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.LinearAlgebra.AffineSpace.Basis | {
"line": 369,
"column": 2
} | {
"line": 371,
"column": 49
} | [
{
"pp": "ι : Type u_1\nk : Type u_5\nV : Type u_6\nP : Type u_7\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : DivisionRing k\ninst✝¹ : Module k V\ninst✝ : CharZero k\nb : AffineBasis ι k P\ns : Finset ι\ni : ι\nhi : i ∈ s\n⊢ (b.coord i) (Finset.centroid k s ⇑b) = (↑s.card)⁻¹",
"usedConstants"... | rw [Finset.centroid,
b.coord_apply_combination_of_mem hi (s.sum_centroidWeights_eq_one_of_nonempty _ ⟨i, hi⟩),
Finset.centroidWeights, Function.const_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.AffineSpace.Basis | {
"line": 369,
"column": 2
} | {
"line": 371,
"column": 49
} | [
{
"pp": "ι : Type u_1\nk : Type u_5\nV : Type u_6\nP : Type u_7\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : DivisionRing k\ninst✝¹ : Module k V\ninst✝ : CharZero k\nb : AffineBasis ι k P\ns : Finset ι\ni : ι\nhi : i ∈ s\n⊢ (b.coord i) (Finset.centroid k s ⇑b) = (↑s.card)⁻¹",
"usedConstants"... | rw [Finset.centroid,
b.coord_apply_combination_of_mem hi (s.sum_centroidWeights_eq_one_of_nonempty _ ⟨i, hi⟩),
Finset.centroidWeights, Function.const_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Basis | {
"line": 369,
"column": 2
} | {
"line": 371,
"column": 49
} | [
{
"pp": "ι : Type u_1\nk : Type u_5\nV : Type u_6\nP : Type u_7\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : DivisionRing k\ninst✝¹ : Module k V\ninst✝ : CharZero k\nb : AffineBasis ι k P\ns : Finset ι\ni : ι\nhi : i ∈ s\n⊢ (b.coord i) (Finset.centroid k s ⇑b) = (↑s.card)⁻¹",
"usedConstants"... | rw [Finset.centroid,
b.coord_apply_combination_of_mem hi (s.sum_centroidWeights_eq_one_of_nonempty _ ⟨i, hi⟩),
Finset.centroidWeights, Function.const_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Seminorm | {
"line": 1345,
"column": 2
} | {
"line": 1345,
"column": 47
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nr : ℝ\n⊢ Balanced 𝕜 ((normSeminorm 𝕜 E).ball 0 r)",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"NormedSpace.toModule",
"Seminorm.balanced_ball_zero",... | exact (normSeminorm _ _).balanced_ball_zero r | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Seminorm | {
"line": 1344,
"column": 2
} | {
"line": 1345,
"column": 47
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nr : ℝ\n⊢ Balanced 𝕜 (Metric.ball 0 r)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"congrArg",
"DistribMulAction.toDistr... | rw [← ball_normSeminorm 𝕜]
exact (normSeminorm _ _).balanced_ball_zero r | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Seminorm | {
"line": 1344,
"column": 2
} | {
"line": 1345,
"column": 47
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nr : ℝ\n⊢ Balanced 𝕜 (Metric.ball 0 r)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"congrArg",
"DistribMulAction.toDistr... | rw [← ball_normSeminorm 𝕜]
exact (normSeminorm _ _).balanced_ball_zero r | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic | {
"line": 406,
"column": 4
} | {
"line": 408,
"column": 72
} | [
{
"pp": "case h.refine_2\nk : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nI : Set k\nm n : ℕ\ns : Simplex k P n\ne : Fin (n + 1) ≃ Fin (m + 1)\nw : Fin (n + 1) → k\nhw : ∑ i, w i = 1\nhwI : ∀ (i : Fin (n + 1)), w i ∈ I\nx✝ : (affi... | rw [← Function.comp_id w, ← Function.comp_id s.points, ← e.symm_comp_self,
← Function.comp_assoc, ← Function.comp_assoc, ← e.coe_toEmbedding,
← Finset.univ.affineCombination_map e.toEmbedding, map_univ_equiv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.NhdsKer | {
"line": 127,
"column": 32
} | {
"line": 128,
"column": 53
} | [
{
"pp": "X : Type u_2\ninst✝¹ : TopologicalSpace X\nY : Type u_3\ninst✝ : TopologicalSpace Y\ns : Set X\nt : Set Y\n⊢ (⋃ x ∈ s, nhdsKer {x}) ×ˢ ⋃ y ∈ t, nhdsKer {y} = nhdsKer s ×ˢ nhdsKer t",
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
"_private.Mathlib.Topology.NhdsKer.0.nhdsKer_prod._s... | by
simp_rw [← nhdsKer_biUnion, biUnion_of_singleton] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.NhdsKer | {
"line": 142,
"column": 42
} | {
"line": 143,
"column": 53
} | [
{
"pp": "ι : Type u_3\nX : ι → Type u_4\ninst✝ : (i : ι) → TopologicalSpace (X i)\ns : (i : ι) → Set (X i)\n⊢ (univ.pi fun i ↦ ⋃ x ∈ s i, nhdsKer {x}) = univ.pi fun i ↦ nhdsKer (s i)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.univ",
"Membership.mem",
"Set.biUnion_of_single... | by
simp_rw [← nhdsKer_biUnion, biUnion_of_singleton] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.AlexandrovDiscrete | {
"line": 192,
"column": 12
} | {
"line": 199,
"column": 33
} | [
{
"pp": "α : Type u_3\ninst✝ : TopologicalSpace α\nhα : ∀ (a : α), 𝓝 a = 𝓟 (nhdsKer {a})\n⊢ AlexandrovDiscrete α",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Topology.AlexandrovDiscrete.0.alexandrovDiscrete_iff_nhds._simp_1_1",
"congrArg",
"Filter.inf_principal",
"Filter.... | by
simp only [alexandrovDiscrete_iff_isClosed, isClosed_iff_clusterPt, ClusterPt, funext hα,
inf_principal, principal_neBot_iff]
intro S hS a ha
rw [sUnion_eq_biUnion, inter_iUnion₂, nonempty_biUnion] at ha
obtain ⟨s, hs, has⟩ := ha
specialize hS s hs a has
exact mem_sUnion_of_mem hS hs | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Connected.LocPathConnected | {
"line": 205,
"column": 6
} | {
"line": 205,
"column": 25
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : LocPathConnectedSpace X\nY : Type u_4\nf : X → Y\nx✝ : TopologicalSpace Y := TopologicalSpace.coinduced f inst✝¹\nhf : Continuous[inst✝¹, x✝] f\ny : Y\nu : Set Y\nhu : IsOpen[TopologicalSpace.coinduced f inst✝¹] u\nx : X\nhx : x ∈ f ⁻¹' pathComponentIn... | ← image_subset_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.Convex | {
