c‖\nR : ℝ\nhR : ‖c‖ < R\nh : ¬FiniteDimensional 𝕜 E\ns : Finset E\nF : Submodule 𝕜 E := Submodule.span 𝕜 ↑s\nhF : F.FG\nthis✝ : FiniteDi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 808,
"column": 4
} | {
"line": 809,
"column": 53
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nf g : α → E\nhT : DominatedFinMeasAdditive μ T C\nh : f =ᶠ[ae μ] g\nhF... | have hgi : ¬Integrable g μ := by rw [integrable_congr h] at hfi; exact hfi
rw [setToFun_undef hT hfi, setToFun_undef hT hgi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 808,
"column": 4
} | {
"line": 809,
"column": 53
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nf g : α → E\nhT : DominatedFinMeasAdditive μ T C\nh : f =ᶠ[ae μ] g\nhF... | have hgi : ¬Integrable g μ := by rw [integrable_congr h] at hfi; exact hfi
rw [setToFun_undef hT hfi, setToFun_undef hT hgi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 455,
"column": 41
} | {
"line": 455,
"column": 52
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : CompleteSpace 𝕜\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : Module 𝕜 V\ninst✝ : ContinuousSMul 𝕜 V\nr : ℝ\nrpos : 0 < r\nc : V\nh : IsCompact (closedBall c r)\n⊢ IsCompact (closedBall 0 r)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 464,
"column": 4
} | {
"line": 464,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : LocallyCompactSpace E\nr : ℝ\nrpos : 0 < r\nhr : IsCompact (closedBall 0 r)\nc : 𝕜\nhc : 1 < ‖c‖\nn : ℕ\nthis : c ^ n ≠ 0\n⊢ IsCompact (closedBall 0 (‖c‖ ^ n * r))",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 460,
"column": 2
} | {
"line": 467,
"column": 57
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : LocallyCompactSpace E\n⊢ ProperSpace E",
"usedConstants": [
"instWeaklyLocallyCompactSpaceOfLocallyCompactSpace",
"NormedCommRing.toNormedRing",
... | rcases exists_isCompact_closedBall (0 : E) with ⟨r, rpos, hr⟩
rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
have hC : ∀ n, IsCompact (closedBall (0 : E) (‖c‖ ^ n * r)) := fun n ↦ by
have : c ^ n ≠ 0 := pow_ne_zero _ <| fun h ↦ by simp [h, zero_le_one.not_gt] at hc
simpa [_root_.smul_closedBall' this... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 460,
"column": 2
} | {
"line": 467,
"column": 57
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : LocallyCompactSpace E\n⊢ ProperSpace E",
"usedConstants": [
"instWeaklyLocallyCompactSpaceOfLocallyCompactSpace",
"NormedCommRing.toNormedRing",
... | rcases exists_isCompact_closedBall (0 : E) with ⟨r, rpos, hr⟩
rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
have hC : ∀ n, IsCompact (closedBall (0 : E) (‖c‖ ^ n * r)) := fun n ↦ by
have : c ^ n ≠ 0 := pow_ne_zero _ <| fun h ↦ by simp [h, zero_le_one.not_gt] at hc
simpa [_root_.smul_closedBall' this... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 591,
"column": 4
} | {
"line": 591,
"column": 15
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nf : α → E\nhf : Summable f\nthis : ∀ {N : ℕ} {g : α → Fin N → ℝ}, Summable g → Summable fun x ↦ ‖g x‖\nv : Basis (Fin (finrank ℝ E)) ℝ E\ne : E ≃L[ℝ] Fin (finrank ℝ E) → ℝ := v.equivFunL\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 910,
"column": 4
} | {
"line": 911,
"column": 63
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nhT : DominatedFinMeasAdditive μ T C\nι : Type u_7\nf : α → E\nhf : AEStronglyMea... | obtain ⟨i, hi, h'i⟩ : ∃ i, ∫⁻ x, ‖fs i x - f x‖ₑ ∂μ < 1 ∧ Integrable (fs i) μ :=
(((tendsto_order.1 hfs).2 _ zero_lt_one).and hfsi).exists | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 641,
"column": 6
} | {
"line": 641,
"column": 32
} | [
{
"pp": "F : Type u_1\ninst✝² : NormedRing F\ninst✝¹ : NormOneClass F\ninst✝ : NormMulClass F\nk : ℕ\nr : F\nhr : ‖r‖ < 1\nu : ℕ → F\nhu : u =O[atTop] fun n ↦ ↑(n ^ k)\nr' : ℝ\nhrr' : ‖r‖ < r'\nh : r' < 1\n⊢ (fun n ↦ ‖u n‖ * ‖r‖ ^ n) =O[atTop] fun n ↦ ‖↑n ^ k‖ * ‖r‖ ^ n",
"usedConstants": [
"Norm.norm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 670,
