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values | kind stringclasses 379
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Mathlib.RepresentationTheory.Invariants | {
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Mathlib.Probability.StrongLaw | {
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Mathlib.Probability.StrongLaw | {
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Mathlib.RepresentationTheory.Invariants | {
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Mathlib.RepresentationTheory.Invariants | {
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Mathlib.RepresentationTheory.Invariants | {
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{
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Mathlib.RepresentationTheory.Invariants | {
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{
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Mathlib.RepresentationTheory.Invariants | {
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} | {
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{
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Mathlib.RepresentationTheory.Invariants | {
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} | {
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} | {
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{
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"ppTerm": "?m.29",
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... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Invariants | {
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} | {
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} | {
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{
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Mathlib.RepresentationTheory.Invariants | {
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} | {
"line": 102,
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} | {
"line": 103,
"column": 2
} | [
{
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Mathlib.Probability.StrongLaw | {
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} | {
"line": 539,
"column": 35
} | {
"line": 540,
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} | [
{
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Mathlib.RepresentationTheory.Invariants | {
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} | {
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} | {
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{
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"assign... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Invariants | {
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} | {
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} | {
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{
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"ppTerm": "?m.38",
"assigned": ... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
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} | {
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{
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Mathlib.RepresentationTheory.Invariants | {
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} | {
"line": 156,
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} | {
"line": 157,
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{
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Mathlib.RepresentationTheory.Invariants | {
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{
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Mathlib.Probability.StrongLaw | {
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} | {
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} | {
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{
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Mathlib.Probability.StrongLaw | {
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} | {
"line": 555,
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} | {
"line": 556,
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{
"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⊢ ∀ (n : ℕ), 0 < ↑⌊c ^ n⌋₊",
... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RepresentationTheory.Invariants | {
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} | {
"line": 191,
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} | {
"line": 192,
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} | [
{
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Mathlib.RepresentationTheory.Invariants | {
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} | {
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} | {
"line": 221,
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} | [
{
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... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Invariants | {
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} | {
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{
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Mathlib.RepresentationTheory.Invariants | {
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} | {
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} | {
