awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality (goal)
Goal:
k G : Type u
inst : CommRing k
inst_1 : Group G
A : Rep k G
S : Subgroup G
inst_2 : S.Normal
⊢ Eq (α := Submodule k (↑(shortComplexH1 (A.coinvariantsShortComplex S).X₁).X₂ : Type u))
    (Submodule.comap (M₂ := (↑(shortComplexH1 A).opcycles : Type u))
      (ModuleCat.Hom.hom
        (Hom.τ₂ (mapShortComplexH1 (MonoidHom.id G) (A.coinvariantsShortComplex S).f) ≫ (shortComplexH1 A).pOpcycles))
      (LinearMap.range
        (ModuleCat.Hom.hom ((mapShortComplexH1 S.subtype (𝟙 (res S.subtype A))).τ₂ ≫ (shortComplexH1 A).pOpcycles))))
    ⊤
have rw : LE.le (α := Submodule k (↑(shortComplexH1 (A.coinvariantsShortComplex S).X₁).X₂ : Type u)) ⊤
  (Submodule.comap (M₂ := (↑(shortComplexH1 A).opcycles : Type u))
    (ModuleCat.Hom.hom
      (Hom.τ₂ (mapShortComplexH1 (MonoidHom.id G) (A.coinvariantsShortComplex S).f) ≫ (shortComplexH1 A).pOpcycles))
    (LinearMap.range
      (ModuleCat.Hom.hom ((mapShortComplexH1 S.subtype (𝟙 (res S.subtype A))).τ₂ ≫ (shortComplexH1 A).pOpcycles))))
Tactic: rw [eq_top_iff]
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality (goal)
Goal:
k G : Type u
inst : CommRing k
inst_1 : Group G
A : Rep k G
S : Subgroup G
inst_2 : S.Normal
x : (↑(shortComplexH1 (A.coinvariantsShortComplex S).X₁).X₂ : Type u)
h : Membership.mem (γ := Submodule k (↑(shortComplexH1 (A.coinvariantsShortComplex S).X₁).X₂ : Type u)) ⊤ x
⊢ Membership.mem (γ := Submodule k (↑(shortComplexH1 (A.coinvariantsShortComplex S).X₁).X₂ : Type u))
    (Submodule.comap (M₂ := (↑(shortComplexH1 A).opcycles : Type u))
      (ModuleCat.Hom.hom
        (Hom.τ₂ (mapShortComplexH1 (MonoidHom.id G) (A.coinvariantsShortComplex S).f) ≫ (shortComplexH1 A).pOpcycles))
      (LinearMap.range
        (ModuleCat.Hom.hom ((mapShortComplexH1 S.subtype (𝟙 (res S.subtype A))).τ₂ ≫ (shortComplexH1 A).pOpcycles))))
    x
have rcases : ∀ (X : G →₀ ↥S →₀ (↑A : Type u))
  (hX :
    mapRange
        (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
              ModuleCat.carrier) (chains₁ToCoinvariantsKer (res S.subtype A))) :
          (a : (↑(ModuleCat.of k (↥S →₀ (↑(res S.subtype A) : Type u))) : Type u)) →
            (↑(ModuleCat.of k ↥(Coinvariants.ker (res S.subtype A).ρ)) : Type u))
        sorry X =
      x),
  Membership.mem (γ := Submodule k (↑(shortComplexH1 (A.coinvariantsShortComplex S).X₁).X₂ : Type u))
    (Submodule.comap (M₂ := (↑(shortComplexH1 A).opcycles : Type u))
      (ModuleCat.Hom.hom
        (Hom.τ₂ (mapShortComplexH1 (MonoidHom.id G) (A.coinvariantsShortComplex S).f) ≫ (shortComplexH1 A).pOpcycles))
      (LinearMap.range
        (ModuleCat.Hom.hom ((mapShortComplexH1 S.subtype (𝟙 (res S.subtype A))).τ₂ ≫ (shortComplexH1 A).pOpcycles))))
    x
Tactic: rcases mapRange_surjective _ (map_zero _) (chains₁ToCoinvariantsKer_surjective
    (res S.subtype A)) x with ⟨(X : G →₀ S →₀ A), hX⟩
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality (goal)
