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.Topology.Algebra.Module.Spaces.UniformConvergenceCLM | {
"line": 506,
"column": 4
} | {
"line": 506,
"column": 71
} | [
{
"pp": "𝕜₁✝ : Type u_1\n𝕜₂✝ : Type u_2\ninst✝²³ : NormedField 𝕜₁✝\ninst✝²² : NormedField 𝕜₂✝\nσ✝ : 𝕜₁✝ →+* 𝕜₂✝\nE✝ : Type u_3\nF✝ : Type u_4\nG✝ : Type u_5\ninst✝²¹ : AddCommGroup E✝\ninst✝²⁰ : Module 𝕜₁✝ E✝\ninst✝¹⁹ : TopologicalSpace E✝\ninst✝¹⁸ : AddCommGroup F✝\ninst✝¹⁷ : Module 𝕜₂✝ F✝\n𝕜₁ : Type ... | rw [(UniformConvergenceCLM.isEmbedding_coeFn _ _ _).continuous_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 1073,
"column": 2
} | {
"line": 1073,
"column": 69
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_6\nι : Type u_9\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : Countable ι\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\nhp : WithSeminorms p\nthis : IsTopologicalAddGroup E\n⊢ FirstCountableTopology E",
"usedConstan... | let _ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Normed.Operator.NNNorm | {
"line": 205,
"column": 21
} | {
"line": 205,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SeminormedAddCommGroup F\ninst✝⁵ : DenselyNormedField 𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜₂\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹ : RingHomIsometric σ₁₂\ninst... | simpa using congrArg NNReal.toReal hx | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Normed.Operator.NNNorm | {
"line": 205,
"column": 21
} | {
"line": 205,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SeminormedAddCommGroup F\ninst✝⁵ : DenselyNormedField 𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜₂\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹ : RingHomIsometric σ₁₂\ninst... | simpa using congrArg NNReal.toReal hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Operator.NNNorm | {
"line": 205,
"column": 21
} | {
"line": 205,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SeminormedAddCommGroup F\ninst✝⁵ : DenselyNormedField 𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜₂\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹ : RingHomIsometric σ₁₂\ninst... | simpa using congrArg NNReal.toReal hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Germ.Basic | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 21
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nl : Filter α\nf : α → β\ng : β → γ\nh : (↑f).IsConstant\n⊢ (↑(g ∘ f)).IsConstant",
"usedConstants": []
}
] | obtain ⟨b, hb⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Order.Filter.ENNReal | {
"line": 193,
"column": 2
} | {
"line": 194,
"column": 22
} | [
{
"pp": "case pos\nα : Type u_1\nf : Filter α\ninst✝ : f.NeBot\nu : α → ℝ≥0∞\na : ℝ≥0∞\nha_top : a ≠ ∞\nha₀ : a = 0\n⊢ liminf (fun x ↦ a * u x) f = a * liminf u f",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Filter.liminf",
"MulZeroClass.toMul",
"congrArg",
"CommSemiring.... | · simp_rw [ha₀, zero_mul, ← ENNReal.bot_eq_zero]
apply liminf_const | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Filter.ENNReal | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 27
} | [
{
"pp": "α : Type u_1\nf : Filter α\nu : α → ℝ\nh₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nh₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nr : ℝ≥0\n⊢ ENNReal.ofReal (limsup u f) ≤ ↑r ↔ limsup (fun a ↦ ENNReal.ofReal (u a)) f ≤ ↑r",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
... | simp only [ofReal_le_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.ConditionalProbability | {
"line": 229,
"column": 21
} | {
"line": 232,
"column": 34
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : Measure Ω\ns t : Set Ω\nhms : MeasurableSet s\nhcst : μ[t | s] ≠ 0\n⊢ 0 < μ (s ∩ t)",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Iff.mpr",
"Eq.mpr",
"instHSMul",
"MeasureTheory.Measure",
"Preorder.toLT",
... | by
refine pos_iff_ne_zero.mpr (right_ne_zero_of_mul (a := (μ s)⁻¹) ?_)
convert! hcst
simp [hms, Set.inter_comm, cond] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.UniformOn | {
"line": 229,
"column": 34
} | {
"line": 229,
"column": 53
} | [
{
"pp": "Ω : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : MeasurableSingletonClass Ω\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : Finite Ω\nf t : ι → Set Ω\nht : ∀ (i : ι), MeasurableSet (t i)\n⊢ ∀ (i : ι), (f i).Finite",
"usedConstants": [
"Subtype.finite",
"Set.Finite",
"Membership.mem",... | simp [Set.toFinite] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.UniformOn | {
"line": 229,
"column": 34
} | {
"line": 229,
"column": 53
} | [
{
"pp": "Ω : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : MeasurableSingletonClass Ω\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : Finite Ω\nf t : ι → Set Ω\nht : ∀ (i : ι), MeasurableSet (t i)\n⊢ ∀ (i : ι), (f i).Finite",
"usedConstants": [
"Subtype.finite",
"Set.Finite",
"Membership.mem",... | simp [Set.toFinite] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.UniformOn | {
"line": 229,
"column": 34
} | {
"line": 229,
"column": 53
} | [
{
"pp": "Ω : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : MeasurableSingletonClass Ω\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : Finite Ω\nf t : ι → Set Ω\nht : ∀ (i : ι), MeasurableSet (t i)\n⊢ ∀ (i : ι), (f i).Finite",