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} | [
{
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Mathlib.RepresentationTheory.Invariants | {
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} | {
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} | {
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{
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"S... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Invariants | {
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} | {
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} | {
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{
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Mathlib.RepresentationTheory.Irreducible | {
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} | {
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{
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Mathlib.RepresentationTheory.Invariants | {
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} | {
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{
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Mathlib.RepresentationTheory.Irreducible | {
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} | {
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} | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
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} | {
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} | {
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{
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Mathlib.RepresentationTheory.Irreducible | {
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{
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Mathlib.RepresentationTheory.Irreducible | {
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} | {
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{
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Mathlib.RepresentationTheory.Irreducible | {
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} | {
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{
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Mathlib.RepresentationTheory.Irreducible | {
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{
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"ppTerm": "?m.6... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Irreducible | {
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} | {
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} | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Irreducible | {
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} | {
"line": 89,
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} | {
"line": 90,
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{
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"ppTerm": "... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Character | {
"line": 61,
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} | {
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} | {
"line": 61,
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{
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... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Character | {
"line": 64,
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} | {
"line": 64,
"column": 86
} | {
"line": 65,
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{
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"ppTerm": "?m.36",
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"usedConstants": [
"LinearMap.t... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Character | {
"line": 69,
"column": 75
} | {
"line": 69,
"column": 77
} | {
"line": 70,
"column": 2
} | [
{
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Mathlib.Probability.StrongLaw | {
"line": 554,
"column": 94
} | {
"line": 554,
"column": 96
} | {
"line": 555,
"column": 2
} | [
{
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Mathlib.RepresentationTheory.Character | {
"line": 75,
"column": 64
} | {
"line": 75,
"column": 66
} | {
"line": 76,
"column": 2
} | [
{
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"ppTerm": "?m.44",... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Character | {
"line": 90,
"column": 81
} | {
"line": 90,
"column": 83
} | {
"line": 91,
"column": 2
} | [
{
"pp": "G : Type u_1\nk : Type u_2\nV : Type u_3\ninst✝³ : Group G\ninst✝² : Field k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nρ : Representation k G V\ng h : G\n⊢ ρ.character (h * g * h⁻¹) = ρ.character g",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMo... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Character | {
"line": 99,
"column": 66
} | {
"line": 99,
"column": 68
} | {
"line": 100,
"column": 2
} | [
{
"pp": "G : Type u_1\nk : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁷ : Group G\ninst✝⁶ : Field k\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : FiniteDimensional k V\ninst✝² : AddCommGroup W\ninst✝¹ : Module k W\ninst✝ : FiniteDimensional k W\nρ : Representation k G V\nσ : Representation k G W\ng : ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Character | {
"line": 106,
"column": 44
} | {
"line": 106,
"column": 46
} | {
