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.Complement | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝³ : Ring R\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np : Submodule R M\nf : M →L[R] ↥p\nhf : ∀ (x : ↥p), f ↑x = x\n⊢ IsTopCompl p (↑f).ker",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Lemmas | {
"line": 99,
"column": 36
} | {
"line": 99,
"column": 78
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : StronglyMeasurable g'\nhg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x\nA : MeasurableSet {x | f x ≠ 0}\na : α\nha : ↑(f a) ≠ 0 →... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Lemmas | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 45
} | [
{
"pp": "case h\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : StronglyMeasurable g'\nhg' : (fun x ↦ ↑(f x) • g x) =ᶠ[ae μ] g'\nx : α\nhx : ↑(f x) • g x = g' x\nh'x : ¬f x = 0\n⊢ ↑(... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Real | {
"line": 395,
"column": 2
} | {
"line": 395,
"column": 43
} | [
{
"pp": "α : Type u_1\nι : Type u_3\nx✝ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\ns : Finset ι\nt : ι → Set α\nh : ∀ i ∈ s, MeasurableSet (t i)\nH : (↑s).PairwiseDisjoint t\n⊢ ∑ x ∈ s, (μ (t x)).toReal ≤ (μ univ).toReal",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Re... | rw [← ENNReal.toReal_sum (by finiteness)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.LpSpace.Indicator | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 84
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nhp : p ≠ ∞\nc : E\nε : ℝ≥0∞\nhε : ε ≠ 0\nh'p : p ≠ 0\nhp₀ : 0 < p\nhp₀' : 0 ≤ 1 / p.toReal\nhp₀'' : 0 < p.toReal\nthis : Tendsto (fun x ↦ ↑(‖c‖₊ * x ^ (1 / p.toReal))) (𝓝 0) (𝓝 ↑0)\nhε' : 0 < ε\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSpace.Indicator | {
"line": 158,
"column": 2
} | {
"line": 159,
"column": 9
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\ns : Set α\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nc : E\n⊢ ‖indicatorConstLp p hs hμs c‖ₑ ≤ ‖c‖ₑ * μ s ^ (1 / p.toReal)",
"usedConstants": [
"Eq.mpr",
"Real",
"DivInvMonoid.toIn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 13
} | [
{
"pp": "case intro\nα : Type u_1\nβ : Type u_2\nε : Type u_3\nmα : MeasurableSpace α\nf : α → ε\nμ : Measure α\ninst✝³ : TopologicalSpace ε\ninst✝² : ContinuousENorm ε\ninst✝¹ : PseudoMetrizableSpace ε\ninst✝ : Finite β\nt : β → Set α\nval✝ : Fintype β\n⊢ IntegrableOn f (⋃ i, t i) μ ↔ ∀ (i : β), IntegrableOn f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 13
} | [
{
"pp": "α : Type u_1\nE : Type u_5\nmα : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : MeasurableSingletonClass α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\ns : Set α\nhs : s.Finite\nf : α → E\n⊢ IntegrableOn f s μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 414,
"column": 18
} | {
"line": 414,
"column": 29
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\ns t : Set α\nμ : Measure α\nε' : Type u_7\ninst✝² : TopologicalSpace ε'\ninst✝¹ : ENormedAddMonoid ε'\ninst✝ : PseudoMetrizableSpace ε'\nf : α → ε'\nhf : IntegrableOn f s μ\nht : NullMeasurableSet t μ\nh't : ∀ᵐ (x : α) ∂μ, x ∈ t \\ s → f x = 0\nu : Set α := {x | x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 451,
"column": 2
} | {
"line": 451,
"column": 13
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\ns : Set α\nμ : Measure α\nε' : Type u_7\ninst✝² : TopologicalSpace ε'\ninst✝¹ : ENormedAddMonoid ε'\ninst✝ : PseudoMetrizableSpace ε'\nf : α → ε'\nhs : MeasurableSet s\nhf : IntegrableOn f (s ∩ support f) μ\n⊢ IntegrableOn f s μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 590,
"column": 4
} | {
"line": 590,
"column": 20
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_5\nmα : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\nμ : Measure α\nl : Filter α\n𝕜 : Type u_7\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 E\nf : α → E\nc : 𝕜\nhc : ¬c = 0\n⊢ IntegrableAtFilter (c • f) l μ ↔ c = 0 ∨ IntegrableAtFilter f l μ",
"usedCons... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 786,
"column": 2
} | {
"line": 786,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : OpensMeasurableSpace α\ninst✝² : CompactSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : PseudoMetrizableSpace β\nμ : Measure α\nf : α → β\nhf : Continuous[inst✝⁴, inst✝¹] f\n⊢ AEStronglyMeasurable f μ",
"usedCon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 36
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nE : ι → Type wE\nG : Type wG\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹ : SeminormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : MultilinearMap 𝕜 E G\nhf : Continuous[Pi.topologicalSpa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 80
