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.MeasureTheory.Function.LpSeminorm.Count | {
"line": 42,
"column": 4
} | {
"line": 42,
"column": 69
} | [
{
"pp": "case refine_2\nα : Type u_1\nε : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nf : α → ε\np : ℝ≥0∞\ninst✝ : Finite α\nh : ∀ (i : α), ‖f i‖ₑ < ∞\nthis : Fintype α\n⊢ eLpNorm (fun x ↦ Finset.univ.sup fun x ↦ ‖f x‖ₑ) p count < ∞",
"usedConstants": [
... | exact (memLp_const_enorm <| by simp [h, LT.lt.ne]).eLpNorm_lt_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.Count | {
"line": 42,
"column": 4
} | {
"line": 42,
"column": 69
} | [
{
"pp": "case refine_2\nα : Type u_1\nε : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nf : α → ε\np : ℝ≥0∞\ninst✝ : Finite α\nh : ∀ (i : α), ‖f i‖ₑ < ∞\nthis : Fintype α\n⊢ eLpNorm (fun x ↦ Finset.univ.sup fun x ↦ ‖f x‖ₑ) p count < ∞",
"usedConstants": [
... | exact (memLp_const_enorm <| by simp [h, LT.lt.ne]).eLpNorm_lt_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 258,
"column": 8
} | {
"line": 258,
"column": 47
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁴ : IsProbabilityMeasure μ'\nmE : MeasurableSpace E\nZ : Ω' → E\nl : Filter ι\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SecondCountableTopology E\ninst✝¹ : BorelSpace E\ninst✝ : l.IsCountablyGenerated\nX : ι → ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 284,
"column": 4
} | {
"line": 284,
"column": 43
} | [
{
"pp": "case refine_2\nι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ'' : Type u_4\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁸ : IsProbabilityMeasure μ'\nm'' : MeasurableSpace Ω''\nμ'' : Measure Ω''\ninst✝⁷ : IsProbabilityMeasure μ''\nmE : MeasurableSpace E\nl : Filter ι\ninst✝⁶ : SeminormedAddCommGroup E\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Intersectivity | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 15
} | [
{
"pp": "α : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Intersectivity | {
"line": 107,
"column": 8
} | {
"line": 107,
"column": 19
} | [
{
"pp": "case refine_3.refine_1.refine_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Intersectivity | {
"line": 110,
"column": 8
} | {
"line": 111,
"column": 15
} | [
{
"pp": "case refine_3.refine_1.refine_2\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Piecewise | {
"line": 31,
"column": 17
} | {
"line": 31,
"column": 49
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : MeasurableSpace α\ns : ι → Set α\nf : ι → α → β\ninst✝¹ : MeasurableSpace β\ninst✝ : Countable ι\nhs : IndexedPartition s\nhm : ∀ (i : ι), MeasurableSet (s i)\nhf : ∀ (i : ι), Measurable (f i)\nt : Set β\nht : MeasurableSet t\n⊢ MeasurableSet (hs.piece... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Piecewise | {
"line": 69,
"column": 6
} | {
"line": 69,
"column": 42
} | [
{
"pp": "case pos\nι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : MeasurableSpace α\ns : ι → Set α\nf : ι → α → β\ninst✝¹ : Countable ι\nhs : IndexedPartition s\nhm : ∀ (i : ι), MeasurableSet (s i)\ninst✝ : TopologicalSpace β\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nFi : Infinite ι\ne : ℕ ≃ ι\ng : (n : ℕ)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Piecewise | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 46
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : MeasurableSpace α\ns : ι → Set α\nf : ι → α → β\ninst✝¹ : Countable ι\nhs : IndexedPartition s\nhm : ∀ (i : ι), MeasurableSet (s i)\ninst✝ : TopologicalSpace β\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nFi : Infinite ι\ne : ℕ ≃ ι\ng : (n : ℕ) → ι → Fin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 33,
"column": 2
} | {
"line": 33,
"column": 88
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhf : MemLp f p μ\n⊢ ENNReal.ofReal (lpNorm f p μ) = eLpNorm f p μ",
"usedConstants": [
"Eq.mpr",
"Real",
"ENNReal.ofReal",
"congrArg",
"PseudoMetricSpac... | rw [← toReal_eLpNorm hf.aestronglyMeasurable, ENNReal.ofReal_toReal hf.eLpNorm_ne_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 33,
"column": 2
} | {
"line": 33,
"column": 88
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhf : MemLp f p μ\n⊢ ENNReal.ofReal (lpNorm f p μ) = eLpNorm f p μ",
"usedConstants": [
"Eq.mpr",
"Real",
"ENNReal.ofReal",
"congrArg",
"PseudoMetricSpac... | rw [← toReal_eLpNorm hf.aestronglyMeasurable, ENNReal.ofReal_toReal hf.eLpNorm_ne_top] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 33,
"column": 2
} | {
"line": 33,
"column": 88
