mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RepresentationTheory.Rep.Iso

Goal state before:
pp: k : Type u
G : Type v
inst✝¹ : CommRing k
inst✝ : Monoid G
⊢ ∀ {X Y : ModuleCat k[G]} (f : X ⟶ Y),
    (ofModuleMonoidAlgebra ⋙ toModuleMonoidAlgebra).map f ≫ (counitIso Y).hom =
      (counitIso X).hom ≫ (𝟭 (ModuleCat k[G])).map f

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Iso

Goal state before:
pp: k : Type u
G✝ : Type v
inst✝² : CommRing k
inst✝¹ : Monoid G✝
G : Type u
inst✝ : Group G
n : ℕ
⊢ Projective (diagonal k G (n + 1))

Tactic:
by

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
ε : ℝ
εpos : 0 < ε
⊢ ∀ᵐ (ω : Ω),
    ∀ᶠ (n : ℕ) in atTop,
      |∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a| <
        ε * ↑⌊c ^ n⌋₊

Tactic:
by

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
⊢ ∀ᵐ (ω : Ω),
    (fun n ↦
        ∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω -
          ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a) =o[atTop]
      fun n ↦ ↑⌊c ^ n⌋₊

Tactic:
by

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
⊢ Tendsto (fun i ↦ ∫ (a : Ω), truncation (X i) (↑i) a) atTop (𝓝 (∫ (a : Ω), X 0 a))

Tactic:
by

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
⊢ (fun n ↦ (∫ (a : Ω), (∑ i ∈ range n, truncation (X i) ↑i) a) - ↑n * ∫ (a : Ω), X 0 a) =o[atTop] Nat.cast

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝³ : CommSemiring k
inst✝² : Group G
inst✝¹ : Fintype G
inst✝ : Invertible ↑(Fintype.card G)
g : G
⊢ single g 1 * average k G = average k G

Tactic:
by

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
⊢ ∀ᵐ (ω : Ω),
    (fun n ↦ ∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ↑⌊c ^ n⌋₊ * ∫ (a : Ω), X 0 a) =o[atTop] fun n ↦ ↑⌊c ^ n⌋₊

Tactic:
by

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
⊢ ∑' (j : ℕ), ℙ {ω | X j ω ∈ Set.Ioi ↑j} < ∞

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝³ : CommSemiring k
inst✝² : Group G
inst✝¹ : Fintype G
inst✝ : Invertible ↑(Fintype.card G)
g : G
⊢ average k G * single g 1 = average k G

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: k : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
inst✝⁵ : CommRing k
inst✝⁴ : Group G
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AddCommGroup W
inst✝ : Module k W
ρ : Representation k G V
σ : Representation k G W
a✝ b✝ : V
hv : a✝ ∈ {v | ∀ (g : G), (ρ g) v = v}
hw : b✝ ∈ {v | ∀ (g : G), (ρ g) v = v}
g : G
⊢ (ρ g) (a✝ + b✝) = a✝ + b✝

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: k : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
inst✝⁵ : CommRing k
inst✝⁴ : Group G
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AddCommGroup W
inst✝ : Module k W
ρ : Representation k G V
σ : Representation k G W
g : G
⊢ (ρ g) 0 = 0

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: k : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
inst✝⁵ : CommRing k
inst✝⁴ : Group G
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AddCommGroup W
inst✝ : Module k W
ρ : Representation k G V
σ : Representation k G W
r : k
v : V
hv : v ∈ {v | ∀ (g : G), (ρ g) v = v}
g : G
⊢ (ρ g) (r • v) = r • v

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: k : Type u_1
G : Type u_2
V : Type u_3
inst✝³ : CommRing k
inst✝² : Group G
inst✝¹ : AddCommGroup V
inst✝ : Module k V
ρ : Representation k G V
v : V
⊢ v ∈ ρ.invariants ↔ ∀ (g : G), (ρ g) v = v

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: k : Type u_1
G : Type u_2
V : Type u_3
inst✝³ : CommRing k
inst✝² : Group G
inst✝¹ : AddCommGroup V
inst✝ : Module k V
ρ : Representation k G V
⊢ ρ.invariants.carrier = ⋂ g, Function.fixedPoints ⇑(ρ g)