"usedConstants": [
"Subtype.finite",
"Set.Finite",
"Membership.mem",... | simp [Set.toFinite] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.UniformOn | {
"line": 230,
"column": 34
} | {
"line": 230,
"column": 53
} | [
{
"pp": "Ω : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : MeasurableSingletonClass Ω\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : Finite Ω\nt : ι → Set Ω\nht : ∀ (i : ι), MeasurableSet (t i)\nf : ι → Finset Ω\n⊢ ∀ (i : ι), (t i).Finite",
"usedConstants": [
"Subtype.finite",
"Set.Finite",
"... | simp [Set.toFinite] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.UniformOn | {
"line": 230,
"column": 34
} | {
"line": 230,
"column": 53
} | [
{
"pp": "Ω : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : MeasurableSingletonClass Ω\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : Finite Ω\nt : ι → Set Ω\nht : ∀ (i : ι), MeasurableSet (t i)\nf : ι → Finset Ω\n⊢ ∀ (i : ι), (t i).Finite",
"usedConstants": [
"Subtype.finite",
"Set.Finite",
"... | simp [Set.toFinite] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.UniformOn | {
"line": 230,
"column": 34
} | {
"line": 230,
"column": 53
} | [
{
"pp": "Ω : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : MeasurableSingletonClass Ω\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : Finite Ω\nt : ι → Set Ω\nht : ∀ (i : ι), MeasurableSet (t i)\nf : ι → Finset Ω\n⊢ ∀ (i : ι), (t i).Finite",
"usedConstants": [
"Subtype.finite",
"Set.Finite",
"... | simp [Set.toFinite] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 34
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_3\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ni : ι\ns : Set ((a : ι) → α a)\nhs : MeasurableSet s\n⊢ MeasurableSet s",
"usedConstants": [
"MeasurableSpace.comap",
"MeasurableSet",
"congrArg",
"Measu... | rw [MeasurableSpace.comap] at hs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.ENNReal.Holder | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 12
} | [
{
"pp": "case inl\np q : ℝ≥0∞\nhq : q ≠ 0\ninst✝ : p.HolderTriple q 0\nhp : p ≠ 0\nthis : 0⁻¹ < ∞\n⊢ False",
"usedConstants": [
"False",
"Preorder.toLT",
"congrArg",
"False.elim",
"PartialOrder.toPreorder",
"lt_self_iff_false._simp_1",
"Eq.mp",
"ENNReal.inv_ze... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.ENNReal.Holder | {
"line": 178,
"column": 26
} | {
"line": 178,
"column": 34
} | [
{
"pp": "p q : ℝ≥0∞\ninst✝ : p.HolderConjugate q\nhp : p ≠ ∞\nthis : p ≠ 0\n⊢ 0 < 1 → 1 < p → p⁻¹ ≠ ∞",
"usedConstants": [
"ENNReal.instIsOrderedRing",
"False",
"Preorder.toLT",
"eq_false",
"IsOrderedRing.toZeroLEOneClass",
"congrArg",
"CommSemiring.toSemiring",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.ENNReal.Holder | {
"line": 178,
"column": 26
} | {
"line": 178,
"column": 34
} | [
{
"pp": "p q : ℝ≥0∞\ninst✝ : p.HolderConjugate q\nhp : p ≠ ∞\nthis : p ≠ 0\n⊢ 0 < 1 → 1 < p → p⁻¹ ≠ ∞",
"usedConstants": [
"ENNReal.instIsOrderedRing",
"False",
"Preorder.toLT",
"eq_false",
"IsOrderedRing.toZeroLEOneClass",
"congrArg",
"CommSemiring.toSemiring",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENNReal.Holder | {
"line": 178,
"column": 26
} | {
"line": 178,
"column": 34
} | [
{
"pp": "p q : ℝ≥0∞\ninst✝ : p.HolderConjugate q\nhp : p ≠ ∞\nthis : p ≠ 0\n⊢ 0 < 1 → 1 < p → p⁻¹ ≠ ∞",
"usedConstants": [
"ENNReal.instIsOrderedRing",
"False",
"Preorder.toLT",
"eq_false",
"IsOrderedRing.toZeroLEOneClass",
"congrArg",
"CommSemiring.toSemiring",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 36,
"column": 64
} | {
"line": 36,
"column": 81
} | [
{
"pp": "α : Type u_1\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\ng : α → G\nc : ℝ≥0\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊\np : ℝ\nhp : 0 < p\n⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ) ^ 1 ≤ ↑c ^ p * (∫⁻ (a : α), ‖g a‖ₑ ^ p ∂μ)... | ENNReal.rpow_one, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 51,
"column": 64
} | {
"line": 51,
"column": 81
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε : Type u_5\nε' : Type u_6\ninst✝³ : TopologicalSpace ε\ninst✝² : ContinuousENorm ε\ninst✝¹ : TopologicalSpace ε'\ninst✝ : ContinuousENorm ε'\nf : α → ε\ng : α → ε'\nc : ℝ≥0\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ ↑c * ‖g x‖ₑ\np : ℝ\nhp : 0 < p\n⊢ (∫⁻ (a : α), ... | ENNReal.rpow_one, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 75,
"column": 79
} | {
"line": 75,
"column": 87
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε' : Type u_6\ninst✝³ : TopologicalSpace ε'\ninst✝² : ContinuousENorm ε'\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\nc : ℝ≥0∞\ng : α → ε'\np : ℝ\nhg : AEStronglyMeasurable g μ\nh✝ : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ c ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 75,
"column": 79
} | {
"line": 75,
"column": 87
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε' : Type u_6\ninst✝³ : TopologicalSpace ε'\ninst✝² : ContinuousENorm ε'\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\nc : ℝ≥0∞\ng : α → ε'\np : ℝ\nhg : AEStronglyMeasurable g μ\nh✝ : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ c ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 75,
"column": 79
} | {
"line": 75,
"column": 87