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhf : MemLp f p μ\n⊢ ENNReal.ofReal (lpNorm f p μ) = eLpNorm f p μ",
"usedConstants": [
"Eq.mpr",
"Real",
"ENNReal.ofReal",
"congrArg",
"PseudoMetricSpac... | rw [← toReal_eLpNorm hf.aestronglyMeasurable, ENNReal.ofReal_toReal hf.eLpNorm_ne_top] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 47,
"column": 2
} | {
"line": 53,
"column": 84
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhp₀ : p ≠ 0\nhp : p ≠ ∞\nhf : AEStronglyMeasurable f μ\n⊢ lpNorm f p μ = (∫ (x : α), ‖f x‖ ^ p.toReal ∂μ) ^ p.toReal⁻¹",
"usedConstants": [
"MeasureTheory.ae",
"Norm.norm... | rw [← toReal_eLpNorm hf, eLpNorm_eq_lintegral_rpow_enorm_toReal hp₀ hp, ← ENNReal.toReal_rpow,
← integral_toReal]
· simp [← ENNReal.toReal_rpow]
· simp_rw [← ofReal_norm]
borelize E
fun_prop
· exact .of_forall fun x ↦ ENNReal.rpow_lt_top_of_nonneg (by positivity) (by simp) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 47,
"column": 2
} | {
"line": 53,
"column": 84
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhp₀ : p ≠ 0\nhp : p ≠ ∞\nhf : AEStronglyMeasurable f μ\n⊢ lpNorm f p μ = (∫ (x : α), ‖f x‖ ^ p.toReal ∂μ) ^ p.toReal⁻¹",
"usedConstants": [
"MeasureTheory.ae",
"Norm.norm... | rw [← toReal_eLpNorm hf, eLpNorm_eq_lintegral_rpow_enorm_toReal hp₀ hp, ← ENNReal.toReal_rpow,
← integral_toReal]
· simp [← ENNReal.toReal_rpow]
· simp_rw [← ofReal_norm]
borelize E
fun_prop
· exact .of_forall fun x ↦ ENNReal.rpow_lt_top_of_nonneg (by positivity) (by simp) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 67,
"column": 2
} | {
"line": 68,
"column": 22
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhf : MemLp f ∞ μ\n⊢ ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ lpNorm f ∞ μ",
"usedConstants": [
"MeasureTheory.ae",
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"MeasureTheor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 37
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhf : MemLp f ∞ μ\n⊢ ∀ᶠ (x : ℝ) in map (fun i ↦ ‖f i‖ₑ.toReal) (ae μ), (fun x1 x2 ↦ x1 ≤ x2) x (lpNorm f ∞ μ)",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"Real.instLE",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 95,
"column": 58
} | {
"line": 95,
"column": 69
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\nhf : ¬AEStronglyMeasurable f μ\nh : AEStronglyMeasurable (-f) μ\n⊢ AEStronglyMeasurable f μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 20
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : Module 𝕜 E\ninst✝ : NormSMulClass 𝕜 E\nc : 𝕜\nf : α → E\nμ : Measure α\nhf : ¬AEStronglyMeasurable f μ\nhc : c ≠ 0\nh : AEStronglyMeasurable (c • f) μ\n⊢ AEStr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 38
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nn : ℕ\nf : α → E\nμ : Measure α\n⊢ lpNorm (n • f) p μ = n • lpNorm f p μ",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"in... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 33
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\n𝕜 : Type u_3\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace ℝ 𝕜\nn : ℕ\nf : α → 𝕜\np : ℝ≥0∞\nμ : Measure α\n⊢ lpNorm (↑n * f) p μ = ↑n * lpNorm f p μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 29
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\n𝕜 : Type u_3\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace ℝ 𝕜\nf : α → 𝕜\nn : ℕ\np : ℝ≥0∞\nμ : Measure α\n⊢ lpNorm (f * ↑n) p μ = lpNorm f p μ * ↑n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 182,
"column": 47
} | {
"line": 182,
"column": 58
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhf : MemLp f p μ\nhp : 1 ≤ p\nhg : ¬MemLp g p μ\nhfg : MemLp (f + g) p μ\n⊢ MemLp g p μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 24
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm (f + g) p μ ≤ lpNorm f p μ + lpNorm g p μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 30
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhf : MemLp f p μ\nhp : 1 ≤ p\n⊢ lpNorm (f - g) p μ ≤ lpNorm f p μ + lpNorm g p μ",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"MeasureTheory.Measur... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 13
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm f p μ ≤ lpNorm g p μ + lpNorm (f - g) p μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 30
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm f p μ ≤ lpNorm g p μ + lpNorm (g - f) p μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 13
