awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (hypothesis)
Goal:
f : ℂ →ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
z : ℂ
this :
  HSub.hSub (β := ℂ) (↑(HPow.hPow (β := ℕ) ‖z‖ 2) : ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ) -
      HSub.hSub (β := ℂ) (HMul.hMul (α := ℂ) (β := ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) 1 : ℂ))
        ((starRingEnd ℂ : (a : ℂ) → ℂ) 1 : ℂ) =
    HSub.hSub (β := ℂ) (HMul.hMul (β := ℂ) z ((starRingEnd ℂ : (a : ℂ) → ℂ) z : ℂ))
        ((starRingEnd ℂ : (a : ℂ) → ℂ) z : ℂ) -
      HSub.hSub (β := ℂ) (HMul.hMul (β := ℂ) z ((starRingEnd ℂ : (a : ℂ) → ℂ) 1 : ℂ))
        ((starRingEnd ℂ : (a : ℂ) → ℂ) 1 : ℂ)
⊢ HAdd.hAdd (β := ℂ) z ((starRingEnd ℂ : (a : ℂ) → ℂ) z : ℂ) =
    HAdd.hAdd (α := ℂ) (β := ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ)
have this₁ : HSub.hSub (β := ℂ) (↑(HPow.hPow (β := ℕ) ‖z‖ 2) : ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ) -
    HSub.hSub (β := ℂ) (HMul.hMul (α := ℂ) (β := ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) 1 : ℂ))
      ((starRingEnd ℂ : (a : ℂ) → ℂ) 1 : ℂ) =
  HSub.hSub (β := ℂ) (↑(HPow.hPow (β := ℕ) ‖z‖ 2) : ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) z : ℂ) -
    HSub.hSub (β := ℂ) (HMul.hMul (β := ℂ) z ((starRingEnd ℂ : (a : ℂ) → ℂ) 1 : ℂ))
      ((starRingEnd ℂ : (a : ℂ) → ℂ) 1 : ℂ)
Tactic: rw [mul_conj, normSq_eq_norm_sq] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (hypothesis)
Goal:
f : ℂ →ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
z : ℂ
this :
  HSub.hSub (β := ℂ) (↑(HPow.hPow (β := ℕ) ‖z‖ 2) : ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ) -
      HSub.hSub (β := ℂ) (HMul.hMul (α := ℂ) (β := ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) 1 : ℂ))
        ((starRingEnd ℂ : (a : ℂ) → ℂ) 1 : ℂ) =
    HSub.hSub (β := ℂ) (↑(HPow.hPow (β := ℕ) ‖z‖ 2) : ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) z : ℂ) -
      HSub.hSub (β := ℂ) (HMul.hMul (β := ℂ) z ((starRingEnd ℂ : (a : ℂ) → ℂ) 1 : ℂ))
        ((starRingEnd ℂ : (a : ℂ) → ℂ) 1 : ℂ)
⊢ HAdd.hAdd (β := ℂ) z ((starRingEnd ℂ : (a : ℂ) → ℂ) z : ℂ) =
    HAdd.hAdd (α := ℂ) (β := ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ)
have this₁ : HAdd.hAdd (α := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ)
    (HSub.hSub (α := ℂ) (β := ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) (1 : ℂ)) =
  HAdd.hAdd (α := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) z : ℂ) (z - 1)
Tactic: simp only [sub_sub, sub_right_inj, mul_one, ofReal_pow, map_one] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (hypothesis)
Goal:
f : ℂ →ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
z : ℂ
this :
  HAdd.hAdd (α := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ)