"line": 106,
"column": 47
} | [
{
"pp": "G : Type u_1\nk : Type u_2\nV : Type u_3\ninst✝⁶ : Group G\ninst✝⁵ : Field k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : FiniteDimensional k V\nρ : Representation k G V\ninst✝¹ : Fintype G\ninst✝ : Invertible ↑(Nat.card G)\n⊢ Invertible ↑(Fintype.card G)",
"ppTerm": "?m.57",
"assign... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RepresentationTheory.Character | {
"line": 105,
"column": 78
} | {
"line": 105,
"column": 80
} | {
"line": 106,
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} | [
{
"pp": "G : Type u_1\nk : Type u_2\nV : Type u_3\ninst✝⁶ : Group G\ninst✝⁵ : Field k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : FiniteDimensional k V\nρ : Representation k G V\ninst✝¹ : Fintype G\ninst✝ : Invertible ↑(Nat.card G)\n⊢ (↑(Nat.card G))⁻¹ * ∑ g, ρ.character g = ↑(finrank k ↥ρ.invariant... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Character | {
"line": 117,
"column": 41
} | {
"line": 117,
"column": 43
} | {
"line": 118,
"column": 2
} | [
{
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Mathlib.RepresentationTheory.Character | {
"line": 136,
"column": 49
} | {
"line": 136,
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} | {
"line": 137,
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{
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Mathlib.RepresentationTheory.Character | {
"line": 160,
"column": 49
} | {
"line": 160,
"column": 51
} | {
"line": 160,
"column": 52
} | [
{
"pp": "k : Type u\ninst✝¹ : Field k\nG : Type v\ninst✝ : Monoid G\nV : FDRep k G\ng h : G\n⊢ V.character (h * g) = V.character (g * h)",
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"assigned": true,
"usedConstants": [
"LinearMap.trace",
"MonoidHom.instMonoidHomClass",
"MonoidHom.instFunLike",
"Semi... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Character | {
"line": 163,
"column": 73
} | {
"line": 163,
"column": 75
} | {
"line": 164,
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} | [
{
"pp": "k : Type u\ninst✝¹ : Field k\nG : Type v\ninst✝ : Monoid G\nV : FDRep k G\n⊢ V.character 1 = ↑(finrank k ↑V.V)",
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"usedConstants": [
"LinearMap.trace",
"MonoidHom.instMonoidHomClass",
"MulOne.toOne",
"MonoidHom.instFunLike",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Character | {
"line": 168,
"column": 89
} | {
"line": 168,
"column": 91
} | {
"line": 169,
"column": 2
} | [
{
"pp": "k : Type u\ninst✝¹ : Field k\nG : Type v\ninst✝ : Monoid G\nV W : FDRep k G\n⊢ (V ⊗ W).character = V.character * W.character",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"LinearMap.trace",
"Eq.mpr",
"MonoidHom.instFunLike",
"Semiring.toModule",
"HM... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Character | {
"line": 172,
"column": 78
} | {
"line": 172,
"column": 80
} | {
"line": 173,
"column": 2
} | [
{
"pp": "k : Type u\ninst✝¹ : Field k\nG : Type v\ninst✝ : Monoid G\nV W : FDRep k G\ni : V ≅ W\n⊢ V.character = W.character",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"LinearMap.trace",
"Eq.mpr",
"MonoidHom.instFunLike",
"Semiring.toModule",
"MonoidHom",... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Character | {
"line": 185,
"column": 97
} | {
"line": 185,
"column": 99
} | {
"line": 186,
"column": 2
} | [
{
"pp": "k : Type u\ninst✝¹ : Field k\nG : Type v\ninst✝ : Group G\nV : FDRep k G\ng h : G\n⊢ V.character (h * g * h⁻¹) = V.character g",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Character | {
"line": 194,
"column": 75
} | {
"line": 194,
"column": 77
} | {
"line": 195,
"column": 2
} | [
{
"pp": "k : Type u\ninst✝¹ : Field k\nG : Type v\ninst✝ : Group G\nV W : FDRep k G\ng : G\n⊢ (of (linHom V.ρ W.ρ)).character g = V.character g⁻¹ * W.character g",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"FDRep.char_tensor",
"Algebra.to_smulCommClass",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Character | {
"line": 201,
"column": 44
} | {
"line": 201,
"column": 46
} | {
"line": 202,
"column": 4
} | [
{
"pp": "k : Type u\ninst✝³ : Field k\nG : Type v\ninst✝² : Group G\ninst✝¹ : Fintype G\ninst✝ : Invertible ↑(Nat.card G)\nV : FDRep k G\n⊢ Invertible ↑(Fintype.card G)",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddGroupWithOne.toAddMonoidWithO... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RepresentationTheory.Character | {
"line": 200,
"column": 80
} | {
"line": 200,
"column": 82
} | {
"line": 201,
"column": 2
} | [
{
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"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"LinearMap.trace",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Character | {
"line": 213,
"column": 32
} | {
"line": 213,
"column": 34
} | {
"line": 214,
"column": 2
} | [
{
"pp": "k : Type u\ninst✝³ : Field k\nG : Type v\ninst✝² : Group G\ninst✝¹ : Fintype G\ninst✝ : Invertible ↑(Nat.card G)\nV W : FDRep k G\n⊢ (↑(Nat.card G))⁻¹ * ∑ g, W.character g * V.character g⁻¹ = ↑(finrank k (V ⟶ W))",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"NonUnitalNonA... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Character | {
"line": 235,
"column": 45
} | {
"line": 235,
