awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.LinearAlgebra.Alternating.DomCoprod (goal)
Goal:
ιa : Type u_1
ιb : Type u_2
inst : Fintype ιa
inst_1 : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst_2 : CommSemiring R'
inst_3 : AddCommGroup N₁
inst_4 : Module R' N₁
inst_5 : AddCommGroup N₂
inst_6 : Module R' N₂
inst_7 : AddCommMonoid Mᵢ
inst_8 : Module R' Mᵢ
inst_9 : DecidableEq ιa
inst_10 : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
v : ιa ⊕ ιb → Mᵢ
σ : Perm (ιa ⊕ ιb)
sl : Perm ιa
sr : Perm ιb
i' j' : ιa
hv : v ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inl i')) = v ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inl j'))
hij :
  Ne (α := ιa ⊕ ιb) ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inl i') : ιa ⊕ ιb)
    ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inl j') : ιa ⊕ ιb)
hσ_1 :
  HSMul.hSMul (α := Perm (ιa ⊕ ιb)) (β := Perm.ModSumCongr ιa ιb)
      (swap ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inl i')) ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inl j')) : Perm (ιa ⊕ ιb))
      (Quotient.mk'' σ) =
    Quotient.mk'' σ
hσ :
  (Perm.sumCongrHom ιa ιb : (a : Perm ιa × Perm ιb) → Perm (ιa ⊕ ιb)) (sl, sr) =
    ((fun (x : Perm (ιa ⊕ ιb)) ↦
            HSMul.hSMul (α := Perm (ιa ⊕ ιb))
              (swap ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inl i')) ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inl j')) :
                Perm (ιa ⊕ ιb))
              x)
          σ)⁻¹ *
      σ
⊢ HSMul.hSMul (α := ℤˣ) (β := N₁ ⊗[R'] N₂) (((Perm.sign : Perm (ιa ⊕ ιb) →* ℤˣ) : (a : Perm (ιa ⊕ ιb)) → ℤˣ) σ : ℤˣ)
      ((((↑a : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁) : (a : (i : ιa) → Mᵢ) → N₁) fun (i₁ : ιa) ↦
            v ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inl i₁))) ⊗ₜ[R']
          ((↑b : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂) : (a : (i : ιb) → Mᵢ) → N₂) fun (i₂ : ιb) ↦
            v ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inr i₂)) :
        N₁ ⊗[R'] N₂) =
    0
have suffices : Eq.{u_5 + 1} (α := N₁) ((a : (a : ιa → Mᵢ) → N₁) fun (i : ιa) ↦ v ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inl i)) : N₁)
  (0 : N₁)
Tactic: suffices (a fun i ↦ v (σ (Sum.inl i))) = 0 by simp_all
-- Lean 4 proof step from Mathlib.LinearAlgebra.Alternating.DomCoprod (goal)
Goal:
ιa : Type u_1
ιb : Type u_2
inst : Fintype ιa
inst_1 : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst_2 : CommSemiring R'
inst_3 : AddCommGroup N₁
inst_4 : Module R' N₁
inst_5 : AddCommGroup N₂
inst_6 : Module R' N₂
inst_7 : AddCommMonoid Mᵢ
inst_8 : Module R' Mᵢ
inst_9 : DecidableEq ιa
inst_10 : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
v : ιa ⊕ ιb → Mᵢ
σ : Perm (ιa ⊕ ιb)
sl : Perm ιa
sr : Perm ιb
i' j' : ιb
hv : v ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inr i')) = v ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inr j'))
