Module: Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality

Goal state before:
pp: case h.refine_2
k G : Type u
inst✝² : CommRing k
inst✝¹ : Group G
A : Rep k G
S : Subgroup G
inst✝ : S.Normal
x : ↥(cocycles₁ A)
hx :
  (ConcreteCategory.hom (H1π (res S.subtype A)))
      ((ConcreteCategory.hom (mapCocycles₁ S.subtype (𝟙 (res S.subtype A)))) x) =
    0
y : ↑A
hy :
  (ModuleCat.Hom.hom (d₀₁ (res S.subtype A))) y =
    ⇑((ConcreteCategory.hom (mapCocycles₁ S.subtype (𝟙 (res S.subtype A)))) x)
h1 : ∀ (g h : G), x (g * h) = (A.ρ g) (x h) + x g
h2 : ∀ s ∈ S, x s = (A.ρ s) y - y
g : G ⧸ S
⊢ ∀ (a b : G), (QuotientGroup.leftRel S) a b → ⟨↑x a - (A.ρ a) y + y, ⋯⟩ = ⟨↑x b - (A.ρ b) y + y, ⋯⟩

pp: case h.refine_3
k G : Type u
inst✝² : CommRing k
inst✝¹ : Group G
A : Rep k G
S : Subgroup G
inst✝ : S.Normal
x : ↥(cocycles₁ A)
hx :
  (ConcreteCategory.hom (H1π (res S.subtype A)))
      ((ConcreteCategory.hom (mapCocycles₁ S.subtype (𝟙 (res S.subtype A)))) x) =
    0
y : ↑A
hy :
  (ModuleCat.Hom.hom (d₀₁ (res S.subtype A))) y =
    ⇑((ConcreteCategory.hom (mapCocycles₁ S.subtype (𝟙 (res S.subtype A)))) x)
h1 : ∀ (g h : G), x (g * h) = (A.ρ g) (x h) + x g
h2 : ∀ s ∈ S, x s = (A.ρ s) y - y
⊢ (fun g ↦ Quotient.liftOn' g (fun g ↦ ⟨↑x g - (A.ρ g) y + y, ⋯⟩) ⋯) ∈ cocycles₁ (A.quotientToInvariants S)

pp: case h.refine_4
k G : Type u
inst✝² : CommRing k
inst✝¹ : Group G
A : Rep k G
S : Subgroup G
inst✝ : S.Normal
x : ↥(cocycles₁ A)
hx :
  (ConcreteCategory.hom (H1π (res S.subtype A)))
      ((ConcreteCategory.hom (mapCocycles₁ S.subtype (𝟙 (res S.subtype A)))) x) =
    0
y : ↑A
hy :
  (ModuleCat.Hom.hom (d₀₁ (res S.subtype A))) y =
    ⇑((ConcreteCategory.hom (mapCocycles₁ S.subtype (𝟙 (res S.subtype A)))) x)
h1 : ∀ (g h : G), x (g * h) = (A.ρ g) (x h) + x g
h2 : ∀ s ∈ S, x s = (A.ρ s) y - y
⊢ (ModuleCat.Hom.hom (map (QuotientGroup.mk' S) (ofHom (A.ρ.quotientToInvariants_lift S)) 1))
      ((ConcreteCategory.hom (H1π (A.quotientToInvariants S)))
        ⟨fun g ↦ Quotient.liftOn' g (fun g ↦ ⟨↑x g - (A.ρ g) y + y, ⋯⟩) ⋯, ?h.refine_3⟩) =
    (ConcreteCategory.hom (H1π A)) x

Tactic:
· intro g h hgh
    have := congr(A.ρ g $(h2 (g⁻¹ * h) <| QuotientGroup.leftRel_apply.1 hgh))
    simp_all [← sub_eq_add_neg, sub_eq_sub_iff_sub_eq_sub]

Goal state after:
case h.refine_3
k G : Type u
inst✝² : CommRing k
inst✝¹ : Group G
A : Rep k G
S : Subgroup G
inst✝ : S.Normal
x : ↥(cocycles₁ A)
hx :
  (ConcreteCategory.hom (H1π (res S.subtype A)))
      ((ConcreteCategory.hom (mapCocycles₁ S.subtype (𝟙 (res S.subtype A)))) x) =
    0
y : ↑A
hy :
  (ModuleCat.Hom.hom (d₀₁ (res S.subtype A))) y =
    ⇑((ConcreteCategory.hom (mapCocycles₁ S.subtype (𝟙 (res S.subtype A)))) x)
h1 : ∀ (g h : G), x (g * h) = (A.ρ g) (x h) + x g
h2 : ∀ s ∈ S, x s = (A.ρ s) y - y
⊢ (fun g ↦ Quotient.liftOn' g (fun g ↦ ⟨↑x g - (A.ρ g) y + y, ⋯⟩) ⋯) ∈ cocycles₁ (A.quotientToInvariants S)

case h.refine_4
k G : Type u
inst✝² : CommRing k
inst✝¹ : Group G
A : Rep k G
S : Subgroup G
inst✝ : S.Normal
x : ↥(cocycles₁ A)
hx :
  (ConcreteCategory.hom (H1π (res S.subtype A)))
      ((ConcreteCategory.hom (mapCocycles₁ S.subtype (𝟙 (res S.subtype A)))) x) =
    0
y : ↑A
hy :
  (ModuleCat.Hom.hom (d₀₁ (res S.subtype A))) y =
    ⇑((ConcreteCategory.hom (mapCocycles₁ S.subtype (𝟙 (res S.subtype A)))) x)
h1 : ∀ (g h : G), x (g * h) = (A.ρ g) (x h) + x g
h2 : ∀ s ∈ S, x s = (A.ρ s) y - y
⊢ (ModuleCat.Hom.hom (map (QuotientGroup.mk' S) (ofHom (A.ρ.quotientToInvariants_lift S)) 1))
      ((ConcreteCategory.hom (H1π (A.quotientToInvariants S)))
        ⟨fun g ↦ Quotient.liftOn' g (fun g ↦ ⟨↑x g - (A.ρ g) y + y, ⋯⟩) ⋯, ?h.refine_3⟩) =
    (ConcreteCategory.hom (H1π A)) x