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε' : Type u_6\ninst✝³ : TopologicalSpace ε'\ninst✝² : ContinuousENorm ε'\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\nc : ℝ≥0∞\ng : α → ε'\np : ℝ\nhg : AEStronglyMeasurable g μ\nh✝ : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ c ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 77,
"column": 6
} | {
"line": 77,
"column": 14
} | [
{
"pp": "case neg\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε' : Type u_6\ninst✝³ : TopologicalSpace ε'\ninst✝² : ContinuousENorm ε'\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\nc : ℝ≥0∞\ng : α → ε'\np : ℝ\nhg : AEStronglyMeasurable g μ\nh : ∀ᵐ (x : α) ∂μ, ‖f... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 77,
"column": 6
} | {
"line": 77,
"column": 14
} | [
{
"pp": "case neg\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε' : Type u_6\ninst✝³ : TopologicalSpace ε'\ninst✝² : ContinuousENorm ε'\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\nc : ℝ≥0∞\ng : α → ε'\np : ℝ\nhg : AEStronglyMeasurable g μ\nh : ∀ᵐ (x : α) ∂μ, ‖f... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 77,
"column": 6
} | {
"line": 77,
"column": 14
} | [
{
"pp": "case neg\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε' : Type u_6\ninst✝³ : TopologicalSpace ε'\ninst✝² : ContinuousENorm ε'\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\nc : ℝ≥0∞\ng : α → ε'\np : ℝ\nhg : AEStronglyMeasurable g μ\nh : ∀ᵐ (x : α) ∂μ, ‖f... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 81,
"column": 64
} | {
"line": 81,
"column": 81
} | [
{
"pp": "case neg\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε' : Type u_6\ninst✝³ : TopologicalSpace ε'\ninst✝² : ContinuousENorm ε'\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\nc : ℝ≥0∞\ng : α → ε'\np : ℝ\nhg : AEStronglyMeasurable g μ\nh : ∀ᵐ (x : α) ∂μ, ‖f... | ENNReal.rpow_one, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 93,
"column": 30
} | {
"line": 93,
"column": 59
} | [
{
"pp": "x y z : ℝ\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nh : 0 < y - x\nhxy' : 0 < x / y\nhxy'' : x / y ≠ 1\n⊢ -log (x / y) = -(log x - log y)",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"Real.instZero",
"congrArg",
"Real.instDivInvMonoid",... | by rw [log_div hx.ne' hy.ne'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 208,
"column": 53
} | {
"line": 208,
"column": 70
} | [
{
"pp": "α : Type u_1\nF : Type u_3\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup F\nhp : p ≠ 0\nhp' : p ≠ ∞\nf : α → F\nC : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nhf : ∀ᵐ (x : α) ∂μ, x ∈ s → C ≤ ‖f x‖₊\n⊢ ↑C ^ p.toReal * μ s ^ 1 ≤ ∫⁻ (x : α), ‖f x‖ₑ ^ p.toReal ∂μ",
"usedConstan... | ENNReal.rpow_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Real.ConjExponents | {
"line": 124,
"column": 20
} | {
"line": 124,
"column": 73
} | [
{
"pp": "p q r : ℝ\nh : p.HolderTriple q r\n⊢ r < p",
"usedConstants": [
"Real.partialOrder",
"Real",
"Preorder.toLT",
"GroupWithZero.toDivisionMonoid",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"MulZeroClass.toMul",
"IsStrictOrde... | by simpa using inv_strictAnti₀ h.inv_pos h.inv_lt_inv | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Real.ConjExponents | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 48
} | [
{
"pp": "p q : ℝ\nhp : 1 < p\nh : p⁻¹ + q⁻¹ = 1\nhp' : 0 < p\n⊢ p.HolderConjugate q",
"usedConstants": [
"Real",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"inv_one",
"Real.instInv",
"DivisionMonoid.toDivInvOneMonoid... | refine ⟨inv_one (G := ℝ) |>.symm ▸ h, hp', ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 159,
"column": 4
} | {
"line": 164,
"column": 41
} | [
{
"pp": "case inr.a.inr\ns : ℝ\nhs✝ : -1 ≤ s\nhs'✝ : s ≠ 0\np : ℝ\nhp1 : 0 < p\nhp2 : p < 1\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhs3 : 1 + s ≠ 1\nhs4 : 1 + p * s ≠ 1\nhs' : 0 < s\n⊢ log (1 + s) * p < log (1 + p * s)",
"usedConstants": [
"Eq.mpr",
"add_lt_add_right",
"GroupWi... | rw [← lt_div_iff₀ hp1, ← div_lt_div_iff_of_pos_right hs']
convert! strictConcaveOn_log_Ioi.secant_strict_mono (zero_lt_one' ℝ) hs2 hs1 hs4 hs3 _ using 1
· rw [add_sub_cancel_left, log_one, sub_zero]
· rw [add_sub_cancel_left, div_div, log_one, sub_zero]
· gcongr
exact mul_lt_of_lt_one_left hs' hp2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 159,
"column": 4
} | {
"line": 164,
"column": 41
} | [
{
"pp": "case inr.a.inr\ns : ℝ\nhs✝ : -1 ≤ s\nhs'✝ : s ≠ 0\np : ℝ\nhp1 : 0 < p\nhp2 : p < 1\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhs3 : 1 + s ≠ 1\nhs4 : 1 + p * s ≠ 1\nhs' : 0 < s\n⊢ log (1 + s) * p < log (1 + p * s)",
"usedConstants": [
"Eq.mpr",
"add_lt_add_right",
"GroupWi... | rw [← lt_div_iff₀ hp1, ← div_lt_div_iff_of_pos_right hs']
convert! strictConcaveOn_log_Ioi.secant_strict_mono (zero_lt_one' ℝ) hs2 hs1 hs4 hs3 _ using 1
· rw [add_sub_cancel_left, log_one, sub_zero]
· rw [add_sub_cancel_left, div_div, log_one, sub_zero]
· gcongr
exact mul_lt_of_lt_one_left hs' hp2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Real.ConjExponents | {
"line": 503,
"column": 4
} | {
"line": 503,
"column": 35
} | [
{
"pp": "p q : ℝ≥0∞\nh : p.HolderConjugate q\nhp : p ≤ 1\n⊢ p.toReal ≤ 1",
"usedConstants": [
"ENNReal",
"One.toOfNat1",
"ENNReal.instOne",
"OfNat.ofNat",
"ENNReal.toReal_mono",