} | [
{
"pp": "case neg\n𝕜 : 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\nhf₀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 13
} | [
{
"pp": "α : Type u_1\nε : Type u_5\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nf : α → ε\np : ℕ\nhf : MemLp f (↑p) μ\nhp : p ≠ 0\n⊢ Integrable (fun x ↦ ‖f x‖ₑ ^ p) μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf : α → β\np : ℕ\nhf : MemLp f (↑p) μ\nhp : p ≠ 0\n⊢ Integrable (fun x ↦ ‖f x‖ ^ p) μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 213,
"column": 48
} | {
"line": 213,
"column": 59
} | [
{
"pp": "α : Type u_1\nε : Type u_5\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\ninst✝ : IsFiniteMeasure μ\nf : α → ε\np : ℕ\nhf : MemLp f (↑p) μ\n⊢ Integrable (fun x ↦ ‖f x‖ₑ ^ p) μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 216,
"column": 48
} | {
"line": 216,
"column": 59
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\ninst✝ : IsFiniteMeasure μ\nf : α → β\np : ℕ\nhf : MemLp f (↑p) μ\n⊢ Integrable (fun x ↦ ‖f x‖ ^ p) μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 328,
"column": 4
} | {
"line": 328,
"column": 72
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε : Type u_8\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\nc : ℝ≥0∞\nh₁ : c ≠ 0\nh₂ : c ≠ ∞\nh : Integrable f (c • μ)\n⊢ Integrable f μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 334,
"column": 30
} | {
"line": 334,
"column": 41
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε : Type u_8\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\nc : ℝ≥0∞\nh₁ : c ≠ 0\nh₂ : c ≠ ∞\n⊢ c⁻¹ ≠ 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"ENNReal.inv_eq_zero._simp_1",
"id",
"Ne",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 334,
"column": 50
} | {
"line": 334,
"column": 61
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε : Type u_8\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\nc : ℝ≥0∞\nh₁ : c ≠ 0\nh₂ : c ≠ ∞\n⊢ c⁻¹ ≠ ∞",
"usedConstants": [
"Eq.mpr",
"congrArg",
"ENNReal.inv_eq_top._simp_1",
"id",
"Ne",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 395,
"column": 2
} | {
"line": 395,
"column": 31
} | [
{
"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 : ContinuousMultilinearMap 𝕜 E G\nm :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 449,
"column": 2
} | {
"line": 449,
"column": 39
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε' : Type u_8\ninst✝² : TopologicalSpace ε'\ninst✝¹ : ESeminormedAddCommMonoid ε'\ninst✝ : ContinuousAdd ε'\nι : Type u_9\ns : Finset ι\nf : ι → α → ε'\nhf : ∀ i ∈ s, Integrable (f i) μ\n⊢ Integrable (fun a ↦ ∑ i ∈ s, f i a) μ",
"usedConstants": [... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 400,
"column": 2
} | {
"line": 400,
"column": 37
} | [
{
"pp": "𝕜 : Type u\nG : Type wG\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : SeminormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nn : ℕ\nEi : Fin n → Type u_1\ninst✝¹ : (i : Fin n) → SeminormedAddCommGroup (Ei i)\ninst✝ : (i : Fin n) → NormedSpace 𝕜 (Ei i)\nf : ContinuousMultilinearMap 𝕜 Ei G\nm : (i : Fi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 541,
"column": 31
} | {
"line": 541,
"column": 64
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : α → β\nhf : Integrable f μ\nhg : Integrable g μ\n⊢ Integrable (f - g) μ",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"AddMonoid.toAddZeroClass",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 545,
"column": 43
} | {
"line": 545,
"column": 76
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : α → β\nhf : Integrable f μ\nhg : Integrable g μ\n⊢ Integrable (fun a ↦ f a - g a) μ",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"AddMonoid.toAddZeroC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 592,
"column": 6
} | {
"line": 592,
"column": 17
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedAddCommGroup β\nR : Type u_8\ninst✝² : NormedRing R\ninst✝¹ : Module R β\ninst✝ : IsBoundedSMul R β\nf : α → β\nhf : MemLp f 1 μ\ng : α → R\ng_aestronglyMeasurable : AEStronglyMeasurable g μ\ness_sup_g : essSup (fun x ↦ ‖g... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LocallyIntegrable | {
"line": 168,
"column": 51
} | {
"line": 168,
"column": 93
} | [
{
"pp": "X : Type u_1\nε : Type u_3\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nf : X → ε\nμ : Measure X\ns : Set X\ninst✝ : SecondCountableTopology X\nhf : LocallyIntegrableOn f s μ\nT : Set (Set X)\nT_count : T.Countable\nT_open : ∀ u ∈ T,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 607,
"column": 6
} | {
"line": 607,
"column": 17
