module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Probability.Distributions.Poisson.Basic | {
"line": 146,
"column": 58
} | {
"line": 161,
"column": 13
} | {
"line": 163,
"column": 0
} | [
{
"pp": "r : ℝ≥0\nt : ℝ\n⊢ charFun Po(ℝ, r) t = cexp (↑↑r * (cexp (↑t * I) - 1))",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Real.inner_apply",
"Mathlib.Tactic.Ring.Common.neg_zero",
... | [] | by
rw [charFun_apply, integral_map .of_discrete (by fun_prop), integral_poissonMeasure r]
simp_rw [Real.inner_apply]
calc ∑' a, (rexp (-r) * r ^ a / a ! : ℝ) * cexp ((a * t : ℝ) * I)
_ = ∑' a, (rexp (-r)) * ((r * cexp (t * I)) ^ a / a !) := by
congr with a
push_cast
rw [mul_pow, ← Complex.exp_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.ProbabilityMassFunction.Monad | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 19
} | {
"line": 263,
"column": 20
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\nf : (a : α) → a ∈ p.support → PMF β\ng : (b : β) → b ∈ (p.bindOnSupport f).support → PMF γ\na : γ\na' : α\nb : β\nh : ¬p a' = 0\nh_1 : ∀ (i : α), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\nH : ¬(f a' h) b = 0\n⊢ (f a' h) b = 0",
... | [
"case neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\nf : (a : α) → a ∈ p.support → PMF β\ng : (b : β) → b ∈ (p.bindOnSupport f).support → PMF γ\na : γ\na' : α\nb : β\nh : ¬p a' = 0\nh_1 : ∀ (i : α), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\nH : ¬(f a' h) b = 0\n⊢ (f a' h) b = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Poisson.Basic | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 31
} | {
"line": 241,
"column": 32
} | [
{
"pp": "r : ℝ≥0\nn : ℕ\n⊢ ENNReal.ofReal (poissonPMFReal r n) = (poissonPMF r) n",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"ENNReal.ofReal",
"PMF",
"ProbabilityTheory.poissonPMF",
"PMF.instFunLike",
"id",
"Nat",
"ENNReal",
"ProbabilityT... | [
"r : ℝ≥0\nn : ℕ\n⊢ ENNReal.ofReal (poissonPMFReal r n) = ⟨fun n ↦ ENNReal.ofReal (poissonPMFReal r n), ⋯⟩ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.TwoValued | {
"line": 46,
"column": 12
} | {
"line": 46,
"column": 70
} | {
"line": 46,
"column": 71
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nhXmeas : AEMeasurable X μ\nhX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1\n⊢ ∀ᵐ (ω : Ω) ∂μ, 1 - X ω = 0 ∨ 1 - X ω = 1",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"AddGroup.toSubtractionMonoid... | [
"Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nhXmeas : AEMeasurable X μ\nhX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1\n⊢ ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.TwoValued | {
"line": 98,
"column": 2
} | {
"line": 99,
"column": 12
} | {
"line": 99,
"column": 13
} | [
{
"pp": "case inr\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsZeroOrProbabilityMeasure μ\nhXmeas : AEMeasurable X μ\nhX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1\nhμ : IsProbabilityMeasure μ\n⊢ Var[X; μ] = μ.real {ω | X ω = 0} * μ.real {ω | X ω = 1}",
"ppTerm": "?inr",
"assigned":... | [
"case inr\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsZeroOrProbabilityMeasure μ\nhXmeas : AEMeasurable X μ\nhX : ∀ᵐ (ω : Ω) ∂μ, X ω = 0 ∨ X ω = 1\nhμ : IsProbabilityMeasure μ\n⊢ Var[X; μ] = μ.real {ω | X ω = 0} * μ.real {ω | X ω = 1}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Uniform | {
"line": 111,
"column": 21
} | {
"line": 111,
"column": 47
} | {
"line": 111,
"column": 48
} | [
{
"pp": "E : Type u_1\ninst✝ : MeasurableSpace E\nμ : Measure E\nΩ : Type u_2\nx✝ : MeasurableSpace Ω\nℙ : Measure Ω\nX : Ω → E\ns : Set E\nhns : μ s ≠ 0\nhnt : μ s ≠ ∞\nhu : IsUniform X s ℙ μ\nt : Set E := toMeasurable μ s\n⊢ μ[|s] = (μ t)⁻¹ • μ.restrict (toMeasurable μ s)",
"ppTerm": "?m.109",
"assign... | [
"E : Type u_1\ninst✝ : MeasurableSpace E\nμ : Measure E\nΩ : Type u_2\nx✝ : MeasurableSpace Ω\nℙ : Measure Ω\nX : Ω → E\ns : Set E\nhns : μ s ≠ 0\nhnt : μ s ≠ ∞\nhu : IsUniform X s ℙ μ\nt : Set E := toMeasurable μ s\n⊢ μ[|s] = (μ t)⁻¹ • μ.restrict s"
] | restrict_toMeasurable hnt, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Uniform | {
"line": 221,
"column": 6
} | {
"line": 221,
"column": 66
} | {
"line": 222,
"column": 8
} | [
{
"pp": "α : Type u_1\ns : Finset α\nhs : s.Nonempty\n⊢ ↑(#s) ≠ 0",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"PMF.uniformOfFinset._simp_2",
"Eq.mpr",
"congrArg",
"Finset",
"ENNReal.instCharZero",
"AddMonoid.toAddZeroClass",
"AddZeroClass.toAdd... | [
"α : Type u_1\ns : Finset α\nhs : s.Nonempty\n⊢ ¬s = ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Uniform | {
"line": 267,
"column": 35
} | {
"line": 267,
"column": 46
} | {
"line": 267,
"column": 47
} | [
{
"pp": "α : Type u_1\ns : Finset α\nhs : s.Nonempty\nt : Set α\nx : α\nhx : x ∈ {x ∈ s | x ∈ t}\n⊢ x ∈ s ∧ x ∈ t",
"ppTerm": "?m.188",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ns : Finset α\nhs : s.Nonempty\nt : Set α\nx : α\nhx : x ∈ {x ∈ s | x ∈ t}\n⊢ x ∈ s ∧ x ∈ t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Distributions.Uniform | {
"line": 359,
"column": 2
} | {
"line": 360,
"column": 43
} | {
"line": 360,
"column": 44
} | [
{
"pp": "α : Type u_1\ns : Multiset α\nhs : s ≠ 0\na : α\nha : a ∉ s\n⊢ (ofMultiset s hs) a = 0",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"instHDiv",
"congrArg",
"PMF.ofMultiset",
"PMF",
"ENNReal.instCharZero",
"AddMo... | [
"α : Type u_1\ns : Multiset α\nhs : s ≠ 0\na : α\nha : a ∉ s\n⊢ a ∉ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Condexp | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 34
} | {
"line": 90,
"column": 35
} | [
{
"pp": "case inr\nΩ : Type u_1\nF : Type u_2\nm mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh : Nonempty Ω\n⊢ IsMarkovKernel (condExpKernel μ m)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"Eq.mpr",
... | [
"case inr\nΩ : Type u_1\nF : Type u_2\nm mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh : Nonempty Ω\n⊢ IsMarkovKernel ((condDistrib id id μ).comap id ⋯)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.ZeroOne | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 51
} | {
"line": 57,
"column": 52
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμα : Measure α\nh : ∀ᵐ (a : α) ∂μα, IsFiniteMeasure (κ a)\nt : Set Ω\nh_indep : IndepSet t t κ μα\na : α\nh_0_1_top : (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ∞\nh' : IsFiniteMeasure (κ a)\n⊢ (κ a) t = 0 ∨ (κ a) t... | [
"α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμα : Measure α\nh : ∀ᵐ (a : α) ∂μα, IsFiniteMeasure (κ a)\nt : Set Ω\nh_indep : IndepSet t t κ μα\na : α\nh_0_1_top : (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ∞\nh' : IsFiniteMeasure (κ a)\n⊢ (κ a) t = 0 ∨ (κ a) t = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.ZeroOne | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 13
} | {
"line": 78,
"column": 14
} | [
{
"pp": "Ω : Type u_2\nm m0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nhm : Indep m m μ\nt : Set Ω\nht : MeasurableSet t\n⊢ μ t = 0 ∨ μ t = 1",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"Ω : Type u_2\nm m0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nhm : Indep m m μ\nt : Set Ω\nht : MeasurableSet t\n⊢ μ t = 0 ∨ μ t = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Invariance | {
"line": 71,
"column": 50
} | {
"line": 71,
"column": 77
} | {
"line": 71,
"column": 78
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nκ : Kernel α α\ninst✝ : IsMarkovKernel κ\nπ : Measure α\nh_rev : κ.IsReversible π\ns : Set α\nhs : MeasurableSet s\n⊢ ∫⁻ (x : α), (κ x) s ∂π = ∫⁻ (x : α) in s, (κ x) Set.univ ∂π",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"MeasureThe... | [
"α : Type u_1\nmα : MeasurableSpace α\nκ : Kernel α α\ninst✝ : IsMarkovKernel κ\nπ : Measure α\nh_rev : κ.IsReversible π\ns : Set α\nhs : MeasurableSet s\n⊢ ∫⁻ (x : α), (κ x) s ∂π = π s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Irreducible | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 15
