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.Moments.SubGaussian | {
"line": 851,
"column": 27
} | {
"line": 851,
"column": 29
} | {
"line": 851,
"column": 30
} | [
{
"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\nhc : ∫ (x : Ω), X x ∂μ = 0\nt : ℝ\nht : t < 0\n⊢ mgf X μ t = mgf (-X) μ (-t)",
"ppTerm": "?m.119",
"assigned": true,
"usedCo... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 854,
"column": 35
} | {
"line": 854,
"column": 43
} | {
"line": 854,
"column": 44
} | [
{
"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\nhc : ∫ (x : Ω), X x ∂μ = 0\nt : ℝ\nht : t < 0\nω : Ω\n⊢ X ω ∈ Set.Icc a b → (-X) ω ∈ Set.Icc (-b) (-a)",
"ppTerm": "?m.256",
"as... | [
"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\nhc : ∫ (x : Ω), X x ∂μ = 0\nt : ℝ\nht : t < 0\nω : Ω\nhl : a ≤ X ω\nhr : X ω ≤ b\n⊢ (-X) ω ∈ Set.Icc (-b) (-a)"
] | ⟨hl, hr⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Term.anonymousCtor |
Mathlib.RepresentationTheory.Intertwining | {
"line": 504,
"column": 27
} | {
"line": 504,
"column": 29
} | {
"line": 504,
"column": 30
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G ... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 428,
"column": 80
} | {
"line": 428,
"column": 82
} | {
"line": 429,
"column": 8
} | [
{
"pp": "Ω : 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 < ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 852,
"column": 53
} | {
"line": 852,
"column": 55
} | {
"line": 853,
"column": 6
} | [
{
"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\nhc : ∫ (x : Ω), X x ∂μ = 0\nt : ℝ\nht : t < 0\n⊢ mgf (-X) μ (-t) ≤ rexp ((↑‖-a - -b‖₊ / 2) ^ 2 * (-t) ^ 2 / 2)",
"ppTerm": "?m.164",... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 857,
"column": 50
} | {
"line": 857,
"column": 52
} | {
"line": 857,
"column": 53
} | [
{
"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\nhc : ∫ (x : Ω), X x ∂μ = 0\nt : ℝ\nht : t < 0\n⊢ rexp ((↑‖-a - -b‖₊ / 2) ^ 2 * (-t) ^ 2 / 2) = rexp ((↑‖b - a‖₊ / 2) ^ 2 * t ^ 2 / 2)",
... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 846,
"column": 14
} | {
"line": 846,
"column": 16
} | {
"line": 847,
"column": 4
} | [
{
"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\nhc : ∫ (x : Ω), X x ∂μ = 0\nt : ℝ\n⊢ mgf X μ t ≤ rexp (↑((‖b - a‖₊ / 2) ^ 2) * t ^ 2 / 2)",
"ppTerm": "?m.71",
"assigned": true,... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 515,
"column": 7
} | {
"line": 515,
"column": 9
} | {
"line": 515,
"column": 10
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G ... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 437,
"column": 89
} | {
"line": 437,
"column": 91
} | {
"line": 438,
"column": 8
} | [
{
"pp": "Ω : 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 < ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 865,
"column": 41
} | {
"line": 865,
"column": 43
} | {
"line": 865,
"column": 44
} | [
{
"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.... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 516,
"column": 22
} | {
"line": 516,
"column": 24
} | {
"line": 516,
"column": 25
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G ... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 439,
"column": 50
} | {
"line": 439,
"column": 52
} | {
"line": 440,
"column": 8
} | [
{
"pp": "Ω : 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 < ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 862,
"column": 69
} | {
"line": 862,
"column": 71
} | {
"line": 863,
"column": 2
} | [
{
"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\n⊢ HasSubgaussianMGF (fun ω ↦ X ω - ∫ (x : Ω), X x ∂μ) ((‖b - a‖₊ / 2) ^ 2) μ",
"ppTerm": "?m.61",
"assigned": true,