"ENNReal.one_ne_top"
]
}
] | exact toReal_mono one_ne_top hp | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Slope | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\n⊢ (z - x) * f y ≤ (z - y) * f x + (y - x) * f z",
"usedConstant... | have hxz' : 0 < z - x := by linarith | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.Slope | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\n⊢ (f y - f x) / (y - x) ≤ (f z - f x) / (z - x)",
"usedConstants": [
"Mathl... | have hxz' : 0 < z - x := by linarith | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.Slope | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhyz' : 0 < z - y\n⊢ (f z - f x) / (z - x) ≤ (f z - f y) / (z - y)",
"usedConstants": [
"Mathl... | have hxz' : 0 < z - x := by linarith | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.Slope | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\nhyz' : 0 < z - y\n⊢ (z - x) * f y < (z - y) * f x + (y - x) * f z",
"usedCo... | have hxz' : 0 < z - x := by linarith | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.Slope | {
"line": 231,
"column": 2
} | {
"line": 231,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhxy' : 0 < y - x\n⊢ (f y - f x) / (y - x) < (f z - f x) / (z - x)",
"usedConstants": [
... | have hxz' : 0 < z - x := by linarith | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.Slope | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : StrictConvexOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nhyz' : 0 < z - y\n⊢ (f z - f x) / (z - x) < (f z - f y) / (z - y)",
"usedConstants": [
... | have hxz' : 0 < z - x := by linarith | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.Slope | {
"line": 280,
"column": 4
} | {
"line": 281,
"column": 22
} | [
{
"pp": "case inr.refine_2\n𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\nstep1 : ∀ {z : �... | · rw [openSegment_eq_Ioo (hu2.trans huv)]
exact ⟨hu2, huv⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.Monovary | {
"line": 300,
"column": 54
} | {
"line": 300,
"column": 95
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : LinearOrder α\ninst✝² : Semifield β\ninst✝¹ : LinearOrder β\ninst✝ : IsStrictOrderedRing β\ns : Set ι\nf : ι → α\ng : ι → β\nhg : ∀ (i : ι), i ∈ s → 0 < g i\ni : ι\nhi : i ∈ s\nj : ι\nhj : j ∈ s\n⊢ g⁻¹ j < g⁻¹ i → f j ≤ f i ↔ g i < g j → f j ≤ f i",
... | by simp [inv_lt_inv₀ (hg _ hj) (hg _ hi)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Monovary | {
"line": 304,
"column": 54
} | {
"line": 304,
"column": 95
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : LinearOrder α\ninst✝² : Semifield β\ninst✝¹ : LinearOrder β\ninst✝ : IsStrictOrderedRing β\ns : Set ι\nf : ι → α\ng : ι → β\nhg : ∀ (i : ι), i ∈ s → 0 < g i\ni : ι\nhi : i ∈ s\nj : ι\nhj : j ∈ s\n⊢ g⁻¹ j < g⁻¹ i → f i ≤ f j ↔ g i < g j → f i ≤ f j",
... | by simp [inv_lt_inv₀ (hg _ hj) (hg _ hi)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 110,
"column": 2
} | {
"line": 125,
"column": 86
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.HolderConjugate q\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhf_nontop : ∫⁻ (a : α), f a ^ p ∂μ ≠ ∞\nhg_nontop : ∫⁻ (a : α), g a ^ q ∂μ ≠ ∞\nhf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0\nhg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0\n⊢ ∫⁻ (a : ... | let npf := (∫⁻ c : α, f c ^ p ∂μ) ^ (1 / p)
let nqg := (∫⁻ c : α, g c ^ q ∂μ) ^ (1 / q)
calc
(∫⁻ a : α, (f * g) a ∂μ) =
∫⁻ a : α, (funMulInvSnorm f p μ * funMulInvSnorm g q μ) a * (npf * nqg) ∂μ := by
refine lintegral_congr fun a => ?_
rw [Pi.mul_apply, fun_eq_funMulInvSnorm_mul_eLpNorm f hf... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 110,
"column": 2
} | {
"line": 125,
"column": 86
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.HolderConjugate q\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhf_nontop : ∫⁻ (a : α), f a ^ p ∂μ ≠ ∞\nhg_nontop : ∫⁻ (a : α), g a ^ q ∂μ ≠ ∞\nhf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0\nhg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0\n⊢ ∫⁻ (a : ... | let npf := (∫⁻ c : α, f c ^ p ∂μ) ^ (1 / p)
let nqg := (∫⁻ c : α, g c ^ q ∂μ) ^ (1 / q)
calc
(∫⁻ a : α, (f * g) a ∂μ) =
∫⁻ a : α, (funMulInvSnorm f p μ * funMulInvSnorm g q μ) a * (npf * nqg) ∂μ := by
refine lintegral_congr fun a => ?_
rw [Pi.mul_apply, fun_eq_funMulInvSnorm_mul_eLpNorm f hf... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 173,
"column": 26
} | {
"line": 173,
"column": 36
} | [
{
"pp": "p q : ℝ\na b : ℝ≥0\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0), a ^ q = (a ^ p) ^ (q / p)\nh_rpow_add_rpow_le_add : ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p\n⊢ ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)",
"usedConstants": [
... | one_div p, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 249,
"column": 53
} | {
"line": 249,
"column": 72
} | [
{
"pp": "case refine_2\nι : 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_... | toNNReal_sum h_top, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 308,
"column": 26
} | {
"line": 308,
"column": 36
} | [
{
"pp": "p q : ℝ\na b : ℝ≥0∞\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0∞), a ^ q = (a ^ p) ^ (q / p)\nh_rpow_add_rpow_le_add : ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p\n⊢ ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)",