} | {
"line": 72,
"column": 16
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nc : ℝ≥0∞\nφ : Measure α\nκ : Kernel α α\nhκ : IsIrreducible φ κ\ns : Set α\nhs : MeasurableSet s\nhsp : (c • φ) s > 0\n⊢ ∀ (a : α), ∃ n, ((κ ^ n) a) s > 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nc : ℝ≥0∞\nφ : Measure α\nκ : Kernel α α\nhκ : IsIrreducible φ κ\ns : Set α\nhs : MeasurableSet s\nhsp : (c • φ) s > 0\n⊢ ∀ (a : α), ∃ n, 0 < ((κ ^ n) a) s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Irreducible | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 15
} | {
"line": 78,
"column": 16
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nφ₁ φ₂ : Measure α\nhφ : φ₁ ≤ φ₂\nκ : Kernel α α\nhκ : IsIrreducible φ₂ κ\ns : Set α\nhs : MeasurableSet s\nhsp : φ₁ s > 0\n⊢ ∀ (a : α), ∃ n, ((κ ^ n) a) s > 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ProbabilityTh... | [
"α : Type u_1\nmα : MeasurableSpace α\nφ₁ φ₂ : Measure α\nhφ : φ₁ ≤ φ₂\nκ : Kernel α α\nhκ : IsIrreducible φ₂ κ\ns : Set α\nhs : MeasurableSet s\nhsp : φ₁ s > 0\n⊢ ∀ (a : α), ∃ n, 0 < ((κ ^ n) a) s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 95,
"column": 4
} | {
"line": 96,
"column": 48
} | {
"line": 97,
"column": 4
} | [
{
"pp": "X : SFinKer\nf₁ : X.carrier → PUnit.{?u.96 + 1} × X.carrier := fun x ↦ (PUnit.unit, x)\nhf₁ : Measurable f₁\nhf₂ : Measurable Prod.snd\n⊢ { carrier := { carrier := PUnit.{u + 1}, str := PUnit.instMeasurableSpace }.carrier × X.carrier,\n str := Prod.instMeasurableSpace } ≅\n X",
"ppTerm": "?... | [
"case refine_1\nX : SFinKer\nf₁ : X.carrier → PUnit.{u + 1} × X.carrier := fun x ↦ (PUnit.unit, x)\nhf₁ : Measurable f₁\nhf₂ : Measurable Prod.snd\n⊢ { hom := Kernel.id.map Prod.snd, property := ⋯ } ≫ { hom := Kernel.id.map f₁, property := ⋯ } =\n 𝟙\n { carrier := { carrier := PUnit.{u + 1}, str := PUnit.i... | refine ⟨⟨Kernel.id.map Prod.snd, inferInstance⟩,
⟨Kernel.id.map f₁, inferInstance⟩, ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Kernel.Proper | {
"line": 55,
"column": 42
} | {
"line": 55,
"column": 89
} | {
"line": 55,
"column": 90
} | [
{
"pp": "case refine_1\nX : Type u_1\n𝓑 𝓧 : MeasurableSpace X\nπ : Kernel X X\nh𝓑𝓧 : 𝓑 ≤ 𝓧\nx✝ : π.IsProper\nh : ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • π x\n⊢ ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • ... | [
"case refine_1\nX : Type u_1\n𝓑 𝓧 : MeasurableSpace X\nπ : Kernel X X\nh𝓑𝓧 : 𝓑 ≤ 𝓧\nx✝ : π.IsProper\nh : ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • π x\n⊢ ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • π x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Proper | {
"line": 55,
"column": 42
} | {
"line": 55,
"column": 89
} | {
"line": 55,
"column": 90
} | [
{
"pp": "case refine_2\nX : Type u_1\n𝓑 𝓧 : MeasurableSpace X\nπ : Kernel X X\nh𝓑𝓧 : 𝓑 ≤ 𝓧\nh : ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • π x\n⊢ ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • π x",
"ppTerm... | [
"case refine_2\nX : Type u_1\n𝓑 𝓧 : MeasurableSpace X\nπ : Kernel X X\nh𝓑𝓧 : 𝓑 ≤ 𝓧\nh : ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • π x\n⊢ ∀ ⦃B : Set X⦄ (hB : MeasurableSet B) (x : X), (π.restrict ⋯) x = B.indicator (fun x ↦ 1) x • π x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Kernel.Proper | {
"line": 78,
"column": 6
} | {
"line": 78,
"column": 38
} | {
"line": 78,
"column": 38
} | [
{
"pp": "X : Type u_1\n𝓑 𝓧 : MeasurableSpace X\nπ : Kernel X X\nA B : Set X\nhπ : π.IsProper\nh𝓑𝓧 : 𝓑 ≤ 𝓧\nμ : Measure X\nhA : MeasurableSet A\nhB : MeasurableSet B\n⊢ ∫⁻ (a : X) in B, (π a) A ∂μ = ∫⁻ (a : X), B.indicator (fun x ↦ (π a) A) a ∂μ",
"ppTerm": "?m.54",
"assigned": true,
"usedConst... | [
"X : Type u_1\n𝓑 𝓧 : MeasurableSpace X\nπ : Kernel X X\nA B : Set X\nhπ : π.IsProper\nh𝓑𝓧 : 𝓑 ≤ 𝓧\nμ : Measure X\nhA : MeasurableSet A\nhB : MeasurableSet B\n⊢ ∫⁻ (a : X), B.indicator (fun a ↦ (π a) A) a ∂μ = ∫⁻ (a : X), B.indicator (fun x ↦ (π a) A) a ∂μ"
] | ← lintegral_indicator (h𝓑𝓧 _ hB) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 59,
"column": 26
} | {
"line": 59,
"column": 67
} | {
"line": 59,
"column": 67
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\nι : Type u_3\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\ninst✝² : FirstCountableTopology ι\nℱ : Filtration ι m\ninst✝¹... | [
"Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\nι : Type u_3\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\ninst✝² : FirstCountableTopology ι\nℱ : Filtration ι m\ninst✝¹ : SigmaFini... | IsStoppingTime.measurableSet_inter_eq_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 69,
"column": 28
} | {
"line": 69,
"column": 69
} | {
"line": 69,
"column": 69
} | [
{
"pp": "case pos\nΩ : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\nι : Type u_3\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\ninst✝² : FirstCountableTopology ι\nℱ : Filtration ι... | [
"case pos\nΩ : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\nι : Type u_3\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\ninst✝² : FirstCountableTopology ι\nℱ : Filtration ι m\ninst✝¹ :... | IsStoppingTime.measurableSet_inter_eq_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 102,
"column": 4
} | {
"line": 105,
"column": 22
} | {
"line": 106,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : CompleteSpace E\nι : Type u_3\ninst✝⁶ : LinearOrder ι\ninst✝⁵ : TopologicalSpace ι\ninst✝⁴ : OrderTopology ι\ninst✝³ : FirstCountableTopology ι\nℱ : Filtration ι m\ninst✝²... | [] | simp only [h, Set.mem_range] at hi
obtain ⟨ω, hω⟩ := hi
specialize hτ_le ω
simp [hω] at hτ_le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 102,
"column": 4
} | {
"line": 105,
"column": 22
} | {
"line": 106,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : CompleteSpace E\nι : Type u_3\ninst✝⁶ : LinearOrder ι\ninst✝⁵ : TopologicalSpace ι\ninst✝⁴ : OrderTopology ι\ninst✝³ : FirstCountableTopology ι\nℱ : Filtration ι m\ninst✝²... | [] | simp only [h, Set.mem_range] at hi
obtain ⟨ω, hω⟩ := hi
specialize hτ_le ω
simp [hω] at hτ_le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Conditional | {
"line": 884,
"column": 6
} | {
"line": 884,
"column": 17
} | {
"line": 884,
"column": 18
} | [
{
"pp": "case e_f\nΩ : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : N... | [
"case e_f\nΩ : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 13
} | {
"line": 154,
"column": 14
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c κ ν\nh_int : Integrable (fun ω ↦ rexp (1 * X ω)) (⇑κ ∘ₘ ν)\n⊢ AEStronglyMeasurable X (⇑κ ∘ₘ ν)",
"ppTerm": "?m.30",
"assigned": false,
... | [
"Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c κ ν\nh_int : Integrable (fun ω ↦ rexp (1 * X ω)) (⇑κ ∘ₘ ν)\n⊢ AEStronglyMeasurable X (⇑κ ∘ₘ ν)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 13
} | {
"line": 164,
"column": 14
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c κ ν\nh_int✝ : ∀ᵐ (ω' : Ω') ∂ν, Integrable (fun y ↦ rexp (1 * X y)) (κ ω')\nω : Ω'\nh_int : Integrable (fun y ↦ rexp (1 * X y)) (κ ω)\n⊢ AEStrongl... | [
"Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c κ ν\nh_int✝ : ∀ᵐ (ω' : Ω') ∂ν, Integrable (fun y ↦ rexp (1 * X y)) (κ ω')\nω : Ω'\nh_int : Integrable (fun y ↦ rexp (1 * X y)) (κ ω)\n⊢ AEStronglyMeasurable ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.Tilted | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 20
} | {
"line": 144,
"column": 21
} | [
{
"pp": "case pos\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\nht : t ∈ interior (integrableExpSet X μ)\np : ℝ≥0\nhX : AEMeasurable X μ\nhp : p = 0\n⊢ MemLp X (↑p) (μ.tilted fun x ↦ t * X x)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"N... | [
"case pos\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\nht : t ∈ interior (integrableExpSet X μ)\np : ℝ≥0\nhX : AEMeasurable X μ\nhp : p = 0\n⊢ AEStronglyMeasurable X (μ.tilted fun x ↦ t * X x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 192,
"column": 4
} | {