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm f p μ ≤ lpNorm (f + g) p μ + lpNorm g p μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 13
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g h : α → E\nhf : MemLp f p μ\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm (f - h) p μ ≤ lpNorm (f - g) p μ + lpNorm (g - h) p μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 13
} | [
{
"pp": "case h₁.hb\nα : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nι : Type u_3\ns : Finset ι\nf : ι → α → E\nhf : ∀ i ∈ s, MemLp (f i) p μ\nhp : 1 ≤ p\n⊢ ∑ i ∈ s, eLpNorm (f i) p μ ≠ ∞",
"usedConstants": [
"Eq.mpr",
"not_exists._simp_1... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 240,
"column": 6
} | {
"line": 240,
"column": 22
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nc : ℝ≥0\nhc : c ≠ 0\nhf : ¬AEStronglyMeasurable f μ\nh : AEStronglyMeasurable f (c • μ)\n⊢ AEStronglyMeasurable f μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 254,
"column": 4
} | {
"line": 254,
"column": 20
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nhp : p ≠ ∞\nf : α → E\nc : ℝ≥0\nhf : ¬AEStronglyMeasurable f μ\nhp₀ : p ≠ 0\nhc : c ≠ 0\nh : AEStronglyMeasurable f (c • μ)\n⊢ AEStronglyMeasurable f μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 262,
"column": 6
} | {
"line": 262,
"column": 95
} | [
{
"pp": "case pos.hf\nα : Type u_1\nm : MeasurableSpace α\nK : Type u_3\ninst✝ : RCLike K\nf : α → K\np : ℝ≥0∞\nμ : Measure α\nhf : AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable ((starRingEnd (α → K)) f) μ",
"usedConstants": [
"AEMeasurable.aestronglyMeasurable",
"NormedCommRing.toSeminormedC... | exact (continuous_star.measurable.comp_aemeasurable hf.aemeasurable).aestronglyMeasurable | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 262,
"column": 6
} | {
"line": 262,
"column": 95
} | [
{
"pp": "case pos.hf\nα : Type u_1\nm : MeasurableSpace α\nK : Type u_3\ninst✝ : RCLike K\nf : α → K\np : ℝ≥0∞\nμ : Measure α\nhf : AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable ((starRingEnd (α → K)) f) μ",
"usedConstants": [
"AEMeasurable.aestronglyMeasurable",
"NormedCommRing.toSeminormedC... | exact (continuous_star.measurable.comp_aemeasurable hf.aemeasurable).aestronglyMeasurable | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 262,
"column": 6
} | {
"line": 262,
"column": 95
} | [
{
"pp": "case pos.hf\nα : Type u_1\nm : MeasurableSpace α\nK : Type u_3\ninst✝ : RCLike K\nf : α → K\np : ℝ≥0∞\nμ : Measure α\nhf : AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable ((starRingEnd (α → K)) f) μ",
"usedConstants": [
"AEMeasurable.aestronglyMeasurable",
"NormedCommRing.toSeminormedC... | exact (continuous_star.measurable.comp_aemeasurable hf.aemeasurable).aestronglyMeasurable | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 264,
"column": 4
} | {
"line": 265,
"column": 11
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nK : Type u_3\ninst✝ : RCLike K\nf : α → K\np : ℝ≥0∞\nμ : Measure α\nhf : ¬AEStronglyMeasurable f μ\nh : AEStronglyMeasurable ((starRingEnd (α → K)) f) μ\n⊢ AEStronglyMeasurable f μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FoelnerFilter | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 43
} | [
{
"pp": "case h\nG : Type u_1\nX : Type u_2\ninst✝² : MeasurableSpace X\nμ : Measure X\ninst✝¹ : Group G\ninst✝ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\nhfoel : IsFoelner G μ (↑u) F\ns : Set X\ni : ι\nhi : μ (F i) ≠ 0\nhi' : μ (F i) ≠ ∞\n⊢ μ (s ∩ F i) / μ (F i) ∈ Icc 0 1",
"usedConst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 68,
"column": 9
} | {
"line": 68,
"column": 41
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : NormedAddCommGroup β\nx✝ : MeasurableSpace α\nf : ι → α → β\np : ℝ≥0∞\nμ : Measure α\n⊢ μ ∅ ≠ ∞",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"id",
"MeasureTheory.measure_empty",
"Ne"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FoelnerFilter | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 27
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝³ : MeasurableSpace X\nμ : Measure X\ninst✝² : Group G\ninst✝¹ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\ninst✝ : SMulInvariantMeasure G X μ\nhfoel : IsFoelner G μ (↑u) F\ng h : G\n⊢ Tendsto (fun i ↦ μ ((g • F i) ∆ (h • F i)) / μ (F i)) (↑u) (𝓝 0)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FoelnerFilter | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 69
} | [
{
"pp": "case h\nG : Type u_1\nX : Type u_2\ninst✝³ : MeasurableSpace X\nμ : Measure X\ninst✝² : Group G\ninst✝¹ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\ninst✝ : SMulInvariantMeasure G X μ\nhfoel : IsFoelner G μ (↑u) F\ng✝ h✝ : G\ns : Set X\ng h : G\ni : ι\nhi : μ (F i) ≠ 0\n⊢ μ (g • s ∩... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FoelnerFilter | {
"line": 190,
"column": 2
} | {
"line": 190,
"column": 13
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝³ : MeasurableSpace X\nμ : Measure X\ninst✝² : Group G\ninst✝¹ : MulAction G X\nι : Type u_3\nu : Ultrafilter ι\nF : ι → Set X\ninst✝ : SMulInvariantMeasure G X μ\nhfoel : IsFoelner G μ (↑u) F\ng : G\ns : Set X\n⊢ mean μ u F (g • s) = mean μ u F s",
"usedConstants":... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 67
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\ninst✝ : Finite ι\nhp_top : p ≠ ∞\nf : ι → α → β\nhf : ∀ (i : ι), MemLp (f i) p μ\nε : ℝ≥0\nhε : 0 < ε\nn : ℕ\nhn : Nonempty (ι ≃ Fin n)\ng : Fin n → α → β := f ∘ ⇑hn.some.symm\nhg : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.GeometryOfNumbers | {
"line": 97,
"column": 76
} | {
"line": 97,
"column": 92
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : MeasurableSpace E\nμ : Measure E\nF s : Set E\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\ninst✝³ : Nontrivial E\ninst✝² : μ.IsAddHaarMeasure\nL : AddSubgroup E\ninst✝¹ : Countable ↥L\ninst✝ : DiscreteTopology ↥L... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleTransform | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 25
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nz w : ℂ\nf : ℂ → E\nthis : ∀ (c : ℂ) (R : ℝ), Periodic (circleMap c R) (2 * π)\n⊢ Periodic (circleTransformDeriv R z w f) (2 * π)",
"usedConstants": [
"Real",
"Real.pi",
"HMul.hMul",
"Nat.instAtLeas... | simp_rw [Periodic] at * | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Integral.CircleTransform | {
"line": 109,
"column": 6
} | {
"line": 109,
"column": 32
} | [
{
"pp": "case hg.hg.hf\nR r : ℝ\nhr : r < R\nz : ℂ\n⊢ ContinuousOn (fun x ↦ ((circleMap z R x.2 - x.1) ^ 2)⁻¹) (closedBall z r ×ˢ univ)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleTransform | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 27
} | [
{
"pp": "R r : ℝ\nhr : r < R\nhr' : 0 ≤ r\nz : ℂ\ncts : ContinuousOn ((fun x ↦ ‖x‖) ∘ circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ)\ncomp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]])\nnone : (closedBall z r ×ˢ [[0, 2 * π]]).Nonempty\nthis :\n ∃ x ∈ closedBall z r ×ˢ [[0, 2 * π]],\n IsMaxOn (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleTransform | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 83
} | [
{
"pp": "R : ℝ\nhR : 0 < R\nz x : ℂ\nf : ℂ → ℂ\nhx : x ∈ ball z R\nhf : ContinuousOn f (sphere z R)\nr : ℝ\nhr : r < R\nhrx : x ∈ ball z r\nε' : ℝ\nhε' : ε' > 0\nH : ball x ε' ⊆ ball z r\na : ℂ\nb : ℝ\nha : (a, b).1 ∈ closedBall z r\nhb : (a, b).2 ∈ [[0, 2 * π]]\nhab :\n ∀ (y : ↑(closedBall z r ×ˢ [[0, 2 * π]]... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Gamma | {
"line": 64,
"column": 60
} | {
"line": 67,
"column": 64
} | [
{
"pp": "p : ℝ\nhp : 0 < p\n⊢ ∫ (x : ℝ) in Ioi 0, rexp (-x ^ p) = Gamma (1 / p + 1)",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"InnerProductSpace.toNormedSpace",
"MulOne.toOne",
"Real.instPow",
"Real.partialOrder",
"Real",
"Set.Ioi",
... | by
convert! (integral_rpow_mul_exp_neg_rpow hp neg_one_lt_zero) using 1
· simp_rw [rpow_zero, one_mul]
· rw [zero_add, Gamma_add_one (one_div_ne_zero (ne_of_gt hp))] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Indicator | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 67
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nA : Set α\nι : Type u_2\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nAs : ι → Set α\nμ : Measure α\nA_mble : MeasurableSet A\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nB : Set α\nB_mble : MeasurableSet B\nB_finmeas : μ B ≠ ∞\nAs_le_B : ∀ᶠ (i :... | exact fun i ↦ Measurable.indicator measurable_const (As_mble i) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Indicator | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 67
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nA : Set α\nι : Type u_2\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nAs : ι → Set α\nμ : Measure α\nA_mble : MeasurableSet A\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nB : Set α\nB_mble : MeasurableSet B\nB_finmeas : μ B ≠ ∞\nAs_le_B : ∀ᶠ (i :... | exact fun i ↦ Measurable.indicator measurable_const (As_mble i) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Indicator | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 67
