awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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 :=
    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)
⊢ Eq (α := (shortComplexH2 A).X₁ ⟶ (shortComplexH2 B).X₁) (Hom.τ₁ (mapShortComplexH2 f 0)) (Hom.τ₁ 0)
have refine_1₁ : ∀ (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: { simp only [shortComplexH2]
    ext
    simp }
-- 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 →₀ (↑A : Type u))) (ModuleCat.of k (H × H →₀ (↑B : Type u))))
    (Hom.τ₂ (mapShortComplexH2 f 0)) (Hom.τ₂ 0)
have refine_2 : ∀ (a : G × G) (x : (↑A : Type u)) (a_1 : H × H),
  Eq (α := (↑B : Type u))
    ((((ModuleCat.Hom.hom (Hom.τ₂ (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 }.X₂ :
                    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) }.X₂ :
              Type u)) :
          (a : H × H) → (↑B : Type u))
        a_1 :
      (↑B : Type u))
    (((((ModuleCat.Hom.hom (Hom.τ₂ 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) }.X₂ :
                        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) }.X₂ :
                        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 }.X₂ :
                    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) }.X₂ :
              Type u)) :
          (a : H × H) → (↑B : Type u))
        a_1 :
      (↑B : Type u))
Tactic: 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_2 :
  Eq (α :=
    Quiver.Hom (V := ModuleCat k) (ModuleCat.of k (G × G →₀ (↑A : Type u))) (ModuleCat.of k (H × H →₀ (↑B : Type u))))
    (Hom.τ₂ (mapShortComplexH2 f 0)) (Hom.τ₂ 0)
⊢ Eq (α := (shortComplexH2 A).X₂ ⟶ (shortComplexH2 B).X₂) (Hom.τ₂ (mapShortComplexH2 f 0)) (Hom.τ₂ 0)
have refine_2₁ : ∀ (a : G × G) (x : (↑A : Type u)) (a_1 : H × H),
  Eq (α := (↑B : Type u))
    ((((ModuleCat.Hom.hom (Hom.τ₂ (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 }.X₂ :
                    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) }.X₂ :
              Type u)) :
          (a : H × H) → (↑B : Type u))
        a_1 :
      (↑B : Type u))
    (((((ModuleCat.Hom.hom (Hom.τ₂ 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) }.X₂ :
                        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) }.X₂ :
                        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 }.X₂ :
                    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) }.X₂ :
              Type u)) :
          (a : H × H) → (↑B : Type u))
        a_1 :
      (↑B : Type u))
Tactic: { simp only [shortComplexH2]
    ext
    simp }
-- 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 →₀ (↑A : Type u))) (ModuleCat.of k (H →₀ (↑B : Type u))))
    (Hom.τ₃ (mapShortComplexH2 f 0)) (Hom.τ₃ 0)
have refine_3 : ∀ (a : G) (x : (↑A : Type u)) (a_1 : H),
  Eq (α := (↑B : Type u))
    ((((ModuleCat.Hom.hom (Hom.τ₃ (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 }.X₃ :
                    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) }.X₃ :
              Type u)) :
          (a : H) → (↑B : Type u))
        a_1 :
      (↑B : Type u))
    (((((ModuleCat.Hom.hom (Hom.τ₃ 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) }.X₃ :
                        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) }.X₃ :
                        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 }.X₃ :
                    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) }.X₃ :
              Type u)) :
          (a : H) → (↑B : Type u))
        a_1 :
      (↑B : Type u))
Tactic: 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_3 :
  Eq (α := Quiver.Hom (V := ModuleCat k) (ModuleCat.of k (G →₀ (↑A : Type u))) (ModuleCat.of k (H →₀ (↑B : Type u))))
    (Hom.τ₃ (mapShortComplexH2 f 0)) (Hom.τ₃ 0)
⊢ Eq (α := (shortComplexH2 A).X₃ ⟶ (shortComplexH2 B).X₃) (Hom.τ₃ (mapShortComplexH2 f 0)) (Hom.τ₃ 0)
have refine_3₁ : ∀ (a : G) (x : (↑A : Type u)) (a_1 : H),
  Eq (α := (↑B : Type u))
    ((((ModuleCat.Hom.hom (Hom.τ₃ (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 }.X₃ :
                    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) }.X₃ :
              Type u)) :
          (a : H) → (↑B : Type u))
        a_1 :
      (↑B : Type u))
    (((((ModuleCat.Hom.hom (Hom.τ₃ 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) }.X₃ :
                        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) }.X₃ :
                        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 }.X₃ :
                    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) }.X₃ :
              Type u)) :
          (a : H) → (↑B : Type u))
        a_1 :
      (↑B : Type u))
Tactic: { simp only [shortComplexH2]
    ext
    simp }
-- 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
⊢ mapShortComplexH2 (MonoidHom.id G) (𝟙 A) = 𝟙 (shortComplexH2 A)
have refine_1 : Hom.τ₁ (mapShortComplexH2 (MonoidHom.id G) (𝟙 A)) = Hom.τ₁ (𝟙 (shortComplexH2 A))
Tactic: refine ShortComplex.hom_ext _ _ ?_ ?_ ?_
-- 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
refine_1 : Hom.τ₁ (mapShortComplexH2 (MonoidHom.id G) (𝟙 A)) = Hom.τ₁ (𝟙 (shortComplexH2 A))
⊢ mapShortComplexH2 (MonoidHom.id G) (𝟙 A) = 𝟙 (shortComplexH2 A)