      (HSub.hSub (α := ℂ) (β := ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) (1 : ℂ)) =
    HAdd.hAdd (α := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) z : ℂ) (z - 1)
⊢ HAdd.hAdd (β := ℂ) z ((starRingEnd ℂ : (a : ℂ) → ℂ) z : ℂ) =
    HAdd.hAdd (α := ℂ) (β := ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ)
have this₁ : HAdd.hAdd (α := ℂ) (β := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) =
  HAdd.hAdd (α := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) z : ℂ) z
Tactic: simp only [add_sub, sub_left_inj] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
f : ℂ →ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
z : ℂ
this :
  HAdd.hAdd (α := ℂ) (β := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) =
    HAdd.hAdd (α := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) z : ℂ) z
⊢ HAdd.hAdd (β := ℂ) z ((starRingEnd ℂ : (a : ℂ) → ℂ) z : ℂ) =
    HAdd.hAdd (α := ℂ) (β := ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ)
have rw : HAdd.hAdd (α := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) z : ℂ) z =
  HAdd.hAdd (α := ℂ) (β := ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ)
Tactic: rw [add_comm, ← this, add_comm]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
f : ℂ →ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
z : ℂ
this :
  HAdd.hAdd (α := ℂ) (β := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) =
    HAdd.hAdd (α := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) z : ℂ) z
rw :
  HAdd.hAdd (α := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) z : ℂ) z =
    HAdd.hAdd (α := ℂ) (β := ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ)
⊢ HAdd.hAdd (β := ℂ) z ((starRingEnd ℂ : (a : ℂ) → ℂ) z : ℂ) =
    HAdd.hAdd (α := ℂ) (β := ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ)
have rw₁ : HAdd.hAdd (α := ℂ) (β := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) =
  HAdd.hAdd (α := ℂ) (β := ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ)
Tactic: rw [add_comm, ← this, add_comm]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
f : ℂ →ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
z : ℂ
this :
  HAdd.hAdd (α := ℂ) (β := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) =
    HAdd.hAdd (α := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) z : ℂ) z
rw :
  HAdd.hAdd (α := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) z : ℂ) z =
    HAdd.hAdd (α := ℂ) (β := ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ)
rw₁ :
  HAdd.hAdd (α := ℂ) (β := ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) =
    HAdd.hAdd (α := ℂ) (β := ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ)
⊢ HAdd.hAdd (β := ℂ) z ((starRingEnd ℂ : (a : ℂ) → ℂ) z : ℂ) =