"column": 47
} | {
"line": 236,
"column": 2
} | [
{
"pp": "k : Type u\ninst✝⁶ : Field k\nG : Type v\ninst✝⁵ : Group G\ninst✝⁴ : IsAlgClosed k\ninst✝³ : Fintype G\ninst✝² : Invertible ↑(Nat.card G)\nV W : FDRep k G\ninst✝¹ : Simple V\ninst✝ : Simple W\n⊢ (↑(Nat.card G))⁻¹ * ∑ g, V.character g * W.character g⁻¹ = if Nonempty (V ≅ W) then 1 else 0",
"ppTerm":... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 578,
"column": 80
} | {
"line": 578,
"column": 82
} | {
"line": 579,
"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\n⊢ ∀ᵐ (ω : Ω), Tendsto (fun n ↦ (∑ i ∈ range n, X i ω) /... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 606,
"column": 37
} | {
"line": 606,
"column": 39
} | {
"line": 607,
"column": 6
} | [
{
"pp": "Ω : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\nX : ℕ → Ω → ℝ\nhint : Integrable (X 0) μ\nhindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)\nhident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ\nmΩ : MeasureSpace Ω := { toMeasurableSpace := m, volume := μ }\nh : ∀ᵐ (ω : Ω), X 0 ω = 0\n⊢ ∀ᵐ (ω : Ω), ∀ ... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.StrongLaw | {
"line": 602,
"column": 83
} | {
"line": 602,
"column": 85
} | {
"line": 603,
"column": 2
} | [
{
"pp": "Ω : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\nX : ℕ → Ω → ℝ\nhint : Integrable (X 0) μ\nhindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)\nhident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ\n⊢ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ (∑ i ∈ range n, X i ω) / ↑n) atTop (𝓝 (∫ (x : Ω), X 0 x ∂μ))",
"ppTe... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 662,
"column": 28
} | {
"line": 662,
"column": 30
} | {
"line": 663,
"column": 6
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsProbabilityMeasure μ\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\ninst✝ : MeasurableSpace E\nX : ℕ → Ω → E\nh' : Measurable (X 0)\nhindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)\nhid... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.StrongLaw | {
"line": 671,
"column": 93
} | {
"line": 671,
"column": 95
} | {
"line": 672,
"column": 6
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsProbabilityMeasure μ\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\ninst✝ : MeasurableSpace E\nX : ℕ → Ω → E\nh' : Measurable (X 0)\nhindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)\nhid... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.StrongLaw | {
"line": 650,
"column": 93
} | {
"line": 650,
"column": 95
} | {
"line": 653,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsProbabilityMeasure μ\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\ninst✝ : MeasurableSpace E\nX : ℕ → Ω → E\nh' : Measurable (X 0)\nhindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)\nhid... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 707,
"column": 25
} | {
"line": 707,
"column": 27
} | {
"line": 707,
"column": 28
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\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\nh' : StronglyMeasurable (X 0... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.StrongLaw | {
"line": 720,
"column": 52
} | {
"line": 720,
"column": 54
} | {
"line": 721,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\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\nh' : StronglyMeasurable (X 0... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.StrongLaw | {
"line": 741,
"column": 61
} | {
"line": 741,
"column": 63
} | {
"line": 741,
"column": 64
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\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\nh' : StronglyMeasurable (X 0... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinduced | {
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"column": 22
} | {
"line": 64,
"column": 24
} | {
"line": 64,
"column": 25
} | [
{
"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\na✝ b✝ : H → A\nx✝... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 741,
"column": 76
} | {
"line": 741,
"column": 78
} | {
"line": 741,
"column": 79
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\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\nh' : StronglyMeasurable (X 0... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinduced | {
"line": 65,
"column": 15
} | {
"line": 65,
"column": 17
} | {
"line": 65,
"column": 18
} | [
{
"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\n⊢ 0 ∈ {f | ∀ (g :... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinduced | {
"line": 66,
"column": 21
} | {
"line": 66,
"column": 23
} | {
"line": 66,
"column": 24
} | [
{
"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\nx✝² : k\nx✝¹ : H ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinduced | {
"line": 83,
"column": 71
} | {
"line": 83,
"column": 73
} | {
"line": 84,
"column": 4
} | [
{
"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\... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinduced | {
"line": 85,
"column": 14
} | {
"line": 85,
"column": 16
} | {
"line": 85,
"column": 17
} | [
{
"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\n⊢ (LinearMap.funL... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 744,