hij :
  Ne (α := ιa ⊕ ιb) ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inr i') : ιa ⊕ ιb)
    ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inr j') : ιa ⊕ ιb)
hσ_1 :
  HSMul.hSMul (α := Perm (ιa ⊕ ιb)) (β := Perm.ModSumCongr ιa ιb)
      (swap ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inr i')) ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inr j')) : Perm (ιa ⊕ ιb))
      (Quotient.mk'' σ) =
    Quotient.mk'' σ
hσ :
  (Perm.sumCongrHom ιa ιb : (a : Perm ιa × Perm ιb) → Perm (ιa ⊕ ιb)) (sl, sr) =
    ((fun (x : Perm (ιa ⊕ ιb)) ↦
            HSMul.hSMul (α := Perm (ιa ⊕ ιb))
              (swap ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inr i')) ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inr j')) :
                Perm (ιa ⊕ ιb))
              x)
          σ)⁻¹ *
      σ
⊢ HSMul.hSMul (α := ℤˣ) (β := N₁ ⊗[R'] N₂) (((Perm.sign : Perm (ιa ⊕ ιb) →* ℤˣ) : (a : Perm (ιa ⊕ ιb)) → ℤˣ) σ : ℤˣ)
      ((((↑a : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁) : (a : (i : ιa) → Mᵢ) → N₁) fun (i₁ : ιa) ↦
            v ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inl i₁))) ⊗ₜ[R']
          ((↑b : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂) : (a : (i : ιb) → Mᵢ) → N₂) fun (i₂ : ιb) ↦
            v ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inr i₂)) :
        N₁ ⊗[R'] N₂) =
    0
have suffices : Eq.{u_6 + 1} (α := N₂) ((b : (a : ιb → Mᵢ) → N₂) fun (i : ιb) ↦ v ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inr i)) : N₂)
  (0 : N₂)
Tactic: suffices (b fun i ↦ v (σ (Sum.inr i))) = 0 by simp_all
-- Lean 4 proof step from Mathlib.LinearAlgebra.Alternating.DomCoprod (goal)
Goal:
ιa : Type u_1
ιb : Type u_2
inst : Fintype ιa
inst_1 : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst_2 : CommSemiring R'
inst_3 : AddCommGroup N₁
inst_4 : Module R' N₁
inst_5 : AddCommGroup N₂
inst_6 : Module R' N₂
inst_7 : AddCommMonoid Mᵢ
inst_8 : Module R' Mᵢ
inst_9 : DecidableEq ιa
inst_10 : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
v : ιa ⊕ ιb → Mᵢ
σ : Perm (ιa ⊕ ιb)
sl : Perm ιa
sr : Perm ιb
i' j' : ιb
hv : v ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inr i')) = v ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inr j'))
hij :
  Ne (α := ιa ⊕ ιb) ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inr i') : ιa ⊕ ιb)
    ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inr j') : ιa ⊕ ιb)
hσ_1 :
  HSMul.hSMul (α := Perm (ιa ⊕ ιb)) (β := Perm.ModSumCongr ιa ιb)
      (swap ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inr i')) ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inr j')) : Perm (ιa ⊕ ιb))
      (Quotient.mk'' σ) =
    Quotient.mk'' σ
hσ :
  (Perm.sumCongrHom ιa ιb : (a : Perm ιa × Perm ιb) → Perm (ιa ⊕ ιb)) (sl, sr) =
