case refine'_1
θ ψ : Angle
⊢ θ = ψ → sign θ = sign ψ ∧ abs (toReal θ) = abs (toReal ψ)

case refine'_2
θ ψ : Angle
h : sign θ = sign ψ ∧ abs (toReal θ) = abs (toReal ψ)
⊢ θ = ψ
θ ψ : Angle
⊢ θ = ψ ↔ sign θ = sign ψ ∧ abs (toReal θ) = abs (toReal ψ)
refine' ⟨_, fun h => _⟩
case refine'_2.intro
θ ψ : Angle
hs : sign θ = sign ψ
hr : abs (toReal θ) = abs (toReal ψ)
⊢ θ = ψ
case refine'_2
θ ψ : Angle
h : sign θ = sign ψ ∧ abs (toReal θ) = abs (toReal ψ)
⊢ θ = ψ
rcases h with ⟨hs, hr⟩
case refine'_2.intro
θ ψ : Angle
hs : sign θ = sign ψ
hr : toReal θ = toReal ψ ∨ toReal θ = -toReal ψ
⊢ θ = ψ
case refine'_2.intro
θ ψ : Angle
hs : sign θ = sign ψ
hr : abs (toReal θ) = abs (toReal ψ)
⊢ θ = ψ
rw [abs_eq_abs] at hr
case refine'_2.intro.inl
θ ψ : Angle
hs : sign θ = sign ψ
hr : toReal θ = toReal ψ
⊢ θ = ψ

case refine'_2.intro.inr
θ ψ : Angle
hs : sign θ = sign ψ
hr : toReal θ = -toReal ψ
⊢ θ = ψ
case refine'_2.intro
θ ψ : Angle
hs : sign θ = sign ψ
hr : toReal θ = toReal ψ ∨ toReal θ = -toReal ψ
⊢ θ = ψ
rcases hr with (hr | hr)
case refine'_1
θ : Angle
⊢ sign θ = sign θ ∧ abs (toReal θ) = abs (toReal θ)
case refine'_1
θ ψ : Angle
⊢ θ = ψ → sign θ = sign ψ ∧ abs (toReal θ) = abs (toReal ψ)
rintro rfl
no goals
case refine'_1
θ : Angle
⊢ sign θ = sign θ ∧ abs (toReal θ) = abs (toReal θ)
exact ⟨rfl, rfl⟩
no goals
case refine'_2.intro.inl
θ ψ : Angle
hs : sign θ = sign ψ
hr : toReal θ = toReal ψ
⊢ θ = ψ
exact toReal_injective hr
case pos
θ ψ : Angle
hs : sign θ = sign ψ
hr : toReal θ = -toReal ψ
h : θ = ↑π
⊢ θ = ψ

case neg
θ ψ : Angle
hs : sign θ = sign ψ
hr : toReal θ = -toReal ψ
h : ¬θ = ↑π
⊢ θ = ψ
case refine'_2.intro.inr
θ ψ : Angle
hs : sign θ = sign ψ
hr : toReal θ = -toReal ψ
⊢ θ = ψ
by_cases h : θ = π
case pos
θ ψ : Angle
hs : sign θ = sign ψ
hr : -π = toReal ψ
h : θ = ↑π
⊢ θ = ψ
case pos
θ ψ : Angle
hs : sign θ = sign ψ
hr : toReal θ = -toReal ψ
h : θ = ↑π
⊢ θ = ψ
rw [h, toReal_pi, ← neg_eq_iff_eq_neg] at hr
no goals
case pos
θ ψ : Angle
hs : sign θ = sign ψ
hr : -π = toReal ψ
h : θ = ↑π
⊢ θ = ψ
exact False.elim ((neg_pi_lt_toReal ψ).ne hr)
case pos
θ ψ : Angle
hs : sign θ = sign ψ
hr : toReal θ = -toReal ψ
h : ¬θ = ↑π
h' : ψ = ↑π
⊢ θ = ψ

case neg
θ ψ : Angle
hs : sign θ = sign ψ
hr : toReal θ = -toReal ψ
h : ¬θ = ↑π
h' : ¬ψ = ↑π
⊢ θ = ψ
case neg
θ ψ : Angle
hs : sign θ = sign ψ
hr : toReal θ = -toReal ψ
h : ¬θ = ↑π
⊢ θ = ψ
by_cases h' : ψ = π
case pos
θ ψ : Angle
hs : sign θ = sign ψ
hr : toReal θ = -π
h : ¬θ = ↑π
h' : ψ = ↑π
⊢ θ = ψ
case pos
θ ψ : Angle
hs : sign θ = sign ψ
hr : toReal θ = -toReal ψ
h : ¬θ = ↑π
h' : ψ = ↑π
⊢ θ = ψ
rw [h', toReal_pi] at hr
no goals
case pos
θ ψ : Angle
hs : sign θ = sign ψ
hr : toReal θ = -π
h : ¬θ = ↑π
h' : ψ = ↑π
⊢ θ = ψ
exact False.elim ((neg_pi_lt_toReal θ).ne hr.symm)
case neg
θ ψ : Angle
hs : ψ = 0
hr : toReal θ = -toReal ψ
h : ¬θ = ↑π
h' : ¬ψ = ↑π
⊢ θ = ψ
case neg
θ ψ : Angle
hs : sign θ = sign ψ
hr : toReal θ = -toReal ψ
h : ¬θ = ↑π
h' : ¬ψ = ↑π
⊢ θ = ψ
rw [← sign_toReal h, ← sign_toReal h', hr, Left.sign_neg, SignType.neg_eq_self_iff,
  _root_.sign_eq_zero_iff, toReal_eq_zero_iff] at hs
case neg
θ ψ : Angle
hs : ψ = 0
hr : θ = 0
h : ¬θ = ↑π
h' : ¬ψ = ↑π
⊢ θ = ψ
case neg
θ ψ : Angle
hs : ψ = 0
hr : toReal θ = -toReal ψ
h : ¬θ = ↑π
h' : ¬ψ = ↑π
⊢ θ = ψ
rw [hs, toReal_zero, neg_zero, toReal_eq_zero_iff] at hr
no goals
case neg
θ ψ : Angle
hs : ψ = 0
hr : θ = 0
h : ¬θ = ↑π
h' : ¬ψ = ↑π
⊢ θ = ψ
rw [hr, hs]