Goal:
k G : Type u
inst : CommRing k
inst_1 : Group G
A : Rep k G
S : Subgroup G
inst_2 : S.Normal
x : (↑(shortComplexH1 (A.coinvariantsShortComplex S).X₁).X₂ : Type u)
h : Membership.mem (γ := Submodule k (↑(shortComplexH1 (A.coinvariantsShortComplex S).X₁).X₂ : Type u)) ⊤ x
X : G →₀ ↥S →₀ (↑A : Type u)
hX :
  mapRange
      (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
            ModuleCat.carrier) (chains₁ToCoinvariantsKer (res S.subtype A))) :
        (a : (↑(ModuleCat.of k (↥S →₀ (↑(res S.subtype A) : Type u))) : Type u)) →
          (↑(ModuleCat.of k ↥(Coinvariants.ker (res S.subtype A).ρ)) : Type u))
      ⋯ X =
    x
Y : ↥S →₀ (↑A : Type u) :=
  X.sum fun (g : G) (f : ↥S →₀ (↑A : Type u)) ↦
    HSub.hSub (α := ↥S →₀ (↑A : Type u))
      ((mapRange.linearMap ((A.ρ : (a : G) → (↑A : Type u) →ₗ[k] (↑A : Type u)) g⁻¹) :
            (a : ↥S →₀ (↑A : Type u)) → ↥S →₀ (↑A : Type u))
          ((lmapDomain (↑A : Type u) k fun (s : ↥S) ↦
                (((MulAut.conjNormal : (a : G) → MulAut ↥S) g⁻¹ : MulAut ↥S) : (a : ↥S) → ↥S) s :
              (a : ↥S →₀ (↑A : Type u)) → ↥S →₀ (↑A : Type u))
            f) :
        ↥S →₀ (↑A : Type u))
      f
Z : G × G →₀ (↑A : Type u) :=
  X.sum fun (g : G) (f : ↥S →₀ (↑A : Type u)) ↦
    HSub.hSub (α := G × G →₀ (↑A : Type u)) (β := G × G →₀ (↑A : Type u))
      ((lmapDomain (↑A : Type u) k fun (s : ↥S) ↦ (g, g⁻¹ * (↑s : G) * g) :
            (a : ↥S →₀ (↑A : Type u)) → G × G →₀ (↑A : Type u))
          f :
        G × G →₀ (↑A : Type u))
      ((lmapDomain (↑A : Type u) k fun (s : ↥S) ↦ ((↑s : G), g) : (a : ↥S →₀ (↑A : Type u)) → G × G →₀ (↑A : Type u))
          f :
        G × G →₀ (↑A : Type u))
⊢ Membership.mem (γ := Submodule k (↑(shortComplexH1 (A.coinvariantsShortComplex S).X₁).X₂ : Type u))
    (Submodule.comap (M₂ := (↑(shortComplexH1 A).opcycles : Type u))
      (ModuleCat.Hom.hom
        (Hom.τ₂ (mapShortComplexH1 (MonoidHom.id G) (A.coinvariantsShortComplex S).f) ≫ (shortComplexH1 A).pOpcycles))
      (LinearMap.range
        (ModuleCat.Hom.hom ((mapShortComplexH1 S.subtype (𝟙 (res S.subtype A))).τ₂ ≫ (shortComplexH1 A).pOpcycles))))
    x
have h : Eq (α := (↑(shortComplexH1 A).opcycles : Type u))
  ((ModuleCat.Hom.hom ((mapShortComplexH1 S.subtype (𝟙 (res S.subtype A))).τ₂ ≫ (shortComplexH1 A).pOpcycles) :
        (a : (↑(shortComplexH1 (res S.subtype A)).X₂ : Type u)) → (↑(shortComplexH1 A).opcycles : Type u))
      Y :
    (↑(shortComplexH1 A).opcycles : Type u))
  ((ModuleCat.Hom.hom
          (Hom.τ₂ (mapShortComplexH1 (MonoidHom.id G) (A.coinvariantsShortComplex S).f) ≫
            (shortComplexH1 A).pOpcycles) :
        (a : (↑(shortComplexH1 (A.coinvariantsShortComplex S).X₁).X₂ : Type u)) →
          (↑(shortComplexH1 A).opcycles : Type u))
      x :
    (↑(shortComplexH1 A).opcycles : Type u))
Tactic: use Y
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality (goal)
Goal:
k G H : Type u
inst : CommRing k
inst_1 : Group G