"usedCon... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 517,
"column": 23
} | {
"line": 517,
"column": 25
} | {
"line": 517,
"column": 26
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G ... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 409,
"column": 72
} | {
"line": 409,
"column": 74
} | {
"line": 410,
"column": 4
} | [
{
"pp": "Ω : 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 < ... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 518,
"column": 18
} | {
"line": 518,
"column": 20
} | {
"line": 518,
"column": 21
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 885,
"column": 62
} | {
"line": 885,
"column": 64
} | {
"line": 886,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝¹ : StandardBorelSpace Ω\ninst✝ : IsFiniteMeasure μ\nY : Ω → ℝ\ncX cY : ℝ≥0\nhm : m ≤ mΩ\nhX : HasSubgaussianMGF X cX (μ.trim hm)\nhY : HasCondSubgaussianMGF m hm Y cY μ\nthis : HasSubgaussianMGF (fun p ↦ X p.1 + Y p.2) (cX + cY) (M... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 519,
"column": 19
} | {
"line": 519,
"column": 21
} | {
"line": 519,
"column": 22
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 893,
"column": 61
} | {
"line": 893,
"column": 63
} | {
"line": 893,
"column": 64
} | [
{
"pp": "Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝¹ : StandardBorelSpace Ω\ninst✝ : IsFiniteMeasure μ\nY : Ω → ℝ\ncX cY : ℝ≥0\nhm : m ≤ mΩ\nhX : Kernel.HasSubgaussianMGF X cX (Kernel.const Unit (μ.trim hm)) (Measure.dirac ())\nhY : HasCondSubgaussianMGF m hm Y cY μ\n⊢ Kernel.HasSub... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 525,
"column": 37
} | {
"line": 525,
"column": 39
} | {
"line": 525,
"column": 40
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 894,
"column": 2
} | {
"line": 894,
"column": 26
} | {
"line": 895,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝¹ : StandardBorelSpace Ω\ninst✝ : IsFiniteMeasure μ\nY : Ω → ℝ\ncX cY : ℝ≥0\nhm : m ≤ mΩ\nhX : Kernel.HasSubgaussianMGF X cX (Kernel.const Unit (μ.trim hm)) (Measure.dirac ())\nhY : HasCondSubgaussianMGF m hm Y cY μ\nhY' : Kernel.Ha... | [
"case e'_7\nΩ : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝¹ : StandardBorelSpace Ω\ninst✝ : IsFiniteMeasure μ\nY : Ω → ℝ\ncX cY : ℝ≥0\nhm : m ≤ mΩ\nhX : Kernel.HasSubgaussianMGF X cX (Kernel.const Unit (μ.trim hm)) (Measure.dirac ())\nhY : HasCondSubgaussianMGF m hm Y cY μ\nhY' : Kernel.Has... | convert! hX.add_comp hY' | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.RepresentationTheory.Intertwining | {
"line": 528,
"column": 37
} | {
"line": 528,
"column": 39
} | {
"line": 528,
"column": 40
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 551,
"column": 16
} | {
"line": 551,
"column": 18
} | {
"line": 552,
"column": 6
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 883,
"column": 45
} | {
"line": 883,
"column": 47
} | {
"line": 884,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\ninst✝¹ : StandardBorelSpace Ω\ninst✝ : IsFiniteMeasure μ\nY : Ω → ℝ\ncX cY : ℝ≥0\nhm : m ≤ mΩ\nhX : HasSubgaussianMGF X cX (μ.trim hm)\nhY : HasCondSubgaussianMGF m hm Y cY μ\n⊢ HasSubgaussianMGF (X + Y) (cX + cY) μ",
"ppTerm": "?m.2... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 563,
"column": 5
} | {
"line": 563,
"column": 7
} | {
"line": 564,
"column": 6
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 571,
"column": 52
} | {
"line": 571,
"column": 54
} | {
"line": 571,
"column": 55
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G ... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
"line": 918,
"column": 41
} | {
"line": 918,
"column": 43
} | {
"line": 918,
"column": 44
