-- Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean
theorem oangle_eq_iff_eq_norm_div_norm_smul_rotation_or_eq_zero {x y : V} (θ : Real.Angle) :
    o.oangle x y = θ ↔
      x ≠ 0 ∧ y ≠ 0 ∧ y = (‖y‖ / ‖x‖) • o.rotation θ x ∨ θ = 0 ∧ (x = 0 ∨ y = 0)

Proof state:
case neg
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
θ : Real.Angle
hx : ¬x = 0
hy : ¬y = 0
⊢ y = (‖y‖ / ‖x‖) • (o.rotation θ) x ↔ x ≠ 0 ∧ y ≠ 0 ∧ y = (‖y‖ / ‖x‖) • (o.rotation θ) x ∨ θ = 0 ∧ (x = 0 ∨ y = 0)

Tactic:
simp [hx, hy]

Result:
no goals

Proof state:
case neg
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
θ : Real.Angle
hx : ¬x = 0
⊢ o.oangle x y = θ ↔ x ≠ 0 ∧ y ≠ 0 ∧ y = (‖y‖ / ‖x‖) • (o.rotation θ) x ∨ θ = 0 ∧ (x = 0 ∨ y = 0)

Tactic:
by_cases hy : y = 0

Result:
case pos
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
θ : Real.Angle
hx : ¬x = 0
hy : y = 0
⊢ o.oangle x y = θ ↔ x ≠ 0 ∧ y ≠ 0 ∧ y = (‖y‖ / ‖x‖) • (o.rotation θ) x ∨ θ = 0 ∧ (x = 0 ∨ y = 0)

case neg
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
θ : Real.Angle
hx : ¬x = 0
hy : ¬y = 0
⊢ o.oangle x y = θ ↔ x ≠ 0 ∧ y ≠ 0 ∧ y = (‖y‖ / ‖x‖) • (o.rotation θ) x ∨ θ = 0 ∧ (x = 0 ∨ y = 0)

Proof state:
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
θ : Real.Angle
⊢ o.oangle x y = θ ↔ x ≠ 0 ∧ y ≠ 0 ∧ y = (‖y‖ / ‖x‖) • (o.rotation θ) x ∨ θ = 0 ∧ (x = 0 ∨ y = 0)

Tactic:
by_cases hx : x = 0

Result:
case pos
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
θ : Real.Angle
hx : x = 0
⊢ o.oangle x y = θ ↔ x ≠ 0 ∧ y ≠ 0 ∧ y = (‖y‖ / ‖x‖) • (o.rotation θ) x ∨ θ = 0 ∧ (x = 0 ∨ y = 0)

case neg
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
θ : Real.Angle
hx : ¬x = 0
⊢ o.oangle x y = θ ↔ x ≠ 0 ∧ y ≠ 0 ∧ y = (‖y‖ / ‖x‖) • (o.rotation θ) x ∨ θ = 0 ∧ (x = 0 ∨ y = 0)

Proof state:
case pos
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
θ : Real.Angle
hx : x = 0
⊢ o.oangle x y = θ ↔ x ≠ 0 ∧ y ≠ 0 ∧ y = (‖y‖ / ‖x‖) • (o.rotation θ) x ∨ θ = 0 ∧ (x = 0 ∨ y = 0)

Tactic:
simp [hx, eq_comm]

Result:
no goals

Proof state:
case neg
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
θ : Real.Angle
hx : ¬x = 0
hy : ¬y = 0
⊢ o.oangle x y = θ ↔ x ≠ 0 ∧ y ≠ 0 ∧ y = (‖y‖ / ‖x‖) • (o.rotation θ) x ∨ θ = 0 ∧ (x = 0 ∨ y = 0)

Tactic:
rw [o.oangle_eq_iff_eq_norm_div_norm_smul_rotation_of_ne_zero hx hy]

Result:
case neg
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
θ : Real.Angle
hx : ¬x = 0
hy : ¬y = 0
⊢ y = (‖y‖ / ‖x‖) • (o.rotation θ) x ↔ x ≠ 0 ∧ y ≠ 0 ∧ y = (‖y‖ / ‖x‖) • (o.rotation θ) x ∨ θ = 0 ∧ (x = 0 ∨ y = 0)

Proof state:
case pos
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
θ : Real.Angle
hx : ¬x = 0
hy : y = 0
⊢ o.oangle x y = θ ↔ x ≠ 0 ∧ y ≠ 0 ∧ y = (‖y‖ / ‖x‖) • (o.rotation θ) x ∨ θ = 0 ∧ (x = 0 ∨ y = 0)

Tactic:
simp [hy, eq_comm]

Result:
no goals