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: k : Type u_1
G : Type u_2
V : Type u_3
inst✝³ : CommRing k
inst✝² : Group G
inst✝¹ : AddCommGroup V
inst✝ : Module k V
ρ : Representation k G V
g : G
hg : ∀ (x : G), x ∈ Subgroup.zpowers g
x : V
hx : (ρ g) x = x
γ : G
⊢ (ρ γ) x = x

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: k : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
inst✝⁵ : CommRing k
inst✝⁴ : Group G
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AddCommGroup W
inst✝ : Module k W
ρ : Representation k G V
σ : Representation k G W
f : V →ₗ[k] W
⊢ (∀ (g : G), σ g ∘ₗ f ∘ₗ ρ g⁻¹ = f) ↔ ρ.IsIntertwiningMap σ f

Tactic:
by

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : ∑' (j : ℕ), ℙ {ω | X j ω ∈ Set.Ioi ↑j} < ∞
ω : Ω
hω : ∀ᶠ (n : ℕ) in atTop, X n ω ∉ Set.Ioi ↑n
n : ℕ
hn : X n ω ∉ Set.Ioi ↑n
npos : n ∈ Set.Ioi 0
h : ¬(-↑n < X n ω ∧ X n ω ≤ ↑n)
⊢ -↑n < X n ω

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: k : Type u_1
G : Type u_2
V : Type u_3
inst✝⁵ : CommRing k
inst✝⁴ : Group G
inst✝³ : AddCommGroup V
inst✝² : Module k V
ρ : Representation k G V
inst✝¹ : Fintype G
inst✝ : Invertible ↑(Fintype.card G)
v : V
g : G
⊢ (ρ g) (ρ.averageMap v) = ρ.averageMap v

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: k : Type u_1
G : Type u_2
V : Type u_3
inst✝⁵ : CommRing k
inst✝⁴ : Group G
inst✝³ : AddCommGroup V
inst✝² : Module k V
ρ : Representation k G V
inst✝¹ : Fintype G
inst✝ : Invertible ↑(Fintype.card G)
v : V
hv : v ∈ ρ.invariants
⊢ ρ.averageMap v = v

Tactic:
by

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : ∑' (j : ℕ), ℙ {ω | X j ω ∈ Set.Ioi ↑j} < ∞
⊢ ∀ᵐ (ω : Ω), Tendsto (fun n ↦ truncation (X n) (↑n) ω - X n ω) atTop (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: k : Type u_1
G : Type u_2
inst✝⁴ : CommRing k
inst✝³ : Group G
V : Type u_5
inst✝² : AddCommGroup V
inst✝¹ : Module k V
ρ : Representation k G V
S : Subgroup G
inst✝ : S.Normal
g : G
x : V
hx : x ∈ invariants (MonoidHom.comp ρ S.subtype)
x✝ : ↥S
s : G
hs : s ∈ S
⊢ ((MonoidHom.comp ρ S.subtype) ⟨s, hs⟩) ((ρ g) x) = (ρ g) x

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: k : Type u_1
G : Type u_2
V✝ : Type u_3
W : Type u_4
inst✝⁸ : CommRing k
inst✝⁷ : Group G
inst✝⁶ : AddCommGroup V✝
inst✝⁵ : Module k V✝
inst✝⁴ : AddCommGroup W
inst✝³ : Module k W
ρ✝ : Representation k G V✝
σ : Representation k G W
V : Type u_5
inst✝² : AddCommGroup V
inst✝¹ : Module k V
ρ : Representation k G V
S : Subgroup G
inst✝ : S.Normal
g : ↥S
x✝ : ↥(invariants (MonoidHom.comp ρ S.subtype))
x : V
hx : x ∈ invariants (MonoidHom.comp ρ S.subtype)
⊢ ↑(((MonoidHom.comp (ρ.toInvariants S) S.subtype) g) ⟨x, hx⟩) = ↑(LinearMap.id ⟨x, hx⟩)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
⊢ ∀ᵐ (ω : Ω), (fun n ↦ ∑ i ∈ range n, truncation (X i) (↑i) ω - ∑ i ∈ range n, X i ω) =o[atTop] fun n ↦ ↑n