    ((fun (x : Perm (ιa ⊕ ιb)) ↦
            HSMul.hSMul (α := Perm (ιa ⊕ ιb))
              (swap ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inr i')) ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inr j')) :
                Perm (ιa ⊕ ιb))
              x)
          σ)⁻¹ *
      σ
⊢ HSMul.hSMul (α := ℤˣ) (β := N₁ ⊗[R'] N₂) (((Perm.sign : Perm (ιa ⊕ ιb) →* ℤˣ) : (a : Perm (ιa ⊕ ιb)) → ℤˣ) σ : ℤˣ)
      ((((↑a : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁) : (a : (i : ιa) → Mᵢ) → N₁) fun (i₁ : ιa) ↦
            v ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inl i₁))) ⊗ₜ[R']
          ((↑b : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂) : (a : (i : ιb) → Mᵢ) → N₂) fun (i₂ : ιb) ↦
            v ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inr i₂)) :
        N₁ ⊗[R'] N₂) =
    0
have case : Eq.{u_6 + 1} (α := N₂) ((b : (a : ιb → Mᵢ) → N₂) fun (i : ιb) ↦ v ((σ : (a : ιa ⊕ ιb) → ιa ⊕ ιb) (Sum.inr i)) : N₂)
  (0 : N₂)
Tactic: case inr.inr =>
    suffices (b fun i ↦ v (σ (Sum.inr i))) = 0 by simp_all
    exact b.map_eq_zero_of_eq _ hv fun hij' => hij (hij' ▸ rfl)
-- Lean 4 proof step from Mathlib.LinearAlgebra.Alternating.DomCoprod (goal)
Goal:
ιa : Type u_1
ιb : Type u_2
inst : Fintype ιa
inst_1 : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst_2 : CommSemiring R'
inst_3 : AddCommGroup N₁
inst_4 : Module R' N₁
inst_5 : AddCommGroup N₂
inst_6 : Module R' N₂
inst_7 : AddCommMonoid Mᵢ
inst_8 : Module R' Mᵢ
inst_9 : DecidableEq ιa
inst_10 : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
v : ιa ⊕ ιb → Mᵢ
i j : ιa ⊕ ιb
hv : v i = v j
hij : i ≠ j
⊢ Eq (α := N₁ ⊗[R'] N₂)
    ((∑ σ : Perm.ModSumCongr ιa ιb, domCoprod.summand a b σ : (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂) v : N₁ ⊗[R'] N₂)
    (0 : N₁ ⊗[R'] N₂)
have rw : Eq (α := N₁ ⊗[R'] N₂)
  (∑ a_1 : Perm.ModSumCongr ιa ιb, (domCoprod.summand a b a_1 : (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂) v) 0
Tactic: rw [MultilinearMap.sum_apply]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Alternating.DomCoprod (goal)
Goal:
ιa : Type u_1
ιb : Type u_2
inst : Fintype ιa
inst_1 : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst_2 : CommSemiring R'
inst_3 : AddCommGroup N₁
inst_4 : Module R' N₁
inst_5 : AddCommGroup N₂
inst_6 : Module R' N₂
inst_7 : AddCommMonoid Mᵢ
inst_8 : Module R' Mᵢ
inst_9 : DecidableEq ιa
inst_10 : DecidableEq ιb
⊢ Mᵢ [⋀^ιa]→ₗ[R'] N₁ →ₗ[R'] Mᵢ [⋀^ιb]→ₗ[R'] N₂ →ₗ[R'] Mᵢ [⋀^ιa ⊕ ιb]→ₗ[R'] N₁ ⊗[R'] N₂
have refine_1 : ∀ (m₁ m₂ : Mᵢ [⋀^ιa]→ₗ[R'] N₁) (n : Mᵢ [⋀^ιb]→ₗ[R'] N₂), (m₁ + m₂).domCoprod n = m₁.domCoprod n + m₂.domCoprod n
Tactic: refine
      LinearMap.mk₂ R' domCoprod (fun m₁ m₂ n => ?_) (fun c m n => ?_) (fun m n₁ n₂ => ?_)
        fun c m n => ?_
-- Lean 4 proof step from Mathlib.LinearAlgebra.Alternating.DomCoprod (goal)
Goal:
ιa : Type u_1
ιb : Type u_2
inst : Fintype ιa
inst_1 : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst_2 : CommSemiring R'
inst_3 : AddCommGroup N₁
inst_4 : Module R' N₁
inst_5 : AddCommGroup N₂
inst_6 : Module R' N₂
inst_7 : AddCommMonoid Mᵢ
inst_8 : Module R' Mᵢ
inst_9 : DecidableEq ιa
inst_10 : DecidableEq ιb
refine_1 :
  ∀ (m₁ m₂ : Mᵢ [⋀^ιa]→ₗ[R'] N₁) (n : Mᵢ [⋀^ιb]→ₗ[R'] N₂), (m₁ + m₂).domCoprod n = m₁.domCoprod n + m₂.domCoprod n
⊢ Mᵢ [⋀^ιa]→ₗ[R'] N₁ →ₗ[R'] Mᵢ [⋀^ιb]→ₗ[R'] N₂ →ₗ[R'] Mᵢ [⋀^ιa ⊕ ιb]→ₗ[R'] N₁ ⊗[R'] N₂
have refine_2 : ∀ (c : R') (m : Mᵢ [⋀^ιa]→ₗ[R'] N₁) (n : Mᵢ [⋀^ιb]→ₗ[R'] N₂), (c • m).domCoprod n = c • m.domCoprod n
Tactic: refine
      LinearMap.mk₂ R' domCoprod (fun m₁ m₂ n => ?_) (fun c m n => ?_) (fun m n₁ n₂ => ?_)
        fun c m n => ?_
-- Lean 4 proof step from Mathlib.LinearAlgebra.Alternating.DomCoprod (goal)
Goal:
ιa : Type u_1
ιb : Type u_2
inst : Fintype ιa
inst_1 : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst_2 : CommSemiring R'
inst_3 : AddCommGroup N₁
inst_4 : Module R' N₁
inst_5 : AddCommGroup N₂
inst_6 : Module R' N₂
inst_7 : AddCommMonoid Mᵢ
inst_8 : Module R' Mᵢ
inst_9 : DecidableEq ιa
inst_10 : DecidableEq ιb
refine_1 :
  ∀ (m₁ m₂ : Mᵢ [⋀^ιa]→ₗ[R'] N₁) (n : Mᵢ [⋀^ιb]→ₗ[R'] N₂), (m₁ + m₂).domCoprod n = m₁.domCoprod n + m₂.domCoprod n
refine_2 : ∀ (c : R') (m : Mᵢ [⋀^ιa]→ₗ[R'] N₁) (n : Mᵢ [⋀^ιb]→ₗ[R'] N₂), (c • m).domCoprod n = c • m.domCoprod n
⊢ Mᵢ [⋀^ιa]→ₗ[R'] N₁ →ₗ[R'] Mᵢ [⋀^ιb]→ₗ[R'] N₂ →ₗ[R'] Mᵢ [⋀^ιa ⊕ ιb]→ₗ[R'] N₁ ⊗[R'] N₂
have refine_3 : ∀ (m : Mᵢ [⋀^ιa]→ₗ[R'] N₁) (n₁ n₂ : Mᵢ [⋀^ιb]→ₗ[R'] N₂), m.domCoprod (n₁ + n₂) = m.domCoprod n₁ + m.domCoprod n₂
Tactic: refine
      LinearMap.mk₂ R' domCoprod (fun m₁ m₂ n => ?_) (fun c m n => ?_) (fun m n₁ n₂ => ?_)
        fun c m n => ?_
-- Lean 4 proof step from Mathlib.LinearAlgebra.Alternating.DomCoprod (goal)
Goal:
ιa : Type u_1
ιb : Type u_2
inst : Fintype ιa
inst_1 : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst_2 : CommSemiring R'
inst_3 : AddCommGroup N₁
inst_4 : Module R' N₁
inst_5 : AddCommGroup N₂
inst_6 : Module R' N₂
inst_7 : AddCommMonoid Mᵢ
inst_8 : Module R' Mᵢ
inst_9 : DecidableEq ιa
inst_10 : DecidableEq ιb
refine_1 :
  ∀ (m₁ m₂ : Mᵢ [⋀^ιa]→ₗ[R'] N₁) (n : Mᵢ [⋀^ιb]→ₗ[R'] N₂), (m₁ + m₂).domCoprod n = m₁.domCoprod n + m₂.domCoprod n
refine_2 : ∀ (c : R') (m : Mᵢ [⋀^ιa]→ₗ[R'] N₁) (n : Mᵢ [⋀^ιb]→ₗ[R'] N₂), (c • m).domCoprod n = c • m.domCoprod n
refine_3 :
  ∀ (m : Mᵢ [⋀^ιa]→ₗ[R'] N₁) (n₁ n₂ : Mᵢ [⋀^ιb]→ₗ[R'] N₂), m.domCoprod (n₁ + n₂) = m.domCoprod n₁ + m.domCoprod n₂
⊢ Mᵢ [⋀^ιa]→ₗ[R'] N₁ →ₗ[R'] Mᵢ [⋀^ιb]→ₗ[R'] N₂ →ₗ[R'] Mᵢ [⋀^ιa ⊕ ιb]→ₗ[R'] N₁ ⊗[R'] N₂
have refine_4 : ∀ (c : R') (m : Mᵢ [⋀^ιa]→ₗ[R'] N₁) (n : Mᵢ [⋀^ιb]→ₗ[R'] N₂), m.domCoprod (c • n) = c • m.domCoprod n
Tactic: refine
      LinearMap.mk₂ R' domCoprod (fun m₁ m₂ n => ?_) (fun c m n => ?_) (fun m n₁ n₂ => ?_)
        fun c m n => ?_
-- Lean 4 proof step from Mathlib.LinearAlgebra.Alternating.DomCoprod (goal)
Goal:
ιa : Type u_1
ιb : Type u_2
inst : Fintype ιa
inst_1 : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst_2 : CommSemiring R'
inst_3 : AddCommGroup N₁
inst_4 : Module R' N₁
inst_5 : AddCommGroup N₂
inst_6 : Module R' N₂
inst_7 : AddCommMonoid Mᵢ
inst_8 : Module R' Mᵢ
inst_9 : DecidableEq ιa
inst_10 : DecidableEq ιb
m₁ m₂ : Mᵢ [⋀^ιa]→ₗ[R'] N₁
n : Mᵢ [⋀^ιb]→ₗ[R'] N₂
⊢ (m₁ + m₂).domCoprod n = m₁.domCoprod n + m₂.domCoprod n
have refine_1 : ∀ (x : ιa ⊕ ιb → Mᵢ),
  Eq (α := N₁ ⊗[R'] N₂) (((m₁ + m₂).domCoprod n : (a : ιa ⊕ ιb → Mᵢ) → N₁ ⊗[R'] N₂) x : N₁ ⊗[R'] N₂)
    ((m₁.domCoprod n + m₂.domCoprod n : (a : ιa ⊕ ιb → Mᵢ) → N₁ ⊗[R'] N₂) x : N₁ ⊗[R'] N₂)
Tactic: ext
-- Lean 4 proof step from Mathlib.LinearAlgebra.Alternating.DomCoprod (goal)
Goal:
ιa : Type u_1
ιb : Type u_2
inst : Fintype ιa
inst_1 : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst_2 : CommSemiring R'
inst_3 : AddCommGroup N₁
inst_4 : Module R' N₁
inst_5 : AddCommGroup N₂
inst_6 : Module R' N₂
inst_7 : AddCommMonoid Mᵢ
inst_8 : Module R' Mᵢ
inst_9 : DecidableEq ιa
inst_10 : DecidableEq ιb
m₁ m₂ : Mᵢ [⋀^ιa]→ₗ[R'] N₁
n : Mᵢ [⋀^ιb]→ₗ[R'] N₂
x : ιa ⊕ ιb → Mᵢ
⊢ Eq (α := N₁ ⊗[R'] N₂) (((m₁ + m₂).domCoprod n : (a : ιa ⊕ ιb → Mᵢ) → N₁ ⊗[R'] N₂) x : N₁ ⊗[R'] N₂)
    ((m₁.domCoprod n + m₂.domCoprod n : (a : ιa ⊕ ιb → Mᵢ) → N₁ ⊗[R'] N₂) x : N₁ ⊗[R'] N₂)
have refine_1 : Eq (α := N₁ ⊗[R'] N₂)
  (∑ a : Perm.ModSumCongr ιa ιb,
    (Quotient.liftOn' a
          (fun (σ : Perm (ιa ⊕ ιb)) ↦
            HSMul.hSMul (α := ℤˣ) (((Perm.sign : Perm (ιa ⊕ ιb) →* ℤˣ) : (a : Perm (ιa ⊕ ιb)) → ℤˣ) σ : ℤˣ)
              (MultilinearMap.domDomCongr σ
                ((↑(m₁ + m₂) : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                  (↑n : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂))))
          sorry :
        (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
      x)
  (∑ x_1 : Perm.ModSumCongr ιa ιb,
    HAdd.hAdd (α := N₁ ⊗[R'] N₂) (β := N₁ ⊗[R'] N₂)
      ((Quotient.liftOn' x_1
              (fun (σ : Perm (ιa ⊕ ιb)) ↦
                HSMul.hSMul (α := ℤˣ) (((Perm.sign : Perm (ιa ⊕ ιb) →* ℤˣ) : (a : Perm (ιa ⊕ ιb)) → ℤˣ) σ : ℤˣ)
                  (MultilinearMap.domDomCongr σ
                    ((↑m₁ : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                      (↑n : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂))))
              sorry :
            (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
          x :
        N₁ ⊗[R'] N₂)
      ((Quotient.liftOn' x_1
              (fun (σ : Perm (ιa ⊕ ιb)) ↦
                HSMul.hSMul (α := ℤˣ) (((Perm.sign : Perm (ιa ⊕ ιb) →* ℤˣ) : (a : Perm (ιa ⊕ ιb)) → ℤˣ) σ : ℤˣ)
                  (MultilinearMap.domDomCongr σ
                    ((↑m₂ : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                      (↑n : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂))))
              sorry :
            (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
          x :
        N₁ ⊗[R'] N₂))
Tactic: simp only [domCoprod_apply, add_apply, smul_apply, ← Finset.sum_add_distrib,
        Finset.smul_sum, MultilinearMap.sum_apply, domCoprod.summand]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Alternating.DomCoprod (goal)
Goal:
ιa : Type u_1
ιb : Type u_2
inst : Fintype ιa
inst_1 : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst_2 : CommSemiring R'
inst_3 : AddCommGroup N₁
inst_4 : Module R' N₁
inst_5 : AddCommGroup N₂
inst_6 : Module R' N₂
inst_7 : AddCommMonoid Mᵢ
inst_8 : Module R' Mᵢ
inst_9 : DecidableEq ιa
inst_10 : DecidableEq ιb
c : R'
m : Mᵢ [⋀^ιa]→ₗ[R'] N₁
n : Mᵢ [⋀^ιb]→ₗ[R'] N₂
⊢ (c • m).domCoprod n = c • m.domCoprod n
have refine_2 : ∀ (x : ιa ⊕ ιb → Mᵢ),
  Eq (α := N₁ ⊗[R'] N₂) (((c • m).domCoprod n : (a : ιa ⊕ ιb → Mᵢ) → N₁ ⊗[R'] N₂) x : N₁ ⊗[R'] N₂)
    ((c • m.domCoprod n : (a : ιa ⊕ ιb → Mᵢ) → N₁ ⊗[R'] N₂) x : N₁ ⊗[R'] N₂)
Tactic: ext
-- Lean 4 proof step from Mathlib.LinearAlgebra.Alternating.DomCoprod (goal)
Goal:
ιa : Type u_1
ιb : Type u_2
inst : Fintype ιa
inst_1 : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst_2 : CommSemiring R'
inst_3 : AddCommGroup N₁
inst_4 : Module R' N₁
inst_5 : AddCommGroup N₂
inst_6 : Module R' N₂
inst_7 : AddCommMonoid Mᵢ
inst_8 : Module R' Mᵢ
inst_9 : DecidableEq ιa
inst_10 : DecidableEq ιb
c : R'
m : Mᵢ [⋀^ιa]→ₗ[R'] N₁
n : Mᵢ [⋀^ιb]→ₗ[R'] N₂
x : ιa ⊕ ιb → Mᵢ
⊢ Eq (α := N₁ ⊗[R'] N₂) (((c • m).domCoprod n : (a : ιa ⊕ ιb → Mᵢ) → N₁ ⊗[R'] N₂) x : N₁ ⊗[R'] N₂)
    ((c • m.domCoprod n : (a : ιa ⊕ ιb → Mᵢ) → N₁ ⊗[R'] N₂) x : N₁ ⊗[R'] N₂)
have refine_2 : Eq (α := N₁ ⊗[R'] N₂)
  (∑ a : Perm.ModSumCongr ιa ιb,
    (Quotient.liftOn' a
          (fun (σ : Perm (ιa ⊕ ιb)) ↦
            HSMul.hSMul (α := ℤˣ) (((Perm.sign : Perm (ιa ⊕ ιb) →* ℤˣ) : (a : Perm (ιa ⊕ ιb)) → ℤˣ) σ : ℤˣ)
              (MultilinearMap.domDomCongr σ
                ((↑(c • m) : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                  (↑n : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂))))
          sorry :
        (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
      x)
  (∑ x_1 : Perm.ModSumCongr ιa ιb,
    HSMul.hSMul (β := N₁ ⊗[R'] N₂) c
      ((Quotient.liftOn' x_1
              (fun (σ : Perm (ιa ⊕ ιb)) ↦
                HSMul.hSMul (α := ℤˣ) (((Perm.sign : Perm (ιa ⊕ ιb) →* ℤˣ) : (a : Perm (ιa ⊕ ιb)) → ℤˣ) σ : ℤˣ)
                  (MultilinearMap.domDomCongr σ
                    ((↑m : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                      (↑n : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂))))
              sorry :
            (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
          x :
        N₁ ⊗[R'] N₂))
Tactic: simp only [domCoprod_apply, add_apply, smul_apply, ← Finset.sum_add_distrib,
        Finset.smul_sum, MultilinearMap.sum_apply, domCoprod.summand]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Alternating.DomCoprod (goal)
Goal:
ιa : Type u_1
ιb : Type u_2
inst : Fintype ιa
inst_1 : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst_2 : CommSemiring R'
inst_3 : AddCommGroup N₁
inst_4 : Module R' N₁
inst_5 : AddCommGroup N₂
inst_6 : Module R' N₂
inst_7 : AddCommMonoid Mᵢ
inst_8 : Module R' Mᵢ
inst_9 : DecidableEq ιa
inst_10 : DecidableEq ιb
c : R'
m : Mᵢ [⋀^ιa]→ₗ[R'] N₁
n : Mᵢ [⋀^ιb]→ₗ[R'] N₂
x : ιa ⊕ ιb → Mᵢ
⊢ Eq (α := N₁ ⊗[R'] N₂)
    (∑ a : Perm.ModSumCongr ιa ιb,
      (Quotient.liftOn' a
            (fun (σ : Perm (ιa ⊕ ιb)) ↦
              HSMul.hSMul (α := ℤˣ) (((Perm.sign : Perm (ιa ⊕ ιb) →* ℤˣ) : (a : Perm (ιa ⊕ ιb)) → ℤˣ) σ : ℤˣ)
                (MultilinearMap.domDomCongr σ
                  ((↑(c • m) : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                    (↑n : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂))))
            ⋯ :
          (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
        x)
    (∑ x_1 : Perm.ModSumCongr ιa ιb,
      HSMul.hSMul (β := N₁ ⊗[R'] N₂) c
        ((Quotient.liftOn' x_1
                (fun (σ : Perm (ιa ⊕ ιb)) ↦
                  HSMul.hSMul (α := ℤˣ) (((Perm.sign : Perm (ιa ⊕ ιb) →* ℤˣ) : (a : Perm (ιa ⊕ ιb)) → ℤˣ) σ : ℤˣ)
                    (MultilinearMap.domDomCongr σ
                      ((↑m : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                        (↑n : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂))))
                ⋯ :
              (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
            x :
          N₁ ⊗[R'] N₂))
have refine_2 : Eq (α := Perm.ModSumCongr ιa ιb → N₁ ⊗[R'] N₂)
  (fun (a : Perm.ModSumCongr ιa ιb) ↦
    (Quotient.liftOn' a
          (fun (σ : Perm (ιa ⊕ ιb)) ↦
            HSMul.hSMul (α := ℤˣ) (((Perm.sign : Perm (ιa ⊕ ιb) →* ℤˣ) : (a : Perm (ιa ⊕ ιb)) → ℤˣ) σ : ℤˣ)
              (MultilinearMap.domDomCongr σ
                ((↑(c • m) : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                  (↑n : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂))))
          sorry :
        (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
      x)
  fun (x_1 : Perm.ModSumCongr ιa ιb) ↦
  HSMul.hSMul (β := N₁ ⊗[R'] N₂) c
    ((Quotient.liftOn' x_1
            (fun (σ : Perm (ιa ⊕ ιb)) ↦
              HSMul.hSMul (α := ℤˣ) (((Perm.sign : Perm (ιa ⊕ ιb) →* ℤˣ) : (a : Perm (ιa ⊕ ιb)) → ℤˣ) σ : ℤˣ)
                (MultilinearMap.domDomCongr σ
                  ((↑m : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                    (↑n : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂))))