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedAddCommGroup β\n𝕜 : Type u_8\ninst✝² : NormedRing 𝕜\ninst✝¹ : MulActionWithZero 𝕜 β\ninst✝ : IsBoundedSMul 𝕜 β\nf : α → 𝕜\nhf : MemLp f 1 μ\ng : α → β\ng_aestronglyMeasurable : AEStronglyMeasurable g μ\ness_sup_g : es... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LocallyIntegrable | {
"line": 170,
"column": 41
} | {
"line": 170,
"column": 69
} | [
{
"pp": "X : Type u_1\nε : Type u_3\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nf : X → ε\nμ : Measure X\ns : Set X\ninst✝ : SecondCountableTopology X\nhf : LocallyIntegrableOn f s μ\nT : Set (Set X)\nT_count : T.Countable\nT_open : ∀ u ∈ T,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 670,
"column": 4
} | {
"line": 671,
"column": 11
} | [
{
"pp": "case a\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_8\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousENorm E\nf : α → E\nhf : Integrable f μ\nε : ℝ≥0∞\nhε : 0 < ε\nhε' : ε ≠ ∞\n⊢ ε⁻¹ ^ ENNReal.toReal 1 < ∞",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 473,
"column": 42
} | {
"line": 473,
"column": 77
} | [
{
"pp": "case h1\n𝕜 : 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\nA : ∀ (f : ContinuousMultilinea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 672,
"column": 4
} | {
"line": 672,
"column": 59
} | [
{
"pp": "case a\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_8\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousENorm E\nf : α → E\nhf : Integrable f μ\nε : ℝ≥0∞\nhε : 0 < ε\nhε' : ε ≠ ∞\n⊢ eLpNorm f 1 μ ^ ENNReal.toReal 1 < ∞",
"usedConstants": [
"Eq.mpr",
"Real",
"Preor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 478,
"column": 4
} | {
"line": 478,
"column": 71
} | [
{
"pp": "case refine_2\n𝕜 : 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\nA : ∀ (f : ContinuousMult... | simp only [Seminorm.mem_ball_zero, mem_closedBall_zero_iff] at hf ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 772,
"column": 4
} | {
"line": 773,
"column": 11
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup β\ninst✝ : MeasurableSingletonClass α\nf : α → β\nhs : Summable fun x ↦ ‖f x‖\n⊢ (Function.support f).Countable",
"usedConstants": [
"id",
"SubtractionMonoid.toSubNegZeroMonoid",
"SubNegZeroMonoid.toNeg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 778,
"column": 4
} | {
"line": 778,
"column": 67
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup β\ninst✝ : MeasurableSingletonClass α\nf : α → β\nhs : Summable fun x ↦ ‖f x‖\nhs' : (Function.support f).Countable\nthis✝ : MeasurableSpace β := borel β\nthis : BorelSpace β\n⊢ {x | 0 x ≠ f x}.Countable",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 798,
"column": 6
} | {
"line": 798,
"column": 71
} | [
{
"pp": "case pos\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_8\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng : α → E\nH : AEStronglyMeasurable (fun x ↦ ↑(f x) • g x) μ\n⊢ AEMeasurable (fun a ↦ ‖g a‖ₑ) (μ.withDensity fun x ↦ ↑(f x))",
"usedCon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 535,
"column": 2
} | {
"line": 535,
"column": 38
} | [
{
"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 : ContinuousMultilinearMap 𝕜 E G\n⊢ I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 549,
"column": 33
} | {
"line": 549,
"column": 80
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nι' : Type v'\nE : ι → Type wE\nE' : ι' → Type wE'\ninst✝⁶ : Fintype ι'\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝³ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝² : Fintype ι\ninst✝¹ : (i' : ι') → SeminormedAddCommGroup (E' i')\ninst✝ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 598,
"column": 4
} | {
"line": 598,
"column": 15
} | [
{
"pp": "case a\n𝕜 : Type u\nι : Type v\nE : ι → Type wE\nG : Type wG\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝³ : SeminormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : Fintype ι\ninst✝ : IsEmpty ι\nx : G\n⊢ ‖x‖... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LocallyIntegrable | {
"line": 291,
"column": 2
} | {
"line": 291,
"column": 91
} | [
{
"pp": "X : Type u_1\nε : Type u_3\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nf : X → ε\nμ : Measure X\ns : Set X\ninst✝ : OpensMeasurableSpace X\nhf : LocallyIntegrable f (μ.restrict s)\nx : X\na✝ : x ∈ s\nt : Set X\nht_mem : t ∈ 𝓝 x\nht... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LocallyIntegrable | {
"line": 301,
"column": 4
} | {
"line": 305,
"column": 32
} | [
{
"pp": "case pos\nX : Type u_1\nε : Type u_3\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nf : X → ε\nμ : Measure X\ns : Set X\ninst✝ : OpensMeasurableSpace X\nhs : IsClosed[inst✝³] s\nhf : LocallyIntegrableOn f s μ\nx : X\nh : x ∈ s\n⊢ Integ... | obtain ⟨t, ht_nhds, ht_int⟩ := hf x h