"usedConstants": [
... | one_div p, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 38
} | [
{
"pp": "case inr\nα : Type u_1\nε : Type u_3\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nμ : Measure α\nf g : α → ε\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\np : ℝ≥0∞\nhp : p ≠ 0\n⊢ eLpNorm (f + g) p μ ≤ p.LpAddConst * (eLpNorm f p μ + eLpNorm g p μ... | rcases lt_or_ge p 1 with (h'p | h'p) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality | {
"line": 71,
"column": 6
} | {
"line": 71,
"column": 57
} | [
{
"pp": "case h.e'_4.h.e'_5\nα : Type u_1\nε : Type u_3\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nμ : Measure α\nf g : α → ε\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\np : ℝ≥0∞\nhp : p ≠ 0\nh'p : p < 1\n⊢ p.LpAddConst = 2 ^ (1 / p.toReal - 1)",
... | have : p ∈ Set.Ioo (0 : ℝ≥0∞) 1 := ⟨hp.bot_lt, h'p⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 235,
"column": 4
} | {
"line": 235,
"column": 12
} | [
{
"pp": "case inl\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nμ : Measure α\nf : α → E\ng : α → F\nq r : ℝ≥0∞\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nb : E → F →... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 235,
"column": 4
} | {
"line": 235,
"column": 12
} | [
{
"pp": "case inl\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nμ : Measure α\nf : α → E\ng : α → F\nq r : ℝ≥0∞\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nb : E → F →... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 235,
"column": 4
} | {
"line": 235,
"column": 12
} | [
{
"pp": "case inl\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nμ : Measure α\nf : α → E\ng : α → F\nq r : ℝ≥0∞\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nb : E → F →... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 239,
"column": 4
} | {
"line": 239,
"column": 12
} | [
{
"pp": "case inr.inr.inl\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nμ : Measure α\nf : α → E\ng : α → F\np r : ℝ≥0∞\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nb :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 239,
"column": 4
} | {
"line": 239,
"column": 12
} | [
{
"pp": "case inr.inr.inl\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nμ : Measure α\nf : α → E\ng : α → F\np r : ℝ≥0∞\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nb :... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 239,
"column": 4
} | {
"line": 239,
"column": 12
} | [
{
"pp": "case inr.inr.inl\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nμ : Measure α\nf : α → E\ng : α → F\np r : ℝ≥0∞\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nb :... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 39
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\n𝕜 : Type u_3\nx✝ : MeasurableSpace α\ninst✝ : NormedCommRing 𝕜\nμ : Measure α\nf : ι → α → 𝕜\np : ι → ℝ≥0∞\ns : Finset ι\nhf : ∀ i ∈ s, MemLp (f i) (p i) μ\n⊢ MemLp (∏ i ∈ s, f i) (∑ i ∈ s, (p i)⁻¹)⁻¹ μ",
"usedConstants": [
"NormedCommRing.toNormedRing",
"... | induction s using cons_induction with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.MeasureTheory.Function.LpSpace.Complete | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 24
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nE : Type u_3\ninst✝ : NormedAddCommGroup E\nhp : Fact (1 ≤ p)\nH :\n ∀ (f : ℕ → α → E),\n (∀ (n : ℕ), MemLp (f n) p μ) →\n ∀ (B : ℕ → ℝ≥0∞),\n ∑' (i : ℕ), B i < ∞ →\n (∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → eLpNorm (f n ... | rw [hB1_has.tsum_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.ConvergenceInMeasure | {
"line": 66,
"column": 2
} | {
"line": 71,
"column": 8
} | [
{
"pp": "case pos\nα : Type u_1\nι : Type u_2\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : EDist E\nf : ι → α → E\nl : Filter ι\ng : α → E\nh : ∀ (ε : ℝ≥0∞), 0 < ε → ε ≠ ∞ → Tendsto (fun i ↦ μ {x | ε ≤ edist (f i x) (g x)}) l (𝓝 0)\nε : ℝ≥0∞\nhε : 0 < ε\nhε_top : ε = ∞\n⊢ Tendsto (fun i ↦ μ {x ... | · have h1 : Tendsto (fun n ↦ μ {ω | 1 ≤ edist (f n ω) (g ω)}) l (𝓝 0) := h 1 (by simp) (by simp)
refine tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds h1 (fun _ ↦ zero_le) ?_
intro n
simp only [hε_top]
gcongr
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.ConvergenceInMeasure | {
"line": 167,
"column": 23
} | {
"line": 167,
"column": 31
} | [
{
"pp": "case pos\nα : Type u_1\nι : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nl : Filter ι\nF : Type u_5\ninst✝¹ : PseudoEMetricSpace F\ninst✝ : Zero F\nf : ι → α → F\ng : α → F\nhg : TendstoInMeasure μ f l g\ns : Set α\nε : ℝ≥0∞\nhε : 0 < ε\nn : ι\nx : α\nhx : x ∈ {x | ε ≤ edist ((fun i ↦ s.indicator (f... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Function.ConvergenceInMeasure | {
"line": 167,
"column": 23
} | {
"line": 167,
"column": 31
} | [
{
"pp": "case neg\nα : Type u_1\nι : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nl : Filter ι\nF : Type u_5\ninst✝¹ : PseudoEMetricSpace F\ninst✝ : Zero F\nf : ι → α → F\ng : α → F\nhg : TendstoInMeasure μ f l g\ns : Set α\nε : ℝ≥0∞\nhε : 0 < ε\nn : ι\nx : α\nhx : x ∈ {x | ε ≤ edist ((fun i ↦ s.indicator (f... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.MeanInequalities | {
"line": 1051,