Tactic:
by

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
⊢ ∀ (n : ℕ), 0 < ↑⌊c ^ n⌋₊

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: k : Type u
inst✝¹ : CommRing k
G : Type v
inst✝ : Group G
X Y : Rep k G
f : ↑X →ₗ[k] ↑Y
g : G
⊢ ((X.ρ.linHom Y.ρ) g) f = f ↔ f ∘ₗ X.ρ g = Y.ρ g ∘ₗ f

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: k : Type u
inst✝¹ : Field k
G : Type v
inst✝ : Group G
X Y : FDRep k G
⊢ ↥(linHom X.ρ Y.ρ).invariants ≃ₗ[k] X ⟶ Y

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: k : Type u
G : Type v
inst✝² : CommRing k
inst✝¹ : Group G
A✝ : Rep k G
S : Subgroup G
inst✝ : S.Normal
A B : Rep k G
f : A ⟶ B
x✝ : ↥A.ρ.invariants
g : G
c : ↑A
hc : c ∈ A.ρ.invariants
⊢ (B.ρ g) ((↑(Hom.hom f) ∘ₗ A.ρ.invariants.subtype) ⟨c, hc⟩) = (↑(Hom.hom f) ∘ₗ A.ρ.invariants.subtype) ⟨c, hc⟩

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: k : Type u
G : Type v
inst✝³ : CommRing k
inst✝² : Group G
A : Rep k G
S✝ : Subgroup G
inst✝¹ : S✝.Normal
S : Subgroup G
inst✝ : S.Normal
X Y : Rep k G
f : X ⟶ Y
g✝ : G ⧸ S
g : G
⊢ ModuleCat.Hom.hom ((invariantsFunctor k ↥S).map ((resFunctor S.subtype).map f)) ∘ₗ (X.ρ.quotientToInvariants S) ↑g =
    (Y.ρ.quotientToInvariants S) ↑g ∘ₗ ModuleCat.Hom.hom ((invariantsFunctor k ↥S).map ((resFunctor S.subtype).map f))

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: k : Type u
G : Type v
inst✝² : CommRing k
inst✝¹ : Group G
A : Rep k G
S : Subgroup G
inst✝ : S.Normal
x✝ : ModuleCat k
⊢ ∀ (c : ↑x✝), LinearMap.id c ∈ ((trivialFunctor k G).obj x✝).ρ.invariants

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: k : Type u
G : Type v
inst✝² : CommRing k
inst✝¹ : Group G
A : Rep k G
S : Subgroup G
inst✝ : S.Normal
X : Rep k G
g : G
⊢ X.ρ.invariants.subtype ∘ₗ (Representation.trivial k G ↑((invariantsFunctor k G).obj X)) g =
    X.ρ g ∘ₗ X.ρ.invariants.subtype

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Irreducible

Goal state before:
pp: G : Type u_1
k : Type u_2
V : Type u_3
inst✝³ : Monoid G
inst✝² : Field k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
ρ : Representation k G V
⊢ ρ.IsIrreducible ↔ IsSimpleModule k[G] ρ.asModule

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: k : Type u
G : Type v
inst✝² : CommRing k
inst✝¹ : Group G
A : Rep k G
S : Subgroup G
inst✝ : S.Normal
X : ModuleCat k
Y : Rep k G
f : (trivialFunctor k G).obj X ⟶ Y
⊢ ∀ (c : ↑((trivialFunctor k G).obj X)), (Hom.hom f).toLinearMap c ∈ Y.ρ.invariants

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Irreducible

Goal state before:
pp: G : Type u_1
k : Type u_2
inst✝³ : Monoid G
inst✝² : Field k
M : Type u_5
inst✝¹ : AddCommGroup M
inst✝ : Module k[G] M
⊢ IsSimpleModule k[G] M ↔ (ofModule M).IsIrreducible