"line": 96,
"column": 2
} | {
"line": 98,
"column": 56
} | [
{
"pp": "F : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : Nontrivial F\nx : F\nr : ℝ\nhr : 0 ≤ r\n⊢ (convexHull ℝ) (sphere x r) = closedBall x r",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"ChainCompletePartialOrder.instOfCompleteLattice"... | suffices convexHull ℝ (sphere (0 : F) r) = closedBall 0 r by
rw [← add_zero x, ← vadd_eq_add, ← vadd_sphere, convexHull_vadd,
this, vadd_closedBall_zero, vadd_eq_add, add_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.Normed.Module.Convex | {
"line": 127,
"column": 83
} | {
"line": 129,
"column": 32
} | [
{
"pp": "E : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns : Set E\nhs : Convex ℝ s\nδ : ℝ\n⊢ Convex ℝ (Metric.thickening δ s)",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"congrArg",
"DistribMulAction.toDistribSMul",
"AddCommGroup... | by
rw [← add_ball_zero]
exact hs.add (convex_ball 0 _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 518,
"column": 12
} | {
"line": 518,
"column": 13
} | [
{
"pp": "case mp\n𝕜 : Type u_2\nE : Type u_6\nι : Type u_9\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\nh : ∀ i ∈ p.basisSets, Absorbs 𝕜 (id i) s\n⊢ ∀ (I : Finset ι), ∃ r > 0, ∀ x ∈ ... | I | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 550,
"column": 11
} | {
"line": 550,
"column": 12
} | [
{
"pp": "case mpr\n𝕜 : Type u_2\nE : Type u_6\nι : Type u_9\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\nhi : ∀ (i : ι), ∃ r > 0, ∀ x ∈ s, (p i) x < r\n⊢ ∀ (I : Finset ι), ∃ r > 0, ∀ ... | I | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap | {
"line": 293,
"column": 40
} | {
"line": 293,
"column": 65
} | [
{
"pp": "R : Type u_1\n𝕜₂ : Type u_3\n𝕜₃ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹³ : Semiring R\ninst✝¹² : NormedField 𝕜₂\ninst✝¹¹ : NormedField 𝕜₃\ninst✝¹⁰ : AddCommMonoid E\ninst✝⁹ : Module R E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : Topol... | rw [f.map_add, add_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap | {
"line": 293,
"column": 40
} | {
"line": 293,
"column": 65
} | [
{
"pp": "R : Type u_1\n𝕜₂ : Type u_3\n𝕜₃ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹³ : Semiring R\ninst✝¹² : NormedField 𝕜₂\ninst✝¹¹ : NormedField 𝕜₃\ninst✝¹⁰ : AddCommMonoid E\ninst✝⁹ : Module R E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : Topol... | rw [f.map_add, add_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap | {
"line": 293,
"column": 40
} | {
"line": 293,
"column": 65
} | [
{
"pp": "R : Type u_1\n𝕜₂ : Type u_3\n𝕜₃ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹³ : Semiring R\ninst✝¹² : NormedField 𝕜₂\ninst✝¹¹ : NormedField 𝕜₃\ninst✝¹⁰ : AddCommMonoid E\ninst✝⁹ : Module R E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : Topol... | rw [f.map_add, add_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap | {
"line": 385,
"column": 2
} | {
"line": 385,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹³ : AddCommGroup E\ninst✝¹² : TopologicalSpace E\ninst✝¹¹ : Module 𝕜 E\ninst✝¹⁰ : ContinuousSMul 𝕜 E\nF : Type u_3\ninst✝⁹ : AddCommGroup F\ninst✝⁸ : UniformSpace F\ninst✝⁷ : IsUniformAddGroup F\ninst✝⁶ : Module 𝕜 F\n𝕜' : Type... | rw [← isUniformEmbedding_toUniformOnFun.of_comp_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.AEEqFun | {
"line": 589,
"column": 2
} | {
"line": 589,
"column": 28
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeSup β\ninst✝ : ContinuousSup β\nf g f' : α →ₘ[μ] β\nhf : f ≤ f'\nhg : g ≤ f'\n⊢ f ⊔ g ≤ f'",
"usedConstants": [
"MeasureTheory.ae",
"MeasureTheory.Measure",
"con... | rw [← coeFn_le] at hf hg ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.AEEqFun | {
"line": 590,
"column": 52
} | {
"line": 590,
"column": 55
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeSup β\ninst✝ : ContinuousSup β\nf g f' : α →ₘ[μ] β\nhf : ↑f ≤ᶠ[ae μ] ↑f'\nhg : ↑g ≤ᶠ[ae μ] ↑f'\na✝ : α\nhaf : ↑f a✝ ≤ ↑f' a✝\n⊢ ↑g a✝ ≤ ↑f' a✝ → ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝ → ... | hag | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.AEEqFun | {
"line": 618,
"column": 2
} | {
"line": 618,
"column": 28
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf' f g : α →ₘ[μ] β\nhf : f' ≤ f\nhg : f' ≤ g\n⊢ f' ≤ f ⊓ g",
"usedConstants": [
"MeasureTheory.ae",
"MeasureTheory.Measure",
"con... | rw [← coeFn_le] at hf hg ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.AEEqFun | {
"line": 619,
"column": 52
} | {
"line": 619,
"column": 55
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf' f g : α →ₘ[μ] β\nhf : ↑f' ≤ᶠ[ae μ] ↑f\nhg : ↑f' ≤ᶠ[ae μ] ↑g\na✝ : α\nhaf : ↑f' a✝ ≤ ↑f a✝\n⊢ ↑f' a✝ ≤ ↑g a✝ → ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝ → ... | hag | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.SpecialFunctions.Basic | {
"line": 297,
"column": 2
} | {
"line": 297,
"column": 61
} | [
{
"pp": "⊢ MeasurablePow ℝ≥0∞ ℝ",
"usedConstants": [
"Real",
"ENNReal.instPowReal",
"ENNReal.measurableSpace",
"MeasurablePow.mk",
"ENNReal.measurable_of_measurable_nnreal_prod",
"Prod.fst",
"Real.measurableSpace",
"HPow.hPow",
"ENNReal",
"instHPow... | refine ⟨ENNReal.measurable_of_measurable_nnreal_prod ?_ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.LpSeminorm.Indicator | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 34
} | [
{
"pp": "case inr\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nc : ε\ns : Set α\np : ℝ≥0∞\nhp : p ≠ 0\n⊢ eLpNorm (s.indicator fun x ↦ c) p μ ≤ ‖c‖ₑ * μ s ^ (1 / p.toReal)",
"usedConstants": [
"ENNReal",