have refine_2 : Hom.τ₂ (mapShortComplexH2 (MonoidHom.id G) (𝟙 A)) = Hom.τ₂ (𝟙 (shortComplexH2 A))
Tactic: refine ShortComplex.hom_ext _ _ ?_ ?_ ?_
-- 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
refine_1 : Hom.τ₁ (mapShortComplexH2 (MonoidHom.id G) (𝟙 A)) = Hom.τ₁ (𝟙 (shortComplexH2 A))
refine_2 : Hom.τ₂ (mapShortComplexH2 (MonoidHom.id G) (𝟙 A)) = Hom.τ₂ (𝟙 (shortComplexH2 A))
⊢ mapShortComplexH2 (MonoidHom.id G) (𝟙 A) = 𝟙 (shortComplexH2 A)
have refine_3 : Hom.τ₃ (mapShortComplexH2 (MonoidHom.id G) (𝟙 A)) = Hom.τ₃ (𝟙 (shortComplexH2 A))
Tactic: refine ShortComplex.hom_ext _ _ ?_ ?_ ?_
-- 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
⊢ Hom.τ₁ (mapShortComplexH2 (MonoidHom.id G) (𝟙 A)) =
    Hom.τ₁
      (𝟙
        { 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 := ⋯ : ShortComplex (ModuleCat k) })
have refine_1 : ∀ (a : G × G × G) (x : (↑A : Type u)) (a_1 : G × G × G),
  Eq (α := (↑A : Type u))
    ((((ModuleCat.Hom.hom (Hom.τ₁ (mapShortComplexH2 (MonoidHom.id G) (𝟙 A))) ∘ₗ lsingle a :
                (a : (↑A : Type u)) →
                  (↑{ 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 }.X₁ :
                    Type u))
              x :
            (↑{ 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)) :
          (a : G × G × G) → (↑A : Type u))
        a_1 :
      (↑A : Type u))
    ((((ModuleCat.Hom.hom
                    (Hom.τ₁
                      (𝟙
                        { 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) })) ∘ₗ
                  lsingle a :
                (a : (↑A : Type u)) →
                  (↑{ 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 }.X₁ :
                    Type u))
              x :
            (↑{ 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)) :
          (a : G × G × G) → (↑A : Type u))
        a_1 :
      (↑A : Type u))
Tactic: ext
-- 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
⊢ Hom.τ₁ (mapShortComplexH2 (MonoidHom.id G) (𝟙 A)) = Hom.τ₁ (𝟙 (shortComplexH2 A))
have refine_1 : Hom.τ₁ (mapShortComplexH2 (MonoidHom.id G) (𝟙 A)) =
  Hom.τ₁
    (𝟙
      { 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) })
Tactic: { simp only [shortComplexH2]
    ext
    simp }
-- 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
refine_1 :
  Hom.τ₁ (mapShortComplexH2 (MonoidHom.id G) (𝟙 A)) =
    Hom.τ₁
      (𝟙
        { 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 := ⋯ : ShortComplex (ModuleCat k) })
⊢ Hom.τ₁ (mapShortComplexH2 (MonoidHom.id G) (𝟙 A)) = Hom.τ₁ (𝟙 (shortComplexH2 A))
have refine_1₁ : ∀ (a : G × G × G) (x : (↑A : Type u)) (a_1 : G × G × G),
  Eq (α := (↑A : Type u))
    ((((ModuleCat.Hom.hom (Hom.τ₁ (mapShortComplexH2 (MonoidHom.id G) (𝟙 A))) ∘ₗ lsingle a :
                (a : (↑A : Type u)) →
                  (↑{ 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 }.X₁ :
                    Type u))
              x :
            (↑{ 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)) :
          (a : G × G × G) → (↑A : Type u))
        a_1 :
      (↑A : Type u))
    ((((ModuleCat.Hom.hom
                    (Hom.τ₁
                      (𝟙
                        { 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) })) ∘ₗ
                  lsingle a :
                (a : (↑A : Type u)) →
                  (↑{ 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 }.X₁ :
                    Type u))
              x :
            (↑{ 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)) :
          (a : G × G × G) → (↑A : Type u))
        a_1 :
      (↑A : Type u))
Tactic: { simp only [shortComplexH2]
    ext
    simp }
-- 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
⊢ Hom.τ₂ (mapShortComplexH2 (MonoidHom.id G) (𝟙 A)) =
    Hom.τ₂
      (𝟙
        { 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 := ⋯ : ShortComplex (ModuleCat k) })
have refine_2 : ∀ (a : G × G) (x : (↑A : Type u)) (a_1 : G × G),
  Eq (α := (↑A : Type u))
    ((((ModuleCat.Hom.hom (Hom.τ₂ (mapShortComplexH2 (MonoidHom.id G) (𝟙 A))) ∘ₗ lsingle a :
                (a : (↑A : Type u)) →
                  (↑{ 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 }.X₂ :
                    Type u))
              x :
            (↑{ 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) }.X₂ :
              Type u)) :
          (a : G × G) → (↑A : Type u))
        a_1 :
      (↑A : Type u))
    ((((ModuleCat.Hom.hom
                    (Hom.τ₂
                      (𝟙
                        { 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) })) ∘ₗ
                  lsingle a :
                (a : (↑A : Type u)) →
                  (↑{ 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 }.X₂ :
                    Type u))
              x :
            (↑{ 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) }.X₂ :
              Type u)) :
          (a : G × G) → (↑A : Type u))
        a_1 :
      (↑A : Type u))
Tactic: ext
-- 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
⊢ Hom.τ₂ (mapShortComplexH2 (MonoidHom.id G) (𝟙 A)) = Hom.τ₂ (𝟙 (shortComplexH2 A))
have refine_2 : Hom.τ₂ (mapShortComplexH2 (MonoidHom.id G) (𝟙 A)) =
  Hom.τ₂
    (𝟙
      { 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) })
Tactic: { simp only [shortComplexH2]
    ext
    simp }