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.RepresentationTheory.Rep.Basic | {
"line": 791,
"column": 66
} | {
"line": 791,
"column": 68
} | {
"line": 792,
"column": 4
} | [
{
"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... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 821,
"column": 28
} | {
"line": 821,
"column": 30
} | {
"line": 822,
"column": 2
} | [
{
"pp": "k : Type u\ninst✝¹ : CommRing k\nG : Type v\ninst✝ : Group G\nA B : Rep k G\n⊢ (Hom.hom ((ihom.ev A).app B)).toLinearMap =\n (TensorProduct.uncurry (RingHom.id k) (↑A) (↑A →ₗ[k] ↑B) ↑B) LinearMap.id.flip",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
"LinearMap.id",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 856,
"column": 68
} | {
"line": 856,
"column": 70
} | {
"line": 857,
"column": 2
} | [
{
"pp": "k : Type u\ninst✝¹ : CommRing k\nG : Type v\ninst✝ : Group G\nA B C : Rep k G\nf : B ⟶ (ihom A).obj C\n⊢ (Hom.hom ((linearHomEquiv A B C).symm f)).toLinearMap =\n (TensorProduct.uncurry (RingHom.id k) ↑A ↑B ↑C) (Hom.hom f).flip",
"ppTerm": "?m.113",
"assigned": true,
"usedConstants": [
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 874,
"column": 12
} | {
"line": 874,
"column": 14
} | {
"line": 874,
"column": 15
} | [
{
"pp": "k✝ : Type u\nG✝ : Type v\ninst✝⁴ : CommRing k✝\ninst✝³ : Monoid G✝\nk : Type u\ninst✝² : Semiring k\nG : Type v\ninst✝¹ : Group G\ninst✝ : Fintype G\nA : Rep k G\ng : G\n⊢ A.ρ.norm ∘ₗ A.ρ g = A.ρ g ∘ₗ A.ρ.norm",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Rep.V",
"R... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 880,
"column": 73
} | {
"line": 880,
"column": 75
} | {
"line": 881,
"column": 2
} | [
{
"pp": "k : Type u\ninst✝² : Semiring k\nG : Type v\ninst✝¹ : Group G\ninst✝ : Fintype G\nA B : Rep k G\nf : A ⟶ B\n⊢ f ≫ B.norm = A.norm ≫ f",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Rep.V",
"Representation.IntertwiningMap.instLinearMapClass",
"Representation",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 928,
"column": 94
} | {
"line": 928,
"column": 96
} | {
"line": 929,
"column": 2
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : CommRing k\ninst✝ : Monoid G\nα : Type u'\nA : Rep k G\nf g : free k G α ⟶ A\nh : ∀ (i : α), (Hom.hom f) (single i (MonoidAlgebra.single 1 1)) = (Hom.hom g) (single i (MonoidAlgebra.single 1 1))\n⊢ f = g",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants"... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 973,
"column": 25
} | {
"line": 973,
"column": 27
} | {
"line": 973,
"column": 28
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : CommRing k\ninst✝ : Monoid G\nX✝ Y✝ Z : Action (Type u) G\nf : X✝ ⟶ Y✝\n⊢ (linearization k G).obj Z ◁ (linearization k G).map f ≫ ofHom (μ Z Y✝) =\n ofHom (μ Z X✝) ≫ (linearization k G).map (Z ◁ f)",
"ppTerm": "?m.100",
"assigned": true,
"usedConstants": ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 974,
"column": 25
} | {
"line": 974,
"column": 27
} | {
"line": 974,
"column": 28
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : CommRing k\ninst✝ : Monoid G\nX Y Z : Action (Type u) G\n⊢ ofHom (μ X Y) ▷ (linearization k G).obj Z ≫ ofHom (μ (X ⊗ Y) Z) ≫ (linearization k G).map (α_ X Y Z).hom =\n (α_ ((linearization k G).obj X) ((linearization k G).obj Y) ((linearization k G).obj Z)).hom ≫\n ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 984,
"column": 42
} | {
"line": 984,
"column": 44
} | {
"line": 984,
"column": 45
} | [
{
"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 ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 1048,
"column": 68
} | {
"line": 1048,
"column": 70
} | {
"line": 1049,
"column": 2
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : CommRing k\ninst✝ : Monoid G\nA : Rep k G\nx : ↑A\ng : G\n⊢ (Hom.hom (A.leftRegularHomEquiv.symm x)) (MonoidAlgebra.single g 1) = (A.ρ g) x",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Rep.V",
"LinearEquiv.symm",
"instHSMul",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.ContCohomology.Basic | {
"line": 76,
"column": 69
} | {
"line": 76,
"column": 71
} | {
"line": 77,
"column": 2
} | [
{
"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\nn : ℕ\n⊢ X.d n ≫ X.d (n + 1) = 0",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"ContinuousLinearMap.comp",... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.ContCohomology.Basic | {
"line": 119,
"column": 51
} | {
"line": 119,
"column": 53
} | {
"line": 120,
"column": 2
} | [
{
"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\ni : ℕ\n⊢ X.homogeneousCochains.d i (i + 1) = (invariantsFunctor k G).map (X.d (i + 1))",
"ppTerm": "?m.47",
"assigned": true,
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.ContCohomology.Basic | {
"line": 125,
"column": 73
} | {
"line": 125,
"column": 75
} | {
"line": 126,
"column": 2