"eq_or_ne... | obtain rfl | h'p := eq_or_ne p ∞ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 829,
"column": 4
} | {
"line": 830,
"column": 61
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Fintype ι\nX : ι → Type u_4\ninst✝ : Unique ι\nm : (i : ι) → MeasurableSpace (X i)\nμ : (i : ι) → Measure (X i)\ne : ((i : ι) → X i) ≃ᵐ X default := MeasurableEquiv.piUnique X\nthis : (piPremeasure fun i ↦ (μ i).toOuterMeasure) = ⇑(Measure.map (⇑e.symm) (μ default))\n⊢ Measure.ma... | simp_rw [Measure.pi, OuterMeasure.pi, this, ← coe_toOuterMeasure, boundedBy_eq_self,
toOuterMeasure_toMeasure, MeasurableEquiv.map_map_symm] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 143,
"column": 8
} | {
"line": 143,
"column": 29
} | [
{
"pp": "s : ℝ\nhs✝ : -1 ≤ s\nhs' : s ≠ 0\np : ℝ\nhp1 : 0 < p\nhp2 : p < 1\nhs : -1 < s\nhs1 : 0 < 1 + s\n⊢ 0 < 1 + p * s",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.partialOrder",
"Real",
"HMul.hMul",
"Real.instZero",
... | ← neg_lt_iff_pos_add' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Real.ConjExponents | {
"line": 174,
"column": 2
} | {
"line": 176,
"column": 25
} | [
{
"pp": "p q : ℝ\nh : p.HolderConjugate q\n⊢ (ENNReal.ofReal p)⁻¹ + (ENNReal.ofReal q)⁻¹ = 1",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"Real",
"ENNReal.ofReal",
"congrArg",
"Real.instInv",
"Real.HolderTriple.inv_nonneg",
"Real.HolderConjugate.symm",
... | rw [← ENNReal.ofReal_one, ← ENNReal.ofReal_inv_of_pos h.pos,
← ENNReal.ofReal_inv_of_pos h.symm.pos, ← ENNReal.ofReal_add h.inv_nonneg h.symm.inv_nonneg,
h.inv_add_inv_eq_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Real.ConjExponents | {
"line": 174,
"column": 2
} | {
"line": 176,
"column": 25
} | [
{
"pp": "p q : ℝ\nh : p.HolderConjugate q\n⊢ (ENNReal.ofReal p)⁻¹ + (ENNReal.ofReal q)⁻¹ = 1",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"Real",
"ENNReal.ofReal",
"congrArg",
"Real.instInv",
"Real.HolderTriple.inv_nonneg",
"Real.HolderConjugate.symm",
... | rw [← ENNReal.ofReal_one, ← ENNReal.ofReal_inv_of_pos h.pos,
← ENNReal.ofReal_inv_of_pos h.symm.pos, ← ENNReal.ofReal_add h.inv_nonneg h.symm.inv_nonneg,
h.inv_add_inv_eq_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Real.ConjExponents | {
"line": 174,
"column": 2
} | {
"line": 176,
"column": 25
} | [
{
"pp": "p q : ℝ\nh : p.HolderConjugate q\n⊢ (ENNReal.ofReal p)⁻¹ + (ENNReal.ofReal q)⁻¹ = 1",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"Real",
"ENNReal.ofReal",
"congrArg",
"Real.instInv",
"Real.HolderTriple.inv_nonneg",
"Real.HolderConjugate.symm",
... | rw [← ENNReal.ofReal_one, ← ENNReal.ofReal_inv_of_pos h.pos,
← ENNReal.ofReal_inv_of_pos h.symm.pos, ← ENNReal.ofReal_add h.inv_nonneg h.symm.inv_nonneg,
h.inv_add_inv_eq_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Real.ConjExponents | {
"line": 555,
"column": 2
} | {
"line": 555,
"column": 33
} | [
{
"pp": "case inr\np q : ℝ≥0∞\nh : p.HolderConjugate q\nhp : p ≠ ∞\n⊢ p * q = p + q",
"usedConstants": [
"ENNReal",
"eq_or_ne",
"ENNReal.instTop",
"Top.top"
]
}
] | obtain rfl | hq := eq_or_ne q ∞ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Data.Real.ConjExponents | {
"line": 561,
"column": 2
} | {
"line": 561,
"column": 33
} | [
{
"pp": "p q : ℝ≥0∞\nh : p.HolderConjugate q\n⊢ p / q = p - 1",
"usedConstants": [
"ENNReal",
"eq_or_ne",
"ENNReal.instTop",
"Top.top"
]
}
] | obtain rfl | hq := eq_or_ne q ∞ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 238,
"column": 13
} | {
"line": 238,
"column": 43
} | [
{
"pp": "t : ℝ\nht : -1 ≤ t ∧ t ≤ 1\nx : ℝ\n⊢ (1 + t) / 2 * rexp x + (1 - t) / 2 * rexp (-x) = cosh x + t * sinh x",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Real.sinh_eq",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAss... | by rw [cosh_eq, sinh_eq]; ring | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 910,
"column": 75
} | {
"line": 927,
"column": 92
} | [
{
"pp": "α : Type u_1\nE : Type u_4\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhbdd : ∀ (n : ℕ), eLpNorm (f n) p μ ≤ ↑R\n⊢ ∀ᵐ (x : α) ∂μ, liminf (fun n ... | by
by_cases hp0 : p.toReal = 0
· simp only [hp0, ENNReal.rpow_zero]
filter_upwards with _
rw [liminf_const (1 : ℝ≥0∞)]
exact ENNReal.one_lt_top
have hp : p ≠ 0 := fun h => by simp [h] at hp0
have hp' : p ≠ ∞ := fun h => by simp [h] at hp0
refine
ae_lt_top (.liminf fun n => (hfmeas n).nnnorm.co... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 241,
"column": 8
} | {
"line": 241,
"column": 56
} | [
{
"pp": "ι : Type u\ns : Finset ι\nw z : ι → ℝ≥0∞\nhw' : ∑ i ∈ s, w i = 1\np : ℝ\nhp : 1 ≤ p\nhp_pos : 0 < p\nhp_nonneg : 0 ≤ p\nhp_not_neg : ¬p < 0\nh_top_iff_rpow_top : ∀ i ∈ s, w i * z i = ∞ ↔ w i * z i ^ p = ∞\nh_top_rpow_sum : (∑ i ∈ s, w i * z i) ^ p ≠ ∞\na✝ : ∑ i ∈ s, w i * z i ^ p ≠ ∞\nh : ∑ i ∈ s, w i ... | rw [h, top_rpow_of_pos hp_pos] at h_top_rpow_sum | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 341,
"column": 34
} | {
"line": 341,
"column": 53
} | [
{
"pp": "p : ℝ≥0∞\nh : p ∈ Set.Ioo 0 1\n⊢ p⁻¹ ≠ ∞",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"ENNReal.inv_eq_top._simp_1",
"id",
"Ne",
"LT.lt.ne'",
"Inv.inv",
"And.left",
"LT.lt",
"ENNReal",
... | simpa using h.1.ne' | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 341,
"column": 34
} | {
"line": 341,
"column": 53
} | [
{
"pp": "p : ℝ≥0∞\nh : p ∈ Set.Ioo 0 1\n⊢ p⁻¹ ≠ ∞",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"ENNReal.inv_eq_top._simp_1",
"id",
"Ne",
"LT.lt.ne'",
"Inv.inv",
"And.left",
"LT.lt",
"ENNReal",
... | simpa using h.1.ne' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 341,
"column": 34
} | {
"line": 341,
"column": 53
} | [
{