"line": 192,
"column": 21
} | {
"line": 192,
"column": 22
} | [
{
"pp": "case pos\nΩ : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c κ ν\nt : ℝ\np : ℝ≥0\nhp0 : p = 0\n⊢ MemLp (fun ω ↦ rexp (t * X ω)) (↑p) (⇑κ ∘ₘ ν)",
"ppTerm": "?pos✝",
"assigned": true,
"u... | [
"case pos\nΩ : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c κ ν\nt : ℝ\np : ℝ≥0\nhp0 : p = 0\n⊢ AEStronglyMeasurable (fun ω ↦ rexp (t * X ω)) (⇑κ ∘ₘ ν)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.ProbabilityMassFunction.Binomial | {
"line": 72,
"column": 35
} | {
"line": 72,
"column": 46
} | {
"line": 72,
"column": 47
} | [
{
"pp": "k b : ℕ\nhb : k ≤ b\nx : ℝ≥0\nh : x ≤ 1\n⊢ k % (b + 1) = k",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instOne",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"DistribLattice.toLattice",
"i... | [
"k b : ℕ\nhb : k ≤ b\nx : ℝ≥0\nh : x ≤ 1\n⊢ k ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 211,
"column": 6
} | {
"line": 211,
"column": 22
} | {
"line": 211,
"column": 23
} | [
{
"pp": "case pos\nΩ : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh✝ : HasSubgaussianMGF X c κ ν\nω' : Ω'\nh : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)\nh_int : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')\nt ... | [
"case pos\nΩ : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh✝ : HasSubgaussianMGF X c κ ν\nω' : Ω'\nh : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)\nh_int : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')\nt : ℝ\nh0 : κ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 36
} | {
"line": 220,
"column": 37
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh✝ : HasSubgaussianMGF X c κ ν\nω' : Ω'\nh : Integrable (fun y ↦ rexp (0 * X y)) (κ ω')\nh_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)\n⊢ IsFiniteMeasure (κ ω'... | [
"Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh✝ : HasSubgaussianMGF X c κ ν\nω' : Ω'\nh : Integrable (fun y ↦ rexp (0 * X y)) (κ ω')\nh_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)\n⊢ IsFiniteMeasure (κ ω')"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 227,
"column": 2
} | {
"line": 227,
"column": 19
} | {
"line": 227,
"column": 20
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh✝ : HasSubgaussianMGF X c κ ν\nω' : Ω'\nh : IsFiniteMeasure (κ ω')\nh_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)\n⊢ (κ ω').real Set.univ ≤ 1",
"ppTerm": ... | [
"Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh✝ : HasSubgaussianMGF X c κ ν\nω' : Ω'\nh : IsFiniteMeasure (κ ω')\nh_mgf : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)\n⊢ (κ ω').real Set.univ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Kolmogorov | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 49
} | {
"line": 156,
"column": 50
} | [
{
"pp": "T : Type u_1\nΩ : Type u_2\nE : Type u_3\ninst✝¹ : PseudoEMetricSpace T\nmΩ : MeasurableSpace Ω\ninst✝ : PseudoEMetricSpace E\np q : ℝ\nM : ℝ≥0\nP : Measure Ω\nX : T → Ω → E\nhX : IsAEKolmogorovProcess X P p q M\ns t : T\nh : edist s t = 0\nthis : (fun ω ↦ edist (X s ω) (X t ω) ^ p) =ᵐ[P] 0\nω : Ω\nhω ... | [
"T : Type u_1\nΩ : Type u_2\nE : Type u_3\ninst✝¹ : PseudoEMetricSpace T\nmΩ : MeasurableSpace Ω\ninst✝ : PseudoEMetricSpace E\np q : ℝ\nM : ℝ≥0\nP : Measure Ω\nX : T → Ω → E\nhX : IsAEKolmogorovProcess X P p q M\ns t : T\nh : edist s t = 0\nthis : (fun ω ↦ edist (X s ω) (X t ω) ^ p) =ᵐ[P] 0\nω : Ω\nhω : edist (X s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Kolmogorov | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 49
} | {
"line": 172,
"column": 50
} | [
{
"pp": "T : Type u_1\nΩ : Type u_2\nE : Type u_3\ninst✝¹ : PseudoEMetricSpace T\nmΩ : MeasurableSpace Ω\ninst✝ : PseudoEMetricSpace E\np q : ℝ\nP : Measure Ω\nX : T → Ω → E\nhX : IsAEKolmogorovProcess X P p q 0\ns t : T\nthis : (fun ω ↦ edist (X s ω) (X t ω) ^ p) =ᵐ[P] 0\nω : Ω\nhω : edist (X s ω) (X t ω) ^ p ... | [
"T : Type u_1\nΩ : Type u_2\nE : Type u_3\ninst✝¹ : PseudoEMetricSpace T\nmΩ : MeasurableSpace Ω\ninst✝ : PseudoEMetricSpace E\np q : ℝ\nP : Measure Ω\nX : T → Ω → E\nhX : IsAEKolmogorovProcess X P p q 0\ns t : T\nthis : (fun ω ↦ edist (X s ω) (X t ω) ^ p) =ᵐ[P] 0\nω : Ω\nhω : edist (X s ω) (X t ω) ^ p = 0 ω\n⊢ edi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 265,
"column": 29
} | {
"line": 265,
"column": 40
} | {
"line": 265,
"column": 41
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c κ ν\nt : ℝ\n⊢ Integrable (fun ω ↦ rexp (t * (-X) ω)) (⇑κ ∘ₘ ν)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mp... | [
"Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c κ ν\nt : ℝ\n⊢ Integrable (fun ω ↦ rexp (-(t * X ω))) (⇑κ ∘ₘ ν)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 266,
"column": 63
} | {
"line": 266,
"column": 80
} | {
"line": 266,
"column": 81
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c κ ν\nω' : Ω'\nhm : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)\nt : ℝ\n⊢ mgf (-X) (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)",
"ppTerm": "?m.72",... | [
"Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c κ ν\nω' : Ω'\nhm : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)\nt : ℝ\n⊢ ∫ (x : Ω), rexp (-(t * X x)) ∂κ ω' ≤ rexp (↑c * t ^ 2 / 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 306,
"column": 4
} | {
"line": 306,
"column": 67
} | {
"line": 306,
"column": 68
} | [
{
"pp": "case refine_3\nΩ : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nhX : Measurable X\nh : HasSubgaussianMGF X c κ ν\n⊢ ∀ᵐ (ω' : Ω') ∂ν, ∀ (t : ℝ), mgf id ((κ.map X) ω') t ≤ rexp (↑c * t ^ 2 / 2)",
"ppTerm": "?refine_3",... | [
"case refine_3\nΩ : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX : Ω → ℝ\nc : ℝ≥0\nhX : Measurable X\nh : HasSubgaussianMGF X c κ ν\n⊢ ∀ᵐ (ω' : Ω') ∂ν, ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑c * t ^ 2 / 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.LocalProperty | {
"line": 156,
"column": 70
} | {
"line": 156,
"column": 85
} | {
"line": 157,
"column": 4
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁴ : LinearOrder ι\n𝓕 : Filtration ι mΩ\np : (ι → Ω → E) → Prop\ninst✝³ : OrderBot ι\ninst✝² : Zero E\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\nhp : IsStable 𝓕 p\nX : ι → Ω → E\nhX : (fun Y ↦ Locally p �... | [
"ι : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁴ : LinearOrder ι\n𝓕 : Filtration ι mΩ\np : (ι → Ω → E) → Prop\ninst✝³ : OrderBot ι\ninst✝² : Zero E\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\nhp : IsStable 𝓕 p\nX : ι → Ω → E\nhX : (fun Y ↦ Locally p 𝓕 Y P) X\nτ ... | Set.inter_comm, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RepresentationTheory.Subrepresentation | {
"line": 129,
"column": 10
} | {
"line": 129,
"column": 22
} | {
"line": 129,
"column": 23
} | [
{
"pp": "case single\nA : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nσ : Subrepresentation (Representation.ofModule M)\nm : M\nhm : m ∈ σ... | [
"case single\nA : Type u_1\nG : Type u_2\nW : Type u_3\nM : Type u_4\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid W\ninst✝² : Module A W\nρ : Representation A G W\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A[G] M\nσ : Subrepresentation (Representation.ofModule M)\nm : M\nhm : m ∈ σ.toSubmodule... | ← mul_one a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 437,
"column": 10
} | {
"line": 437,
"column": 21
} | {
"line": 437,
"column": 22
} | [
{
"pp": "case hf\nΩ : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX κ ν\nhY : HasSubgaussianMGF Y cY κ ν\nhX0 : ¬cX = 0\nhY0 : ¬cY = 0\np : ℝ≥0 := ⋯\nq : ℝ≥0 := ⋯\nω' : Ω'\nhmX : ∀ (t : ℝ), mgf X ... | [
"case hf\nΩ : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX κ ν\nhY : HasSubgaussianMGF Y cY κ ν\nhX0 : ¬cX = 0\nhY0 : ¬cY = 0\np : ℝ≥0 := (NNReal.sqrt cX + NNReal.sqrt cY) / NNReal.sqrt cX\nq : ℝ≥0 :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 438,
"column": 10
} | {
"line": 438,
"column": 21
} | {
"line": 438,
