awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality (goal)
Goal:
k G H : Type u
inst : CommRing k
inst_1 : Group G
inst_2 : Group H
A_1 : Rep k H
B : Rep k G
f : G →* H
φ : Quiver.Hom.{?u.266374, max (?u.266374 + 1) u} (res f A_1) B
n : ℕ
A : Rep k G
S : Subgroup G
inst_3 : S.Normal
x : (↑(H1InfRes A S).X₁ : Type u)
hx :
  Eq (α := (↑(H1InfRes A S).X₂ : Type u))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
            ModuleCat.carrier) (H1InfRes A S).f :
          (a : (↑(H1InfRes A S).X₁ : Type u)) → (↑(H1InfRes A S).X₂ : Type u))
        x :
      (↑(H1InfRes A S).X₂ : Type u))
    (0 : (↑(H1InfRes A S).X₂ : Type u))
⊢ x = 0
have h : ∀ (x : ↥(cocycles₁ (A.quotientToInvariants S)))
  (hx :
    Eq (α := (↑(H1InfRes A S).X₂ : Type u))
      ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
              ModuleCat.carrier) (H1InfRes A S).f :
            (a : (↑(H1InfRes A S).X₁ : Type u)) → (↑(H1InfRes A S).X₂ : 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 ↥(cocycles₁ (A.quotientToInvariants S)))
                  (H1π (A.quotientToInvariants S)) :
                (a : (↑(ModuleCat.of k ↥(cocycles₁ (A.quotientToInvariants S))) : Type u)) →
                  (↑(H1 (A.quotientToInvariants S)) : Type u))
              x :
            (↑(H1 (A.quotientToInvariants S)) : Type u)) :
        (↑(H1InfRes A S).X₂ : Type u))
      (0 : (↑(H1InfRes A S).X₂ : Type u))),
  Eq (α := (↑(H1InfRes A S).X₁ : 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 ↥(cocycles₁ (A.quotientToInvariants S)))
            (H1π (A.quotientToInvariants S)) :
          (a : (↑(ModuleCat.of k ↥(cocycles₁ (A.quotientToInvariants S))) : Type u)) →
            (↑(H1 (A.quotientToInvariants S)) : Type u))
        x :
      (↑(H1 (A.quotientToInvariants S)) : Type u))
    (0 : (↑(H1InfRes A S).X₁ : Type u))
Tactic: induction x using H1_induction_on with | @h x =>
  simp_all only [H1InfRes_X₂, H1InfRes_X₁, H1InfRes_f, H1π_comp_map_apply (QuotientGroup.mk' S)]
  rcases (H1π_eq_zero_iff _).1 hx with ⟨y, hy⟩
  refine (H1π_eq_zero_iff _).2 ⟨⟨y, fun s => ?_⟩, funext fun g => QuotientGroup.induction_on g
    fun g => Subtype.ext <| by simpa [-SetLike.coe_eq_coe] using congr_fun hy g⟩
  simpa [coe_mapCocycles₁ (x := x), sub_eq_zero, (QuotientGroup.eq_one_iff s.1).2 s.2] using
    congr_fun hy s.1
Cleared hypotheses: x, hx
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupCohomology.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
⊢ (H1InfRes A S).Exact
have rw : (ModuleCat.Hom.hom (H1InfRes A S).g).ker ≤ (ModuleCat.Hom.hom (H1InfRes A S).f).range
Tactic: rw [moduleCat_exact_iff_ker_sub_range]
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality (hypothesis)
Goal:
k G : Type u
inst : CommRing k
inst_1 : Group G
A : Rep k G
S : Subgroup G
inst_2 : S.Normal
x : ↥(cocycles₁ A)
hx :
  Membership.mem (α := (↑(H1InfRes A S).X₂ : Type u)) (ModuleCat.Hom.hom (H1InfRes A S).g).ker
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
            ModuleCat.carrier) (H1π A) :
          (a : (↑(ModuleCat.of k ↥(cocycles₁ A)) : Type u)) → (↑(H1 A) : Type u))
        x :
      (↑(H1 A) : Type u))
⊢ Membership.mem (α := (↑(H1InfRes A S).X₂ : Type u)) (ModuleCat.Hom.hom (H1InfRes A S).f).range
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