"pp": "p : ℝ≥0∞\nh : p ∈ Set.Ioo 0 1\n⊢ p⁻¹ ≠ ∞",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"ENNReal.inv_eq_top._simp_1",
"id",
"Ne",
"LT.lt.ne'",
"Inv.inv",
"And.left",
"LT.lt",
"ENNReal",
... | simpa using h.1.ne' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 352,
"column": 13
} | {
"line": 352,
"column": 32
} | [
{
"pp": "case right\nz₁ z₂ : ℝ≥0∞\np : ℝ\nhp : 0 ≤ p\nh : 1 < p\n⊢ (ENNReal.ofReal p)⁻¹ < 1",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"ENNReal.ofReal",
"congrArg",
"PartialOrder.toPreorder",
"id",
"ENNReal.inv_lt_one",
"Inv.inv",
"LT.lt",
"EN... | ENNReal.inv_lt_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.MeanInequalities | {
"line": 592,
"column": 2
} | {
"line": 597,
"column": 12
} | [
{
"pp": "ι : Type u\nf g : ι → ℝ≥0\np q r : ℝ\nhpqr : p.HolderTriple q r\nhf : Summable fun i ↦ f i ^ p\nhg : Summable fun i ↦ g i ^ q\n⊢ (∑' (i : ι), (f i * g i) ^ r) ^ (1 / r) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)",
"usedConstants": [
"NNReal.instTopologicalSpace",
... | convert!
rpow_le_rpow_iff (inv_eq_one_div r ▸ inv_pos.mpr hpqr.pos') |>.mpr <|
Lr_rpow_le_Lp_mul_Lq_tsum hpqr hf hg
have hr := hpqr.pos'.ne'
simp only [← rpow_mul, mul_rpow]
field_simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.MeanInequalities | {
"line": 592,
"column": 2
} | {
"line": 597,
"column": 12
} | [
{
"pp": "ι : Type u\nf g : ι → ℝ≥0\np q r : ℝ\nhpqr : p.HolderTriple q r\nhf : Summable fun i ↦ f i ^ p\nhg : Summable fun i ↦ g i ^ q\n⊢ (∑' (i : ι), (f i * g i) ^ r) ^ (1 / r) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)",
"usedConstants": [
"NNReal.instTopologicalSpace",
... | convert!
rpow_le_rpow_iff (inv_eq_one_div r ▸ inv_pos.mpr hpqr.pos') |>.mpr <|
Lr_rpow_le_Lp_mul_Lq_tsum hpqr hf hg
have hr := hpqr.pos'.ne'
simp only [← rpow_mul, mul_rpow]
field_simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 27
} | [
{
"pp": "α : Type u_2\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\np2 : ℝ := q / p\n... | let q2 := p2.conjExponent | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Convex.Mul | {
"line": 168,
"column": 16
} | {
"line": 168,
"column": 81
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : CommRing 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nn : ℕ\nx : 𝕜\nx✝¹ : x ∈ univ\ny : 𝕜\nx✝ : y ∈ univ\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a * b * (x - y) ^ 2 = a • x ^ 2 + b • y ^ 2 - (a • x + b • y) ^ 2",
"usedConstants": [
"Math... | obtain rfl := eq_sub_of_add_eq hab; simp only [smul_eq_mul]; ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Mul | {
"line": 168,
"column": 16
} | {
"line": 168,
"column": 81
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : CommRing 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nn : ℕ\nx : 𝕜\nx✝¹ : x ∈ univ\ny : 𝕜\nx✝ : y ∈ univ\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a * b * (x - y) ^ 2 = a • x ^ 2 + b • y ^ 2 - (a • x + b • y) ^ 2",
"usedConstants": [
"Math... | obtain rfl := eq_sub_of_add_eq hab; simp only [smul_eq_mul]; ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.MeanInequalities | {
"line": 980,
"column": 4
} | {
"line": 984,
"column": 27
} | [
{
"pp": "case pos\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nH : (∑ i ∈ s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i ∈ s, g i ^ q) ^ (1 / q) = 0\n⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)",
"usedConstants": [
"ENNReal.instCanonicallyOrde... | replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0 := by
simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by rcases H with H | H <;> simp [H i hi]
simp [sum_eq_zero this] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.MeanInequalities | {
"line": 980,
"column": 4
} | {
"line": 984,
"column": 27
} | [
{
"pp": "case pos\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nH : (∑ i ∈ s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i ∈ s, g i ^ q) ^ (1 / q) = 0\n⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)",
"usedConstants": [
"ENNReal.instCanonicallyOrde... | replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0 := by
simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := fun i hi => by rcases H with H | H <;> simp [H i hi]
simp [sum_eq_zero this] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConvergenceInMeasure | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 74
} | [
{
"pp": "case zero\nα : Type u_1\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : PseudoEMetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nk : ℕ\nhn : seqTendstoAeSeq hfg 0 ≤ k\n⊢ μ {x | 2⁻¹ ^ 0 ≤ edist (f k x) (g x)} ≤ 2⁻¹ ^ 0",
"usedConstants": [
"PseudoEMetric... | exact Classical.choose_spec (exists_nat_measure_lt_two_inv hfg 0) k hn | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.ConvergenceInMeasure | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 74
} | [
{
"pp": "case zero\nα : Type u_1\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : PseudoEMetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nk : ℕ\nhn : seqTendstoAeSeq hfg 0 ≤ k\n⊢ μ {x | 2⁻¹ ^ 0 ≤ edist (f k x) (g x)} ≤ 2⁻¹ ^ 0",
"usedConstants": [
"PseudoEMetric... | exact Classical.choose_spec (exists_nat_measure_lt_two_inv hfg 0) k hn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConvergenceInMeasure | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 74
} | [
{
"pp": "case zero\nα : Type u_1\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : PseudoEMetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nk : ℕ\nhn : seqTendstoAeSeq hfg 0 ≤ k\n⊢ μ {x | 2⁻¹ ^ 0 ≤ edist (f k x) (g x)} ≤ 2⁻¹ ^ 0",
"usedConstants": [
