awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.LinearAlgebra.Complex.Module (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : Algebra ℝ A
I' : A
hf : I' * I' = -1
x₁ y₁ x₂ y₂ : ℝ
e_a :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A) (y₂ • I') +
      HMul.hMul (β := A) (y₁ • I') ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)
e_a₁ :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A)
        (HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) y₂ : A) I') +
      HMul.hMul (β := A) (y₁ • I') ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)
e_a₂ :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A)
        (HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) y₂ : A) I') +
      HMul.hMul (β := A) (HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) y₁ : A) I')
        ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)
e_a₃ :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A)
        (HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) y₂ : A) I') +
      HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) y₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A) * I'
e_a₄ :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) y₂ : A) * I' +
      HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) y₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A) * I'
⊢ (x₁ * y₂ + y₁ * x₂) • I' =
    HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A) (y₂ • I') +
      HMul.hMul (β := A) (y₁ • I') ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)
have e_a₅ : HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
  (HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) y₂ : A) +
      HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) y₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)) *
    I'
Tactic: rw [smul_def, smul_def, smul_def, ← right_comm _ x₂,
      ← mul_assoc, ← add_mul, ← map_mul, ← map_mul, ← map_add]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Complex.Module (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : Algebra ℝ A
I' : A
hf : I' * I' = -1
x₁ y₁ x₂ y₂ : ℝ
e_a :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A) (y₂ • I') +
      HMul.hMul (β := A) (y₁ • I') ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)
e_a₁ :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A)
        (HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) y₂ : A) I') +
      HMul.hMul (β := A) (y₁ • I') ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)
e_a₂ :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A)
        (HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) y₂ : A) I') +
      HMul.hMul (β := A) (HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) y₁ : A) I')
        ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)
e_a₃ :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A)
        (HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) y₂ : A) I') +
      HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) y₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A) * I'
e_a₄ :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) y₂ : A) * I' +
      HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) y₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A) * I'
e_a₅ :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    (HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) y₂ : A) +
        HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) y₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)) *
      I'
⊢ (x₁ * y₂ + y₁ * x₂) • I' =
    HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A) (y₂ • I') +
      HMul.hMul (β := A) (y₁ • I') ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)
have e_a₆ : HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
  HAdd.hAdd (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂) : A)
      (HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) y₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)) *
    I'
Tactic: rw [smul_def, smul_def, smul_def, ← right_comm _ x₂,
      ← mul_assoc, ← add_mul, ← map_mul, ← map_mul, ← map_add]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Complex.Module (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : Algebra ℝ A