            ModuleCat.carrier) (H1π A) :
          (a : (↑(ModuleCat.of k ↥(cocycles₁ A)) : Type u)) → (↑(H1 A) : Type u))
        x :
      (↑(H1 A) : Type u))
have hx₁ : Eq (α := (↑(groupCohomology (res S.subtype A) 1) : Type u))
  ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
          ModuleCat.carrier) (H1π (res S.subtype A)) :
        (a : ↥(cocycles₁ (res S.subtype A))) → (↑(groupCohomology (res S.subtype A) 1) : Type u))
      ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
              ModuleCat.carrier) (mapCocycles₁ S.subtype (𝟙 (res S.subtype A))) :
            (a : ↥(cocycles₁ A)) → ↥(cocycles₁ (res S.subtype A)))
          x :
        ↥(cocycles₁ (res S.subtype A))) :
    (↑(groupCohomology (res S.subtype A) 1) : Type u))
  (0 : (↑(groupCohomology (res S.subtype A) 1) : Type u))
Tactic: simp_all only [H1InfRes_X₂, H1InfRes_X₃, H1InfRes_g, H1InfRes_X₁, LinearMap.mem_ker,
    H1π_comp_map_apply S.subtype, H1InfRes_f]
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupCohomology.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 : ↥(cocycles₁ A)
hx :
  Eq (α := (↑(groupCohomology (res S.subtype A) 1) : Type u))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
            ModuleCat.carrier) (H1π (res S.subtype A)) :
          (a : ↥(cocycles₁ (res S.subtype A))) → (↑(groupCohomology (res S.subtype A) 1) : Type u))
        ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
                ModuleCat.carrier) (mapCocycles₁ S.subtype (𝟙 (res S.subtype A))) :
              (a : ↥(cocycles₁ A)) → ↥(cocycles₁ (res S.subtype A)))
            x :
          ↥(cocycles₁ (res S.subtype A))) :
      (↑(groupCohomology (res S.subtype A) 1) : Type u))
    (0 : (↑(groupCohomology (res S.subtype A) 1) : Type u))
⊢ Membership.mem (α := (↑(groupCohomology A 1) : Type u)) (γ := Submodule k (↑(groupCohomology A 1) : Type u))
    (LinearMap.range (M := (↑(groupCohomology (A.quotientToInvariants S) 1) : Type u))
      (ModuleCat.Hom.hom (map (QuotientGroup.mk' S) (ofHom (A.ρ.quotientToInvariants_lift S)) 1)))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
            ModuleCat.carrier) (H1π A) :
          (a : ↥(cocycles₁ A)) → (↑(H1 A) : Type u))
        x :
      (↑(H1 A) : Type u))
have h : ∀ (y : (↑A : Type u))
  (hy :
    Eq (α := ↥S → (↑(res S.subtype A) : Type u))
      ((ModuleCat.Hom.hom (d₀₁ (res S.subtype A)) :
            (a : (↑(ModuleCat.of k (↑(res S.subtype A) : Type u)) : Type u)) → ↥S → (↑(res S.subtype A) : Type u))
          y :
        ↥S → (↑(res S.subtype A) : Type u))
      ((⇑((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
                    ModuleCat.carrier) (mapCocycles₁ S.subtype (𝟙 (res S.subtype A))) :
                  (a : ↥(cocycles₁ A)) → ↥(cocycles₁ (res S.subtype A)))
                x :
              ↥(cocycles₁ (res S.subtype A))) :
          (a : ↥S) → (↑(res S.subtype A) : Type u)) :
        (a : ↥S) → (↑(res S.subtype A) : Type u))),
  Membership.mem (α := (↑(groupCohomology A 1) : Type u)) (γ := Submodule k (↑(groupCohomology A 1) : Type u))
    (LinearMap.range (M := (↑(groupCohomology (A.quotientToInvariants S) 1) : Type u))
      (ModuleCat.Hom.hom (map (QuotientGroup.mk' S) (ofHom (A.ρ.quotientToInvariants_lift S)) 1)))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