inst_2 : Group H
A : Rep k G
B : Rep k H
f : G →* H
φ : A ⟶ res f B
n : ℕ
S : Subgroup G
inst_3 : S.Normal
⊢ Epi (H1CoresCoinf A S).g
have rw : Function.Surjective (α := (↑(H1CoresCoinf A S).X₂ : Type u)) (β := (↑(H1CoresCoinf A S).X₃ : Type u))
  (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
        ModuleCat.carrier) (H1CoresCoinf A S).g) :
    (a : (↑(H1CoresCoinf A S).X₂ : Type u)) → (↑(H1CoresCoinf A S).X₃ : Type u))
Tactic: rw [ModuleCat.epi_iff_surjective]
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality (goal)
Goal:
k G H : Type u
inst : CommRing k
inst_1 : Group G
inst_2 : Group H
A : Rep k G
B : Rep k H
f : G →* H
φ : A ⟶ res f B
n : ℕ
S : Subgroup G
inst_3 : S.Normal
x : ↥(cycles₁ (A.quotientToCoinvariants S))
⊢ ∃ (a : (↑(H1CoresCoinf A S).X₂ : Type u)),
    Eq (α := (↑(H1CoresCoinf A S).X₃ : Type u))
      ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
              ModuleCat.carrier) (H1CoresCoinf A S).g :
            (a : (↑(H1CoresCoinf A S).X₂ : Type u)) → (↑(H1CoresCoinf A S).X₃ : Type u))
          a :
        (↑(H1CoresCoinf A S).X₃ : Type u))
      ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
              ModuleCat.carrier) (X := ModuleCat.of k ↥(cycles₁ (A.quotientToCoinvariants S)))
              (H1π (A.quotientToCoinvariants S)) :
            (a : (↑(ModuleCat.of k ↥(cycles₁ (A.quotientToCoinvariants S))) : Type u)) →
              (↑(H1 (A.quotientToCoinvariants S)) : Type u))
          x :
        (↑(H1 (A.quotientToCoinvariants S)) : Type u))
have h : ∀ (y : (↑(ModuleCat.of k ↥(cycles₁ (A.toCoinvariants S))) : Type u))
  (hy :
    (ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
            ModuleCat.carrier) (Y := ModuleCat.of k ↥(cycles₁ ((A.toCoinvariants S).ofQuotient S)))
            (mapCycles₁ (QuotientGroup.mk' S) (Iso.inv ((A.toCoinvariants S).resOfQuotientIso S))) :
          (a : (↑(ModuleCat.of k ↥(cycles₁ (A.toCoinvariants S))) : Type u)) →
            (↑(ModuleCat.of k ↥(cycles₁ ((A.toCoinvariants S).ofQuotient S))) : Type u))
        y =
      x),
  ∃ (a : (↑(H1CoresCoinf A S).X₂ : Type u)),
    Eq (α := (↑(H1CoresCoinf A S).X₃ : Type u))
      ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
              ModuleCat.carrier) (H1CoresCoinf A S).g :
            (a : (↑(H1CoresCoinf A S).X₂ : Type u)) → (↑(H1CoresCoinf A S).X₃ : Type u))
          a :
        (↑(H1CoresCoinf A S).X₃ : Type u))
      ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
              ModuleCat.carrier) (X := ModuleCat.of k ↥(cycles₁ (A.quotientToCoinvariants S)))
              (H1π (A.quotientToCoinvariants S)) :
            (a : (↑(ModuleCat.of k ↥(cycles₁ (A.quotientToCoinvariants S))) : Type u)) →
              (↑(H1 (A.quotientToCoinvariants S)) : Type u))
          x :
        (↑(H1 (A.quotientToCoinvariants S)) : Type u))
Tactic: rcases (ModuleCat.epi_iff_surjective _).1
    (mapCycles₁_quotientGroupMk'_epi (A.toCoinvariants S) S) x with ⟨y, hy⟩