    HAdd.hAdd (α := ℂ) (β := ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ)
have rw₂ : HAdd.hAdd (α := ℂ) (β := ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ) =
  HAdd.hAdd (α := ℂ) (β := ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ)
Tactic: rw [add_comm, ← this, add_comm]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
f : ℂ →ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
z : ℂ
⊢ re ((f : (a : ℂ) → ℂ) z) = z.re
have h₃ : ∀ (z : ℂ),
  HAdd.hAdd (β := ℂ) z ((starRingEnd ℂ : (a : ℂ) → ℂ) z : ℂ) =
    HAdd.hAdd (α := ℂ) (β := ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) ((starRingEnd ℂ : (a : ℂ) → ℂ) ((f : (a : ℂ) → ℂ) z : ℂ) : ℂ)
Tactic: apply LinearIsometry.re_apply_eq_re_of_add_conj_eq
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
⊢ (f : (a : ℂ) → ℂ) I = I ∨ (f : (a : ℂ) → ℂ) I = -I
have simp : re ((f : (a : ℂ) → ℂ) I) = 0 ∧ (im ((f : (a : ℂ) → ℂ) I) = I.im ∨ im ((f : (a : ℂ) → ℂ) I) = -I.im)
Tactic: simp only [Complex.ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
⊢ re ((f : (a : ℂ) → ℂ) I) = 0 ∧ (im ((f : (a : ℂ) → ℂ) I) = I.im ∨ im ((f : (a : ℂ) → ℂ) I) = -I.im)
have left : re ((f : (a : ℂ) → ℂ) I) = 0
Tactic: constructor
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
left : re ((f : (a : ℂ) → ℂ) I) = 0
⊢ re ((f : (a : ℂ) → ℂ) I) = 0 ∧ (im ((f : (a : ℂ) → ℂ) I) = I.im ∨ im ((f : (a : ℂ) → ℂ) I) = -I.im)
have right : im ((f : (a : ℂ) → ℂ) I) = I.im ∨ im ((f : (a : ℂ) → ℂ) I) = -I.im
Tactic: constructor
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
⊢ im ((f : (a : ℂ) → ℂ) I) = I.im ∨ im ((f : (a : ℂ) → ℂ) I) = -I.im
have right : ∀ (z : ℂ), re ((f.toLinearIsometry : (a : ℂ) → ℂ) z) = z.re
Tactic: apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
z : ℂ
⊢ re ((f.toLinearIsometry : (a : ℂ) → ℂ) z) = z.re
have right : z.re = z.re
Tactic: rw [@LinearIsometry.re_apply_eq_re f.toLinearIsometry h]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (hypothesis)
Goal:
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
⊢ f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLIE
have h0 : (f : (a : ℂ) → ℂ) I = I ∨ (f : (a : ℂ) → ℂ) I = -I
Tactic: have h0 : f I = I ∨ f I = -I := by
    simp only [Complex.ext_iff, ← and_or_left, neg_re, I_re, neg_im, neg_zero]
    constructor
    · rw [← I_re]
      exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I
    · apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry
      intro z
      rw [@LinearIsometry.re_apply_eq_re f.toLinearIsometry h]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
h0 : (f : (a : ℂ) → ℂ) I = I ∨ (f : (a : ℂ) → ℂ) I = -I
⊢ f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLIE
have refine_1 : ∀ (h' : (f : (a : ℂ) → ℂ) I = I), f = LinearIsometryEquiv.refl ℝ ℂ
Tactic: refine h0.imp (fun h' : f I = I => ?_) fun h' : f I = -I => ?_
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
h0 : (f : (a : ℂ) → ℂ) I = I ∨ (f : (a : ℂ) → ℂ) I = -I
refine_1 : ∀ (h' : (f : (a : ℂ) → ℂ) I = I), f = LinearIsometryEquiv.refl ℝ ℂ
⊢ f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = conjLIE
have refine_2 : ∀ (h' : (f : (a : ℂ) → ℂ) I = -I), f = conjLIE
Tactic: refine h0.imp (fun h' : f I = I => ?_) fun h' : f I = -I => ?_
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
h0 : (f : (a : ℂ) → ℂ) I = I ∨ (f : (a : ℂ) → ℂ) I = -I
h' : (f : (a : ℂ) → ℂ) I = I
⊢ f = LinearIsometryEquiv.refl ℝ ℂ
have refine_1 : f.toLinearEquiv = LinearIsometryEquiv.toLinearEquiv (LinearIsometryEquiv.refl ℝ ℂ)
Tactic: apply LinearIsometryEquiv.toLinearEquiv_injective
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
h0 : (f : (a : ℂ) → ℂ) I = I ∨ (f : (a : ℂ) → ℂ) I = -I
h' : (f : (a : ℂ) → ℂ) I = I
⊢ f.toLinearEquiv = LinearIsometryEquiv.toLinearEquiv (LinearIsometryEquiv.refl ℝ ℂ)
have refine_1 : ∀ (i : Fin 2),
  Eq (α := ℂ) ((f.toLinearEquiv : (a : ℂ) → ℂ) ((basisOneI : (a : Fin 2) → ℂ) i : ℂ) : ℂ)
    ((LinearIsometryEquiv.toLinearEquiv (LinearIsometryEquiv.refl ℝ ℂ) : (a : ℂ) → ℂ)
        ((basisOneI : (a : Fin 2) → ℂ) i : ℂ) :
      ℂ)
Tactic: apply Complex.basisOneI.ext'
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
h0 : (f : (a : ℂ) → ℂ) I = I ∨ (f : (a : ℂ) → ℂ) I = -I
h' : (f : (a : ℂ) → ℂ) I = I
i : Fin 2
⊢ Eq (α := ℂ) ((f.toLinearEquiv : (a : ℂ) → ℂ) ((basisOneI : (a : Fin 2) → ℂ) i : ℂ) : ℂ)
    ((LinearIsometryEquiv.toLinearEquiv (LinearIsometryEquiv.refl ℝ ℂ) : (a : ℂ) → ℂ)
        ((basisOneI : (a : Fin 2) → ℂ) i : ℂ) :
      ℂ)
have refine_1 : Eq (α := ℂ) ((f.toLinearEquiv : (a : ℂ) → ℂ) ((basisOneI : (a : Fin 2) → ℂ) ((fun (i : Fin 2) ↦ i) ⟨0, sorry⟩) : ℂ) : ℂ)
  ((LinearIsometryEquiv.toLinearEquiv (LinearIsometryEquiv.refl ℝ ℂ) : (a : ℂ) → ℂ)
      ((basisOneI : (a : Fin 2) → ℂ) ((fun (i : Fin 2) ↦ i) ⟨0, sorry⟩) : ℂ) :
    ℂ)
Tactic: fin_cases i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
h0 : (f : (a : ℂ) → ℂ) I = I ∨ (f : (a : ℂ) → ℂ) I = -I
h' : (f : (a : ℂ) → ℂ) I = I
i : Fin 2
refine_1 :
  Eq (α := ℂ) ((f.toLinearEquiv : (a : ℂ) → ℂ) ((basisOneI : (a : Fin 2) → ℂ) ((fun (i : Fin 2) ↦ i) ⟨0, ⋯⟩) : ℂ) : ℂ)
    ((LinearIsometryEquiv.toLinearEquiv (LinearIsometryEquiv.refl ℝ ℂ) : (a : ℂ) → ℂ)
        ((basisOneI : (a : Fin 2) → ℂ) ((fun (i : Fin 2) ↦ i) ⟨0, ⋯⟩) : ℂ) :
      ℂ)
⊢ Eq (α := ℂ) ((f.toLinearEquiv : (a : ℂ) → ℂ) ((basisOneI : (a : Fin 2) → ℂ) i : ℂ) : ℂ)
    ((LinearIsometryEquiv.toLinearEquiv (LinearIsometryEquiv.refl ℝ ℂ) : (a : ℂ) → ℂ)
        ((basisOneI : (a : Fin 2) → ℂ) i : ℂ) :
      ℂ)
have refine_1₁ : Eq (α := ℂ) ((f.toLinearEquiv : (a : ℂ) → ℂ) ((basisOneI : (a : Fin 2) → ℂ) ((fun (i : Fin 2) ↦ i) ⟨1, sorry⟩) : ℂ) : ℂ)