"column": 57
} | {
"line": 744,
"column": 59
} | {
"line": 745,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\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\nh' : StronglyMeasurable (X 0... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinduced | {
"line": 86,
"column": 18
} | {
"line": 86,
"column": 20
} | {
"line": 86,
"column": 21
} | [
{
"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\nx✝¹ x✝ : H\n⊢ (Li... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinduced | {
"line": 93,
"column": 66
} | {
"line": 93,
"column": 68
} | {
"line": 94,
"column": 4
} | [
{
"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... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinduced | {
"line": 98,
"column": 23
} | {
"line": 98,
"column": 25
} | {
"line": 98,
"column": 26
} | [
{
"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... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinduced | {
"line": 147,
"column": 54
} | {
"line": 147,
"column": 56
} | {
"line": 147,
"column": 57
} | [
{
"pp": "k : Type u\nG✝ : Type v\nH : Type w\ninst✝⁴ : CommRing k\ninst✝³ : Monoid G✝\ninst✝² : Monoid H\nφ : G✝ →* H\nA : Rep k G✝\nG : Type v'\ninst✝¹ : Group G\nS : Subgroup G\nX Y : Rep k ↥S\nf : X ⟶ Y\ninst✝ : Epi f\ny : ↑((coindFunctor k S.subtype).obj Y)\nthis : Setoid G := QuotientGroup.rightRel S\ns✝ :... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinduced | {
"line": 148,
"column": 55
} | {
"line": 148,
"column": 57
} | {
"line": 148,
"column": 58
} | [
{
"pp": "k : Type u\nG✝ : Type v\nH : Type w\ninst✝⁴ : CommRing k\ninst✝³ : Monoid G✝\ninst✝² : Monoid H\nφ : G✝ →* H\nA : Rep k G✝\nG : Type v'\ninst✝¹ : Group G\nS : Subgroup G\nX Y : Rep k ↥S\nf : X ⟶ Y\ninst✝ : Epi f\ny : ↑((coindFunctor k S.subtype).obj Y)\nthis : Setoid G := QuotientGroup.rightRel S\ns✝ :... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.Probability.StrongLaw | {
"line": 748,
"column": 36
} | {
"line": 748,
"column": 38
} | {
"line": 749,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\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\nh' : StronglyMeasurable (X 0... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RepresentationTheory.Coinduced | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 65
} | {
"line": 150,
"column": 4
} | [
{
"pp": "k : Type u\nG✝ : Type v\nH : Type w\ninst✝⁴ : CommRing k\ninst✝³ : Monoid G✝\ninst✝² : Monoid H\nφ : G✝ →* H\nA : Rep k G✝\nG : Type v'\ninst✝¹ : Group G\nS : Subgroup G\nX Y : Rep k ↥S\nf : X ⟶ Y\ninst✝ : Epi f\ny : ↑((coindFunctor k S.subtype).obj Y)\nthis : Setoid G := QuotientGroup.rightRel S\ns : ... | [
"k : Type u\nG✝ : Type v\nH : Type w\ninst✝⁴ : CommRing k\ninst✝³ : Monoid G✝\ninst✝² : Monoid H\nφ : G✝ →* H\nA : Rep k G✝\nG : Type v'\ninst✝¹ : Group G\nS : Subgroup G\nX Y : Rep k ↥S\nf : X ⟶ Y\ninst✝ : Epi f\ny : ↑((coindFunctor k S.subtype).obj Y)\nthis : Setoid G := QuotientGroup.rightRel S\ns : ↑Y → ↑X\nhs ... | let x (g : G) : X := X.ρ (γ g) (s (y.1 (i (Quotient.mk' g)))) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Probability.StrongLaw | {
"line": 755,
"column": 89
} | {
"line": 755,
"column": 91
} | {
"line": 756,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\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\nh' : StronglyMeasurable (X 0... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RepresentationTheory.Coinduced | {
"line": 140,
"column": 57
} | {
"line": 140,
"column": 59
} | {
"line": 141,
"column": 4
} | [
{
"pp": "k : Type u\nG✝ : Type v\nH : Type w\ninst✝⁴ : CommRing k\ninst✝³ : Monoid G✝\ninst✝² : Monoid H\nφ : G✝ →* H\nA : Rep k G✝\nG : Type v'\ninst✝¹ : Group G\nS : Subgroup G\nX Y : Rep k ↥S\nf : X ⟶ Y\ninst✝ : Epi f\ny : ↑((coindFunctor k S.subtype).obj Y)\n⊢ ∃ a, (Hom.hom ((coindFunctor k S.subtype).map f... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinduced | {
"line": 173,
"column": 14
} | {
"line": 173,
"column": 16
} | {
"line": 174,
"column": 4
} | [
{
"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\n⊢ { toFun := fun f ↦ (resFunctor φ).map (↑(leftRegular k H).leftRegularHomEquiv.symm (MonoidAlgebra.single 1 1)) ≫ f,\n map_add' := ⋯, map_smul' := ⋯ } =\n 1",
"ppTerm": "?... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinduced | {
"line": 176,
"column": 18
} | {
"line": 176,
"column": 20
} | {
"line": 177,
"column": 4
} | [
{
"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\nx✝¹ x✝ : H\n⊢ {\n toFun := fun f ↦\n (resFunctor φ).map (↑(leftRegular k H).leftRegularHomEquiv.symm (MonoidAlgebra.single (x✝¹ * x✝) 1)) ≫ f,\n map_add' := ⋯, map_smu... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinduced | {
"line": 191,
"column": 17
} | {
"line": 191,
"column": 19
} | {
"line": 191,
"column": 20
} | [
{
"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)\n⊢ Hom.hom f = Hom.hom g",
"ppTerm": "... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinduced | {
"line": 198,
"column": 25
} | {
"line": 198,
"column": 27
} | {
"line": 198,
"column": 28
} | [
{