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality (goal)
Goal:
k G H : Type u
inst : CommRing k
inst_1 : Group G
inst_2 : Group H
A : Rep k G
B : Rep k H
f : G →* H
φ : A ⟶ res f B
n : ℕ
S : Subgroup G
inst_3 : S.Normal
x : ↥(cycles₁ (A.quotientToCoinvariants S))
y : (↑(ModuleCat.of k ↥(cycles₁ (A.toCoinvariants S))) : Type u)
hy :
  (ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
          ModuleCat.carrier) (Y := ModuleCat.of k ↥(cycles₁ ((A.toCoinvariants S).ofQuotient S)))
          (mapCycles₁ (QuotientGroup.mk' S) (Iso.inv ((A.toCoinvariants S).resOfQuotientIso S))) :
        (a : (↑(ModuleCat.of k ↥(cycles₁ (A.toCoinvariants S))) : Type u)) →
          (↑(ModuleCat.of k ↥(cycles₁ ((A.toCoinvariants S).ofQuotient S))) : Type u))
      y =
    x
⊢ ∃ (a : (↑(H1CoresCoinf A S).X₂ : Type u)),
    Eq (α := (↑(H1CoresCoinf A S).X₃ : Type u))
      ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
              ModuleCat.carrier) (H1CoresCoinf A S).g :
            (a : (↑(H1CoresCoinf A S).X₂ : Type u)) → (↑(H1CoresCoinf A S).X₃ : Type u))
          a :
        (↑(H1CoresCoinf A S).X₃ : Type u))
      ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
              ModuleCat.carrier) (X := ModuleCat.of k ↥(cycles₁ (A.quotientToCoinvariants S)))
              (H1π (A.quotientToCoinvariants S)) :
            (a : (↑(ModuleCat.of k ↥(cycles₁ (A.quotientToCoinvariants S))) : Type u)) →
              (↑(H1 (A.quotientToCoinvariants S)) : Type u))
          x :
        (↑(H1 (A.quotientToCoinvariants S)) : Type u))
have h : ∀ (Y : G →₀ (↑A : Type u))
  (hY :
    mapRange
        (⇑(Coinvariants.mk (MonoidHom.comp A.ρ S.subtype)) :
          (a : (↑A : Type u)) → Coinvariants (MonoidHom.comp A.ρ S.subtype))
        sorry Y =
      (↑y : G →₀ (↑(A.toCoinvariants S) : Type u))),
  ∃ (a : (↑(H1CoresCoinf A S).X₂ : Type u)),
    Eq (α := (↑(H1CoresCoinf A S).X₃ : Type u))
      ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
              ModuleCat.carrier) (H1CoresCoinf A S).g :
            (a : (↑(H1CoresCoinf A S).X₂ : Type u)) → (↑(H1CoresCoinf A S).X₃ : Type u))
          a :
        (↑(H1CoresCoinf A S).X₃ : Type u))
      ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
              ModuleCat.carrier) (X := ModuleCat.of k ↥(cycles₁ (A.quotientToCoinvariants S)))
              (H1π (A.quotientToCoinvariants S)) :
            (a : (↑(ModuleCat.of k ↥(cycles₁ (A.quotientToCoinvariants S))) : Type u)) →
              (↑(H1 (A.quotientToCoinvariants S)) : Type u))
          x :
        (↑(H1 (A.quotientToCoinvariants S)) : Type u))
Tactic: rcases mapRange_surjective _ (map_zero _)
    (Coinvariants.mk_surjective (A.ρ.comp S.subtype)) y.1 with ⟨Y, hY⟩
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality (hypothesis)
Goal:
k G H : Type u
inst : CommRing k
inst_1 : Group G
inst_2 : Group H
A : Rep k G
B : Rep k H
f : G →* H
φ : A ⟶ res f B
n : ℕ
S : Subgroup G