obtain ⟨u, hu_o, hu_x, hu_sub⟩ := mem_nhdsWithin.mp ht_nhds
refine ⟨u, hu_o.mem_nhds hu_x, ?_⟩
rw [IntegrableOn, restrict_restrict hu_o.measurableSet]
exact ht_int.mono_set hu_sub | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LocallyIntegrable | {
"line": 301,
"column": 4
} | {
"line": 305,
"column": 32
} | [
{
"pp": "case pos\nX : Type u_1\nε : Type u_3\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nf : X → ε\nμ : Measure X\ns : Set X\ninst✝ : OpensMeasurableSpace X\nhs : IsClosed[inst✝³] s\nhf : LocallyIntegrableOn f s μ\nx : X\nh : x ∈ s\n⊢ Integ... | obtain ⟨t, ht_nhds, ht_int⟩ := hf x h
obtain ⟨u, hu_o, hu_x, hu_sub⟩ := mem_nhdsWithin.mp ht_nhds
refine ⟨u, hu_o.mem_nhds hu_x, ?_⟩
rw [IntegrableOn, restrict_restrict hu_o.measurableSet]
exact ht_int.mono_set hu_sub | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 743,
"column": 12
} | {
"line": 743,
"column": 32
} | [
{
"pp": "case refine_1\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nn : ℕ\nA : Type u_1\ninst✝¹ : SeminormedRing A\ninst✝ : NormedAlgebra 𝕜 A\nm : Fin n.succ → A\n⊢ ¬List.map m (List.finRange n.succ) = []",
"usedConstants": [
"Eq.mpr",
"congrArg",
"List.map",
"List.map_eq_nil_... | List.map_eq_nil_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 879,
"column": 6
} | {
"line": 879,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nε' : Type u_6\nε'' : Type u_7\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁹ : MeasurableSpace δ\ninst✝⁸ : NormedAddCommGroup β\ninst✝⁷ : NormedAddCommGroup γ\ninst✝⁶ : TopologicalSpace ε\ninst✝⁵ : ContinuousENorm ε\ninst✝⁴ : Topolo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 892,
"column": 6
} | {
"line": 892,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nε' : Type u_6\nε'' : Type u_7\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁹ : MeasurableSpace δ\ninst✝⁸ : NormedAddCommGroup β\ninst✝⁷ : NormedAddCommGroup γ\ninst✝⁶ : TopologicalSpace ε\ninst✝⁵ : ContinuousENorm ε\ninst✝⁴ : Topolo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LocallyIntegrable | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 34
} | [
{
"pp": "X : Type u_1\nε : Type u_3\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : TopologicalSpace X\ninst✝³ : TopologicalSpace ε\ninst✝² : ContinuousENorm ε\nf : X → ε\nμ : Measure X\ninst✝¹ : PseudoMetrizableSpace ε\ninst✝ : SecondCountableTopology X\nhf : LocallyIntegrable f μ\n⊢ AEStronglyMeasurable f μ",
"used... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LocallyIntegrable | {
"line": 349,
"column": 2
} | {
"line": 349,
"column": 31
} | [
{
"pp": "X : Type u_1\nε : Type u_3\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nf : X → ε\nμ : Measure X\ninst✝ : SecondCountableTopology X\nhf : LocallyIntegrable f μ\nu : ℕ → Set X\nu_open : ∀ (n : ℕ), IsOpen[inst✝³] (u n)\nu_union : univ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.FiniteDimension | {
"line": 162,
"column": 6
} | {
"line": 162,
"column": 58
} | [
{
"pp": "𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\nE : Type v\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : TopologicalSpace E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\nl : E →ₗ[𝕜] 𝕜\nhl : IsClosed[inst✝²] ↑l.ker\nH : ¬finrank 𝕜 ↥l.range = 0\nthis : finrank 𝕜 ↥l.range... | rw [← LinearMap.range_eq_top, Submodule.range_liftQ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.LocallyIntegrable | {
"line": 441,
"column": 2
} | {
"line": 441,
"column": 39
} | [
{
"pp": "X : Type u_1\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace X\nμ : Measure X\nε''' : Type u_9\ninst✝² : TopologicalSpace ε'''\ninst✝¹ : ESeminormedAddCommMonoid ε'''\ninst✝ : ContinuousAdd ε'''\nι : Type u_10\ns : Finset ι\nf : ι → X → ε'''\nhf : ∀ i ∈ s, LocallyIntegrable (f i) μ\n⊢ LocallyInt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 865,
"column": 6
} | {
"line": 865,
"column": 17
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nι' : Type v'\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nG : Type wG\nG' : Type wG'\ninst✝¹⁰ : Fintype ι'\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁶ : (i : ι) → SeminormedAd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 1124,
"column": 89
} | {
"line": 1126,
"column": 13
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_8\ninst✝ : RCLike 𝕜\nf : α → 𝕜\nhf : Integrable f μ\n⊢ Integrable (fun x ↦ RCLike.im (f x)) μ",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"AddMonoid.toAddSemigroup",
"Re... | by
rw [← memLp_one_iff_integrable] at hf ⊢
exact hf.im | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 1145,
"column": 2