Tactic:
by

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
H : ∀ (n : ℕ), 0 < ↑⌊c ^ n⌋₊
ω : Ω
hω : (fun n ↦ ∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ↑⌊c ^ n⌋₊ * ∫ (a : Ω), X 0 a) =o[atTop] fun n ↦ ↑⌊c ^ n⌋₊
h'ω : (fun n ↦ ∑ i ∈ range n, truncation (X i) (↑i) ω - ∑ i ∈ range n, X i ω) =o[atTop] fun n ↦ ↑n
⊢ (fun n ↦ ∑ i ∈ range ⌊c ^ n⌋₊, X i ω - ↑⌊c ^ n⌋₊ * ∫ (a : Ω), X 0 a) =o[atTop] fun n ↦ ↑⌊c ^ n⌋₊

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Irreducible

Goal state before:
pp: G : Type u_1
k : Type u_2
V : Type u_3
W : Type u_4
inst✝⁶ : Monoid G
inst✝⁵ : Field k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AddCommGroup W
inst✝¹ : Module k W
ρ : Representation k G V
σ : Representation k G W
f : ρ.IntertwiningMap σ
inst✝ : ρ.IsIrreducible
⊢ Injective ⇑f ∨ f = 0

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Irreducible

Goal state before:
pp: G : Type u_1
k : Type u_2
V : Type u_3
W : Type u_4
inst✝⁶ : Monoid G
inst✝⁵ : Field k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AddCommGroup W
inst✝¹ : Module k W
ρ : Representation k G V
σ : Representation k G W
inst✝ : ρ.IsIrreducible
g : σ.IntertwiningMap ρ
⊢ Surjective ⇑g ∨ g = 0

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Irreducible

Goal state before:
pp: G : Type u_1
k : Type u_2
V : Type u_3
W : Type u_4
inst✝⁷ : Monoid G
inst✝⁶ : Field k
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
inst✝³ : AddCommGroup W
inst✝² : Module k W
ρ : Representation k G V
σ : Representation k G W
f : ρ.IntertwiningMap σ
inst✝¹ : ρ.IsIrreducible
inst✝ : σ.IsIrreducible
⊢ Bijective ⇑f ∨ f = 0

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Irreducible

Goal state before:
pp: G : Type u_1
k : Type u_2
V : Type u_3
inst✝⁶ : Monoid G
inst✝⁵ : Field k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
ρ : Representation k G V
inst✝² : ρ.IsIrreducible
inst✝¹ : FiniteDimensional k V
inst✝ : IsAlgClosed k
⊢ Bijective ⇑(algebraMap k (ρ.IntertwiningMap ρ))

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Irreducible

Goal state before:
pp: G : Type u_1
k : Type u_2
V : Type u_3
inst✝⁶ : Monoid G
inst✝⁵ : Field k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
ρ : Representation k G V
inst✝² : ρ.IsIrreducible
inst✝¹ : FiniteDimensional k V
inst✝ : IsAlgClosed k
⊢ Module.finrank k (ρ.IntertwiningMap ρ) = 1

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Irreducible

Goal state before:
pp: G : Type u_1
k : Type u_2
V : Type u_3
inst✝⁷ : Monoid G
inst✝⁶ : Field k
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
ρ : Representation k G V
inst✝³ : ρ.IsIrreducible
inst✝² : FiniteDimensional k V
inst✝¹ : IsAlgClosed k
inst✝ : IsMulCommutative G
⊢ Module.finrank k V = 1

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Character

Goal state before:
pp: G : Type u_1
k : Type u_2
V : Type u_3
inst✝³ : Monoid G
inst✝² : Field k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
ρ : Representation k G V
g h : G
⊢ ρ.character (h * g) = ρ.character (g * h)

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Character

Goal state before:
pp: G : Type u_1
k : Type u_2
V : Type u_3
inst✝⁴ : Monoid G
inst✝³ : Field k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : FiniteDimensional k V
ρ : Representation k G V
⊢ ρ.character 1 = ↑(finrank k V)