inst_3 : S.Normal
x : ↥(cycles₁ (A.quotientToCoinvariants S))
y : (↑(ModuleCat.of k ↥(cycles₁ (A.toCoinvariants S))) : Type u)
hy :
  (ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
          ModuleCat.carrier) (Y := ModuleCat.of k ↥(cycles₁ ((A.toCoinvariants S).ofQuotient S)))
          (mapCycles₁ (QuotientGroup.mk' S) (Iso.inv ((A.toCoinvariants S).resOfQuotientIso S))) :
        (a : (↑(ModuleCat.of k ↥(cycles₁ (A.toCoinvariants S))) : Type u)) →
          (↑(ModuleCat.of k ↥(cycles₁ ((A.toCoinvariants S).ofQuotient S))) : Type u))
      y =
    x
Y : G →₀ (↑A : Type u)
hY :
  mapRange
      (⇑(Coinvariants.mk (MonoidHom.comp A.ρ S.subtype)) :
        (a : (↑A : Type u)) → Coinvariants (MonoidHom.comp A.ρ S.subtype))
      ⋯ Y =
    (↑y : G →₀ (↑(A.toCoinvariants S) : Type u))
⊢ Membership.mem (α := (↑(ModuleCat.of k (↑A : Type u)) : Type u)) (Coinvariants.ker (MonoidHom.comp A.ρ S.subtype))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
            ModuleCat.carrier) (d₁₀ A) :
          (a : (↑(ModuleCat.of k (G →₀ (↑A : Type u))) : Type u)) → (↑(ModuleCat.of k (↑A : Type u)) : Type u))
        Y :
      (↑(ModuleCat.of k (↑A : Type u)) : Type u))
have h' : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Tactic: have h' := congr($((mapShortComplexH1 (B := toCoinvariants A S)
      (MonoidHom.id G) (toCoinvariantsMkQ A S)).comm₂₃) Y)
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality (hypothesis)
Goal:
k G H : Type u
inst : CommRing k
inst_1 : Group G
inst_2 : Group H
A : Rep k G
B : Rep k H
f : G →* H
φ : A ⟶ res f B
n : ℕ
S : Subgroup G
inst_3 : S.Normal
x : ↥(cycles₁ (A.quotientToCoinvariants S))
y : (↑(ModuleCat.of k ↥(cycles₁ (A.toCoinvariants S))) : Type u)
hy :
  (ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
          ModuleCat.carrier) (Y := ModuleCat.of k ↥(cycles₁ ((A.toCoinvariants S).ofQuotient S)))
          (mapCycles₁ (QuotientGroup.mk' S) (Iso.inv ((A.toCoinvariants S).resOfQuotientIso S))) :
        (a : (↑(ModuleCat.of k ↥(cycles₁ (A.toCoinvariants S))) : Type u)) →
          (↑(ModuleCat.of k ↥(cycles₁ ((A.toCoinvariants S).ofQuotient S))) : Type u))
      y =
    x
Y : G →₀ (↑A : Type u)
hY :
  mapRange
      (⇑(Coinvariants.mk (MonoidHom.comp A.ρ S.subtype)) :
        (a : (↑A : Type u)) → Coinvariants (MonoidHom.comp A.ρ S.subtype))
      ⋯ Y =
    (↑y : G →₀ (↑(A.toCoinvariants S) : Type u))
⊢ ∃ (a : (↑(H1CoresCoinf A S).X₂ : Type u)),
    Eq (α := (↑(H1CoresCoinf A S).X₃ : Type u))
      ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
              ModuleCat.carrier) (H1CoresCoinf A S).g :
            (a : (↑(H1CoresCoinf A S).X₂ : Type u)) → (↑(H1CoresCoinf A S).X₃ : Type u))
          a :
        (↑(H1CoresCoinf A S).X₃ : Type u))
      ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
              ModuleCat.carrier) (X := ModuleCat.of k ↥(cycles₁ (A.quotientToCoinvariants S)))
              (H1π (A.quotientToCoinvariants S)) :
