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
c : R'
m : Mᵢ [⋀^ιa]→ₗ[R'] N₁
n : Mᵢ [⋀^ιb]→ₗ[R'] N₂
x : ιa ⊕ ιb → Mᵢ
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)
have 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))))
          sorry :
        (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))))
              sorry :
            (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
          x :
        N₁ ⊗[R'] N₂)
Tactic: induction σ using Quotient.inductionOn'
Cleared hypotheses: σ
-- 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
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)
have 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)
Tactic: simp only [Quotient.liftOn'_mk'', coe_add, coe_smul, MultilinearMap.smul_apply,
        ← MultilinearMap.domCoprod'_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
m : Mᵢ [⋀^ιa]→ₗ[R'] N₁
n₁ n₂ : Mᵢ [⋀^ιb]→ₗ[R'] N₂
⊢ m.domCoprod (n₁ + n₂) = m.domCoprod n₁ + m.domCoprod n₂
have refine_3 : ∀ (x : ιa ⊕ ιb → Mᵢ),
  Eq (α := N₁ ⊗[R'] N₂) ((m.domCoprod (n₁ + n₂) : (a : ιa ⊕ ιb → Mᵢ) → N₁ ⊗[R'] N₂) x : N₁ ⊗[R'] N₂)
    ((m.domCoprod n₁ + m.domCoprod n₂ : (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
m : Mᵢ [⋀^ιa]→ₗ[R'] N₁
n₁ n₂ : Mᵢ [⋀^ιb]→ₗ[R'] N₂
x : ιa ⊕ ιb → Mᵢ
⊢ Eq (α := N₁ ⊗[R'] N₂) ((m.domCoprod (n₁ + n₂) : (a : ιa ⊕ ιb → Mᵢ) → N₁ ⊗[R'] N₂) x : N₁ ⊗[R'] N₂)
    ((m.domCoprod n₁ + m.domCoprod n₂ : (a : ιa ⊕ ιb → Mᵢ) → N₁ ⊗[R'] N₂) x : N₁ ⊗[R'] N₂)
have refine_3 : 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
                ((↑m : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                  (↑(n₁ + n₂) : 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
                    ((↑m : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                      (↑n₁ : 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
                    ((↑m : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                      (↑n₂ : 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
⊢ m.domCoprod (c • n) = c • m.domCoprod n
have refine_4 : ∀ (x : ιa ⊕ ιb → Mᵢ),
  Eq (α := N₁ ⊗[R'] N₂) ((m.domCoprod (c • 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) ((m.domCoprod (c • 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_4 : 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
                ((↑m : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                  (↑(c • 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
                  ((↑m : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                    (↑(c • 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_4 : 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
                ((↑m : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                  (↑(c • 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
                  ((↑m : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                    (↑(c • 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_4 : ∀ (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
                ((↑m : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                  (↑(c • 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
-- 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
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
                ((↑m : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                  (↑(c • 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)
have refine_4 : ∀ (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
                ((↑m : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                  (↑(c • 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' (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))))
              sorry :
            (a : (i : ιa ⊕ ιb) → Mᵢ) → N₁ ⊗[R'] N₂)
          x :
        N₁ ⊗[R'] N₂)
Tactic: induction σ using Quotient.inductionOn'
Cleared hypotheses: σ
-- 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
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
                ((↑m : MultilinearMap R' (fun (x : ιa) ↦ Mᵢ) N₁).domCoprod
                  (↑(c • 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)
have refine_4 : 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']
                (c • (↑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)
Tactic: simp only [Quotient.liftOn'_mk'', coe_add, coe_smul, MultilinearMap.smul_apply,
        ← MultilinearMap.domCoprod'_apply]