"column": 4
} | {
"line": 1051,
"column": 50
} | [
{
"pp": "case pos\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0∞\np : ℝ\nhp : 1 ≤ p\nH' : (∑ i ∈ s, f i ^ p) ^ (1 / p) = ∞ ∨ (∑ i ∈ s, g i ^ p) ^ (1 / p) = ∞\n⊢ (∑ i ∈ s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) + (∑ i ∈ s, g i ^ p) ^ (1 / p)",
"usedConstants": [
"ENNReal.instAdd",
"... | rcases H' with H' | H' <;> simp [H', -one_div] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.MeanInequalities | {
"line": 1051,
"column": 4
} | {
"line": 1051,
"column": 50
} | [
{
"pp": "case pos\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0∞\np : ℝ\nhp : 1 ≤ p\nH' : (∑ i ∈ s, f i ^ p) ^ (1 / p) = ∞ ∨ (∑ i ∈ s, g i ^ p) ^ (1 / p) = ∞\n⊢ (∑ i ∈ s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) + (∑ i ∈ s, g i ^ p) ^ (1 / p)",
"usedConstants": [
"ENNReal.instAdd",
"... | rcases H' with H' | H' <;> simp [H', -one_div] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.MeanInequalities | {
"line": 1051,
"column": 4
} | {
"line": 1051,
"column": 50
} | [
{
"pp": "case pos\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0∞\np : ℝ\nhp : 1 ≤ p\nH' : (∑ i ∈ s, f i ^ p) ^ (1 / p) = ∞ ∨ (∑ i ∈ s, g i ^ p) ^ (1 / p) = ∞\n⊢ (∑ i ∈ s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) + (∑ i ∈ s, g i ^ p) ^ (1 / p)",
"usedConstants": [
"ENNReal.instAdd",
"... | rcases H' with H' | H' <;> simp [H', -one_div] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.MeanInequalities | {
"line": 1055,
"column": 2
} | {
"line": 1058,
"column": 11
} | [
{
"pp": "case neg\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0∞\np : ℝ\nhp : 1 ≤ p\npos : 0 < p\nH' : (∀ i ∈ s, f i ≠ ∞) ∧ ∀ i ∈ s, g i ≠ ∞\n⊢ (∑ i ∈ s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) + (∑ i ∈ s, g i ^ p) ^ (1 / p)",
"usedConstants": [
"Iff.mpr",
"Real",
"ENNReal.ofN... | have :=
ENNReal.coe_le_coe.2
(@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
hp) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.MeanInequalities | {
"line": 1060,
"column": 2
} | {
"line": 1060,
"column": 10
} | [
{
"pp": "case neg\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0∞\np : ℝ\nhp : 1 ≤ p\npos : 0 < p\nH' : (∀ i ∈ s, f i ≠ ∞) ∧ ∀ i ∈ s, g i ≠ ∞\nthis :\n (∑ x ∈ s, (↑(f x).toNNReal + ↑(g x).toNNReal) ^ p) ^ (1 / p) ≤\n (∑ x ∈ s, ↑(f x).toNNReal ^ p) ^ (1 / p) + (∑ x ∈ s, ↑(g x).toNNReal ^ p) ^ (1 / p)\n⊢ (∑ i ∈ s, ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 79
} | [
{
"pp": "α : Type u_1\nE : Type u_4\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : ↥(Lp E p μ)\n⊢ dist f g = (eLpNorm (↑↑f - ↑↑g) p μ).toReal",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"SubtractionMonoid.toInvolutiveNeg",
"Real",
... | rw [dist_eq_eLpNorm_neg_add, ← eLpNorm_neg, neg_add, neg_neg, sub_eq_add_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 79
} | [
{
"pp": "α : Type u_1\nE : Type u_4\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : ↥(Lp E p μ)\n⊢ dist f g = (eLpNorm (↑↑f - ↑↑g) p μ).toReal",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"SubtractionMonoid.toInvolutiveNeg",
"Real",
... | rw [dist_eq_eLpNorm_neg_add, ← eLpNorm_neg, neg_add, neg_neg, sub_eq_add_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 79
} | [
{
"pp": "α : Type u_1\nE : Type u_4\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : ↥(Lp E p μ)\n⊢ dist f g = (eLpNorm (↑↑f - ↑↑g) p μ).toReal",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"SubtractionMonoid.toInvolutiveNeg",
"Real",
... | rw [dist_eq_eLpNorm_neg_add, ← eLpNorm_neg, neg_add, neg_neg, sub_eq_add_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 894,
"column": 10
} | {
"line": 894,
"column": 13
} | [
{
"pp": "case h\nα : Type u_1\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nf : ↥(Lp ℝ p μ)\na✝ : α\nh₁ : ↑↑(posPart (-f)) a✝ = max (↑↑(-f) a✝) 0\nh₂ : ↑↑(-f) a✝ = (-↑↑f) a✝\n⊢ max (↑↑(-f) a✝) 0 = max (-↑↑f a✝) 0",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"Pi.ins... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 903,
"column": 2
} | {
"line": 903,
"column": 16
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : Fact (1 ≤ p)\n⊢ Continuous fun f ↦ negPart f",
"usedConstants": [
"Real",
"Continuous",
"PseudoMetricSpace.toUniformSpace",
"MeasureTheory.Lp.negPart",
"AddCommGroup.toAddGroup",
"Membership.me... | unfold negPart | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.MeasureTheory.Measure.Real | {
"line": 331,
"column": 48
} | {
"line": 336,
"column": 64
} | [
{
"pp": "α : Type u_1\nx✝ : MeasurableSpace α\nμ : Measure α\ns₁ s₂ : Set α\nh : μ s₂ ≠ ∞\n⊢ μ.real s₁ - μ.real s₂ ≤ μ.real (s₁ \\ s₂)",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Trans.trans",
"_private.Mathlib.MeasureTheory.Measure.Real.0.MeasureTheory.le_measureRea... | by
simp only [tsub_le_iff_left]
calc
μ.real s₁ ≤ μ.real (s₂ ∪ s₁) := measureReal_le_measureReal_union_right h
_ = μ.real (s₂ ∪ s₁ \ s₂) := congr_arg μ.real union_diff_self.symm
_ ≤ μ.real s₂ + μ.real (s₁ \ s₂) := measureReal_union_le _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 699,
"column": 33
} | {
"line": 699,
"column": 73
} | [
{