  ((LinearIsometryEquiv.toLinearEquiv (LinearIsometryEquiv.refl ℝ ℂ) : (a : ℂ) → ℂ)
      ((basisOneI : (a : Fin 2) → ℂ) ((fun (i : Fin 2) ↦ i) ⟨1, sorry⟩) : ℂ) :
    ℂ)
Tactic: fin_cases i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
h0 : (f : (a : ℂ) → ℂ) I = I ∨ (f : (a : ℂ) → ℂ) I = -I
h' : (f : (a : ℂ) → ℂ) I = -I
⊢ f = conjLIE
have refine_2 : f.toLinearEquiv = conjLIE.toLinearEquiv
Tactic: apply LinearIsometryEquiv.toLinearEquiv_injective
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
h0 : (f : (a : ℂ) → ℂ) I = I ∨ (f : (a : ℂ) → ℂ) I = -I
h' : (f : (a : ℂ) → ℂ) I = -I
⊢ f.toLinearEquiv = conjLIE.toLinearEquiv
have refine_2 : ∀ (i : Fin 2),
  Eq (α := ℂ) ((f.toLinearEquiv : (a : ℂ) → ℂ) ((basisOneI : (a : Fin 2) → ℂ) i : ℂ) : ℂ)
    ((conjLIE.toLinearEquiv : (a : ℂ) → ℂ) ((basisOneI : (a : Fin 2) → ℂ) i : ℂ) : ℂ)
Tactic: apply Complex.basisOneI.ext'
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
h0 : (f : (a : ℂ) → ℂ) I = I ∨ (f : (a : ℂ) → ℂ) I = -I
h' : (f : (a : ℂ) → ℂ) I = -I
i : Fin 2
⊢ Eq (α := ℂ) ((f.toLinearEquiv : (a : ℂ) → ℂ) ((basisOneI : (a : Fin 2) → ℂ) i : ℂ) : ℂ)
    ((conjLIE.toLinearEquiv : (a : ℂ) → ℂ) ((basisOneI : (a : Fin 2) → ℂ) i : ℂ) : ℂ)
have refine_2 : Eq (α := ℂ) ((f.toLinearEquiv : (a : ℂ) → ℂ) ((basisOneI : (a : Fin 2) → ℂ) ((fun (i : Fin 2) ↦ i) ⟨0, sorry⟩) : ℂ) : ℂ)
  ((conjLIE.toLinearEquiv : (a : ℂ) → ℂ) ((basisOneI : (a : Fin 2) → ℂ) ((fun (i : Fin 2) ↦ i) ⟨0, sorry⟩) : ℂ) : ℂ)
Tactic: fin_cases i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
f : ℂ ≃ₗᵢ[ℝ] ℂ
h : Eq (α := ℂ) ((f : (a : ℂ) → ℂ) 1 : ℂ) (1 : ℂ)
h0 : (f : (a : ℂ) → ℂ) I = I ∨ (f : (a : ℂ) → ℂ) I = -I
h' : (f : (a : ℂ) → ℂ) I = -I
i : Fin 2
refine_2 :
  Eq (α := ℂ) ((f.toLinearEquiv : (a : ℂ) → ℂ) ((basisOneI : (a : Fin 2) → ℂ) ((fun (i : Fin 2) ↦ i) ⟨0, ⋯⟩) : ℂ) : ℂ)
    ((conjLIE.toLinearEquiv : (a : ℂ) → ℂ) ((basisOneI : (a : Fin 2) → ℂ) ((fun (i : Fin 2) ↦ i) ⟨0, ⋯⟩) : ℂ) : ℂ)
⊢ Eq (α := ℂ) ((f.toLinearEquiv : (a : ℂ) → ℂ) ((basisOneI : (a : Fin 2) → ℂ) i : ℂ) : ℂ)
    ((conjLIE.toLinearEquiv : (a : ℂ) → ℂ) ((basisOneI : (a : Fin 2) → ℂ) i : ℂ) : ℂ)
have refine_2₁ : Eq (α := ℂ) ((f.toLinearEquiv : (a : ℂ) → ℂ) ((basisOneI : (a : Fin 2) → ℂ) ((fun (i : Fin 2) ↦ i) ⟨1, sorry⟩) : ℂ) : ℂ)
  ((conjLIE.toLinearEquiv : (a : ℂ) → ℂ) ((basisOneI : (a : Fin 2) → ℂ) ((fun (i : Fin 2) ↦ i) ⟨1, sorry⟩) : ℂ) : ℂ)
Tactic: fin_cases i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
f : ℂ ≃ₗᵢ[ℝ] ℂ
a : Circle :=
  Subtype.mk (p := fun (x : ℂ) ↦ Membership.mem (γ := Submonoid ℂ) (Submonoid.unitSphere ℂ) x) ((f : (a : ℂ) → ℂ) 1) ⋯
⊢ ∃ (a : Circle),
    f = (rotation : (a : Circle) → ℂ ≃ₗᵢ[ℝ] ℂ) a ∨ f = conjLIE.trans ((rotation : (a : Circle) → ℂ ≃ₗᵢ[ℝ] ℂ) a)
have h : f = (rotation : (a : Circle) → ℂ ≃ₗᵢ[ℝ] ℂ) a ∨ f = conjLIE.trans ((rotation : (a : Circle) → ℂ ≃ₗᵢ[ℝ] ℂ) a)
Tactic: use a
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (hypothesis)
Goal:
f : ℂ ≃ₗᵢ[ℝ] ℂ
a : Circle :=
  Subtype.mk (p := fun (x : ℂ) ↦ Membership.mem (γ := Submonoid ℂ) (Submonoid.unitSphere ℂ) x) ((f : (a : ℂ) → ℂ) 1) ⋯
⊢ f = (rotation : (a : Circle) → ℂ ≃ₗᵢ[ℝ] ℂ) a ∨ f = conjLIE.trans ((rotation : (a : Circle) → ℂ ≃ₗᵢ[ℝ] ℂ) a)
have this : Eq (α := ℂ) ((f.trans (LinearIsometryEquiv.symm ((rotation : (a : Circle) → ℂ ≃ₗᵢ[ℝ] ℂ) a)) : (a : ℂ) → ℂ) 1 : ℂ)
  (1 : ℂ)
Tactic: have : (f.trans (rotation a).symm) 1 = 1 := by simpa [a] using rotation_apply a⁻¹ (f 1)
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
f : ℂ ≃ₗᵢ[ℝ] ℂ
a : Circle :=
  Subtype.mk (p := fun (x : ℂ) ↦ Membership.mem (γ := Submonoid ℂ) (Submonoid.unitSphere ℂ) x) ((f : (a : ℂ) → ℂ) 1) ⋯
this :
  Eq (α := ℂ) ((f.trans (LinearIsometryEquiv.symm ((rotation : (a : Circle) → ℂ ≃ₗᵢ[ℝ] ℂ) a)) : (a : ℂ) → ℂ) 1 : ℂ)
    (1 : ℂ)
⊢ f = (rotation : (a : Circle) → ℂ ≃ₗᵢ[ℝ] ℂ) a ∨ f = conjLIE.trans ((rotation : (a : Circle) → ℂ ≃ₗᵢ[ℝ] ℂ) a)
have h : ∀ (h₁ : f.trans (LinearIsometryEquiv.symm ((rotation : (a : Circle) → ℂ ≃ₗᵢ[ℝ] ℂ) a)) = LinearIsometryEquiv.refl ℝ ℂ),
  f = (rotation : (a : Circle) → ℂ ≃ₗᵢ[ℝ] ℂ) a
Tactic: refine (linear_isometry_complex_aux this).imp (fun h₁ => ?_) fun h₂ => ?_
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
f : ℂ ≃ₗᵢ[ℝ] ℂ
a : Circle :=
  Subtype.mk (p := fun (x : ℂ) ↦ Membership.mem (γ := Submonoid ℂ) (Submonoid.unitSphere ℂ) x) ((f : (a : ℂ) → ℂ) 1) ⋯
this :
  Eq (α := ℂ) ((f.trans (LinearIsometryEquiv.symm ((rotation : (a : Circle) → ℂ ≃ₗᵢ[ℝ] ℂ) a)) : (a : ℂ) → ℂ) 1 : ℂ)
    (1 : ℂ)
h :
  ∀ (h₁ : f.trans (LinearIsometryEquiv.symm ((rotation : (a : Circle) → ℂ ≃ₗᵢ[ℝ] ℂ) a)) = LinearIsometryEquiv.refl ℝ ℂ),
    f = (rotation : (a : Circle) → ℂ ≃ₗᵢ[ℝ] ℂ) a
⊢ f = (rotation : (a : Circle) → ℂ ≃ₗᵢ[ℝ] ℂ) a ∨ f = conjLIE.trans ((rotation : (a : Circle) → ℂ ≃ₗᵢ[ℝ] ℂ) a)
have h₁ : ∀ (h₂ : f.trans (LinearIsometryEquiv.symm ((rotation : (a : Circle) → ℂ ≃ₗᵢ[ℝ] ℂ) a)) = conjLIE),
  f = conjLIE.trans ((rotation : (a : Circle) → ℂ ≃ₗᵢ[ℝ] ℂ) a)
Tactic: refine (linear_isometry_complex_aux this).imp (fun h₁ => ?_) fun h₂ => ?_
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
a : Circle
⊢ (LinearMap.toMatrix basisOneI basisOneI : (a : ℂ →ₗ[ℝ] ℂ) → Matrix (Fin 2) (Fin 2) ℝ)
      (↑(LinearIsometryEquiv.toLinearEquiv ((rotation : (a : Circle) → ℂ ≃ₗᵢ[ℝ] ℂ) a)) : ℂ →ₗ[ℝ] ℂ) =
    (↑(Matrix.planeConformalMatrix (↑a : ℂ).re (↑a : ℂ).im ⋯) : Matrix (Fin 2) (Fin 2) ℝ)
have a₁ : ∀ (i j : Fin 2),
  (LinearMap.toMatrix basisOneI basisOneI : (a : ℂ →ₗ[ℝ] ℂ) → Matrix (Fin 2) (Fin 2) ℝ)
      (↑(LinearIsometryEquiv.toLinearEquiv ((rotation : (a : Circle) → ℂ ≃ₗᵢ[ℝ] ℂ) a)) : ℂ →ₗ[ℝ] ℂ) i j =
    (↑(Matrix.planeConformalMatrix (↑a : ℂ).re (↑a : ℂ).im sorry) : Matrix (Fin 2) (Fin 2) ℝ) i j
Tactic: ext i j
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
a : Circle
i j : Fin 2
⊢ (LinearMap.toMatrix basisOneI basisOneI : (a : ℂ →ₗ[ℝ] ℂ) → Matrix (Fin 2) (Fin 2) ℝ)
      (↑(LinearIsometryEquiv.toLinearEquiv ((rotation : (a : Circle) → ℂ ≃ₗᵢ[ℝ] ℂ) a)) : ℂ →ₗ[ℝ] ℂ) i j =
    (↑(Matrix.planeConformalMatrix (↑a : ℂ).re (↑a : ℂ).im ⋯) : Matrix (Fin 2) (Fin 2) ℝ) i j
have a₁ : ![(↑a : ℂ).re * (![1, I] j).re - (↑a : ℂ).im * (![1, I] j).im,
      (↑a : ℂ).re * (![1, I] j).im + (↑a : ℂ).im * (![1, I] j).re]
    i =
  Matrix.vecCons (![(↑a : ℂ).re, -(↑a : ℂ).im] j) (fun (i : Fin 1) ↦ ![(↑a : ℂ).im, (↑a : ℂ).re] j) i
Tactic: simp only [LinearMap.toMatrix_apply, coe_basisOneI, LinearEquiv.coe_coe,
    LinearIsometryEquiv.coe_toLinearEquiv, rotation_apply, coe_basisOneI_repr, mul_re, mul_im,
    Matrix.val_planeConformalMatrix, Matrix.of_apply, Matrix.cons_val', Matrix.empty_val',
    Matrix.cons_val_fin_one]
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
a : Circle
i j : Fin 2
⊢ ![(↑a : ℂ).re * (![1, I] j).re - (↑a : ℂ).im * (![1, I] j).im,
        (↑a : ℂ).re * (![1, I] j).im + (↑a : ℂ).im * (![1, I] j).re]
      i =
    Matrix.vecCons (![(↑a : ℂ).re, -(↑a : ℂ).im] j) (fun (i : Fin 1) ↦ ![(↑a : ℂ).im, (↑a : ℂ).re] j) i
have a₁ : ![(↑a : ℂ).re * (![1, I] j).re - (↑a : ℂ).im * (![1, I] j).im,
      (↑a : ℂ).re * (![1, I] j).im + (↑a : ℂ).im * (![1, I] j).re]
    ((fun (i : Fin 2) ↦ i) (⟨0, sorry⟩ : Fin 2)) =
  Matrix.vecCons (n := 1) (![(↑a : ℂ).re, -(↑a : ℂ).im] j) (fun (i : Fin 1) ↦ ![(↑a : ℂ).im, (↑a : ℂ).re] j)
    ((fun (i : Fin 2) ↦ i) ⟨0, sorry⟩)
Tactic: fin_cases i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
a : Circle
i j : Fin 2
a₁ :
  ![(↑a : ℂ).re * (![1, I] j).re - (↑a : ℂ).im * (![1, I] j).im,
        (↑a : ℂ).re * (![1, I] j).im + (↑a : ℂ).im * (![1, I] j).re]
      ((fun (i : Fin 2) ↦ i) (⟨0, ⋯⟩ : Fin 2)) =
    Matrix.vecCons (n := 1) (![(↑a : ℂ).re, -(↑a : ℂ).im] j) (fun (i : Fin 1) ↦ ![(↑a : ℂ).im, (↑a : ℂ).re] j)
      ((fun (i : Fin 2) ↦ i) ⟨0, ⋯⟩)
⊢ ![(↑a : ℂ).re * (![1, I] j).re - (↑a : ℂ).im * (![1, I] j).im,
        (↑a : ℂ).re * (![1, I] j).im + (↑a : ℂ).im * (![1, I] j).re]
      i =
    Matrix.vecCons (![(↑a : ℂ).re, -(↑a : ℂ).im] j) (fun (i : Fin 1) ↦ ![(↑a : ℂ).im, (↑a : ℂ).re] j) i
have a₂ : ![(↑a : ℂ).re * (![1, I] j).re - (↑a : ℂ).im * (![1, I] j).im,
      (↑a : ℂ).re * (![1, I] j).im + (↑a : ℂ).im * (![1, I] j).re]
    ((fun (i : Fin 2) ↦ i) (⟨1, sorry⟩ : Fin 2)) =
  Matrix.vecCons (n := 1) (![(↑a : ℂ).re, -(↑a : ℂ).im] j) (fun (i : Fin 1) ↦ ![(↑a : ℂ).im, (↑a : ℂ).re] j)
    ((fun (i : Fin 2) ↦ i) ⟨1, sorry⟩)
Tactic: fin_cases i
Cleared hypotheses: i
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
a : Circle
j : Fin 2
⊢ ![(↑a : ℂ).re * (![1, I] j).re - (↑a : ℂ).im * (![1, I] j).im,
        (↑a : ℂ).re * (![1, I] j).im + (↑a : ℂ).im * (![1, I] j).re]
      ((fun (i : Fin 2) ↦ i) (⟨0, ⋯⟩ : Fin 2)) =
    Matrix.vecCons (n := 1) (![(↑a : ℂ).re, -(↑a : ℂ).im] j) (fun (i : Fin 1) ↦ ![(↑a : ℂ).im, (↑a : ℂ).re] j)
      ((fun (i : Fin 2) ↦ i) ⟨0, ⋯⟩)
have a₁ : ![(↑a : ℂ).re * (![1, I] ((fun (i : Fin 2) ↦ i) (⟨0, sorry⟩ : Fin 2))).re -
        (↑a : ℂ).im * (![1, I] ((fun (i : Fin 2) ↦ i) (⟨0, sorry⟩ : Fin 2))).im,
      (↑a : ℂ).re * (![1, I] ((fun (i : Fin 2) ↦ i) (⟨0, sorry⟩ : Fin 2))).im +
        (↑a : ℂ).im * (![1, I] ((fun (i : Fin 2) ↦ i) (⟨0, sorry⟩ : Fin 2))).re]
    ((fun (i : Fin 2) ↦ i) (⟨0, sorry⟩ : Fin 2)) =
  Matrix.vecCons (n := 1) (![(↑a : ℂ).re, -(↑a : ℂ).im] ((fun (i : Fin 2) ↦ i) ⟨0, sorry⟩))
    (fun (i : Fin 1) ↦ ![(↑a : ℂ).im, (↑a : ℂ).re] ((fun (i : Fin 2) ↦ i) ⟨0, sorry⟩)) ((fun (i : Fin 2) ↦ i) ⟨0, sorry⟩)
Tactic: fin_cases j
Cleared hypotheses: j
-- Lean 4 proof step from Mathlib.Analysis.Complex.Isometry (goal)
Goal:
a : Circle
j : Fin 2
a₁ :
  ![(↑a : ℂ).re * (![1, I] ((fun (i : Fin 2) ↦ i) (⟨0, ⋯⟩ : Fin 2))).re -
          (↑a : ℂ).im * (![1, I] ((fun (i : Fin 2) ↦ i) (⟨0, ⋯⟩ : Fin 2))).im,
        (↑a : ℂ).re * (![1, I] ((fun (i : Fin 2) ↦ i) (⟨0, ⋯⟩ : Fin 2))).im +
          (↑a : ℂ).im * (![1, I] ((fun (i : Fin 2) ↦ i) (⟨0, ⋯⟩ : Fin 2))).re]
      ((fun (i : Fin 2) ↦ i) (⟨0, ⋯⟩ : Fin 2)) =
    Matrix.vecCons (n := 1) (![(↑a : ℂ).re, -(↑a : ℂ).im] ((fun (i : Fin 2) ↦ i) ⟨0, ⋯⟩))
      (fun (i : Fin 1) ↦ ![(↑a : ℂ).im, (↑a : ℂ).re] ((fun (i : Fin 2) ↦ i) ⟨0, ⋯⟩)) ((fun (i : Fin 2) ↦ i) ⟨0, ⋯⟩)
⊢ ![(↑a : ℂ).re * (![1, I] j).re - (↑a : ℂ).im * (![1, I] j).im,
        (↑a : ℂ).re * (![1, I] j).im + (↑a : ℂ).im * (![1, I] j).re]
      ((fun (i : Fin 2) ↦ i) (⟨0, ⋯⟩ : Fin 2)) =
    Matrix.vecCons (n := 1) (![(↑a : ℂ).re, -(↑a : ℂ).im] j) (fun (i : Fin 1) ↦ ![(↑a : ℂ).im, (↑a : ℂ).re] j)
      ((fun (i : Fin 2) ↦ i) ⟨0, ⋯⟩)
have a₂ : ![(↑a : ℂ).re * (![1, I] ((fun (i : Fin 2) ↦ i) (⟨1, sorry⟩ : Fin 2))).re -
        (↑a : ℂ).im * (![1, I] ((fun (i : Fin 2) ↦ i) (⟨1, sorry⟩ : Fin 2))).im,
      (↑a : ℂ).re * (![1, I] ((fun (i : Fin 2) ↦ i) (⟨1, sorry⟩ : Fin 2))).im +
        (↑a : ℂ).im * (![1, I] ((fun (i : Fin 2) ↦ i) (⟨1, sorry⟩ : Fin 2))).re]
    ((fun (i : Fin 2) ↦ i) (⟨0, sorry⟩ : Fin 2)) =
  Matrix.vecCons (n := 1) (![(↑a : ℂ).re, -(↑a : ℂ).im] ((fun (i : Fin 2) ↦ i) ⟨1, sorry⟩))
    (fun (i : Fin 1) ↦ ![(↑a : ℂ).im, (↑a : ℂ).re] ((fun (i : Fin 2) ↦ i) ⟨1, sorry⟩)) ((fun (i : Fin 2) ↦ i) ⟨0, sorry⟩)
Tactic: fin_cases j
Cleared hypotheses: j