"PseudoEMetric... | exact Classical.choose_spec (exists_nat_measure_lt_two_inv hfg 0) k hn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpOrder | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 14
} | [
{
"pp": "case h\nα : Type u_1\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedAddCommGroup E\ninst✝ : PartialOrder E\nf : ↥(Lp E p μ)\nh0 : ↑↑0 =ᶠ[ae μ] 0\nh : ↑↑0 ≤ᶠ[ae μ] ↑↑f\na✝¹ : α\na✝ : ↑↑0 a✝¹ ≤ ↑↑f a✝¹\nh2 : ↑↑0 a✝¹ = 0 a✝¹\n⊢ 0 a✝¹ ≤ ↑↑f a✝¹",
"usedConstants": [
... | rwa [← h2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.Function.LpOrder | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 14
} | [
{
"pp": "case h\nα : Type u_1\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedAddCommGroup E\ninst✝ : PartialOrder E\nf : ↥(Lp E p μ)\nh0 : ↑↑0 =ᶠ[ae μ] 0\nh : ↑↑0 ≤ᶠ[ae μ] ↑↑f\na✝¹ : α\na✝ : ↑↑0 a✝¹ ≤ ↑↑f a✝¹\nh2 : ↑↑0 a✝¹ = 0 a✝¹\n⊢ 0 a✝¹ ≤ ↑↑f a✝¹",
"usedConstants": [
... | rwa [← h2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpOrder | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 14
} | [
{
"pp": "case h\nα : Type u_1\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedAddCommGroup E\ninst✝ : PartialOrder E\nf : ↥(Lp E p μ)\nh0 : ↑↑0 =ᶠ[ae μ] 0\nh : ↑↑0 ≤ᶠ[ae μ] ↑↑f\na✝¹ : α\na✝ : ↑↑0 a✝¹ ≤ ↑↑f a✝¹\nh2 : ↑↑0 a✝¹ = 0 a✝¹\n⊢ 0 a✝¹ ≤ ↑↑f a✝¹",
"usedConstants": [
... | rwa [← h2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConvergenceInMeasure | {
"line": 327,
"column": 2
} | {
"line": 328,
"column": 68
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : PseudoEMetricSpace E\nu : Filter ι\ninst✝¹ : u.NeBot\ninst✝ : u.IsCountablyGenerated\nf : ι → α → E\ng : α → E\nhfg : TendstoInMeasure μ f u g\n⊢ ∃ ns, Tendsto ns atTop u ∧ TendstoInMeasure μ (fun n ↦ f (ns n)) atT... | obtain ⟨ns, h_tendsto_ns⟩ : ∃ ns : ℕ → ι, Tendsto ns atTop u := exists_seq_tendsto u
exact ⟨ns, h_tendsto_ns, fun ε hε => (hfg ε hε).comp h_tendsto_ns⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConvergenceInMeasure | {
"line": 327,
"column": 2
} | {
"line": 328,
"column": 68
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : PseudoEMetricSpace E\nu : Filter ι\ninst✝¹ : u.NeBot\ninst✝ : u.IsCountablyGenerated\nf : ι → α → E\ng : α → E\nhfg : TendstoInMeasure μ f u g\n⊢ ∃ ns, Tendsto ns atTop u ∧ TendstoInMeasure μ (fun n ↦ f (ns n)) atT... | obtain ⟨ns, h_tendsto_ns⟩ : ∃ ns : ℕ → ι, Tendsto ns atTop u := exists_seq_tendsto u
exact ⟨ns, h_tendsto_ns, fun ε hε => (hfg ε hε).comp h_tendsto_ns⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 488,
"column": 4
} | {
"line": 488,
"column": 59
} | [
{
"pp": "case pos\nα : Type u_1\nE : Type u_4\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhf : MemLp f p μ\nq : ℝ≥0∞\nq_top : ¬q = ∞\nq_zero : q = 0\n⊢ eLpNorm (fun x ↦ ‖f x‖ ^ q.toReal) (p / q) μ < ∞",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
... | simp only [q_zero, ENNReal.toReal_zero, Real.rpow_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 493,
"column": 2
} | {
"line": 493,
"column": 60
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_4\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhf : MemLp f p μ\nq : ℝ≥0∞\nq_top : ¬q = ∞\nq_zero : ¬q = 0\n⊢ eLpNorm (fun x ↦ ‖f x‖ ^ q.toReal) (p / q) μ < ∞",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
... | rw [eLpNorm_norm_rpow _ (ENNReal.toReal_pos q_zero q_top)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 838,
"column": 53
} | {
"line": 838,
"column": 58
} | [
{
"pp": "case h\nα : Type u_1\n𝕜✝ : Type u_2\n𝕜'✝ : Type u_3\nE : Type u_4\nF✝ : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F✝\ng✝ : E → F✝\nc✝ : ℝ≥0\n𝕜 : Type u_6\n𝕜' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : Nontriv... | hsmul | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Normed.Operator.NormedSpace | {
"line": 249,
"column": 27
} | {
"line": 249,
"column": 54
} | [
{
"pp": "case h.e'_3\n𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NontriviallyNormedField 𝕜₂\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₁ : 𝕜₂ →+* 𝕜\nin... | ContinuousLinearMap.norm_id | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Operator.NormedSpace | {
"line": 347,
"column": 6
} | {
"line": 348,
"column": 93
} | [
{
"pp": "case mpr\n𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\nι : Type u_8\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁴ : RingHomIsometric σ₁₂\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\... | refine ⟨C.toNNReal • normSeminorm 𝕜 E,
((norm_withSeminorms 𝕜 E).continuous_seminorm 0).const_smul C.toNNReal, fun i x ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Algebra.Module.Multilinear.Topology | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\nι : Type u_2\nE : ι → Type u_3\nF : Type u_4\ninst✝¹⁴ : NormedField 𝕜\ninst✝¹³ : (i : ι) → TopologicalSpace (E i)\ninst✝¹² : (i : ι) → AddCommGroup (E i)\ninst✝¹¹ : (i : ι) → Module 𝕜 (E i)\ninst✝¹⁰ : AddCommGroup F\ninst✝⁹ : Module 𝕜 F\ninst✝⁸ : UniformSpace F\ninst✝⁷ : IsUniformAddG... | rw [← isUniformEmbedding_toUniformOnFun.of_comp_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Real | {
"line": 42,
"column": 2
} | {
"line": 43,
"column": 21
} | [
{
"pp": "α : Type u_1\nx✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nh : μ s ≠ ∞\n⊢ μ.real s = 0 ↔ μ s = 0",
"usedConstants": [
"Eq.mpr",
"Real",
"MeasureTheory.Measure",
"Real.instZero",
"congrArg",
"MeasureTheory.Measure.real",
"id",
"Iff",
"ENNRea... | rw [Measure.real, ENNReal.toReal_eq_zero_iff]
exact or_iff_left h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Real | {
"line": 42,
"column": 2
} | {
"line": 43,
"column": 21
} | [
{