"column": 22
} | [
{
"pp": "case hg\nΩ : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX κ ν\nhY : HasSubgaussianMGF Y cY κ ν\nhX0 : ¬cX = 0\nhY0 : ¬cY = 0\np : ℝ≥0 := ⋯\nq : ℝ≥0 := ⋯\nω' : Ω'\nhmX : ∀ (t : ℝ), mgf X ... | [
"case hg\nΩ : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX κ ν\nhY : HasSubgaussianMGF Y cY κ ν\nhX0 : ¬cX = 0\nhY0 : ¬cY = 0\np : ℝ≥0 := (NNReal.sqrt cX + NNReal.sqrt cY) / NNReal.sqrt cX\nq : ℝ≥0 :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.StrongLaw | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 13
} | {
"line": 173,
"column": 14
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : AEStronglyMeasurable f μ\nA : ℝ\nhA : 0 ≤ A\n⊢ ∫ (x : α), truncation f A x ∂μ = ∫ (y : ℝ) in -A..A, y ∂Measure.map f μ",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : AEStronglyMeasurable f μ\nA : ℝ\nhA : 0 ≤ A\n⊢ ∫ (x : α), truncation f A x ∂μ = ∫ (y : ℝ) in -A..A, y ∂Measure.map f μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.StrongLaw | {
"line": 177,
"column": 2
} | {
"line": 177,
"column": 13
} | {
"line": 177,
"column": 14
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : AEStronglyMeasurable f μ\nA : ℝ\nh'f : 0 ≤ f\n⊢ ∫ (x : α), truncation f A x ∂μ = ∫ (y : ℝ) in 0..A, y ∂Measure.map f μ",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : AEStronglyMeasurable f μ\nA : ℝ\nh'f : 0 ≤ f\n⊢ ∫ (x : α), truncation f A x ∂μ = ∫ (y : ℝ) in 0..A, y ∂Measure.map f μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 461,
"column": 4
} | {
"line": 461,
"column": 15
} | {
"line": 461,
"column": 16
} | [
{
"pp": "case pos.integrable_exp_mul\nΩ : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nΩ'' : Type u_3\nmΩ'' : MeasurableSpace Ω''\nY : Ω'' → ℝ\ncY : ℝ≥0\nη : Kernel Ω Ω''\nh : HasSubgaussianMGF Y cY η (⇑κ ∘ₘ ν)\nhν : SFinite ν\nhκ : IsSFiniteKernel ... | [
"case pos.integrable_exp_mul\nΩ : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nν : Measure Ω'\nκ : Kernel Ω' Ω\nΩ'' : Type u_3\nmΩ'' : MeasurableSpace Ω''\nY : Ω'' → ℝ\ncY : ℝ≥0\nη : Kernel Ω Ω''\nh : HasSubgaussianMGF Y cY η (⇑κ ∘ₘ ν)\nhν : SFinite ν\nhκ : IsSFiniteKernel κ\n⊢ ∀ (t : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.StrongLaw | {
"line": 265,
"column": 13
} | {
"line": 265,
"column": 67
} | {
"line": 265,
"column": 67
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK N : ℕ\nhKN : K ≤ N\nρ : Measure ℝ := Measure.map X ℙ\nthis : IsProbabilityMeasure ρ\n⊢ (∫ (a : Ω), truncation X (↑N) a) + ∫ (x : ℝ) in 0..↑N, 1 ∂ρ ≤ (∫ (a : Ω), X a) + ∫ (x : ℝ) i... | [
"Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK N : ℕ\nhKN : K ≤ N\nρ : Measure ℝ := Measure.map X ℙ\nthis : IsProbabilityMeasure ρ\n⊢ (∫ (x : Ω), X x) + ∫ (x : ℝ) in 0..↑N, 1 ∂ρ ≤ (∫ (a : Ω), X a) + ∫ (x : ℝ) in 0..↑N, 1 ∂ρ"
] | integral_truncation_le_integral_of_nonneg hint hnonneg | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 613,
"column": 59
} | {
"line": 613,
"column": 70
} | {
"line": 613,
"column": 71
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nx✝ : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())\nh1 : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (⇑(Kernel.const Unit μ) ∘ₘ Measure.dirac ())\nh2 : ∀ᵐ (ω' : Unit) ∂Measure.dirac (), ∀ (t : ℝ), mgf X ... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nx✝ : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())\nh1 : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (⇑(Kernel.const Unit μ) ∘ₘ Measure.dirac ())\nh2 : ∀ᵐ (ω' : Unit) ∂Measure.dirac (), ∀ (t : ℝ), mgf X ((Kernel.con... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 613,
"column": 78
} | {
"line": 613,
"column": 89
} | {
"line": 613,
"column": 90
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nx✝ : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())\nh1 : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (⇑(Kernel.const Unit μ) ∘ₘ Measure.dirac ())\nh2 : ∀ᵐ (ω' : Unit) ∂Measure.dirac (), ∀ (t : ℝ), mgf X ... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nx✝ : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())\nh1 : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (⇑(Kernel.const Unit μ) ∘ₘ Measure.dirac ())\nh2 : ∀ᵐ (ω' : Unit) ∂Measure.dirac (), ∀ (t : ℝ), mgf X ((Kernel.con... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 613,
"column": 78
} | {
"line": 613,
"column": 92
} | {
"line": 613,
"column": 92
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nx✝ : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())\nh1 : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (⇑(Kernel.const Unit μ) ∘ₘ Measure.dirac ())\nh2 : ∀ᵐ (ω' : Unit) ∂Measure.dirac (), ∀ (t : ℝ), mgf X ... | [] | simpa using h2 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Probability.Moments.SubGaussian | {
"line": 613,
"column": 78
} | {
"line": 613,
"column": 92
} | {
"line": 613,
"column": 92
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nx✝ : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())\nh1 : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (⇑(Kernel.const Unit μ) ∘ₘ Measure.dirac ())\nh2 : ∀ᵐ (ω' : Unit) ∂Measure.dirac (), ∀ (t : ℝ), mgf X ... | [] | simpa using h2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.SubGaussian | {
"line": 613,
"column": 78
} | {
"line": 613,
"column": 92
} | {
"line": 613,
"column": 92
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nx✝ : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())\nh1 : ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (⇑(Kernel.const Unit μ) ∘ₘ Measure.dirac ())\nh2 : ∀ᵐ (ω' : Unit) ∂Measure.dirac (), ∀ (t : ℝ), mgf X ... | [] | simpa using h2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.SubGaussian | {
"line": 619,
"column": 2
} | {
"line": 619,
"column": 13
} | {
"line": 619,
"column": 14
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c μ\nh_int : Integrable (fun ω ↦ rexp (1 * X ω)) μ\n⊢ AEStronglyMeasurable X μ",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c μ\nh_int : Integrable (fun ω ↦ rexp (1 * X ω)) μ\n⊢ AEStronglyMeasurable X μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 633,
"column": 2
} | {
"line": 633,
"column": 13
} | {
"line": 633,
"column": 14
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())\nt : ℝ\np : ℝ≥0\n⊢ MemLp (fun ω ↦ rexp (t * X ω)) (↑p) μ",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())\nt : ℝ\np : ℝ≥0\n⊢ MemLp (fun ω ↦ rexp (t * X ω)) (↑p) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 637,
"column": 2
} | {
"line": 637,
"column": 13
} | {
"line": 637,
"column": 14
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())\nt : ℝ\n⊢ cgf X μ t ≤ ↑c * t ^ 2 / 2",
"ppTerm": "?m.41",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())\nt : ℝ\n⊢ cgf X μ t ≤ ↑c * t ^ 2 / 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 647,
"column": 2
} | {
"line": 647,
"column": 44
} | {
"line": 647,
"column": 45
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c μ\n⊢ HasSubgaussianMGF (-X) c μ",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Unit.unit",
"Real",
"Pi.instNeg",
"_private.Mathlib.Probabilit... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : HasSubgaussianMGF X c μ\n⊢ Kernel.HasSubgaussianMGF (-X) c (Kernel.const Unit μ) (Measure.dirac ())"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Equiv | {
"line": 173,
"column": 24
} | {
"line": 173,
"column": 49
} | {
"line": 174,
"column": 2
} | [
{
"pp": "k✝ : Type u\ninst✝¹¹ : Semiring k✝\nG : Type v\ninst✝¹⁰ : Monoid G\nV : Type v'\ninst✝⁹ : AddCommMonoid V\ninst✝⁸ : Module k✝ V\nW : Type w'\ninst✝⁷ : AddCommMonoid W\ninst✝⁶ : Module k✝ W\nH : Type w\ninst✝⁵ : Subsingleton H\ninst✝⁴ : MulOneClass H\ninst✝³ : MulAction G H\nk : Type u\ninst✝² : CommSem... | [] | simp [← x.isIntertwining] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Moments.SubGaussian | {
"line": 666,
"column": 8
} | {
"line": 666,
"column": 21
} | {
"line": 666,
"column": 21
} | [
{