            (a : (↑(ModuleCat.of k ↥(cycles₁ (A.quotientToCoinvariants S))) : Type u)) →
              (↑(H1 (A.quotientToCoinvariants S)) : Type u))
          x :
        (↑(H1 (A.quotientToCoinvariants S)) : Type u))
have this : Membership.mem (α := (↑(ModuleCat.of k (↑A : Type u)) : Type u)) (Coinvariants.ker (MonoidHom.comp A.ρ S.subtype))
  ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
          ModuleCat.carrier) (d₁₀ A) :
        (a : (↑(ModuleCat.of k (G →₀ (↑A : Type u))) : Type u)) → (↑(ModuleCat.of k (↑A : Type u)) : Type u))
      Y :
    (↑(ModuleCat.of k (↑A : Type u)) : Type u))
Tactic: have : d₁₀ _ Y ∈ Coinvariants.ker (A.ρ.comp S.subtype) := by
    have h' := congr($((mapShortComplexH1 (B := toCoinvariants A S)
      (MonoidHom.id G) (toCoinvariantsMkQ A S)).comm₂₃) Y)
    simp_all [shortComplexH1, ← Coinvariants.mk_eq_zero]
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality (goal)
Goal:
k G H : Type u
inst : CommRing k
inst_1 : Group G
inst_2 : Group H
A : Rep k G
B : Rep k H
f : G →* H
φ : A ⟶ res f B
n : ℕ
⊢ chainsMap₃ f φ ≫ d₃₂ B = d₃₂ A ≫ chainsMap₂ f φ
have hf : ∀ (a : G × G × G) (x : (↑A : Type u)),
  Eq (α := (↑(ModuleCat.of k (H × H →₀ (↑B : Type u))) : Type u))
    ((ModuleCat.Hom.hom (chainsMap₃ f φ ≫ d₃₂ B) ∘ₗ lsingle a :
          (a : (↑A : Type u)) → (↑(ModuleCat.of k (H × H →₀ (↑B : Type u))) : Type u))
        x :
      (↑(ModuleCat.of k (H × H →₀ (↑B : Type u))) : Type u))
    ((ModuleCat.Hom.hom (d₃₂ A ≫ chainsMap₂ f φ) ∘ₗ lsingle a :
          (a : (↑A : Type u)) → (↑(ModuleCat.of k (H × H →₀ (↑B : Type u))) : Type u))
        x :
      (↑(ModuleCat.of k (H × H →₀ (↑B : Type u))) : Type u))
Tactic: ext : 3
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality (goal)
Goal:
k G H : Type u
inst : CommRing k
inst_1 : Group G
inst_2 : Group H
A : Rep k G
B : Rep k H
f : G →* H
⊢ Eq (α := shortComplexH2 A ⟶ shortComplexH2 B) (mapShortComplexH2 f 0) 0
have refine_1 : Eq (α := (shortComplexH2 A).X₁ ⟶ (shortComplexH2 B).X₁) (Hom.τ₁ (mapShortComplexH2 f 0)) (Hom.τ₁ 0)
Tactic: refine ShortComplex.hom_ext _ _ ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality (goal)
Goal:
k G H : Type u
inst : CommRing k
inst_1 : Group G
inst_2 : Group H
A : Rep k G
B : Rep k H
f : G →* H
refine_1 : Eq (α := (shortComplexH2 A).X₁ ⟶ (shortComplexH2 B).X₁) (Hom.τ₁ (mapShortComplexH2 f 0)) (Hom.τ₁ 0)
⊢ Eq (α := shortComplexH2 A ⟶ shortComplexH2 B) (mapShortComplexH2 f 0) 0
have refine_2 : Eq (α := (shortComplexH2 A).X₂ ⟶ (shortComplexH2 B).X₂) (Hom.τ₂ (mapShortComplexH2 f 0)) (Hom.τ₂ 0)
Tactic: refine ShortComplex.hom_ext _ _ ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality (goal)
Goal:
k G H : Type u
inst : CommRing k
inst_1 : Group G
inst_2 : Group H
A : Rep k G
B : Rep k H
f : G →* H
refine_1 : Eq (α := (shortComplexH2 A).X₁ ⟶ (shortComplexH2 B).X₁) (Hom.τ₁ (mapShortComplexH2 f 0)) (Hom.τ₁ 0)
refine_2 : Eq (α := (shortComplexH2 A).X₂ ⟶ (shortComplexH2 B).X₂) (Hom.τ₂ (mapShortComplexH2 f 0)) (Hom.τ₂ 0)