"pp": "α : Type u_1\nx✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nh : μ s ≠ ∞\n⊢ μ.real s = 0 ↔ μ s = 0",
"usedConstants": [
"Eq.mpr",
"Real",
"MeasureTheory.Measure",
"Real.instZero",
"congrArg",
"MeasureTheory.Measure.real",
"id",
"Iff",
"ENNRea... | rw [Measure.real, ENNReal.toReal_eq_zero_iff]
exact or_iff_left h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 78,
"column": 45
} | {
"line": 78,
"column": 81
} | [
{
"pp": "α : Type u_1\nE : Type u_5\nmα : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\ns : Set α\nμ : Measure α\nC : ℝ\nhs : μ s < ∞\nhf : ∀ᵐ (x : α) ∂μ.restrict s, ‖f x‖ ≤ C\n⊢ (μ.restrict s) univ < ∞",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"Preorder.toLT",
... | by rwa [Measure.restrict_apply_univ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 891,
"column": 59
} | {
"line": 892,
"column": 91
} | [
{
"pp": "α : Type u_1\nε' : Type u_4\nmα : MeasurableSpace α\ninst✝⁴ : PartialOrder α\ninst✝³ : MeasurableSingletonClass α\ninst✝² : TopologicalSpace ε'\ninst✝¹ : ESeminormedAddMonoid ε'\ninst✝ : PseudoMetrizableSpace ε'\nf : α → ε'\nμ : Measure α\nb : α\nhb : μ {b} ≠ ∞\nhb' : ‖f b‖ₑ ≠ ∞\n⊢ IntegrableOn f (Ici ... | by
rw [← Ioi_union_left, integrableOn_union, eq_true (integrableOn_singleton hb'), and_true] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 218,
"column": 6
} | {
"line": 242,
"column": 44
} | [
{
"pp": "case insert\n𝕜 : Type u\nι : Type v\nE : ι → Type wE\nG : Type wG\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝³ : SeminormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf :... | have I :
‖f (s.piecewise m₂ m₁) - f ((insert i s).piecewise m₂ m₁)‖ ≤
C * ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ := by
have A : (insert i s).piecewise m₂ m₁ = Function.update (s.piecewise m₂ m₁) i (m₂ i) :=
s.piecewise_insert _ _ _
have B : s.piecewise m₂ m₁ ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 218,
"column": 6
} | {
"line": 242,
"column": 44
} | [
{
"pp": "case insert\n𝕜 : Type u\nι : Type v\nE : ι → Type wE\nG : Type wG\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝³ : SeminormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf :... | have I :
‖f (s.piecewise m₂ m₁) - f ((insert i s).piecewise m₂ m₁)‖ ≤
C * ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖ := by
have A : (insert i s).piecewise m₂ m₁ = Function.update (s.piecewise m₂ m₁) i (m₂ i) :=
s.piecewise_insert _ _ _
have B : s.piecewise m₂ m₁ ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 507,
"column": 37
} | {
"line": 507,
"column": 40
} | [
{
"pp": "case h\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ\nh_meas : AEStronglyMeasurable f μ\nhf : 0 ≤ᶠ[ae μ] f\nhg : 0 ≤ᶠ[ae μ] g\nh_int : Integrable (f + g) μ\na : α\nhaf : 0 a ≤ f a\n⊢ 0 a ≤ g a → ‖f a‖ ≤ (f + g) a",
"usedConstants": [
"Real.instLE",
"Real",
"Real... | hag | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 72
} | [
{
"pp": "case pos\nβ : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\np : ℝ≥0∞\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nhp_ne_top : p ≠ ∞\nμ : Measure β\nhμ : ∀ᵐ (x :... | simpa only [hp_zero, eLpNorm_exponent_zero] using tendsto_const_nhds | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 72
} | [
{
"pp": "case pos\nβ : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\np : ℝ≥0∞\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nhp_ne_top : p ≠ ∞\nμ : Measure β\nhμ : ∀ᵐ (x :... | simpa only [hp_zero, eLpNorm_exponent_zero] using tendsto_const_nhds | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 72
} | [
{
"pp": "case pos\nβ : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\np : ℝ≥0∞\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nhp_ne_top : p ≠ ∞\nμ : Measure β\nhμ : ∀ᵐ (x :... | simpa only [hp_zero, eLpNorm_exponent_zero] using tendsto_const_nhds | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 349,
"column": 4
} | {
"line": 349,
"column": 48
} | [
{
"pp": "α : Type u_1\nF : Type u_3\nF' : Type u_4\nG : Type u_5\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → F →L[ℝ] F'\nh_add : FinMeasAdditive μ T\nf : α →ₛ ... | rw [← Finset.set_biUnion_preimage_singleton] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 369,
"column": 4
} | {
"line": 369,
"column": 39
} | [
{
"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\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable (⇑f) μ\nhg : Integrable (⇑g)... | rcases mem_range.1 hp with ⟨a, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Algebra.Module.FiniteDimension | {
"line": 631,
"column": 2
} | {
"line": 631,
"column": 67
} | [
{
"pp": "𝕜 : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : CompleteSpace 𝕜\nEᵤ : Type u_6\ninst✝⁵ : AddCommGroup Eᵤ\ninst✝⁴ : Module 𝕜 Eᵤ\ninst✝³ : UniformSpace Eᵤ\ninst✝² : T2Space Eᵤ\ninst✝¹ : IsUniformAddGroup Eᵤ\ninst✝ : ContinuousSMul 𝕜 Eᵤ\nU : Set Eᵤ\nhU_nhds : U ∈ 𝓝 0\nhU_tb : TotallyBound... | letI : FiniteDimensional 𝕜 M := Finite.span_of_finite 𝕜 hF_finite | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.Topology.Algebra.Module.FiniteDimension | {
"line": 718,
"column": 51
} | {
"line": 718,
"column": 62
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : CompleteSpace 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : ContinuousSMul 𝕜 E\np q : Submodule 𝕜 E\nh : IsCompl p q\nhp : IsClosed[inst✝⁴] ↑p\ninst✝... | by simp [φ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 749,
"column": 2
} | {
"line": 750,
"column": 60
} | [
{
