awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.LinearAlgebra.Alternating.DomCoprod
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
⊢ (↑(a.domCoprod b) : MultilinearMap R' (fun (x : ιa ⊕ ιb) ↦ Mᵢ) (N₁ ⊗[R'] N₂)) =
    ∑ σ : Perm.ModSumCongr ιa ιb, domCoprod.summand a b σ

Tactic:
MultilinearMap.ext fun _ => rfl

Premises used: MultilinearMap.ext, rfl
-- Module: Mathlib.LinearAlgebra.Alternating.DomCoprod
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
refine_4 : ∀ (c : R') (m : Mᵢ [⋀^ιa]→ₗ[R'] N₁) (n : Mᵢ [⋀^ιb]→ₗ[R'] N₂), m.domCoprod (c • 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

Tactic:
refine
      LinearMap.mk₂ R' domCoprod (fun m₁ m₂ n => ?_) (fun c m n => ?_) (fun m n₁ n₂ => ?_)
        fun c m n => ?_

Premises used: LinearMap.mk₂, AlternatingMap.domCoprod
-- Module: Mathlib.LinearAlgebra.Alternating.DomCoprod
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
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)
⊢ (m₁ + m₂).domCoprod n = m₁.domCoprod n + m₂.domCoprod n

Tactic:
ext
-- Module: Mathlib.LinearAlgebra.Alternating.DomCoprod
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
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))))
            ⋯ :
          (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))))
                ⋯ :
              (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))))
                ⋯ :
              (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
            x :
          N₁ ⊗[R'] N₂))
⊢ 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:
simp only [domCoprod_apply, add_apply, smul_apply, ← Finset.sum_add_distrib,
        Finset.smul_sum, MultilinearMap.sum_apply, domCoprod.summand]

Premises used: AlternatingMap.add_apply, AlternatingMap.smul_apply, Finset.sum_add_distrib, AlternatingMap.domCoprod.summand, Finset.smul_sum, MultilinearMap.sum_apply, AlternatingMap.domCoprod_apply
-- Module: Mathlib.LinearAlgebra.Alternating.DomCoprod
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
a : Perm (ιa ⊕ ιb)
⊢ HSMul.hSMul (α := ℤˣ) (β := N₁ ⊗[R'] N₂) (((Perm.sign : Perm (ιa ⊕ ιb) →* ℤˣ) : (a : Perm (ιa ⊕ ιb)) → ℤˣ) a : ℤˣ)
      ((MultilinearMap.domDomCongr a
              (HAdd.hAdd (α := MultilinearMap R' (fun (x : ιa ⊕ ιb) ↦ Mᵢ) (N₁ ⊗[R'] N₂)) (β :=
                MultilinearMap R' (fun (x : ιa ⊕ ιb) ↦ Mᵢ) (N₁ ⊗[R'] N₂))
                ((MultilinearMap.domCoprod' :
                      (a : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁ ⊗[R'] MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂) →
                        MultilinearMap R' (fun (x : ιa ⊕ ιb) ↦ Mᵢ) (N₁ ⊗[R'] N₂))
                    ((↑m₁ : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁) ⊗ₜ[R']
                      (↑n : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂)) :
                  MultilinearMap R' (fun (x : ιa ⊕ ιb) ↦ Mᵢ) (N₁ ⊗[R'] N₂))
                ((MultilinearMap.domCoprod' :
                      (a : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁ ⊗[R'] MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂) →
                        MultilinearMap R' (fun (x : ιa ⊕ ιb) ↦ Mᵢ) (N₁ ⊗[R'] N₂))
                    ((↑m₂ : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁) ⊗ₜ[R']
                      (↑n : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂)) :
                  MultilinearMap R' (fun (x : ιa ⊕ ιb) ↦ Mᵢ) (N₁ ⊗[R'] N₂))) :
            (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
          x :
        N₁ ⊗[R'] N₂) =
    HSMul.hSMul (α := ℤˣ) (((Perm.sign : Perm (ιa ⊕ ιb) →* ℤˣ) : (a : Perm (ιa ⊕ ιb)) → ℤˣ) a : ℤˣ)
      (HAdd.hAdd (α := N₁ ⊗[R'] N₂) (β := N₁ ⊗[R'] N₂)
        ((MultilinearMap.domDomCongr a
                ((MultilinearMap.domCoprod' :
                    (a : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁ ⊗[R'] MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂) →
                      MultilinearMap R' (fun (x : ιa ⊕ ιb) ↦ Mᵢ) (N₁ ⊗[R'] N₂))
                  ((↑m₁ : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁) ⊗ₜ[R']
                    (↑n : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂))) :
              (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
            x :
          N₁ ⊗[R'] N₂)
        ((MultilinearMap.domDomCongr a
                ((MultilinearMap.domCoprod' :
                    (a : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁ ⊗[R'] MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂) →
                      MultilinearMap R' (fun (x : ιa ⊕ ιb) ↦ Mᵢ) (N₁ ⊗[R'] N₂))
                  ((↑m₂ : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁) ⊗ₜ[R']
                    (↑n : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂))) :
              (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
            x :
          N₁ ⊗[R'] N₂))

Tactic:
rfl
-- Module: Mathlib.LinearAlgebra.Alternating.DomCoprod
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
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)
⊢ (c • m).domCoprod n = c • m.domCoprod n

Tactic:
ext
-- Module: Mathlib.LinearAlgebra.Alternating.DomCoprod
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
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))))
            ⋯ :
          (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))
⊢ 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:
simp only [domCoprod_apply, add_apply, smul_apply, ← Finset.sum_add_distrib,
        Finset.smul_sum, MultilinearMap.sum_apply, domCoprod.summand]

Premises used: AlternatingMap.add_apply, AlternatingMap.smul_apply, Finset.sum_add_distrib, AlternatingMap.domCoprod.summand, Finset.smul_sum, MultilinearMap.sum_apply, AlternatingMap.domCoprod_apply
-- Module: Mathlib.LinearAlgebra.Alternating.DomCoprod
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
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))))
            ⋯ :
          (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)
⊢ 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))

Tactic:
congr
-- Module: Mathlib.LinearAlgebra.Alternating.DomCoprod
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
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))))
            ⋯ :
          (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))))
                ⋯ :
              (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
            x :
          N₁ ⊗[R'] N₂)
⊢ 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)

Tactic:
ext σ
-- Module: Mathlib.LinearAlgebra.Alternating.DomCoprod
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
σ : Perm.ModSumCongr ιa ιb
refine_2 :
  ∀ (a : Perm (ιa ⊕ ιb)),
    (Quotient.liftOn' (s₁ := QuotientGroup.leftRel (Perm.sumCongrHom ιa ιb).range) (Quotient.mk'' 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 =
      HSMul.hSMul (β := N₁ ⊗[R'] N₂) c
        ((Quotient.liftOn' (s₁ := QuotientGroup.leftRel (Perm.sumCongrHom ιa ιb).range) (Quotient.mk'' 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
                      ((↑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)
⊢ (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))))
          ⋯ :
        (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))))
              ⋯ :
            (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
          x :
        N₁ ⊗[R'] N₂)