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝¹ : StandardBorelSpace Ω\nY : ℕ → Ω → ℝ\ncY : ℕ → ℝ≥0\nℱ : Filtration ℕ mΩ\ninst✝ : IsZeroOrProbabilityMeasure μ\nh_adapted : StronglyAdapted ℱ Y\nh0 : HasSubgaussianMGF (Y 0) (cY 0) μ\nn : ℕ\na✝ :\n ((∀ i < n - 1, HasCondSubgaussianMGF (↑ℱ i) ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 581,
"column": 5
} | {
"line": 581,
"column": 7
} | {
"line": 581,
"column": 8
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G ... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 448,
"column": 74
} | {
"line": 448,
"column": 76
} | {
"line": 449,
"column": 8
} | [
{
"pp": "Ω : 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 < ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 584,
"column": 35
} | {
"line": 584,
"column": 37
} | {
"line": 585,
"column": 2
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommSemiring A\ninst✝² : Monoid G\ninst✝¹ : AddCommMonoid V\ninst✝ : Module A V\nρ : Representation A G V\ng : G\nhg : g ∈ Submonoid.center G\n⊢ ρ.IsIntertwiningMap ρ (ρ g)",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq... | [] | by | [anonymous] | by |
Mathlib.Probability.Moments.SubGaussian | {
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Mathlib.RepresentationTheory.Intertwining | {
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Mathlib.RepresentationTheory.Intertwining | {
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Mathlib.RepresentationTheory.Intertwining | {
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Mathlib.RepresentationTheory.Intertwining | {
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Mathlib.Probability.StrongLaw | {
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Mathlib.RepresentationTheory.Intertwining | {
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Mathlib.RepresentationTheory.Intertwining | {
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Mathlib.RepresentationTheory.Intertwining | {
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Mathlib.RepresentationTheory.Intertwining | {
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Mathlib.RepresentationTheory.Intertwining | {
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Mathlib.RepresentationTheory.Intertwining | {
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Mathlib.Probability.Moments.SubGaussian | {
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Mathlib.RepresentationTheory.Intertwining | {
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Mathlib.RepresentationTheory.Intertwining | {
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Mathlib.RepresentationTheory.Intertwining | {
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Mathlib.RepresentationTheory.Intertwining | {
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Mathlib.RepresentationTheory.FDRep | {
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Mathlib.RepresentationTheory.FDRep | {
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Mathlib.RepresentationTheory.FDRep | {
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Mathlib.RepresentationTheory.FDRep | {
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Mathlib.RepresentationTheory.FDRep | {
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Mathlib.RepresentationTheory.FDRep | {
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Mathlib.RepresentationTheory.Intertwining | {
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Mathlib.RepresentationTheory.FDRep | {
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Mathlib.RepresentationTheory.Intertwining | {
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Mathlib.RepresentationTheory.Intertwining | {