            ModuleCat.carrier) (H1π A) :
          (a : ↥(cocycles₁ A)) → (↑(H1 A) : Type u))
        x :
      (↑(H1 A) : Type u))
Tactic: rcases (H1π_eq_zero_iff _).1 hx with ⟨(y : A), hy⟩
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality (hypothesis)
Goal:
k G : Type u
inst : CommRing k
inst_1 : Group G
A : Rep k G
S : Subgroup G
inst_2 : S.Normal
x : ↥(cocycles₁ A)
hx :
  Eq (α := (↑(groupCohomology (res S.subtype A) 1) : Type u))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
            ModuleCat.carrier) (H1π (res S.subtype A)) :
          (a : ↥(cocycles₁ (res S.subtype A))) → (↑(groupCohomology (res S.subtype A) 1) : Type u))
        ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
                ModuleCat.carrier) (mapCocycles₁ S.subtype (𝟙 (res S.subtype A))) :
              (a : ↥(cocycles₁ A)) → ↥(cocycles₁ (res S.subtype A)))
            x :
          ↥(cocycles₁ (res S.subtype A))) :
      (↑(groupCohomology (res S.subtype A) 1) : Type u))
    (0 : (↑(groupCohomology (res S.subtype A) 1) : Type u))
y : (↑A : Type u)
hy :
  Eq (α := ↥S → (↑(res S.subtype A) : Type u))
    ((ModuleCat.Hom.hom (d₀₁ (res S.subtype A)) :
          (a : (↑(ModuleCat.of k (↑(res S.subtype A) : Type u)) : Type u)) → ↥S → (↑(res S.subtype A) : Type u))
        y :
      ↥S → (↑(res S.subtype A) : Type u))
    ((⇑((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
                  ModuleCat.carrier) (mapCocycles₁ S.subtype (𝟙 (res S.subtype A))) :
                (a : ↥(cocycles₁ A)) → ↥(cocycles₁ (res S.subtype A)))
              x :
            ↥(cocycles₁ (res S.subtype A))) :
        (a : ↥S) → (↑(res S.subtype A) : Type u)) :
      (a : ↥S) → (↑(res S.subtype A) : Type u))
⊢ Membership.mem (α := (↑(groupCohomology A 1) : Type u)) (γ := Submodule k (↑(groupCohomology A 1) : Type u))
    (LinearMap.range (M := (↑(groupCohomology (A.quotientToInvariants S) 1) : Type u))
      (ModuleCat.Hom.hom (map (QuotientGroup.mk' S) (ofHom (A.ρ.quotientToInvariants_lift S)) 1)))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
            ModuleCat.carrier) (H1π A) :
          (a : ↥(cocycles₁ A)) → (↑(H1 A) : Type u))
        x :
      (↑(H1 A) : Type u))
have h1 : ∀ (g h : G),
  (x : (a : G) → (↑A : Type u)) (g * h) =
    HAdd.hAdd (α := (↑A : Type u)) (β := (↑A : Type u))
      ((((A.ρ : (a : G) → (↑A : Type u) →ₗ[k] (↑A : Type u)) g : (↑A : Type u) →ₗ[k] (↑A : Type u)) :
            (a : (↑A : Type u)) → (↑A : Type u))
          ((x : (a : G) → (↑A : Type u)) h : (↑A : Type u)) :
        (↑A : Type u))
      ((x : (a : G) → (↑A : Type u)) g : (↑A : Type u))
Tactic: have h1 := (mem_cocycles₁_iff x).1 x.2
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality (hypothesis)
Goal:
k G : Type u
inst : CommRing k
inst_1 : Group G
A : Rep k G
S : Subgroup G
inst_2 : S.Normal
x : ↥(cocycles₁ A)
hx :
  Eq (α := (↑(groupCohomology (res S.subtype A) 1) : Type u))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
            ModuleCat.carrier) (H1π (res S.subtype A)) :
          (a : ↥(cocycles₁ (res S.subtype A))) → (↑(groupCohomology (res S.subtype A) 1) : Type u))
        ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
                ModuleCat.carrier) (mapCocycles₁ S.subtype (𝟙 (res S.subtype A))) :
              (a : ↥(cocycles₁ A)) → ↥(cocycles₁ (res S.subtype A)))