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nA : Set α\nι : Type u_2\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nAs : ι → Set α\nμ : Measure α\nA_mble : MeasurableSet A\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nB : Set α\nB_mble : MeasurableSet B\nB_finmeas : μ B ≠ ∞\nAs_le_B : ∀ᶠ (i :... | exact fun i ↦ Measurable.indicator measurable_const (As_mble i) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Indicator | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 79
} | [
{
"pp": "case refine_4\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nA : Set α\nι : Type u_2\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nAs : ι → Set α\nμ : Measure α\nA_mble : MeasurableSet A\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nB : Set α\nB_mble : MeasurableSet B\nB_finmeas : μ B ≠ ∞\nAs_le_B : ∀ᶠ (i :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 45
} | [
{
"pp": "case pos\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nγ✝ : Type u_4\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : Zero β\nγ : Type u_5\ninst✝¹ : TopologicalSpace γ\ninst✝ : Zero γ\ng : C(β, γ)\nf : α →C_c β\nhg : g 0 = 0\n⊢ HasCompactSupport (g.comp ↑f).toFun",
"usedConstants": ... | · exact f.hasCompactSupport'.comp_left hg | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral | {
"line": 64,
"column": 8
} | {
"line": 64,
"column": 69
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nB : ℝ → ℝ\nhab : a ≤ b\nhfc : ContinuousOn f (Icc a b)\nhfd : DifferentiableOn ℝ f (Ioo a b)\nhfB : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ B t\nhBi : IntervalIntegrable B volume a b\nthis :\n ∀ {E : Type u_1} [i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 281,
"column": 38
} | {
"line": 281,
"column": 79
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nx : α\ninst✝¹ : AddGroup β\ninst✝ : IsTopologicalAddGroup β\nf✝ g f : α →C_c β\n⊢ HasCompactSupport (-⇑f.toContinuousMap)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 294,
"column": 17
} | {
"line": 294,
"column": 45
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nx : α\ninst✝¹ : AddGroup β\ninst✝ : IsTopologicalAddGroup β\nf✝ g✝ f g : α →C_c β\n⊢ HasCompactSupport (⇑f.toContinuousMap - ⇑g.toContinuousMap)",
"usedConstants": [
"AddGroup.toS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 39
} | [
{
"pp": "case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nB : ℝ → ℝ\nhab : a ≤ b\nhfc : ContinuousOn f (Icc a b)\nhfd : DifferentiableOn ℝ f (Ioo a b)\nhfB : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ B t\nhBi : IntervalIntegrable B volume a b\nthis :\n ∀ {E : Ty... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 682,
"column": 4
} | {
"line": 682,
"column": 34
} | [
{
"pp": "case refine_2.h.a\nα : Type u_2\ninst✝ : TopologicalSpace α\nf₁ f₂ : α →C_c ℝ≥0\nh : f₁ ≤ f₂\nx : α\n⊢ ↑((f₁ + { toContinuousMap := f₂.toContinuousMap - f₁.toContinuousMap, hasCompactSupport' := ⋯ }) x) = ↑(f₂ x)",
"usedConstants": [
"NNReal.instTopologicalSpace",
"Eq.mpr",
"NonAs... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 31
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\nh : CurveIntegrable ω γ\n⊢ CurveIntegrable ω γ.symm",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 697,
"column": 2
} | {
"line": 697,
"column": 13
} | [
{
"pp": "case h.a\nα : Type u_2\ninst✝ : TopologicalSpace α\nf : α →C_c ℝ\nhf : 0 ≤ f\nx : α\n⊢ ↑((-f).nnrealPart x) = ↑(0 x)",
"usedConstants": [
"NNReal.instTopologicalSpace",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"Real.instZero",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 211,
"column": 14
} | {
"line": 211,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\nh : CurveIntegrable ω γ.symm\n⊢ CurveIntegrable ω γ",
"usedConstants": []
}
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 720,
"column": 2
} | {
"line": 720,
"column": 13
} | [
{
"pp": "α : Type u_2\ninst✝ : TopologicalSpace α\nf g : α →C_c ℝ\nx : α\n⊢ (f + g).nnrealPart x ≤ (f.nnrealPart + g.nnrealPart) x",
"usedConstants": [
"NNReal.instTopologicalSpace",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 261,
"column": 2
} | {
"line": 261,
"column": 35
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b c : E\nω : E → E →L[𝕜] F\nγab : Path a b\nh : CurveIntegrable ω γab\nγbc : Path b c\n⊢ IntervalIntegrable (fun t ↦ 2 • c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 267,