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Mathlib.RepresentationTheory.Intertwining | {
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Mathlib.RepresentationTheory.FDRep | {
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Mathlib.RepresentationTheory.FDRep | {
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Mathlib.RepresentationTheory.FDRep | {
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Mathlib.RepresentationTheory.FDRep | {
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Mathlib.RepresentationTheory.Intertwining | {
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Mathlib.RepresentationTheory.FDRep | {
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Mathlib.RepresentationTheory.Intertwining | {
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} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 790,
"column": 48
} | {
"line": 790,
"column": 50
} | {
"line": 790,
"column": 51
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G ... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 460,
"column": 42
} | {
"line": 460,
"column": 44
} | {
"line": 461,
"column": 8
} | [
{
"pp": "Ω : 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 < ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Intertwining | {
"line": 818,
"column": 23
} | {
"line": 818,
"column": 25
} | {
"line": 819,
"column": 4
} | [
{
"pp": "A : Type u_1\nG✝ : Type u_2\nV✝ : Type u_3\nW✝ : Type u_4\nU : Type u_5\ninst✝¹⁵ : CommSemiring A\ninst✝¹⁴ : Monoid G✝\ninst✝¹³ : AddCommMonoid V✝\ninst✝¹² : AddCommMonoid W✝\ninst✝¹¹ : AddCommMonoid U\ninst✝¹⁰ : Module A V✝\ninst✝⁹ : Module A W✝\ninst✝⁸ : Module A U\nρ✝ : Representation A G✝ V✝\nσ✝ : ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FDRep | {
"line": 168,
"column": 84
} | {
"line": 168,
"column": 86
} | {
"line": 169,
"column": 4
} | [
{
"pp": "R k : Type u\nG : Type v\ninst✝² : CommRing R\ninst✝¹ : Field k\ninst✝ : Monoid G\nX Y : FDRep R G\nf : (forget₂ (FDRep R G) (Rep R G)).obj X ⟶ (forget₂ (FDRep R G) (Rep R G)).obj Y\ng : G\n⊢ X.ρ g ≫ InducedCategory.homMk (ModuleCat.ofHom (Rep.Hom.hom f).toLinearMap) =\n InducedCategory.homMk (Modul... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 444,
"column": 38
} | {
"line": 444,
"column": 40
} | {
"line": 445,
"column": 4
} | [
{
"pp": "Ω : 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 < ... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RepresentationTheory.FDRep | {
"line": 174,
"column": 90
} | {
"line": 174,
"column": 92
} | {
"line": 175,
"column": 4
} | [
{
"pp": "R k : Type u\nG : Type v\ninst✝² : CommRing R\ninst✝¹ : Field k\ninst✝ : Monoid G\nX Y : FDRep R G\nf : X ⟶ Y\ng : G\n⊢ ModuleCat.Hom.hom ((forget₂ (FGModuleCat R) (ModuleCat R)).map f.hom) ∘ₗ\n ((((forget₂ (FGModuleCat R) (ModuleCat R)).mapAction G).obj X).V.endRingEquiv.toMonoidHom.comp\n ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FDRep | {
"line": 178,
"column": 51
} | {
"line": 178,
"column": 53
} | {
"line": 179,
"column": 2
} | [
{
"pp": "R k : Type u\nG : Type v\ninst✝² : CommRing R\ninst✝¹ : Field k\ninst✝ : Monoid G\n⊢ (forget₂ (FDRep R G) (Rep R G)).Full",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Rep.V",
"Action.instFunLikeHomSubtypeV",
"CategoryTheory.Functor.IsEquivalence.full",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FDRep | {
"line": 182,
"column": 55
} | {
"line": 182,
"column": 57
} | {
"line": 183,
"column": 2
} | [
{
"pp": "R k : Type u\nG : Type v\ninst✝² : CommRing R\ninst✝¹ : Field k\ninst✝ : Monoid G\n⊢ (forget₂ (FDRep R G) (Rep R G)).Faithful",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Rep.V",
"Action.instFunLikeHomSubtypeV",
"ModuleCat",
"CategoryTheory.ObjectProper... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FDRep | {