"pp": "α : Type u_1\ninst✝² : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nG : Type u_7\ninst✝¹ : NormedAddCommGroup G\ninst✝ : PartialOrder G\nhp : Fact (1 ≤ p)\nhp_ne_top : p ≠ ∞\ng : { g // 0 ≤ g }\nthis✝¹ : MeasurableSpace G := borel G\nthis✝ : BorelSpace G\nhg_memLp : MemLp (↑↑↑g) p μ\nzero_mem : 0 ∈ (Set... | let x n := SimpleFunc.approxOn (g : α → G) g_meas
((range (g : α → G) ∪ {0}) ∩ { y | 0 ≤ y }) 0 zero_mem n | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 604,
"column": 2
} | {
"line": 607,
"column": 7
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : PartialOrder E\ninst✝² : IsOrderedAddMonoid E\ninst✝¹ : IsOrderedModule ℝ E\ninst✝ : ClosedIciTopology E\nf : α → E\nhf : f ≤ᶠ[ae μ] 0\n⊢ ∫ (x : α), f x ∂μ ≤ 0",
"used... | rw [← neg_nonneg, ← integral_neg]
refine integral_nonneg_of_ae ?_
filter_upwards [hf] with x hx
simpa | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 604,
"column": 2
} | {
"line": 607,
"column": 7
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : PartialOrder E\ninst✝² : IsOrderedAddMonoid E\ninst✝¹ : IsOrderedModule ℝ E\ninst✝ : ClosedIciTopology E\nf : α → E\nhf : f ≤ᶠ[ae μ] 0\n⊢ ∫ (x : α), f x ∂μ ≤ 0",
"used... | rw [← neg_nonneg, ← integral_neg]
refine integral_nonneg_of_ae ?_
filter_upwards [hf] with x hx
simpa | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 682,
"column": 11
} | {
"line": 683,
"column": 31
} | [
{
"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 = ENNReal.ofReal (∫ (a : α), ↑(f a) ∂μ)",
"usedConstants": [
"MeasureTheory.ae",
"MeasureTheory.Integrable.aestronglyMeasurable",
"Eq.mpr",
"NormedCo... | integral_eq_lintegral_of_nonneg_ae (Eventually.of_forall fun x => (f x).coe_nonneg)
hfi.aestronglyMeasurable, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 710,
"column": 4
} | {
"line": 710,
"column": 37
} | [
{
"pp": "case neg\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0\nb : ℝ≥0\nh : ∫⁻ (a : α), ↑(f a) ∂μ ≤ ↑b\nhf : ¬Integrable (fun a ↦ ↑(f a)) μ\n⊢ ∫ (a : α), ↑(f a) ∂μ ≤ ↑b",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
... | rw [integral_undef hf]; exact b.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 710,
"column": 4
} | {
"line": 710,
"column": 37
} | [
{
"pp": "case neg\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0\nb : ℝ≥0\nh : ∫⁻ (a : α), ↑(f a) ∂μ ≤ ↑b\nhf : ¬Integrable (fun a ↦ ↑(f a)) μ\n⊢ ∫ (a : α), ↑(f a) ∂μ ≤ ↑b",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
... | rw [integral_undef hf]; exact b.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 164,
"column": 7
} | {
"line": 164,
"column": 54
} | [
{
"pp": "case H\nα : Type u_1\ninst✝ : MeasurableSpace α\nf : α →ₛ ℝ\na✝ : α\n⊢ (map norm f.posPart) a✝ = f.posPart a✝",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"Real.lattice",
"MeasureTheory.SimpleFunc.posPart",
"Real.instZero",
"abs",
"congrArg",
... | rw [map_apply, Real.norm_eq_abs, abs_of_nonneg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 116,
"column": 45
} | {
"line": 116,
"column": 72
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : NormedAddCommGroup β\nu : α → β\nl : Filter α\nh : u =O[l] 0\n⊢ u =ᶠ[l] 0",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"congrArg",
"Membership.mem",
"Exists",
"Filter.EventuallyEq",
"id",
"Pi.instZero",
... | eventuallyEq_iff_exists_mem | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 233,
"column": 2
} | {
"line": 234,
"column": 78
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nf : α →ₛ F\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\n⊢ integral μ (piecewise s hs f 0) = integral (μ.restrict s) f",
"usedConstants": [
"instDecidableNot",
"Real",
"in... | refine (integral_eq_sum_of_subset ?_).trans
((sum_congr rfl fun y hy => ?_).trans (integral_eq_sum_filter _ _).symm) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 272,
"column": 2
} | {
"line": 272,
"column": 19
} | [
{
"pp": "case cons\nα : Type u_1\nι : Type u_2\nβ : Type u_3\ninst✝ : NormedField β\nl : Filter α\nf g : ι → α → β\ni : ι\nL : List ι\nihL :\n (∀ i ∈ L, f i ~[l] g i) →\n (fun x ↦ (List.map (fun x_1 ↦ f x_1 x) L).prod) ~[l] fun x ↦ (List.map (fun x_1 ↦ g x_1 x) L).prod\nh : ∀ i_1 ∈ i :: L, f i_1 ~[l] g i_1\... | | cons i L ihL => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 593,
"column": 24
} | {
"line": 593,
"column": 47
} | [
{
"pp": "α : Type u_1\nE : Type u_2\n𝕜 : Type u_4\ninst✝⁶ : NormedAddCommGroup E\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedRing 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : IsBoundedSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : CompleteSpace E\nc : 𝕜\nf : ↥(Lp E 1 μ)\n⊢ integra... | integral_eq' 𝕜 (c • f), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1022,
"column": 4
} | {
"line": 1022,
"column": 98
} | [
{
"pp": "case neg\nα : Type u_1\nG : Type u_5\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_6\ninst✝ : MeasurableSpace β\nφ : α → β\nhφ : Measurable φ\nf : β → G\nhfm : StronglyMeasurable f\nhfi : ¬Integrable f (Measure.map φ μ)\n⊢ ¬Integrable (fun x ... | exact fun hfφ => hfi ((integrable_map_measure hfm.aestronglyMeasurable hφ.aemeasurable).2 hfφ) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Module.RCLike.Basic | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\nhx : x ≠ 0\n⊢ ‖(↑‖x‖)⁻¹ • x‖ = 1",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"False",
"Real",
"eq_false",
"norm_eq_zero._simp_1"... | have : ‖x‖ ≠ 0 := by simp [hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Module.RCLike.Basic | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nr : ℝ\nr_nonneg : 0 ≤ r\nx : E\nhx : x ≠ 0\n⊢ ‖(↑r * (↑‖x‖)⁻¹) • x‖ = r",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"False",
"Real",
"eq_fals... | have : ‖x‖ ≠ 0 := by simp [hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Module.RieszLemma | {