} | {
"line": 1145,
"column": 72
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nH : Type u_8\ninst✝ : NormedAddCommGroup H\nm0 : MeasurableSpace α\nμ' : Measure α\nf : α → H\nhm : m ≤ m0\nhf_meas_ae : AEStronglyMeasurable f (μ'.trim hm)\nhf : HasFiniteIntegral f (μ'.trim hm)\n⊢ HasFiniteIntegral f μ'",
"usedConstants": [
"PseudoMetric... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 1201,
"column": 14
} | {
"line": 1201,
"column": 25
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_8\nH : Type u_9\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedAddCommGroup H\n𝕜 : Type u_10\n𝕜' : Type u_11\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NontriviallyNormedField 𝕜'\ninst✝⁵ : NormedSpace 𝕜' E\ninst✝⁴ : NormedSpace 𝕜 H... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LocallyIntegrable | {
"line": 541,
"column": 4
} | {
"line": 542,
"column": 59
} | [
{
"pp": "case mpr\nX : Type u_1\nε : Type u_3\ninst✝⁸ : MeasurableSpace X\ninst✝⁷ : TopologicalSpace X\ninst✝⁶ : TopologicalSpace ε\ninst✝⁵ : ContinuousENorm ε\nf : X → ε\nμ : Measure X\ninst✝⁴ : PseudoMetrizableSpace ε\na : X\ninst✝³ : LinearOrder X\ninst✝² : CompactIccSpace X\ninst✝¹ : NoMinOrder X\ninst✝ : O... | exact integrableOn_Iic_iff_integrableAtFilter_atBot.mpr
⟨hbot, hlocal.mono_set (Iic_subset_Iio.mpr hs'_mono)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.L1Space.AEEqFun | {
"line": 93,
"column": 7
} | {
"line": 93,
"column": 18
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedAddCommGroup β\n𝕜 : Type u_5\ninst✝² : NormedRing 𝕜\ninst✝¹ : Module 𝕜 β\ninst✝ : IsBoundedSMul 𝕜 β\nc : 𝕜\nf : α →ₘ[μ] β\n_f : α → β\nhfm : AEStronglyMeasurable _f μ\nhfi : (mk _f hfm).Integrable\n⊢ MeasureTheory.Int... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LocallyIntegrable | {
"line": 655,
"column": 18
} | {
"line": 655,
"column": 29
} | [
{
"pp": "X : Type u_1\nE : Type u_6\ninst✝⁸ : MeasurableSpace X\ninst✝⁷ : TopologicalSpace X\ninst✝⁶ : NormedAddCommGroup E\nμ : Measure X\ns : Set X\ninst✝⁵ : BorelSpace X\ninst✝⁴ : ConditionallyCompleteLinearOrder X\ninst✝³ : ConditionallyCompleteLinearOrder E\ninst✝² : OrderTopology X\ninst✝¹ : OrderTopology... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 81,
"column": 2
} | {
"line": 82,
"column": 9
} | [
{
"pp": "β : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nn : ℕ\n⊢ ‖(approxOn f hf s y₀ h₀ n) x - y₀‖ ≤ ‖f x - y₀‖ + ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 81,
"column": 2
} | {
"line": 82,
"column": 40
} | [
{
"pp": "β : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nn : ℕ\n⊢ ‖(approxOn f hf s y₀ h₀ n) x - y₀‖ ≤ ‖f x - y₀‖ + ... | simpa [enorm, edist_eq_enorm_sub, ← ENNReal.coe_add, norm_sub_rev]
using edist_approxOn_y0_le hf h₀ x n | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 81,
"column": 2
} | {
"line": 82,
"column": 40
} | [
{
"pp": "β : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nn : ℕ\n⊢ ‖(approxOn f hf s y₀ h₀ n) x - y₀‖ ≤ ‖f x - y₀‖ + ... | simpa [enorm, edist_eq_enorm_sub, ← ENNReal.coe_add, norm_sub_rev]
using edist_approxOn_y0_le hf h₀ x n | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 81,
"column": 2
} | {
"line": 82,
"column": 40
} | [
{
"pp": "β : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nn : ℕ\n⊢ ‖(approxOn f hf s y₀ h₀ n) x - y₀‖ ≤ ‖f x - y₀‖ + ... | simpa [enorm, edist_eq_enorm_sub, ← ENNReal.coe_add, norm_sub_rev]
using edist_approxOn_y0_le hf h₀ x n | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 87,
"column": 2
} | {
"line": 88,
"column": 9
} | [
{
"pp": "β : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\nh₀ : 0 ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nn : ℕ\n⊢ ‖(approxOn f hf s 0 h₀ n) x‖ ≤ ‖f x‖ + ‖f x‖",
"usedCon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 87,
"column": 2
} | {
"line": 88,
"column": 40
} | [
{
"pp": "β : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\nh₀ : 0 ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nn : ℕ\n⊢ ‖(approxOn f hf s 0 h₀ n) x‖ ≤ ‖f x‖ + ‖f x‖",
"usedCon... | simpa [enorm, edist_eq_enorm_sub, ← ENNReal.coe_add, norm_sub_rev]
using edist_approxOn_y0_le hf h₀ x n | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 87,
"column": 2
} | {
"line": 88,
"column": 40
} | [
{
"pp": "β : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\nh₀ : 0 ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nn : ℕ\n⊢ ‖(approxOn f hf s 0 h₀ n) x‖ ≤ ‖f x‖ + ‖f x‖",
"usedCon... | simpa [enorm, edist_eq_enorm_sub, ← ENNReal.coe_add, norm_sub_rev]
using edist_approxOn_y0_le hf h₀ x n | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 87,
"column": 2
} | {
"line": 88,
"column": 40
} | [
{