"line": 211,
"column": 91
} | {
"line": 211,
"column": 93
} | {
"line": 212,
"column": 2
} | [
{
"pp": "k : Type u\nG : Type v\nV : Type u\ninst✝⁴ : Field k\ninst✝³ : Group G\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : FiniteDimensional k V\nρV : Representation k G V\nW : FDRep k G\n⊢ of ρV.dual ⊗ W ≅ of (ρV.linHom W.ρ)",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Res | {
"line": 34,
"column": 24
} | {
"line": 34,
"column": 26
} | {
"line": 34,
"column": 27
} | [
{
"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": [
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Res | {
"line": 69,
"column": 21
} | {
"line": 69,
"column": 23
} | {
"line": 70,
"column": 4
} | [
{
"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": [
"Rep.... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Res | {
"line": 75,
"column": 48
} | {
"line": 75,
"column": 50
} | {
"line": 75,
"column": 51
} | [
{
"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\ng : G\n⊢ (Hom.hom f').toLinearMap ∘ₗ X.ρ g = Y.ρ g ∘ₗ (Hom.hom f').toLinearMap",
"ppTerm": "?m.73",
"assig... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Res | {
"line": 83,
"column": 53
} | {
"line": 83,
"column": 55
} | {
"line": 83,
"column": 56
} | [
{
"pp": "k : Type u\ninst✝² : Semiring k\nG : Type v1\nH : Type v2\ninst✝¹ : Monoid G\ninst✝ : Monoid H\nf : H →* G\nhf : Function.Surjective ⇑f\nX Y : Rep k G\nf' : (resFunctor f).obj X ⟶ (resFunctor f).obj Y\n⊢ (resFunctor f).map (liftHomOfSurj f hf f') = f'",
"ppTerm": "?m.35",
"assigned": true,
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Res | {
"line": 86,
"column": 23
} | {
"line": 86,
"column": 25
} | {
"line": 86,
"column": 26
} | [
{
"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✝³ x✝² : Rep k G\nx✝¹ x✝ : x✝³ ⟶ x✝²\n⊢ (resFunctor f).map (x✝¹ + x✝) = (resFunctor f).map x✝¹ + (resFunctor f).map x✝",
"ppTerm": "?m.27",
"assigned": true,
"usedConsta... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Res | {
"line": 89,
"column": 24
} | {
"line": 89,
"column": 26
} | {
"line": 89,
"column": 27
} | [
{
"pp": "k✝ : Type u\ninst✝³ : Semiring k✝\nG : Type v1\nH : Type v2\ninst✝² : Monoid G\ninst✝¹ : Monoid H\nf : H →* G\nM : Rep k✝ G\nk : Type u\ninst✝ : CommSemiring k\nx✝³ x✝² : Rep k G\nx✝¹ : x✝³ ⟶ x✝²\nx✝ : k\n⊢ (resFunctor f).map (x✝ • x✝¹) = x✝ • (resFunctor f).map x✝¹",
"ppTerm": "?m.34",
"assign... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Res | {
"line": 110,
"column": 35
} | {
"line": 110,
"column": 37
} | {
"line": 111,
"column": 2
} | [
{
"pp": "G : Type v1\nH : Type v2\ninst✝² : Monoid G\ninst✝¹ : Monoid H\nf : H →* G\nk : Type u\ninst✝ : Ring k\nM : Rep k G\n⊢ IsZero (res f M) ↔ IsZero M",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Rep.V",
"congrArg",
"Iff.rfl",
"Rep.instCateg... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Res | {
"line": 118,
"column": 46
} | {
"line": 118,
"column": 48
} | {
"line": 119,
"column": 2
} | [
{
"pp": "G : Type v1\nH : Type v2\ninst✝² : Monoid G\ninst✝¹ : Monoid H\nf : H →* G\nk : Type u\ninst✝ : CommRing k\nS : ShortComplex (Rep k G)\n⊢ (S.map (resFunctor f)).Exact ↔ S.Exact",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Rep.instPreservesLimitsResFunctor",
"Eq.mpr... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Iso | {
"line": 78,
"column": 54
} | {
"line": 78,
"column": 56
} | {
"line": 79,
"column": 2
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝⁵ : CommRing k\ninst✝⁴ : Monoid G\nV : Type u_3\nW : Type u_4\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup W\ninst✝¹ : Module k V\ninst✝ : Module k W\nρ : G →* V →ₗ[k] V\nσ : G →* W →ₗ[k] W\nf : V →ₗ[k] W\nw : ∀ (g : G), f ∘ₗ ρ g = σ g ∘ₗ f\nr : k[G]\nx : V\n⊢ f ((((... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Iso | {