"column": 29
} | {
"line": 670,
"column": 66
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace ℝ F\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : FiniteDimensional ℝ F\nf : ℕ → E\ng : ℕ → F\nh : f =Θ[atTop] g\n⊢ f =Θ[cofinite] g",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Content | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 13
} | [
{
"pp": "G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\nK₁ K₂ : Compacts G\nh : ↑K₁ ⊆ ↑K₂\n⊢ μ K₁ ≤ μ K₂",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Content | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 56
} | [
{
"pp": "G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\nK₁ K₂ : Compacts G\nh : Disjoint ↑K₁ ↑K₂\nh₁ : IsClosed ↑K₁\nh₂ : IsClosed ↑K₂\n⊢ μ (K₁ ⊔ K₂) = μ K₁ + μ K₂",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Content | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 30
} | [
{
"pp": "G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\nK₁ K₂ : Compacts G\n⊢ μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Content | {
"line": 117,
"column": 30
} | {
"line": 117,
"column": 64
} | [
{
"pp": "G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\n⊢ μ ⊥ = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1093,
"column": 2
} | {
"line": 1095,
"column": 34
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC C' : ℝ\nμ' : Measure α\nhT_add : DominatedFinMeasAdditive (μ + μ') T C'\nhT : Dominat... | refine setToFun_congr_measure_of_integrable 1 one_ne_top ?_ hT_add hT f hf
rw [one_smul]
exact Measure.le_add_left le_rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1093,
"column": 2
} | {
"line": 1095,
"column": 34
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC C' : ℝ\nμ' : Measure α\nhT_add : DominatedFinMeasAdditive (μ + μ') T C'\nhT : Dominat... | refine setToFun_congr_measure_of_integrable 1 one_ne_top ?_ hT_add hT f hf
rw [one_smul]
exact Measure.le_add_left le_rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1129,
"column": 4
} | {
"line": 1129,
"column": 70
} | [
{
"pp": "case cons\nα : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nf : α → E\nι : Type u_7\ns✝ : Finset ι\nμ : ι → Measure α\nT : ι → Set α → E →L[ℝ] F\nC : ι → ℝ\nhTs : ∀ (i : ι), ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1176,
"column": 71
} | {
"line": 1176,
"column": 82
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ μ' μ'' : Measure α\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf : α → E\nhT : DominatedFinMeasAdditive μ T C\nhT' : Domin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Marginal | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 31
} | [
{
"pp": "δ : Type u_1\nX : δ → Type u_3\ninst✝² : (i : δ) → MeasurableSpace (X i)\nμ : (i : δ) → Measure (X i)\ninst✝¹ : DecidableEq δ\ns : Finset δ\ninst✝ : ∀ (i : δ), SigmaFinite (μ i)\nf : ((i : δ) → X i) → ℝ≥0∞\nhf : Measurable f\ni : δ\nhi : i ∈ s\nx : (i : δ) → X i\n⊢ (∫⋯∫⁻_s, f ∂μ) x = ∫⁻ (xᵢ : X i), (∫⋯... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 58
} | [
{
"pp": "case hm\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK : Set G\nhK : IsCompact K\nV : Set G\nhV : (interior V).Nonempty\ng : G\ns : Finset G\nh1s : K ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V\nh2s : s.card = index K V\n⊢ (fun h ↦ g * h) '' ⋃ g ∈ s, (fun h ↦ g * h) ... | rintro _ ⟨g₁, ⟨_, ⟨g₂, rfl⟩, ⟨_, ⟨hg₂, rfl⟩, hg₁⟩⟩, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.MeasureTheory.Integral.Marginal | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 31
} | [
{
"pp": "δ : Type u_1\nX : δ → Type u_3\ninst✝² : (i : δ) → MeasurableSpace (X i)\nμ : (i : δ) → Measure (X i)\ninst✝¹ : DecidableEq δ\ns : Finset δ\ninst✝ : ∀ (i : δ), SigmaFinite (μ i)\nf : ((i : δ) → X i) → ℝ≥0∞\nhf : Measurable f\ni : δ\nhi : i ∈ s\n⊢ ∫⋯∫⁻_s, f ∂μ = ∫⋯∫⁻_s.erase i, fun x ↦ ∫⁻ (xᵢ : X i), f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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