"pp": "case refine_3\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nhX : AEMeasurable X μ\nh : HasSubgaussianMGF X c μ\nt : ℝ\n⊢ mgf id (Measure.map X μ) t ≤ rexp (↑c * t ^ 2 / 2)",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.i... | [
"case refine_3\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nhX : AEMeasurable X μ\nh : HasSubgaussianMGF X c μ\nt : ℝ\n⊢ mgf X μ t ≤ rexp (↑c * t ^ 2 / 2)"
] | mgf_id_map hX | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 707,
"column": 2
} | {
"line": 707,
"column": 13
} | {
"line": 707,
"column": 14
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())\nε : ℝ\nhε : 0 ≤ ε\n⊢ μ.real {ω | ε ≤ X ω} ≤ rexp (-ε ^ 2 / (2 * ↑c))",
"ppTerm": "?m.51",
"assigned": false,
"usedConstants": [],
"usedFVar... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nc : ℝ≥0\nh : Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())\nε : ℝ\nhε : 0 ≤ ε\n⊢ μ.real {ω | ε ≤ X ω} ≤ rexp (-ε ^ 2 / (2 * ↑c))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 714,
"column": 2
} | {
"line": 714,
"column": 13
} | {
"line": 714,
"column": 14
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nh : HasSubgaussianMGF X 0 μ\n⊢ X =ᵐ[μ] 0",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nh : HasSubgaussianMGF X 0 μ\n⊢ X =ᵐ[μ] 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 724,
"column": 2
} | {
"line": 724,
"column": 44
} | {
"line": 724,
"column": 45
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX μ\nhY : HasSubgaussianMGF Y cY μ\nthis :\n Kernel.HasSubgaussianMGF (fun ω ↦ X ω + Y ω) ((NNReal.sqrt cX + NNReal.sqrt cY) ^ 2) (Kernel.const Unit μ)\n (Measure.dirac ())\n⊢ HasSubgaussianMGF ... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX Y : Ω → ℝ\ncX cY : ℝ≥0\nhX : HasSubgaussianMGF X cX μ\nhY : HasSubgaussianMGF Y cY μ\nthis :\n Kernel.HasSubgaussianMGF (fun ω ↦ X ω + Y ω) ((NNReal.sqrt cX + NNReal.sqrt cY) ^ 2) (Kernel.const Unit μ)\n (Measure.dirac ())\n⊢ Kernel.HasSubgaussianMGF (fun ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Intertwining | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 13
} | {
"line": 55,
"column": 14
} | [
{
"pp": "case mk.mk\nA : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module A V\ninst✝ : Module A W\nρ : Representation A G V\nσ : Representation A G W\ntoLinearMap✝¹ : V →ₗ[A] W\nisIntertwining'✝¹ : ∀ (... | [
"case mk.mk\nA : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module A V\ninst✝ : Module A W\nρ : Representation A G V\nσ : Representation A G W\ntoLinearMap✝¹ : V →ₗ[A] W\nisIntertwining'✝¹ : ∀ (g : G), toLi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 792,
"column": 2
} | {
"line": 792,
"column": 27
} | {
"line": 792,
"column": 28
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : ℕ → Ω → ℝ\nh_indep : iIndepFun X μ\nc : ℝ≥0\nn : ℕ\nh_subG : ∀ i < n, HasSubgaussianMGF (X i) c μ\nε : ℝ\nhε : 0 ≤ ε\nh : μ.real {ω | ε ≤ ∑ i ∈ Finset.range n, X i ω} ≤ rexp (-ε ^ 2 / (2 * ↑(∑ i ∈ Finset.range n, c)))\n⊢ μ.real {ω | ε ≤ ∑ i ∈ Fin... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : ℕ → Ω → ℝ\nh_indep : iIndepFun X μ\nc : ℝ≥0\nn : ℕ\nh_subG : ∀ i < n, HasSubgaussianMGF (X i) c μ\nε : ℝ\nhε : 0 ≤ ε\nh : μ.real {ω | ε ≤ ∑ i ∈ Finset.range n, X i ω} ≤ rexp (-ε ^ 2 / (2 * ↑(∑ i ∈ Finset.range n, c)))\n⊢ μ.real {ω | ε ≤ ∑ i ∈ Finset.range n,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Intertwining | {
"line": 342,
"column": 35
} | {
"line": 342,
"column": 72
} | {
"line": 342,
"column": 73
} | [
{
"pp": "case mk'.mk'\nA : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module A V\ninst✝ : Module A W\nρ : Representation A G V\nσ : Representation A G W\ntoIntertwiningMap✝¹ : σ.IntertwiningMap ρ\ninvFu... | [
"case mk'.mk'\nA : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module A V\ninst✝ : Module A W\nρ : Representation A G V\nσ : Representation A G W\ntoIntertwiningMap✝¹ : σ.IntertwiningMap ρ\ninvFun✝¹ : V → W\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Intertwining | {
"line": 367,
"column": 2
} | {
"line": 367,
"column": 13
} | {
"line": 367,
"column": 14
} | [
{
"pp": "case mk'.mk'\nA : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module A V\ninst✝ : Module A W\nρ : Representation A G V\nσ : Representation A G W\ntoIntertwiningMap✝¹ : ρ.IntertwiningMap σ\ninvFu... | [
"case mk'.mk'\nA : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : Semiring A\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : AddCommMonoid W\ninst✝¹ : Module A V\ninst✝ : Module A W\nρ : Representation A G V\nσ : Representation A G W\ntoIntertwiningMap✝¹ : ρ.IntertwiningMap σ\ninvFun✝¹ : W → V\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.SubGaussian | {
"line": 865,
"column": 44
} | {
"line": 865,
"column": 55
} | {
"line": 865,
"column": 56
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝ : IsProbabilityMeasure μ\na b : ℝ\nhm : AEMeasurable X μ\nhb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b\nω : Ω\nhab : X ω ∈ Set.Icc a b\n⊢ X ω - ∫ (x : Ω), X x ∂μ ∈ Set.Icc (a - ∫ (x : Ω), X x ∂μ) (b - ∫ (x : Ω), X x ∂μ)",
"ppTerm": "?m.... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝ : IsProbabilityMeasure μ\na b : ℝ\nhm : AEMeasurable X μ\nhb : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b\nω : Ω\nhab : X ω ∈ Set.Icc a b\n⊢ a ≤ X ω ∧ X ω ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.StrongLaw | {
"line": 465,
"column": 8
} | {
"line": 465,
"column": 34
} | {
"line": 465,
"column": 35
} | [
{
"pp": "case hbc\nΩ : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : ℕ → Ω → ℝ\nhint : Integrable (X 0) ℙ\nhindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)\nhident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ\nhnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω\nc : ℝ\nc_one : 1 < c\nε : ℝ\nεpos : 0 < ε\nc_... | [
"case hbc\nΩ : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : ℕ → Ω → ℝ\nhint : Integrable (X 0) ℙ\nhindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)\nhident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ\nhnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω\nc : ℝ\nc_one : 1 < c\nε : ℝ\nεpos : 0 < ε\nc_pos : 0 < c\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Rep.Res | {
"line": 34,
"column": 27
} | {
"line": 34,
"column": 38
} | {
"line": 34,
"column": 39
} | [
{
"pp": "k : Type u\ninst✝² : Semiring k\nG : Type v1\nH : Type v2\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nX Y : Rep k G\nf : H →* G\np : X ⟶ Y\nh : H\n⊢ ↑(Hom.hom p) ∘ₗ (MonoidHom.comp X.ρ f) h = (MonoidHom.comp Y.ρ f) h ∘ₗ ↑(Hom.hom p)",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
... | [
"k : Type u\ninst✝² : Semiring k\nG : Type v1\nH : Type v2\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nX Y : Rep k G\nf : H →* G\np : X ⟶ Y\nh : H\n⊢ (Hom.hom p).toLinearMap ∘ₗ X.ρ (f h) = Y.ρ (f h) ∘ₗ (Hom.hom p).toLinearMap"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Rep.Res | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 73
} | {
"line": 70,
"column": 74
} | [
{
"pp": "k : Type u\ninst✝² : Semiring k\nG : Type v1\nH : Type v2\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nf : H →* G\nM : Rep k G\nX✝ Y✝ : Rep k G\na₁✝ a₂✝ : X✝ ⟶ Y✝\nh : (resFunctor f).map a₁✝ = (resFunctor f).map a₂✝\n⊢ a₁✝ = a₂✝",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"k : Type u\ninst✝² : Semiring k\nG : Type v1\nH : Type v2\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nf : H →* G\nM : Rep k G\nX✝ Y✝ : Rep k G\na₁✝ a₂✝ : X✝ ⟶ Y✝\nh : (resFunctor f).map a₁✝ = (resFunctor f).map a₂✝\n⊢ (Hom.hom a₁✝).toLinearMap = (Hom.hom a₂✝).toLinearMap"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Rep.Res | {
"line": 75,
"column": 76
} | {
"line": 75,
"column": 87
} | {
"line": 75,
"column": 88
} | [
{
"pp": "k : Type u\ninst✝² : Semiring k\nG : Type v1\nH : Type v2\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nf : H →* G\nM : Rep k G\nX Y : Rep k G\nhf : Function.Surjective ⇑f\nf' : res f X ⟶ res f Y\nh : H\n⊢ (Hom.hom f').toLinearMap ∘ₗ X.ρ (f h) = Y.ρ (f h) ∘ₗ (Hom.hom f').toLinearMap",