"line": 107,
"column": 25
} | {
"line": 107,
"column": 27
} | {
"line": 107,
"column": 28
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : CommRing k\ninst✝ : Monoid G\nX✝ Y✝ : ModuleCat k[G]\nf : X✝ ⟶ Y✝\ng : G\n⊢ { toAddHom := (ModuleCat.Hom.hom f).toAddHom, map_smul' := ⋯ } ∘ₗ (Representation.ofModule ↑X✝) g =\n (Representation.ofModule ↑Y✝) g ∘ₗ { toAddHom := (ModuleCat.Hom.hom f).toAddHom, map_smul... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Iso | {
"line": 120,
"column": 66
} | {
"line": 120,
"column": 68
} | {
"line": 121,
"column": 2
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : CommRing k\ninst✝ : Monoid G\nM : ModuleCat k[G]\n⊢ ↑((ofModuleMonoidAlgebra ⋙ toModuleMonoidAlgebra).obj M) ≃+ ↑M",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"MonoidAlgebra.semiring",
"RestrictScalars.module",
"Rep.ofModuleMo... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Res | {
"line": 135,
"column": 15
} | {
"line": 135,
"column": 17
} | {
"line": 135,
"column": 18
} | [
{
"pp": "G : Type v1\nH : Type v2\ninst✝² : Monoid G\ninst✝¹ : Monoid H\nk : Type u\ninst✝ : CommRing k\nφ : H →* G\nS : ShortComplex (Rep k G)\nexact✝ : S.Exact\nmono_f : Mono S.f\nepi_g : Epi S.g\n⊢ (S.map (resFunctor φ)).Exact",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants": [
"Rep.... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Iso | {
"line": 128,
"column": 36
} | {
"line": 128,
"column": 38
} | {
"line": 129,
"column": 2
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : CommRing k\ninst✝ : Monoid G\nV : Rep k G\n⊢ ↑V ≃+ ↑((toModuleMonoidAlgebra ⋙ ofModuleMonoidAlgebra).obj V)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Rep.V",
"LinearEquiv.symm",
"Rep.ofModuleMonoidAlgebra",
"ModuleCat... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Iso | {
"line": 138,
"column": 30
} | {
"line": 138,
"column": 32
} | {
"line": 139,
"column": 8
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : CommRing k\ninst✝ : Monoid G\nM : ModuleCat k[G]\nr : k[G]\nx : (ofModuleMonoidAlgebra.obj M).ρ.asModule\n⊢ counitIsoAddEquiv.toFun (r • x) = (RingHom.id k[G]) r • counitIsoAddEquiv.toFun x",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Mon... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Res | {
"line": 136,
"column": 16
} | {
"line": 136,
"column": 18
} | {
"line": 136,
"column": 19
} | [
{
"pp": "G : Type v1\nH : Type v2\ninst✝² : Monoid G\ninst✝¹ : Monoid H\nk : Type u\ninst✝ : CommRing k\nφ : H →* G\nS : ShortComplex (Rep k G)\nexact✝ : S.Exact\nmono_f : Mono S.f\nepi_g : Epi S.g\n⊢ Mono (S.map (resFunctor φ)).f",
"ppTerm": "?m.119",
"assigned": true,
"usedConstants": [
"Rep... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Iso | {
"line": 145,
"column": 71
} | {
"line": 145,
"column": 73
} | {
"line": 146,
"column": 2
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : CommRing k\ninst✝ : Monoid G\nV : Rep k G\ng : G\nx : ↑V\n⊢ unitIsoAddEquiv ((V.ρ g).toFun x) =\n ((ofModuleMonoidAlgebra.obj (toModuleMonoidAlgebra.obj V)).ρ g).toFun (unitIsoAddEquiv x)",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Mo... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Res | {
"line": 137,
"column": 15
} | {
"line": 137,
"column": 17
} | {
"line": 137,
"column": 18
} | [
{
"pp": "G : Type v1\nH : Type v2\ninst✝² : Monoid G\ninst✝¹ : Monoid H\nk : Type u\ninst✝ : CommRing k\nφ : H →* G\nS : ShortComplex (Rep k G)\nexact✝ : S.Exact\nmono_f : Mono S.f\nepi_g : Epi S.g\n⊢ Epi (S.map (resFunctor φ)).g",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants": [