            x :
          ↥(cocycles₁ (res S.subtype A))) :
      (↑(groupCohomology (res S.subtype A) 1) : Type u))
    (0 : (↑(groupCohomology (res S.subtype A) 1) : Type u))
y : (↑A : Type u)
hy :
  Eq (α := ↥S → (↑(res S.subtype A) : Type u))
    ((ModuleCat.Hom.hom (d₀₁ (res S.subtype A)) :
          (a : (↑(ModuleCat.of k (↑(res S.subtype A) : Type u)) : Type u)) → ↥S → (↑(res S.subtype A) : Type u))
        y :
      ↥S → (↑(res S.subtype A) : Type u))
    ((⇑((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
                  ModuleCat.carrier) (mapCocycles₁ S.subtype (𝟙 (res S.subtype A))) :
                (a : ↥(cocycles₁ A)) → ↥(cocycles₁ (res S.subtype A)))
              x :
            ↥(cocycles₁ (res S.subtype A))) :
        (a : ↥S) → (↑(res S.subtype A) : Type u)) :
      (a : ↥S) → (↑(res S.subtype A) : Type u))
h1 :
  ∀ (g h : G),
    (x : (a : G) → (↑A : Type u)) (g * h) =
      HAdd.hAdd (α := (↑A : Type u)) (β := (↑A : Type u))
        ((((A.ρ : (a : G) → (↑A : Type u) →ₗ[k] (↑A : Type u)) g : (↑A : Type u) →ₗ[k] (↑A : Type u)) :
              (a : (↑A : Type u)) → (↑A : Type u))
            ((x : (a : G) → (↑A : Type u)) h : (↑A : Type u)) :
          (↑A : Type u))
        ((x : (a : G) → (↑A : Type u)) g : (↑A : Type u))
⊢ Membership.mem (α := (↑(groupCohomology A 1) : Type u)) (γ := Submodule k (↑(groupCohomology A 1) : Type u))
    (LinearMap.range (M := (↑(groupCohomology (A.quotientToInvariants S) 1) : Type u))
      (ModuleCat.Hom.hom (map (QuotientGroup.mk' S) (ofHom (A.ρ.quotientToInvariants_lift S)) 1)))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
            ModuleCat.carrier) (H1π A) :
          (a : ↥(cocycles₁ A)) → (↑(H1 A) : Type u))
        x :
      (↑(H1 A) : Type u))
have h2 : ∀ s ∈ S,
  (x : (a : G) → (↑A : Type u)) s =
    HSub.hSub (α := (↑A : Type u))
      ((((A.ρ : (a : G) → (↑A : Type u) →ₗ[k] (↑A : Type u)) s : (↑A : Type u) →ₗ[k] (↑A : Type u)) :
            (a : (↑A : Type u)) → (↑A : Type u))
          y :
        (↑A : Type u))
      y
Tactic: have h2 : ∀ s ∈ S, x s = A.ρ s y - y :=
    fun s hs => funext_iff.1 hy.symm ⟨s, hs⟩
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupCohomology.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 : (↑(H1InfRes A S).X₂ : Type u)
hx : x ∈ (ModuleCat.Hom.hom (H1InfRes A S).g).ker
⊢ x ∈ (ModuleCat.Hom.hom (H1InfRes A S).f).range
have h : ∀ (x : ↥(cocycles₁ A))
  (hx :
    Membership.mem (α := (↑(H1InfRes A S).X₂ : Type u)) (ModuleCat.Hom.hom (H1InfRes A S).g).ker
      ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
              ModuleCat.carrier) (H1π A) :
            (a : (↑(ModuleCat.of k ↥(cocycles₁ A)) : Type u)) → (↑(H1 A) : Type u))
          x :
        (↑(H1 A) : Type u))),
  Membership.mem (α := (↑(H1InfRes A S).X₂ : Type u)) (ModuleCat.Hom.hom (H1InfRes A S).f).range
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CC :=
            ModuleCat.carrier) (H1π A) :
          (a : (↑(ModuleCat.of k ↥(cocycles₁ A)) : Type u)) → (↑(H1 A) : Type u))
        x :
      (↑(H1 A) : Type u))
Tactic: induction x using H1_induction_on with | @h x =>
  simp_all only [H1InfRes_X₂, H1InfRes_X₃, H1InfRes_g, H1InfRes_X₁, LinearMap.mem_ker,
    H1π_comp_map_apply S.subtype, H1InfRes_f]
  rcases (H1π_eq_zero_iff _).1 hx with ⟨(y : A), hy⟩
  have h1 := (mem_cocycles₁_iff x).1 x.2
  have h2 : ∀ s ∈ S, x s = A.ρ s y - y :=
    fun s hs => funext_iff.1 hy.symm ⟨s, hs⟩
  refine ⟨H1π _ ⟨fun g => Quotient.liftOn' g (fun g => ⟨x.1 g - A.ρ g y + y, ?_⟩) ?_, ?_⟩, ?_⟩
  · intro s
    calc
      _ = x (s * g) - x s - A.ρ s (A.ρ g y) + (x s + y) := by
        simp [add_eq_of_eq_sub (h2 s s.2), sub_eq_of_eq_add (h1 s g)]
      _ = x (g * (g⁻¹ * s * g)) - A.ρ g (A.ρ (g⁻¹ * s * g) y - y) - A.ρ g y + y := by
        simp only [mul_assoc, mul_inv_cancel_left, map_mul, Module.End.mul_apply, map_sub,
          Representation.self_inv_apply]
        abel
      _ = x g - A.ρ g y + y := by
        simp [eq_sub_of_add_eq' (h1 g (g⁻¹ * s * g)).symm,
          h2 (g⁻¹ * s * g) (Subgroup.Normal.conj_mem' ‹_› _ s.2 _)]
  · 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]
  · rw [mem_cocycles₁_iff]
    intro g h
    induction g using QuotientGroup.induction_on with | @H g =>
    induction h using QuotientGroup.induction_on with | @H h =>
    apply Subtype.ext
    simp [← QuotientGroup.mk_mul, h1 g h, sub_add_eq_add_sub, add_assoc]
  · symm
    simp only [H1π_comp_map_apply, H1π_eq_iff (A := A)]
    use y
    ext g
    simp [coe_mapCocycles₁ (QuotientGroup.mk' S),
      cocycles₁.coe_mk (A := A.quotientToInvariants S), ← sub_sub]
Cleared hypotheses: x, hx
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality (goal)
Goal:
k G H : Type u
inst : CommRing k
inst_1 : Group G
inst_2 : Group H
A : Rep.{u_1, u, u} k H
B : Rep.{u_1, u, u} k G
f : G →* H
φ : res f A ⟶ B
n : ℕ
⊢ cochainsMap₁ f φ ≫ (shortComplexH2 B).f = (shortComplexH2 A).f ≫ cochainsMap₂ f φ
have hf : ∀ (x : (↑(shortComplexH2 A).X₁ : Type (max u u_1))),
  Eq (α := (↑(shortComplexH2 B).X₂ : Type (max u u_1)))
    ((ModuleCat.Hom.hom (cochainsMap₁ f φ ≫ (shortComplexH2 B).f) :
          (a : (↑(shortComplexH2 A).X₁ : Type (max u u_1))) → (↑(shortComplexH2 B).X₂ : Type (max u u_1)))
        x :
      (↑(shortComplexH2 B).X₂ : Type (max u u_1)))
    ((ModuleCat.Hom.hom ((shortComplexH2 A).f ≫ cochainsMap₂ f φ) :
          (a : (↑(shortComplexH2 A).X₁ : Type (max u u_1))) → (↑(shortComplexH2 B).X₂ : Type (max u u_1)))
        x :
      (↑(shortComplexH2 B).X₂ : Type (max u u_1)))
Tactic: ext x
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality (goal)
Goal:
k G H : Type u
inst : CommRing k
inst_1 : Group G
inst_2 : Group H
A : Rep.{u_1, u, u} k H
B : Rep.{u_1, u, u} k G
f : G →* H
φ : res f A ⟶ B
n : ℕ
x : (↑(shortComplexH2 A).X₁ : Type (max u u_1))
⊢ Eq (α := (↑(shortComplexH2 B).X₂ : Type (max u u_1)))
    ((ModuleCat.Hom.hom (cochainsMap₁ f φ ≫ (shortComplexH2 B).f) :
          (a : (↑(shortComplexH2 A).X₁ : Type (max u u_1))) → (↑(shortComplexH2 B).X₂ : Type (max u u_1)))
        x :
      (↑(shortComplexH2 B).X₂ : Type (max u u_1)))
    ((ModuleCat.Hom.hom ((shortComplexH2 A).f ≫ cochainsMap₂ f φ) :
          (a : (↑(shortComplexH2 A).X₁ : Type (max u u_1))) → (↑(shortComplexH2 B).X₂ : Type (max u u_1)))
        x :
      (↑(shortComplexH2 B).X₂ : Type (max u u_1)))
have hf : ∀ (g : G × G),
  Eq.{u_1 + 1} (α := (↑B : Type u_1))
    ((ModuleCat.Hom.hom (cochainsMap₁ f φ ≫ (shortComplexH2 B).f) :
          (a : (↑(shortComplexH2 A).X₁ : Type (max u u_1))) → (↑(shortComplexH2 B).X₂ : Type (max u u_1)))
        x g :
      (↑B : Type u_1))
    ((ModuleCat.Hom.hom ((shortComplexH2 A).f ≫ cochainsMap₂ f φ) :
          (a : (↑(shortComplexH2 A).X₁ : Type (max u u_1))) → (↑(shortComplexH2 B).X₂ : Type (max u u_1)))
        x g :
      (↑B : Type u_1))
Tactic: funext g
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality (goal)
Goal:
k G H : Type u
inst : CommRing k
inst_1 : Group G
inst_2 : Group H
A : Rep.{u_1, u, u} k H
B : Rep.{u_1, u, u} k G
f : G →* H
φ : res f A ⟶ B
n : ℕ
⊢ cochainsMap₂ f φ ≫ (shortComplexH2 B).g = (shortComplexH2 A).g ≫ cochainsMap₃ f φ
have hf : ∀ (x : (↑(shortComplexH2 A).X₂ : Type (max u u_1))),
  Eq (α := (↑(shortComplexH2 B).X₃ : Type (max u u_1)))
    ((ModuleCat.Hom.hom (cochainsMap₂ f φ ≫ (shortComplexH2 B).g) :
          (a : (↑(shortComplexH2 A).X₂ : Type (max u u_1))) → (↑(shortComplexH2 B).X₃ : Type (max u u_1)))
        x :
      (↑(shortComplexH2 B).X₃ : Type (max u u_1)))
    ((ModuleCat.Hom.hom ((shortComplexH2 A).g ≫ cochainsMap₃ f φ) :
          (a : (↑(shortComplexH2 A).X₂ : Type (max u u_1))) → (↑(shortComplexH2 B).X₃ : Type (max u u_1)))
        x :
      (↑(shortComplexH2 B).X₃ : Type (max u u_1)))
Tactic: ext x
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality (goal)
Goal:
k G H : Type u
inst : CommRing k
inst_1 : Group G
inst_2 : Group H
A : Rep.{u_1, u, u} k H
B : Rep.{u_1, u, u} k G
f : G →* H
φ : res f A ⟶ B
n : ℕ
x : (↑(shortComplexH2 A).X₂ : Type (max u u_1))
⊢ Eq (α := (↑(shortComplexH2 B).X₃ : Type (max u u_1)))
    ((ModuleCat.Hom.hom (cochainsMap₂ f φ ≫ (shortComplexH2 B).g) :
          (a : (↑(shortComplexH2 A).X₂ : Type (max u u_1))) → (↑(shortComplexH2 B).X₃ : Type (max u u_1)))
        x :
      (↑(shortComplexH2 B).X₃ : Type (max u u_1)))
    ((ModuleCat.Hom.hom ((shortComplexH2 A).g ≫ cochainsMap₃ f φ) :
          (a : (↑(shortComplexH2 A).X₂ : Type (max u u_1))) → (↑(shortComplexH2 B).X₃ : Type (max u u_1)))
        x :
      (↑(shortComplexH2 B).X₃ : Type (max u u_1)))
have hf : ∀ (g : G × G × G),
  Eq.{u_1 + 1} (α := (↑B : Type u_1))
    ((ModuleCat.Hom.hom (cochainsMap₂ f φ ≫ (shortComplexH2 B).g) :
          (a : (↑(shortComplexH2 A).X₂ : Type (max u u_1))) → (↑(shortComplexH2 B).X₃ : Type (max u u_1)))
        x g :
      (↑B : Type u_1))
    ((ModuleCat.Hom.hom ((shortComplexH2 A).g ≫ cochainsMap₃ f φ) :
          (a : (↑(shortComplexH2 A).X₂ : Type (max u u_1))) → (↑(shortComplexH2 B).X₃ : Type (max u u_1)))
        x g :
      (↑B : Type u_1))
Tactic: funext g
-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.GroupCohomology.Functoriality (goal)
Goal:
k G H : Type u
inst : CommRing k
inst_1 : Group G
inst_2 : Group H
A : Rep k H
B : Rep k G
f : G →* H
φ : Quiver.Hom.{?u.445319, max (?u.445319 + 1) u} (res f A) B
n_1 n : ℕ
X Y Z : Rep k G
x : X ⟶ Y
x_1 : Y ⟶ Z
⊢ Eq (α := groupCohomology X n ⟶ groupCohomology Z n) (map (MonoidHom.id G) (x ≫ x_1) n)
    (CategoryStruct.comp (Y := groupCohomology Y n) (map (MonoidHom.id G) x n) (map (MonoidHom.id G) x_1 n))
have simp : Eq (α := groupCohomology X n ⟶ groupCohomology Z n) (map (MonoidHom.id G) (x ≫ x_1) n)
  (HomologicalComplex.homologyMap
    (CategoryStruct.comp (Y := inhomogeneousCochains Y) (cochainsMap (MonoidHom.id G) x)
      (cochainsMap (MonoidHom.id G) x_1))
    n)
Tactic: simp only [← HomologicalComplex.homologyMap_comp]