} | [
{
"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\ni : ℕ\nσ : ↑(X.homogeneousCochains.X i).toModuleCat\n⊢ ↑((TopModuleCat.Hom.hom (X.homogeneousCochains.d i (i + 1))) σ) = (Hom.hom (X.d (... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 46,
"column": 19
} | {
"line": 46,
"column": 21
} | {
"line": 46,
"column": 22
} | [
{
"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\ng g₁ : G\nh : (QuotientGroup.rightRel S) g₁ g\n⊢ g₁ * g⁻¹ ∈ S",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"mul_inv_can... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 52,
"column": 33
} | {
"line": 52,
"column": 35
} | {
"line": 53,
"column": 2
} | [
{
"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\ng : G\na : ↑A\n⊢ (A.indToCoindAux g) a g = a",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 58,
"column": 34
} | {
"line": 58,
"column": 36
} | {
"line": 59,
"column": 2
} | [
{
"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\ng g₁ : G\na : ↑A\nh : ¬(QuotientGroup.rightRel S) g₁ g\n⊢ (A.indToCoindAux g) a g₁ = 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstan... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 69,
"column": 42
} | {
"line": 69,
"column": 44
} | {
"line": 69,
"column": 45
} | [
{
"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\ng g₁ : G\na : ↑A\ns : ↥S\nh : ¬(QuotientGroup.rightRel S) g₁ g\nx✝ : (QuotientGroup.rightRel S) (↑s * g₁) g\ns₁ : ↥S\nhs₁ : (fun m ↦ m • g) s₁ = ↑s * ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 63,
"column": 69
} | {
"line": 63,
"column": 71
} | {
"line": 64,
"column": 2
} | [
{
"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\ng g₁ : G\na : ↑A\ns : ↥S\n⊢ (A.indToCoindAux g) a (↑s * g₁) = (A.ρ s) ((A.indToCoindAux g) a g₁)",
"ppTerm": "?m.36",
"assigned": true,
"u... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 76,
"column": 27
} | {
"line": 76,
"column": 29
} | {
"line": 76,
"column": 30
} | [
{
"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 : G\na : ↑A\ns s₁ : ↥S\n⊢ (fun m ↦ m • (↑s * g₁)) (s₁ * s⁻¹) = s₁ • g₁",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 81,
"column": 40
} | {
"line": 81,
"column": 42
} | {
"line": 81,
"column": 43
} | [
{
"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 : G\na : ↑A\ns : ↥S\nh : ¬(QuotientGroup.rightRel S) g₂ g₁\nx✝ : (QuotientGroup.rightRel S) g₂ (↑s * g₁)\ns₁ : ↥S\nhs₁ : (fun m ↦ m • (↑s * g₁))... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 73,
"column": 72
} | {
"line": 73,
"column": 74
} | {
"line": 74,
"column": 2
} | [
{
"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 : G\na : ↑A\ns : ↥S\n⊢ (A.indToCoindAux (↑s * g₁)) ((A.ρ s) a) g₂ = (A.indToCoindAux g₁) a g₂",
"ppTerm": "?m.36",
"assigned": true,
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 89,
"column": 33
} | {
"line": 89,
"column": 35
} | {
"line": 89,
"column": 36
} | [
{
"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\nh : ¬(QuotientGroup.rightRel S) (g₂ * g₃⁻¹) g₁\nx✝ : (QuotientGroup.rightRel S) g₂ (g₁ * g₃)\ns : ↥S\nhs : (fun m ↦ m • (g₁ * g₃... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 85,
"column": 73
} | {
"line": 85,
"column": 75
} | {
"line": 86,
"column": 2
} | [
{
"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) a (g₂ * g₃⁻¹) = (A.indToCoindAux (g₁ * g₃)) a g₂",
"ppTerm": "?m.35",
"assigned": true,
"used... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 93,
"column": 73
} | {
"line": 93,
"column": 75
} | {
"line": 94,
"column": 2
} | [
{
"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": true,
"used... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 97,
"column": 73
} | {
"line": 97,
"column": 75
} | {
"line": 98,
"column": 2
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝² : CommRing k\ninst✝¹ : Group G\nS : Subgroup G\ninst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)\nA B : Rep.{u_1, u, v} k ↥S\nf : A ⟶ B\ng₁ g₂ : G\na : ↑A\n⊢ (B.indToCoindAux g₁) ((Hom.hom f) a) g₂ = (Hom.hom f) ((A.indToCoindAux g₁) a g₂)",
"ppTerm": "?m.45",
"as... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 109,
"column": 61
} | {
"line": 109,
"column": 63
} | {
"line": 109,
"column": 64
} | [
{
"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\ng : G\nx✝² : ↑A\nx✝¹ : ↥S\nx✝ : G\n⊢ (A.indToCoindAux g) x✝² (S.subtype x✝¹ * x✝) = (A.ρ x✝¹) ((A.indToCoindAux g) x✝² x✝)",
"ppTerm": "?m.134",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 110,
"column": 61
} | {
"line": 110,
"column": 63
} | {
"line": 110,
"column": 64
} | [
{
"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\nx✝ : ↥S\n⊢ TensorProduct.lift\n ((linearCombination k fun g ↦ LinearMap.codRestrict (coindV S.subtype A.ρ) (A.indToCoindAux g) ⋯) ∘ₗ\n ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 125,
"column": 41
} | {
"line": 125,
"column": 43
} | {
"line": 125,
"column": 44
} | [
{
"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\nf : ↑(coind.{u, v, v, w} S.subtype A)\ng : Quotient (QuotientGroup.rightRel S)\ng₁ g₂ : G\nx✝ : g₁ ≈ g₂\ns : ↥S\nhs : ↑s * g₂ ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 127,
"column": 70
} | {
"line": 127,
"column": 72
} | {
"line": 127,
"column": 73
} | [
{
"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✝³ x✝² : ↑(coind.{u, v, v, w} S.subtype A)\nz : Quotient (QuotientGroup.rightRel S)\nx✝¹ : z ∈ Finset.univ\nx✝ : G\n⊢ ⟦x✝⟧.li... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 126,
"column": 18
} | {
"line": 126,
"column": 20
} | {
"line": 126,
"column": 21
} | [
{
"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, ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 129,
"column": 36
} | {
"line": 129,
"column": 38
} | {
"line": 129,
"column": 39
} | [
{
"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✝³ : k\nx✝² : ↑(coind.{u, v, v, w} S.subtype A)\nz : Quotient (QuotientGroup.rightRel S)\nx✝¹ : z ∈ Finset.univ\nx✝ : G\n⊢ ⟦x... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 128,
"column": 19
} | {
"line": 128,
"column": 21
} | {
"line": 128,
"column": 22
} | [
{
"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 ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 139,
"column": 24
} | {
"line": 139,
"column": 26
} | {
"line": 140,
"column": 6
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝² : CommRing k\ninst✝¹ : Group G\nS : Subgroup G\nA : Rep.{w, u, v} k ↥S\ninst✝ : S.FiniteIndex\ng : G\nf : ↑(coind.{u, v, v, w} S.subtype A)\nhx : Function.support ↑f ⊆ MulAction.orbit (↥S) g\nb : G\na✝ : ⟦b⟧ ∈ Finset.univ\nhb : ⟦b⟧ ≠ ⟦g⟧\n⊢ ↑f b = 0",
"ppTerm": "?m.97... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 134,
"column": 54
} | {
"line": 134,
"column": 56
} | {
"line": 135,
"column": 2
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝² : CommRing k\ninst✝¹ : Group G\nS : Subgroup G\nA : Rep.{w, u, v} k ↥S\ninst✝ : S.FiniteIndex\ng : G\nf : ↑(coind.{u, v, v, w} S.subtype A)\nhx : Function.support ↑f ⊆ MulAction.orbit (↥S) g\n⊢ A.coindToInd f = (IndV.mk S.subtype A.ρ g) (↑f g)",
"ppTerm": "?m.52",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality | {
"line": 65,
"column": 78
} | {
"line": 65,
"column": 80
} | {
"line": 66,
"column": 2
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝⁴ : Ring k\ninst✝³ : TopologicalSpace k\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nX : TopRep k G\ni : ℕ\n⊢ resolutionMap (ContinuousMonoidHom.id G) (𝟙 X) i = 𝟙 (X.resolutionX i)",
"ppTerm": "?m.46",
"assigned": true,
"usedCo... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality | {
"line": 76,
"column": 83
} | {
"line": 76,
"column": 85
} | {
"line": 77,
"column": 2
} | [
{
"pp": "k : Type u\nG H K : Type v\ninst✝¹⁰ : Ring k\ninst✝⁹ : TopologicalSpace k\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : Group H\ninst✝⁴ : TopologicalSpace H\ninst✝³ : IsTopologicalGroup H\ninst✝² : Group K\ninst✝¹ : TopologicalSpace K\ninst✝ : IsTopologicalGrou... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality | {
"line": 87,
"column": 75
} | {
"line": 87,
"column": 77
} | {
"line": 88,
"column": 2
} | [
{
"pp": "k : Type u\nG H : Type v\ninst✝⁷ : Ring k\ninst✝⁶ : TopologicalSpace k\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalSpace G\ninst✝³ : IsTopologicalGroup G\ninst✝² : Group H\ninst✝¹ : TopologicalSpace H\ninst✝ : IsTopologicalGroup H\nX : TopRep k G\nY : TopRep k H\nφ : H →ₜ* G\nf : res (↑φ) X ⟶ Y\ni : ℕ\n⊢ re... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality | {
"line": 106,
"column": 29
} | {
"line": 106,
"column": 31
} | {
"line": 107,
"column": 4
} | [
{
"pp": "k : Type u\nG H K : Type v\ninst✝¹⁰ : Ring k\ninst✝⁹ : TopologicalSpace k\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : Group H\ninst✝⁴ : TopologicalSpace H\ninst✝³ : IsTopologicalGroup H\ninst✝² : Group K\ninst✝¹ : TopologicalSpace K\ninst✝ : IsTopologicalGrou... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality | {
"line": 114,
"column": 80
} | {
"line": 114,
"column": 82
} | {
"line": 115,
"column": 2
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝⁴ : Ring k\ninst✝³ : TopologicalSpace k\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nX : TopRep k G\n⊢ cochainsMap (ContinuousMonoidHom.id G) (𝟙 X) = 𝟙 X.homogeneousCochains",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality | {
"line": 123,
"column": 43
} | {
"line": 123,
"column": 45
} | {
"line": 124,
"column": 2
} | [
{
"pp": "k : Type u\nG H K : Type v\ninst✝¹⁰ : Ring k\ninst✝⁹ : TopologicalSpace k\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : Group H\ninst✝⁴ : TopologicalSpace H\ninst✝³ : IsTopologicalGroup H\ninst✝² : Group K\ninst✝¹ : TopologicalSpace K\ninst✝ : IsTopologicalGrou... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality | {
"line": 135,
"column": 60
} | {
"line": 135,
"column": 62
} | {
"line": 136,
"column": 2
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝⁴ : Ring k\ninst✝³ : TopologicalSpace k\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nX : TopRep k G\nn : ℕ\n⊢ cocyclesMap (ContinuousMonoidHom.id G) (𝟙 X) n = 𝟙 (cocycles X n)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstan... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality | {
"line": 142,
"column": 47
} | {
"line": 142,
"column": 49
} | {
"line": 143,
"column": 2
} | [
{
"pp": "k : Type u\nG H K : Type v\ninst✝¹⁰ : Ring k\ninst✝⁹ : TopologicalSpace k\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : Group H\ninst✝⁴ : TopologicalSpace H\ninst✝³ : IsTopologicalGroup H\ninst✝² : Group K\ninst✝¹ : TopologicalSpace K\ninst✝ : IsTopologicalGrou... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality | {
"line": 154,
"column": 53
} | {
"line": 154,
"column": 55
} | {
"line": 155,
"column": 2
} | [
{
"pp": "k : Type u\nG H : Type v\ninst✝⁷ : Ring k\ninst✝⁶ : TopologicalSpace k\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalSpace G\ninst✝³ : IsTopologicalGroup G\ninst✝² : Group H\ninst✝¹ : TopologicalSpace H\ninst✝ : IsTopologicalGroup H\nX : TopRep k G\nY : TopRep k H\nφ : H →ₜ* G\nf : res (↑φ) X ⟶ Y\nn : ℕ\n⊢ π ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality | {
"line": 159,
"column": 52
} | {
"line": 159,
"column": 54
} | {
"line": 160,
"column": 2
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝⁴ : Ring k\ninst✝³ : TopologicalSpace k\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nX : TopRep k G\nn : ℕ\n⊢ map (ContinuousMonoidHom.id G) (𝟙 X) n = 𝟙 (continuousCohomology n X)",
"ppTerm": "?m.40",
"assigned": true,
"usedCon... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality | {
"line": 164,
"column": 95
} | {
"line": 164,
"column": 97
} | {
"line": 165,
"column": 2
} | [
{
"pp": "k : Type u\nG H K : Type v\ninst✝¹⁰ : Ring k\ninst✝⁹ : TopologicalSpace k\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : Group H\ninst✝⁴ : TopologicalSpace H\ninst✝³ : IsTopologicalGroup H\ninst✝² : Group K\ninst✝¹ : TopologicalSpace K\ninst✝ : IsTopologicalGrou... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 59,
"column": 41
} | {
"line": 59,
"column": 43
} | {
"line": 59,
"column": 44
} | [
{
"pp": "k : 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\n⊢ (ρ g - LinearMap.id) 0 = (fun gv ↦ (ρ gv.1) gv.2 - gv.2) ((fun x ↦ g ^ x) 0,... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 149,
"column": 77
} | {
"line": 149,
"column": 79
} | {
"line": 150,
"column": 2
} | [
{
"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\n⊢ A.indToCoind ∘ₗ A.coindToInd = LinearMap.id",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Lin... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 55,
"column": 65
} | {
"line": 55,
"column": 67
} | {
"line": 56,
"column": 2
} | [
{
"pp": "k : 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\n⊢ Coinvariants.ker ρ = (ρ g - LinearMap.id).range",
"ppTerm": "?m.45",
"assig... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 81,
"column": 66
} | {
"line": 81,
"column": 68
} | {
"line": 81,
"column": 69
} | [
{
"pp": "k : 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⊢ ∀ (i : G), i ∈ Finset.univ ↔ g⁻¹ * i ∈ Finset.univ",
"ppTerm": "?m.140",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Finset.univ",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 81,
"column": 76
} | {
"line": 81,
"column": 78
} | {
"line": 81,
"column": 79
} | [
{
"pp": "k : 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⊢ ∀ i ∈ Finset.univ, y.coeff (g⁻¹ * i) = y.coeff (g⁻¹ * i)",
"ppTerm": "?m.141",
"assigned": true,
"usedConstants": [
"_private.Mathlib.RepresentationTh... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 82,
"column": 69
} | {
"line": 82,
"column": 71
} | {
"line": 83,
"column": 6
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Finite G\nthis : Fintype G\nx : MonoidAlgebra k G\nhx : x ∈ ((linearCombination k fun x ↦ 1) ∘ₗ ↑(MonoidAlgebra.coeffLinearEquiv k)).ker\n⊢ x = x.coeff.sum fun g r ↦ MonoidAlgebra.single g r - MonoidAlgebra.single 1 r",
"ppT... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 17
} | {
"line": 170,
"column": 4
} | [
{
"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 :\n x ∈\n Function.support\n ↑(A.indToCoind\n ((Coinvariants.mk (tprod (Mo... | [
"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 ((Coinvariant... | contrapose hx | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose_1 | Mathlib.Tactic.Contrapose.contrapose |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 89,
"column": 62
} | {
"line": 89,
"column": 64
} | {
"line": 89,
"column": 65
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Finite G\nthis✝ : Fintype G\nx : MonoidAlgebra k G\nhx : x ∈ ((linearCombination k fun x ↦ 1) ∘ₗ ↑(MonoidAlgebra.coeffLinearEquiv k)).ker\nthis : x = x.coeff.sum fun g r ↦ MonoidAlgebra.single g r - MonoidAlgebra.single 1 r\ng :... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 162,
"column": 77
} | {
"line": 162,
"column": 79
} | {
"line": 163,
"column": 2
} | [
{
"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\n⊢ A.coindToInd ∘ₗ A.indToCoind = LinearMap.id",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Lin... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 76,
"column": 59
} | {
"line": 76,
"column": 61
} | {
"line": 77,
"column": 2
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Finite G\n⊢ Coinvariants.ker (leftRegular k G) = ((linearCombination k fun x ↦ 1) ∘ₗ ↑(MonoidAlgebra.coeffLinearEquiv k)).ker",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Finsupp.... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 104,
"column": 31
} | {
"line": 104,
"column": 33
} | {
"line": 104,
"column": 34
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nx✝¹ : ↑(leftRegular k G)\nx✝ : x✝¹ ∈ (Hom.hom (leftRegular k G).norm).range\nw✝ : ↑(leftRegular k G)\nh : (Hom.hom (leftRegular k G).norm).toLinearMap w✝ = x✝¹\n⊢ x✝¹ ∈ (Hom.hom ((... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 108,
"column": 7
} | {
"line": 108,
"column": 9
} | {
"line": 108,
"column": 10
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nx : ↑(leftRegular k G)\nhx : x ∈ (Hom.hom ((leftRegular k G).applyAsHom g - 𝟙 (leftRegular k G))).ker\nγ : G\n⊢ ((Representation.leftRegular k G) g) x = x",
"ppTerm": "?m.152"... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 181,
"column": 68
} | {
"line": 181,
"column": 70
} | {
"line": 181,
"column": 71
} | [
{
"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\nA : Rep.{max w u, u, v} k ↥S\ng : G\n⊢ ↑(LinearEquiv.ofLinearMap A.indToCoind A.coindToInd ⋯ ⋯) ∘ₗ (Representation.ind S.subt... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 105,
"column": 40
} | {
"line": 105,
"column": 42
} | {
"line": 106,
"column": 4
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nx : ↑(leftRegular k G)\nhx : x ∈ (Hom.hom ((leftRegular k G).applyAsHom g - 𝟙 (leftRegular k G))).ker\n⊢ (Hom.hom (leftRegular k G).norm).toLinearMap (MonoidAlgebra.single 1 (x.co... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 115,
"column": 59
} | {
"line": 115,
"column": 61
} | {
"line": 116,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ng : G\ninst✝ : Finite G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\n⊢ (Hom.hom ((leftRegular k G).applyAsHom g - 𝟙 (leftRegular k G))).range =\n ((linearCombination k fun x ↦ 1) ∘ₗ ↑(MonoidAlgebra.coeffLinearEquiv k)).ker",
"ppTerm": "?m.116... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 122,
"column": 62
} | {
"line": 122,
"column": 64
} | {
"line": 123,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\n⊢ (Hom.hom ((leftRegular k G).applyAsHom g - 𝟙 (leftRegular k G))).range = (Hom.hom (leftRegular k G).norm).ker",
"ppTerm": "?m.92",
"assigned": true,
"usedConstants":... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 138,
"column": 5
} | {
"line": 138,
"column": 7
} | {
"line": 138,
"column": 8
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\nA✝ : Rep k G\ng : G\nA : Rep k G\n⊢ A.norm ≫ (A.applyAsHom g - 𝟙 A) = 0",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Module.End.instRing",
"Representation.IntertwiningMap.instSub",
"... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 191,
"column": 68
} | {
"line": 191,
"column": 70
} | {
"line": 192,
"column": 4
} | [
{
"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✝ Y✝ : Rep.{max u w, u, v} k ↥S\nf : X✝ ⟶ Y✝\n⊢ (indFunctor k S.subtype).map f ≫ Y✝.indCoindIso.hom = X✝.indCoindIso.hom ≫ (c... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 138,
"column": 48
} | {
"line": 138,
"column": 50
} | {
"line": 138,
"column": 51
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\nA✝ : Rep k G\ng : G\nA : Rep k G\n⊢ (A.applyAsHom g - 𝟙 A) ≫ A.norm = 0",
"ppTerm": "?m.106",
"assigned": true,
"usedConstants": [
"Module.End.instRing",
"Representation.IntertwiningMap.instSub",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 225,
"column": 88
} | {
"line": 225,
"column": 90
} | {
"line": 226,
"column": 2
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝³ : CommRing k\ninst✝² : Group G\nS : Subgroup G\ninst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)\ninst✝ : S.FiniteIndex\nA : Rep.{max w u v, u, v} k ↥S\nB : Rep.{max w u v, u, v} k G\nf : res S.subtype B ⟶ A\n⊢ ((resIndAdjunction k S).homEquiv B A) f = (resCoindHomEquiv.... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 142,
"column": 13
} | {
"line": 142,
"column": 15
} | {
"line": 143,
"column": 6
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\nA : Rep k G\ng : G\nX✝ Y✝ : Rep k G\nf : X✝ ⟶ Y✝\n⊢ ∀ (i j : ℕ),\n (ComplexShape.down ℕ).Rel i j →\n f ≫ (HomologicalComplex.alternatingConst Y✝ ⋯ ⋯ ⋯).d i j =\n (HomologicalComplex.alternatingConst X✝ ⋯ ⋯ ⋯).d i... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 158,
"column": 5
} | {
"line": 158,
"column": 7
} | {
"line": 158,
"column": 8
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\nA : Rep k G\ng : G\n⊢ ModuleCat.ofHom (Hom.hom A.norm).toLinearMap ≫ ModuleCat.ofHom (Hom.hom (A.applyAsHom g - 𝟙 A)).toLinearMap = 0",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"LinearMap.id",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 165,
"column": 46
} | {
"line": 165,
"column": 48
} | {
"line": 165,
"column": 49
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\nA : Rep k G\ng : G\n⊢ ModuleCat.ofHom (Hom.hom (A.applyAsHom g - 𝟙 A)).toLinearMap ≫ ModuleCat.ofHom (Hom.hom A.norm).toLinearMap = 0",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"LinearMap.id",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 260,
"column": 68
} | {
"line": 260,
"column": 70
} | {
"line": 261,
"column": 2
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝³ : CommRing k\ninst✝² : Group G\nS : Subgroup G\ninst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)\ninst✝ : S.FiniteIndex\nA : Rep.{max w u v, u, v} k ↥S\n⊢ (coindResAdjunction k S).unit.app A =\n (indResAdjunction k S.subtype).unit.app A ≫ (resFunctor S.subtype).map A.... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 174,
"column": 5
} | {
"line": 174,
"column": 7
} | {
"line": 174,
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} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\nA : Rep k G\ng : G\n⊢ ModuleCat.ofHom (Hom.hom A.norm).toLinearMap ≫ ModuleCat.ofHom (Hom.hom (A.applyAsHom g - 𝟙 A)).toLinearMap = 0",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
"LinearMap.id",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 267,
"column": 76
} | {
"line": 267,
"column": 78
} | {
"line": 268,
"column": 2
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝³ : CommRing k\ninst✝² : Group G\nS : Subgroup G\ninst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)\ninst✝ : S.FiniteIndex\nA : Rep.{max w u v, u, v} k ↥S\nB : Rep.{max (max u v) w, u, v} k G\nf : coind.{u, v, v, max (max u v) w} S.subtype A ⟶ B\n⊢ ((coindResAdjunction k S)... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.FiniteIndex | {
"line": 273,
"column": 81
} | {
"line": 273,
"column": 83
} | {
"line": 274,
"column": 2
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝³ : CommRing k\ninst✝² : Group G\nS : Subgroup G\ninst✝¹ : DecidableRel ⇑(QuotientGroup.rightRel S)\ninst✝ : S.FiniteIndex\nA : Rep.{max w u v, u, v} k ↥S\nB : Rep.{max (max u v) w, u, v} k G\nf : A ⟶ res S.subtype B\n⊢ ((coindResAdjunction k S).homEquiv A B).symm f = A.ind... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 174,
"column": 48
} | {
"line": 174,
"column": 50
} | {
"line": 174,
"column": 51
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\nA : Rep k G\ng : G\n⊢ ModuleCat.ofHom (Hom.hom (A.applyAsHom g - 𝟙 A)).toLinearMap ≫ ModuleCat.ofHom (Hom.hom A.norm).toLinearMap = 0",
"ppTerm": "?m.117",
"assigned": true,
"usedConstants": [
"LinearMap.id",... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 184,
"column": 5
} | {
"line": 184,
"column": 7
} | {
"line": 184,
"column": 8
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\nA : Rep k G\ng : G\n⊢ ModuleCat.ofHom (Hom.hom (A.applyAsHom g - 𝟙 A)).toLinearMap ≫ ModuleCat.ofHom (Hom.hom A.norm).toLinearMap = 0",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
"LinearMap.id",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 184,
"column": 48
} | {
"line": 184,
"column": 50
} | {
"line": 184,
"column": 51
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\nA : Rep k G\ng : G\n⊢ ModuleCat.ofHom (Hom.hom A.norm).toLinearMap ≫ ModuleCat.ofHom (Hom.hom (A.applyAsHom g - 𝟙 A)).toLinearMap = 0",
"ppTerm": "?m.117",
"assigned": true,
"usedConstants": [
"LinearMap.id",... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 202,
"column": 62
} | {
"line": 202,
"column": 64
} | {
"line": 203,
"column": 4
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\nA : Rep k G\ng✝ g : G\n⊢ (trivial k G k).leftRegularHomEquiv\n (((chainComplexFunctor k g).obj (leftRegular k G)).d 1 0 ≫ (trivial k G k).leftRegularHom 1) =\n (trivial k G k).leftRegularHomEquiv 0",
"ppTerm": "?m.7... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.ContCohomology.LowDegree | {
"line": 32,
"column": 83
} | {
"line": 32,
"column": 85
} | {
"line": 33,
"column": 2
} | [
{
"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\nhσ : σ ∈ (↑(TopModuleCat.Hom.hom (X.homogeneousCochains.d 0 1))).ker\n⊢ ↑σ 1 ∈ X.ρ.invaria... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.ContCohomology.LowDegree | {
"line": 49,
"column": 78
} | {
"line": 49,
"column": 80
} | {
"line": 50,
"column": 2
} | [
{
"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\nx : ↑X\nh : ContinuousMap.const G x ∈ (X.resolution'.X 0).ρ.invariants\n⊢ ⟨ContinuousMap.const G x, h⟩ ∈ (↑(TopModuleCat.Hom.hom (X.homo... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.ContCohomology.LowDegree | {
"line": 57,
"column": 81
} | {
"line": 57,
"column": 83
} | {
"line": 57,
"column": 84
} | [
{
"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\n⊢ (ComplexShape.up ℕ).next 0 = 1",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants": [
"Nat.instOne",
"congr... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 225,
"column": 63
} | {
"line": 225,
"column": 65
} | {
"line": 225,
"column": 66
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nm : ℕ\na✝ : QuasiIsoAt (resolution.π k g) m\n⊢ (ComplexShape.down ℕ).prev (m + 1) = m + 2",
"ppTerm": "?m.176",
"assigned": true,
"usedConstants": [
"of_decide_eq... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 225,
"column": 73
} | {
"line": 225,
"column": 75
} | {
"line": 225,
"column": 76
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nm : ℕ\na✝ : QuasiIsoAt (resolution.π k g) m\n⊢ (ComplexShape.down ℕ).next (m + 1) = m",
"ppTerm": "?m.177",
"assigned": true,
"usedConstants": [
"Nat.instOne",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Induced | {
"line": 82,
"column": 13
} | {
"line": 82,
"column": 15
} | {
"line": 82,
"column": 16
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝⁶ : CommRing k\ninst✝⁵ : Group G\ninst✝⁴ : Group H\nφ : G →* H\nA : Type u_4\nB : Type u_5\ninst✝³ : AddCommGroup A\ninst✝² : Module k A\nρ : Representation k G A\ninst✝¹ : AddCommGroup B\ninst✝ : Module k B\nτ : Representation k G B\nh : H\nx✝ : G\n⊢ Line... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Induced | {
"line": 83,
"column": 14
} | {
"line": 83,
"column": 16
} | {
"line": 83,
"column": 17
} | [
{
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Mathlib.RepresentationTheory.Induced | {
"line": 84,
"column": 18
} | {
"line": 84,
"column": 20
} | {
"line": 84,
"column": 21
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝⁶ : CommRing k\ninst✝⁵ : Group G\ninst✝⁴ : Group H\nφ : G →* H\nA : Type u_4\nB : Type u_5\ninst✝³ : AddCommGroup A\ninst✝² : Module k A\nρ : Representation k G A\ninst✝¹ : AddCommGroup B\ninst✝ : Module k B\nτ : Representation k G B\nx✝¹ x✝ : H\n⊢ Coinvar... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 209,
"column": 18
} | {
"line": 209,
"column": 20
} | {
"line": 210,
"column": 4
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\nm : ℕ\n⊢ QuasiIsoAt (resolution.π k g) m",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
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"Iff.mpr",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Induced | {
"line": 87,
"column": 65
} | {
"line": 87,
"column": 67
} | {
"line": 88,
"column": 2
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝⁴ : CommRing k\ninst✝³ : Group G\ninst✝² : Group H\nφ : G →* H\nA : Type u_4\ninst✝¹ : AddCommGroup A\ninst✝ : Module k A\nρ : Representation k G A\nh₁ h₂ : H\na : A\n⊢ ((ind φ ρ) h₁) ((IndV.mk φ ρ h₂) a) = (IndV.mk φ ρ (h₂ * h₁⁻¹)) a",
"ppTerm": "?m.5... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Induced | {
"line": 109,
"column": 69
} | {
"line": 109,
"column": 71
} | {
"line": 110,
"column": 4
} | [
{
"pp": "k : Type u\nG : Type v\nH : Type v'\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nφ : G →* H\nA✝ : Rep k G\nA B : Rep k G\nf : A ⟶ B\n⊢ ∀ (g : G),\n LinearMap.lTensor k[H] (Hom.hom f).toLinearMap ∘ₗ\n (Representation.tprod (MonoidHom.comp (Representation.leftRegular k H) φ) A.ρ) g ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic | {
"line": 92,
"column": 20
} | {
"line": 92,
"column": 22
} | {
"line": 93,
"column": 6
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\nn✝ : ℕ\ninst✝ : Monoid G\nA : Rep k G\nn : ℕ\nf g : (Fin n → G) → ↑A\n⊢ (fun g_1 ↦\n (A.ρ (g_1 0)) ((f + g) fun i ↦ g_1 i.succ) +\n ∑ j, (-1) ^ (↑j + 1) • (f + g) (j.contractNth (fun x1 x2 ↦ x1 * x2) g_1)) =\n (fun g ↦ (A.ρ (g 0)) (f fun i ↦ g i.succ) +... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic | {
"line": 95,
"column": 21
} | {
"line": 95,
"column": 23
} | {
"line": 96,
"column": 6
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\nn✝ : ℕ\ninst✝ : Monoid G\nA : Rep k G\nn : ℕ\nr : k\nf : (Fin n → G) → ↑A\n⊢ (fun g ↦\n (A.ρ (g 0)) ((r • f) fun i ↦ g i.succ) + ∑ j, (-1) ^ (↑j + 1) • (r • f) (j.contractNth (fun x1 x2 ↦ x1 * x2) g)) =\n (RingHom.id k) r • fun g ↦\n (A.ρ (g 0)) (f fun i ... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic | {
"line": 107,
"column": 70
} | {
"line": 107,
"column": 72
} | {
"line": 108,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nn : ℕ\n⊢ d A n =\n (freeLiftLEquiv k G (Fin n → G) A).toModuleIso.inv ≫\n ((barComplex k G).linearYonedaObj k A).d n (n + 1) ≫ (freeLiftLEquiv k G (Fin (n + 1) → G) A).toModuleIso.hom",
"ppTerm": "?m.73",
"assigned": true,... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.ContCohomology.LowDegree | {
"line": 68,
"column": 47
} | {
"line": 68,
"column": 49
} | {
"line": 69,
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{
"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\nx✝ : ↥(↑(TopModuleCat.Hom.hom (X.homogeneousCochains.d 0 1))).ker\nx : C(G, ↑X)\nhx' : x ∈ (X.resolution'.X 0).ρ.invariants\nhx : ⟨x, hx... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Induced | {
"line": 111,
"column": 12
} | {
"line": 111,
"column": 14
} | {
"line": 111,
"column": 15
} | [
{
"pp": "k : Type u\nG : Type v\nH : Type v'\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nφ : G →* H\nA✝ : Rep k G\nA B : Rep k G\nf : A ⟶ B\ng : H\n⊢ Representation.Coinvariants.map (Representation.tprod (MonoidHom.comp (Representation.leftRegular k H) φ) A.ρ)\n (Representation.tprod (Monoid... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Induced | {
"line": 120,
"column": 14
} | {
"line": 120,
"column": 16
} | {
"line": 120,
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} | [
{
"pp": "k : Type u\nG : Type v\nH : Type v'\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nφ : G →* H\nA x✝ : Rep k G\n⊢ indMap φ (𝟙 x✝) = 𝟙 (ind φ x✝)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
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"Rep.V",
"Semiri... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Induced | {
"line": 121,
"column": 18
} | {
"line": 121,
"column": 20
} | {
"line": 121,
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} | [
{
"pp": "k : Type u\nG : Type v\nH : Type v'\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nφ : G →* H\nA X✝ Y✝ Z✝ : Rep k G\nx✝¹ : X✝ ⟶ Y✝\nx✝ : Y✝ ⟶ Z✝\n⊢ indMap φ (x✝¹ ≫ x✝) = indMap φ x✝¹ ≫ indMap φ x✝",
"ppTerm": "?m.140",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.t... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Induced | {
"line": 136,
"column": 70
} | {
"line": 136,
"column": 72
} | {
"line": 137,
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{
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Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic | {
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"column": 52
} | {
"line": 125,
"column": 54
} | {
"line": 126,
"column": 4
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\nn✝ : ℕ\ninst✝ : Group G\nA : Rep k G\nn : ℕ\n⊢ d A n ≫ d A (n + 1) = 0",
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"usedConstants": [
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"Pi.Function.module",
"inhomog... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic | {
"line": 137,
"column": 53
} | {
"line": 137,
"column": 55
} | {
"line": 138,
"column": 2
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{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nn : ℕ\n⊢ (inhomogeneousCochains A).d n (n + 1) = d A n",
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"usedConstants": [
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Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic | {
"line": 142,
"column": 31
} | {
"line": 142,
"column": 33
} | {
"line": 143,
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{
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"inhomogeneousCochains.d",
"Rep.V",
"of_decide_eq_true",
... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Induced | {
"line": 145,
"column": 12
} | {
"line": 145,
"column": 14
} | {
"line": 146,
"column": 6
} | [
{
"pp": "k : Type u\nG : Type v\nH : Type v'\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nφ : G →* H\nA✝ : Rep k G\nB✝ : Rep k H\nA : Rep k G\nB : Rep k H\nf : A ⟶ res φ B\ng : G\n⊢ TensorProduct.lift\n (((lift (↑A →ₗ[k] ↑B) k H) fun h ↦ B.ρ h⁻¹ ∘ₗ (Hom.hom f).toLinearMap) ∘ₗ\n ↑(Mon... | [] | by | [anonymous] | by |
Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic | {
"line": 149,
"column": 70
} | {
"line": 149,
"column": 72
} | {
"line": 150,
"column": 2
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\nn : ℕ\ninst✝ : Group G\nA : Rep k G\n⊢ inhomogeneousCochains A ≅ (barComplex k G).linearYonedaObj k A",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Pi.Function.module",
"inhomogeneousCo... | [] | by | [anonymous] | by |
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