Tactic:
induction σ using Quotient.inductionOn'
-- Module: Mathlib.LinearAlgebra.Alternating.DomCoprod
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
a : Perm (ιa ⊕ ιb)
refine_2 :
  HSMul.hSMul (α := ℤˣ) (β := N₁ ⊗[R'] N₂) (((Perm.sign : Perm (ιa ⊕ ιb) →* ℤˣ) : (a : Perm (ιa ⊕ ιb)) → ℤˣ) a : ℤˣ)
      ((MultilinearMap.domDomCongr a
              ((MultilinearMap.domCoprod' :
                  (a : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁ ⊗[R'] MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂) →
                    MultilinearMap R' (fun (x : ιa ⊕ ιb) ↦ Mᵢ) (N₁ ⊗[R'] N₂))
                ((c • (↑m : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁)) ⊗ₜ[R']
                  (↑n : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂))) :
            (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
          x :
        N₁ ⊗[R'] N₂) =
    c •
      HSMul.hSMul (α := ℤˣ) (β := N₁ ⊗[R'] N₂) (((Perm.sign : Perm (ιa ⊕ ιb) →* ℤˣ) : (a : Perm (ιa ⊕ ιb)) → ℤˣ) a : ℤˣ)
        ((MultilinearMap.domDomCongr a
                ((MultilinearMap.domCoprod' :
                    (a : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁ ⊗[R'] MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂) →
                      MultilinearMap R' (fun (x : ιa ⊕ ιb) ↦ Mᵢ) (N₁ ⊗[R'] N₂))
                  ((↑m : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁) ⊗ₜ[R']
                    (↑n : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂))) :
              (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
            x :
          N₁ ⊗[R'] N₂)
⊢ (Quotient.liftOn' (s₁ := QuotientGroup.leftRel (Perm.sumCongrHom ιa ιb).range) (Quotient.mk'' 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 =
    HSMul.hSMul (β := N₁ ⊗[R'] N₂) c
      ((Quotient.liftOn' (s₁ := QuotientGroup.leftRel (Perm.sumCongrHom ιa ιb).range) (Quotient.mk'' 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
                    ((↑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)

Tactic:
simp only [Quotient.liftOn'_mk'', coe_add, coe_smul, MultilinearMap.smul_apply,
        ← MultilinearMap.domCoprod'_apply]

Premises used: MultilinearMap.smul_apply, Quotient.liftOn'_mk'', AlternatingMap.coe_smul, MultilinearMap.domCoprod'_apply, AlternatingMap.coe_add
-- Module: Mathlib.LinearAlgebra.Alternating.DomCoprod
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
a : Perm (ιa ⊕ ιb)
⊢ HSMul.hSMul (α := ℤˣ) (β := N₁ ⊗[R'] N₂) (((Perm.sign : Perm (ιa ⊕ ιb) →* ℤˣ) : (a : Perm (ιa ⊕ ιb)) → ℤˣ) a : ℤˣ)
      ((MultilinearMap.domDomCongr a
              (HSMul.hSMul (β := MultilinearMap R' (fun (x : ιa ⊕ ιb) ↦ Mᵢ) (N₁ ⊗[R'] N₂)) c
                ((MultilinearMap.domCoprod' :
                      (a : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁ ⊗[R'] MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂) →
                        MultilinearMap R' (fun (x : ιa ⊕ ιb) ↦ Mᵢ) (N₁ ⊗[R'] N₂))
                    ((↑m : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁) ⊗ₜ[R']
                      (↑n : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂)) :
                  MultilinearMap R' (fun (x : ιa ⊕ ιb) ↦ Mᵢ) (N₁ ⊗[R'] N₂))) :
            (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
          x :
        N₁ ⊗[R'] N₂) =
    HSMul.hSMul (α := ℤˣ) (((Perm.sign : Perm (ιa ⊕ ιb) →* ℤˣ) : (a : Perm (ιa ⊕ ιb)) → ℤˣ) a : ℤˣ)
      (HSMul.hSMul (β := N₁ ⊗[R'] N₂) c
        ((MultilinearMap.domDomCongr a
                ((MultilinearMap.domCoprod' :
                    (a : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁ ⊗[R'] MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂) →
                      MultilinearMap R' (fun (x : ιa ⊕ ιb) ↦ Mᵢ) (N₁ ⊗[R'] N₂))
                  ((↑m : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁) ⊗ₜ[R']
                    (↑n : MultilinearMap R' (fun (x : ιb) ↦ Mᵢ) N₂))) :
              (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
            x :
          N₁ ⊗[R'] N₂))

Tactic:
rfl