            sorry :
          (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
        x :
      N₁ ⊗[R'] N₂)
Tactic: congr
-- Lean 4 proof step from Mathlib.LinearAlgebra.Alternating.DomCoprod (goal)
Goal:
ιa : Type u_1
ιb : Type u_2
inst : Fintype ιa
inst_1 : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst_2 : CommSemiring R'
inst_3 : AddCommGroup N₁
inst_4 : Module R' N₁
inst_5 : AddCommGroup N₂
inst_6 : Module R' N₂
inst_7 : AddCommMonoid Mᵢ
inst_8 : Module R' Mᵢ
inst_9 : DecidableEq ιa
inst_10 : DecidableEq ιb
c : R'
m : Mᵢ [⋀^ιa]→ₗ[R'] N₁
n : Mᵢ [⋀^ιb]→ₗ[R'] N₂
x : ιa ⊕ ιb → Mᵢ
⊢ Eq (α := Perm.ModSumCongr ιa ιb → N₁ ⊗[R'] N₂)
    (fun (a : Perm.ModSumCongr ιa ιb) ↦
      (Quotient.liftOn' a
            (fun (σ : Perm (ιa ⊕ ιb)) ↦
              HSMul.hSMul (α := ℤˣ) (((Perm.sign : Perm (ιa ⊕ ιb) →* ℤˣ) : (a : Perm (ιa ⊕ ιb)) → ℤˣ) σ : ℤˣ)
                (MultilinearMap.domDomCongr σ
                  ((↑(c • m) : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                    (↑n : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂))))
            ⋯ :
          (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
        x)
    fun (x_1 : Perm.ModSumCongr ιa ιb) ↦
    HSMul.hSMul (β := N₁ ⊗[R'] N₂) c
      ((Quotient.liftOn' x_1
              (fun (σ : Perm (ιa ⊕ ιb)) ↦
                HSMul.hSMul (α := ℤˣ) (((Perm.sign : Perm (ιa ⊕ ιb) →* ℤˣ) : (a : Perm (ιa ⊕ ιb)) → ℤˣ) σ : ℤˣ)
                  (MultilinearMap.domDomCongr σ
                    ((↑m : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                      (↑n : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂))))
              ⋯ :
            (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
          x :
        N₁ ⊗[R'] N₂)
have refine_2 : ∀ (σ : Perm.ModSumCongr ιa ιb),
  (Quotient.liftOn' σ
          (fun (σ : Perm (ιa ⊕ ιb)) ↦
            HSMul.hSMul (α := ℤˣ) (((Perm.sign : Perm (ιa ⊕ ιb) →* ℤˣ) : (a : Perm (ιa ⊕ ιb)) → ℤˣ) σ : ℤˣ)
              (MultilinearMap.domDomCongr σ
                ((↑(c • m) : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                  (↑n : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂))))
          sorry :
        (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
      x =
    HSMul.hSMul (β := N₁ ⊗[R'] N₂) c
      ((Quotient.liftOn' σ
              (fun (σ : Perm (ιa ⊕ ιb)) ↦
                HSMul.hSMul (α := ℤˣ) (((Perm.sign : Perm (ιa ⊕ ιb) →* ℤˣ) : (a : Perm (ιa ⊕ ιb)) → ℤˣ) σ : ℤˣ)
                  (MultilinearMap.domDomCongr σ
                    ((↑m : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                      (↑n : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂))))
              sorry :
            (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
          x :
        N₁ ⊗[R'] N₂)
Tactic: ext σ