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Character

Goal state before:
pp: G : Type u_1
k : Type u_2
V : Type u_3
W : Type u_4
inst✝⁷ : Monoid G
inst✝⁶ : Field k
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
inst✝³ : FiniteDimensional k V
inst✝² : AddCommGroup W
inst✝¹ : Module k W
inst✝ : FiniteDimensional k W
ρ : Representation k G V
σ : Representation k G W
⊢ (ρ.tprod σ).character = ρ.character * σ.character

Tactic:
by

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
⊢ ∀ᵐ (ω : Ω), Tendsto (fun n ↦ (∑ i ∈ range ⌊c ^ n⌋₊, X i ω) / ↑⌊c ^ n⌋₊) atTop (𝓝 (∫ (a : Ω), X 0 a))

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Character

Goal state before:
pp: G : Type u_1
k : Type u_2
V : Type u_3
W : Type u_4
inst✝⁵ : Monoid G
inst✝⁴ : Field k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AddCommGroup W
inst✝ : Module k W
ρ : Representation k G V
σ : Representation k G W
φ : ρ.Equiv σ
⊢ ρ.character = σ.character

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Character

Goal state before:
pp: G : Type u_1
k : Type u_2
V : Type u_3
inst✝³ : Group G
inst✝² : Field k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
ρ : Representation k G V
g h : G
⊢ ρ.character (h * g * h⁻¹) = ρ.character g

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Character

Goal state before:
pp: G : Type u_1
k : Type u_2
V : Type u_3
W : Type u_4
inst✝⁷ : Group G
inst✝⁶ : Field k
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
inst✝³ : FiniteDimensional k V
inst✝² : AddCommGroup W
inst✝¹ : Module k W
inst✝ : FiniteDimensional k W
ρ : Representation k G V
σ : Representation k G W
g : G
⊢ (ρ.linHom σ).character g = ρ.character g⁻¹ * σ.character g

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Character

Goal state before:
pp: G : Type u_1
k : Type u_2
V : Type u_3
inst✝⁶ : Group G
inst✝⁵ : Field k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : FiniteDimensional k V
ρ : Representation k G V
inst✝¹ : Fintype G
inst✝ : Invertible ↑(Nat.card G)
⊢ Invertible ↑(Fintype.card G)

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Character

Goal state before:
pp: G : Type u_1
k : Type u_2
V : Type u_3
inst✝⁶ : Group G
inst✝⁵ : Field k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : FiniteDimensional k V
ρ : Representation k G V
inst✝¹ : Fintype G
inst✝ : Invertible ↑(Nat.card G)
⊢ (↑(Nat.card G))⁻¹ * ∑ g, ρ.character g = ↑(finrank k ↥ρ.invariants)

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Character

Goal state before:
pp: G : Type u_1
k : Type u_2
V : Type u_3
W : Type u_4
inst✝⁹ : Group G
inst✝⁸ : Field k
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module k V
inst✝⁵ : FiniteDimensional k V
inst✝⁴ : AddCommGroup W
inst✝³ : Module k W
inst✝² : FiniteDimensional k W
ρ : Representation k G V
σ : Representation k G W
inst✝¹ : Fintype G
inst✝ : Invertible ↑(Nat.card G)
⊢ (↑(Nat.card G))⁻¹ * ∑ g, σ.character g * ρ.character g⁻¹ = ↑(finrank k (ρ.IntertwiningMap σ))

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Character

Goal state before:
pp: G : Type u_1
k : Type u_2
V : Type u_3
W : Type u_4
inst✝¹² : Group G
inst✝¹¹ : Field k
inst✝¹⁰ : AddCommGroup V
inst✝⁹ : Module k V
inst✝⁸ : FiniteDimensional k V
inst✝⁷ : AddCommGroup W
inst✝⁶ : Module k W
inst✝⁵ : FiniteDimensional k W
ρ : Representation k G V
σ : Representation k G W
inst✝⁴ : Fintype G
inst✝³ : Invertible ↑(Nat.card G)
inst✝² : IsAlgClosed k
inst✝¹ : ρ.IsIrreducible
inst✝ : σ.IsIrreducible
⊢ (↑(Nat.card G))⁻¹ * ∑ g, ρ.character g * σ.character g⁻¹ = if Nonempty (σ.Equiv ρ) then 1 else 0