I' : A
hf : I' * I' = -1
x₁ y₁ x₂ y₂ : ℝ
e_a :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A) (y₂ • I') +
      HMul.hMul (β := A) (y₁ • I') ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)
e_a₁ :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A)
        (HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) y₂ : A) I') +
      HMul.hMul (β := A) (y₁ • I') ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)
e_a₂ :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A)
        (HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) y₂ : A) I') +
      HMul.hMul (β := A) (HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) y₁ : A) I')
        ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)
e_a₃ :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A)
        (HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) y₂ : A) I') +
      HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) y₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A) * I'
e_a₄ :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) y₂ : A) * I' +
      HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) y₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A) * I'
e_a₅ :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    (HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) y₂ : A) +
        HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) y₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)) *
      I'
e_a₆ :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HAdd.hAdd (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂) : A)
        (HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) y₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)) *
      I'
⊢ (x₁ * y₂ + y₁ * x₂) • I' =
    HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A) (y₂ • I') +
      HMul.hMul (β := A) (y₁ • I') ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)
have e_a₇ : HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
  HAdd.hAdd (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂) : A)
      ((algebraMap ℝ A : (a : ℝ) → A) (y₁ * x₂) : A) *
    I'
Tactic: rw [smul_def, smul_def, smul_def, ← right_comm _ x₂,
      ← mul_assoc, ← add_mul, ← map_mul, ← map_mul, ← map_add]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Complex.Module (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : Algebra ℝ A
I' : A
hf : I' * I' = -1
x₁ y₁ x₂ y₂ : ℝ
e_a :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A) (y₂ • I') +
      HMul.hMul (β := A) (y₁ • I') ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)
e_a₁ :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A)
        (HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) y₂ : A) I') +
      HMul.hMul (β := A) (y₁ • I') ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)
e_a₂ :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A)
        (HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) y₂ : A) I') +
      HMul.hMul (β := A) (HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) y₁ : A) I')
        ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)
e_a₃ :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A)
        (HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) y₂ : A) I') +
      HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) y₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A) * I'
e_a₄ :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) y₂ : A) * I' +
      HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) y₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A) * I'
e_a₅ :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    (HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) y₂ : A) +
        HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) y₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)) *
      I'
e_a₆ :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HAdd.hAdd (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂) : A)
        (HMul.hMul (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) y₁ : A) ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)) *