"pp": "α : Type u_1\nE : Type u_5\nmα : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nμ : Measure α\nl : Filter α\nf : α → E\nh' : ∀ s ∈ l, μ s = ∞\na : E\nhf : Tendsto f l (𝓝 a)\nH : ¬a = 0\nε : ℝ\nεpos : 0 < ε\nhε : ε < ‖a‖\nu : Set α\nul : u ∈ l\nhu : IntegrableOn f u μ\nv : Set α := u ∩ {b | ε < ‖f b‖... | simpa only [Measure.restrict_apply_self] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 699,
"column": 33
} | {
"line": 699,
"column": 73
} | [
{
"pp": "α : Type u_1\nE : Type u_5\nmα : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nμ : Measure α\nl : Filter α\nf : α → E\nh' : ∀ s ∈ l, μ s = ∞\na : E\nhf : Tendsto f l (𝓝 a)\nH : ¬a = 0\nε : ℝ\nεpos : 0 < ε\nhε : ε < ‖a‖\nu : Set α\nul : u ∈ l\nhu : IntegrableOn f u μ\nv : Set α := u ∩ {b | ε < ‖f b‖... | simpa only [Measure.restrict_apply_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 699,
"column": 33
} | {
"line": 699,
"column": 73
} | [
{
"pp": "α : Type u_1\nE : Type u_5\nmα : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nμ : Measure α\nl : Filter α\nf : α → E\nh' : ∀ s ∈ l, μ s = ∞\na : E\nhf : Tendsto f l (𝓝 a)\nH : ¬a = 0\nε : ℝ\nεpos : 0 < ε\nhε : ε < ‖a‖\nu : Set α\nul : u ∈ l\nhu : IntegrableOn f u μ\nv : Set α := u ∩ {b | ε < ‖f b‖... | simpa only [Measure.restrict_apply_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 730,
"column": 6
} | {
"line": 730,
"column": 41
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : OpensMeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : TopologicalSpace β\ninst✝ : BorelSpace β\nf : α → β\ns : Set α\nμ : Measure α\nhf : ContinuousOn f s\nhs : NullMeasurableSet s μ\nt : Set α\nts : t ⊆ s\... | ← Measure.restrict_congr_set t_eq_s | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 291,
"column": 2
} | {
"line": 292,
"column": 63
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nE : ι → Type wE\nG : Type wG\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝³ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝² : SeminormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : Fintype ι\nf : MultilinearMap 𝕜 E G\nC : ℝ\nD : ℝ ... | have : max ‖m'‖ ‖m‖ ≤ ‖m‖ + 1 := by
simp [zero_le_one, norm_le_of_mem_closedBall (le_of_lt h')] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Algebra.Module.FiniteDimension | {
"line": 232,
"column": 8
} | {
"line": 232,
"column": 63
} | [
{
"pp": "𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\ninst✝⁷ : CompleteSpace 𝕜\nn : ℕ\nIH :\n ∀ {E : Type v} [inst : AddCommGroup E] [inst_1 : Module 𝕜 E] [inst_2 : TopologicalSpace E]\n [inst_3 : IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E] [T2Space E] {ι : Type v} [inst_6 : Finite ι]\n (ξ : Bas... | have : Continuous b.equivFun := IH b inferInstance this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 265,
"column": 19
} | {
"line": 265,
"column": 27
} | [
{
"pp": "case singleton\nα : Type u_1\nm : MeasurableSpace α\nβ : Type u_7\ninst✝ : SeminormedAddCommGroup β\nι : Type u_8\ns : Finset ι\nμ : ι → Measure α\nT : ι → Set α → β\nC : ι → ℝ\nhT : ∀ (i : ι), DominatedFinMeasAdditive (μ i) (T i) (C i)\ni : ι\n⊢ DominatedFinMeasAdditive (∑ i ∈ {i}, μ i) (∑ i ∈ {i}, T ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 265,
"column": 19
} | {
"line": 265,
"column": 27
} | [
{
"pp": "case singleton\nα : Type u_1\nm : MeasurableSpace α\nβ : Type u_7\ninst✝ : SeminormedAddCommGroup β\nι : Type u_8\ns : Finset ι\nμ : ι → Measure α\nT : ι → Set α → β\nC : ι → ℝ\nhT : ∀ (i : ι), DominatedFinMeasAdditive (μ i) (T i) (C i)\ni : ι\n⊢ DominatedFinMeasAdditive (∑ i ∈ {i}, μ i) (∑ i ∈ {i}, T ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 265,
"column": 19
} | {
"line": 265,
"column": 27
} | [
{
"pp": "case singleton\nα : Type u_1\nm : MeasurableSpace α\nβ : Type u_7\ninst✝ : SeminormedAddCommGroup β\nι : Type u_8\ns : Finset ι\nμ : ι → Measure α\nT : ι → Set α → β\nC : ι → ℝ\nhT : ∀ (i : ι), DominatedFinMeasAdditive (μ i) (T i) (C i)\ni : ι\n⊢ DominatedFinMeasAdditive (∑ i ∈ {i}, μ i) (∑ i ∈ {i}, T ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.FiniteDimension | {
"line": 667,
"column": 59
} | {
"line": 672,
"column": 77
} | [
{
"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\nx : Eᵤ\nU : Set Eᵤ\nhU_nhds : U ∈ 𝓝 x\nhU_tb : Tota... | by
replace hU_nhds : x +ᵥ (-x) +ᵥ U ∈ 𝓝 x := by simpa
rw [vadd_mem_nhds_self] at hU_nhds
refine .of_totallyBounded_nhds_zero _ hU_nhds ?_
have : -x +ᵥ U = (· - x) '' U := by simp [← Set.image_vadd, neg_add_eq_sub]
exact this ▸ hU_tb.image (uniformContinuous_id.sub uniformContinuous_const) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 343,
"column": 70
} | {
"line": 345,
"column": 79
} | [
{
"pp": "α : Type u_1\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nm : MeasurableSpace α\nμ : Measure α\nf g : α → G\nhf : Integrable f μ\nhg : Integrable g μ\n⊢ edist (∫ (a : α), f a ∂μ) (∫ (a : α), g a ∂μ) ≤ ∫⁻ (a : α), edist (f a) (g a) ∂μ",
"usedConstants": [
"Eq.mpr",
... | by
rw [edist_dist]
exact ENNReal.ofReal_le_of_le_toReal (dist_integral_le_lintegral_edist hf hg) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 464,
"column": 12
} | {
"line": 464,
"column": 15
} | [
{
"pp": "case h\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : Integrable f μ\nf₁ : ↥(Lp ℝ 1 μ) := Integrable.toL1 f hf\na✝ : α\nh₁ : ↑↑(Lp.posPart f₁) a✝ = max (↑↑f₁ a✝) 0\nh₂ : ↑↑(Integrable.toL1 f hf) a✝ = f a✝\n⊢ ENNReal.ofReal (f a✝) = ‖max (↑↑f₁ a✝) 0‖ₑ",