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Character

Goal state before:
pp: k : Type u
inst✝¹ : Field k
G : Type v
inst✝ : Monoid G
V : FDRep k G
g h : G
⊢ V.character (h * g) = V.character (g * h)

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Character

Goal state before:
pp: k : Type u
inst✝¹ : Field k
G : Type v
inst✝ : Monoid G
V : FDRep k G
⊢ V.character 1 = ↑(finrank k ↑V.V)

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Character

Goal state before:
pp: k : Type u
inst✝¹ : Field k
G : Type v
inst✝ : Monoid G
V W : FDRep k G
⊢ (V ⊗ W).character = V.character * W.character

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Character

Goal state before:
pp: k : Type u
inst✝¹ : Field k
G : Type v
inst✝ : Monoid G
V W : FDRep k G
i : V ≅ W
⊢ V.character = W.character

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Character

Goal state before:
pp: k : Type u
inst✝¹ : Field k
G : Type v
inst✝ : Group G
V : FDRep k G
g h : G
⊢ V.character (h * g * h⁻¹) = V.character g

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Character

Goal state before:
pp: k : Type u
inst✝¹ : Field k
G : Type v
inst✝ : Group G
V W : FDRep k G
g : G
⊢ (of (linHom V.ρ W.ρ)).character g = V.character g⁻¹ * W.character g

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Character

Goal state before:
pp: k : Type u
inst✝³ : Field k
G : Type v
inst✝² : Group G
inst✝¹ : Fintype G
inst✝ : Invertible ↑(Nat.card G)
V : FDRep k G
⊢ Invertible ↑(Fintype.card G)

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Character

Goal state before:
pp: k : Type u
inst✝³ : Field k
G : Type v
inst✝² : Group G
inst✝¹ : Fintype G
inst✝ : Invertible ↑(Nat.card G)
V : FDRep k G
⊢ (↑(Nat.card G))⁻¹ * ∑ g, V.character g = ↑(finrank k ↥(invariants V.ρ))

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Character

Goal state before:
pp: k : Type u
inst✝³ : Field k
G : Type v
inst✝² : Group G
inst✝¹ : Fintype G
inst✝ : Invertible ↑(Nat.card G)
V W : FDRep k G
⊢ (↑(Nat.card G))⁻¹ * ∑ g, W.character g * V.character g⁻¹ = ↑(finrank k (V ⟶ W))

Tactic:
by

Goal state after:
Module: Mathlib.RepresentationTheory.Character

Goal state before:
pp: k : Type u
inst✝⁶ : Field k
G : Type v
inst✝⁵ : Group G
inst✝⁴ : IsAlgClosed k
inst✝³ : Fintype G
inst✝² : Invertible ↑(Nat.card G)
V W : FDRep k G
inst✝¹ : Simple V
inst✝ : Simple W
⊢ (↑(Nat.card G))⁻¹ * ∑ g, V.character g * W.character g⁻¹ = if Nonempty (V ≅ W) then 1 else 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
⊢ ∀ᵐ (ω : Ω), Tendsto (fun n ↦ (∑ i ∈ range n, X i ω) / ↑n) atTop (𝓝 (∫ (a : Ω), X 0 a))

Tactic:
by

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
X : ℕ → Ω → ℝ
hint : Integrable (X 0) μ
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
mΩ : MeasureSpace Ω := { toMeasurableSpace := m, volume := μ }
h : ∀ᵐ (ω : Ω), X 0 ω = 0
⊢ ∀ᵐ (ω : Ω), ∀ (i : ℕ), X i ω = 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
X : ℕ → Ω → ℝ
hint : Integrable (X 0) μ
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
⊢ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ (∑ i ∈ range n, X i ω) / ↑n) atTop (𝓝 (∫ (x : Ω), X 0 x ∂μ))

Tactic:
by

Goal state after: