awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.Prod (goal)
Goal:
R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M₁
inst_2 : AddCommGroup M₂
inst_3 : AddCommGroup N
inst_4 : Module R M₁
inst_5 : Module R M₂
inst_6 : Module R N
Q₁ : QuadraticForm R M₁
Q₂ : QuadraticForm R M₂
Qₙ : QuadraticForm R N
f₁ : Q₁ →qᵢ Qₙ
f₂ : Q₂ →qᵢ Qₙ
hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ ((f₁ : (a : M₁) → N) x) ((f₂ : (a : M₂) → N) y)
m₁ : CliffordAlgebra Q₁
m₂ : CliffordAlgebra Q₂
i₂ : ZMod 2
m₁' : CliffordAlgebra Q₁
i₁n i : ℕ
x₁ : CliffordAlgebra Q₁
_hx₁ : x₁ ∈ (ι Q₁).range ^ i
v₁ : M₁
mem :
  HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ) (0 : CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (-1) i₂ •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ) (0 : CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
⊢ HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
      ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ) 0 : CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (-1) i₂ •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
        ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ) 0 : CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
have mem₁ : HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
    ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ) (0 : CliffordAlgebra Qₙ) =
  HSMul.hSMul (β := CliffordAlgebra Qₙ) (HPow.hPow (α := ℤˣ) (-1) i₂) 0
Tactic: rw [map_zero, zero_mul, mul_zero, smul_zero]
-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.Prod (goal)
Goal:
R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M₁
inst_2 : AddCommGroup M₂
inst_3 : AddCommGroup N
inst_4 : Module R M₁
inst_5 : Module R M₂
inst_6 : Module R N
Q₁ : QuadraticForm R M₁
Q₂ : QuadraticForm R M₂
Qₙ : QuadraticForm R N
f₁ : Q₁ →qᵢ Qₙ
f₂ : Q₂ →qᵢ Qₙ
hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ ((f₁ : (a : M₁) → N) x) ((f₂ : (a : M₂) → N) y)
m₁ : CliffordAlgebra Q₁
m₂ : CliffordAlgebra Q₂
i₂ : ZMod 2
m₁' : CliffordAlgebra Q₁
i₁n i : ℕ
x₁ : CliffordAlgebra Q₁
_hx₁ : x₁ ∈ (ι Q₁).range ^ i
v₁ : M₁
mem :
  HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ) (0 : CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (-1) i₂ •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ) (0 : CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
mem₁ :
  HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ) (0 : CliffordAlgebra Qₙ) =
    HSMul.hSMul (β := CliffordAlgebra Qₙ) (HPow.hPow (α := ℤˣ) (-1) i₂) 0
⊢ HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
      ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ) 0 : CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (-1) i₂ •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
        ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ) 0 : CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
have mem₂ : 0 = HSMul.hSMul (β := CliffordAlgebra Qₙ) (HPow.hPow (α := ℤˣ) (-1) i₂) 0
Tactic: rw [map_zero, zero_mul, mul_zero, smul_zero]
-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.Prod (goal)
Goal:
R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M₁
inst_2 : AddCommGroup M₂
inst_3 : AddCommGroup N
inst_4 : Module R M₁
inst_5 : Module R M₂
inst_6 : Module R N
Q₁ : QuadraticForm R M₁
Q₂ : QuadraticForm R M₂
Qₙ : QuadraticForm R N
f₁ : Q₁ →qᵢ Qₙ
f₂ : Q₂ →qᵢ Qₙ
hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ ((f₁ : (a : M₁) → N) x) ((f₂ : (a : M₂) → N) y)
m₁ : CliffordAlgebra Q₁
m₂ : CliffordAlgebra Q₂
i₂ : ZMod 2
m₁' : CliffordAlgebra Q₁
i₁n i : ℕ
x₁ : CliffordAlgebra Q₁
_hx₁ : x₁ ∈ (ι Q₁).range ^ i
v₁ : M₁
mem :
  HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ) (0 : CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (-1) i₂ •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ) (0 : CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
mem₁ :
  HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ) (0 : CliffordAlgebra Qₙ) =
    HSMul.hSMul (β := CliffordAlgebra Qₙ) (HPow.hPow (α := ℤˣ) (-1) i₂) 0
mem₂ : 0 = HSMul.hSMul (β := CliffordAlgebra Qₙ) (HPow.hPow (α := ℤˣ) (-1) i₂) 0
⊢ HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
      ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ) 0 : CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (-1) i₂ •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
        ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ) 0 : CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
have mem₃ : Eq (α := CliffordAlgebra Qₙ) 0 0
Tactic: rw [map_zero, zero_mul, mul_zero, smul_zero]
-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.Prod (goal)
Goal:
R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M₁
inst_2 : AddCommGroup M₂
inst_3 : AddCommGroup N
inst_4 : Module R M₁
inst_5 : Module R M₂
inst_6 : Module R N
Q₁ : QuadraticForm R M₁
Q₂ : QuadraticForm R M₂
Qₙ : QuadraticForm R N
f₁ : Q₁ →qᵢ Qₙ
f₂ : Q₂ →qᵢ Qₙ
hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ ((f₁ : (a : M₁) → N) x) ((f₂ : (a : M₂) → N) y)
m₁ : CliffordAlgebra Q₁
i₂ : ZMod 2
m₁' : CliffordAlgebra Q₁
i₁n i : ℕ
x₁ : CliffordAlgebra Q₁
_hx₁ : x₁ ∈ (ι Q₁).range ^ i
v₁ : M₁
i₂' : { n : ℕ // (↑n : ZMod 2) = i₂ }
m₂' : CliffordAlgebra Q₂
hm₂ : m₂' ∈ (ι Q₂).range ^ (↑i₂' : ℕ)
⊢ HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
      ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ) m₂' : CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (-1) i₂ •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
        ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ) m₂' : CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
have mem : ∀ (i₂n : ℕ)
  (hm₂ :
    m₂' ∈ (ι Q₂).range ^ (↑(Subtype.mk (p := fun (n : ℕ) ↦ Eq (α := ZMod 2) (↑n : ZMod 2) (↑i₂n : ZMod 2)) i₂n sorry) : ℕ)),
  HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
      ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ) m₂' : CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (β := ZMod 2) (-1) (↑i₂n : ZMod 2) •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
        ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ) m₂' : CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
Tactic: obtain ⟨i₂n, rfl⟩ := i₂'
Cleared hypotheses: i₂, i₂', hm₂
-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.Prod (goal)
Goal:
R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M₁
inst_2 : AddCommGroup M₂
inst_3 : AddCommGroup N
inst_4 : Module R M₁
inst_5 : Module R M₂
inst_6 : Module R N
Q₁ : QuadraticForm R M₁
Q₂ : QuadraticForm R M₂
Qₙ : QuadraticForm R N
f₁ : Q₁ →qᵢ Qₙ
f₂ : Q₂ →qᵢ Qₙ
hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ ((f₁ : (a : M₁) → N) x) ((f₂ : (a : M₂) → N) y)
m₁ m₁' : CliffordAlgebra Q₁
i₁n i : ℕ
x₁ : CliffordAlgebra Q₁
_hx₁ : x₁ ∈ (ι Q₁).range ^ i
v₁ : M₁
m₂' : CliffordAlgebra Q₂
i₂n : ℕ
r : R
⊢ HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
      ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ)
          ((algebraMap R (CliffordAlgebra Q₂) : (a : R) → CliffordAlgebra Q₂) r) :
        CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (β := ZMod 2) (-1) (↑0 : ZMod 2) •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
        ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ)
            ((algebraMap R (CliffordAlgebra Q₂) : (a : R) → CliffordAlgebra Q₂) r) :
          CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
have mem : HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
    ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
    ((algebraMap R (CliffordAlgebra Qₙ) : (a : R) → CliffordAlgebra Qₙ) r : CliffordAlgebra Qₙ) =
  HPow.hPow (α := ℤˣ) (β := ZMod 2) (-1) (↑0 : ZMod 2) •
    HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((algebraMap R (CliffordAlgebra Qₙ) : (a : R) → CliffordAlgebra Qₙ) r : CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
Tactic: rw [AlgHom.commutes, Nat.cast_zero, uzpow_zero, one_smul, Algebra.commutes]
-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.Prod (goal)
Goal:
R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M₁
inst_2 : AddCommGroup M₂
inst_3 : AddCommGroup N
inst_4 : Module R M₁
inst_5 : Module R M₂
inst_6 : Module R N
Q₁ : QuadraticForm R M₁
Q₂ : QuadraticForm R M₂
Qₙ : QuadraticForm R N
f₁ : Q₁ →qᵢ Qₙ
f₂ : Q₂ →qᵢ Qₙ
hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ ((f₁ : (a : M₁) → N) x) ((f₂ : (a : M₂) → N) y)
m₁ m₁' : CliffordAlgebra Q₁
i₁n i : ℕ
x₁ : CliffordAlgebra Q₁
_hx₁ : x₁ ∈ (ι Q₁).range ^ i
v₁ : M₁
m₂' : CliffordAlgebra Q₂
i₂n : ℕ
r : R
mem :
  HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
      ((algebraMap R (CliffordAlgebra Qₙ) : (a : R) → CliffordAlgebra Qₙ) r : CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (β := ZMod 2) (-1) (↑0 : ZMod 2) •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
        ((algebraMap R (CliffordAlgebra Qₙ) : (a : R) → CliffordAlgebra Qₙ) r : CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
⊢ HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
      ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ)
          ((algebraMap R (CliffordAlgebra Q₂) : (a : R) → CliffordAlgebra Q₂) r) :
        CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (β := ZMod 2) (-1) (↑0 : ZMod 2) •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
        ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ)
            ((algebraMap R (CliffordAlgebra Q₂) : (a : R) → CliffordAlgebra Q₂) r) :
          CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
have mem₁ : HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
    ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
    ((algebraMap R (CliffordAlgebra Qₙ) : (a : R) → CliffordAlgebra Qₙ) r : CliffordAlgebra Qₙ) =
  HPow.hPow (α := ℤˣ) (β := ZMod 2) (-1) 0 •
    HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((algebraMap R (CliffordAlgebra Qₙ) : (a : R) → CliffordAlgebra Qₙ) r : CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
Tactic: rw [AlgHom.commutes, Nat.cast_zero, uzpow_zero, one_smul, Algebra.commutes]
-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.Prod (goal)
Goal:
R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M₁
inst_2 : AddCommGroup M₂
inst_3 : AddCommGroup N
inst_4 : Module R M₁
inst_5 : Module R M₂
inst_6 : Module R N
Q₁ : QuadraticForm R M₁
Q₂ : QuadraticForm R M₂
Qₙ : QuadraticForm R N
f₁ : Q₁ →qᵢ Qₙ
f₂ : Q₂ →qᵢ Qₙ
hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ ((f₁ : (a : M₁) → N) x) ((f₂ : (a : M₂) → N) y)
m₁ m₁' : CliffordAlgebra Q₁
i₁n i : ℕ
x₁ : CliffordAlgebra Q₁
_hx₁ : x₁ ∈ (ι Q₁).range ^ i
v₁ : M₁
m₂' : CliffordAlgebra Q₂
i₂n : ℕ
r : R
mem :
  HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
      ((algebraMap R (CliffordAlgebra Qₙ) : (a : R) → CliffordAlgebra Qₙ) r : CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (β := ZMod 2) (-1) (↑0 : ZMod 2) •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
        ((algebraMap R (CliffordAlgebra Qₙ) : (a : R) → CliffordAlgebra Qₙ) r : CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
mem₁ :
  HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
      ((algebraMap R (CliffordAlgebra Qₙ) : (a : R) → CliffordAlgebra Qₙ) r : CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (β := ZMod 2) (-1) 0 •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
        ((algebraMap R (CliffordAlgebra Qₙ) : (a : R) → CliffordAlgebra Qₙ) r : CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
⊢ HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
      ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ)
          ((algebraMap R (CliffordAlgebra Q₂) : (a : R) → CliffordAlgebra Q₂) r) :
        CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (β := ZMod 2) (-1) (↑0 : ZMod 2) •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
        ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ)
            ((algebraMap R (CliffordAlgebra Q₂) : (a : R) → CliffordAlgebra Q₂) r) :
          CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
have mem₂ : HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
    ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
    ((algebraMap R (CliffordAlgebra Qₙ) : (a : R) → CliffordAlgebra Qₙ) r : CliffordAlgebra Qₙ) =
  HSMul.hSMul (α := ℤˣ) 1
    (HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((algebraMap R (CliffordAlgebra Qₙ) : (a : R) → CliffordAlgebra Qₙ) r : CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ))
Tactic: rw [AlgHom.commutes, Nat.cast_zero, uzpow_zero, one_smul, Algebra.commutes]
-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.Prod (goal)
Goal:
R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M₁
inst_2 : AddCommGroup M₂
inst_3 : AddCommGroup N
inst_4 : Module R M₁
inst_5 : Module R M₂
inst_6 : Module R N
Q₁ : QuadraticForm R M₁
Q₂ : QuadraticForm R M₂
Qₙ : QuadraticForm R N
f₁ : Q₁ →qᵢ Qₙ
f₂ : Q₂ →qᵢ Qₙ
hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ ((f₁ : (a : M₁) → N) x) ((f₂ : (a : M₂) → N) y)
m₁ m₁' : CliffordAlgebra Q₁
i₁n i_1 : ℕ
x₁ : CliffordAlgebra Q₁
_hx₁ : x₁ ∈ (ι Q₁).range ^ i_1
v₁ : M₁
m₂' : CliffordAlgebra Q₂
i₂n : ℕ
m₂ : CliffordAlgebra Q₂
hm₂ : m₂ ∈ (ι Q₂).range
i : ℕ
x₂ : CliffordAlgebra Q₂
_hx₂ : x₂ ∈ (ι Q₂).range ^ i
ih₂ :
  HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
      ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ) x₂ : CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (β := ZMod 2) (-1) (↑i : ZMod 2) •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
        ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ) x₂ : CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
⊢ HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
      ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ) (m₂ * x₂) : CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (β := ZMod 2) (-1) (↑i.succ : ZMod 2) •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
        ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ) (m₂ * x₂) : CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
have mem : ∀ (v₂ : M₂),
  HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
      ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ)
          (HMul.hMul (α := CliffordAlgebra Q₂) ((ι Q₂ : (a : M₂) → CliffordAlgebra Q₂) v₂ : CliffordAlgebra Q₂) x₂) :
        CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (β := ZMod 2) (-1) (↑i.succ : ZMod 2) •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
        ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ)
            (HMul.hMul (α := CliffordAlgebra Q₂) ((ι Q₂ : (a : M₂) → CliffordAlgebra Q₂) v₂ : CliffordAlgebra Q₂) x₂) :
          CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
Tactic: obtain ⟨v₂, rfl⟩ := hm₂
Cleared hypotheses: m₂, hm₂
-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.Prod (goal)
Goal:
R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M₁
inst_2 : AddCommGroup M₂
inst_3 : AddCommGroup N
inst_4 : Module R M₁
inst_5 : Module R M₂
inst_6 : Module R N
Q₁ : QuadraticForm R M₁
Q₂ : QuadraticForm R M₂
Qₙ : QuadraticForm R N
f₁ : Q₁ →qᵢ Qₙ
f₂ : Q₂ →qᵢ Qₙ
hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ ((f₁ : (a : M₁) → N) x) ((f₂ : (a : M₂) → N) y)
m₁ m₁' : CliffordAlgebra Q₁
i₁n i : ℕ
x₁ : CliffordAlgebra Q₁
_hx₁ : x₁ ∈ (ι Q₁).range ^ i
v₁ : M₁
m₂' : CliffordAlgebra Q₂
i₂n : ℕ
hm₂ : m₂' ∈ (ι Q₂).range ^ i₂n
⊢ HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
      ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ) m₂' : CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (β := ZMod 2) (-1) (↑i₂n : ZMod 2) •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
        ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ) m₂' : CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
have mem : ∀ (r : R),
  HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
      ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ)
          ((algebraMap R (CliffordAlgebra Q₂) : (a : R) → CliffordAlgebra Q₂) r) :
        CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (β := ZMod 2) (-1) (↑0 : ZMod 2) •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
        ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ)
            ((algebraMap R (CliffordAlgebra Q₂) : (a : R) → CliffordAlgebra Q₂) r) :
          CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
Tactic: induction hm₂ using Submodule.pow_induction_on_left' with
        | algebraMap =>
          rw [AlgHom.commutes, Nat.cast_zero, uzpow_zero, one_smul, Algebra.commutes]
        | add _ _ _ _ _ ihx ihy =>
          rw [map_add, add_mul, mul_add, ihx, ihy, smul_add]
        | mem_mul m₂ hm₂ i x₂ _hx₂ ih₂ =>
          obtain ⟨v₂, rfl⟩ := hm₂
          -- This is the second interesting goal.
          rw [map_mul, map_apply_ι, Nat.cast_succ, ← mul_assoc,
            ι_mul_ι_comm_of_isOrtho (hf _ _), neg_mul, mul_assoc, ih₂, mul_smul_comm,
            ← mul_assoc, ← Units.neg_smul, uzpow_add, uzpow_one, mul_neg_one]
Cleared hypotheses: hm₂
-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.Prod (goal)
Goal:
R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M₁
inst_2 : AddCommGroup M₂
inst_3 : AddCommGroup N
inst_4 : Module R M₁
inst_5 : Module R M₂
inst_6 : Module R N
Q₁ : QuadraticForm R M₁
Q₂ : QuadraticForm R M₂
Qₙ : QuadraticForm R N
f₁ : Q₁ →qᵢ Qₙ
f₂ : Q₂ →qᵢ Qₙ
hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ ((f₁ : (a : M₁) → N) x) ((f₂ : (a : M₂) → N) y)
m₁ : CliffordAlgebra Q₁
m₂ : CliffordAlgebra Q₂
i₂ : ZMod 2
hm₂ : m₂ ∈ evenOdd Q₂ i₂
m₁' : CliffordAlgebra Q₁
i₁n i : ℕ
x₁ : CliffordAlgebra Q₁
_hx₁ : x₁ ∈ (ι Q₁).range ^ i
v₁ : M₁
⊢ HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
      ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ) m₂ : CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (-1) i₂ •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
        ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ) m₂ : CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
have mem : HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
    ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
    ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ) 0 : CliffordAlgebra Qₙ) =
  HPow.hPow (α := ℤˣ) (-1) i₂ •
    HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ) 0 : CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
Tactic: induction hm₂ using Submodule.iSup_induction' with
      | zero => rw [map_zero, zero_mul, mul_zero, smul_zero]
      | add _ _ _ _ ihx ihy => rw [map_add, add_mul, mul_add, ihx, ihy, smul_add]
      | mem i₂' m₂' hm₂ =>
        clear m₂
        obtain ⟨i₂n, rfl⟩ := i₂'
        dsimp only at *
        induction hm₂ using Submodule.pow_induction_on_left' with
        | algebraMap =>
          rw [AlgHom.commutes, Nat.cast_zero, uzpow_zero, one_smul, Algebra.commutes]
        | add _ _ _ _ _ ihx ihy =>
          rw [map_add, add_mul, mul_add, ihx, ihy, smul_add]
        | mem_mul m₂ hm₂ i x₂ _hx₂ ih₂ =>
          obtain ⟨v₂, rfl⟩ := hm₂
          -- This is the second interesting goal.
          rw [map_mul, map_apply_ι, Nat.cast_succ, ← mul_assoc,
            ι_mul_ι_comm_of_isOrtho (hf _ _), neg_mul, mul_assoc, ih₂, mul_smul_comm,
            ← mul_assoc, ← Units.neg_smul, uzpow_add, uzpow_one, mul_neg_one]
Cleared hypotheses: hm₂