"ppTerm": "?m.89",
... | [
"k : Type u\ninst✝² : Semiring k\nG : Type v1\nH : Type v2\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nf : H →* G\nM : Rep k G\nX Y : Rep k G\nhf : Function.Surjective ⇑f\nf' : res f X ⟶ res f Y\nh : H\n⊢ (Hom.hom f').toLinearMap ∘ₗ X.ρ (f h) = Y.ρ (f h) ∘ₗ (Hom.hom f').toLinearMap"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Invariants | {
"line": 98,
"column": 4
} | {
"line": 98,
"column": 50
} | {
"line": 98,
"column": 51
} | [
{
"pp": "case pred\nk : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommRing k\ninst✝² : Group G\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nρ : Representation k G V\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nx : V\nhx : (ρ g) x = x\ni : ℕ\nh : (ρ ((fun x ↦ g ^ x) (-↑i))) x = x\n⊢ (ρ ((fun x ↦ g ^ x) ... | [
"case pred\nk : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommRing k\ninst✝² : Group G\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nρ : Representation k G V\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nx : V\nhx : (ρ g) x = x\ni : ℕ\nh : (ρ ((fun x ↦ g ^ x) (-↑i))) x = x\n⊢ (ρ g⁻¹) ((ρ (g ^ i)⁻¹) x) = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Invariants | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 15
} | {
"line": 157,
"column": 16
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝⁴ : CommRing k\ninst✝³ : Group G\nV : Type u_5\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nρ : Representation k G V\nS : Subgroup G\ninst✝ : S.Normal\ng : G\nx : V\nhx : x ∈ invariants (MonoidHom.comp ρ S.subtype)\nx✝ : ↥S\ns : G\nhs : s ∈ S\n⊢ ((MonoidHom.comp ρ S.s... | [
"k : Type u_1\nG : Type u_2\ninst✝⁴ : CommRing k\ninst✝³ : Group G\nV : Type u_5\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nρ : Representation k G V\nS : Subgroup G\ninst✝ : S.Normal\ng : G\nx : V\nhx : x ∈ invariants (MonoidHom.comp ρ S.subtype)\nx✝ : ↥S\ns : G\nhs : s ∈ S\n⊢ (ρ s) ((ρ g) x) = (ρ g) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Invariants | {
"line": 166,
"column": 58
} | {
"line": 166,
"column": 69
} | {
"line": 166,
"column": 70
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV✝ : Type u_3\nW : Type u_4\ninst✝⁸ : CommRing k\ninst✝⁷ : Group G\ninst✝⁶ : AddCommGroup V✝\ninst✝⁵ : Module k V✝\ninst✝⁴ : AddCommGroup W\ninst✝³ : Module k W\nρ✝ : Representation k G V✝\nσ : Representation k G W\nV : Type u_5\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\... | [
"k : Type u_1\nG : Type u_2\nV✝ : Type u_3\nW : Type u_4\ninst✝⁸ : CommRing k\ninst✝⁷ : Group G\ninst✝⁶ : AddCommGroup V✝\ninst✝⁵ : Module k V✝\ninst✝⁴ : AddCommGroup W\ninst✝³ : Module k W\nρ✝ : Representation k G V✝\nσ : Representation k G W\nV : Type u_5\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nρ : Represe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Character | {
"line": 117,
"column": 41
} | {
"line": 119,
"column": 52
} | {
"line": 121,
"column": 0
} | [
{
"pp": "G : Type u_1\nk : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁹ : Group G\ninst✝⁸ : Field k\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : FiniteDimensional k V\ninst✝⁴ : AddCommGroup W\ninst✝³ : Module k W\ninst✝² : FiniteDimensional k W\nρ : Representation k G V\nσ : Representation k G W\nins... | [] | by
simp_rw [mul_comm, ← char_linHom, card_inv_mul_sum_char_eq_finrank,
(invariantsEquivIntertwiningMap ρ σ).finrank_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RepresentationTheory.Coinduced | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 27
} | {
"line": 84,
"column": 28
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝⁶ : Semiring k\ninst✝⁵ : Monoid G\ninst✝⁴ : Monoid H\nφ : G →* H\nA : Type u_4\nB : Type u_5\ninst✝³ : AddCommMonoid A\ninst✝² : Module k A\ninst✝¹ : AddCommMonoid B\ninst✝ : Module k B\nσ : Representation k G A\nρ : Representation k G B\nh : H\nx : H → B\... | [
"k : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝⁶ : Semiring k\ninst✝⁵ : Monoid G\ninst✝⁴ : Monoid H\nφ : G →* H\nA : Type u_4\nB : Type u_5\ninst✝³ : AddCommMonoid A\ninst✝² : Module k A\ninst✝¹ : AddCommMonoid B\ninst✝ : Module k B\nσ : Representation k G A\nρ : Representation k G B\nh : H\nx : H → B\nhx : x ∈ co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Coinduced | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 19
} | {
"line": 97,
"column": 20
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝⁶ : Semiring k\ninst✝⁵ : Monoid G\ninst✝⁴ : Monoid H\nφ : G →* H\nA : Type u_4\nB : Type u_5\ninst✝³ : AddCommMonoid A\ninst✝² : Module k A\ninst✝¹ : AddCommMonoid B\ninst✝ : Module k B\nσ : Representation k G A\nρ : Representation k G B\nf : σ.Intertwinin... | [
"k : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝⁶ : Semiring k\ninst✝⁵ : Monoid G\ninst✝⁴ : Monoid H\nφ : G →* H\nA : Type u_4\nB : Type u_5\ninst✝³ : AddCommMonoid A\ninst✝² : Module k A\ninst✝¹ : AddCommMonoid B\ninst✝ : Module k B\nσ : Representation k G A\nρ : Representation k G B\nf : σ.IntertwiningMap ρ\nx : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Coinduced | {
"line": 191,
"column": 40
} | {
"line": 191,
"column": 51
} | {
"line": 191,
"column": 52
} | [
{
"pp": "k : Type u\nG : Type v\nH : Type w\ninst✝² : CommRing k\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nφ : G →* H\nA : Rep k G\nf g : ↑(coind' φ A)\nhfg : ∀ (h : H), (Hom.hom f).toLinearMap (MonoidAlgebra.single h 1) = (Hom.hom g).toLinearMap (MonoidAlgebra.single h 1)\nh : H\n⊢ ((Hom.hom f).toLinearMap ∘ₗ Mono... | [
"k : Type u\nG : Type v\nH : Type w\ninst✝² : CommRing k\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nφ : G →* H\nA : Rep k G\nf g : ↑(coind' φ A)\nhfg : ∀ (h : H), (Hom.hom f).toLinearMap (MonoidAlgebra.single h 1) = (Hom.hom g).toLinearMap (MonoidAlgebra.single h 1)\nh : H\n⊢ (Hom.hom f) (MonoidAlgebra.single h 1) = (Ho... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 110,
"column": 4
} | {
"line": 111,
"column": 32
} | {
"line": 111,
"column": 33
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nX : Type u_5\ninst✝⁷ : CommRing k\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AddCommGroup W\ninst✝² : Module k W\ninst✝¹ : AddCommGroup X\ninst✝ : Module k X\nρ : Representation k G V\nτ : Representation k G W\nυ : ... | [
"k : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nX : Type u_5\ninst✝⁷ : CommRing k\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AddCommGroup W\ninst✝² : Module k W\ninst✝¹ : AddCommGroup X\ninst✝ : Module k X\nρ : Representation k G V\nτ : Representation k G W\nυ : Representati... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 131,
"column": 34
} | {
"line": 131,
"column": 45
} | {
"line": 131,
"column": 46
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nX : Type u_5\ninst✝⁷ : CommRing k\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AddCommGroup W\ninst✝² : Module k W\ninst✝¹ : AddCommGroup X\ninst✝ : Module k X\nρ : Representation k G V\nτ : Representation k G W\nυ : ... | [
"k : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nX : Type u_5\ninst✝⁷ : CommRing k\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AddCommGroup W\ninst✝² : Module k W\ninst✝¹ : AddCommGroup X\ninst✝ : Module k X\nρ : Representation k G V\nτ : Representation k G W\nυ : Representati... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Coinduced | {
"line": 223,
"column": 15
} | {
"line": 223,
"column": 26
} | {
"line": 223,
"column": 27
} | [
{
"pp": "k : Type u\nG : Type v\nH : Type w\ninst✝² : CommRing k\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nφ : G →* H\nA : Rep k G\nf : res φ (leftRegular k H) ⟶ A\ng : G\nh : H\n⊢ ?m.240",
"ppTerm": "?m.241",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"k : Type u\nG : Type v\nH : Type w\ninst✝² : CommRing k\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nφ : G →* H\nA : Rep k G\nf : res φ (leftRegular k H) ⟶ A\ng : G\nh : H\n⊢ ?m.240"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 161,