"column": 2
} | {
"line": 267,
"column": 35
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b c : E\nω : E → E →L[𝕜] F\nγbc : Path b c\nγab : Path a b\nh : CurveIntegrable ω γbc\n⊢ IntervalIntegrable (fun t ↦ 2 • c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CompactlySupported | {
"line": 852,
"column": 4
} | {
"line": 852,
"column": 51
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : Group α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : R1Space β\ninst✝² : Group β\ninst✝¹ : ContinuousMul β\ninst✝ : NormedAddCommGroup γ\nφ : α →* β\nhφ : Topology.IsClosedEmbedding ⇑φ\nf : β →C_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 359,
"column": 2
} | {
"line": 359,
"column": 31
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : E\nω₁ ω₂ : E → E →L[𝕜] F\nγ : Path a b\nh₁ : CurveIntegrable ω₁ γ\nh₂ : CurveIntegrable ω₂ γ\n⊢ CurveIntegrable (ω₁ + ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 396,
"column": 2
} | {
"line": 396,
"column": 31
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\nh : CurveIntegrable ω γ\n⊢ CurveIntegrable (-ω) γ",
"usedConstants": [
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 400,
"column": 14
} | {
"line": 400,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\nh : CurveIntegrable (-ω) γ\n⊢ CurveIntegrable ω γ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 85
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\na b : E\nC : ℝ\nhfc : ContinuousOn f (segment ℝ a b)\nhfd : ∀ t ∈ Ioo 0 1, LineDifferentiableAt ℝ f ((lineMap a b) t) (b - a)\nhf' : ∀ᵐ (t : ℝ), t ∈ Io... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.MeanValue | {
"line": 81,
"column": 10
} | {
"line": 81,
"column": 21
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\ns : Set α\nf g : α → ℝ\nμ : Measure α\nhs_conn : IsConnected s\nhs_meas : MeasurableSet s\nhf : ContinuousOn f s\nhg : IntegrableOn g s μ\nhfg : IntegrableOn (fun x ↦ f x * g x) s μ\nhg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x\nρ : α → E... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral | {
"line": 145,
"column": 40
} | {
"line": 145,
"column": 79
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\na b : E\nC : ℝ\nhfc : ContinuousOn f (segment ℝ a b)\nhfd : ∀ t ∈ Ioo 0 1, LineDifferentiableAt ℝ f ((lineMap a b) t) (b - a)\nhf' : ∀ᵐ (t : ℝ), t ∈ Io... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 465,
"column": 2
} | {
"line": 465,
"column": 31
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\n𝕝 : Type u_4\ninst✝² : RCLike 𝕝\ninst✝¹ : NormedSpace 𝕝 F\ninst✝ : SMulCommCla... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 473,
"column": 4
} | {
"line": 473,
"column": 20
} | [
{
"pp": "case inr\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\nγ : Path a b\n𝕝 : Type u_4\ninst✝² : RCLike 𝕝\ninst✝¹ : NormedSpace 𝕝 F\ninst✝ : S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.MeanValue | {
"line": 84,
"column": 39
} | {
"line": 84,
"column": 50
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\ns : Set α\nf g : α → ℝ\nμ : Measure α\nhs_conn : IsConnected s\nhs_meas : MeasurableSet s\nhf : ContinuousOn f s\nhg : IntegrableOn g s μ\nhfg : IntegrableOn (fun x ↦ f x * g x) s μ\nhg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x\nρ : α → E... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 531,
"column": 4
} | {
"line": 531,
"column": 30
} | [
{
"pp": "case h\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\na : E\ns : Set E\nω : E → E →L[𝕜] F\nhs : Convex ℝ s\nhω : ∀ᶠ (x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.MeanValue | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 38
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\ns : Set α\nf g : α → ℝ\nμ : Measure α\nhs_conn : IsConnected s\nhs_meas : MeasurableSet s\nhf : ContinuousOn f s\nhg : IntegrableOn g s μ\nhfg : IntegrableOn (fun x ↦ f x * g x) s μ\nhg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x\nρ : α → E... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.MeanValue | {
"line": 62,
"column": 37
} | {
"line": 62,
"column": 53
} | [
{
"pp": "a✝ b✝ : ℝ\nf g : ℝ → ℝ\nμ : Measure ℝ\na b : ℝ\nhf : ContinuousOn f [[a, b]]\nhg : IntervalIntegrable g μ a b\nhg0 : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), 0 ≤ g x\nh : ¬a = b\nhab : a < b\ns : Set ℝ := Ι a b\nhs : s = Ioc a b\nhs' : s ⊆ [[a, b]]\n⊢ IsConnected s",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.MeanValue | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 60
} | [
{