"pp": "β : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\nh₀ : 0 ∈ s\ninst✝ : SeparableSpace ↑s\nx : β\nn : ℕ\n⊢ ‖(approxOn f hf s 0 h₀ n) x‖ ≤ ‖f x‖ + ‖f x‖",
"usedCon... | simpa [enorm, edist_eq_enorm_sub, ← ENNReal.coe_add, norm_sub_rev]
using edist_approxOn_y0_le hf h₀ x n | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 53
} | [
{
"pp": "case pos\nβ : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\np : ℝ≥0∞\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nhp_ne_top : p ≠ ∞\nμ : Measure β\nhμ : ∀ᵐ (x :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 43
} | [
{
"pp": "β : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\np : ℝ≥0∞\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nhp_ne_top : p ≠ ∞\nμ : Measure β\nhμ : ∀ᵐ (x : β) ∂μ, f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 1175,
"column": 6
} | {
"line": 1175,
"column": 58
} | [
{
"pp": "case hbc.h₂.h1\n𝕜 : Type u\nι : Type v\nι' : Type v'\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nG : Type wG\nG' : Type wG'\ninst✝¹² : Fintype ι'\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁹ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁸ : (i :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 13
} | [
{
"pp": "case neg\nβ : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\np : ℝ≥0∞\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nhp_ne_top : p ≠ ∞\nμ : Measure β\nhμ : ∀ᵐ (x :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 1161,
"column": 4
} | {
"line": 1175,
"column": 92
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nι' : Type v'\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nG : Type wG\nG' : Type wG'\ninst✝¹² : Fintype ι'\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁹ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁸ : (i : ι) → Seminormed... | intro x m
simp only [MultilinearMap.iteratedFDerivComponent, MultilinearMap.domDomRestrictₗ,
MultilinearMap.coe_mk, MultilinearMap.domDomRestrict_apply, coe_coe]
apply (f.le_opNorm _).trans _
classical
rw [← prod_compl_mul_prod s.toFinset, mul_assoc]
gcongr
· apply le_of_eq
have : ∀ ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 1161,
"column": 4
} | {
"line": 1175,
"column": 92
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nι' : Type v'\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nG : Type wG\nG' : Type wG'\ninst✝¹² : Fintype ι'\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁹ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁸ : (i : ι) → Seminormed... | intro x m
simp only [MultilinearMap.iteratedFDerivComponent, MultilinearMap.domDomRestrictₗ,
MultilinearMap.coe_mk, MultilinearMap.domDomRestrict_apply, coe_coe]
apply (f.le_opNorm _).trans _
classical
rw [← prod_compl_mul_prod s.toFinset, mul_assoc]
gcongr
· apply le_of_eq
have : ∀ ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 212,
"column": 32
} | {
"line": 212,
"column": 43
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_7\ninst✝ : SeminormedAddCommGroup β\nT : Set α → β\nC : ℝ\nhT : DominatedFinMeasAdditive μ T C\ns : Set α\nhs : MeasurableSet s\nhμs : μ s < ∞\n⊢ ‖(-T) s‖ ≤ C * μ.real s",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 1225,
"column": 4
} | {
"line": 1225,
"column": 15
} | [
{
"pp": "case h\n𝕜 : Type u\nι : Type v\nE₁ : ι → Type wE₁\nG : Type wG\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : (i : ι) → SeminormedAddCommGroup (E₁ i)\ninst✝³ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝² : SeminormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : Fintype ι\nf : ContinuousMultilinearMap... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 267,
"column": 4
} | {
"line": 267,
"column": 45
} | [
{
"pp": "case cons\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\ns : Finset ι\nhis : i ∉ s\nhs' : s.Nonempty\nih : DominatedFin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 163,
"column": 4
} | {
"line": 163,
"column": 15
} | [
{
"pp": "case refine_2\nβ : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\np : ℝ≥0∞\ninst✝¹ : BorelSpace E\nf : β → E\nhp_ne_top : p ≠ ∞\nμ : Measure β\nfmeas : Measurable f\ninst✝ : SeparableSpace ↑(Set.range f ∪ {0})\nhf : eLpNorm f p μ < ∞\n⊢ eL... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 177,
"column": 2
} | {
"line": 177,
"column": 56
} | [
{
"pp": "β : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\np : ℝ≥0∞\ninst✝¹ : BorelSpace E\nf : β → E\nhp : Fact (1 ≤ p)\nhp_ne_top : p ≠ ∞\nμ : Measure β\nfmeas : Measurable f\ninst✝ : SeparableSpace ↑(Set.range f ∪ {0})\nhf : MemLp f p μ\n⊢ Tend... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 49
} | [
{