"column": 4
} | {
"line": 161,
"column": 22
} | {
"line": 161,
"column": 23
} | [
{
"pp": "k : Type u_6\nG : Type u_7\nV : Type u_8\ninst✝⁴ : CommRing k\ninst✝³ : Group G\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nρ : Representation k G V\nS : Subgroup G\ninst✝ : S.Normal\ng : G\nx✝¹ : V\nx✝ : x✝¹ ∈ Set.range fun gv ↦ ((MonoidHom.comp ρ S.subtype) gv.1) gv.2 - gv.2\ns : ↥S\nx : V\nhs : (... | [
"k : Type u_6\nG : Type u_7\nV : Type u_8\ninst✝⁴ : CommRing k\ninst✝³ : Group G\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nρ : Representation k G V\nS : Subgroup G\ninst✝ : S.Normal\ng : G\nx✝¹ : V\nx✝ : x✝¹ ∈ Set.range fun gv ↦ ((MonoidHom.comp ρ S.subtype) gv.1) gv.2 - gv.2\ns : ↥S\nx : V\nhs : (fun gv ↦ ((M... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Coinduced | {
"line": 253,
"column": 45
} | {
"line": 253,
"column": 56
} | {
"line": 253,
"column": 57
} | [
{
"pp": "k : Type u\nG : Type v\nH : Type w\ninst✝² : CommRing k\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nφ : G →* H\nA✝ : Rep k G\nB : Rep k H\nA : Rep k G\nf : res φ B ⟶ A\nx✝² : ↑B\nx✝¹ : G\nx✝ : H\n⊢ (LinearMap.pi fun h ↦ (Hom.hom f).toLinearMap ∘ₗ B.ρ h) x✝² (φ x✝¹ * x✝) =\n (A.ρ x✝¹) ((LinearMap.pi fun h ... | [
"k : Type u\nG : Type v\nH : Type w\ninst✝² : CommRing k\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nφ : G →* H\nA✝ : Rep k G\nB : Rep k H\nA : Rep k G\nf : res φ B ⟶ A\nx✝² : ↑B\nx✝¹ : G\nx✝ : H\n⊢ (Hom.hom f) ((B.ρ (φ x✝¹)) ((B.ρ x✝) x✝²)) = (A.ρ x✝¹) ((Hom.hom f) ((B.ρ x✝) x✝²))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.StrongLaw | {
"line": 798,
"column": 4
} | {
"line": 798,
"column": 20
} | {
"line": 798,
"column": 21
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nX : ℕ → Ω → E\nhint : Integrable (X 0) μ\nhindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)\nhident : ... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nX : ℕ → Ω → E\nhint : Integrable (X 0) μ\nhindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)\nhident : ∀ (i : ℕ), I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Continuous.TopRep | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 37
} | {
"line": 201,
"column": 38
} | [
{
"pp": "k : Type u\nG : Type v\nX✝ Y✝ : Type w\ninst✝¹² : TopologicalSpace k\ninst✝¹¹ : Ring k\ninst✝¹⁰ : Monoid G\ninst✝⁹ : AddCommGroup X✝\ninst✝⁸ : Module k X✝\ninst✝⁷ : TopologicalSpace X✝\ninst✝⁶ : IsTopologicalAddGroup X✝\ninst✝⁵ : ContinuousSMul k X✝\ninst✝⁴ : AddCommGroup Y✝\ninst✝³ : Module k Y✝\ninst... | [
"k : Type u\nG : Type v\nX✝ Y✝ : Type w\ninst✝¹² : TopologicalSpace k\ninst✝¹¹ : Ring k\ninst✝¹⁰ : Monoid G\ninst✝⁹ : AddCommGroup X✝\ninst✝⁸ : Module k X✝\ninst✝⁷ : TopologicalSpace X✝\ninst✝⁶ : IsTopologicalAddGroup X✝\ninst✝⁵ : ContinuousSMul k X✝\ninst✝⁴ : AddCommGroup Y✝\ninst✝³ : Module k Y✝\ninst✝² : Topolog... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 13
} | {
"line": 124,
"column": 14
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : Semiring k\ninst✝ : Monoid G\nA B : Rep k G\nf : A ⟶ B\ng : G\na : ↑A\n⊢ (Hom.hom f) ((A.ρ g) a) = (B.ρ g) ((Hom.hom f) a)",
"ppTerm": "?m.49",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"k : Type u\nG : Type v\ninst✝¹ : Semiring k\ninst✝ : Monoid G\nA B : Rep k G\nf : A ⟶ B\ng : G\na : ↑A\n⊢ (Hom.hom f) ((A.ρ g) a) = (B.ρ g) ((Hom.hom f) a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Continuous.Basic | {
"line": 317,
"column": 35
} | {
"line": 317,
"column": 76
} | {
"line": 317,
"column": 77
} | [
{
"pp": "case mk''.mk''\nR : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁹ : Monoid G\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : TopologicalSpace V\ninst✝⁵ : IsTopologicalAddGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddCommGroup W\ninst✝² : TopologicalSpace W\ninst✝¹ : IsTopologicalAddGroup W... | [
"case mk''.mk''\nR : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁹ : Monoid G\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : TopologicalSpace V\ninst✝⁵ : IsTopologicalAddGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddCommGroup W\ninst✝² : TopologicalSpace W\ninst✝¹ : IsTopologicalAddGroup W\ninst✝ : Mo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Continuous.Basic | {
"line": 343,
"column": 2
} | {
"line": 343,
"column": 13
} | {
"line": 343,
"column": 14
} | [
{
"pp": "case mk''.mk''\nR : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁹ : Monoid G\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : TopologicalSpace V\ninst✝⁵ : IsTopologicalAddGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddCommGroup W\ninst✝² : TopologicalSpace W\ninst✝¹ : IsTopologicalAddGroup W... | [
"case mk''.mk''\nR : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁹ : Monoid G\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : TopologicalSpace V\ninst✝⁵ : IsTopologicalAddGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddCommGroup W\ninst✝² : TopologicalSpace W\ninst✝¹ : IsTopologicalAddGroup W\ninst✝ : Mo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 499,
"column": 40
} | {
"line": 499,
"column": 51
} | {
"line": 499,
"column": 52
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : Ring k\ninst✝ : Monoid G\nA B C : Rep k G\nX✝ Y✝ : Action (ModuleCat k) G\nf : X✝ ⟶ Y✝\ng : G\n⊢ ModuleCat.Hom.hom f.hom ∘ₗ (X✝.V.endRingEquiv.toMonoidHom.comp X✝.ρ) g =\n (Y✝.V.endRingEquiv.toMonoidHom.comp Y✝.ρ) g ∘ₗ ModuleCat.Hom.hom f.hom",
"ppTerm": "?m.88",... | [
"k : Type u\nG : Type v\ninst✝¹ : Ring k\ninst✝ : Monoid G\nA B C : Rep k G\nX✝ Y✝ : Action (ModuleCat k) G\nf : X✝ ⟶ Y✝\ng : G\n⊢ ModuleCat.Hom.hom f.hom ∘ₗ ModuleCat.Hom.hom (X✝.ρ g) = ModuleCat.Hom.hom (Y✝.ρ g) ∘ₗ ModuleCat.Hom.hom f.hom"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.FinGroupCharZero | {
"line": 114,
"column": 6
} | {
"line": 114,
"column": 22
} | {
"line": 114,
"column": 23
} | [
{
"pp": "case refine_2\nk : Type u\ninst✝⁴ : Field k\nG : Type u\ninst✝³ : Finite G\ninst✝² : Group G\ninst✝¹ : IsAlgClosed k\ninst✝ : NeZero ↑(Nat.card G)\nV : FDRep k G\nh : Module.finrank k (V ⟶ V) = 1\nW : FDRep k G\nf : W ⟶ V\nx✝ : Mono f\nι : Abelian.image f ⟶ V := Abelian.image.ι f\nhf : Abelian.factorTh... | [
"case refine_2\nk : Type u\ninst✝⁴ : Field k\nG : Type u\ninst✝³ : Finite G\ninst✝² : Group G\ninst✝¹ : IsAlgClosed k\ninst✝ : NeZero ↑(Nat.card G)\nV : FDRep k G\nh : Module.finrank k (V ⟶ V) = 1\nW : FDRep k G\nf : W ⟶ V\nx✝ : Mono f\nι : Abelian.image f ⟶ V := Abelian.image.ι f\nhf : Abelian.factorThruImage f ≫ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.FinGroupCharZero | {
"line": 125,
"column": 10
} | {
"line": 125,
"column": 68
} | {
"line": 125,
"column": 69
} | [
{
"pp": "case mp\nk : Type u\ninst✝⁴ : Field k\nG : Type u\ninst✝³ : Group G\ninst✝² : IsAlgClosed k\ninst✝¹ : CharZero k\ninst✝ : Fintype G\nV : FDRep k G\nthis : Invertible ↑(Nat.card G)\nh : Simple V\n⊢ ↑(Nat.card G) = ∑ g, V.character g * V.character g⁻¹",
"ppTerm": "?mp",
"assigned": true,
"use... | [
"case mp\nk : Type u\ninst✝⁴ : Field k\nG : Type u\ninst✝³ : Group G\ninst✝² : IsAlgClosed k\ninst✝¹ : CharZero k\ninst✝ : Fintype G\nV : FDRep k G\nthis : Invertible ↑(Nat.card G)\nh : Simple V\n⊢ ↑(Fintype.card G) = ∑ g, V.character g * V.character g⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 788,
"column": 15
} | {
"line": 788,
"column": 26
} | {
"line": 788,
"column": 27
} | [
{
"pp": "k : Type u\nG✝ : Type v\ninst✝² : CommRing k\ninst✝¹ : Monoid G✝\nG : Type v\ninst✝ : Group G\nA✝ B✝ C✝ : Rep k G\nA B C : Rep k G\nf : A ⊗ B ⟶ C\ng : G\nx : ↑B\ny : ↑A\n⊢ ?m.292",
"ppTerm": "?m.293",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"k : Type u\nG✝ : Type v\ninst✝² : CommRing k\ninst✝¹ : Monoid G✝\nG : Type v\ninst✝ : Group G\nA✝ B✝ C✝ : Rep k G\nA B C : Rep k G\nf : A ⊗ B ⟶ C\ng : G\nx : ↑B\ny : ↑A\n⊢ ?m.292"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 792,