"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\nA B : Rep k G\nf : A ⟶ B\nh : H\n⊢ Linear.rightComp k (res φ (leftRegular k H)) f ∘ₗ (Representation.coind' φ A) h =\n (Representation.coind' φ B) h ∘ₗ Linear.rightComp k (res φ (l... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinduced | {
"line": 218,
"column": 12
} | {
"line": 218,
"column": 14
} | {
"line": 218,
"column": 15
} | [
{
"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 : ↥(Representation.coindV φ A.ρ)\ng : G\n⊢ (linearCombination k ↑f ∘ₗ ↑(MonoidAlgebra.coeffLinearEquiv k)) ∘ₗ (MonoidHom.comp (leftRegular k H).ρ φ) g =\n A.ρ g ∘ₗ linearCombinati... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinduced | {
"line": 219,
"column": 34
} | {
"line": 219,
"column": 36
} | {
"line": 219,
"column": 37
} | [
{
"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\nx✝¹ x✝ : ↥(Representation.coindV φ A.ρ)\n⊢ ∀ (h : H),\n (Hom.hom\n (ofHom\n { toLinearMap := linearCombination k ↑(x✝¹ + x✝) ∘ₗ ↑(MonoidAlgebra.coeffLinearEq... | [] | by | [anonymous] | by |
Mathlib.Probability.StrongLaw | {
"line": 764,
"column": 89
} | {
"line": 764,
"column": 91
} | {
"line": 765,
"column": 6
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\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\nh' : StronglyMeasurable (X 0... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinduced | {
"line": 220,
"column": 35
} | {
"line": 220,
"column": 37
} | {
"line": 220,
"column": 38
} | [
{
"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\nx✝¹ : k\nx✝ : ↥(Representation.coindV φ A.ρ)\n⊢ ∀ (h : H),\n (Hom.hom\n (ofHom\n { toLinearMap := linearCombination k ↑(x✝¹ • x✝) ∘ₗ ↑(MonoidAlgebra.coeffLin... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 109,
"column": 67
} | {
"line": 109,
"column": 69
} | {
"line": 110,
"column": 4
} | [
{
"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 : ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 131,
"column": 31
} | {
"line": 131,
"column": 33
} | {
"line": 131,
"column": 34
} | [
{
"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 : ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinduced | {
"line": 223,
"column": 12
} | {
"line": 223,
"column": 14
} | {
"line": 223,
"column": 15
} | [
{
"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": true,
"usedConstants": [
"Rep.V",
"Representation",
"MonoidHom.i... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 142,
"column": 91
} | {
"line": 142,
"column": 93
} | {
"line": 143,
"column": 2
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝³ : CommRing k\ninst✝² : Monoid G\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nρ : Representation k G V\n⊢ map ρ ρ (IntertwiningMap.id ρ) = LinearMap.id",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"Lin... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 162,
"column": 73
} | {
"line": 162,
"column": 75
} | {
"line": 162,
"column": 76
} | [
{
"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 : (... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinduced | {
"line": 221,
"column": 66
} | {
"line": 221,
"column": 68
} | {
"line": 222,
"column": 4
} | [
{
"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⊢ (fun h ↦ (Hom.hom f).toLinearMap (MonoidAlgebra.single h 1)) (φ g * h) =\n (A.ρ g) ((fun h ↦ (Hom.hom f).toLinearMap (MonoidAlgebra.... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinduced | {
"line": 225,
"column": 16
} | {
"line": 225,
"column": 18
} | {
"line": 225,
"column": 19
} | [
{
"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\nx : ↥(Representation.coindV φ A.ρ)\n⊢ (fun f ↦ ⟨fun h ↦ (Hom.hom f).toLinearMap (MonoidAlgebra.single h 1), ⋯⟩)\n ((fun f ↦\n ofHom { toLinearMap := linearCombination k ↑... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 160,
"column": 44
} | {
"line": 160,
"column": 46
} | {
"line": 161,
"column": 4
} | [
{
"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 : (... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinduced | {
"line": 226,
"column": 39
} | {
"line": 226,
"column": 41
} | {
"line": 226,
"column": 42
} | [
{
"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\nx : res φ (leftRegular k H) ⟶ A\nx✝ : H\n⊢ (Hom.hom\n ((fun f ↦\n ofHom\n { toLinearMap := linearCombination k ↑f ∘ₗ ↑(MonoidAlgebra.coeffLinearEqu... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 186,
"column": 11
} | {
"line": 186,
"column": 13
} | {
"line": 187,
"column": 4
} | [
{
"pp": "k✝ : Type u_1\nG✝ : Type u_2\nV✝ : Type u_3\nW : Type u_4\nX : Type u_5\ninst✝¹² : CommRing k✝\ninst✝¹¹ : Monoid G✝\ninst✝¹⁰ : AddCommGroup V✝\ninst✝⁹ : Module k✝ V✝\ninst✝⁸ : AddCommGroup W\ninst✝⁷ : Module k✝ W\ninst✝⁶ : AddCommGroup X\ninst✝⁵ : Module k✝ X\nρ✝ : Representation k✝ G✝ V✝\nτ : Represen... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 210,
"column": 75
} | {
"line": 210,
"column": 77
} | {
"line": 210,
"column": 78
} | [
{
"pp": "k✝ : Type u_1\nG✝ : Type u_2\nV✝ : Type u_3\nW : Type u_4\nX : Type u_5\ninst✝¹¹ : CommRing k✝\ninst✝¹⁰ : Monoid G✝\ninst✝⁹ : AddCommGroup V✝\ninst✝⁸ : Module k✝ V✝\ninst✝⁷ : AddCommGroup W\ninst✝⁶ : Module k✝ W\ninst✝⁵ : AddCommGroup X\ninst✝⁴ : Module k✝ X\nρ✝ : Representation k✝ G✝ V✝\nτ : Represent... | [] | by | [anonymous] | by |
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