"pp": "case h\nβ : Type u_2\ninst✝¹ : MeasurableSpace β\np : ℝ≥0∞\nE : Type u_7\ninst✝ : NormedAddCommGroup E\nf : β → E\nμ : Measure β\nhf : MemLp f p μ\nhp_ne_top : p ≠ ∞\nε : ℝ≥0∞\nhε : ε ≠ 0\nthis✝¹ : MeasurableSpace E := borel E\nthis✝ : BorelSpace E\nf' : β → E := AEStronglyMeasurable.mk f ⋯\ng : β →ₛ E... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 46
} | [
{
"pp": "β : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nμ : Measure β\nhμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure[PseudoMetricSpace... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 217,
"column": 10
} | {
"line": 217,
"column": 54
} | [
{
"pp": "β : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nμ : Measure β\nhμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure[PseudoMetricSpace... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 231,
"column": 4
} | {
"line": 231,
"column": 15
} | [
{
"pp": "case hi\nβ : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nμ : Measure β\ninst✝ : SeparableSpace ↑(Set.range f ∪ {0})\nfmeas : Measurable f\nhf : Integrable f μ\n⊢ HasFiniteIntegral (fun x ↦ f x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 292,
"column": 6
} | {
"line": 292,
"column": 31
} | [
{
"pp": "case pos.inr.inl\nα : Type u_1\nE : Type u_4\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\nhp_pos : p ≠ 0\nhp_ne_top : p ≠ ∞\nf : α →ₛ E\nhf : MemLp (⇑f) p μ\ny : E\nhy_ne : y ≠ 0\nhp_pos_real : 0 < p.toReal\nhyf : y ∈ f.range\nhf_eLpNorm : ‖y‖ₑ ^ p.toReal = 0\n⊢ y... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.FiniteDimension | {
"line": 561,
"column": 6
} | {
"line": 561,
"column": 41
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁴ : NontriviallyNormedField 𝕜\ninst✝¹³ : CompleteSpace 𝕜\ninst✝¹² : AddCommGroup E\ninst✝¹¹ : TopologicalSpace E\ninst✝¹⁰ : IsTopologicalAddGroup E\ninst✝⁹ : Module 𝕜 E\ninst✝⁸ : ContinuousSMul 𝕜 E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : TopologicalSpace ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.FiniteDimension | {
"line": 637,
"column": 14
} | {
"line": 637,
"column": 25
} | [
{
"pp": "case zero\n𝕜 : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : CompleteSpace 𝕜\nEᵤ : Type u_6\ninst✝⁵ : AddCommGroup Eᵤ\ninst✝⁴ : Module 𝕜 Eᵤ\ninst✝³ : UniformSpace Eᵤ\ninst✝² : T2Space Eᵤ\ninst✝¹ : IsUniformAddGroup Eᵤ\ninst✝ : ContinuousSMul 𝕜 Eᵤ\nU : Set Eᵤ\nhU_nhds : U ∈ 𝓝 0\nhU_tb : T... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 333,
"column": 2
} | {
"line": 333,
"column": 47
} | [
{
"pp": "α : Type u_1\nE : Type u_4\nF : Type u_5\ninst✝² : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\nμ : Measure α\nf : α →ₛ E\ng : α →ₛ F\n⊢ Integrable (⇑f) μ → Integrable (⇑g) μ → Integrable (⇑(f.pair g)) μ",
"usedConstants": [
"Eq.mpr",
"Prod.seminormedA... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 360,
"column": 2
} | {
"line": 361,
"column": 50
} | [
{
"pp": "α : Type u_1\nE : Type u_4\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\nhp_pos : p ≠ 0\nhp_ne_top : p ≠ ∞\nc : E\nhc : c ≠ 0\ns : Set α\nhs : MeasurableSet s\nhcs : MemLp (⇑(piecewise s hs (const α c) (const α 0))) p μ\nthis : support ⇑(const α c) = Set.univ\n⊢ μ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 424,
"column": 6
} | {
"line": 424,
"column": 32
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nE : Type u_4\nF : Type u_5\n𝕜 : Type u_6\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\np : ℝ≥0∞\nμ : Measure α\ninst✝² : NormedRing 𝕜\ninst✝¹ : Module 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\nk : 𝕜\nf : ↥(simpleFunc E p ... | rcases f with ⟨f, ⟨s, hs⟩⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Algebra.Module.FiniteDimension | {
"line": 657,
"column": 6
} | {
"line": 657,
"column": 53
} | [
{
"pp": "𝕜 : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : CompleteSpace 𝕜\nEᵤ : Type u_6\ninst✝⁵ : AddCommGroup Eᵤ\ninst✝⁴ : Module 𝕜 Eᵤ\ninst✝³ : UniformSpace Eᵤ\ninst✝² : T2Space Eᵤ\ninst✝¹ : IsUniformAddGroup Eᵤ\ninst✝ : ContinuousSMul 𝕜 Eᵤ\nU : Set Eᵤ\nhU_nhds : U ∈ 𝓝 0\nhU_tb : TotallyBound... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.FiniteDimension | {
"line": 719,
"column": 2
} | {
"line": 719,
"column": 17
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : CompleteSpace 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : ContinuousSMul 𝕜 E\np q : Submodule 𝕜 E\nh : IsCompl p q\nhp : IsClosed[inst✝⁴] ↑p\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 588,
"column": 2
} | {
"line": 588,
"column": 33
} | [
{
"pp": "α : Type u_1\nE : Type u_4\ninst✝² : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E\np : ℝ≥0∞\nμ : Measure α\ninst✝ : Fact (1 ≤ p)\nf : ↥(simpleFunc E p μ)\n⊢ ‖f‖ = (eLpNorm (⇑(toSimpleFunc f)) p μ).toReal",