"column": 4
} | {
"line": 792,
"column": 15
} | {
"line": 792,
"column": 16
} | [
{
"pp": "k : Type u\nG✝ : Type v\ninst✝² : CommRing k\ninst✝¹ : Monoid G✝\nG : Type v\ninst✝ : Group G\nA✝ B✝ C✝ : Rep k G\nA B C : Rep k G\nf : B ⟶ A.ihom.obj C\ng : G\nx : ↑A\ny : ↑B\n⊢ ((TensorProduct.uncurry (RingHom.id k) ↑A ↑B ↑C) (Hom.hom f).flip ∘ₗ (A.ρ.tprod B.ρ) g) (x ⊗ₜ[k] y) =\n (C.ρ g ∘ₗ (Tensor... | [
"k : Type u\nG✝ : Type v\ninst✝² : CommRing k\ninst✝¹ : Monoid G✝\nG : Type v\ninst✝ : Group G\nA✝ B✝ C✝ : Rep k G\nA B C : Rep k G\nf : B ⟶ A.ihom.obj C\ng : G\nx : ↑A\ny : ↑B\n⊢ ((Hom.hom f) ((B.ρ g) y)) ((A.ρ g) x) = (C.ρ g) (((Hom.hom f) y) x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 984,
"column": 45
} | {
"line": 984,
"column": 56
} | {
"line": 984,
"column": 57
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : CommRing k\ninst✝ : Monoid G\nX Y Z : Action (Type u) G\n⊢ Hom.hom\n (ofHom (δ (X ⊗ Y) Z) ≫\n ofHom (δ X Y) ▷ (linearization k G).obj Z ≫\n (α_ ((linearization k G).obj X) ((linearization k G).obj Y) ((linearization k G).obj Z)).hom) =\n Hom.hom ... | [
"k : Type u\nG : Type v\ninst✝¹ : CommRing k\ninst✝ : Monoid G\nX Y Z : Action (Type u) G\n⊢ ((↑(Representation.TensorProduct.assoc (Representation.linearize k G X) (Representation.linearize k G Y)\n (Representation.linearize k G Z))).comp\n (Representation.IntertwiningMap.rTensor (Represent... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 13
} | {
"line": 94,
"column": 14
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝² : CommRing k\ninst✝¹ : Group G\nS : Subgroup G\ninst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ng₁ g₂ g₃ : G\na : ↑A\n⊢ (A.indToCoindAux (g₁ * g₂⁻¹)) a g₃ = (A.indToCoindAux g₁) a (g₃ * g₂)",
"ppTerm": "?m.35",
"assigned": false,
"use... | [
"k : Type u\nG : Type v\ninst✝² : CommRing k\ninst✝¹ : Group G\nS : Subgroup G\ninst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ng₁ g₂ g₃ : G\na : ↑A\n⊢ (A.indToCoindAux (g₁ * g₂⁻¹)) a g₃ = (A.indToCoindAux g₁) a (g₃ * g₂)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 126,
"column": 21
} | {
"line": 126,
"column": 83
} | {
"line": 127,
"column": 6
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝³ : CommRing k\ninst✝² : Group G\nS : Subgroup G\ninst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ninst✝ : S.FiniteIndex\nx✝¹ x✝ : ↑(coind.{u, v, v, w} S.subtype A)\n⊢ ∑ g, g.liftOn (fun g ↦ (IndV.mk S.subtype A.ρ g) (↑(x✝¹ + x✝) g)) ⋯ =\n ∑ g, ... | [
"k : Type u\nG : Type v\ninst✝³ : CommRing k\ninst✝² : Group G\nS : Subgroup G\ninst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ninst✝ : S.FiniteIndex\nx✝¹ x✝ : ↑(coind.{u, v, v, w} S.subtype A)\n⊢ ∑ x,\n x.liftOn\n (fun g ↦\n (Coinvariants.mk (tprod (MonoidHom.comp (R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 128,
"column": 22
} | {
"line": 128,
"column": 51
} | {
"line": 128,
"column": 52
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝³ : CommRing k\ninst✝² : Group G\nS : Subgroup G\ninst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ninst✝ : S.FiniteIndex\nx✝¹ : k\nx✝ : ↑(coind.{u, v, v, w} S.subtype A)\n⊢ ∑ g, g.liftOn (fun g ↦ (IndV.mk S.subtype A.ρ g) (↑(x✝¹ • x✝) g)) ⋯ =\n ... | [
"k : Type u\nG : Type v\ninst✝³ : CommRing k\ninst✝² : Group G\nS : Subgroup G\ninst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ninst✝ : S.FiniteIndex\nx✝¹ : k\nx✝ : ↑(coind.{u, v, v, w} S.subtype A)\n⊢ ∑ x,\n x.liftOn\n (fun g ↦\n x✝¹ •\n (Coinvariants.mk (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 158,
"column": 4
} | {
"line": 158,
"column": 15
} | {
"line": 158,
"column": 16
} | [
{
"pp": "case h₀.h\nk : Type u\nG : Type v\ninst✝³ : CommRing k\ninst✝² : Group G\nS : Subgroup G\ninst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ninst✝ : S.FiniteIndex\ng : ↑(coind.{u, v, v, w} S.subtype A)\na b : G\na✝ : ⟦b⟧ ∈ Finset.univ\nhb : ⟦b⟧ ≠ ⟦a⟧\n⊢ ↑(A.indToCoind (⟦b⟧.liftO... | [
"case h₀.h\nk : Type u\nG : Type v\ninst✝³ : CommRing k\ninst✝² : Group G\nS : Subgroup G\ninst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ninst✝ : S.FiniteIndex\ng : ↑(coind.{u, v, v, w} S.subtype A)\na b : G\na✝ : ⟦b⟧ ∈ Finset.univ\nhb : ⟦b⟧ ≠ ⟦a⟧\n⊢ (A.indToCoindAux b) (↑g b) a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 15
} | {
"line": 61,
"column": 16
} | [
{
"pp": "case h\nk : Type u_1\nG : Type u_2\ninst✝⁴ : CommRing k\ninst✝³ : Group G\nV : Type u_4\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nρ : Representation k G V\ng : G\ninst✝ : Finite G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nα : V\nn : ℕ\na✝ : (fun gv ↦ (ρ gv.1) gv.2 - gv.2) ((fun x ↦ g ^ x) n, α) ∈ ↑... | [
"case h\nk : Type u_1\nG : Type u_2\ninst✝⁴ : CommRing k\ninst✝³ : Group G\nV : Type u_4\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nρ : Representation k G V\ng : G\ninst✝ : Finite G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nα : V\nn : ℕ\na✝ : (fun gv ↦ (ρ gv.1) gv.2 - gv.2) ((fun x ↦ g ^ x) n, α) ∈ ↑(ρ g - Linea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 15
} | {
"line": 63,
"column": 16
} | [
{
"pp": "case refine_2\nk : Type u_1\nG : Type u_2\ninst✝⁴ : CommRing k\ninst✝³ : Group G\nV : Type u_4\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nρ : Representation k G V\ng : G\ninst✝ : Finite G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\ny : V\n⊢ (ρ g - LinearMap.id) y ∈ Coinvariants.ker ρ",
"ppTerm": "... | [
"case refine_2\nk : Type u_1\nG : Type u_2\ninst✝⁴ : CommRing k\ninst✝³ : Group G\nV : Type u_4\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nρ : Representation k G V\ng : G\ninst✝ : Finite G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\ny : V\n⊢ (ρ g) y - y ∈ Coinvariants.ker ρ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 80,
"column": 4
} | {
"line": 81,
"column": 11
} | {
"line": 81,
"column": 12
} | [
{
"pp": "case refine_1\nk : Type u_1\nG : Type u_2\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Finite G\nthis : Fintype G\ng : G\ny : MonoidAlgebra k G\n⊢ (fun gv ↦ ((leftRegular k G) gv.1) gv.2 - gv.2) (g, y) ∈\n ↑((linearCombination k fun x ↦ 1) ∘ₗ ↑(MonoidAlgebra.coeffLinearEquiv k)).ker",
"ppTerm... | [
"case refine_1\nk : Type u_1\nG : Type u_2\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Finite G\nthis : Fintype G\ng : G\ny : MonoidAlgebra k G\n⊢ ∑ x, y.coeff (g⁻¹ * x) = ∑ x, y.coeff x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 15
} | {
"line": 170,
"column": 16
} | [
{
"pp": "case hx\nk : Type u\nG : Type v\ninst✝³ : CommRing k\ninst✝² : Group G\nS : Subgroup G\ninst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ninst✝ : S.FiniteIndex\ng : G\na : ↑A\nx : G\nhx : x ∉ MulAction.orbit (↥S) g\n⊢ x ∉\n Function.support\n ↑(A.indToCoind\n (... | [
"case hx\nk : Type u\nG : Type v\ninst✝³ : CommRing k\ninst✝² : Group G\nS : Subgroup G\ninst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA : Rep.{w, u, v} k ↥S\ninst✝ : S.FiniteIndex\ng : G\na : ↑A\nx : G\nhx : x ∉ MulAction.orbit (↥S) g\n⊢ (A.indToCoindAux g) a x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RepresentationTheory.Homological.ContCohomology.LowDegree | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 18
} | {
"line": 42,
"column": 19
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝⁴ : Ring k\ninst✝³ : Group G\ninst✝² : TopologicalSpace k\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nX : TopRep k G\nσ : ↑(X.homogeneousCochains.X 0).toModuleCat\ng : G\nhσ : ↑σ g⁻¹ = ↑σ 1\n⊢ (X.ρ g) (↑σ 1) = ↑σ 1",
"ppTerm": "?m.237",
"assigned... | [
"k : Type u_1\nG : Type u_2\ninst✝⁴ : Ring k\ninst✝³ : Group G\ninst✝² : TopologicalSpace k\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nX : TopRep k G\nσ : ↑(X.homogeneousCochains.X 0).toModuleCat\ng : G\nhσ : ↑σ g⁻¹ = ↑σ 1\n⊢ (X.ρ g) (↑σ 1) = ↑σ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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