"line": 113,
"column": 26
} | {
"line": 113,
"column": 90
} | [
{
"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\nthis : ‖c‖ / R < 1\nx : E\n... | by gcongr; exact hx y' (by simp [y', Submodule.smul_mem _ _ hy]) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.SumMeasure | {
"line": 61,
"column": 8
} | {
"line": 61,
"column": 55
} | [
{
"pp": "case h.e'_3.h.e'_5.h\nι : Type u_1\nX : Type u_2\nE : Type u_3\ninst✝¹ : Countable ι\nmX : MeasurableSpace X\ninst✝ : NormedAddCommGroup E\nμ : ι → Measure X\nf : X → E\nhf : ∀ (i : ι), Integrable f (μ i)\nh : Summable fun i ↦ ∫ (x : X), ‖f x‖ ∂μ i\ni : ι\n⊢ ∫⁻ (a : X), ‖f a‖ₑ ∂μ i = ENNReal.ofReal (∫ ... | ofReal_integral_eq_lintegral_ofReal (hf i).norm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 251,
"column": 2
} | {
"line": 257,
"column": 95
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nG'' : Type u_7\nG' : Type u_8\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : PartialOrder G'\ninst✝⁴ : IsOrderedAddMonoid G'\ninst✝³ : NormedSpace ℝ G'\ninst✝² : NormedAddCommGroup G''\ninst✝¹ : PartialOrder G''\ninst✝ : NormedSpace ℝ G''\nT : Set α → G'' ... | simp_rw [setToL1S]
obtain ⟨f', hf', hff'⟩ := exists_simpleFunc_nonneg_ae_eq hf
replace hff' : simpleFunc.toSimpleFunc f =ᵐ[μ] f' :=
(Lp.simpleFunc.toSimpleFunc_eq_toFun f).trans hff'
rw [SimpleFunc.setToSimpleFunc_congr _ h_zero h_add (SimpleFunc.integrable _) hff']
exact
SimpleFunc.setToSimpleFunc_nonn... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 251,
"column": 2
} | {
"line": 257,
"column": 95
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nG'' : Type u_7\nG' : Type u_8\ninst✝⁶ : NormedAddCommGroup G'\ninst✝⁵ : PartialOrder G'\ninst✝⁴ : IsOrderedAddMonoid G'\ninst✝³ : NormedSpace ℝ G'\ninst✝² : NormedAddCommGroup G''\ninst✝¹ : PartialOrder G''\ninst✝ : NormedSpace ℝ G''\nT : Set α → G'' ... | simp_rw [setToL1S]
obtain ⟨f', hf', hff'⟩ := exists_simpleFunc_nonneg_ae_eq hf
replace hff' : simpleFunc.toSimpleFunc f =ᵐ[μ] f' :=
(Lp.simpleFunc.toSimpleFunc_eq_toFun f).trans hff'
rw [SimpleFunc.setToSimpleFunc_congr _ h_zero h_add (SimpleFunc.integrable _) hff']
exact
SimpleFunc.setToSimpleFunc_nonn... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 66
} | [
{
"pp": "case pos\n𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : CompleteSpace 𝕜\ns : Finset E\nb : Basis (↥s) 𝕜 E\n⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun f ↦ LinearMap.det ↑f",
"usedConstan... | haveI : FiniteDimensional 𝕜 E := b.finiteDimensional_of_finite | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.Topology.ContinuousMap.Bounded.Normed | {
"line": 157,
"column": 52
} | {
"line": 158,
"column": 76
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : SeminormedAddCommGroup β\nf : α →ᵇ β\n⊢ ‖f‖ = ⨆ x, ‖f x‖",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"BoundedContinuousFunction.dist_eq_iSup",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
... | by
simp_rw [norm_def, dist_eq_iSup, coe_zero, Pi.zero_apply, dist_zero_right] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 667,
"column": 61
} | {
"line": 672,
"column": 33
} | [
{
"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 T' : Set α → E →L[ℝ] F\nC C' : ℝ\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ T' C... | by
by_cases hF : CompleteSpace F; swap
· simp [setToFun, hF]
by_cases hf : Integrable f μ
· simp_rw [setToFun_eq _ hf, L1.setToL1_congr_left' T T' hT hT' h]
· simp_rw [setToFun_undef _ hf] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 270,
"column": 67
} | {
"line": 270,
"column": 88
} | [
{
"pp": "𝕜 : Type u\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : CompleteSpace 𝕜\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\n⊢ Continuous[Pi.topologicalSp... | LinearEquiv.symm_symm | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 328,
"column": 4
} | {
"line": 328,
"column": 29
} | [
{
"pp": "case inr\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : CompleteSpace 𝕜\nι : Type u_1\ninst✝ : Finite ι\ni₀ : ι\n⊢ IsOpen[Pi.topologicalSpace] {p | LinearIndependent 𝕜 fun i ↦ p ↑i -ᵥ p i₀}",
"usedConstants": [
... | let ι' := { x // x ≠ i₀ } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Topology.MetricSpace.ThickenedIndicator | {
"line": 287,
"column": 6
} | {
"line": 287,
"column": 32
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx y : α\nh : infEDist x E ≤ infEDist y E\nh_le : ¬infEDist y E * (↑δ.toNNReal)⁻¹ ≤ 1\n⊢ infEDist y E * (↑δ.toNNReal)⁻¹ ≤ edist x y * (↑δ.toNNReal)⁻¹ + infEDist x E * (↑δ.toNNReal)⁻¹",
"usedConstants": [
"PseudoEMetri... | rw [← add_mul, edist_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 381,
"column": 2
} | {
"line": 381,
"column": 29
} | [
{
"pp": "𝕜 : Type u\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nF : Type w\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : CompleteSpace 𝕜\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : SecondCountableTopology F\n⊢ SecondCountableT... | let d := Module.finrank 𝕜 E | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1075,
"column": 2
} | {
"line": 1078,
"column": 64
} | [
{
"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 C' : ℝ\nμ' : Measure α\nc c' : ℝ≥0∞\nhc : c ≠ ∞\nhc' : c' ≠ ∞\nhμ_le : μ ≤ ... | · -- if `f` is not integrable, both `setToFun` are 0.
have h_int : ∀ g : α → E, ¬Integrable g μ → ¬Integrable g μ' := fun g =>
mt fun h => h.of_measure_le_smul hc hμ_le
simp_rw [setToFun_undef _ hf, setToFun_undef _ (h_int f hf)] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
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