"Rep.... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Res | {
"line": 123,
"column": 56
} | {
"line": 123,
"column": 58
} | {
"line": 124,
"column": 2
} | [
{
"pp": "G : Type v1\nH : Type v2\ninst✝² : Monoid G\ninst✝¹ : Monoid H\nk : Type u\ninst✝ : CommRing k\nφ : H →* G\nS : ShortComplex (Rep k G)\n⊢ (S.map (resFunctor φ)).ShortExact ↔ S.ShortExact",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Rep.instPreservesLimitsResFunctor",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Iso | {
"line": 156,
"column": 20
} | {
"line": 156,
"column": 22
} | {
"line": 156,
"column": 23
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : CommRing k\ninst✝ : Monoid G\nV : Rep k G\ng : G\n⊢ ↑(let __src := unitIsoAddEquiv;\n { toFun := __src.toFun, map_add' := ⋯, map_smul' := ⋯, invFun := __src.invFun, left_inv := ⋯,\n right_inv := ⋯ }) ∘ₗ\n V.ρ g =\n (Representation.ofModule ↑(toMo... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Iso | {
"line": 162,
"column": 55
} | {
"line": 162,
"column": 57
} | {
"line": 162,
"column": 58
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : CommRing k\ninst✝ : Monoid G\n⊢ ∀ {X Y : Rep k G} (f : X ⟶ Y),\n (𝟭 (Rep k G)).map f ≫ Y.unitIso.hom = X.unitIso.hom ≫ (toModuleMonoidAlgebra ⋙ ofModuleMonoidAlgebra).map f",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"CategoryTheory.C... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Iso | {
"line": 163,
"column": 59
} | {
"line": 163,
"column": 61
} | {
"line": 163,
"column": 62
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : CommRing k\ninst✝ : Monoid G\n⊢ ∀ {X Y : ModuleCat k[G]} (f : X ⟶ Y),\n (ofModuleMonoidAlgebra ⋙ toModuleMonoidAlgebra).map f ≫ (counitIso Y).hom =\n (counitIso X).hom ≫ (𝟭 (ModuleCat k[G])).map f",
"ppTerm": "?m.50",
"assigned": true,
"usedConstant... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Iso | {
"line": 189,
"column": 41
} | {
"line": 189,
"column": 43
} | {
"line": 190,
"column": 2
} | [
{
"pp": "k : Type u\nG✝ : Type v\ninst✝² : CommRing k\ninst✝¹ : Monoid G✝\nG : Type u\ninst✝ : Group G\nn : ℕ\n⊢ Projective (diagonal k G (n + 1))",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Rep.diagonal",
"Rep.free_projective",
"Rep.instCategory",
"inferInstan... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 380,
"column": 20
} | {
"line": 380,
"column": 22
} | {
"line": 390,
"column": 2
} | [
{
"pp": "Ω : 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\n⊢ ∀ᵐ (ω : Ω)... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 479,
"column": 92
} | {
"line": 479,
"column": 94
} | {
"line": 480,
"column": 2
} | [
{
"pp": "Ω : 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\n⊢ ∀ᵐ (ω : Ω),\n (fun n ↦\n ... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 496,
"column": 72
} | {
"line": 496,
"column": 74
} | {
"line": 497,
"column": 4
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : ℕ → Ω → ℝ\nhint : Integrable (X 0) ℙ\nhident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ\n⊢ Tendsto (fun i ↦ ∫ (a : Ω), truncation (X i) (↑i) a) atTop (𝓝 (∫ (a : Ω), X 0 a))",
"ppTerm": "?m.136",
"assigned": true,
"us... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.StrongLaw | {
"line": 495,
"column": 92
} | {
"line": 495,
"column": 94
} | {
"line": 496,
"column": 2
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : ℕ → Ω → ℝ\nhint : Integrable (X 0) ℙ\nhident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ\n⊢ (fun n ↦ (∫ (a : Ω), (∑ i ∈ range n, truncation (X i) ↑i) a) - ↑n * ∫ (a : Ω), X 0 a) =o[atTop] Nat.cast",
"ppTerm": "?m.103",
"as... | [] | by | [anonymous] | by |
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