      I'
e_a₇ :
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
    HAdd.hAdd (α := A) (β := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂) : A)
        ((algebraMap ℝ A : (a : ℝ) → A) (y₁ * x₂) : A) *
      I'
⊢ (x₁ * y₂ + y₁ * x₂) • I' =
    HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A) (y₂ • I') +
      HMul.hMul (β := A) (y₁ • I') ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A)
have e_a₈ : HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I' =
  HMul.hMul (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * y₂ + y₁ * x₂) : A) I'
Tactic: rw [smul_def, smul_def, smul_def, ← right_comm _ x₂,
      ← mul_assoc, ← add_mul, ← map_mul, ← map_mul, ← map_add]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Complex.Module (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : Algebra ℝ A
I' : A
hf : I' * I' = -1
⊢ ∀ (x y : ℂ),
    (HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) :
          (a : ℂ) → A)
        (x * y) =
      HMul.hMul (α := A) (β := A)
        ((HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) :
              (a : ℂ) → A)
            x :
          A)
        ((HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) :
              (a : ℂ) → A)
            y :
          A)
have rintro : ∀ (x₁ y₁ x₂ y₂ : ℝ),
  (HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) : (a : ℂ) → A)
      ({ re := x₁, im := y₁ : ℂ } * { re := x₂, im := y₂ : ℂ }) =
    HMul.hMul (α := A) (β := A)
      ((HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) :
            (a : ℂ) → A)
          { re := x₁, im := y₁ : ℂ } :
        A)
      ((HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) :
            (a : ℂ) → A)
          { re := x₂, im := y₂ : ℂ } :
        A)
Tactic: finally
  rintro ⟨x₁, y₁⟩ ⟨x₂, y₂⟩
  rw [mk_mul_mk]
  change
    algebraMap ℝ A (x₁ * x₂ - y₁ * y₂) + (x₁ * y₂ + y₁ * x₂) • I' =
      (algebraMap ℝ A x₁ + y₁ • I') * (algebraMap ℝ A x₂ + y₂ • I')
  rw [add_mul, mul_add, mul_add, add_comm _ (y₁ • I' * y₂ • I'), add_add_add_comm]
  congr 1
  -- equate "real" and "imaginary" parts
  · rw [smul_mul_smul_comm, hf, smul_neg, ← algebraMap_eq_smul_one, ← sub_eq_add_neg,
      ← map_mul, ← map_sub]
  · rw [smul_def, smul_def, smul_def, ← right_comm _ x₂,
      ← mul_assoc, ← add_mul, ← map_mul, ← map_mul, ← map_add]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Complex.Module (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : Algebra ℝ A
I' : A
hf : I' * I' = -1
rintro :
  ∀ (x₁ y₁ x₂ y₂ : ℝ),
    (HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) :
          (a : ℂ) → A)
        ({ re := x₁, im := y₁ : ℂ } * { re := x₂, im := y₂ : ℂ }) =
      HMul.hMul (α := A) (β := A)
        ((HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) :
              (a : ℂ) → A)
            { re := x₁, im := y₁ : ℂ } :
          A)
        ((HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) :
              (a : ℂ) → A)
            { re := x₂, im := y₂ : ℂ } :
          A)
⊢ ∀ (x y : ℂ),
    (HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) :
          (a : ℂ) → A)
        (x * y) =
      HMul.hMul (α := A) (β := A)
        ((HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) :
              (a : ℂ) → A)
            x :
          A)
        ((HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) :
              (a : ℂ) → A)
            y :
          A)
have rintro₁ : ∀ (x₁ y₁ x₂ y₂ : ℝ),
  (HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) : (a : ℂ) → A)
      { re := x₁ * x₂ - y₁ * y₂, im := x₁ * y₂ + y₁ * x₂ : ℂ } =
    HMul.hMul (α := A) (β := A)
      ((HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) :
            (a : ℂ) → A)
          { re := x₁, im := y₁ : ℂ } :
        A)
      ((HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) :
            (a : ℂ) → A)
          { re := x₂, im := y₂ : ℂ } :
        A)
Tactic: finally
  rintro ⟨x₁, y₁⟩ ⟨x₂, y₂⟩
  rw [mk_mul_mk]
  change
    algebraMap ℝ A (x₁ * x₂ - y₁ * y₂) + (x₁ * y₂ + y₁ * x₂) • I' =
      (algebraMap ℝ A x₁ + y₁ • I') * (algebraMap ℝ A x₂ + y₂ • I')
  rw [add_mul, mul_add, mul_add, add_comm _ (y₁ • I' * y₂ • I'), add_add_add_comm]
  congr 1
  -- equate "real" and "imaginary" parts
  · rw [smul_mul_smul_comm, hf, smul_neg, ← algebraMap_eq_smul_one, ← sub_eq_add_neg,
      ← map_mul, ← map_sub]
  · rw [smul_def, smul_def, smul_def, ← right_comm _ x₂,
      ← mul_assoc, ← add_mul, ← map_mul, ← map_mul, ← map_add]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Complex.Module (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : Algebra ℝ A
I' : A
hf : I' * I' = -1
rintro :
  ∀ (x₁ y₁ x₂ y₂ : ℝ),
    (HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) :
          (a : ℂ) → A)
        ({ re := x₁, im := y₁ : ℂ } * { re := x₂, im := y₂ : ℂ }) =
      HMul.hMul (α := A) (β := A)
        ((HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) :
              (a : ℂ) → A)
            { re := x₁, im := y₁ : ℂ } :
          A)
        ((HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) :
              (a : ℂ) → A)
            { re := x₂, im := y₂ : ℂ } :
          A)
rintro₁ :
  ∀ (x₁ y₁ x₂ y₂ : ℝ),
    (HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) :
          (a : ℂ) → A)
        { re := x₁ * x₂ - y₁ * y₂, im := x₁ * y₂ + y₁ * x₂ : ℂ } =
      HMul.hMul (α := A) (β := A)
        ((HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) :
              (a : ℂ) → A)
            { re := x₁, im := y₁ : ℂ } :
          A)
        ((HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) :
              (a : ℂ) → A)
            { re := x₂, im := y₂ : ℂ } :
          A)
⊢ ∀ (x y : ℂ),
    (HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) :
          (a : ℂ) → A)
        (x * y) =
      HMul.hMul (α := A) (β := A)
        ((HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) :
              (a : ℂ) → A)
            x :
          A)
        ((HAdd.hAdd (α := ℂ →ₗ[ℝ] A) (Algebra.linearMap ℝ A ∘ₗ reLm) (LinearMap.toSpanSingleton ℝ A I' ∘ₗ imLm) :
              (a : ℂ) → A)
            y :
          A)
have rintro₂ : ∀ (x₁ y₁ x₂ y₂ : ℝ),
  HAdd.hAdd (α := A) ((algebraMap ℝ A : (a : ℝ) → A) (x₁ * x₂ - y₁ * y₂) : A) ((x₁ * y₂ + y₁ * x₂) • I') =
    HAdd.hAdd (α := A) ((algebraMap ℝ A : (a : ℝ) → A) x₁ : A) (y₁ • I') *
      HAdd.hAdd (α := A) ((algebraMap ℝ A : (a : ℝ) → A) x₂ : A) (y₂ • I')
Tactic: finally
  rintro ⟨x₁, y₁⟩ ⟨x₂, y₂⟩
  rw [mk_mul_mk]
  change
    algebraMap ℝ A (x₁ * x₂ - y₁ * y₂) + (x₁ * y₂ + y₁ * x₂) • I' =
      (algebraMap ℝ A x₁ + y₁ • I') * (algebraMap ℝ A x₂ + y₂ • I')
  rw [add_mul, mul_add, mul_add, add_comm _ (y₁ • I' * y₂ • I'), add_add_add_comm]
  congr 1
  -- equate "real" and "imaginary" parts
  · rw [smul_mul_smul_comm, hf, smul_neg, ← algebraMap_eq_smul_one, ← sub_eq_add_neg,
      ← map_mul, ← map_sub]
  · rw [smul_def, smul_def, smul_def, ← right_comm _ x₂,
      ← mul_assoc, ← add_mul, ← map_mul, ← map_mul, ← map_add]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Complex.Module (goal)
Goal:
⊢ Eq (α := Subalgebra ℝ ℂ) ℝ[I] ⊤
have refine : ∀ (x : ℂ) (hx : Membership.mem (γ := Subalgebra ℝ ℂ) ⊤ x), Membership.mem (γ := Subalgebra ℝ ℂ) ℝ[I] x
Tactic: refine top_unique fun x hx => ?_
-- Lean 4 proof step from Mathlib.LinearAlgebra.Complex.Module (goal)
Goal:
x : ℂ
⊢ Membership.mem (γ := Subalgebra ℝ ℂ) ℝ[I] x
have rw : Membership.mem (γ := Subalgebra ℝ ℂ) ℝ[I]
  (HAdd.hAdd (α := ℂ) ((algebraMap ℝ ℂ : (a : ℝ) → ℂ) x.re : ℂ)
    (HSMul.hSMul (α := ℂ) ((algebraMap ℝ ℂ : (a : ℝ) → ℂ) x.im : ℂ) I))
Tactic: rw [← x.re_add_im, ← smul_eq_mul, ← Complex.coe_algebraMap]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Complex.Module (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : Algebra ℝ A
I' : A
hI' : I' * I' = -1
⊢ Eq (α := Subalgebra ℝ A) (AlgHom.range (liftAux I' hI')) ℝ[I']
have simp_rw : Eq (α := Subalgebra ℝ A) (Subalgebra.map (liftAux I' hI') ⊤) ℝ[I']
Tactic: simp_rw [← Algebra.map_top, ← adjoin_I, AlgHom.map_adjoin, Set.image_singleton, liftAux_apply_I]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Complex.Module (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : Algebra ℝ A
I' : A
hI' : I' * I' = -1
simp_rw : Eq (α := Subalgebra ℝ A) (Subalgebra.map (liftAux I' hI') ⊤) ℝ[I']