"usedConstants": [
... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 475,
"column": 12
} | {
"line": 475,
"column": 15
} | [
{
"pp": "case h\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : Integrable f μ\nf₁ : ↥(Lp ℝ 1 μ) := Integrable.toL1 f hf\neq₁ : (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal = ‖Lp.posPart f₁‖\na✝ : α\nh₁ : ↑↑(Lp.negPart f₁) a✝ = -min (↑↑f₁ a✝) 0\nh₂ : ↑↑(Integrable.toL1 f hf) a✝ = f a✝\n⊢ EN... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 75
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : DecidablePred fun x ↦ x ≠ 0\nm : MeasurableSpace α\nf : α →ₛ F\nμ : Measure α\n⊢ integral μ f = ∑ x ∈ f.range with x ≠ 0, μ.real (⇑f ⁻¹' {x}) • x",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.we... | simp_rw [integral_def, setToSimpleFunc_eq_sum_filter, weightedSMul_apply] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 75
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : DecidablePred fun x ↦ x ≠ 0\nm : MeasurableSpace α\nf : α →ₛ F\nμ : Measure α\n⊢ integral μ f = ∑ x ∈ f.range with x ≠ 0, μ.real (⇑f ⁻¹' {x}) • x",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.we... | simp_rw [integral_def, setToSimpleFunc_eq_sum_filter, weightedSMul_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 75
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : DecidablePred fun x ↦ x ≠ 0\nm : MeasurableSpace α\nf : α →ₛ F\nμ : Measure α\n⊢ integral μ f = ∑ x ∈ f.range with x ≠ 0, μ.real (⇑f ⁻¹' {x}) • x",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.we... | simp_rw [integral_def, setToSimpleFunc_eq_sum_filter, weightedSMul_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 318,
"column": 56
} | {
"line": 318,
"column": 74
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedSpace ℝ E\nν : Measure α\nf : α →ₛ E\nhf : Integrable (⇑f) (μ + ν)\ns : Set α\nx✝ : MeasurableSet s\nhμνs : μ s + ν s ≠ ∞\n⊢ weightedSMul (μ + ν) s = weightedSMul μ s + weightedSMul ν s",
... | ENNReal.add_ne_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 352,
"column": 4
} | {
"line": 352,
"column": 35
} | [
{
"pp": "case pos\nα : 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... | obtain (hx | hx) := hx.eq_or_lt | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 362,
"column": 4
} | {
"line": 362,
"column": 12
} | [
{
"pp": "case neg\nα : 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... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 563,
"column": 4
} | {
"line": 563,
"column": 43
} | [
{
"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\nf : ↥(Lp E 1 μ)\n⊢ Continuous ⇑int... | exact (integralCLM (E := E)).continuous | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 626,
"column": 2
} | {
"line": 627,
"column": 50
} | [
{
"pp": "case refine_2\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ↥(Lp ℝ 1 μ)\n⊢ Continuous fun x ↦ ‖Lp.posPart x‖ - ‖Lp.negPart x‖",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Real",
"Real.instSub",
"Continuous.comp",
"PseudoMetricSpace.t... | · refine Continuous.sub (continuous_norm.comp Lp.continuous_posPart)
(continuous_norm.comp Lp.continuous_negPart) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 912,
"column": 6
} | {
"line": 912,
"column": 40
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nH : Type u_6\ninst✝ : NormedAddCommGroup H\nf : α → H\np : ℝ≥0∞\nhp1 : p ≠ 0\nhp2 : p ≠ ∞\nhf : MemLp f p μ\nA : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ ^ p.toReal) ∂μ = ∫⁻ (a : α), ‖f a‖ₑ ^ p.toReal ∂μ\n⊢ (∫⁻ (x : α), ‖f x‖ₑ ^ p.toReal ∂μ) ^ p.toReal⁻¹ = E... | integral_eq_lintegral_of_nonneg_ae | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1001,
"column": 48
} | {
"line": 1011,
"column": 79
} | [
{
"pp": "α : Type u_1\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nm : MeasurableSpace α\nμ : Measure α\nf : α → G\nc : ℝ≥0∞\n⊢ ∫ (x : α), f x ∂c • μ = c.toReal • ∫ (x : α), f x ∂μ",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"Nor... | by
-- First we consider the “degenerate” case `c = ∞`
rcases eq_or_ne c ∞ with (rfl | hc)
· rw [ENNReal.toReal_top, zero_smul, integral_eq_setToFun, setToFun_top_smul_measure]
-- Main case: `c ≠ ∞`
simp_rw [integral_eq_setToFun, ← setToFun_smul_left]
have hdfma : DominatedFinMeasAdditive μ (weightedSMul (c ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1021,
"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⊢ ∫ (y : β), f y ∂Mea... | rw [integral_undef hfi, integral_undef]
exact fun hfφ => hfi ((integrable_map_measure hfm.aestronglyMeasurable hφ.aemeasurable).2 hfφ) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1021,
"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⊢ ∫ (y : β), f y ∂Mea... | rw [integral_undef hfi, integral_undef]
exact fun hfφ => hfi ((integrable_map_measure hfm.aestronglyMeasurable hφ.aemeasurable).2 hfφ) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1136,
"column": 4
} | {
"line": 1136,
"column": 38
} | [
{
"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\n⊢ (∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ).toReal ≤\n (∫⁻ (a : α), ENNReal.ofR... | integral_eq_lintegral_of_nonneg_ae | Lean.Elab.Tactic.evalRewriteSeq | null |
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