"usedConstants": [
"Norm.norm",
"Real",
"MeasureTheory.Lp.simpleFunc.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 687,
"column": 2
} | {
"line": 687,
"column": 43
} | [
{
"pp": "α : Type u_1\nE : Type u_4\ninst✝² : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E\np : ℝ≥0∞\nμ : Measure α\ninst✝ : Fact (1 ≤ p)\nhp_ne_top : p ≠ ∞\n⊢ Dense ↑(simpleFunc E p μ)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Extend | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 15
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_3\nEₗ : Type u_4\nF : Type u_5\nFₗ : Type u_6\ninst✝⁷ : DivisionRing 𝕜\ninst✝⁶ : DivisionRing 𝕜₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : SeminormedAddCommGroup Eₗ\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜₂ F\ninst✝ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Extend | {
"line": 157,
"column": 31
} | {
"line": 162,
"column": 17
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_3\nEₗ : Type u_4\nF : Type u_5\nFₗ : Type u_6\ninst✝⁷ : DivisionRing 𝕜\ninst✝⁶ : DivisionRing 𝕜₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : SeminormedAddCommGroup Eₗ\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜₂ F\ninst✝ ... | by
obtain ⟨C, h⟩ := h
intro x hx
specialize h x
rw [hx] at h
simpa using h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 724,
"column": 4
} | {
"line": 724,
"column": 15
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝² : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nG : Type u_7\ninst✝¹ : NormedAddCommGroup G\ninst✝ : PartialOrder G\ng : α →ₛ G\nhp : AEEqFun.mk ⇑g ⋯ ∈ Lp G p μ\nhf : 0 ≤ᶠ[ae μ] ⇑g\nx : α\n⊢ 0 x ≤ (SimpleFunc.map ({x | 0 ≤ x}.piecewise id 0) g) x",
"usedConstants":... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Extend | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 23
} | [
{
"pp": "case h\n𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_3\nEₗ : Type u_4\nF : Type u_5\nFₗ : Type u_6\ninst✝⁷ : DivisionRing 𝕜\ninst✝⁶ : DivisionRing 𝕜₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : SeminormedAddCommGroup Eₗ\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜₂ F... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Extend | {
"line": 197,
"column": 8
} | {
"line": 197,
"column": 19
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_3\nEₗ : Type u_4\nF : Type u_5\ninst✝¹⁰ : NormedDivisionRing 𝕜\ninst✝⁹ : NormedDivisionRing 𝕜₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : SeminormedAddCommGroup Eₗ\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 726,
"column": 4
} | {
"line": 726,
"column": 15
} | [
{
"pp": "case h\nα : Type u_1\ninst✝² : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nG : Type u_7\ninst✝¹ : NormedAddCommGroup G\ninst✝ : PartialOrder G\ng : α →ₛ G\nhp : AEEqFun.mk ⇑g ⋯ ∈ Lp G p μ\nhf : 0 ≤ᶠ[ae μ] ⇑g\nx : α\nhx : 0 ≤ g x\n⊢ g x = (SimpleFunc.map ({x | 0 ≤ x}.piecewise id 0) g) x",
"usedCons... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Extend | {
"line": 206,
"column": 4
} | {
"line": 206,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_3\nEₗ : Type u_4\nF : Type u_5\ninst✝¹⁰ : NormedDivisionRing 𝕜\ninst✝⁹ : NormedDivisionRing 𝕜₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : SeminormedAddCommGroup Eₗ\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Extend | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 15
} | [
{
"pp": "case h_dense\n𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_3\nEₗ : Type u_4\nF : Type u_5\ninst✝¹⁰ : NormedDivisionRing 𝕜\ninst✝⁹ : NormedDivisionRing 𝕜₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : SeminormedAddCommGroup Eₗ\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : Module �... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Extend | {
"line": 324,
"column": 61
} | {
"line": 324,
"column": 72
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_3\nEₗ : Type u_4\nF : Type u_5\nFₗ : Type u_6\ninst✝¹³ : NormedField 𝕜\ninst✝¹² : NormedField 𝕜₂\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : Module 𝕜 E\ninst✝⁹ : AddCommGroup F\ninst✝⁸ : Module 𝕜₂ F\ninst✝⁷ : NormedAddCommGroup Eₗ\ninst✝⁶ : NormedSpace 𝕜 Eₗ\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 772,
"column": 6
} | {
"line": 772,
"column": 17
} | [
{
"pp": "case hμ\nα : Type u_1\ninst✝² : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nG : Type u_7\ninst✝¹ : NormedAddCommGroup G\ninst✝ : PartialOrder G\nhp : Fact (1 ≤ p)\nhp_ne_top : p ≠ ∞\ng : { g // 0 ≤ g }\nthis✝¹ : MeasurableSpace G := borel G\nthis✝ : BorelSpace G\nhg_memLp : MemLp (↑↑↑g) p μ\nzero_mem :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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