⊢ Eq (α := Subalgebra ℝ A) (AlgHom.range (liftAux I' hI')) ℝ[I']
have simp_rw₁ : Eq (α := Subalgebra ℝ A) (Subalgebra.map (liftAux I' hI') ℝ[I]) ℝ[I']
Tactic: simp_rw [← Algebra.map_top, ← adjoin_I, AlgHom.map_adjoin, Set.image_singleton, liftAux_apply_I]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Complex.Module (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : Algebra ℝ A
I' : A
hI' : I' * I' = -1
simp_rw : Eq (α := Subalgebra ℝ A) (Subalgebra.map (liftAux I' hI') ⊤) ℝ[I']
simp_rw₁ : Eq (α := Subalgebra ℝ A) (Subalgebra.map (liftAux I' hI') ℝ[I]) ℝ[I']
⊢ Eq (α := Subalgebra ℝ A) (AlgHom.range (liftAux I' hI')) ℝ[I']
have simp_rw₂ : Eq (α := Subalgebra ℝ A) (adjoin ℝ (Set.image (α := ℂ) (⇑(liftAux I' hI') : (a : ℂ) → A) {I})) ℝ[I']
Tactic: simp_rw [← Algebra.map_top, ← adjoin_I, AlgHom.map_adjoin, Set.image_singleton, liftAux_apply_I]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Complex.Module (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : Algebra ℝ A
I' : A
hI' : I' * I' = -1
simp_rw : Eq (α := Subalgebra ℝ A) (Subalgebra.map (liftAux I' hI') ⊤) ℝ[I']
simp_rw₁ : Eq (α := Subalgebra ℝ A) (Subalgebra.map (liftAux I' hI') ℝ[I]) ℝ[I']
simp_rw₂ : Eq (α := Subalgebra ℝ A) (adjoin ℝ (Set.image (α := ℂ) (⇑(liftAux I' hI') : (a : ℂ) → A) {I})) ℝ[I']
⊢ Eq (α := Subalgebra ℝ A) (AlgHom.range (liftAux I' hI')) ℝ[I']
have simp_rw₃ : Eq (α := Subalgebra ℝ A) ℝ[((liftAux I' hI' : (a : ℂ) → A) I : A)] ℝ[I']
Tactic: simp_rw [← Algebra.map_top, ← adjoin_I, AlgHom.map_adjoin, Set.image_singleton, liftAux_apply_I]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Complex.Module (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : Algebra ℝ A
F : ℂ →ₐ[ℝ] A
⊢ HMul.hMul (α := A) (β := A) ((F : (a : ℂ) → A) I : A) ((F : (a : ℂ) → A) I : A) = -1
have rw : Eq (α := A) ((F : (a : ℂ) → A) (I * I) : A) (-1 : A)
Tactic: rw [← map_mul, I_mul_I, map_neg, map_one]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Complex.Module (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : Algebra ℝ A
F : ℂ →ₐ[ℝ] A
rw : Eq (α := A) ((F : (a : ℂ) → A) (I * I) : A) (-1 : A)
⊢ HMul.hMul (α := A) (β := A) ((F : (a : ℂ) → A) I : A) ((F : (a : ℂ) → A) I : A) = -1
have rw₁ : Eq (α := A) ((F : (a : ℂ) → A) (-1) : A) (-1 : A)
Tactic: rw [← map_mul, I_mul_I, map_neg, map_one]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Complex.Module (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : Algebra ℝ A
F : ℂ →ₐ[ℝ] A
rw : Eq (α := A) ((F : (a : ℂ) → A) (I * I) : A) (-1 : A)
rw₁ : Eq (α := A) ((F : (a : ℂ) → A) (-1) : A) (-1 : A)
⊢ HMul.hMul (α := A) (β := A) ((F : (a : ℂ) → A) I : A) ((F : (a : ℂ) → A) I : A) = -1
have rw₂ : Eq (α := A) (-(F : (a : ℂ) → A) 1 : A) (-1 : A)
Tactic: rw [← map_mul, I_mul_I, map_neg, map_one]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Complex.Module (goal)
Goal:
A : Type u_1
inst : Ring A
inst_1 : Algebra ℝ A
F : ℂ →ₐ[ℝ] A
rw : Eq (α := A) ((F : (a : ℂ) → A) (I * I) : A) (-1 : A)
rw₁ : Eq (α := A) ((F : (a : ℂ) → A) (-1) : A) (-1 : A)
rw₂ : Eq (α := A) (-(F : (a : ℂ) → A) 1 : A) (-1 : A)
⊢ HMul.hMul (α := A) (β := A) ((F : (a : ℂ) → A) I : A) ((F : (a : ℂ) → A) I : A) = -1
have rw₃ : Eq.{u_1 + 1} (α := A) (-1) (-1)
Tactic: rw [← map_mul, I_mul_I, map_neg, map_one]
-- Lean 4 proof step from Mathlib.LinearAlgebra.Complex.Module (goal)
Goal:
A : Type u_1
inst : AddCommGroup A
inst_1 : Module ℂ A
inst_2 : StarAddMonoid A
inst_3 : StarModule ℂ A
a b : ↥(skewAdjoint A)
⊢ Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := AddSubgroup A) (selfAdjoint A) x) (-I • (↑(a + b) : A)) ⋯ =
    Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := AddSubgroup A) (selfAdjoint A) x) (-I • (↑a : A)) ⋯ +
      Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := AddSubgroup A) (selfAdjoint A) x) (-I • (↑b : A)) ⋯
have a₁ : (↑(Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := AddSubgroup A) (selfAdjoint A) x) (-I • (↑(a + b) : A)) sorry) : A) =
  (↑(Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := AddSubgroup A) (selfAdjoint A) x) (-I • (↑a : A)) sorry +
        Subtype.mk (p := fun (x : A) ↦ Membership.mem (γ := AddSubgroup A) (selfAdjoint A) x) (-I • (↑b : A)) sorry) :
    A)
Tactic: ext