⊢ Eq (α := shortComplexH2 A ⟶ shortComplexH2 B) (mapShortComplexH2 f 0) 0
have refine_3 : Eq (α := (shortComplexH2 A).X₃ ⟶ (shortComplexH2 B).X₃) (Hom.τ₃ (mapShortComplexH2 f 0)) (Hom.τ₃ 0)
Tactic: refine ShortComplex.hom_ext _ _ ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality (goal)
Goal:
k G H : Type u
inst : CommRing k
inst_1 : Group G
inst_2 : Group H
A : Rep k G
B : Rep k H
f : G →* H
⊢ Eq (α :=
    Quiver.Hom (V := ModuleCat k) (ModuleCat.of k (G × G × G →₀ (↑A : Type u)))
      (ModuleCat.of k (H × H × H →₀ (↑B : Type u))))
    (Hom.τ₁ (mapShortComplexH2 f 0)) (Hom.τ₁ 0)
have refine_1 : ∀ (a : G × G × G) (x : (↑A : Type u)) (a_1 : H × H × H),
  Eq (α := (↑B : Type u))
    ((((ModuleCat.Hom.hom (Hom.τ₁ (mapShortComplexH2 f 0)) ∘ₗ lsingle a :
                (a : (↑A : Type u)) →
                  (↑{ X₁ := ModuleCat.of k (H × H × H →₀ (↑B : Type u)), X₂ := ModuleCat.of k (H × H →₀ (↑B : Type u)),
                          X₃ := ModuleCat.of k (H →₀ (↑B : Type u)), f := d₃₂ B, g := d₂₁ B, zero := sorry }.X₁ :
                    Type u))
              x :
            (↑{ X₁ := ModuleCat.of k (H × H × H →₀ (↑B : Type u)), X₂ := ModuleCat.of k (H × H →₀ (↑B : Type u)),
                    X₃ := ModuleCat.of k (H →₀ (↑B : Type u)), f := d₃₂ B, g := d₂₁ B, zero := sorry :
                    ShortComplex (ModuleCat k) }.X₁ :
              Type u)) :
          (a : H × H × H) → (↑B : Type u))
        a_1 :
      (↑B : Type u))
    (((((ModuleCat.Hom.hom (Hom.τ₁ 0) :
                    (↑{ X₁ := ModuleCat.of k (G × G × G →₀ (↑A : Type u)),
                              X₂ := ModuleCat.of k (G × G →₀ (↑A : Type u)), X₃ := ModuleCat.of k (G →₀ (↑A : Type u)),
                              f := d₃₂ A, g := d₂₁ A, zero := sorry : ShortComplex (ModuleCat k) }.X₁ :
                        Type u) →ₗ[k]
                      (↑{ X₁ := ModuleCat.of k (H × H × H →₀ (↑B : Type u)),
                              X₂ := ModuleCat.of k (H × H →₀ (↑B : Type u)), X₃ := ModuleCat.of k (H →₀ (↑B : Type u)),
                              f := d₃₂ B, g := d₂₁ B, zero := sorry : ShortComplex (ModuleCat k) }.X₁ :
                        Type u)) ∘ₗ
                  lsingle a :
                (a : (↑A : Type u)) →
                  (↑{ X₁ := ModuleCat.of k (H × H × H →₀ (↑B : Type u)), X₂ := ModuleCat.of k (H × H →₀ (↑B : Type u)),
                          X₃ := ModuleCat.of k (H →₀ (↑B : Type u)), f := d₃₂ B, g := d₂₁ B, zero := sorry }.X₁ :
                    Type u))
              x :
            (↑{ X₁ := ModuleCat.of k (H × H × H →₀ (↑B : Type u)), X₂ := ModuleCat.of k (H × H →₀ (↑B : Type u)),
                    X₃ := ModuleCat.of k (H →₀ (↑B : Type u)), f := d₃₂ B, g := d₂₁ B, zero := sorry :
                    ShortComplex (ModuleCat k) }.X₁ :
              Type u)) :
          (a : H × H × H) → (↑B : Type u))
        a_1 :
      (↑B : Type u))
Tactic: ext