"pp": "a✝ b✝ : ℝ\nf g : ℝ → ℝ\nμ : Measure ℝ\na b : ℝ\nhf : ContinuousOn f [[a, b]]\nhg : IntervalIntegrable g μ a b\nhg0 : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), 0 ≤ g x\nh✝ : ¬a = b\nhab : a < b\ns : Set ℝ := Ι a b\nhs : s = Ioc a b\nhs' : s ⊆ [[a, b]]\nhs_conn : IsConnected s\nhfg : IntegrableOn (fun x ↦ f x * g ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.LebesgueNormedSpace | {
"line": 34,
"column": 6
} | {
"line": 34,
"column": 26
} | [
{
"pp": "case mp.ht\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : ∀ᵐ (x ... | filter_upwards [hg'] | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Integral.LebesgueNormedSpace | {
"line": 36,
"column": 36
} | {
"line": 36,
"column": 78
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : ∀ᵐ (x : α) ∂μ, ↑(f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 19
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\na : E\nr : ℝ\nhr : 0 ≤ r\nhdf : ∀ᶠ (x : E) in 𝓝 a, DifferentiableAt ℝ f x\nhderiv : fderiv ℝ f =O[𝓝 a] fun x ↦ ‖x - a‖ ^ r\nhf₀ : f a = 0\n⊢ f =O[𝓝 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.LebesgueNormedSpace | {
"line": 45,
"column": 4
} | {
"line": 45,
"column": 24
} | [
{
"pp": "case mpr\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : (fun x ↦... | filter_upwards [hg'] | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Integral.LebesgueNormedSpace | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 45
} | [
{
"pp": "case h\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : SecondCountableTopology E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : Measurable g'\nhg' : (fun x ↦ ↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Union | {
"line": 31,
"column": 16
} | {
"line": 31,
"column": 50
} | [
{
"pp": "X : Type u_1\ninst✝ : LinearOrder X\na : ℕ → X\nN : ℕ\nih : Ioc (a 0) (a N) ⊆ ⋃ i ∈ Finset.range N, Ioc (a i) (a (i + 1))\n⊢ Ioc (a 0) (a N) ∪ Ioc (a N) (a (N + 1)) ⊆ ⋃ i ∈ Finset.range (N + 1), Ioc (a i) (a (i + 1))",
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"Finset.mem_range._sim... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Union | {
"line": 41,
"column": 16
} | {
"line": 41,
"column": 50
} | [
{
"pp": "X : Type u_1\ninst✝ : LinearOrder X\na : ℕ → X\nN : ℕ\nih : Ico (a 0) (a N) ⊆ ⋃ i ∈ Finset.range N, Ico (a i) (a (i + 1))\n⊢ Ico (a 0) (a N) ∪ Ico (a N) (a (N + 1)) ⊆ ⋃ i ∈ Finset.range (N + 1), Ico (a i) (a (i + 1))",
"usedConstants": [
"Eq.mpr",
"Finset.mem_range._simp_1",
"and_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 21
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na✝ b✝ c d : E\nγ₁ : Path a✝ b✝\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countab... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 112,
"column": 6
} | {
"line": 112,
"column": 32
} | [
{
"pp": "case inl\nf : ℝ → ℝ\nN : ℕ\na h : ℝ\nN_nonzero : 0 < N\nh_f_int : IntervalIntegrable f volume a (a + ↑N * h)\nk✝ : ℕ\nhk✝ : k✝ < N\nh_neg : h ≤ 0\nk : ℕ\nhk : ↑k ≤ ↑N\n⊢ a + ↑k * h ∈ [[a, a + ↑N * h]]",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"add_le_add_if... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic | {
"line": 187,
"column": 4
} | {
"line": 188,
"column": 11
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\ns₀ s₁ t : Set X\ns₀_compact : IsCompact s₀\ns₁_compact : IsCompact s₁\nt_compact : IsCompact t\ndisj : Disjoint s₀ s₁\nhst : s₀ ∪ s₁ ⊆ t\nso : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ\nhso : so = fu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 39
} | [
{
"pp": "f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\ng : ℝ → ℝ := fun t ↦ trapezoidal_error f 1 a t\ndg : ℝ → ℝ := fun t ↦ 1 /... | rw [iteratedDerivWithin_eq_iterate] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 38
} | [
{
"pp": "f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\ng : ℝ → ℝ := fun t ↦ trapezoidal_error f 1 a t\ndg : ℝ → ℝ := fun t ↦ 1 /... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.NNReal | {
"line": 77,
"column": 2
} | {
"line": 81,
"column": 5
} | [
{
"pp": "X : Type u_1\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : T2Space X\ninst✝⁴ : LocallyCompactSpace X\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\nμ ν : Measure X\ninst✝¹ : μ.Regular\ninst✝ : ν.Regular\nhμν : ∀ (f : X →C_c ℝ≥0), ∫ (x : X), ↑(f x) ∂μ = ∫ (x : X), ↑(f x) ∂ν\n⊢ μ = ν",
"usedConstants":... | apply Measure.ext_of_integral_eq_on_compactlySupported
intro f
repeat rw [integral_eq_integral_pos_part_sub_integral_neg_part f.integrable]
erw [hμν f.nnrealPart, hμν (-f).nnrealPart]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.NNReal | {
"line": 77,
"column": 2
} | {
"line": 81,
"column": 5
} | [
{
"pp": "X : Type u_1\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : T2Space X\ninst✝⁴ : LocallyCompactSpace X\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\nμ ν : Measure X\ninst✝¹ : μ.Regular\ninst✝ : ν.Regular\nhμν : ∀ (f : X →C_c ℝ≥0), ∫ (x : X), ↑(f x) ∂μ = ∫ (x : X), ↑(f x) ∂ν\n⊢ μ = ν",
"usedConstants":... | apply Measure.ext_of_integral_eq_on_compactlySupported
intro f
repeat rw [integral_eq_integral_pos_part_sub_integral_neg_part f.integrable]
erw [hμν f.nnrealPart, hμν (-f).nnrealPart]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.NNReal | {
"line": 89,
"column": 9
} | {
"line": 89,
"column": 20
} | [
{
"pp": "X : Type u_1\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : T2Space X\ninst✝⁴ : LocallyCompactSpace X\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\nμ ν : Measure X\ninst✝¹ : μ.Regular\ninst✝ : ν.Regular\nhμν : integralLinearMap μ = integralLinearMap ν\nf : X →C_c ℝ≥0\n⊢ ∫ (x : X), ↑(f x) ∂μ = ∫ (x : X), ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.TorusIntegral | {
"line": 126,
"column": 23
} | {
"line": 126,
"column": 43
} | [
{
"pp": "n : ℕ\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : (Fin n → ℂ) → E\nc : Fin n → ℂ\n⊢ IntegrableOn (fun θ ↦ f (torusMap c 0 θ)) (Icc 0 fun x ↦ 2 * π) volume",
"usedConstants": [
"Eq.mpr",
"Real",
"Pi.preorder",
"Real.pi",
"HMul.hMul",
"Real.instZero",
"c... | torusMap_zero_radius | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.TorusIntegral | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 58
} | [
{
"pp": "n : ℕ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : (Fin n → ℂ) → E\nc : Fin n → ℂ\nR : Fin n → ℝ\nhf : TorusIntegrable f c R\nhg : TorusIntegrable g c R\n⊢ (∯ (x : Fin n → ℂ) in T(c, R), f x + g x) = (∯ (x : Fin n → ℂ) in T(c, R), f x) + ∯ (x : Fin n → ℂ) in T(c, R), g x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.TorusIntegral | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 56
} | [
{
"pp": "n : ℕ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : (Fin n → ℂ) → E\nc : Fin n → ℂ\nR : Fin n → ℝ\nhf : TorusIntegrable f c R\nhg : TorusIntegrable g c R\n⊢ (∯ (x : Fin n → ℂ) in T(c, R), f x - g x) = (∯ (x : Fin n → ℂ) in T(c, R), f x) - ∯ (x : Fin n → ℂ) in T(c, R), g x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Card | {
"line": 142,
"column": 2
} | {
"line": 152,
"column": 45
} | [
{
"pp": "α : Type u\ns : Set (Set α)\nt : Set α\nht : t ∈ {t | GenerateMeasurable s t}\n⊢ t ∈ generateMeasurableRec s (ω_ 1)",
"usedConstants": [
"Preorder.toLT",
"Order.succ",
"Ordinal.partialOrder",
"MeasurableSpace.generateMeasurableRec",
"congrArg",
"MeasurableSpace.s... | induction ht with
| basic u hu => exact self_subset_generateMeasurableRec s _ hu
| empty => exact empty_mem_generateMeasurableRec s _
| compl u _ IH =>
rw [generateMeasurableRec_omega_one, mem_iUnion₂] at IH
obtain ⟨i, hi, hi'⟩ := IH
exact generateMeasurableRec_mono _ ((isSuccLimit_omega 1).succ_lt hi... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.MeasureTheory.Integral.TorusIntegral | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 13
} | [
{
"pp": "n : ℕ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : (Fin (n + 1) → ℂ) → E\nc : Fin (n + 1) → ℂ\nR : Fin (n + 1) → ℝ\nhf : TorusIntegrable f c R\n⊢ (∯ (x : Fin (n + 1) → ℂ) in T(c, R), f x) =\n ∮ (x : ℂ) in C(c 0, R 0), ∯ (y : Fin n → ℂ) in T(c ∘ Fin.succ, R ∘ Fin.succ), ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Card | {
"line": 217,
"column": 6
} | {
"line": 217,
"column": 30
} | [
{
"pp": "α : Type u\ns : Set (Set α)\nhs : #↑s ≤ 𝔠\n⊢ max (#↑s) 2 ^ ℵ₀ ≤ 𝔠",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Cardinal.instPowCardinal",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
"Nat.instAtLeastTwoHAddOfNat",
"Preorder.toLE... | ← continuum_power_aleph0 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 34
} | [
{
"pp": "X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : T2Space X\ninst✝² : MeasurableSpace X\ninst✝¹ : BorelSpace X\nΛ : (X →C_c ℝ) →ₚ[ℝ] ℝ\ninst✝ : LocallyCompactSpace X\nf : X →C_c ℝ\nhf : ∀ (x : X), 0 ≤ f x ∧ f x ≤ 1\nV : Set X\nhV : tsupport ⇑f ⊆ V\nthis :\n (rieszContent (toNNRealLinear Λ)).measure ... | refine (Λ.mono ?_).trans hg.2.le | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
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